Surface Fractal Dimension of Bentonite and its Application in Calculation of Swelling Deformation
NASA Astrophysics Data System (ADS)
Xiang, G. S.; Xu, Y. F.; Jiang, H.
2014-09-01
The correlation between the void ratio of swelled montmorillonite and the vertical overburden pressure can be expressed as {e}{ m} = Kp{ s}{D{ s}-3}. The surface fractal dimension Ds of five bentonites were estimated from the swelling deformation tests according to this fractal correlation. The reliability of surface fractal dimension obtained from the swelling deformation test was confirmed by nitrogen adsorption test, with identical values of surface fractal dimension obtained from both tests. The surface fractal dimension can also be used to estimate the swelling deformation of bentonite, after calculating the swelling coefficient K from the parameters of diffuse double layer (DDL) model in the osmotic swelling phase. Comparison of the model predictions with a number of experimental results on swelling deformation of both Na dominant and Ca dominant bentonites suggests that the surface fractal model works excellent in the cases tested.
The Calculation of Fractal Dimension in the Presence of Non-Fractal Clutter
NASA Technical Reports Server (NTRS)
Herren, Kenneth A.; Gregory, Don A.
1999-01-01
The area of information processing has grown dramatically over the last 50 years. In the areas of image processing and information storage the technology requirements have far outpaced the ability of the community to meet demands. The need for faster recognition algorithms and more efficient storage of large quantities of data has forced the user to accept less than lossless retrieval of that data for analysis. In addition to clutter that is not the object of interest in the data set, often the throughput requirements forces the user to accept "noisy" data and to tolerate the clutter inherent in that data. It has been shown that some of this clutter, both the intentional clutter (clouds, trees, etc) as well as the noise introduced on the data by processing requirements can be modeled as fractal or fractal-like. Traditional methods using Fourier deconvolution on these sources of noise in frequency space leads to loss of signal and can, in many cases, completely eliminate the target of interest. The parameters that characterize fractal-like noise (predominately the fractal dimension) have been investigated and a technique to reduce or eliminate noise from real scenes has been developed. Examples of clutter reduced images are presented.
Calculation of multi-fractal dimensions in spin chains
Atas, Y. Y.; Bogomolny, E.
2014-01-01
It was demonstrated in Atas & Bogomolny (2012 Phys. Rev. E 86, 021104) that the ground-state wave functions for a large variety of one-dimensional spin- models are multi-fractals in the natural spin-z basis. We present here the details of analytical derivations and numerical confirmations of this statement. PMID:24344342
a New Method for Calculating the Fractal Dimension of Surface Topography
NASA Astrophysics Data System (ADS)
Zuo, Xue; Zhu, Hua; Zhou, Yuankai; Li, Yan
2015-06-01
A new method termed as three-dimensional root-mean-square (3D-RMS) method, is proposed to calculate the fractal dimension (FD) of machined surfaces. The measure of this method is the root-mean-square value of surface data, and the scale is the side length of square in the projection plane. In order to evaluate the calculation accuracy of the proposed method, the isotropic surfaces with deterministic FD are generated based on the fractional Brownian function and Weierstrass-Mandelbrot (WM) fractal function, and two kinds of anisotropic surfaces are generated by stretching or rotating a WM fractal curve. Their FDs are estimated by the proposed method, as well as differential boxing-counting (DBC) method, triangular prism surface area (TPSA) method and variation method (VM). The results show that the 3D-RMS method performs better than the other methods with a lower relative error for both isotropic and anisotropic surfaces, especially for the surfaces with dimensions higher than 2.5, since the relative error between the estimated value and its theoretical value decreases with theoretical FD. Finally, the electrodeposited surface, end-turning surface and grinding surface are chosen as examples to illustrate the application of 3D-RMS method on the real machined surfaces. This method gives a new way to accurately calculate the FD from the surface topographic data.
Fractal dimension in software networks
NASA Astrophysics Data System (ADS)
Concas, G.; Locci, M. F.; Marchesi, M.; Pinna, S.; Turnu, I.
2006-12-01
A large number of real networks are characterized by two fundamental properties: they are small world and scale-free. A recent paper demonstrated that the structure of many complex networks is also self-similar under a length-scale transformation, and calculated their fractal dimension using the “box counting” method. We studied nine large object-oriented software systems, finding that the graphs associated to these networks are self-similar. We also studied the time evolution of the fractal dimension during system growth, finding a significant correlation between the fractal dimension and object-oriented complexity metrics known to be correlated with software fault-proneness. Thus, in software systems the fractal dimension could be considered as a measure of internal complexity, and consequently of the system quality.
Video fire detection based on three-state Markov modal and fractal dimension calculation
NASA Astrophysics Data System (ADS)
Lei, Bo; Zhang, Zhijie; Wang, Chensheng
2012-11-01
Fire detection based on video surveillance is a very effective method for large area outdoor fire prevention, but the unpredictable place and time makes automatic fire detection a difficult problem. This paper adopts a loose color selection and frame differential to narrow down possible fire regions, where every pixel's temporal color variations are analyzed by 3-state Markov modals. One of the Markov modal is used for brightness variation examination and the other one is used for fire color likeness that is measured by color difference. In order to eliminate false detections, the fractal dimension calculation and texture match are performed. Experimental results prove the proposed method is feasible and suitable for outdoor or indoor fire detection in surveillance videos.
Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series
NASA Astrophysics Data System (ADS)
Sánchez-Granero, M. J.; Fernández-Martínez, M.; Trinidad-Segovia, J. E.
2012-03-01
In this paper, three new algorithms are introduced in order to explore long memory in financial time series. They are based on a new concept of fractal dimension of a curve. A mathematical support is provided for each algorithm and its accuracy is tested for different length time series by Monte Carlo simulations. In particular, in the case of short length series, the introduced algorithms perform much better than the classical methods. Finally, an empirical application for some stock market indexes as well as some individual stocks is presented.
Dimension of fractal basin boundaries
Park, B.S.
1988-01-01
In many dynamical systems, multiple attractors coexist for certain parameter ranges. The set of initial conditions that asymptotically approach each attractor is its basin of attraction. These basins can be intertwined on arbitrary small scales. Basin boundary can be either smooth or fractal. Dynamical systems that have fractal basin boundary show final state sensitivity of the initial conditions. A measure of this sensitivity (uncertainty exponent {alpha}) is related to the dimension of the basin boundary d = D - {alpha}, where D is the dimension of the phase space and d is the dimension of the basin boundary. At metamorphosis values of the parameter, there might happen a conversion from smooth to fractal basin boundary (smooth-fractal metamorphosis) or a conversion from fractal to another fractal basin boundary characteristically different from the previous fractal one (fractal-fractal metamorphosis). The dimension changes continuously with the parameter except at the metamorphosis values where the dimension of the basin boundary jumps discontinuously. We chose the Henon map and the forced damped pendulum to investigate this. Scaling of the basin volumes near the metamorphosis values of the parameter is also being studied for the Henon map. Observations are explained analytically by using low dimensional model map.
Exterior dimension of fat fractals
NASA Technical Reports Server (NTRS)
Grebogi, C.; Mcdonald, S. W.; Ott, E.; Yorke, J. A.
1985-01-01
Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which is called the exterior dimension. In addition, it is shown that the exterior dimension is related to the 'uncertainty exponent' previously used in studies of fractal basin boundaries, and it is shown how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.
Fractal dimension applied in highway surface crack detection
NASA Astrophysics Data System (ADS)
Chen, Bei; Cao, Wenlun; He, Yuyao
2012-04-01
This paper presents a fractal dimension calculation technique of highway pavement which is used to detect the pavement cracking. Firstly, the impulse noises generated by pavement unevenness are eliminated by the use of median filtering of pavement image. Secondly, the fractal dimension of fissure is calculated according to the crack's fractal feature. Finally, the pavement damage is detected and located with respect to fractal dimension. These three steps can realize the location detection and damage rate computation of pavement damage successfully. To a number of random images, the experiment results show that the fractal dimension of pavement crack region is from 2.25 to 2.99 basically.
Box-covering algorithm for fractal dimension of weighted networks
NASA Astrophysics Data System (ADS)
Wei, Dai-Jun; Liu, Qi; Zhang, Hai-Xin; Hu, Yong; Deng, Yong; Mahadevan, Sankaran
2013-10-01
Box-covering algorithm is a widely used method to measure the fractal dimension of complex networks. Existing researches mainly deal with the fractal dimension of unweighted networks. Here, the classical box covering algorithm is modified to deal with the fractal dimension of weighted networks. Box size length is obtained by accumulating the distance between two nodes connected directly and graph-coloring algorithm is based on the node strength. The proposed method is applied to calculate the fractal dimensions of the ``Sierpinski'' weighted fractal networks, the E.coli network, the Scientific collaboration network, the C.elegans network and the USAir97 network. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks. We find that the fractal property is influenced by the edge-weight in weighted networks. The possible variation of fractal dimension due to changes in edge-weights of weighted networks is also discussed.
Fractal Dimension in Epileptic EEG Signal Analysis
NASA Astrophysics Data System (ADS)
Uthayakumar, R.
Fractal Analysis is the well developed theory in the data analysis of non-linear time series. Especially Fractal Dimension is a powerful mathematical tool for modeling many physical and biological time signals with high complexity and irregularity. Fractal dimension is a suitable tool for analyzing the nonlinear behaviour and state of the many chaotic systems. Particularly in analysis of chaotic time series such as electroencephalograms (EEG), this feature has been used to identify and distinguish specific states of physiological function.Epilepsy is the main fatal neurological disorder in our brain, which is analyzed by the biomedical signal called Electroencephalogram (EEG). The detection of Epileptic seizures in the EEG Signals is an important tool in the diagnosis of epilepsy. So we made an attempt to analyze the EEG in depth for knowing the mystery of human consciousness. EEG has more fluctuations recorded from the human brain due to the spontaneous electrical activity. Hence EEG Signals are represented as Fractal Time Series.The algorithms of fractal dimension methods have weak ability to the estimation of complexity in the irregular graphs. Divider method is widely used to obtain the fractal dimension of curves embedded into a 2-dimensional space. The major problem is choosing initial and final step length of dividers. We propose a new algorithm based on the size measure relationship (SMR) method, quantifying the dimensional behaviour of irregular rectifiable graphs with minimum time complexity. The evidence for the suitability (equality with the nature of dimension) of the algorithm is illustrated graphically.We would like to demonstrate the criterion for the selection of dividers (minimum and maximum value) in the calculation of fractal dimension of the irregular curves with minimum time complexity. For that we design a new method of computing fractal dimension (FD) of biomedical waveforms. Compared to Higuchi's algorithm, advantages of this method include greater speed and the criterion to choose the maximum and minimum values for time intervals. Comparisons with the other waveform fractal dimension algorithms are also demonstrated. In order to discriminate the Healthy and the Epileptic EEGs, an improved method of Multifractal Measure such as Generalized Fractal Dimensions (GFD) is also proposed. Finally we conclude that there are significant differences between the Healthy and Epileptic Signals in the designed method than the GFD through graphical and statistical tools. The improved multifractal measure is very efficient technique to analyze the EEG Signals and to compute the state of illness of the Epileptic patients.
Nieckarz, Zenon; Tatoń, Grzegorz; Kozerska, Magdalena; Skrzat, Janusz; Sioma, Andrzej
2015-01-01
We presented a novel approach to studies of the vascular grooves located on the inner surface of the cranial vault. A three-dimensional vision system that acquired the endocranial surface topography was used for this purpose. The acquired data were used to generate images showing the branching pattern of the middle meningeal artery. Fractal dimension was used to characterize and analyze branching pattern complexity. We discussed the usefulness of the latter method and indicated difficulties and potential errors connected to the fractal dimension application. The technique introduced for recording traits of the object surface appears to be helpful in anatomical study of morphological variation of dural vascularization. It may also be applicable in paleoneurological research based on analysis of the cranial remnants. Fractal dimension should be used carefully as a method sensitive to many aspects of data acquisition and processing. PMID:25807002
Fractal Dimension of Bioconvection Patterns
NASA Astrophysics Data System (ADS)
Noever, David A.
1990-10-01
Shallow cultures of the motile algal strain, Euglena gracilis, were concentrated to 2× 106 organisms per ml and placed in constant temperature water baths at 24 and 38 C. Bioconvective patterns formed an open two-dimensional structure with random branches, similar to clusters encountered in the diffusion-limited aggregation (DLA) model. When averaged over several example cultures, the pattern was found to have no natural length scale, self-similar branching and a fractal dimension (d˜1.7). These agree well with the two-dimensional DLA.
Fractal dimension of bioconvection patterns
NASA Technical Reports Server (NTRS)
Noever, David A.
1990-01-01
Shallow cultures of the motile algal strain, Euglena gracilis, were concentrated to 2 x 10 to the 6th organisms per ml and placed in constant temperature water baths at 24 and 38 C. Bioconvective patterns formed an open two-dimensional structure with random branches, similar to clusters encountered in the diffusion-limited aggregation (DLA) model. When averaged over several example cultures, the pattern was found to have no natural length scale, self-similar branching, and a fractal dimension (d about 1.7). These agree well with the two-dimensional DLA.
A Fractal Dimension Survey of Active Region Complexity
NASA Technical Reports Server (NTRS)
McAteer, R. T. James; Gallagher, Peter; Ireland, Jack
2005-01-01
A new approach to quantifying the magnetic complexity of active regions using a fractal dimension measure is presented. This fully-automated approach uses full disc MDI magnetograms of active regions from a large data set (2742 days of the SoHO mission; 9342 active regions) to compare the calculated fractal dimension to both Mount Wilson classification and flare rate. The main Mount Wilson classes exhibit no distinct fractal dimension distribution, suggesting a self-similar nature of all active regions. Solar flare productivity exhibits an increase in both the frequency and GOES X-ray magnitude of flares from regions with higher fractal dimensions. Specifically a lower threshold fractal dimension of 1.2 and 1.25 exists as a necessary, but not sufficient, requirement for an active region to produce M- and X-class flares respectively .
NASA Astrophysics Data System (ADS)
Yan, Kun
2007-04-01
In this paper, by discussing the basic hypotheses about the continuous orbit and discrete orbit in two research directions of the background medium theory for celestial body motion, the concrete equation forms and their summary of the theoretic frame of celestial body motion are introduced. Future more, by discussing the general form of Binet's equation of celestial body motion orbit and it's solution of the advance of the perihelion of planets, the relations and differences between the continuous orbit theory and Newton's gravitation theory and Einstein's general relativity are given. And by discussing the fractional-dimension expanded equation for the celestial body motion orbits, the concrete equations and the prophesy data of discrete orbit or stable orbits of celestial bodies which included the planets in the Solar system, satellites in the Uranian system, satellites in the Earth system and satellites obtaining the Moon obtaining from discrete orbit theory are given too. Especially, as the preliminary exploration and inference to the gravitation curve of celestial bodies in broadly range, the concept for the ideal black hole with trend to infinite in mass density difficult to be formed by gravitation only is explored. By discussing the position hypothesis of fractional-dimension derivative about general function and the formula form the hypothesis of fractional-dimension derivative about power function, the concrete equation formulas of fractional-dimension derivative, differential and integral are described distinctly further, and the difference between the fractional-dimension derivative and the fractional-order derivative are given too. Subsequently, the concrete forms of measure calculation equations of self-similar fractal obtaining by based on the definition of form in fractional-dimension calculus about general fractal measure are discussed again, and the differences with Hausdorff measure method or the covering method at present are given. By applying the measure calculation equations, the me asure of self-similar fractals which include middle-third Cantor set, Koch curve, Sierpinski gasket and orthogonal cross star are calculated and analyzed.
Dimension of a fractal streamer structure
NASA Astrophysics Data System (ADS)
Lehtinen, Nikolai G.; Østgaard, Nikolai
2015-04-01
Streamer corona plays an important role in formation of leader steps in lightning. In order to understand its dynamics, the streamer front velocity is calculated in a 1D model with curvature. We concentrate on the role of photoionization mechanism in the propagation of the streamer ionization front, the other important mechanisms being electron drift and electron diffusion. The results indicate, in particular, that the effect of photoionization on the streamer velocity for both positive and negative streamers is mostly determined by the photoionization length, with a weaker dependence on the amount of photoionization, and that the velocity is decreased for positive curvature, i.e., convex fronts. These results are used in a fractal model in which the front propagation velocity is simulated as the cluster growth probability [Niemeyer et al, 1984, doi:10.1103/PhysRevLett.52.1033]. Monte Carlo simulations of the cluster growth for various ratios of background electric field E to the breakdown field Eb show that the emerging transverse size of the streamers is of the order of the photoionization length, and at the larger scale the streamer structure is a fractal similar to the one obtained in a diffusion-limited aggregation (DLA) system. In the absence of electron attachment (Eb = 0), the fractal dimension is the same (D ˜ 1.67) as in the DLA model, and is reduced, i.e., the fractal has less branching, for Eb > 0.
Fractal dimension of cerebral surfaces using magnetic resonance images
Majumdar, S.; Prasad, R.R.
1988-11-01
The calculation of the fractal dimension of the surface bounded by the grey matter in the normal human brain using axial, sagittal, and coronal cross-sectional magnetic resonance (MR) images is presented. The fractal dimension in this case is a measure of the convolutedness of this cerebral surface. It is proposed that the fractal dimension, a feature that may be extracted from MR images, may potentially be used for image analysis, quantitative tissue characterization, and as a feature to monitor and identify cerebral abnormalities and developmental changes.
The use of fractal dimension for texture classification
Dixon, K.E.
1989-04-01
This paper addresses the idea of using fractal dimension as a measure of image texture. The computation of the fractal dimension of a grey-scale image and also of the ''fractal signature'' of the image is presented. Two methods of scanning the image for these calculations are introduced: a line scan and a window scan. Several subsets of features extracted from the calculations are investigated as features for classification of the texture. Results from various classification experiments are presented. 5 refs., 8 tabs.
MASS FRACTAL DIMENSION OF SHRINKING SOIL AGGREGATES
Technology Transfer Automated Retrieval System (TEKTRAN)
Fractal scaling for mass of dry soil aggregates has been documented in literature. This scaling results in power-law dependencies of aggregate porosity or bulk density on aggregate size. Such dependencies if measured are used to estimate mass fractal dimensions. Changes in water content are known to...
Measurement of fractal dimension using 3-D technique
NASA Astrophysics Data System (ADS)
Chuang, Keh-Shih; Valentino, Daniel J.; Huang, H. K.
1991-06-01
Fractal dimension is a measure of complexity of a fractal object. Its application in medical image analysis is widely accepted. In this paper we use 3-D surface tracking technique to calculate the fractal dimension of the brain. The fractal dimension in this case is a measure of the convolution of the cerebral surface. Series of MR images of the brain are read and interpolated into a 3-D volume such that each voxel is a cube. Each cube is equivalent to a box in the box-counting method. A surface tracking routine is applied to this 3-D volume. The number of faces (surface area) on the surface as well as volume inside the object can be obtained. Then we can change the voxel size and do the interpolation and surface tracking again. Based on these measurements for various voxel sizes we can calculate the fractal dimension. The fractal dimension measured for a brain specimen using the 3-D box counting technique is equal to 2.207.
Application of Fractal Dimension on Palsar Data
NASA Astrophysics Data System (ADS)
Singh, Dharmendra; Pant, Triloki
Study of land cover is the primal task of remote sensing where microwave imaging plays an important role. As an alternate of optical imaging, microwave, in particular, Synthetic Aperture Radar (SAR) imaging is very popular. With the advancement of technology, multi-polarized images are now available, e.g., ALOS-PALSAR (Phased Array type L-band SAR), which are beneficial because each of the polarization channel shows different sensitivity to various land features. Further, using the textural features, various land classes can be classified on the basis of the textural measures. One of the textural measure is fractal dimension. It is noteworthy that fractal dimension is a measure of roughness and thus various land classes can be distinguished on the basis of their roughness. The value of fractal dimension for the surfaces lies between 2.0 and 3.0 where 2.0 represents a smooth surface while 3.0 represents drastically rough surface. The study area covers subset images lying between 2956'53"N, 7750'32"E and 2950'40"N, 7757'19"E. The PALSAR images of the year 2007 and 2009 are considered for the study. In present paper a fractal based classification of PALSAR images has been performed for identification of Water, Urban and Agricultural Area. Since fractals represent the image texture, hence the present study attempts to find the fractal properties of land covers to distinguish them from one another. For the purpose a context has been defined on the basis of a moving window, which is used to estimate the local fractal dimension and then moved over the whole image. The size of the window is an important issue for estimation of textural measures which is considered to be 55 in present study. This procedure, in response, produces a textural map called fractal map. The fractal map is constituted with the help of local fractal dimension values and can be used for contextual classification. In order to study the fractal properties of PALSAR images, the three polarization images viz. HH (Horizontal-Horizontal Polarization), VV (Vertical-Vertical Polarization) and HV (Horizontal-Vertical Polarization) are considered individually. First of all each polarized image is classified in an unsupervised way and various clusters, i.e., four clusters are identified with the help of reference data as Water, Urban and Agricultural Area. For each cluster, the fractal dimension is obtained from the fractal map. Based on the study the ranges of fractal dimension for three classes are Water: 2.0-2.17, Agricultural Area: 2.24-2.72, Urban Area: 2.63-2.92 for HH polarized image; Water: 2.0-2.21, Agricultural Area: 2.20-2.64, Urban; 2.58-2.94 for VV polarized image and Water: 2.0-2.14, Agricultural Area: 2.18-2.58, Urban: 2.46-2.94 for HV polarized image. Since the class Others represents a mixture of various classes, an explicit range of D for this class can not be determined. A closer look at the ranges of fractal dimension indicates that there is an overlapping of the values for different classes, despite of which the classes can be distinguished. Also, the class Water having low value of fractal dimension can be treated as smooth and Urban Area having higher values of fractal dimension can be considered rough in structure while the class Agricultural Area shows an intermediate roughness.
Fractal dimension structure of Cosmos and its mathematical foundations
NASA Astrophysics Data System (ADS)
Yan, Kun
Fractal dimension structure of the Cosmos are explored, and the mathematical foundation, which include the expressions of fractal dimension differential and calculus, regular space integral solutions of fractal dimension differential equations, the fractal calculus definitions of fractal measure as well as the measure computational equation of self-similar fractal, of fractal dimension calculus and fractal measure are given. As annotation, an equation of the relation between neutrons and protons in nuclei and its periodical solutions as well as atomic number limit are discussed.
Trabecular Bone Mechanical Properties and Fractal Dimension
NASA Technical Reports Server (NTRS)
Hogan, Harry A.
1996-01-01
Countermeasures for reducing bone loss and muscle atrophy due to extended exposure to the microgravity environment of space are continuing to be developed and improved. An important component of this effort is finite element modeling of the lower extremity and spinal column. These models will permit analysis and evaluation specific to each individual and thereby provide more efficient and effective exercise protocols. Inflight countermeasures and post-flight rehabilitation can then be customized and targeted on a case-by-case basis. Recent Summer Faculty Fellowship participants have focused upon finite element mesh generation, muscle force estimation, and fractal calculations of trabecular bone microstructure. Methods have been developed for generating the three-dimensional geometry of the femur from serial section magnetic resonance images (MRI). The use of MRI as an imaging modality avoids excessive exposure to radiation associated with X-ray based methods. These images can also detect trabecular bone microstructure and architecture. The goal of the current research is to determine the degree to which the fractal dimension of trabecular architecture can be used to predict the mechanical properties of trabecular bone tissue. The elastic modulus and the ultimate strength (or strain) can then be estimated from non-invasive, non-radiating imaging and incorporated into the finite element models to more accurately represent the bone tissue of each individual of interest. Trabecular bone specimens from the proximal tibia are being studied in this first phase of the work. Detailed protocols and procedures have been developed for carrying test specimens through all of the steps of a multi-faceted test program. The test program begins with MRI and X-ray imaging of the whole bones before excising a smaller workpiece from the proximal tibia region. High resolution MRI scans are then made and the piece further cut into slabs (roughly 1 cm thick). The slabs are X-rayed again and also scanned using dual-energy X-ray absorptiometry (DEXA). Cube specimens are then cut from the slabs and tested mechanically in compression. Correlations between mechanical properties and fractal dimension will then be examined to assess and quantify the predictive capability of the fractal calculations.
Edge extraction of optical subaperture based on fractal dimension method
NASA Astrophysics Data System (ADS)
Wang, Yunqi; Hui, Mei; Liu, Ming; Dong, Liquan; Liu, Xiaohua; Zhao, Yuejin
2015-09-01
Optical synthetic aperture imaging technology is an effective approach to increase the aperture diameter of optical system for purpose of improving resolution. In optical synthetic aperture imaging system, the edge is more complex than that of traditional optical imaging system, and the relatively large size of the gaps between the subapertures makes cophasing a difficult problem. So it is significant to extract edge phase of each subaperture for achieving phase stitching and avoiding the loss of effective frequency. Fractal dimension as a measure feature of image surface irregularities can statistically evaluate the complexity which is consistent with human visual image perception of rough surface texture. Therefore, fractal dimension provides a powerful tool to describe surface characteristics of image and can be applied to edge extraction. In our research, the box-counting dimension was used to calculate fractal dimension of the whole image. Then the calculated fractal dimension is mapped to grayscale image. The region with large fractal dimension represents a sharper change of the gray scale in original image, which was accurately extracted as the edge region. Subaperture region and interference fringe edge was extracted from interference pattern of optical subaperture, which has laid the foundation for the subaperture edge phase detection in the future work.
Trabecular Pattern Analysis Using Fractal Dimension
NASA Astrophysics Data System (ADS)
Ishida, Takayuki; Yamashita, Kazuya; Takigawa, Atsushi; Kariya, Komyo; Itoh, Hiroshi
1993-04-01
Feature extraction from a digitized image is advantageous for the detection of signs of disease. In this work, we attempted to evaluate bone trabecular pattern changes in osteoporosis using the fractal dimension and the root mean square (RMS) values. The relationship between the fractal dimension and the 1st moment of the power spectrum is explored, and we investigated the relationship between the results of this analysis and the bone mineral density (BMD) value which was measured using dual-energy X-ray absorptiometry (DEXA). As a result, we were able to extract useful information, using the fractal dimension and the RMS value of the radiographs (lateral view of the lumbar vertebrae), for the diagnosis of osteoporosis. Abnormal clinical cases were separated from normal cases based on the evaluation values. Negligible correlation between the BMD value and these indexes was observed.
Imai, K; Ikeda, M; Enchi, Y; Niimi, T
2009-12-01
The purposes of our studies are to examine whether or not fractal-feature distance deduced from virtual volume method can simulate observer performance indices and to investigate the physical meaning of pseudo fractal dimension and complexity. Contrast-detail (C-D) phantom radiographs were obtained at various mAs values (0.5 - 4.0 mAs) and 140 kVp with a computed radiography system, and the reference image was acquired at 13 mAs. For all C-D images, fractal analysis was conducted using the virtual volume method that was devised with a fractional Brownian motion model. The fractal-feature distances between the considered and reference images were calculated using pseudo fractal dimension and complexity. Further, we have performed the C-D analysis in which ten radiologists participated, and compared the fractal-feature distances with the image quality figures (IQF). To clarify the physical meaning of the pseudo fractal dimension and complexity, contrast-to-noise ratio (CNR) and standard deviation (SD) of images noise were calculated for each mAs and compared with the pseudo fractal dimension and complexity, respectively. A strong linear correlation was found between the fractal-feature distance and IQF. The pseudo fractal dimensions became large as CNR increased. Further, a linear correlation was found between the exponential complexity and image noise SD. PMID:20169837
Activity dependence of solar supergranular fractal dimension
NASA Astrophysics Data System (ADS)
Paniveni, U.; Krishan, V.; Singh, Jagdev; Srikanth, R.
2010-02-01
We study the complexity of supergranular cells using the intensity patterns obtained at the Kodaikanal Solar Observatory during the solar maximum. Our data consist of visually identified supergranular cells, from which a fractal dimension D for supergranulation is obtained according to the relation P ~ AD/2, where A is the area and P the perimeter of the supergranular cells. We find a fractal dimension of about 1.12 for active region cells and about 1.25 for quiet region cells, a difference that could be attributed to the inhibiting effect of the magnetic field.
ERIC Educational Resources Information Center
Esbenshade, Donald H., Jr.
1991-01-01
Develops the idea of fractals through a laboratory activity that calculates the fractal dimension of ordinary white bread. Extends use of the fractal dimension to compare other complex structures as other breads and sponges. (MDH)
The Correlation Fractal Dimension of Complex Networks
NASA Astrophysics Data System (ADS)
Wang, Xingyuan; Liu, Zhenzhen; Wang, Mogei
2013-05-01
The fractality of complex networks is studied by estimating the correlation dimensions of the networks. Comparing with the previous algorithms of estimating the box dimension, our algorithm achieves a significant reduction in time complexity. For four benchmark cases tested, that is, the Escherichia coli (E. Coli) metabolic network, the Homo sapiens protein interaction network (H. Sapiens PIN), the Saccharomyces cerevisiae protein interaction network (S. Cerevisiae PIN) and the World Wide Web (WWW), experiments are provided to demonstrate the validity of our algorithm.
Fractal dimension of alumina aggregates grown in two dimensions
NASA Technical Reports Server (NTRS)
Larosa, Judith L.; Cawley, James D.
1992-01-01
The concepts of fractal geometry are applied to the analysis of 0.4-micron alumina constrained to agglomerate in two dimensions. Particles were trapped at the bottom surface of a drop of a dilute suspension, and the agglomeration process was directly observed, using an inverted optical microscope. Photographs were digitized and analyzed, using three distinct approaches. The results indicate that the agglomerates are fractal, having a dimension of approximately 1.5, which agrees well with the predictions of the diffusion-limited cluster-cluster aggregation model.
Estimation of fractal dimensions from transect data
Loehle, C.
1994-04-01
Fractals are a useful tool for analyzing the topology of objects such as coral reefs, forest canopies, and landscapes. Transects are often studied in these contexts, and fractal dimensions computed from them. An open question is how representative a single transect is. Transects may also be used to estimate the dimensionality of a surface. Again the question of representativeness of the transect arises. These two issues are related. This note qualifies the conditions under which transect data may be considered to be representative or may be extrapolated, based on both theoretical and empirical results.
Fractal dimension analyses of lava surfaces and flow boundaries
NASA Technical Reports Server (NTRS)
Cleghorn, Timothy F.
1993-01-01
An improved method of estimating fractal surface dimensions has been developed. The accuracy of this method is illustrated using artificially generated fractal surfaces. A slightly different from usual concept of linear dimension is developed, allowing a direct link between that and the corresponding surface dimension estimate. These methods are applied to a series of images of lava flows, representing a variety of physical and chemical conditions. These include lavas from California, Idaho, and Hawaii, as well as some extraterrestrial flows. The fractal surface dimension estimations are presented, as well as the fractal line dimensions where appropriate.
Segmentation of magnetic resonance image using fractal dimension
NASA Astrophysics Data System (ADS)
Yau, Joseph K. K.; Wong, Sau-hoi; Chan, Kwok-Leung
1997-04-01
In recent years, much research has been conducted in the three-dimensional visualization of medical image. This requires a good segmentation technique. Many early works use first-order and second-order statistics. First-order statistical parameters can be calculated quickly but their effectiveness is influenced by many factors such as illumination, contrast and random noise of the image. Second-order statistical parameters, such as spatial gray level co-occurrence matrices statistics, take longer time to compute but can extract the textural information. In this investigating, two different parameters, namely the entropy and the fractal dimension, are employed to perform segmentation of the magnetic resonance images of the head of a male cadaver. The entropy is calculated from the spatial gray level co-occurrence matrices. The fractal dimension is calculated by the reticular cell counting method. Several regions of the human head are chosen for analysis. They are the bone, gyrus and lobe. Results show that the parameters are able to segment different types of tissue. The entropy gives very good result but it requires very long computation time and large amount of memory. The performance of the fractal dimension is comparable with the entropy. It is simple to estimate and demands lesser memory space.
Local Earth's gravity field in view of fractal dimension
NASA Astrophysics Data System (ADS)
Mészárosová, Katarína; Minarechová, Zuzana; Janák, Juraj
2013-04-01
The poster presents the relative roughness of chosen characteristics of the Earth's gravity field in several small regions in area of Slovakia (e.g. free-air anomaly, Bouguer anomaly, gravity disturbance...) using the values of fractal dimension. In this approach, a three dimensional box counting method and the Hurst analysis method are applied to estimate the values of fractal dimensions. Then the computed fractal dimension values are used to compare all 3D models of all chosen characteristics.
Fractal dimension in nonhyperbolic chaotic scattering
NASA Technical Reports Server (NTRS)
Lau, Yun-Tung; Finn, John M.; Ott, Edward
1991-01-01
In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic (e.g., there are Kolmogorov-Arnol'd-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic case, although the singular set has zero total length, strong evidence is presented to show that its fractal dimension is 1.
Statistics of Active Region Complexity: A Large-Scale Fractal Dimension Survey
NASA Astrophysics Data System (ADS)
McAteer, R. T. James; Gallagher, Peter T.; Ireland, Jack
2005-09-01
A quantification of the magnetic complexity of active regions using a fractal dimension measure is presented. This fully automated approach uses full-disk MDI magnetograms of active regions from a large data set (2742 days of the SOHO mission, 9342 active region images) to compare the calculated fractal dimension of each region to both its Mount Wilson classification and flare rate. Each Mount Wilson class exhibits a similar fractal dimension frequency distribution, possibly suggesting a self-similar nature of all active regions. Solar flare productivity exhibits an increase in both the frequency and GOES X-ray magnitude of flares from regions with higher fractal dimension. Specifically, a lower threshold fractal dimension of 1.2 and 1.25 exists as a necessary, but not sufficient, requirement for an active region to produce M- and X-class flares, respectively, within 24 hr of the observation.
Fractal dimension based corneal fungal infection diagnosis
NASA Astrophysics Data System (ADS)
Balasubramanian, Madhusudhanan; Perkins, A. Louise; Beuerman, Roger W.; Iyengar, S. Sitharama
2006-08-01
We present a fractal measure based pattern classification algorithm for automatic feature extraction and identification of fungus associated with an infection of the cornea of the eye. A white-light confocal microscope image of suspected fungus exhibited locally linear and branching structures. The pixel intensity variation across the width of a fungal element was gaussian. Linear features were extracted using a set of 2D directional matched gaussian-filters. Portions of fungus profiles that were not in the same focal plane appeared relatively blurred. We use gaussian filters of standard deviation slightly larger than the width of a fungus to reduce discontinuities. Cell nuclei of cornea and nerves also exhibited locally linear structure. Cell nuclei were excluded by their relatively shorter lengths. Nerves in the cornea exhibited less branching compared with the fungus. Fractal dimensions of the locally linear features were computed using a box-counting method. A set of corneal images with fungal infection was used to generate class-conditional fractal measure distributions of fungus and nerves. The a priori class-conditional densities were built using an adaptive-mixtures method to reflect the true nature of the feature distributions and improve the classification accuracy. A maximum-likelihood classifier was used to classify the linear features extracted from test corneal images as 'normal' or 'with fungal infiltrates', using the a priori fractal measure distributions. We demonstrate the algorithm on the corneal images with culture-positive fungal infiltrates. The algorithm is fully automatic and will help diagnose fungal keratitis by generating a diagnostic mask of locations of the fungal infiltrates.
Single cell correlation fractal dimension of chromatin
Récamier, Vincent; Izeddin, Ignacio; Bosanac, Lana; Dahan, Maxime; Proux, Florence; Darzacq, Xavier
2014-01-01
Chromatin is a major nuclear component, and it is an active matter of debate to understand its different levels of spatial organization, as well as its implication in gene regulation. Measurements of nuclear chromatin compaction were recently used to understand how DNA is folded inside the nucleus and to detect cellular dysfunctions such as cancer. Super-resolution imaging opens new possibilities to measure chromatin organization in situ. Here, we performed a direct measure of chromatin compaction at the single cell level. We used histone H2B, one of the 4 core histone proteins forming the nucleosome, as a chromatin density marker. Using photoactivation localization microscopy (PALM) and adaptive optics, we measured the three-dimensional distribution of H2B with nanometric resolution. We computed the distribution of distances between every two points of the chromatin structure, namely the Ripley K(r) distribution. We found that the K(r) distribution of H2B followed a power law, leading to a precise measurement of the correlation fractal dimension of chromatin of 2.7. Moreover, using photoactivable GFP fused to H2B, we observed dynamic evolution of chromatin sub-regions compaction. As a result, the correlation fractal dimension of chromatin reported here can be interpreted as a dynamically maintained non-equilibrium state. PMID:24637833
Fractal dimension analysis of complexity in Ligeti piano pieces
NASA Astrophysics Data System (ADS)
Bader, Rolf
2005-04-01
Fractal correlation dimensional analysis has been performed with whole solo piano pieces by Gyrgy Ligeti at every 50ms interval of the pieces. The resulting curves of development of complexity represented by the fractal dimension showed up a very reasonable correlation with the perceptional density of events during these pieces. The seventh piece of Ligeti's ``Musica ricercata'' was used as a test case. Here, each new part of the piece was followed by an increase of the fractal dimension because of the increase of information at the part changes. The second piece ``Galamb borong,'' number seven of the piano Etudes was used, because Ligeti wrote these Etudes after studying fractal geometry. Although the piece is not fractal in the strict mathematical sense, the overall structure of the psychoacoustic event-density as well as the detailed event development is represented by the fractal dimension plot.
Fractal Dimension of Earthquakes From Relocated Seismicity
NASA Astrophysics Data System (ADS)
Nadaeau, R. M.
2001-12-01
Fractal dimension (D) describing the distribution of earthquakes has been shown to be a very useful parameter for understanding earthquakes on many levels. Estimates of D have been used to infer the state of stress in Earth's crust, the degree of predictability of earthquakes, scaling relationships for earthquake source parameters recurrence times and b-values, and for estimating kernels used in earthquake hazard estimation. Most estimates of D (D2 and Do) for earthquake distributions are based on earthquake epicentral (2-D) or hypocenters (3-D) locations from standard location catalogs derived using routine location methods. Location uncertainties using these methods are typically on the order of 1 km or more in both relative terms. Relative uncertainties can be viewed as the scatter of earthquakes from their true locations, and in 2 and 3 dimensions this property preferentially increases separation distances (offsets) between events. This imposes serious limitations on the accuracy with which D can be determined, since it limits the usable range of event separations and introduces a bias in D estimates towards larger values by diminishing the numbers of small offsets in favor of larger offsets. Location uncertainties can also mask second order effects in fractal structure such as log-periodic fluctuations indicative of discrete rescaling, hierarchal clustering, and lineations in earthquake quake distributions. In this study, various catalogs of relocated events derived using cross-correlation and double-difference techniques on earthquakes occurring along the SAF system in Central CA are used to estimate D (3-D) and the estimates are compared with those derived using corresponding routine catalogs and relocated catalogs with random scatter added. Implications of the results are discussed. Initial results indicate that D using relocated events is generally between 1 and 1.5, compared to 2 or greater using routine catalogs and relocated catalogs with 1 km of added scatter. The effect of masking second order fractal structure is also observed. The lower values of D suggest a lower stress criticality in Earth's crust in this region and greater predictability of earthquake occurrence. Lower D is also in accordance with scaling relationships derived using repeating earthquakes from this region.
Fractal dimension and mechanism of aggregation of apple juice particles.
Benítez, E I; Lozano, J E; Genovese, D B
2010-04-01
Turbidity of freshly squeezed apple juice is produced by a polydisperse suspension of particles coming from the cellular tissue. After precipitation of coarse particles by gravity, only fine-colloidal particles remain in suspension. Aggregation of colloidal particles leads to the formation of fractal structures. The fractal dimension is a measure of the internal density of these aggregates and depends on their mechanism of aggregation. Digitized images of primary particles and aggregates of depectinized, diafiltered cloudy apple juice were obtained by scanning electron microscopy (SEM). Average radius of the primary particles was found to be a = 40 ± 11 nm. Maximum radius of the aggregates, R(L), ranged between 250 and 7750 nm. Fractal dimension of the aggregates was determined by analyzing SEM images with the variogram method, obtaining an average value of D(f) = 2.3 ± 0.1. This value is typical of aggregates formed by rapid flocculation or diffusion limited aggregation. Diafiltration process was found to reduce the average size and polydispersity of the aggregates, determined by photon correlation spectroscopy. Average gyration radius of the aggregates before juice diafiltration was found to be R(g) = 629 ± 87 nm. Average number of primary particles per aggregate was calculated to be N = 1174. PMID:21339133
NASA Astrophysics Data System (ADS)
Ulyanov, Alexander S.; Lyapina, Anna M.; Ulianova, Onega V.; Feodorova, Valentina A.
2010-10-01
New field of application of fractal dimensions is proposed. A technique, based on the calculation of fractal dimension, was used for express-diagnostics and identification of bacteria of the vaccine strain Yersinia pestis EV line NIIEG. Purpose of this study was the experimental investigation of properties of speckle patterns, formed under laser illumination of a single colony of the strain that was grown on different agars.
NASA Astrophysics Data System (ADS)
Ulyanov, Alexander S.; Lyapina, Anna M.; Ulianova, Onega V.; Feodorova, Valentina A.
2011-03-01
New field of application of fractal dimensions is proposed. A technique, based on the calculation of fractal dimension, was used for express-diagnostics and identification of bacteria of the vaccine strain Yersinia pestis EV line NIIEG. Purpose of this study was the experimental investigation of properties of speckle patterns, formed under laser illumination of a single colony of the strain that was grown on different agars.
Shape analysis using fractal dimension: a curvature based approach.
Backes, André R; Florindo, João B; Bruno, Odemir M
2012-12-01
The present work shows a novel fractal dimension method for shape analysis. The proposed technique extracts descriptors from a shape by applying a multi-scale approach to the calculus of the fractal dimension. The fractal dimension is estimated by applying the curvature scale-space technique to the original shape. By applying a multi-scale transform to the calculus, we obtain a set of descriptors which is capable of describing the shape under investigation with high precision. We validate the computed descriptors in a classification process. The results demonstrate that the novel technique provides highly reliable descriptors, confirming the efficiency of the proposed method. PMID:23278038
Archaeon and archaeal virus diversity classification via sequence entropy and fractal dimension
NASA Astrophysics Data System (ADS)
Tremberger, George, Jr.; Gallardo, Victor; Espinoza, Carola; Holden, Todd; Gadura, N.; Cheung, E.; Schneider, P.; Lieberman, D.; Cheung, T.
2010-09-01
Archaea are important potential candidates in astrobiology as their metabolism includes solar, inorganic and organic energy sources. Archaeal viruses would also be expected to be present in a sustainable archaeal exobiological community. Genetic sequence Shannon entropy and fractal dimension can be used to establish a two-dimensional measure for classification and phylogenetic study of these organisms. A sequence fractal dimension can be calculated from a numerical series consisting of the atomic numbers of each nucleotide. Archaeal 16S and 23S ribosomal RNA sequences were studied. Outliers in the 16S rRNA fractal dimension and entropy plot were found to be halophilic archaea. Positive correlation (R-square ~ 0.75, N = 18) was observed between fractal dimension and entropy across the studied species. The 16S ribosomal RNA sequence entropy correlates with the 23S ribosomal RNA sequence entropy across species with R-square 0.93, N = 18. Entropy values correspond positively with branch lengths of a published phylogeny. The studied archaeal virus sequences have high fractal dimensions of 2.02 or more. A comparison of selected extremophile sequences with archaeal sequences from the Humboldt Marine Ecosystem database (Wood-Hull Oceanography Institute, MIT) suggests the presence of continuous sequence expression as inferred from distributions of entropy and fractal dimension, consistent with the diversity expected in an exobiological archaeal community.
Fractal dimension and unscreened angles measured for radial viscous fingering
NASA Astrophysics Data System (ADS)
Praud, Olivier; Swinney, Harry L.
2005-07-01
We have examined fractal patterns formed by the injection of air into oil in a thin (0.127mm) layer contained between two cylindrical glass plates of 288mm diameter (a Hele-Shaw cell), for pressure differences in the range 0.25⩽ΔP⩽1.75atm . We find that an asymptotic structure is reached at large values of the ratio r/b , where r is the pattern radius and b the gap between the plates. Both the driving force and the size of the pattern, which reaches r/b=900 , are far larger than in past experiments. The fractal dimension D0 of the pattern for large r/b is 1.70±0.02 . Further, the generalized dimensions Dq of the pattern are independent of q , Dq≃1.70 for the range examined, -11calculations for diffusion-limited aggregation (DLA) clusters. We have also measured the probability distribution of unscreened angles. At late times, the distribution approaches a universal (i.e., forcing and size-independent) asymptotic form that has mean 145° and standard deviation 36°. These results indicate that the distribution function for the unscreened angle is an invariant property of the growth process.
Fractal dimensions of rampart impact craters on Mars
NASA Technical Reports Server (NTRS)
Ching, Delwyn; Taylor, G. Jeffrey; Mouginis-Mark, Peter; Bruno, Barbara C.
1993-01-01
Ejecta blanket morphologies of Martian rampart craters may yield important clues to the atmospheric densities during impact, and the nature of target materials (e.g., hard rock, fine-grained sediments, presence of volatiles). In general, the morphologies of such craters suggest emplacement by a fluidized, ground hugging flow instead of ballistic emplacement by dry ejecta. We have quantitatively characterized the shape of the margins of the ejecta blankets of 15 rampart craters using fractal geometry. Our preliminary results suggest that the craters are fractals and are self-similar over scales of approximately 0.1 km to 30 km. Fractal dimensions (a measure of the extent to which a line fills a plane) range from 1.06 to 1.31. No correlations of fractal dimension with target type, elevation, or crater size were observed, though the data base is small. The range in fractal dimension and lack of correlation may be due to a complex interplay of target properties (grain size, volatile content), atmospheric pressure, and crater size. The mere fact that the ejecta margins are fractals, however, indicates that viscosity and yield strength of the ejecta were at least as low as those of basalts, because silicic lava flows are not generally fractals.
Relationship between the fractal dimension and the width to length ratio of mass movements
NASA Astrophysics Data System (ADS)
Sezer, Ebru
2009-04-01
Mass movements have some typical geometrical dimensions. One of these typical geometrical dimensions is the width to length ratio. Also, the fractal dimensions of mass movements from the inventory maps of natural mass movements can be used for their geometrical description and characterization. For this reason, in the present study, development of a computer programme for digitizing and determining the fractal dimensions of mass movements, and investigation of the relationship between the fractal dimensions and the width to length (W/L) tario of the mass movements are aimed. For the purpose of the study, a computer programme namely FRACEK for determination of fractal dimensions of amorphous areas is developed by using the JAVA computer language at first. Secondly, a database including the shapes of the mass movements was compiled from the literature and digitized. Then, their width to length ratios and fractal dimensions were calculated. Finally, a series of simple statistical analyses were performed on the data obtained and the results were interpreted. To investigate the relationships between the fractal dimensions and W/L ratios of the mass movements, a series of simple regression analysis is performed. During the regression analyses, linear, power, logarithmic and exponential functions are employed. According to the results obtained, there are some correlations between the D and the W/L ratio. When considering only debris flow data, a power relationship between the D and the W/L ratio was found and its coefficient of correlation was obtained as 0.85. The lowest coefficient of correlations were obtained from the rotational failure data. The coefficients of correlations of the power and exponential funtions were same, 0.53. A similar result was obtained for the translational failure data. Their coefficient of correlations was 0.74. When all data is evaluated together, a relatively strong correlation between the D and the W/L ratio was obtained. These results revealed that to make a differantiation among the mass movements using the fractal dimension is possible.
NASA Astrophysics Data System (ADS)
Gao, Wei; Zakharov, Valery P.; Myakinin, Oleg O.; Bratchenko, Ivan A.; Artemyev, Dmitry N.; Kornilin, Dmitry V.
2015-07-01
Optical coherence tomography (OCT) is usually employed for the measurement of retinal thickness characterizing the structural changes of tissue. However, fractal dimension (FD) could also character the structural changes of tissue. Therefore, fractal dimension changes may provide further information regarding cellular layers and early damage in ocular diseases. We investigated the possibility of OCT in detecting changes in fractal dimension from layered retinal structures. OCT images were obtained from diabetic patients without retinopathy (DM, n = 38 eyes) or mild diabetic retinopathy (MDR, n = 43 eyes) and normal healthy subjects (Controls, n = 74 eyes). Fractal dimension was calculated using the differentiate box counting methodology. We evaluated the usefulness of quantifying fractal dimension of layered structures in the detection of retinal damage. Generalized estimating equations considering within-subject intereye relations were used to test for differences between the groups. A modified p value of <0.001 was considered statistically significant. Receiver operating characteristic (ROC) curves were constructed to describe the ability of fractal dimension to discriminate between the eyes of DM, MDR and healthy eyes. Significant decreases of fractal dimension were observed in all layers in the MDR eyes compared with controls except in the inner nuclear layer (INL). Significant decreases of fractal dimension were also observed in all layers in the MDR eyes compared with DM eyes. The highest area under receiver operating characteristic curve (AUROC) values estimated for fractal dimension were observed for the outer plexiform layer (OPL) and outer segment photoreceptors (OS) when comparing MDR eyes with controls. The highest AUROC value estimated for fractal dimension were also observed for the retinal nerve fiber layer (RNFL) and OS when comparing MDR eyes with DM eyes. Our results suggest that fractal dimension of the intraretinal layers may provide useful information to differentiate pathological from healthy eyes. Further research is warranted to determine how this approach may be used to improve diagnosis of early retinal neurodegeneration.
Smitha, K A; Gupta, A K; Jayasree, R S
2015-09-01
Glioma, the heterogeneous tumors originating from glial cells, generally exhibit varied grades and are difficult to differentiate using conventional MR imaging techniques. When this differentiation is crucial in the disease prognosis and treatment, even the advanced MR imaging techniques fail to provide a higher discriminative power for the differentiation of malignant tumor from benign ones. A powerful image processing technique applied to the imaging techniques is expected to provide a better differentiation. The present study focuses on the fractal analysis of fluid attenuation inversion recovery MR images, for the differentiation of glioma. For this, we have considered the most important parameters of fractal analysis, fractal dimension and lacunarity. While fractal analysis assesses the malignancy and complexity of a fractal object, lacunarity gives an indication on the empty space and the degree of inhomogeneity in the fractal objects. Box counting method with the preprocessing steps namely binarization, dilation and outlining was used to obtain the fractal dimension and lacunarity in glioma. Statistical analysis such as one-way analysis of variance and receiver operating characteristic (ROC) curve analysis helped to compare the mean and to find discriminative sensitivity of the results. It was found that the lacunarity of low and high grade gliomas vary significantly. ROC curve analysis between low and high grade glioma for fractal dimension and lacunarity yielded 70.3% sensitivity and 66.7% specificity and 70.3% sensitivity and 88.9% specificity, respectively. The study observes that fractal dimension and lacunarity increases with an increase in the grade of glioma and lacunarity is helpful in identifying most malignant grades. PMID:26305773
Fractal dimension analysis of cerebellum in Chiari Malformation type I.
Akar, Engin; Kara, Sadık; Akdemir, Hidayet; Kırış, Adem
2015-09-01
Chiari Malformation type I (CM-I) is a serious neurological disorder that is characterized by hindbrain herniation. Our aim was to evaluate the usefulness of fractal analysis in CM-I patients. To examine the morphological complexity features of this disorder, fractal dimension (FD) of cerebellar regions were estimated from magnetic resonance images (MRI) of 17 patients with CM-I and 16 healthy control subjects in this study. The areas of white matter (WM), gray matter (GM) and cerebrospinal fluid (CSF) were calculated and the corresponding FD values were computed using a 2D box-counting method in both groups. The results indicated that CM-I patients had significantly higher (p<0.05) FD values of GM, WM and CSF tissues compared to control group. According to the results of correlation analysis between FD values and the corresponding area values, FD and area values of GM tissues in the patients group were found to be correlated. The results of the present study suggest that FD values of cerebellar regions may be a discriminative feature and a useful marker for investigation of abnormalities in the cerebellum of CM-I patients. Further studies to explore the changes in cerebellar regions with the help of 3D FD analysis and volumetric calculations should be performed as a future work. PMID:26189156
Measuring fractal dimension of metro systems
NASA Astrophysics Data System (ADS)
Deng, S.; Li, W.; Gu, J.; Zhu, Y.; Zhao, L.; Han, J.
2015-04-01
We discuss cluster growing method and box-covering method as well as their connection to fractal geometry. Our measurements show that for small network systems, box-covering method gives a better scaling relation. We then measure both unweighted and weighted metro networks with optimal box-covering method.
Mukherjee, Anika; Chan, Adrian D C; Keating, Sarah; Redline, Raymond W; Fritsch, Michael K; Machin, Geoffrey A; Cornejo-Palma, Daniel; de Nanassy, Joseph; El-Demellawy, Dina; von Dadelszen, Peter; Benton, Samantha J; Grynspan, David
2016-01-01
The distal villous hypoplasia (DVH) pattern is a placental correlate of fetal growth restriction. Because the pattern seems to involve less complexity than do appropriately developed placental villi, we postulated that it may be associated with lower fractal dimension-a mathematical measure of complexity. Our study objectives were to evaluate interobserver agreement related to the DVH pattern among expert pathologists and to determine whether pathologist classification of DVH correlates with fractal dimension. A study set of 30 images of placental parenchyma at ×4 magnification was created by a single pathologist from a digital slide archive. The images were graded for the DVH pattern according to pre-specified definitions and included 10 images graded as "no DVH" (grade = 0), 10 with mild to moderate DVH (grade = 1), and 10 with severe DVH (grade = 2). The images were randomly sorted and shown to a panel of 4 international experts who similarly graded the images for DVH. Weighted kappas were calculated. For each image, fractal dimension was calculated by the Box Counting method. The correlation coefficient between (1) the averaged DVH scores obtained by the 5 pathologists and (2) fractal dimension was calculated. The mean weighted kappa score among the observers was 0.59 (range: 0.42-0.70). The correlation coefficient between fractal dimension and the averaged DVH score was -0.915 (P < 0.001). Expert pathologists achieve fair to substantial agreement in grading DVH, indicating consensus on the definition of DVH. Distal villous hypoplasia correlates extremely well with fractal dimension and represents an objective measure for DVH. PMID:26275121
Electroencephalographic Fractal Dimension in Healthy Ageing and Alzheimer's Disease.
Smits, Fenne Margreeth; Porcaro, Camillo; Cottone, Carlo; Cancelli, Andrea; Rossini, Paolo Maria; Tecchio, Franca
2016-01-01
Brain activity is complex; a reflection of its structural and functional organization. Among other measures of complexity, the fractal dimension is emerging as being sensitive to neuronal damage secondary to neurological and psychiatric diseases. Here, we calculated Higuchi's fractal dimension (HFD) in resting-state eyes-closed electroencephalography (EEG) recordings from 41 healthy controls (age: 20-89 years) and 67 Alzheimer's Disease (AD) patients (age: 50-88 years), to investigate whether HFD is sensitive to brain activity changes typical in healthy aging and in AD. Additionally, we considered whether AD-accelerating effects of the copper fraction not bound to ceruloplasmin (also called "free" copper) are reflected in HFD fluctuations. The HFD measure showed an inverted U-shaped relationship with age in healthy people (R2 = .575, p < .001). Onset of HFD decline appeared around the age of 60, and was most evident in central-parietal regions. In this region, HFD decreased with aging stronger in the right than in the left hemisphere (p = .006). AD patients demonstrated reduced HFD compared to age- and education-matched healthy controls, especially in temporal-occipital regions. This was associated with decreasing cognitive status as assessed by mini-mental state examination, and with higher levels of non-ceruloplasmin copper. Taken together, our findings show that resting-state EEG complexity increases from youth to maturity and declines in healthy, aging individuals. In AD, brain activity complexity is further reduced in correlation with cognitive impairment. In addition, elevated levels of non-ceruloplasmin copper appear to accelerate the reduction of neural activity complexity. Overall, HDF appears to be a proper indicator for monitoring EEG-derived brain activity complexity in healthy and pathological aging. PMID:26872349
Comparison of Two Numerical Methods for Computing Fractal Dimensions
NASA Astrophysics Data System (ADS)
Shiozawa, Yui; Miller, Bruce; Rouet, Jean-Louis
2012-10-01
From cosmology to economics, the examples of fractals can be found virtually everywhere. However, since few fractals permit the analytical evaluation of generalized fractal dimensions or R'enyi dimensions, the search for effective numerical methods is inevitable. In this project two promising numerical methods for obtaining generalized fractal dimensions, based on the distribution of distances within a set, are examined. They can be applied, in principle, to any set even if no closed-form expression is available. The biggest advantage of these methods is their ability to generate a spectrum of generalized dimensions almost simultaneously. It should be noted that this feature is essential to the analysis of multifractals. As a test of their effectiveness, here the methods were applied to the generalized Cantor set and the multiplicative binomial process. The generalized dimensions of both sets can be readily derived analytically, thus enabling the accuracy of the numerical methods to be verified. Here we will present a comparison of the analytical results and the predictions of the methods. We will show that, while they are effective, care must be taken in their interpretation.
On the Numerical Study of the Complexity and Fractal Dimension of CMB Anisotropies
NASA Astrophysics Data System (ADS)
Allahverdyan, A. E.; Gurzadyan, V. G.; Soghoyan, A. A.
We consider the problem of numerical computation of the Kolmogorov complexity and the fractal dimension of the anisotropy spots of Cosmic Microwave Background (CMB) radiation. Namely, we describe an algorithm of estimation of the complexity of spots given by certain pixel configuration on a grid and represent the results of computations for a series of structures of different complexity. Thus, we demonstrate the calculability of such an abstract descriptor as the Kolmogorov complexity for CMB digitized maps. The correlation of complexity of the anisotropy spots with their fractal dimension is revealed as well. This technique can be especially important while analyzing the data of the forthcoming space experiments.
NASA Astrophysics Data System (ADS)
Feng, Yongjiu; Liu, Miaolong; Tong, Xiaohua
2007-06-01
An improved fractal measurement, the weighted radial dimension, is put forward for highway transportation networks distribution. The radial dimension (DL), originated from subway investigation in Stuttgart, is a fractal measurement for transportation systems under ideal assumption considering all the network lines to be homogeneous curves, ignoring the difference on spatial structure, quality and level, especially the highway networks. Considering these defects of radial dimension, an improved fractal measurement called weighted radial dimension (D WL) is introduced and the transportation system in Guangdong province is studied in detail using this novel method. Weighted radial dimensions are measured and calculated, and the spatial structure, intensity and connectivity of transportation networks are discussed in Guangdong province and the four sub-areas: the Pearl River Delta area, the East Costal area, the West Costal area and the Northern Guangdong area. In Guangdong province, the fractal spatial pattern characteristics of transportation system vary remarkably: it is the highest in the Pearl River Delta area, moderate in Costal area and lowest in the Northern Guangdong area. With the Pearl River Delta area as the centre, the weighted radial dimensions decrease with the distance increasing, while the decline level is smaller in the costal area and greater in the Northern Guangdong province. By analysis of the conic of highway density, it is recognized that the density decrease with the distance increasing from the calculation centre (Guangzhou), demonstrating the same trend as weighted radial dimensions shown. Evidently, the improved fractal measurement, weighted radial dimension, is an indictor describing the characteristics of highway transportation system more effectively and accurately.
NASA Astrophysics Data System (ADS)
Lin, Naiming; Guo, Junwen; Xie, Faqin; Zou, Jiaojuan; Tian, Wei; Yao, Xiaofei; Zhang, Hongyan; Tang, Bin
2014-08-01
In the field of corrosion research, mass gain/loss, electrochemical tests and comparing the surface elemental distributions, phase constitutions as well as surface morphologies before and after corrosion are extensively applied to investigate the corrosion behavior or estimate the corrosion resistance of materials that operated in various environments. Most of the above methods are problem oriented, complex and longer-period time-consuming. However from an object oriented point of view, the corroded surfaces of materials often have self-similar characterization: fractal property which can be employed to efficiently achieve damaged surface analysis. The present work describes a strategy of comparison of the surface fractal dimensions for corrosion resistance estimation: chromizing coating was synthesized on P110 steel surface to improve its performance via pack cementation. Scanning electron microscope (SEM) was used to investigate the surface morphologies of the original and corroded samples. Surface fractal dimensions of the detected samples were calculated by binary images related to SEM images of surface morphologies with box counting algorithm method. The results showed that both surface morphologies and surface fractal dimensions of P110 steel varied greatly before and after corrosion test, but the chromizing coating changed slightly. The chromizing coating indicated better corrosion resistance than P110 steel. Comparison of surface fractal dimensions of original and corroded samples can rapidly and exactly realize the estimation of corrosion resistance.
Fractal dimension computation from equal mass partitions.
Shiozawa, Yui; Miller, Bruce N; Rouet, Jean-Louis
2014-09-01
Numerical methods which utilize partitions of equal-size, including the box-counting method, remain the most popular choice for computing the generalized dimension of multifractal sets. However, it is known that mass-oriented methods generate relatively good results for computing generalized dimensions for important cases where the box-counting method is known to fail. Here, we revisit two mass-oriented methods and discuss their strengths and limitations. PMID:25273186
Effect of 3D fractal dimension on contact area and asperity interactions in elastoplastic contact
NASA Astrophysics Data System (ADS)
Jourani, Abdeljalil
2016-05-01
Few models are devoted to investigate the effect of 3D fractal dimension Ds on contact area and asperity interactions. These models used statistical approaches or two-dimensional deterministic simulations without considering the asperity interactions and elastic-plastic transition regime. In this study, a complete 3D deterministic model is adopted to simulate the contact between fractal surfaces which are generated using a modified two-variable Weierstrass-Mandelbrot function. This model incorporates the asperity interactions and considers the different deformation modes of surface asperities which range from entirely elastic through elastic-plastic to entirely plastic contact. The simulations reveal that the elastoplastic model is more appropriate to calculate the contact area ratio and pressure field. It is also shown that the influence of the asperity interactions cannot be neglected, especially at lower fractal dimension Ds and higher load.
Fractal dimension of steady nonequilibrium flows
Hoover, W.G. ); Posch, H.A. ); Hoover, C.G. )
1992-04-01
The Kaplan--Yorke information dimension of phase-space attractors for two kinds of steady nonequilibrium many-body flows is evaluated. In both cases a set of Newtonian particles is considered which interacts with boundary particles. Time-averaged boundary temperatures are imposed by Nose--Hoover thermostat forces. For both kinds of nonequilibrium systems, it is demonstrated numerically that external isothermal boundaries can drive the otherwise purely Newtonian flow onto a {ital multifractal} {ital attractor} with a phase-space information dimension significantly less than that of the corresponding equilibrium flow. Thus the Gibbs' entropy of such nonequilibrium flows can diverge.
Chain conformation dependence of the fractal dimension of polymer aggregates
NASA Astrophysics Data System (ADS)
Wu, Chi; Peng, Shufu
2001-03-01
The Ca++/COO- complexation induced aggregation of linear poly(N-vinylcaprolactam-co--sodium acrylate) (P(VCL-co-NaA) chains in water was investigated by a combination of static and dynamic laser light scattering. Using thermally sensitive PVCL, we were able alternate the chain conformation by temperature and study its effect on the structures of the resultant aggregates. We found, for the first time, that the fractal dimension of the aggregates decreased from 2.4 to 1.6 when the chain conformation changed from a crumpled coil to a collapsed globule. Such a change of the fractal dimension could be attributed to the competition between the intrachain and interchain complexation.
Fractal dimension analysis of grey matter in multiple sclerosis.
Esteban, Francisco J; Sepulcre, Jorge; de Miras, Juan Ruiz; Navas, Juan; de Mendizábal, Nieves Vélez; Goñi, Joaquín; Quesada, José M A; Bejarano, Bartolome; Villoslada, Pablo
2009-07-15
The fractal dimension (FD) is a quantitative parameter that characterizes the morphometric variability of a complex object. Among other applications, FD has been used to identify abnormalities of the human brain in conventional magnetic resonance imaging (MRI), including white matter abnormalities in patients with Multiple Sclerosis (MS). Extensive grey matter (GM) pathology has been recently identified in MS and it appears to be a key factor in long-term disability. The aim of the present work was to assess whether FD measurement of GM in T1 MRI sequences can identify GM abnormalities in patients with MS in the early phase of the disease. A voxel-based morphometry approach optimized for MS was used to obtain the segmented brain, where we later calculated the three-dimensional FD of the GM in MS patients and healthy controls. We found that patients with MS had a significant increase in the FD of the GM compared to controls. Such differences were present even in patients with short disease durations, including patients with first attacks of MS. In addition, the FD of the GM correlated with T1 and T2 lesion load, but not with GM atrophy or disability. The FD abnormalities of the GM here detected differed from the previously published FD of the white matter in MS, suggesting that different pathological processes were taking place in each structure. These results indicate that GM morphology is abnormal in patients with MS and that this alteration appears early in the course of the disease. PMID:19167728
Fractal dimensions of flocs between clay particles and HAB organisms
NASA Astrophysics Data System (ADS)
Wang, Hongliang; Yu, Zhiming; Cao, Xihua; Song, Xiuxian
2011-05-01
The impact of harmful algal blooms (HABs) on public health and related economics have been increasing in many coastal regions of the world. Sedimentation of algal cells through flocculation with clay particles is a promising strategy for controlling HABs. Previous studies found that removal efficiency (RE) was influenced by many factors, including clay type and concentration, algal growth stage, and physiological aspects of HAB cells. To estimate the effect of morphological characteristics of the aggregates on HAB cell removal, fractal dimensions were measured and the RE of three species of HAB organism, Heterosigma akashiwo, Alexandrium tamarense, and Skeletonema costatum, by original clay and modified clay, was determined. For all HAB species, the modified clay had a higher RE than original clay. For the original clay, the two-dimensional fractal dimension ( D 2) was 1.92 and three-dimensional fractal dimension ( D 3) 2.81, while for the modified clay, D 2 was 1.84 and D 3 was 2.50. The addition of polyaluminum chloride (PACl) lead to a decrease of the repulsive barrier between clay particles, and resulted in lower D 2 and D 3. Due to the decrease of D 3, and the increase of the effective sticking coefficient, the flocculation rate between modified clay particles and HAB organisms increased, and thus resulted in a high RE. The fractal dimensions of flocs differed in HAB species with different cell morphologies. For example, Alexandrium tamarense cells are ellipsoidal, and the D 3 and D 2 of flocs were the highest, while for Skeletonema costatum, which has filamentous cells, the D 3 and D 2 of flocs were the lowest.
Estimation of Fractal Dimension in Differential Diagnosis of Pigmented Skin Lesions
NASA Astrophysics Data System (ADS)
Aralica, Gorana; Milošević, Danko; Konjevoda, Paško; Seiwerth, Sven; Štambuk, Nikola
Medical differential diagnosis is a method of identifying the presence of a particular entity (disease) within a set of multiple possible alternatives. The significant problem in dermatology and pathology is the differential diagnosis of malignant melanoma and other pigmented skin lesions, especially of dysplastic nevi. Malignant melanoma is the most malignant skin neoplasma, with increasing incidence in various parts of the world. It is hoped that the methods of quantitative pathology, i.e. morphometry, can help objectification of the diagnostic process, since early discovery of melanoma results in 10-year survival rate of 90%. The aim of the study was to use fractal dimension calculated from the perimeter-area relation of the cell nuclei as a tool for the differential diagnosis of pigmented skin lesions. We analyzed hemalaun-eosin stained pathohistological slides of pigmented skin lesions: intradermal naevi (n = 45), dysplastic naevi (n = 47), and malignant melanoma (n = 50). It was found that fractal dimension of malignant melanoma cell nuclei differs significantly from the intradermal and dysplastic naevi (p ≤ 0. 001, Steel-Dwass Multiple Comparison Test). Additionaly, ROC analysis confirmed the value of fractal dimension based evaluation. It is suggested that the estimation of fractal dimension from the perimeter-area relation of the cell nuclei may be a potentially useful morphometric parameter in the medical differential diagnosis of pigmented skin lesions.
Fractal dimension analysis of malignant and benign endobronchial ultrasound nodes
2014-01-01
Background Endobronchial ultrasonography (EBUS) has been applied as a routine procedure for the diagnostic of hiliar and mediastinal nodes. The authors assessed the relationship between the echographic appearance of mediastinal nodes, based on endobronchial ultrasound images, and the likelihood of malignancy. Methods The images of twelve malignant and eleven benign nodes were evaluated. A previous processing method was applied to improve the quality of the images and to enhance the details. Texture and morphology parameters analyzed were: the image texture of the echographies and a fractal dimension that expressed the relationship between area and perimeter of the structures that appear in the image, and characterizes the convoluted inner structure of the hiliar and mediastinal nodes. Results Processed images showed that relationship between log perimeter and log area of hilar nodes was lineal (i.e. perimeter vs. area follow a power law). Fractal dimension was lower in the malignant nodes compared with non-malignant nodes (1.47(0.09), 1.53(0.10) mean(SD), Mann–Whitney U test p < 0.05)). Conclusion Fractal dimension of ultrasonographic images of mediastinal nodes obtained through endobronchial ultrasound differ in malignant nodes from non-malignant. This parameter could differentiate malignat and non-malignat mediastinic and hiliar nodes. PMID:24920158
Fractal dimension as a measure of the scale of homogeneity
NASA Astrophysics Data System (ADS)
Yadav, Jaswant K.; Bagla, J. S.; Khandai, Nishikanta
2010-07-01
In the multifractal analysis of the large-scale matter distribution, the scale of the transition to homogeneity is defined as the scale above which the fractal dimension (Dq) of the underlying point distribution is equal to the ambient dimension (D) of the space in which points are distributed. With the finite sized weakly clustered distribution of tracers obtained from galaxy redshift surveys it is difficult to achieve this equality. Recently Bagla et al. have defined the scale of homogeneity to be the scale above which the deviation (ΔDq) of the fractal dimension from the ambient dimension becomes smaller than the statistical dispersion of ΔDq, i.e. . In this paper we use the relation between the fractal dimensions and the correlation function to compute for any given model in the limit of weak clustering amplitude. We compare ΔDq and for the Λ cold dark matter (ΛCDM) model and discuss the implication of this comparison for the expected scale of homogeneity in the concordant model of cosmology. We estimate the upper limit to the scale of homogeneity to be close to 260h-1Mpc for the ΛCDM model. Actual estimates of the scale of homogeneity should be smaller than this as we have considered only the statistical contribution to and we have ignored cosmic variance and contributions due to survey geometry and the selection function. Errors arising due to these factors enhance and as ΔDq decreases with increasing scale, we expect to measure a smaller scale of homogeneity. We find that as long as non-linear corrections to the computation of ΔDq are insignificant, the scale of homogeneity does not change with epoch. The scale of homogeneity depends very weakly on the choice of tracer of the density field. Thus the suggested definition of the scale of homogeneity is fairly robust.
Surface evaluation by estimation of fractal dimension and statistical tools.
Hotar, Vlastimil; Salac, Petr
2014-01-01
Structured and complex data can be found in many applications in research and development, and also in industrial practice. We developed a methodology for describing the structured data complexity and applied it in development and industrial practice. The methodology uses fractal dimension together with statistical tools and with software modification is able to analyse data in a form of sequence (signals, surface roughness), 2D images, and dividing lines. The methodology had not been tested for a relatively large collection of data. For this reason, samples with structured surfaces produced with different technologies and properties were measured and evaluated with many types of parameters. The paper intends to analyse data measured by a surface roughness tester. The methodology shown compares standard and nonstandard parameters, searches the optimal parameters for a complete analysis, and specifies the sensitivity to directionality of samples for these types of surfaces. The text presents application of fractal geometry (fractal dimension) for complex surface analysis in combination with standard roughness parameters (statistical tool). PMID:25250380
Surface Evaluation by Estimation of Fractal Dimension and Statistical Tools
Salac, Petr
2014-01-01
Structured and complex data can be found in many applications in research and development, and also in industrial practice. We developed a methodology for describing the structured data complexity and applied it in development and industrial practice. The methodology uses fractal dimension together with statistical tools and with software modification is able to analyse data in a form of sequence (signals, surface roughness), 2D images, and dividing lines. The methodology had not been tested for a relatively large collection of data. For this reason, samples with structured surfaces produced with different technologies and properties were measured and evaluated with many types of parameters. The paper intends to analyse data measured by a surface roughness tester. The methodology shown compares standard and nonstandard parameters, searches the optimal parameters for a complete analysis, and specifies the sensitivity to directionality of samples for these types of surfaces. The text presents application of fractal geometry (fractal dimension) for complex surface analysis in combination with standard roughness parameters (statistical tool). PMID:25250380
Fractal dimension of faults network in the upper silesian coal basin (Poland): Preliminary studies
NASA Astrophysics Data System (ADS)
Idziak, Adam; Teper, Lesław
1996-07-01
Fractal analysis of faults network, tremor foci spatial distribution as well as the Gutenberg-Richter relationship could further explain whether the biggest seismic events are connected with recent tectonic activity. Fractality of fault systems geometry, as a first step of the analysis, was tested fro a part of the USCB embodying the main structural units. The cluster analysis and the box counting methods were employed. The calculated fractal dimension of fault network was 1.98 for the whole area yet for considered structural units it was close to 1.6. The results point to similarity of studied fault pattern to river network. Faults within selected tectonic units make separate sets which have a distinct geometry and origin. The value of 1.6 is an upper limit to the fracture geometry of rocks that can be explained on the basis of Griffith energy balance concept.
Effect of mobile phone radiation on brain using EEG analysis by Higuichi's fractal dimension method
NASA Astrophysics Data System (ADS)
Smitha, C. K.; Narayanan, N. K.
2013-01-01
venient window on the mind, revealing synaptic action that is moderately to strongly co-relate with brain state. Fractal dimension, measure of signal complexity can be used to characterize the physiological conditions of the brain. As the EEG signal is non linear, non stationary and noisy, non linear methods will be suitable for the analysis. In this paper Higuichi's fractal method is applied to find the fractal dimension. EEGs of 5 volunteers were recorded at rest and on exposure to radiofrequency (RF) emissions from mobile phones having different SAR values. Mobiles were positioned near the ears and then near the cz position. Fractal dimensions for all conditions are calculated using Higuich's FD estimation algorithm. The result shows that there are some changes in the FD while using mobile phone. The change in FD of the signal varies from person to person. The changes in FD show the variations in EEG signal while using mobile phone, which demonstrate transformation in the activities of brain due to radiation.
Estimating the level of dynamical noise in time series by using fractal dimensions
NASA Astrophysics Data System (ADS)
Sase, Takumi; Ramírez, Jonatán Peña; Kitajo, Keiichi; Aihara, Kazuyuki; Hirata, Yoshito
2016-03-01
We present a method for estimating the dynamical noise level of a 'short' time series even if the dynamical system is unknown. The proposed method estimates the level of dynamical noise by calculating the fractal dimensions of the time series. Additionally, the method is applied to EEG data to demonstrate its possible effectiveness as an indicator of temporal changes in the level of dynamical noise.
Kan, An-Kang; Cao, Dan; Zhang, Xue-Lai
2015-04-01
Accurately predicting the effective thermal conductivity of the fibrous materials is highly desirable but remains to be a challenging work. In this paper, the microstructure of the porous fiber materials is analyzed, approximated and modeled on basis of the statistical self-similarity of fractal theory. A fractal model is presented to accurately calculate the effective thermal conductivity of fibrous porous materials. Taking the two-phase heat transfer effect into account, the existing statistical microscopic geometrical characteristics are analyzed and the Hertzian Contact solution is introduced to calculate the thermal resistance of contact points. Using the fractal method, the impacts of various factors, including the porosity, fiber orientation, fractal diameter and dimension, rarified air pressure, bulk thermal conductivity coefficient, thickness and environment condition, on the effective thermal conductivity, are analyzed. The calculation results show that the fiber orientation angle caused the material effective thermal conductivity to be anisotropic, and normal distribution is introduced into the mathematic function. The effective thermal conductivity of fibrous material increases with the fiber fractal diameter, fractal dimension and rarefied air pressure within the materials, but decreases with the increase of vacancy porosity. PMID:26353563
NASA Astrophysics Data System (ADS)
Tijera, Manuel; Maqueda, Gregorio; Cano, José L.; López, Pilar; Yagüe, Carlos
2010-05-01
The wind velocity series of the atmospheric turbulent flow in the planetary boundary layer (PBL), in spite of being highly erratic, present a self-similarity structure (Frisch, 1995; Peitgen et., 2004; Falkovich et., 2006). So, the wind velocity can be seen as a fractal magnitude. We calculate the fractal dimension (Komolgorov capacity or box-counting dimension) of the wind perturbation series (u' = u- ) in the physical spaces (namely velocity-time). It has been studied the time evolution of the fractal dimension along different days and at three levels above the ground (5.8 m, 13.5 m, 32 m). The data analysed was recorded in the experimental campaign SABLES-98 (Cuxart et al., 2000) at the Research Centre for the Lower Atmosphere (CIBA) located in Valladolid (Spain). In this work the u, v and w components of wind velocity series have been measured by sonic anemometers (20 Hz sampling rate). The fractal dimension versus the integral length scales of the mean wind series have been studied, as well as the influence of different turbulent parameters. A method for estimating these integral scales is developed using the normalized autocorrelation function and a Gaussian fit. Finally, it will be analysed the variation of the fractal dimension versus stability parameters (as Richardson number) in order to explain some of the dominant features which are likely immersed in the fractal nature of these turbulent flows. References - Cuxart J, Yagüe C, Morales G, Terradellas E, Orbe J, Calvo J, Fernández A, Soler MR, Infante C, Buenestado P, Espinalt A, Joergensen HE, Rees JM, Vilá J, Redondo JM, Cantalapiedra IR and Conangla L (2000) Stable atmospheric boundary-layer experiment in Spain (SABLES98): a report. Boundary- Layer Meteorol 96:337-370 - Falkovich G and Kattepalli R. Sreenivasan (2006) Lessons from Hidrodynamic Turbulence. Physics Today 59: 43-49 - Frisch U (1995) Turbulence the legacy of A.N. Kolmogorov Cambridge University Press 269pp - Peitgen H, Jürgens H and Saupe D (2004) Chaos and Fractals Springer-Verlag 971pp
Fractal Dimensions for Radioisotope Pollution Patterns by Nuclear Power Plant Accidents
NASA Astrophysics Data System (ADS)
Saito, K.; Ogawa, S.
2015-04-01
The radioisotope pollution shows two types of patterns: dry and wet deposits for nuclear power plant accidents. Two surface pollution patterns were analysed by fractal. In Fukushima nuclear power plant accident, surface pollution by wet deposits was estimated to occur. However, actually it was no rain and white crystals were observed on the surface. Then, fractal analysis was carried out for the spatial distribution patterns of radio isotopes on the surface to judge the types of deposits. As a reference, Chernobyl nuclear power plant accident was checked for the spatial distribution patterns of radioisotopes on the surface. The objective patterns by fractal analysis were the surface pollution maps in Fukushima and Chernobyl, Abukuma river watershed map, and NOAA/AVHRR. The calculation of fractal dimensions was carried out with the box counting for binarized images. Fractal analysis results suggested the next conclusions. The radioisotope pollution in Fukushima might occur in both dry and wet deposits. The dry deposit might make the pollution pattern similar to the watershed, while the wet deposit might make the pollution pattern similar to cloud images. Moreover, most radioisotope contaminants might flow on the road in the forest valley and deposit on forest with and without rainfall in Fukushima.
Heat treatment parameters effecting the fractal dimensions of AuGe metallization on GaAs
NASA Astrophysics Data System (ADS)
Mojzes, Imre; Dominkovics, Csaba; Harsányi, Gábor; Nagy, Szilvia; Pipek, János; Dobos, László
2007-08-01
Correlation was detected between the thermal treatment parameters of the AuGe-GaAs system and surface fractal structure. Structural entropic calculations were used to confirm the results obtained by fractal calculations.
Speech Emotion Recognition Based on Parametric Filter and Fractal Dimension
NASA Astrophysics Data System (ADS)
Mao, Xia; Chen, Lijiang
In this paper, we propose a new method that employs two novel features, correlation density (Cd) and fractal dimension (Fd), to recognize emotional states contained in speech. The former feature obtained by a list of parametric filters reflects the broad frequency components and the fine structure of lower frequency components, contributed by unvoiced phones and voiced phones, respectively; the latter feature indicates the non-linearity and self-similarity of a speech signal. Comparative experiments based on Hidden Markov Model and K Nearest Neighbor methods are carried out. The results show that Cd and Fd are much more closely related with emotional expression than the features commonly used.
Mu, Teh-Jing; Lee, Dong-Won; Park, Kwang-Ho
2013-01-01
Purpose To assess bony trabecular changes potentially caused by loading stress around dental implants using fractal dimension analysis. Methods Fractal dimensions were measured in 48 subjects by comparing radiographs taken immediately after prosthesis delivery with those taken 1 year after functional loading. Regions of interest were isolated, and fractal analysis was performed using the box-counting method with Image J 1.42 software. Wilcoxon signed-rank test was used to analyze the difference in fractal dimension before and after implant loading. Results The mean fractal dimension before loading (1.4213±0.0525) increased significantly to 1.4329±0.0479 at 12 months after loading (P<0.05). Conclusions Fractal dimension analysis might be helpful in detecting changes in peri-implant alveolar trabecular bone patterns in clinical situations. PMID:24236242
NASA Astrophysics Data System (ADS)
Donadio, Carlo; Magdaleno, Fernando; Mazzarella, Adriano; Mathias Kondolf, G.
2015-07-01
By applying fractal geometry analysis to the drainage network of three large watercourses in America and Europe, we have calculated for the first time their fractal dimension. The aim is to interpret the geomorphologic characteristics to better understand the morphoevolutionary processes of these fluvial morphotypes; to identify and discriminate geomorphic phenomena responsible for any difference or convergence of a fractal dimension; to classify hydrographic patterns, and finally to compare the fractal degree with some geomorphic-quantitative indexes. The analyzed catchment of Russian (California, USA), Ebro (Spain), and Volturno (Italy) rivers are situated in Mediterranean-climate regions sensu Köppen, but with different geologic context and tectonic styles. Results show fractal dimensions ranging from 1.08 to 1.50. According to the geological setting and geomorphic indexes of these basins, the lower fractal degree indicates a prevailing tectonics, active or not, while the higher degree indicates the stronger erosion processes on inherited landscapes.
Hyper-Fractal Analysis: A visual tool for estimating the fractal dimension of 4D objects
NASA Astrophysics Data System (ADS)
Grossu, I. V.; Grossu, I.; Felea, D.; Besliu, C.; Jipa, Al.; Esanu, T.; Bordeianu, C. C.; Stan, E.
2013-04-01
This work presents a new version of a Visual Basic 6.0 application for estimating the fractal dimension of images and 3D objects (Grossu et al. (2010) [1]). The program was extended for working with four-dimensional objects stored in comma separated values files. This might be of interest in biomedicine, for analyzing the evolution in time of three-dimensional images. New version program summaryProgram title: Hyper-Fractal Analysis (Fractal Analysis v03) Catalogue identifier: AEEG_v3_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEG_v3_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 745761 No. of bytes in distributed program, including test data, etc.: 12544491 Distribution format: tar.gz Programming language: MS Visual Basic 6.0 Computer: PC Operating system: MS Windows 98 or later RAM: 100M Classification: 14 Catalogue identifier of previous version: AEEG_v2_0 Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 831-832 Does the new version supersede the previous version? Yes Nature of problem: Estimating the fractal dimension of 4D images. Solution method: Optimized implementation of the 4D box-counting algorithm. Reasons for new version: Inspired by existing applications of 3D fractals in biomedicine [3], we extended the optimized version of the box-counting algorithm [1, 2] to the four-dimensional case. This might be of interest in analyzing the evolution in time of 3D images. The box-counting algorithm was extended in order to support 4D objects, stored in comma separated values files. A new form was added for generating 2D, 3D, and 4D test data. The application was tested on 4D objects with known dimension, e.g. the Sierpinski hypertetrahedron gasket, Df=ln(5)/ln(2) (Fig. 1). The algorithm could be extended, with minimum effort, to higher number of dimensions. Easy integration with other applications by using the very simple comma separated values file format for storing multi-dimensional images. Implementation of χ2 test as a criterion for deciding whether an object is fractal or not. User friendly graphical interface. Hyper-Fractal Analysis-Test on the Sierpinski hypertetrahedron 4D gasket (Df=ln(5)/ln(2)≅2.32). Running time: In a first approximation, the algorithm is linear [2]. References: [1] V. Grossu, D. Felea, C. Besliu, Al. Jipa, C.C. Bordeianu, E. Stan, T. Esanu, Computer Physics Communications, 181 (2010) 831-832. [2] I.V. Grossu, C. Besliu, M.V. Rusu, Al. Jipa, C. C. Bordeianu, D. Felea, Computer Physics Communications, 180 (2009) 1999-2001. [3] J. Ruiz de Miras, J. Navas, P. Villoslada, F.J. Esteban, Computer Methods and Programs in Biomedicine, 104 Issue 3 (2011) 452-460.
Aggregation of liposomes in presence of La3+: a study of the fractal dimension.
Sabín, Juan; Prieto, Gerardo; Ruso, Juan M; Messina, Paula; Sarmiento, Félix
2007-07-01
A study of the fractal dimension of the aggregation of three different types of large unilamellar vesicles, formed by egg yolk phosphatidylcholine (EYPC), dimyristoyl-phosphocholine (DMPC), and dipalmitoyl-phosphocholine (DPPC), in the presence of La3+, is presented. Aggregate liposome fractal dimensions were calculated by two methods, aggregation kinetics, using the approaches diffusion-limited cluster aggregation (DLCA) and reaction-limited cluster aggregation (RLCA) and angle-scattering light dispersion. Electrophoretic measurements show a similar variation of the zeta potential (zeta potential) for EYPC and DPPC, with a small increase of initial positive values. However, the zeta potential of DMPC changes from a initial negative value to near zero with increasing La3+ concentration. The evolution of the aggregate sizes was followed by light scattering. DPPC and DMPC show a RLCA regimen growth at low La3+ concentrations and a DLCA regimen at higher concentrations. In the case of EYPC, the final size of aggregation strongly depends on La3+ concentration. The calculated fractal dimension is in the range 1.8 to 2.1. PMID:17677442
Lu, Xiao-long; Zheng, Qin; Yin, Xian-zhen; Xiao, Guang-qing; Liao, Zu-hua; Yang, Ming; Zhang, Ji-wen
2015-06-01
The shape and structure of granules are controlled by the granulation process, which is one of the main factors to determine the nature of the solid dosage forms. In this article, three kinds of granules of a traditional Chinese medicine for improving appetite and promoting digestion, namely, Jianwei Granules, were prepared using granulation technologies as pendular granulation, high speed stirring granulation, and fluidized bed granulation and the powder properties of them were investigated. Meanwhile, synchrotron radiation X-ray computed micro tomography (SR-µCT) was applied to quantitatively determine the irregular internal structures of the granules. The three-dimensional (3D) structure models were obtained by 3D reconstruction, which were more accurately to characterize the three-dimensional structures of the particles through the quantitative data. The models were also used to quantitatively compare the structural differences of granules prepared by different granulation processes with the same formula, so as to characterize how the production process plays a role in the pharmaceutical behaviors of the granules. To focus on the irregularity of the particle structure, the box counting method was used to calculate the fractal dimensions of the granules. The results showed that the fractal dimension is more sensitive to reflect the minor differences in the structure features than the conventional parameters, and capable to specifically distinct granules in structure. It is proved that the fractal dimension could quantitatively characterize the structural information of irregular granules. It is the first time suggested by our research that the fractal dimension difference (Df,c) between two fractal dimension parameters, namely, the volume matrix fractal dimension and the surface matrix fractal dimension, is a new index to characterize granules with irregular structures and evaluate the effects of production processes on the structures of granules as a new indicator for the granulating process control and optimization. PMID:26521451
Scaling Particle Size in Fault Gouge: Variable Fractal Dimension or Non-Fractal Distribution?
NASA Astrophysics Data System (ADS)
Dewers, T.; Wilson, B.; Reches, Z.
2003-12-01
The particulate nature of fault-gouges is believed to be the product as well as the control of earthquake rupture and fault slip. It is expected that the particle-size distribution (PSD) will display a fractal dimension that develops by grain comminution and progressive fault slip. We examine this expectation by measuring the PSD with laser particle size analysis (with 0.04 to 2,000.00 microns range), and observations with scanning and transmission electron microscopy. The gouges of two faults were studied: (1) The exhumed fault-zone of the San Andreas at Tejon Pass, California, with > 80 samples collected along a 70 m fault-normal traverse and a few sub-meter exposures within the pulverized Cretaceous Tejon Lookout granite (Wilson et al, 2003, Fall meeting, AGU); and (2) A "new-born" fault formed during the M=3.7 1997 earthquake in Hartebeestfontein gold mine, Klerksdorp, South Africa. The quartzitic gouge of this fault was collected at the focal zone, which was mined more than one year after the earthquake. We ran the laser particle size analysis for continuous periods up to three days while conducting multiple PSD measurements of a single sample. The main results are: (1) The PSD of the gouge powders from both faults systematically vary with measurement time due to progressive grain disaggregation; e.g., the mean grain size (by volume) drops from an initial value of 5.9+/-22.8 micron to 0.5+/-0.2 microns after 72 hours. The submicron nature of the gouges is verified by SEM and TEM. (2) PSD data for a wide, relevant range (0.04-2,000 microns) revealed that fractal dimensions of a single sample could range from 1.7 to 3.6 during the initial measurement. (3) The grain disaggregation (with running time in the laser analyzer) led to bi-modal fractal distributions with anomalous values as well as non-fractal distributions. We conclude that the frequently observed fractal nature of a gouge reflects the particulate agglomeration of finely fragmented grains, and does not represent the true gouge dimensionality or the gouge comminution associated with earthquake rupture. Finally, the intense pulverization with generation of large surface area in the studied gouges could contribute significantly to the earthquake energy balance.
Application of fractal dimensions to study the structure of flocs formed in lime softening process.
Vahedi, Arman; Gorczyca, Beata
2011-01-01
The use of fractal dimensions to study the internal structure and settling of flocs formed in lime softening process was investigated. Fractal dimensions of flocs were measured directly on floc images and indirectly from their settling velocity. An optical microscope with a motorized stage was used to measure the fractal dimensions of lime softening flocs directly on their images in 2 and 3D space. The directly determined fractal dimensions of the lime softening flocs were 1.11-1.25 for floc boundary, 1.82-1.99 for cross-sectional area and 2.6-2.99 for floc volume. The fractal dimension determined indirectly from the flocs settling rates was 1.87 that was different from the 3D fractal dimension determined directly on floc images. This discrepancy is due to the following incorrect assumptions used for fractal dimensions determined from floc settling rates: linear relationship between square settling velocity and floc size (Stokes' Law), Euclidean relationship between floc size and volume, constant fractal dimensions and one primary particle size describing entire population of flocs. Floc settling model incorporating variable floc fractal dimensions as well as variable primary particle size was found to describe the settling velocity of large (>50 μm) lime softening flocs better than Stokes' Law. Settling velocities of smaller flocs (<50 μm) could still be quite well predicted by Stokes' Law. The variation of fractal dimensions with lime floc size in this study indicated that two mechanisms are involved in the formation of these flocs: cluster-cluster aggregation for small flocs (<50 μm) and diffusion-limited aggregation for large flocs (>50 μm). Therefore, the relationship between the floc fractal dimension and floc size appears to be determined by floc formation mechanisms. PMID:20937512
Laaksonen, Ari; Malila, Jussi; Nenes, Athanasios; Hung, Hui-Ming; Chen, Jen-Ping
2016-01-01
Surface porosity affects the ability of a substance to adsorb gases. The surface fractal dimension D is a measure that indicates the amount that a surface fills a space, and can thereby be used to characterize the surface porosity. Here we propose a new method for determining D, based on measuring both the water vapour adsorption isotherm of a given substance, and its ability to act as a cloud condensation nucleus when introduced to humidified air in aerosol form. We show that our method agrees well with previous methods based on measurement of nitrogen adsorption. Besides proving the usefulness of the new method for general surface characterization of materials, our results show that the surface fractal dimension is an important determinant in cloud drop formation on water insoluble particles. We suggest that a closure can be obtained between experimental critical supersaturation for cloud drop activation and that calculated based on water adsorption data, if the latter is corrected using the surface fractal dimension of the insoluble cloud nucleus. PMID:27138171
Laaksonen, Ari; Malila, Jussi; Nenes, Athanasios; Hung, Hui-Ming; Chen, Jen-Ping
2016-01-01
Surface porosity affects the ability of a substance to adsorb gases. The surface fractal dimension D is a measure that indicates the amount that a surface fills a space, and can thereby be used to characterize the surface porosity. Here we propose a new method for determining D, based on measuring both the water vapour adsorption isotherm of a given substance, and its ability to act as a cloud condensation nucleus when introduced to humidified air in aerosol form. We show that our method agrees well with previous methods based on measurement of nitrogen adsorption. Besides proving the usefulness of the new method for general surface characterization of materials, our results show that the surface fractal dimension is an important determinant in cloud drop formation on water insoluble particles. We suggest that a closure can be obtained between experimental critical supersaturation for cloud drop activation and that calculated based on water adsorption data, if the latter is corrected using the surface fractal dimension of the insoluble cloud nucleus. PMID:27138171
Analysis of fractal dimensions of rat bones from film and digital images
NASA Technical Reports Server (NTRS)
Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Yamauchi, M.
2001-01-01
OBJECTIVES: (1) To compare the effect of two different intra-oral image receptors on estimates of fractal dimension; and (2) to determine the variations in fractal dimensions between the femur, tibia and humerus of the rat and between their proximal, middle and distal regions. METHODS: The left femur, tibia and humerus from 24 4-6-month-old Sprague-Dawley rats were radiographed using intra-oral film and a charge-coupled device (CCD). Films were digitized at a pixel density comparable to the CCD using a flat-bed scanner. Square regions of interest were selected from proximal, middle, and distal regions of each bone. Fractal dimensions were estimated from the slope of regression lines fitted to plots of log power against log spatial frequency. RESULTS: The fractal dimensions estimates from digitized films were significantly greater than those produced from the CCD (P=0.0008). Estimated fractal dimensions of three types of bone were not significantly different (P=0.0544); however, the three regions of bones were significantly different (P=0.0239). The fractal dimensions estimated from radiographs of the proximal and distal regions of the bones were lower than comparable estimates obtained from the middle region. CONCLUSIONS: Different types of image receptors significantly affect estimates of fractal dimension. There was no difference in the fractal dimensions of the different bones but the three regions differed significantly.
Wang Xujing; Becker, Frederick F.; Gascoyne, Peter R. C.
2010-12-15
The scale-invariant property of the cytoplasmic membrane of biological cells is examined by applying the Minkowski-Bouligand method to digitized scanning electron microscopy images of the cell surface. The membrane is found to exhibit fractal behavior, and the derived fractal dimension gives a good description of its morphological complexity. Furthermore, we found that this fractal dimension correlates well with the specific membrane dielectric capacitance derived from the electrorotation measurements. Based on these findings, we propose a new fractal single-shell model to describe the dielectrics of mammalian cells, and compare it with the conventional single-shell model (SSM). We found that while both models fit with experimental data well, the new model is able to eliminate the discrepancy between the measured dielectric property of cells and that predicted by the SSM.
Fractal Dimension Analysis of the Cortical Ribbon in Mild Alzheimer’s Disease
King, Richard D.; Brown, Brandon; Hwang, Michael; Jeon, Tina; George, Anuh T.
2010-01-01
Fractal analysis methods are used to quantify the complexity of the human cerebral cortex. Many recent studies have focused on high resolution three-dimensional reconstructions of either the outer (pial) surface of the brain or the junction between the grey and white matter, but ignore the structure between these surfaces. This study uses a new method to incorporate the entire cortical thickness. Data were obtained from the Alzheimer’s Disease (AD) Neuroimaging Initiative database (Control N=35, Mild AD N=35). Image segmentation was performed using a semi-automated analysis program. The fractal dimensions of three cortical models (the pial surface, grey/white surface and entire cortical ribbon) were calculated using a custom cube-counting triangle-intersection algorithm. The fractal dimension of the cortical ribbon showed highly significant differences between control and AD subjects (p<0.001). The inner surface analysis also found smaller but significant differences (p< 0.05). The pial surface dimensionality was not significantly different between the two groups. All three models had a significant positive correlation with the cortical gyrification index (r > 0.55, p<0.001). Only the cortical ribbon had a significant correlation with cortical thickness (r = 0.832, p< 0.001) and the Alzheimer’s Disease Assessment Scale cognitive battery (r = −0.513, p = 0.002). The cortical ribbon dimensionality showed a larger effect size (d=1.12) in separating control and mild AD subjects than cortical thickness (d=1.01) or gyrification index (d=0.84). The methodological change shown in this paper may allow for further clinical application of cortical fractal dimension as a biomarker for structural changes that accrue with neurodegenerative diseases. PMID:20600974
Fractal dimension analysis of the cortical ribbon in mild Alzheimer's disease.
King, Richard D; Brown, Brandon; Hwang, Michael; Jeon, Tina; George, Anuh T
2010-11-01
Fractal analysis methods are used to quantify the complexity of the human cerebral cortex. Many recent studies have focused on high resolution three-dimensional reconstructions of either the outer (pial) surface of the brain or the junction between the gray and white matter, but ignore the structure between these surfaces. This study uses a new method to incorporate the entire cortical thickness. Data were obtained from the Alzheimer's Disease (AD) Neuroimaging Initiative database (Control N=35, Mild AD N=35). Image segmentation was performed using a semi-automated analysis program. The fractal dimension of three cortical models (the pial surface, gray/white surface and entire cortical ribbon) were calculated using a custom cube-counting triangle-intersection algorithm. The fractal dimension of the cortical ribbon showed highly significant differences between control and AD subjects (p<0.001). The inner surface analysis also found smaller but significant differences (p<0.05). The pial surface dimensionality was not significantly different between the two groups. All three models had a significant positive correlation with the cortical gyrification index (r>0.55, p<0.001). Only the cortical ribbon had a significant correlation with cortical thickness (r=0.832, p<0.001) and the Alzheimer's Disease Assessment Scale cognitive battery (r=-0.513, p=0.002). The cortical ribbon dimensionality showed a larger effect size (d=1.12) in separating control and mild AD subjects than cortical thickness (d=1.01) or gyrification index (d=0.84). The methodological change shown in this paper may allow for further clinical application of cortical fractal dimension as a biomarker for structural changes that accrue with neurodegenerative diseases. PMID:20600974
NASA Astrophysics Data System (ADS)
Tian, F.; Wang, B.; Hu, H.
2009-12-01
The natural river networks are usually featured by a self-similar tree-like structure which represents a deep sense statistical symmetry and can be described by the fractal geometry theory in a decent way (Rodriguez-Iturbe and Rinaldo, 1997). This self-similarity provides a basis to investigate the scale invariance of many hydrological phenomena and has received extensive attention. Many researchers use box-counting method to estimate the fractal dimension of river networks due to now widely available DEM data and easy-to-use river network extraction packages from DEM such as in ESRI’s ArcGIS software package. Zhou et al.’s (2008) study, however, shows that the box-counting dimension is subject to considerable degree of uncertainty, which depends strongly on: 1) the threshold area used to delineate the river network; 2) the range of calculated box sizes to overlay and intersect the river network. This study is devoted to examine the sources of this uncertainty and develop an improved procedure to calculate the box-counting dimension. The calculation based on real world DEMs and theoretical analysis on the mathematical definition of box-counting dimension show that the uncertainty is evoked by the upper bound of range of calculated box sizes, i.e., given the larger upper bound the box-counting dimension varies with the threshold area while given an appropriate upper bound is the unique value of box-counting dimension could be obtained. A more rigorous procedure is finally proposed to calculate the fractal dimension of river networks with the box-counting method. This procedure effectively eliminates the uncertainties of the box-counting method, and thus gives the fractal dimension with higher confidence.
Scaling exponents for a monkey on a tree: Fractal dimensions of randomly branched polymers
NASA Astrophysics Data System (ADS)
Janssen, Hans-Karl; Stenull, Olaf
2012-05-01
We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large, randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to two-loop order and, where available, compare them to numerical results.
Preparation of activated carbons from walnut wood: a study of microporosity and fractal dimension
NASA Astrophysics Data System (ADS)
Gómez-Serrano, Vicente; Cuerda-Correa, Eduardo M.; Fernández-González, M. Carmen; Alexandre-Franco, María F.; Macías-García, Antonio
2005-04-01
Agricultural and forest residues constitute an extraordinarily important source of precursors for the manufacture of activated carbons. Activated carbons are well known as porous solids with a highly developed apparent surface area. In the present work, activated carbon has been prepared from forest residues of walnut tree wood (a raw material not studied until now) by physical activation. Raw material has been carbonized between 573 and 1073 K and afterwards activated in air at temperatures between 623 and 823 K. The apparent surface area, micropore volume, mesopore volume and fractal dimension of the samples prepared have been calculated.
Ye, S. |; Vijh, A.K.; Dao, L.H.
1997-05-01
A new fuel-cell electrocatalyst based on highly porous carbonized polyacrylonitrile (PAN) microcellular foam with platinum particles was prepared recently in this laboratory. Its surface morphology, one of the most important aspects of a practical electrocatalyst, has been examined in terms of fractal theory and methods. The fractal dimension of the platinum particles dispersed in porous carbonized PAN foam was determined by using chronometric and rotating-disk-electrode methods in oxygen-saturated solutions. A fractal dimension smaller than 2 was obtained, which was attributed to the partially active nature of the surface of this electrocatalytic material. This value of fractal dimension is also proposed to be considered as a reaction dimension. A reaction dimension smaller than 2 may indicate that not all of the platinum particle surface is accessible to the incoming oxygen molecules.
Low fractal dimension cluster-dilute soot aggregates from a premixed flame.
Chakrabarty, Rajan K; Moosmüller, Hans; Arnott, W Patrick; Garro, Mark A; Tian, Guoxun; Slowik, Jay G; Cross, Eben S; Han, Jeong-Ho; Davidovits, Paul; Onasch, Timothy B; Worsnop, Douglas R
2009-06-12
Using a novel morphology segregation technique, we observed minority populations ( approximately 3%) of submicron-sized, cluster-dilute fractal-like aggregates, formed in the soot-formation window (fuel-to-air equivalence ratio of 2.0-3.5) of a premixed flame, to have mass fractal dimensions between 1.2 and 1.51. Our observations disagree with previous observations of a universal mass fractal dimension of approximately 1.8 for fractal-like aerosol aggregates formed in the dilute-limit via three-dimensional diffusion-limited cluster aggregation processes. A hypothesis is presented to explain this observation. Subject to verification of this hypothesis, it may be possible to control the fractal dimension and associated properties of aggregates in the cluster-dilute limit through application of a static electric field during the aggregation process. PMID:19658949
Urschler, Martin; Kullnig, Peter; Stollberger, Rudolf; Kovacs, Gabor; Olschewski, Andrea; Olschewski, Horst; Bálint, Zoltán
2014-01-01
Pulmonary hypertension (PH) can result in vascular pruning and increased tortuosity of the blood vessels. In this study we examined whether automatic extraction of lung vessels from contrast-enhanced thoracic computed tomography (CT) scans and calculation of tortuosity as well as 3D fractal dimension of the segmented lung vessels results in measures associated with PH. In this pilot study, 24 patients (18 with and 6 without PH) were examined with thorax CT following their diagnostic or follow-up right-sided heart catheterisation (RHC). Images of the whole thorax were acquired with a 128-slice dual-energy CT scanner. After lung identification, a vessel enhancement filter was used to estimate the lung vessel centerlines. From these, the vascular trees were generated. For each vessel segment the tortuosity was calculated using distance metric. Fractal dimension was computed using 3D box counting. Hemodynamic data from RHC was used for correlation analysis. Distance metric, the readout of vessel tortuosity, correlated with mean pulmonary arterial pressure (Spearman correlation coefficient: ρ = 0.60) and other relevant parameters, like pulmonary vascular resistance (ρ = 0.59), arterio-venous difference in oxygen (ρ = 0.54), arterial (ρ = −0.54) and venous oxygen saturation (ρ = −0.68). Moreover, distance metric increased with increase of WHO functional class. In contrast, 3D fractal dimension was only significantly correlated with arterial oxygen saturation (ρ = 0.47). Automatic detection of the lung vascular tree can provide clinically relevant measures of blood vessel morphology. Non-invasive quantification of pulmonary vessel tortuosity may provide a tool to evaluate the severity of pulmonary hypertension. Trial Registration ClinicalTrials.gov NCT01607489 PMID:24498123
Wu, Jia-Ming; Kuo, Chung-Ming; Chen, Ching-Jiang
2013-01-01
Purpose. This study describes how to identify the coincidence of desired planning isodose curves with film experimental results by using a mathematical fractal dimension characteristic method to avoid the errors caused by visual inspection in the intensity modulation radiation therapy (IMRT). Methods and Materials. The isodose curves of the films delivered by linear accelerator according to Plato treatment planning system were acquired using Osiris software to aim directly at a single interested dose curve for fractal characteristic analysis. The results were compared with the corresponding planning desired isodose curves for fractal dimension analysis in order to determine the acceptable confidence level between the planning and the measurement. Results. The film measured isodose curves and computer planning curves were deemed identical in dose distribution if their fractal dimensions are within some criteria which suggested that the fractal dimension is a unique fingerprint of a curve in checking the planning and film measurement results. The dose measured results of the film were presumed to be the same if their fractal dimension was within 1%. Conclusions. This quantitative rather than qualitative comparison done by fractal dimension numerical analysis helps to decrease the quality assurance errors in IMRT dosimetry verification. PMID:23956976
On the use of spectral methods for the determination of fractal dimension
Hough, S.E. )
1989-07-01
This paper discusses the use of fractal theory for quantitative analysis of profiles from two-dimensional surfaces. The theoretical relationship between spectral parameters and fractal dimension is examined, including the conditions under which the derived relationship between spectral parameters and fractal dimension by Voss (1985) is valid, and the limitations in making inferences from spectral parameters. Applications of fractal theory to geophysical data are also discussed. In particular, it is shown that an amplitude spectrum with a decay corresponding to a fractal dimension of 1.5 can result from the concatenation of time series with decays corresponding to different fractal dimensions. It has been shown that the spectral density of a fractal distribution will be characterized by a power-law decay, but this paper illustrates that a power-law decay is not sufficient to identify a fractal distribution. Although this paper discusses applications to the study of Earth topography, the results are applicable to the study of any one-dimensional profile.
Fractal Dimension Analysis of Gustatory Electroencephalograms in Humans
NASA Astrophysics Data System (ADS)
Igasaki, Tomohiko; Murayama, Nobuki
To quantify the neural dynamics of the brain responsible for gustatory recognition and discrimination, fractal dimensions (FDs) of electroencephalograms (EEGs), which were measured under resting and three gustatory stimulation states, were investigated. The seven normal subjects sat on a chair with the chin resting on a frame made of plaster bandage and eyes closed. Distilled water (DW), high concentrated taste (HCT) solution (300 mM NaCl, 1 mM quinine-HCl, 40 mM acetic acid and 500 mM sucrose) and low concentrated taste (LCT) solution (51 mM NaCl, 0.026 mM quinine-HCl, 3 mM acetic acid and 14 mM sucrose) were randomly delivered to the anterior region of the tongue which was protruded slightly out of the mouth. FDs of EEGs from Cz in the resting and in the DW stimulation state were 5.43±1.01 and 4.94±1.03, respectively. In the HCT stimulation state, FD significantly decreased to 4.20±1.08 as compared with that in the resting (P<0.001). While, in the LCT stimulation state, FD significantly increased to 5.77±1.02 as compared with that in the HCT stimulation state (P<0.001). These results suggest that information processing of the brain is relatively simple when easily recognized tastes are applied.
Seizure detection method based on fractal dimension and gradient boosting.
Zhang, Yanli; Zhou, Weidong; Yuan, Shasha; Yuan, Qi
2015-02-01
Automatic seizure detection technology is necessary and crucial for the long-term electroencephalography (EEG) monitoring of patients with epilepsy. This article presents a patient-specific method for the detection of epileptic seizures. The fractal dimensions of preprocessed multichannel EEG were firstly estimated using a k-nearest neighbor algorithm. Then, the feature vector constructed for each epoch was fed into a trained gradient boosting classifier. After a series of postprocessing, including smoothing, threshold processing, collar operation, and union of seizure detections in a short time interval, a binary decision was made to determine whether the epoch belonged to seizure status or not. Both the epoch-based and event-based assessments were used for the performance evaluation of this method on the EEG data of 21 patients from the Freiburg dataset. An average epoch-based sensitivity of 91.01% and a specificity of 95.77% were achieved. For the event-based assessment, this method obtained an average sensitivity of 94.05%, with a false detection rate of 0.27/h. PMID:25549952
A Fractal Dimension and Wavelet Transform Based Method for Protein Sequence Similarity Analysis.
Yang, Lina; Tang, Yuan Yan; Lu, Yang; Luo, Huiwu
2015-01-01
One of the key tasks related to proteins is the similarity comparison of protein sequences in the area of bioinformatics and molecular biology, which helps the prediction and classification of protein structure and function. It is a significant and open issue to find similar proteins from a large scale of protein database efficiently. This paper presents a new distance based protein similarity analysis using a new encoding method of protein sequence which is based on fractal dimension. The protein sequences are first represented into the 1-dimensional feature vectors by their biochemical quantities. A series of Hybrid method involving discrete Wavelet transform, Fractal dimension calculation (HWF) with sliding window are then applied to form the feature vector. At last, through the similarity calculation, we can obtain the distance matrix, by which, the phylogenic tree can be constructed. We apply this approach by analyzing the ND5 (NADH dehydrogenase subunit 5) protein cluster data set. The experimental results show that the proposed model is more accurate than the existing ones such as Su's model, Zhang's model, Yao's model and MEGA software, and it is consistent with some known biological facts. PMID:26357222
Fractal Dimension Analysis of Subcortical Gray Matter Structures in Schizophrenia
Sehatpour, Pejman; Long, Jun; Gui, Weihua; Qiao, Jianping; Javitt, Daniel C.; Wang, Zhishun
2016-01-01
A failure of adaptive inference—misinterpreting available sensory information for appropriate perception and action—is at the heart of clinical manifestations of schizophrenia, implicating key subcortical structures in the brain including the hippocampus. We used high-resolution, three-dimensional (3D) fractal geometry analysis to study subtle and potentially biologically relevant structural alterations (in the geometry of protrusions, gyri and indentations, sulci) in subcortical gray matter (GM) in patients with schizophrenia relative to healthy individuals. In particular, we focus on utilizing Fractal Dimension (FD), a compact shape descriptor that can be computed using inputs with irregular (i.e., not necessarily smooth) surfaces in order to quantify complexity (of geometrical properties and configurations of structures across spatial scales) of subcortical GM in this disorder. Probabilistic (entropy-based) information FD was computed based on the box-counting approach for each of the seven subcortical structures, bilaterally, as well as the brainstem from high-resolution magnetic resonance (MR) images in chronic patients with schizophrenia (n = 19) and age-matched healthy controls (n = 19) (age ranges: patients, 22.7–54.3 and healthy controls, 24.9–51.6 years old). We found a significant reduction of FD in the left hippocampus (median: 2.1460, range: 2.07–2.18 vs. median: 2.1730, range: 2.15–2.23, p<0.001; Cohen’s effect size, U3 = 0.8158 (95% Confidence Intervals, CIs: 0.6316, 1.0)), the right hippocampus (median: 2.1430, range: 2.05–2.19 vs. median: 2.1760, range: 2.12–2.21, p = 0.004; U3 = 0.8421 (CIs: 0.5263, 1)), as well as left thalamus (median: 2.4230, range: 2.40–2.44, p = 0.005; U3 = 0.7895 (CIs: 0.5789, 0.9473)) in schizophrenia patients, relative to healthy individuals. Our findings provide in-vivo quantitative evidence for reduced surface complexity of hippocampus, with reduced FD indicating a less complex, less regular GM surface detected in schizophrenia. PMID:27176232
Fractal Dimension Analysis of Subcortical Gray Matter Structures in Schizophrenia.
Zhao, Guihu; Denisova, Kristina; Sehatpour, Pejman; Long, Jun; Gui, Weihua; Qiao, Jianping; Javitt, Daniel C; Wang, Zhishun
2016-01-01
A failure of adaptive inference-misinterpreting available sensory information for appropriate perception and action-is at the heart of clinical manifestations of schizophrenia, implicating key subcortical structures in the brain including the hippocampus. We used high-resolution, three-dimensional (3D) fractal geometry analysis to study subtle and potentially biologically relevant structural alterations (in the geometry of protrusions, gyri and indentations, sulci) in subcortical gray matter (GM) in patients with schizophrenia relative to healthy individuals. In particular, we focus on utilizing Fractal Dimension (FD), a compact shape descriptor that can be computed using inputs with irregular (i.e., not necessarily smooth) surfaces in order to quantify complexity (of geometrical properties and configurations of structures across spatial scales) of subcortical GM in this disorder. Probabilistic (entropy-based) information FD was computed based on the box-counting approach for each of the seven subcortical structures, bilaterally, as well as the brainstem from high-resolution magnetic resonance (MR) images in chronic patients with schizophrenia (n = 19) and age-matched healthy controls (n = 19) (age ranges: patients, 22.7-54.3 and healthy controls, 24.9-51.6 years old). We found a significant reduction of FD in the left hippocampus (median: 2.1460, range: 2.07-2.18 vs. median: 2.1730, range: 2.15-2.23, p<0.001; Cohen's effect size, U3 = 0.8158 (95% Confidence Intervals, CIs: 0.6316, 1.0)), the right hippocampus (median: 2.1430, range: 2.05-2.19 vs. median: 2.1760, range: 2.12-2.21, p = 0.004; U3 = 0.8421 (CIs: 0.5263, 1)), as well as left thalamus (median: 2.4230, range: 2.40-2.44, p = 0.005; U3 = 0.7895 (CIs: 0.5789, 0.9473)) in schizophrenia patients, relative to healthy individuals. Our findings provide in-vivo quantitative evidence for reduced surface complexity of hippocampus, with reduced FD indicating a less complex, less regular GM surface detected in schizophrenia. PMID:27176232
Dewey, T G; Datta, M M
1989-01-01
It is demonstrated that fluorescence resonance energy transfer may be used to determine the fractal dimension of aggregates of membrane-bound proteins. Theoretical and experimental results are presented for two different experimental designs: energy transfer between proteins and energy transfer from lipids to proteins. For energy transfer between proteins the lattice spacing must be known independently for a fractal dimension to be uniquely determined, and this represents a disadvantage to this experimental design. Results are presented for the calcium ATPase and a fractal dimension of 1.9 is estimated for ATPase aggregates by assuming a lattice spacing of 50 A. Energy transfer from lipids to protein provides a means of estimating the length of the "coast-line" of the aggregate. In this case the fractal dimension is uniquely determined from a log-log plot. An analysis of data for bacteriohodopsin reconstituted in phospholipid vesicles gives a fractal dimension of 1.6. The structural basis of the value for the fractal dimension is discussed for these two systems. These techniques provide a means of assessing the nature of protein-protein interactions in membranous systems. PMID:2528385
Assessment of disintegrant efficacy with fractal dimensions from real-time MRI.
Quodbach, Julian; Moussavi, Amir; Tammer, Roland; Frahm, Jens; Kleinebudde, Peter
2014-11-20
An efficient disintegrant is capable of breaking up a tablet in the smallest possible particles in the shortest time. Until now, comparative data on the efficacy of different disintegrants is based on dissolution studies or the disintegration time. Extending these approaches, this study introduces a method, which defines the evolution of fractal dimensions of tablets as surrogate parameter for the available surface area. Fractal dimensions are a measure for the tortuosity of a line, in this case the upper surface of a disintegrating tablet. High-resolution real-time MRI was used to record videos of disintegrating tablets. The acquired video images were processed to depict the upper surface of the tablets and a box-counting algorithm was used to estimate the fractal dimensions. The influence of six different disintegrants, of different relative tablet density, and increasing disintegrant concentration was investigated to evaluate the performance of the novel method. Changing relative densities hardly affect the progression of fractal dimensions, whereas an increase in disintegrant concentration causes increasing fractal dimensions during disintegration, which are also reached quicker. Different disintegrants display only minor differences in the maximal fractal dimension, yet the kinetic in which the maximum is reached allows a differentiation and classification of disintegrants. PMID:25234864
Discriminating between elderly and young using a fractal dimension analysis of centre of pressure
2004-01-01
The aim of this project was to evaluate the use of a new analysis technique, fractal dimension analysis, for quantification of quiet stance centre of pressure (COP). By using a fractal dimension analysis of COP, it might be possible to gain more information about control during quiet stance than traditional analyses have previously allowed. The current project considered a group of young healthy participants and a group of elderly healthy participants to compare traditional measures of COP against a fractal dimension analysis of COP. Results indicated that both types of analyses are able to distinguish between eyes open and eyes closed in the elderly group. However, the fractal dimension analysis more accurately detected differences between the participant groups when standing with their eyes closed. Based on these results it is suggested that fractal dimension analysis is more informative about posture control than traditional measures. It is suggested that a fractal dimension type of analysis can be incorporated into clinical testing to identify patients with pathologies. PMID:15912186
SU-D-BRA-04: Fractal Dimension Analysis of Edge-Detected Rectal Cancer CTs for Outcome Prediction
Zhong, H; Wang, J; Hu, W; Shen, L; Wan, J; Zhou, Z; Zhang, Z
2015-06-15
Purpose: To extract the fractal dimension features from edge-detected rectal cancer CTs, and to examine the predictability of fractal dimensions to outcomes of primary rectal cancer patients. Methods: Ninety-seven rectal cancer patients treated with neo-adjuvant chemoradiation were enrolled in this study. CT images were obtained before chemoradiotherapy. The primary lesions of the rectal cancer were delineated by experienced radiation oncologists. These images were extracted and filtered by six different Laplacian of Gaussian (LoG) filters with different filter values (0.5–3.0: from fine to coarse) to achieve primary lesions in different anatomical scales. Edges of the original images were found at zero-crossings of the filtered images. Three different fractal dimensions (box-counting dimension, Minkowski dimension, mass dimension) were calculated upon the image slice with the largest cross-section of the primary lesion. The significance of these fractal dimensions in survival, recurrence and metastasis were examined by Student’s t-test. Results: For a follow-up time of two years, 18 of 97 patients had experienced recurrence, 24 had metastasis, and 18 were dead. Minkowski dimensions under large filter values (2.0, 2.5, 3.0) were significantly larger (p=0.014, 0.006, 0.015) in patients with recurrence than those without. For metastasis, only box-counting dimensions under a single filter value (2.5) showed differences (p=0.016) between patients with and without. For overall survival, box-counting dimensions (filter values = 0.5, 1.0, 1.5), Minkowski dimensions (filter values = 0.5, 1.5, 2.0, 2,5) and mass dimensions (filter values = 1.5, 2.0) were all significant (p<0.05). Conclusion: It is feasible to extract shape information by edge detection and fractal dimensions analysis in neo-adjuvant rectal cancer patients. This information can be used to prognosis prediction.
Landau-Darrieus instability and the fractal dimension of flame fronts
NASA Astrophysics Data System (ADS)
Blinnikov, Sergei, IV; Sasorov, Pavel V.
1996-05-01
Nonlinear dynamics of a slow laminar flame front subject to the Landau-Darrieus instability is investigated by means of numerical simulations of the Frankel equation, when the expansion degree γ=(ρu-ρb)/ρu is small (here ρu and ρb are the densities of the unburned and burned ``gases,'' respectively). Only burning in two-dimensional space is considered in our simulations. The observed acceleration of a front wrinkled by the instability can be ascribed to the development of a fractal structure along the front surface with typical spatial scales being between the maximum and the minimum truly unstable wavelengths. It is found that the fractal excess ΔD=D-1 decreases rapidly with decreasing of γ, to a first approximation as ΔD=D0γ2, where D is the fractal dimension of the front. Our rough estimation of D0 gives D0~=0.3. The low accuracy of the D0 estimation is caused by certain peculiarities of the Frankel equation that lead to extreme difficulties of its simulation even with the aid of supercomputers when γ<~0.3-0.4. It is shown, however, that D0 can be calculated also from the statistical properties of the Sivashinsky equation, which is easier to simulate, though the fractal excess for the Sivashinsky equation itself is equal to 0 (in a certain sense). The other important result of our simulations is that the front self-intersections play an extremely weak role when γ is small.
Local fractal dimension based approaches for colonic polyp classification.
Hfner, Michael; Tamaki, Toru; Tanaka, Shinji; Uhl, Andreas; Wimmer, Georg; Yoshida, Shigeto
2015-12-01
This work introduces texture analysis methods that are based on computing the local fractal dimension (LFD; or also called the local density function) and applies them for colonic polyp classification. The methods are tested on 8 HD-endoscopic image databases, where each database is acquired using different imaging modalities (Pentax's i-Scan technology combined with or without staining the mucosa) and on a zoom-endoscopic image database using narrow band imaging. In this paper, we present three novel extensions to a LFD based approach. These extensions additionally extract shape and/or gradient information of the image to enhance the discriminativity of the original approach. To compare the results of the LFD based approaches with the results of other approaches, five state of the art approaches for colonic polyp classification are applied to the employed databases. Experiments show that LFD based approaches are well suited for colonic polyp classification, especially the three proposed extensions. The three proposed extensions are the best performing methods or at least among the best performing methods for each of the employed databases. The methods are additionally tested by means of a public texture image database, the UIUCtex database. With this database, the viewpoint invariance of the methods is assessed, an important features for the employed endoscopic image databases. Results imply that most of the LFD based methods are more viewpoint invariant than the other methods. However, the shape, size and orientation adapted LFD approaches (which are especially designed to enhance the viewpoint invariance) are in general not more viewpoint invariant than the other LFD based approaches. PMID:26385078
Fractal Dimension Characterization of in-vivo Laser Doppler Flowmetry signals
NASA Astrophysics Data System (ADS)
Srinivasan, Gayathri; Sujatha, N.
Laser Doppler Blood Flow meter uses tissue backscattered light to non-invasively assess the blood flow rate. qualitatively. As there is large spatial variability and the temporal heterogeneity in tissue microvasculature, the measured blood flow rate is expressed in relative units. A non-linear approach in order to understand the dynamics of the microcirculation led to the fractal characterization of the blood flow signals. The study presented in the paper aims to analyze the fractal behavior of Laser Doppler Flow (LDF) signals and to quantitatively estimate the fractal dimension of waveforms using Box-Counting method. The measured Fractal dimension is an estimate of temporal variability of tissue perfusion. The rate at which fractal dimension varies as a function of location between individuals, exhibits a weak correlation with time. Further studies with a larger number of subjects are necessary to test the generality of the findings and if changes in dimension are reproducible in given individuals. In conclusion, the fractal dimension determined by Box-counting method may be useful for characterizing LDF time series signals. Future experiments evaluating whether the technique can be used to quantify microvascular dysfunction, as commonly occurring in conditions such as Diabetes, Raynaud's phenomenon, Erythromelalgia and Achenbach syndrome needs to be evaluated.
Fractal dimension-bound spatio-temporal analysis of digital mammograms
NASA Astrophysics Data System (ADS)
Shanmugavadivu, P.; Sivakumar, V.; Sudhir, Rashmi
2016-02-01
A new Fractal Dimension-based diagnosis technique for the change detection and time-series analysis of masses in the temporal digital mammogram is presented in this paper. As the digital mammograms are confirmed as a reliable source for the prognosis of breast cancer, the demand for the development of precise computer aided detection techniques is constantly on the increase. This formed the basis for the development of this method using Fractal geometry, which is an efficient mathematical approach that deals with self-similar and irregular geometric objects called fractals. This work comprises of the detection of spatial masses using Fractal Hurst bound enhancement and segmentation of those temporal masses using Fractal Thresholding. The consultant radiologist's assessment of mass lesions forms the baseline for comparison and validation of the detected masses. Further, this research work performs temporal analysis of mass lesions, detected from the mammograms of the current and the respective prior view using the principle of Fractal Dimension. The precision of Fractal-dimension based temporal texture analysis of malignant masses of digital mammograms subsequently attributes to their characterization.
NASA Astrophysics Data System (ADS)
Neves, L. A.; Oliveira, F. R.; Peres, F. A.; Moreira, R. D.; Moriel, A. R.; de Godoy, M. F.; Murta Junior, L. O.
2011-03-01
This paper presents a method for the quantification of cellular rejection in endomyocardial biopsies of patients submitted to heart transplant. The model is based on automatic multilevel thresholding, which employs histogram quantification techniques, histogram slope percentage analysis and the calculation of maximum entropy. The structures were quantified with the aid of the multi-scale fractal dimension and lacunarity for the identification of behavior patterns in myocardial cellular rejection in order to determine the most adequate treatment for each case.
LFN, QPO and fractal dimension of X-ray light curves from black hole binaries
NASA Astrophysics Data System (ADS)
Prosvetov, Art; Grebenev, Sergey
The origin of the low frequency noise (LFN) and quasi-periodic oscillations (QPO) observed in X-ray flux of Galactic black hole binaries is still not recognized in spite of multiple studies and attempts to model this phenomenon. There are known correlations between the QPO frequency, X-ray power density, X-ray flux and spectral state of the system, but there is no model that can do these dependences understandable. For the low frequency (~1 Hz) QPO we still have no even an idea capable to explain their production and don't know even what part of an accretion disc is responsible for them. Here we attempted to measure the fractal dimension of X-ray light curves of several black hole X-ray binaries and to study its correlation with the frequency of quasi periodic oscillations observed in their X-ray light-curves. The fractal dimension is a measure of the space-filling capacity of the light curves' profile. To measure the fractal dimension we used R/S method, which is fast enough and has good reputation in financial analytic and materials sciences. We found that if no QPO were observed in X-ray flux from the particular source, the fractal dimension is equal to the unique value which is independent on the source, its luminosity or its spectral state. On the other hand if QPO were detected in the flux, the fractal dimension deviated from its usual value. Also, we found a clear correlation between the QPO frequency and the fractal dimension of the emission. The relationship between these two parameters is solid but nonlinear. We believe that the analysis of X-ray light curves of black hole binaries using the fractal dimension has a good scientific potential and may provide an addition information on the geometry of accretion flow and fundamental physical parameters of the system.
Fractal dimension analysis of weight-bearing bones of rats during skeletal unloading
NASA Technical Reports Server (NTRS)
Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Sanhueza, A. I.; Yamauchi, M.
2001-01-01
Fractal analysis was used to quantify changes in trabecular bone induced through the use of a rat tail-suspension model to simulate microgravity-induced osteopenia. Fractal dimensions were estimated from digitized radiographs obtained from tail-suspended and ambulatory rats. Fifty 4-month-old male Sprague-Dawley rats were divided into groups of 24 ambulatory (control) and 26 suspended (test) animals. Rats of both groups were killed after periods of 1, 4, and 8 weeks. Femurs and tibiae were removed and radiographed with standard intraoral films and digitized using a flatbed scanner. Square regions of interest were cropped at proximal, middle, and distal areas of each bone. Fractal dimensions were estimated from slopes of regression lines fitted to circularly averaged plots of log power vs. log spatial frequency. The results showed that the computed fractal dimensions were significantly greater for images of trabecular bones from tail-suspended groups than for ambulatory groups (p < 0.01) at 1 week. Periods between 1 and 4 weeks likewise yielded significantly different estimates (p < 0.05), consistent with an increase in bone loss. In the tibiae, the proximal regions of the suspended group produced significantly greater fractal dimensions than other regions (p < 0.05), which suggests they were more susceptible to unloading. The data are consistent with other studies demonstrating osteopenia in microgravity environments and the regional response to skeletal unloading. Thus, fractal analysis could be a useful technique to evaluate the structural changes of bone.
Role of fractal dimension in random walks on scale-free networks
NASA Astrophysics Data System (ADS)
Zhang, Zhongzhi; Yang, Yihang; Gao, Shuyang
2011-11-01
Fractal dimension is central to understanding dynamical processes occurring on networks; however, the relation between fractal dimension and random walks on fractal scale-free networks has been rarely addressed, despite the fact that such networks are ubiquitous in real-life world. In this paper, we study the trapping problem on two families of networks. The first is deterministic, often called ( x,y)-flowers; the other is random, which is a combination of (1,3)-flower and (2,4)-flower and thus called hybrid networks. The two network families display rich behavior as observed in various real systems, as well as some unique topological properties not shared by other networks. We derive analytically the average trapping time for random walks on both the ( x,y)-flowers and the hybrid networks with an immobile trap positioned at an initial node, i.e., a hub node with the highest degree in the networks. Based on these analytical formulae, we show how the average trapping time scales with the network size. Comparing the obtained results, we further uncover that fractal dimension plays a decisive role in the behavior of average trapping time on fractal scale-free networks, i.e., the average trapping time decreases with an increasing fractal dimension.
Fractal dimension analysis for spike detection in low SNR extracellular signals
NASA Astrophysics Data System (ADS)
Salmasi, Mehrdad; Büttner, Ulrich; Glasauer, Stefan
2016-06-01
Objective. Many algorithms have been suggested for detection and sorting of spikes in extracellular recording. Nevertheless, it is still challenging to detect spikes in low signal-to-noise ratios (SNR). We propose a spike detection algorithm that is based on the fractal properties of extracellular signals and can detect spikes in low SNR regimes. Semi-intact spikes are low-amplitude spikes whose shapes are almost preserved. The detection of these spikes can significantly enhance the performance of multi-electrode recording systems. Approach. Semi-intact spikes are simulated by adding three noise components to a spike train: thermal noise, inter-spike noise, and spike-level noise. We show that simulated signals have fractal properties which make them proper candidates for fractal analysis. Then we use fractal dimension as the main core of our spike detection algorithm and call it fractal detector. The performance of the fractal detector is compared with three frequently used spike detectors. Main results. We demonstrate that in low SNR, the fractal detector has the best performance and results in the highest detection probability. It is shown that, in contrast to the other three detectors, the performance of the fractal detector is independent of inter-spike noise power and that variations in spike shape do not alter its performance. Finally, we use the fractal detector for spike detection in experimental data and similar to simulations, it is shown that the fractal detector has the best performance in low SNR regimes. Significance. The detection of low-amplitude spikes provides more information about the neural activity in the vicinity of the recording electrodes. Our results suggest using the fractal detector as a reliable and robust method for detecting semi-intact spikes in low SNR extracellular signals.
Fractal Dimensions of Interstellar Medium : I. The Molecular Clouds in the Antigalactic Center
NASA Astrophysics Data System (ADS)
Lee, Youngung
2004-12-01
We have estimated the fractal dimension of the molecular clouds in the Antigalactic Center based on the \\co(J=1-0) and tco (J=1-0) database obtained using the 14m telescope at Taeduk Radio Astronomy Observatory. Using a developed code within IRAF, we were able to identify slice-clouds, and determined the dispersions of two spatial coordinates as well as perimeters and areas. The fractal dimension of the target region was estimated to be D=1.34 for low resolution \\co(J=1-0) database, and D=1.4 for higher resolution \\co(J=1-0) and tco(J=1-0) database, where P ∝ AD/2. The sampling rate (spatial resolution) of observed data must be an important parameter when estimating fractal dimension. Our database with higher resolution of 1 arcminute, which is corresponding to 0.2 pc at a distance of 1.1 kpc, gives us the same estimate of fractal dimension to that of local dark clouds. Fractal dimension is apparently invariant when varying the threshold temperatures applied to cloud identification. According to the dispersion pattern of longitudes and latitudes of identified slice-clouds, there is no preference of elongation direction.
NASA Astrophysics Data System (ADS)
Kurnianto, Rudi; Murakami, Yoshinobu; Hozumi, Naohiro; Nagao, Masayuki
The structural response of tree growth in epoxy resin blended with silica filler has been investigated. The physical properties of the resin were varied by changing its filler content and exposing to humid air. The fractal dimension of the electrical tree and its relationship with filler content and humidity were determined. The damaged area of tree in various contents of filler was also estimated. It is considered that the filler would create such an obstruction to the tree growth both in humid and dry conditions. At the ambient condition, the more filler content, the more obstruction would be generated, leading to the significant suppression of tree growth. Likewise, the introduction of filler brought a rise in fractal dimension due to the increase of branches. It is concluded that the existence of filler makes the tree structure more complicated by introducing obstacles to tree propagation, leading to the high fractal dimension of the tree. In addition, it was found that the fractal dimension of the tree was very relational to the fractal dimension of the composite material including filler particles.
Stankovic, Marija; Pantic, Igor; DE Luka, Silvio R; Puskas, Nela; Zaletel, Ivan; Milutinovic-Smiljanic, Sanja; Pantic, Senka; Trbovich, Alexander M
2016-03-01
The aim of the study was to examine alteration and possible application of fractal dimension, angular second moment, and correlation for quantification of structural changes in acutely inflamed tissue. Acute inflammation was induced by injection of turpentine oil into the right and left hind limb muscles of mice, whereas control animals received intramuscular saline injection. After 12 h, animals were anesthetised and treated muscles collected. The tissue was stained by hematoxylin and eosin, digital micrographs produced, enabling determination of fractal dimension of the cells, angular second moment and correlation of studied tissue. Histopathological analysis showed presence of inflammatory infiltrate and tissue damage in inflammatory group, whereas tissue structure in control group was preserved, devoid of inflammatory infiltrate. Fractal dimension of the cells, angular second moment and correlation of treated tissue in inflammatory group decreased in comparison to the control group. In this study, we were first to observe and report that fractal dimension of the cells, angular second moment, and correlation were reduced in acutely inflamed tissue, indicating loss of overall complexity of the cells in the tissue, the tissue uniformity and structure regularity. Fractal dimension, angular second moment and correlation could be useful methods for quantification of structural changes in acute inflammation. PMID:26501409
Ramakrishnan; Sadana
1998-12-15
The diffusion-limited binding kinetics of antigen (analyte) in solution to antibody (receptor) immobilized on a biosensor surface is analyzed within a fractal framework. Most of the data presented are adequately described by a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot ("Scientific Graphing Procedure, User's Manual," Jandel Scientific, San Rafael, CA, 1993). A couple of examples of a dual-fractal analysis are also presented. It is of interest to note that the binding rate coefficient and the fractal dimension both exhibit changes in the same direction for the analyte-receptor systems analyzed. Binding rate coefficient expressions as a function of the fractal dimension developed for the analyte-receptor binding systems indicate the high sensitivity of the binding rate coefficient on the fractal dimension when both a single- and a dual-fractal analysis are used. For example, for a single-fractal analysis and for the binding of cell surface proteins from Helicobacter pylori strain in solution to sialyl-(alpha-2,3)-lactose-conjugated (20 mol%) polyacrylamide immobilized on a resonant mirror biosensor (S. Hirmo et al., Anal. Biochem. 257, 63, 1998), the order of dependence of the binding rate coefficient, k, on the fractal dimension, Df, was 14.15. The fractional order of dependence of the binding rate coefficient(s) on the fractal dimension(s) further reinforces the fractal nature of the system. The binding rate coefficient(s) expressions developed as a function of the fractal dimension(s) are of particular value since they provide a means to better control biosensor performance by linking it to the heterogeneity on the surface and further emphasize in a quantitative sense the importance of the nature of the surface in biosensor performance. Copyright 1998 Academic Press. PMID:9845690
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Sig-NganLam, Nina; Quattrochi, Dale A.
2004-01-01
The accuracy of traditional multispectral maximum-likelihood image classification is limited by the skewed statistical distributions of reflectances from the complex heterogenous mixture of land cover types in urban areas. This work examines the utility of local variance, fractal dimension and Moran's I index of spatial autocorrelation in segmenting multispectral satellite imagery. Tools available in the Image Characterization and Modeling System (ICAMS) were used to analyze Landsat 7 imagery of Atlanta, Georgia. Although segmentation of panchromatic images is possible using indicators of spatial complexity, different land covers often yield similar values of these indices. Better results are obtained when a surface of local fractal dimension or spatial autocorrelation is combined as an additional layer in a supervised maximum-likelihood multispectral classification. The addition of fractal dimension measures is particularly effective at resolving land cover classes within urbanized areas, as compared to per-pixel spectral classification techniques.
NASA Astrophysics Data System (ADS)
Avellar, J.; Duarte, L. G. S.; da Mota, L. A. C. P.; de Melo, N.; Skea, J. E. F.
2012-09-01
A set of Maple routines is presented, fully compatible with the new releases of Maple (14 and higher). The package deals with the numerical evolution of dynamical systems and provide flexible plotting of the results. The package also brings an initial conditions generator, a numerical solver manager, and a focusing set of routines that allow for better analysis of the graphical display of the results. The novelty that the package presents an optional C interface is maintained. This allows for fast numerical integration, even for the totally inexperienced Maple user, without any C expertise being required. Finally, the package provides the routines to calculate the fractal dimension of boundaries (via box counting). New version program summary Program Title: Ndynamics Catalogue identifier: %Leave blank, supplied by Elsevier. Licensing provisions: no. Programming language: Maple, C. Computer: Intel(R) Core(TM) i3 CPU M330 @ 2.13 GHz. Operating system: Windows 7. RAM: 3.0 GB Keywords: Dynamical systems, Box counting, Fractal dimension, Symbolic computation, Differential equations, Maple. Classification: 4.3. Catalogue identifier of previous version: ADKH_v1_0. Journal reference of previous version: Comput. Phys. Commun. 119 (1999) 256. Does the new version supersede the previous version?: Yes. Nature of problem Computation and plotting of numerical solutions of dynamical systems and the determination of the fractal dimension of the boundaries. Solution method The default method of integration is a fifth-order Runge-Kutta scheme, but any method of integration present on the Maple system is available via an argument when calling the routine. A box counting [1] method is used to calculate the fractal dimension [2] of the boundaries. Reasons for the new version The Ndynamics package met a demand of our research community for a flexible and friendly environment for analyzing dynamical systems. All the user has to do is create his/her own Maple session, with the system to be studied, and use the commands on the package to (for instance) calculate the fractal dimension of a certain boundary, without knowing or worrying about a single line of C programming. So the package combines the flexibility and friendly aspect of Maple with the fast and robust numerical integration of the compiled (for example C) basin. The package is old, but the problems it was designed to dealt with are still there. Since Maple evolved, the package stopped working, and we felt compelled to produce this version, fully compatible with the latest version of Maple, to make it again available to the Maple user. Summary of revisions Deprecated Maple Packages and Commands: Paraphrasing the Maple in-built help files, "Some Maple commands and packages are deprecated. A command (or package) is deprecated when its functionality has been replaced by an improved implementation. The newer command is said to supersede the older one, and use of the newer command is strongly recommended". So, we have examined our code to see if some of these occurrences could be dangerous for it. For example, the "readlib" command is unnecessary, and we have removed its occurrences from our code. We have checked and changed all the necessary commands in order for us to be safe in respect to danger from this source. Another change we had to make was related to the tools we have implemented in order to use the interface for performing the numerical integration in C, externally, via the use of the Maple command "ssystem". In the past, we had used, for the external C integration, the DJGPP system. But now we present the package with (free) Borland distribution. The compilation and compiling commands are now slightly changed. For example, to compile only, we had used "gcc-c"; now, we use "bcc32-c", etc. All this installation (Borland) is explained on a "README" file we are submitting here to help the potential user. Restrictions Besides the inherent restrictions of numerical integration methods, this version of the package only deals with systems of first-order differential equations. Unusual features This package provides user-friendly software tools for analyzing the character of a dynamical system, whether it displays chaotic behaviour, and so on. Options within the package allow the user to specify characteristics that separate the trajectories into families of curves. In conjunction with the facilities for altering the user's viewpoint, this provides a graphical interface for the speedy and easy identification of regions with interesting dynamics. An unusual characteristic of the package is its interface for performing the numerical integrations in C using a fifth-order Runge-Kutta method (default). This potentially improves the speed of the numerical integration by some orders of magnitude and, in cases where it is necessary to calculate thousands of graphs in regions of difficult integration, this feature is very desirable. Besides that tool, somewhat more experienced users can produce their own C integrator and, by using the commands available in the package, use it as the C integrator provided with the package as long as the new integrator manages the input and output in the same format as the default one does. Running time This depends strongly on the dynamical system. With an Intel® Core™ i3 CPU M330 @ 2.13 GHz, the integration of 50 graphs, for a system of two first-order equations, typically takes less than a second to run (with the C integration interface). Without the C interface, it takes a few seconds. In order to calculate the fractal dimension, where we typically use 10,000 points to integrate, using the C interface it takes from 20 to 30 s. Without the C interface, it becomes really impractical, taking, sometimes, for the same case, almost an hour. For some cases, it takes many hours.
Park, Sang Cheol; Wang, Xiao-Hui; Zheng, Bin
2009-01-01
Rationale and Objectives To investigate whether using fractal dimension as an objective index (quantitative measure) to assess and control the “visual” or “texture” similarity of reference image regions selected by a CBIR (content-based image retrieval) scheme will (or will not) affect the performance of the scheme in classification between image regions depicting suspicious breast masses. Materials and Methods An image dataset depicting 1500 verified mass regions and 1500 false-positive mass regions was used. We computed 14 morphological and intensity distribution based features and a fractal dimension. A CBIR scheme using a k-nearest neighbor classifier was applied and two experiments were conducted. In the first experiment, we evaluated our CBIR scheme using all 15 features. In the second experiment, we used the fractal dimension as a prescreening feature to guide the CBIR scheme to search for the most similar reference images that have similar measure in the fractal dimension. Results The CBIR scheme achieved classification performance with area under ROC curve (AZ) of 0.857 with 95% confidence interval (CI) of [0.844, 0.870] using 14 features and 0.866 with 95% CI of [0.853, 0.879] after adding fractal dimension. The p-value of two classification results was 0. 005. After using fractal dimension as a prescreening feature, the CBIR scheme achieved AZ = 0.851 with 95% CI of [0.837, 0.864] without significant difference as comparing with the previous result using the original 14 features (p = 0.120). The difference of fractal dimension values between the selected similar reference images was reduced by 56.7% indicating the improvement of image texture similarity. In addition, more than half of references were early discarded without similarity comparison indicating the improvement of searching efficiency. Conclusions This study demonstrated the feasibility of applying the fractal dimension as an objective (quantitative) and efficient search index to assess and maintain texture similarity of reference mass regions selected by the CBIR schemes without reducing the scheme performance in classifying between suspicious breast masses. PMID:19524455
Smith, R.L. Mecholsky, J.J.
2011-05-15
Fractal analysis has been used as a method to study fracture surfaces of brittle materials. However, it has not been determined if the fractal characteristics of brittle materials is consistent throughout the fracture surface. Therefore, the fractal dimensional increment of the mirror, mist, and hackle regions of the fracture surface of silica glass was determined using atomic force microscopy. The fractal dimensional increment of the mirror region (0.17-0.26) was determined to be statistically greater than that for the mist (0.08-0.12) and hackle (0.08-0.13) regions. It is thought that the increase in the fractal dimensional increment is caused by a greater tortuosity in the mirror region due to, most likely, the slower crack velocity of the propagating crack in that region and that there is a point between the mirror and mist region at which the fractal dimension decreases and becomes constant. - Research Highlights: {yields} The fracture surface of silica glass does not have a constant fractal dimension. {yields} Mirror region has greater fractal dimension than mist or hackle region. {yields} Fractal dimension decreases between mirror and mist region. {yields} Greater fractal dimension could be due to slower crack velocity in mirror region.
Using fractal dimensions of stained flow patterns in a clay soil to predict bypass flow
NASA Astrophysics Data System (ADS)
Hatano, R.; Booltink, H. W. G.
1992-07-01
Methylene blue staining patterns of five undisturbed unsaturated soil cores, 200 mm long and 200 mm in diameter, taken from a well-structured clay soil classified as a Hydric Fluvaquent, were characterized by using the fractal dimension of structure to predict measured bypass flow. Cumulative outflow curves in all cores were well described by a spherical model. Outflow in each core started after a significant time lag from the start ofirrigation. Outflow rates during irrigation in all cores were almost equal to irrigation rates; nearly all the water applied, after outflow had started, contributed to bypass flow. Total outflow ( Om, mm) was regressed on the time lag ( T1, min) as: Om = -0.181 T1 + 9.85. This time lag was caused by the effect of internal catchment of discontinuous macropores and surface storage. The three-dimensional fractal dimension of structure ( Ds3) was calculated for the upper and lower halves of the core, by using values of Ds2 and stained area in cross-sections. A statistically significant empirical equation, relating the total amount of outflow ( Om) to both upper and lower values of Ds3 and to the volume fraction of stained parts ( Vs) is: Om = -230.6( VsDs3-1 ) upper + 232.4( VsDs3-1 ) lower + 12.6 Thus, a greater Ds3 value in the upper half of the core and a lower Ds3 value in the lower half of the core induce larger amounts of outflow: hence vertically continuous macropores, such as fragments of cracks or tubes, play a significant role in the process of bypass flow.
Electroencephalographic Fractal Dimension in Healthy Ageing and Alzheimer’s Disease
Cottone, Carlo; Cancelli, Andrea; Rossini, Paolo Maria; Tecchio, Franca
2016-01-01
Brain activity is complex; a reflection of its structural and functional organization. Among other measures of complexity, the fractal dimension is emerging as being sensitive to neuronal damage secondary to neurological and psychiatric diseases. Here, we calculated Higuchi’s fractal dimension (HFD) in resting-state eyes-closed electroencephalography (EEG) recordings from 41 healthy controls (age: 20–89 years) and 67 Alzheimer’s Disease (AD) patients (age: 50–88 years), to investigate whether HFD is sensitive to brain activity changes typical in healthy aging and in AD. Additionally, we considered whether AD-accelerating effects of the copper fraction not bound to ceruloplasmin (also called “free” copper) are reflected in HFD fluctuations. The HFD measure showed an inverted U-shaped relationship with age in healthy people (R2 = .575, p < .001). Onset of HFD decline appeared around the age of 60, and was most evident in central-parietal regions. In this region, HFD decreased with aging stronger in the right than in the left hemisphere (p = .006). AD patients demonstrated reduced HFD compared to age- and education-matched healthy controls, especially in temporal-occipital regions. This was associated with decreasing cognitive status as assessed by mini-mental state examination, and with higher levels of non-ceruloplasmin copper. Taken together, our findings show that resting-state EEG complexity increases from youth to maturity and declines in healthy, aging individuals. In AD, brain activity complexity is further reduced in correlation with cognitive impairment. In addition, elevated levels of non-ceruloplasmin copper appear to accelerate the reduction of neural activity complexity. Overall, HDF appears to be a proper indicator for monitoring EEG-derived brain activity complexity in healthy and pathological aging. PMID:26872349
Reconstructing the fractal dimension of granular aggregates from light intensity spectra.
Tang, Fiona H M; Maggi, Federico
2015-12-21
There has been growing interest in using the fractal dimension to study the hierarchical structures of soft materials after realising that fractality is an important property of natural and engineered materials. This work presents a method to quantify the internal architecture and the space-filling capacity of granular fractal aggregates by reconstructing the three-dimensional capacity dimension from their two-dimensional optical projections. Use is made of the light intensity of the two-dimensional aggregate images to describe the aggregate surface asperities (quantified by the perimeter-based fractal dimension) and the internal architecture (quantified by the capacity dimension) within a mathematical framework. This method was tested on control aggregates of diffusion-limited (DLA), cluster-cluster (CCA) and self-correlated (SCA) types, stereolithographically-fabricated aggregates, and experimentally-acquired natural sedimentary aggregates. Statistics of the reconstructed capacity dimension featured correlation coefficients R ≥ 98%, residuals NRMSE ≤ 10% and percent errors PE ≤ 4% as compared to controls, and improved earlier approaches by up to 50%. PMID:26414181
Lee, Dae-Hyun; Rhyu, In-Chul; Hong, Jeong-Ug; Lee, Cheol-Woo; Heo, Min-Suk; Huh, Kyung-Hoe
2010-01-01
Purpose It has been suggested that primary implant stability plays an essential role in successful osseointegration. Resonance frequency analysis (RFA) is widely used to measure the initial stability of implants because it provides superior reproducibility and non-invasiveness. The purpose of this study is to investigate whether the fractal dimension from the panoramic radiograph is related to the primary stability of the implant as represented by RFA. Methods This study included 22 patients who underwent dental implant installation at the Department of Periodontology of Seoul National University Dental Hospital. Morphometric analysis and fractal analysis of the bone trabecular pattern were performed using panoramic radiographs, and the implant stability quotient (ISQ) values were measured after implant installation using RFA. The radiographs of 52 implant sites were analyzed, and the ISQ values were compared with the results from the morphometric analysis and fractal analysis. Results The Pearson correlation showed a linear correlation between the ISQ values of RFA and the parameters of morphometric analysis but not of statistical significance. The fractal dimension had a linear correlation that was statistically significant. The correlation was more pronounced in the mandible. Conclusions In conclusion, we suggest that the fractal dimension acquired from the panoramic radiograph may be a useful predictor of the initial stability of dental implants. PMID:20498755
NASA Astrophysics Data System (ADS)
Zhu, Weiren; Zhao, Xiaopeng
2009-11-01
Left-handed metamaterials (LHMs) with fractal dendritic cells have recently been demonstrated at both microwave and infrared frequencies. Here, we try to utilize only a single parameter, the fractal dimension, to manipulate the geometries of dendritic structures, and then adjust the resonant behaviors and lossy characteristics of the dendritic LHMs. Both the dendritic LHMs for electromagnetic wave parallel and normal incidences are discussed in this paper. As the fractal dimension increases, the left-handed resonant frequency of the dendritic LHMs decreases, and the lossy characteristics could be well adjusted. The method proposed in this paper could be an important guidance for LHMs' design, especially for the dendritic LHMs operating at optical frequencies.
Structural and Fractal Dimensions are Reliable Determinants of Grain Yield in Soybean
Technology Transfer Automated Retrieval System (TEKTRAN)
Reliable models are needed to describe plants with complex geometric structures, quantify the impact of management strategies on the plant’s geometric distribution in space and time, and predict yield as a function of fractal dimension. We measured growth and development variables on single soybean ...
Size and Fractal Dimension of Colloid Deposits in Model Porous Media
NASA Astrophysics Data System (ADS)
Roth, E. J.; Mays, D. C.; Gilbert, B.
2014-12-01
Colloids exert significant influence on subsurface hydrology, geochemistry, and microbiology. In particular, colloid deposits reduce permeability, triggering a reduction or realignment of flow. Since many subsurface processes are transport-limited, this reduction or realignment of flow, in turn, influences numerous chemical and biological processes. This work explores a conceptual model linking permeability with colloid deposit morphology, where deposit morphology is quantified by two metrics of the colloid deposit: (1) characteristic size and (2) fractal dimension. These two metrics are measured using static light scattering (SLS) within refractive index matched (RIM) porous media, into which a suspension of 100 nm carboxylate-modified polystyrene microspheres are eluted at constant flow. Scattering data are fitted with a two-parameter model that includes deposit fractal dimension, and with a three-parameter model that also includes deposit size. For each set of scattering measurements, the appropriate model is selected using the Akaike information criterion, and model errors are estimated using the bootstrap with 100 replicates. Results indicate two key findings. First, fractal dimensions generally decrease with time as additional colloids are eluted into the column, indicating a transition from more uniform to more dendritic deposits. Second, permeability reduction is associated with colloid deposits having smaller fractal dimensions, that is, with more dendritic and space-filling deposits. Modeling efforts are currently underway to correlate permeability with the underlying hydrodynamic and geochemical variables that determine colloid deposit morphology.
Fractal dimensions of soy protein nanoparticle aggregates determined by dynamic mechanical method
Technology Transfer Automated Retrieval System (TEKTRAN)
The fractal dimension of the protein aggregates can be estimated by dynamic mechanical methods when the particle aggregates are imbedded in a polymer matrix. Nanocomposites were formed by mixing hydrolyzed soy protein isolate (HSPI) nanoparticle aggregates with styrene-butadiene (SB) latex, followe...
Fractal dimensions: A new paradigm to assess spatial memory and learning using Morris water maze.
Singh, Surjeet; Kaur, Harpreet; Sandhir, Rajat
2016-02-15
Morris water maze has been widely used for analysis of cognitive functions and relies on the time taken by animal to find the platform i.e. escape latency as a parameter to quantify spatial memory and learning. However, escape latency is confounded by swimming speed which is not necessarily a cognitive factor. Rather, path length may be a more appropriate and reliable parameter to assess spatial learning. This paper presents fractal dimension as a new paradigm to assess spatial memory and learning in animals. Male wistar rats were administrated with pentylenetetrazole and scopolamine to induce chronic epilepsy and dementia respectively. Fractal dimension of the random path followed by the animals on Morris water maze was analyzed and statistically compared among different experimental groups; the results suggest that fractal dimension is more reliable and accurate parameter to assess cognitive deficits compared to escape latency. Thus, the present study suggests that fractal dimensions could be used as an independent parameter to assess spatial memory and learning in animals using Morris water maze. PMID:26592165
Roth, Eric J; Gilbert, Benjamin; Mays, David C
2015-10-20
Experiments reveal a wide discrepancy between the permeability of porous media containing colloid deposits and the available predictive equations. Evidence suggests that this discrepancy results, in part, from the predictive equations failing to account for colloid deposit morphology. This article reports a series of experiments using static light scattering (SLS) to characterize colloid deposit morphology within refractive index matched (RIM) porous media during flow through a column. Real time measurements of permeability, specific deposit, deposit fractal dimension, and deposit radius of gyration, at different vertical positions, were conducted with initially clean porous media at various ionic strengths and fluid velocities. Decreased permeability (i.e., increased clogging) corresponded with higher specific deposit, lower fractal dimension, and smaller radius of gyration. During deposition, fractal dimension, radius of gyration, and permeability decreased with increasing specific deposit. During flushing with colloid-free fluid, these trends reversed, with increased fractal dimension, radius of gyration, and permeability. These observations suggest a deposition scenario in which large and uniform aggregates become deposits, which reduce porosity, lead to higher fluid shear forces, which then decompose the deposits, filling the pore space with small and dendritic fragments of aggregate. PMID:26412205
NASA Astrophysics Data System (ADS)
Jo, Junghyo; Hörnblad, Andreas; Kilimnik, German; Hara, Manami; Ahlgren, Ulf; Periwal, Vipul
2013-06-01
The islets of Langerhans, responsible for controlling blood glucose levels, are dispersed within the pancreas. A universal power law governing the fractal spatial distribution of islets in two-dimensional pancreatic sections has been reported. However, the fractal geometry in the actual three-dimensional pancreas volume, and the developmental process that gives rise to such a self-similar structure, has not been investigated. Here, we examined the three-dimensional spatial distribution of islets in intact mouse pancreata using optical projection tomography and found a power law with a fractal dimension of 2.1. Furthermore, based on two-dimensional pancreatic sections of human autopsies, we found that the distribution of human islets also follows a universal power law with a fractal dimension of 1.5 in adult pancreata, which agrees with the value previously reported in smaller mammalian pancreas sections. Finally, we developed a self-avoiding growth model for the development of the islet distribution and found that the fractal nature of the spatial islet distribution may be associated with the self-avoidance in the branching process of vascularization in the pancreas.
NASA Astrophysics Data System (ADS)
Cervantes, F.; Gonzalez, J.; Real, C.; Hoyos, L.
2012-12-01
ABSTRACT: Chaotic invariants like fractal dimensions are used to characterize non-linear time series. The fractal dimension is an important characteristic of fractals that contains information about their geometrical structure at multiple scales. In this work four fractal dimension estimation algorithms are applied to non-linear time series. The algorithms employed are the Higuchi's algorithm, the Petrosian's algorithm, the Katz's Algorithm and the Box counting method. The analyzed time series are associated with natural phenomena, the Dst a geomagnetic index which monitors the world wide magnetic storm; the Dst index is a global indicator of the state of the Earth's geomagnetic activity. The time series used in this work show a behavior self-similar, which depend on the time scale of measurements. It is also observed that fractal dimensions may not be constant over all time scales.
Odderon and pomeron as fractal dimension in pp and p¯p total cross-sections
NASA Astrophysics Data System (ADS)
Borcsik, F. S.; Campos, S. D.
2016-03-01
In this paper, one presents a naive parametrization to pp and p¯p total cross-sections. The main goal of this parametrization is to study the possible fractal structure present in the total cross-section. The result of the fitting procedure shows two different fractal dimensions: a negative (low-energies) and a positive (high-energies). The negative fractal dimension represents the emptiness of the total cross-section structure and the positive represents the filling up process with the energy increase. Hence, the total cross-section presents a multifractal behavior. At low-energies, the odderon exchange may be associated with the negative fractal dimension and at high-energies, the pomeron may be associated with the positive fractal dimension. Therefore, the exchange of odderons and pomerons may be viewed as a transition from a less well-defined to a more well-defined internal structure, depending on the energy.
Are fractal dimensions of the spatial distribution of mineral deposits meaningful?
Raines, G.L.
2008-01-01
It has been proposed that the spatial distribution of mineral deposits is bifractal. An implication of this property is that the number of deposits in a permissive area is a function of the shape of the area. This is because the fractal density functions of deposits are dependent on the distance from known deposits. A long thin permissive area with most of the deposits in one end, such as the Alaskan porphyry permissive area, has a major portion of the area far from known deposits and consequently a low density of deposits associated with most of the permissive area. On the other hand, a more equi-dimensioned permissive area, such as the Arizona porphyry permissive area, has a more uniform density of deposits. Another implication of the fractal distribution is that the Poisson assumption typically used for estimating deposit numbers is invalid. Based on datasets of mineral deposits classified by type as inputs, the distributions of many different deposit types are found to have characteristically two fractal dimensions over separate non-overlapping spatial scales in the range of 5-1000 km. In particular, one typically observes a local dimension at spatial scales less than 30-60 km, and a regional dimension at larger spatial scales. The deposit type, geologic setting, and sample size influence the fractal dimensions. The consequence of the geologic setting can be diminished by using deposits classified by type. The crossover point between the two fractal domains is proportional to the median size of the deposit type. A plot of the crossover points for porphyry copper deposits from different geologic domains against median deposit sizes defines linear relationships and identifies regions that are significantly underexplored. Plots of the fractal dimension can also be used to define density functions from which the number of undiscovered deposits can be estimated. This density function is only dependent on the distribution of deposits and is independent of the definition of the permissive area. Density functions for porphyry copper deposits appear to be significantly different for regions in the Andes, Mexico, United States, and western Canada. Consequently, depending on which regional density function is used, quite different estimates of numbers of undiscovered deposits can be obtained. These fractal properties suggest that geologic studies based on mapping at scales of 1:24,000 to 1:100,000 may not recognize processes that are important in the formation of mineral deposits at scales larger than the crossover points at 30-60 km. ?? 2008 International Association for Mathematical Geology.
NASA Astrophysics Data System (ADS)
Araki, Kan
Data of scattering coefficient on vertical incidence against ground surface at U-band are obtained by Millimeter Wave Radar Altimeter using FM-CW ranging. Noise suppression by wavelet shrinkage can be utilized to extract feature parameter in high spatial frequency band, in which level of fractal noise is dominated by that of white noise. We propose approximate algorithm for estimation of local fractal dimension in high spatial frequency band which is the most effective parameter for identification for classification of ground surfaces such as vegetation, town area and rice field.
Fractal Dimension of EEG Activity Senses Neuronal Impairment in Acute Stroke
Zappasodi, Filippo; Olejarczyk, Elzbieta; Marzetti, Laura; Assenza, Giovanni; Pizzella, Vittorio; Tecchio, Franca
2014-01-01
The brain is a self-organizing system which displays self-similarities at different spatial and temporal scales. Thus, the complexity of its dynamics, associated to efficient processing and functional advantages, is expected to be captured by a measure of its scale-free (fractal) properties. Under the hypothesis that the fractal dimension (FD) of the electroencephalographic signal (EEG) is optimally sensitive to the neuronal dysfunction secondary to a brain lesion, we tested the FD’s ability in assessing two key processes in acute stroke: the clinical impairment and the recovery prognosis. Resting EEG was collected in 36 patients 4–10 days after a unilateral ischemic stroke in the middle cerebral artery territory and 19 healthy controls. National Health Institute Stroke Scale (NIHss) was collected at T0 and 6 months later. Highuchi FD, its inter-hemispheric asymmetry (FDasy) and spectral band powers were calculated for EEG signals. FD was smaller in patients than in controls (1.447±0.092 vs 1.525±0.105) and its reduction was paired to a worse acute clinical status. FD decrease was associated to alpha increase and beta decrease of oscillatory activity power. Larger FDasy in acute phase was paired to a worse clinical recovery at six months. FD in our patients captured the loss of complexity reflecting the global system dysfunction resulting from the structural damage. This decrease seems to reveal the intimate nature of structure-function unity, where the regional neural multi-scale self-similar activity is impaired by the anatomical lesion. This picture is coherent with neuronal activity complexity decrease paired to a reduced repertoire of functional abilities. FDasy result highlights the functional relevance of the balance between homologous brain structures’ activities in stroke recovery. PMID:24967904
Fractal dimension of EEG activity senses neuronal impairment in acute stroke.
Zappasodi, Filippo; Olejarczyk, Elzbieta; Marzetti, Laura; Assenza, Giovanni; Pizzella, Vittorio; Tecchio, Franca
2014-01-01
The brain is a self-organizing system which displays self-similarities at different spatial and temporal scales. Thus, the complexity of its dynamics, associated to efficient processing and functional advantages, is expected to be captured by a measure of its scale-free (fractal) properties. Under the hypothesis that the fractal dimension (FD) of the electroencephalographic signal (EEG) is optimally sensitive to the neuronal dysfunction secondary to a brain lesion, we tested the FD's ability in assessing two key processes in acute stroke: the clinical impairment and the recovery prognosis. Resting EEG was collected in 36 patients 4-10 days after a unilateral ischemic stroke in the middle cerebral artery territory and 19 healthy controls. National Health Institute Stroke Scale (NIHss) was collected at T0 and 6 months later. Highuchi FD, its inter-hemispheric asymmetry (FDasy) and spectral band powers were calculated for EEG signals. FD was smaller in patients than in controls (1.447±0.092 vs 1.525±0.105) and its reduction was paired to a worse acute clinical status. FD decrease was associated to alpha increase and beta decrease of oscillatory activity power. Larger FDasy in acute phase was paired to a worse clinical recovery at six months. FD in our patients captured the loss of complexity reflecting the global system dysfunction resulting from the structural damage. This decrease seems to reveal the intimate nature of structure-function unity, where the regional neural multi-scale self-similar activity is impaired by the anatomical lesion. This picture is coherent with neuronal activity complexity decrease paired to a reduced repertoire of functional abilities. FDasy result highlights the functional relevance of the balance between homologous brain structures' activities in stroke recovery. PMID:24967904
Landmine detection using IR image segmentation by means of fractal dimension analysis
NASA Astrophysics Data System (ADS)
Abbate, Horacio A.; Gambini, Juliana; Delrieux, Claudio; Castro, Eduardo H.
2009-05-01
This work is concerned with buried landmines detection by long wave infrared images obtained during the heating or cooling of the soil and a segmentation process of the images. The segmentation process is performed by means of a local fractal dimension analysis (LFD) as a feature descriptor. We use two different LFD estimators, box-counting dimension (BC), and differential box counting dimension (DBC). These features are computed in a per pixel basis, and the set of features is clusterized by means of the K-means method. This segmentation technique produces outstanding results, with low computational cost.
Zone Specific Fractal Dimension of Retinal Images as Predictor of Stroke Incidence
Kumar, Dinesh Kant; Hao, Hao; Unnikrishnan, Premith; Kawasaki, Ryo; Mitchell, Paul
2014-01-01
Fractal dimensions (FDs) are frequently used for summarizing the complexity of retinal vascular. However, previous techniques on this topic were not zone specific. A new methodology to measure FD of a specific zone in retinal images has been developed and tested as a marker for stroke prediction. Higuchi's fractal dimension was measured in circumferential direction (FDC) with respect to optic disk (OD), in three concentric regions between OD boundary and 1.5 OD diameter from its margin. The significance of its association with future episode of stroke event was tested using the Blue Mountain Eye Study (BMES) database and compared against spectrum fractal dimension (SFD) and box-counting (BC) dimension. Kruskal-Wallis analysis revealed FDC as a better predictor of stroke (H = 5.80, P = 0.016, α = 0.05) compared with SFD (H = 0.51, P = 0.475, α = 0.05) and BC (H = 0.41, P = 0.520, α = 0.05) with overall lower median value for the cases compared to the control group. This work has shown that there is a significant association between zone specific FDC of eye fundus images with future episode of stroke while this difference is not significant when other FD methods are employed. PMID:25485298
Fractal dimension of cohesive sediment flocs at steady state under seven shear flow conditions
Zhu, Zhongfan; Yu, Jingshan; Wang, Hongrui; Dou, Jie; Wang, Cheng
2015-08-12
The morphological properties of kaolin flocs were investigated in a Couette-flow experiment at the steady state under seven shear flow conditions (shear rates of 5.36, 9.17, 14, 24, 31, 41 and 53 s-1). These properties include a one-dimensional (1-D) fractal dimension (D1), a two-dimensional (2-D) fractal dimension (D2), a perimeter-based fractal dimension (Dpf) and an aspect ratio (AR). They were calculated based on the projected area (A), equivalent size, perimeter (P) and length (L) of the major axis of the floc determined through sample observation and an image analysis system. The parameter D2, which characterizes the relationship between the projectedmore » area and the length of the major axis using a power function, A ∝ LD2, increased from 1.73 ± 0.03, 1.72 ± 0.03, and 1.75 ± 0.04 in the low shear rate group (G = 5.36, 9.17, and 14 s-1) to 1.92 ± 0.03, 1.82 ± 0.02, 1.85 ± 0.02, and 1.81 ± 0.02 in the high shear rate group (24, 31, 41 and 53 s-1), respectively. The parameter D1 characterizes the relationship between the perimeter and length of the major axis by the function P ∝ LD1 and decreased from 1.52 ± 0.02, 1.48 ± 0.02, 1.55 ± 0.02, and 1.63 ± 0.02 in the low shear group (5.36, 9.17, 14 and 24 s-1) to 1.45 ± 0.02, 1.39 ± 0.02, and 1.39 ± 0.02 in the high shear group (31, 41 and 53 s-1), respectively. The results indicate that with increasing shear rates, the flocs become less elongated and that their boundary lines become tighter and more regular, caused by more breakages and possible restructurings of the flocs. The parameter Dpf, which is related to the perimeter and the projected area through the function , decreased as the shear rate increased almost linearly. The parameter AR, which is the ratio of the length of the major axis and equivalent diameter, decreased from 1.56, 1.59, 1.53 and 1.51 in the low shear rate group to 1.43, 1.47 and 1.48 in the high shear rate group. These changes in Dpf and AR show that the flocs become less convoluted and more symmetrical and that their boundaries become smoother and more regular in the high shear rate group than in the low shear rate group due to breakage and possible restructuring processes. To assess the effects of electrolyte and sediment concentration, 0.1 mol/L calcium chloride (CaCl2) and initial sediment concentration from 7.87 × 10-5 to 1.57 × 10-5 were used in this preliminary study. The addition of electrolyte and increasing sediment concentration could produce more symmetrical flocs with less convoluted and simpler boundaries. In addition, some new information on the temporal variation of the median size of the flocs during the flocculation process is presented.« less
Fractal dimension of cohesive sediment flocs at steady state under seven shear flow conditions
Zhu, Zhongfan; Yu, Jingshan; Wang, Hongrui; Dou, Jie; Wang, Cheng
2015-08-12
The morphological properties of kaolin flocs were investigated in a Couette-flow experiment at the steady state under seven shear flow conditions (shear rates of 5.36, 9.17, 14, 24, 31, 41 and 53 s^{-1}). These properties include a one-dimensional (1-D) fractal dimension (D_{1}), a two-dimensional (2-D) fractal dimension (D_{2}), a perimeter-based fractal dimension (D_{pf}) and an aspect ratio (AR). They were calculated based on the projected area (A), equivalent size, perimeter (P) and length (L) of the major axis of the floc determined through sample observation and an image analysis system. The parameter D_{2}, which characterizes the relationship between the projected area and the length of the major axis using a power function, A ∝ L^{D2}, increased from 1.73 ± 0.03, 1.72 ± 0.03, and 1.75 ± 0.04 in the low shear rate group (G = 5.36, 9.17, and 14 s^{-1}) to 1.92 ± 0.03, 1.82 ± 0.02, 1.85 ± 0.02, and 1.81 ± 0.02 in the high shear rate group (24, 31, 41 and 53 s^{-1}), respectively. The parameter D_{1} characterizes the relationship between the perimeter and length of the major axis by the function P ∝ L^{D1} and decreased from 1.52 ± 0.02, 1.48 ± 0.02, 1.55 ± 0.02, and 1.63 ± 0.02 in the low shear group (5.36, 9.17, 14 and 24 s^{-1}) to 1.45 ± 0.02, 1.39 ± 0.02, and 1.39 ± 0.02 in the high shear group (31, 41 and 53 s^{-1}), respectively. The results indicate that with increasing shear rates, the flocs become less elongated and that their boundary lines become tighter and more regular, caused by more breakages and possible restructurings of the flocs. The parameter D_{pf}, which is related to the perimeter and the projected area through the function , decreased as the shear rate increased almost linearly. The parameter AR, which is the ratio of the length of the major axis and equivalent diameter, decreased from 1.56, 1.59, 1.53 and 1.51 in the low shear rate group to 1.43, 1.47 and 1.48 in the high shear rate group. These changes in D_{pf} and AR show that the flocs become less convoluted and more symmetrical and that their boundaries become smoother and more regular in the high shear rate group than in the low shear rate group due to breakage and possible restructuring processes. To assess the effects of electrolyte and sediment concentration, 0.1 mol/L calcium chloride (CaCl_{2}) and initial sediment concentration from 7.87 × 10^{-5} to 1.57 × 10^{-5} were used in this preliminary study. The addition of electrolyte and increasing sediment concentration could produce more symmetrical flocs with less convoluted and simpler boundaries. In addition, some new information on the temporal variation of the median size of the flocs during the flocculation process is presented.
Banerji, Anirban; Ghosh, Indira
2009-01-01
A robust marker to describe mass, hydrophobicity and polarizability distribution holds the key to deciphering structural and folding constraints within proteins. Since each of these distributions is inhomogeneous in nature, the construct should be sensitive in describing the patterns therein. We show, for the first time, that the hydrophobicity and polarizability distributions in protein interior follow fractal scaling. It is found that (barring ‘all-α’) all the major structural classes of proteins have an amount of unused hydrophobicity left in them. This amount of untapped hydrophobicity is observed to be greater in thermophilic proteins, than that in their (structurally aligned) mesophilic counterparts. ‘All-β’(thermophilic, mesophilic alike) proteins are found to have maximum amount of unused hydrophobicity, while ‘all-α’ proteins have been found to have minimum polarizability. A non-trivial dependency is observed between dielectric constant and hydrophobicity distributions within (α+β) and ‘all-α’ proteins, whereas absolutely no dependency is found between them in the ‘all-β’ class. This study proves that proteins are not as optimally packed as they are supposed to be. It is also proved that origin of α-helices are possibly not hydrophobic but electrostatic; whereas β-sheets are predominantly hydrophobic in nature. Significance of this study lies in protein engineering studies; because it quantifies the extent of packing that ensures protein functionality. It shows that myths regarding protein interior organization might obfuscate our knowledge of actual reality. However, if the later is studied with a robust marker of strong mathematical basis, unknown correlations can still be unearthed; which help us to understand the nature of hydrophobicity, causality behind protein folding, and the importance of anisotropic electrostatics in stabilizing a highly complex structure named ‘proteins’. PMID:19834622
Fractal dimension analysis of mandibular bones: toward a morphological compatibility of implants.
Oshida, Y; Hashem, A; Nishihara, T; Yapchulay, M V
1994-01-01
In addition to biological and mechanical compatibilities for promising implant materials, a morphological compatibility is proposed by the authors. It has been reported by many investigators that implant surface with appropriate roughness and pore size exhibit better bone ingrowth activities. However, these parameters cannot characterize the complexity of surface textures. In the present study, dentulous and edentulous mandibular alveolar bones were utilized. Four segments from each mandible were subjected to the Fractal Dimension (DF) analysis. It was found that the dentulous mandible showed the DF of 1.81 +/- 0.03 while the edentulous mandible exhibited DF of 1.55 +/- 0.07, indicating that the former has more complex surface texture. It was also found that there could be a linear relationship between the surface roughness and the fractal dimension. PMID:8000293
Tissue characterization using fractal dimension of high frequency ultrasound RF time series.
Moradi, Mehdi; Mousavi, Parvin; Abolmaesumi, Purang
2007-01-01
This paper is the first report on the analysis of ultrasound RF echo time series acquired using high frequency ultrasound. We show that variations in the intensity of one sample of RF echo over time is correlated with tissue microstructure. To form the RF time series, a high frequency probe and a tissue sample were fixed in position and RF signals backscattered from the tissue were continuously recorded. The fractal dimension of RF time series was used as a feature for tissue classification. Feature values acquired from different areas of one tissue type were statistically similar. For animal tissues with different cellular microstructure, we successfully used the fractal dimension of RF time series to distinguish segments as small as 20 microns with accuracies as high as 98%. The results of this study demonstrate that the analysis of RF time series is a promising approach for distinguishing tissue types with different cellular microstructure. PMID:18044654
Kimori, Yoshitaka; Katayama, Eisaku; Morone, Nobuhiro; Kodama, Takao
2011-10-01
In this work, we examined structural changes of actin filaments interacting with myosin visualized by quick freeze deep-etch replica electron microscopy (EM) by using a new method of image processing/analysis based on mathematical morphology. In order to quantify the degree of structural changes, two characteristic patterns were extracted from the EM images. One is the winding pattern of the filament shape (WP) reflecting flexibility of the filament, and the other is the surface pattern of the filament (SP) reflecting intra-molecular domain-mobility of actin monomers constituting the filament. EM images were processed by morphological filtering followed by box-counting to calculate the fractal dimensions for WP (D(WP)) and SP (D(SP)). The result indicates that D(WP) was larger than D(SP) irrespective of the state of the filament (myosin-free or bound) and that both parameters for myosin-bound filaments were significantly larger than those for myosin-free filaments. Overall, this work provides the first quantitative insight into how conformational disorder of actin monomers is correlated with the myosin-induced increase in flexibility of actin filaments along their length as suggested by earlier studies with different techniques. Our method is yet to be improved in details, but promising as a powerful tool for studying the structural change of protein molecules and their assemblies, which can potentially be applied to a wide range of biological and biomedical images. PMID:21801838
Reljin, Natasa; Reyes, Bersain A.; Chon, Ki H.
2015-01-01
In this paper, we propose the use of blanket fractal dimension (BFD) to estimate the tidal volume from smartphone-acquired tracheal sounds. We collected tracheal sounds with a Samsung Galaxy S4 smartphone, from five (N = 5) healthy volunteers. Each volunteer performed the experiment six times; first to obtain linear and exponential fitting models, and then to fit new data onto the existing models. Thus, the total number of recordings was 30. The estimated volumes were compared to the true values, obtained with a Respitrace system, which was considered as a reference. Since Shannon entropy (SE) is frequently used as a feature in tracheal sound analyses, we estimated the tidal volume from the same sounds by using SE as well. The evaluation of the performed estimation, using BFD and SE methods, was quantified by the normalized root-mean-squared error (NRMSE). The results show that the BFD outperformed the SE (at least twice smaller NRMSE was obtained). The smallest NRMSE error of 15.877% ± 9.246% (mean ± standard deviation) was obtained with the BFD and exponential model. In addition, it was shown that the fitting curves calculated during the first day of experiments could be successfully used for at least the five following days. PMID:25923929
Reljin, Natasa; Reyes, Bersain A; Chon, Ki H
2015-01-01
In this paper, we propose the use of blanket fractal dimension (BFD) to estimate the tidal volume from smartphone-acquired tracheal sounds. We collected tracheal sounds with a Samsung Galaxy S4 smartphone, from five (N = 5) healthy volunteers. Each volunteer performed the experiment six times; first to obtain linear and exponential fitting models, and then to fit new data onto the existing models. Thus, the total number of recordings was 30. The estimated volumes were compared to the true values, obtained with a Respitrace system, which was considered as a reference. Since Shannon entropy (SE) is frequently used as a feature in tracheal sound analyses, we estimated the tidal volume from the same sounds by using SE as well. The evaluation of the performed estimation, using BFD and SE methods, was quantified by the normalized root-mean-squared error (NRMSE). The results show that the BFD outperformed the SE (at least twice smaller NRMSE was obtained). The smallest NRMSE error of 15.877% ± 9.246% (mean ± standard deviation) was obtained with the BFD and exponential model. In addition, it was shown that the fitting curves calculated during the first day of experiments could be successfully used for at least the five following days. PMID:25923929
NASA Astrophysics Data System (ADS)
Imre, Attila R.; Bidder, Owen R.
2015-09-01
There are several cases when tortuous continuous lines have to be discretized in order to measure length and other related quantities. In this methodological paper, we are going to demonstrate the importance of the proper sequencing (ranking) of the discrete points and call attention to some typical errors caused by improper sequencing. For demonstration, the fractal dimensions of an animal track with well-known original sequencing and for some mixed sequencing are estimated and the extent of errors is shown.
Pereda, E; Gamundi, A; Rial, R; González, J
1998-07-01
The question of whether the finite values of the correlation dimension (D2), used as an index of EEG complexity are due to its chaotic nature or they reflect its behaviour as linearly-correlated noise, remains open. This report aims at clarifying this by measuring D2 and analysing the non-linear nature of EEG through the method of surrogate data as well as by calculating the fractal exponent (beta) via coarse graining spectral analysis (CGSA) in nine adult subjects during waking and sleep states. The results show that even if it is possible to get an estimation of D2 in all states, non-linear structure appears to be present only during slow wave sleep (SWS). EEG exhibits random fractal structure with 1/f(-beta) spectrum (1 < beta < 3) and a negative linear correlation between D2 and beta in all states except during SWS. In consequence, in those states, finite D2 values could be attributed to the fractal nature of EEG and not to the presence of low-dimensional chaos, and therefore, it the use of beta would be more appropriate to describe the complexity of EEG, due to its lower computational cost. PMID:9697926
Pancheliuga, V A; Pancheliuga, M S
2013-01-01
In the present work a methodological background for the histogram method of time series analysis is developed. Connection between shapes of smoothed histograms constructed on the basis of short segments of time series of fluctuations and the fractal dimension of the segments is studied. It is shown that the fractal dimension possesses all main properties of the histogram method. Based on it a further development of fractal dimension determination algorithm is proposed. This algorithm allows more precision determination of the fractal dimension by using the "all possible combination" method. The application of the method to noise-like time series analysis leads to results, which could be obtained earlier only by means of the histogram method based on human expert comparisons of histograms shapes. PMID:23755565
Fractal dimension analysis of aluminum oxide particle for sandblasting dental use.
Oshida, Y; Munoz, C A; Winkler, M M; Hashem, A; Itoh, M
1993-01-01
Aluminum oxide particles are commonly used as a sandblasting media, particularly in dentistry, for multiple purposes including divesting the casting investment materials and increasing effective surface area for enhancing the mechanical retention strengths of succeedingly applied fired porcelain or luting cements. Usually fine aluminum oxide particles are recycled within the sandblasting machine. Ceramics such as aluminum oxides are brittle, therefore, some portions of recycling aluminum oxide particles might be brittle fractured. If fractured sandblasting particles are involved in the recycling media, it might result in irregularity metallic materials surface as well as the recycling sandblasting media itself be contaminated. Hence, it is necessary from both clinical and practical reasons to monitor the particle conditions in terms of size/shape and effectiveness of sandblasting, so that sandblasting dental prostheses can be fabricated in optimum and acceptable conditions. In the present study, the effect of recycling aluminum oxide particles on the surface texture of metallic materials was evaluated by Fractal Dimension Analysis (FDA). Every week the alumina powder was sampled and analyzed for weight fraction and contaminants. Surface texture of sandblasted standard samples was also characterized by FDA. Results indicate very little change in particle size, while the fractal dimension increased. Fractal dimension analysis showed that the aluminum oxide particle as a sandblasting media should be replaced after 30 or 40 min of total accumulated operation time. PMID:8193563
NASA Astrophysics Data System (ADS)
Dumouchel, Christophe; Cousin, Jean; Triballier, Kaëlig
2005-10-01
The present paper reports an experimental investigation on atomizing liquid flows produced by simplified cavity nozzles. The Weber number being kept low, the sprays produced by these injectors depend on the liquid flow characteristics only, and more precisely, on the non-axial kinetic energy and of the turbulent kinetic energy at the nozzle exit. The investigation reported here concentrates on the characterization of liquid flows during atomization by measuring the spatial variation of the local interface length and of the local interface fractal dimension. Both parameters were found representative of the physics of atomization process: they depend on the characteristics of the flow issuing from the nozzle and they are related to the subsequent drop size distribution. The local interface length is representative of the amount of liquid gas interface surface area, and is a function of both the non-axial and the turbulent kinetic energies at the nozzle exit. The fractal dimension is representative of the tortuosity of the liquid gas interface and, as expected, is mainly related to the turbulent kinetic energy at the nozzle exit. As far as the drop size distribution is concerned, it is found that the local interface length at the instant of break-up determines a representative drop diameter of some kind, whereas the fractal dimension at the same instant controls the dispersion of the distribution.
Gheonea, Dan Ionuț; Streba, Costin Teodor; Vere, Cristin Constantin; Șerbănescu, Mircea; Pirici, Daniel; Comănescu, Maria; Streba, Letiția Adela Maria; Ciurea, Marius Eugen; Mogoantă, Stelian; Rogoveanu, Ion
2014-01-01
Background and Aims. Hepatocellular carcinoma (HCC) remains a leading cause of death by cancer worldwide. Computerized diagnosis systems relying on novel imaging markers gained significant importance in recent years. Our aim was to integrate a novel morphometric measurement—the fractal dimension (FD)—into an artificial neural network (ANN) designed to diagnose HCC. Material and Methods. The study included 21 HCC and 28 liver metastases (LM) patients scheduled for surgery. We performed hematoxylin staining for cell nuclei and CD31/34 immunostaining for vascular elements. We captured digital images and used an in-house application to segment elements of interest; FDs were calculated and fed to an ANN which classified them as malignant or benign, further identifying HCC and LM cases. Results. User intervention corrected segmentation errors and fractal dimensions were calculated. ANNs correctly classified 947/1050 HCC images (90.2%), 1021/1050 normal tissue images (97.23%), 1215/1400 LM (86.78%), and 1372/1400 normal tissues (98%). We obtained excellent interobserver agreement between human operators and the system. Conclusion. We successfully implemented FD as a morphometric marker in a decision system, an ensemble of ANNs designed to differentiate histological images of normal parenchyma from malignancy and classify HCCs and LMs. PMID:25025042
Fractals with Hyperbolic Scators in 1 + 2 Dimensions
NASA Astrophysics Data System (ADS)
Fernández-Guasti, M.
2015-04-01
A nondistributive scator algebra in 1 + 2 dimensions is used to map the quadratic iteration. The hyperbolic numbers square bound set reveals a rich structure when taken into the three-dimensional (3D) hyperbolic scator space. Self-similar small copies of the larger set are obtained along the real axis. These self-similar sets are located at the same positions and have equivalent relative sizes as the small M-set copies found between the Myrberg-Feigenbaum (MF) point and -2 in the complex Mandelbrot set. Furthermore, these small copies are self similar 3D copies of the larger 3D bound set. The real roots of the respective polynomials exhibit basins of attraction in a 3D space. Slices of the 3D confined scator set, labeled {c2i0}{E}+1+2(s;x,y), are shown at different planes to give an approximate idea of the 3D objects highly complicated boundary.
Ul'yanov, A S; Lyapina, A M; Ulianova, O V; Fedorova, V A; Uianov, S S
2011-04-30
Specific statistical characteristics of biospeckles, emerging under the diffraction of coherent beams on the bacterial colonies, are studied. The dependence of the fractal dimensions of biospeckles on the conditions of both illumination and growth of the colonies is studied theoretically and experimentally. Particular attention is paid to the fractal properties of biospeckles, emerging under the scattering of light by the colonies of the vaccinal strain of the plague microbe. The possibility in principle to classify the colonies of Yersinia pestis EV NIIEG using the fractal dimension analysis is demonstrated. (optical technologies in biophysics and medicine)
NASA Astrophysics Data System (ADS)
Ul'yanov, A. S.; Lyapina, A. M.; Ulianova, O. V.; Fedorova, V. A.; Uianov, S. S.
2011-04-01
Specific statistical characteristics of biospeckles, emerging under the diffraction of coherent beams on the bacterial colonies, are studied. The dependence of the fractal dimensions of biospeckles on the conditions of both illumination and growth of the colonies is studied theoretically and experimentally. Particular attention is paid to the fractal properties of biospeckles, emerging under the scattering of light by the colonies of the vaccinal strain of the plague microbe. The possibility in principle to classify the colonies of Yersinia pestis EV NIIEG using the fractal dimension analysis is demonstrated.
NASA Astrophysics Data System (ADS)
Kikuchi, Tsuneo; Nakazawa, Toshihiro; Furukawa, Tetsuo; Higuchi, Toshiyuki; Maruyama, Yukio; Sato, Sojun
1995-05-01
This paper describes the quantitative measurement of the amount of fibrosis in the rat liver using the fractal dimension of the shape of power spectrum. The shape of the power spectrum of the scattered echo from biotissues is strongly affected by its internal structure. The fractal dimension, which is one of the important parameters of the fractal theory, is useful to express the complexity of shape of figures such as the power spectrum. From in vitro experiments using rat liver, it was found that this method can be used to quantitatively measure the amount of fibrosis in the liver, and has the possibility for use in the diagnosis of human liver cirrhosis.
NASA Astrophysics Data System (ADS)
Alonso, C.; Benito, R. M.; Tarquis, A. M.
2012-04-01
Satellite image data have become an important source of information for monitoring vegetation and mapping land cover at several scales. Beside this, the distribution and phenology of vegetation is largely associated with climate, terrain characteristics and human activity. Various vegetation indices have been developed for qualitative and quantitative assessment of vegetation using remote spectral measurements. In particular, sensors with spectral bands in the red (RED) and near-infrared (NIR) lend themselves well to vegetation monitoring and based on them [(NIR - RED) / (NIR + RED)] Normalized Difference Vegetation Index (NDVI) has been widespread used. Given that the characteristics of spectral bands in RED and NIR vary distinctly from sensor to sensor, NDVI values based on data from different instruments will not be directly comparable. The spatial resolution also varies significantly between sensors, as well as within a given scene in the case of wide-angle and oblique sensors. As a result, NDVI values will vary according to combinations of the heterogeneity and scale of terrestrial surfaces and pixel footprint sizes. Therefore, the question arises as to the impact of differences in spectral and spatial resolutions on vegetation indices like the NDVI. The aim of this study is to establish a comparison between two different sensors in their NDVI values at different spatial resolutions. Scaling analysis and modeling techniques are increasingly understood to be the result of nonlinear dynamic mechanisms repeating scale after scale from large to small scales leading to non-classical resolution dependencies. In the remote sensing framework the main characteristic of sensors images is the high local variability in their values. This variability is a consequence of the increase in spatial and radiometric resolution that implies an increase in complexity that it is necessary to characterize. Fractal and multifractal techniques has been proven to be useful to extract such complexities from remote sensing images and will applied in this study to see the scaling behavior for each sensor in generalized fractal dimensions. The studied area is located in the provinces of Caceres and Salamanca (east of Iberia Peninsula) with an extension of 32 x 32 km2. The altitude in the area varies from 1,560 to 320 m, comprising natural vegetation in the mountain area (forest and bushes) and agricultural crops in the valleys. Scaling analysis were applied to Landsat-5 and MODIS TERRA to the normalized derived vegetation index (NDVI) on the same region with one day of difference, 13 and 12 of July 2003 respectively. From these images the area of interest was selected obtaining 1024 x 1024 pixels for Landsat image and 128 x 128 pixels for MODIS image. This implies that the resolution for MODIS is 250x250 m. and for Landsat is 30x30 m. From the reflectance data obtained from NIR and RED bands, NDVI was calculated for each image focusing this study on 0.2 to 0.5 ranges of values. Once that both NDVI fields were obtained several fractal dimensions were estimated in each one segmenting the values in 0.20-0.25, 0.25-0.30 and so on to rich 0.45-0.50. In all the scaling analysis the scale size length was expressed in meters, and not in pixels, to make the comparison between both sensors possible. Results are discussed. Acknowledgements This work has been supported by the Spanish MEC under Projects No. AGL2010-21501/AGR, MTM2009-14621 and i-MATH No. CSD2006-00032
Fractal dimension of chromatin: potential molecular diagnostic applications for cancer prognosis
Metze, Konradin
2013-01-01
Fractal characteristics of chromatin, revealed by light or electron microscopy, have been reported during the last 20 years. Fractal features can easily be estimated in digitalized microscopic images and are helpful for diagnosis and prognosis of neoplasias. During carcinogenesis and tumor progression, an increase of the fractal dimension (FD) of stained nuclei has been shown in intraepithelial lesions of the uterine cervix and the anus, oral squamous cell carcinomas or adenocarcinomas of the pancreas. Furthermore, an increased FD of chromatin is an unfavorable prognostic factor in squamous cell carcinomas of the oral cavity and the larynx, melanomas and multiple myelomas. High goodness-of-fit of the regression line of the FD is a favorable prognostic factor in acute leukemias and multiple myelomas. The nucleus has fractal and power-law organization in several different levels, which might in part be interrelated. Some possible relations between modifications of the chromatin organization during carcinogenesis and tumor progression and an increase of the FD of stained chromatin are suggested. Furthermore, increased complexity of the chromatin structure, loss of heterochromatin and a less-perfect self-organization of the nucleus in aggressive neoplasias are discussed. PMID:24063399
Modified box dimension and average weighted receiving time on the weighted fractal networks
Dai, Meifeng; Sun, Yanqiu; Shao, Shuxiang; Xi, Lifeng; Su, Weiyi
2015-01-01
In this paper a family of weighted fractal networks, in which the weights of edges have been assigned to different values with certain scale, are studied. For the case of the weighted fractal networks the definition of modified box dimension is introduced, and a rigorous proof for its existence is given. Then, the modified box dimension depending on the weighted factor and the number of copies is deduced. Assuming that the walker, at each step, starting from its current node, moves uniformly to any of its nearest neighbors. The weighted time for two adjacency nodes is the weight connecting the two nodes. Then the average weighted receiving time (AWRT) is a corresponding definition. The obtained remarkable result displays that in the large network, when the weight factor is larger than the number of copies, the AWRT grows as a power law function of the network order with the exponent, being the reciprocal of modified box dimension. This result shows that the efficiency of the trapping process depends on the modified box dimension: the larger the value of modified box dimension, the more efficient the trapping process is. PMID:26666355
Modified box dimension and average weighted receiving time on the weighted fractal networks
NASA Astrophysics Data System (ADS)
Dai, Meifeng; Sun, Yanqiu; Shao, Shuxiang; Xi, Lifeng; Su, Weiyi
2015-12-01
In this paper a family of weighted fractal networks, in which the weights of edges have been assigned to different values with certain scale, are studied. For the case of the weighted fractal networks the definition of modified box dimension is introduced, and a rigorous proof for its existence is given. Then, the modified box dimension depending on the weighted factor and the number of copies is deduced. Assuming that the walker, at each step, starting from its current node, moves uniformly to any of its nearest neighbors. The weighted time for two adjacency nodes is the weight connecting the two nodes. Then the average weighted receiving time (AWRT) is a corresponding definition. The obtained remarkable result displays that in the large network, when the weight factor is larger than the number of copies, the AWRT grows as a power law function of the network order with the exponent, being the reciprocal of modified box dimension. This result shows that the efficiency of the trapping process depends on the modified box dimension: the larger the value of modified box dimension, the more efficient the trapping process is.
Estimate for the fractal dimension of the Apollonian gasket in d dimensions
NASA Astrophysics Data System (ADS)
Farr, R. S.; Griffiths, E.
2010-06-01
We adapt a recent theory for the random close packing of polydisperse spheres in three dimensions [R. S. Farr and R. D. Groot, J. Chem. Phys. 131, 244104 (2009)] in order to predict the Hausdorff dimension dA of the Apollonian gasket in dimensions 2 and above. Our approximate results agree with published values in two and three dimensions to within 0.05% and 0.6%, respectively, and we provide predictions for dimensions 4-8.
Huang, F.; Peng, R. D.; Liu, Y. H.; Chen, Z. Y.; Ye, M. F.; Wang, L.
2012-09-15
Fractal dust grains of different shapes are observed in a radially confined magnetized radio frequency plasma. The fractal dimensions of the dust structures in two-dimensional (2D) horizontal dust layers are calculated, and their evolution in the dust growth process is investigated. It is found that as the dust grains grow the fractal dimension of the dust structure decreases. In addition, the fractal dimension of the center region is larger than that of the entire region in the 2D dust layer. In the initial growth stage, the small dust particulates at a high number density in a 2D layer tend to fill space as a normal surface with fractal dimension D = 2. The mechanism of the formation of fractal dust grains is discussed.
Fractal dimension values of cerebral and cerebellar activity in rats loaded with aluminium.
Kekovic, Goran; Culic, Milka; Martac, Ljiljana; Stojadinovic, Gordana; Capo, Ivan; Lalosevic, Dusan; Sekulic, Slobodan
2010-07-01
Aluminium interferes with a variety of cellular metabolic processes in the mammalian nervous system and its intake might increase a risk of developing Alzheimer's disease (AD). While cerebral involvement even at the early stages of intoxication is well known, the role of cerebellum is underestimated. Our aim was to investigate cerebral and cerebellar electrocortical activity in adult male rats exposed to chronic aluminium treatment by nonlinear analytic tools. The adult rats in an aluminium-treated group were injected by AlCl(3), intraperitoneally (2 mg Al/kg, daily for 4 weeks). Fractal analysis of brain activity was performed off-line using Higuchi's algorithm. The average fractal dimension of electrocortical activity in aluminium-treated animals was lower than the average fractal dimension of electrocortical activity in the control rats, at cerebral but not at cerebellar level. The changes in the stationary and nonlinear properties of time series were more expressed in cerebral electrocortical activity than in cerebellar activity. This can be useful for developing effective diagnostic and therapeutic strategies in neurodegenerative diseases. PMID:20424923
Nuclear Fractal Dimensions as a Tool for Prognostication of Oral Squamous Cell Carcinoma
Yinti, Shanmukha Raviteja; Boaz, Karen; Lewis, Amitha J; Ashokkumar, Pandya Jay; Kapila, Supriya Nikita
2015-01-01
Background Carcinogenesis follows complex molecular alterations, which are triggered by subtle chromatin architectural changes that are imperceptible to the human eye. As the treatment decisions in Oral Squamous Cell Carcinoma (OSCC) are hindered by the imprecise clinical stage determination and inter-observer variability in histological grading, focus in recent years has shifted to discovering identifiers related to neoplastic cell morphology studied through computer-aided image analysis. One such approach is the assessment of fractal geometry, a technique first described by Mandelbrot, which aids in precise assessment of architecture of natural objects. Assessment and quantification of degree of complexity of these fractal objects (self-similarities in structural complexity at different magnifying scales) is described as fractal dimension (FD). Aim To evaluate the nuclear fractal dimension (NFD) in OSCC using computer-aided image analysis. Materials and Methods Histological sections of 14 selected cases of Oral Squamous Cell Carcinoma (OSCC) and 6 samples of normal buccal mucosa (as control) were stained with Haematoxylin-Eosin and Feulgen stain for histopathological examination and evaluation of nuclear complexity respectively. Fifteen HPF at Invasive Tumour Front (ITF) and Tumour Proper (TP) of Feulgen-stained sections were selected and photographed in test and control samples. At ITF, TP and normal buccal mucosa 200 nuclei each were selected and analyzed using Image J software to quantify FD. The test and control groups were compared statistically using Independent sample t-test and One-way ANOVA. Results Nuclear FD increased progressively towards worst tumour staging as compared to normal buccal mucosa. Conclusion Nuclear FD can be considered for quantification of nuclear architectural changes as a prognostic indicator in OSCC. PMID:26674013
Generation and geometrical analysis of dense clusters with variable fractal dimension.
Ehrl, Lyonel; Soos, Miroslav; Lattuada, Marco
2009-08-01
The generation and geometrical analysis of clusters composed of rigid monodisperse primary particles with variable fractal dimension, df, in the range from 2.2 to 3 are presented. For all generated aggregate populations, it was found that the dimensionless aggregate mass, i, and the aggregate size, characterized by the radius of gyration, Rg, normalized by the primary particle radius, Rp, follow a fractal scaling, i = kf(Rg/Rp)df. Furthermore, the obtained prefactor of the fractal scaling, kf, is related to df according to kf = 4.46df-2.08, which is in agreement with literature data. For cases when df cannot be directly determined from light scattering or confocal laser scanning microscopy, it can be estimated from its relation with a perimeter fractal dimension, dpf, or a chord fractal dimension, dcf, both obtained from 2D projection of aggregates. A relation between df and dpf of the form df = or-1.5dpf + 4.4 was developed by fitting data obtained in this work for 2.2 < df < 3 together with data of Lee and Kramer [Adv. Colloid Interface Sci. 2004, 112(1-3), 49-57] for 1.8 < df < 2.4. It was found that the method of determining df via dpf is very robust with respect to an artificially introduced blur. In contrary, a relation between df and dcf could only be established for the case of ideal optical analysis, while the introduction of blur results in a significant effect on the chord length distribution (and its moments), up to the point of impeding the evaluation of dcf. Hence, for compact aggregates, it is recommended to determine df from dpf by applying the proposed relation, which is valid in a broad range of df relevant for industrial praxis, with little effect of blur on it. Apart from scaling relations with respect to aggregate mass and size, it was found that the 3D quantities, i and Rg, can be directly related to the area squared over perimeter, A2/P, and the 2D radius of gyration, Rg,2D, respectively, which are obtained from 2D projections. In particular, the following two relations are provided: i = 4.5(A2/P)0.9 and Rg/Rp = 1.47 (Rg,2D/Rp)0.99. PMID:19594146
A Brief Historical Introduction to Fractals and Fractal Geometry
ERIC Educational Resources Information Center
Debnath, Lokenath
2006-01-01
This paper deals with a brief historical introduction to fractals, fractal dimension and fractal geometry. Many fractals including the Cantor fractal, the Koch fractal, the Minkowski fractal, the Mandelbrot and Given fractal are described to illustrate self-similar geometrical figures. This is followed by the discovery of dynamical systems and
A Brief Historical Introduction to Fractals and Fractal Geometry
ERIC Educational Resources Information Center
Debnath, Lokenath
2006-01-01
This paper deals with a brief historical introduction to fractals, fractal dimension and fractal geometry. Many fractals including the Cantor fractal, the Koch fractal, the Minkowski fractal, the Mandelbrot and Given fractal are described to illustrate self-similar geometrical figures. This is followed by the discovery of dynamical systems and…
THE FRACTAL DIMENSION OF STAR-FORMING REGIONS AT DIFFERENT SPATIAL SCALES IN M33
Sanchez, Nestor; Alfaro, Emilio J.; Anez, Neyda; Odekon, Mary Crone
2010-09-01
We study the distribution of stars, H II regions, molecular gas, and individual giant molecular clouds in M33 over a wide range of spatial scales. The clustering strength of these components is systematically estimated through the fractal dimension. We find scale-free behavior at small spatial scales and a transition to a larger correlation dimension (consistent with a nearly uniform distribution) at larger scales. The transition region lies in the range {approx}500-1000 pc. This transition defines a characteristic size that separates the regime of small-scale turbulent motion from that of large-scale galactic dynamics. At small spatial scales, bright young stars and molecular gas are distributed with nearly the same three-dimensional fractal dimension (D {sub f,3D} {approx}< 1.9), whereas fainter stars and H II regions exhibit higher values, D {sub f,3D} {approx_equal} 2.2-2.5. Our results indicate that the interstellar medium in M33 is on average more fragmented and irregular than in the Milky Way.
Fractal dimension analysis of landscape scale variability in greenhouse gas production potentials
NASA Astrophysics Data System (ADS)
da Silva Bicalho, Elton; Spokas, Kurt; La Scala, Newton, Jr.
2015-04-01
Soil greenhouse gas emission is influenced by tillage and management practices that modify soil attributes directly related to the dynamics of soil carbon in the agricultural environment. The aim of this study was to assess the soil CO2 and N2O production potentials and their spatial variability characterized by fractal dimension in different scales, in addition to their correlation with other soil attributes. The quantification of soil CO2 and N2O production was carried out from dry soil samples collected in a grid of 50 × 50 m containing 133 points arranged symmetrically on a sugarcane area under green residue management in southern Brazil. Laboratory incubations were used to analyze greenhouse gas dynamics by gas chromatography. Soil CO2 and N2O production were correlated significantly (P < 0.05) with microbial biomass, silt and clay content, pH, available phosphorus, sum of metal cations (bases), and cation exchange capacity. Similarly, these soil attributes also were correlated with microbial biomass, supporting their role in soil microbial activity and greenhouse gas production. Furthermore, variations in the fractal dimension over the scale indicate that the pattern of the spatial variability structure of soil CO2 production potential was correlated to that observed for microbial biomass, pH, available phosphorus, sum of bases, and cation exchange capacity. On the other hand, only the spatial structure of the clay content, pH and the sum of bases were correlated with the soil N2O production. Therefore, examining the fractal dimension enables the spatially visualization of altering processes across a landscape at different scales, which highlights properties that influence greenhouse gas production and emission in agricultural areas.
NASA Astrophysics Data System (ADS)
Ahammer, Helmut; DeVaney, Trevor T. J.
2004-03-01
The boundary of a fractal object, represented in a two-dimensional space, is theoretically a line with an infinitely small width. In digital images this boundary or contour is limited to the pixel resolution of the image and the width of the line commonly depends on the edge detection algorithm used. The Minkowski dimension was evaluated by using three different edge detection algorithms (Sobel, Roberts, and Laplace operator). These three operators were investigated because they are very widely used and because their edge detection result is very distinct concerning the line width. Very common fractals (Sierpinski carpet and Koch islands) were investigated as well as the binary images from a cancer invasion assay taken with a confocal laser scanning microscope. The fractal dimension is directly proportional to the width of the contour line and the fact, that in practice very often the investigated objects are fractals only within a limited resolution range is considered too.
ERIC Educational Resources Information Center
McCartney, M.; Myers, D.; Sun, Y.
2008-01-01
The divider dimensions of a range of maps of Ireland dating from 1567 to 1893 are evaluated, and it is shown that for maps produced before 1650 the fractal dimension of the map can be correlated to its date of publication. Various classroom uses and extensions are discussed. (Contains 2 figures.)
Measuring capital market efficiency: long-term memory, fractal dimension and approximate entropy
NASA Astrophysics Data System (ADS)
Kristoufek, Ladislav; Vosvrda, Miloslav
2014-07-01
We utilize long-term memory, fractal dimension and approximate entropy as input variables for the Efficiency Index [L. Kristoufek, M. Vosvrda, Physica A 392, 184 (2013)]. This way, we are able to comment on stock market efficiency after controlling for different types of inefficiencies. Applying the methodology on 38 stock market indices across the world, we find that the most efficient markets are situated in the Eurozone (the Netherlands, France and Germany) and the least efficient ones in the Latin America (Venezuela and Chile).
NASA Astrophysics Data System (ADS)
Monthus, Cécile
2015-09-01
For Gaussian Spin-Glasses in low dimensions, we introduce a simple Strong Disorder renormalization at zero temperature in order to construct ground states for Periodic and Anti-Periodic boundary conditions. The numerical study in dimensions d = 2 (up to sizes 20482) and d = 3 (up to sizes 1283) yields that Domain Walls are fractal of dimensions ds(d = 2) ≃ 1.27 and ds(d = 3) ≃ 2.55, respectively.
Taylor, Adele M.; MacGillivray, Thomas J.; Henderson, Ross D.; Ilzina, Lasma; Dhillon, Baljean; Starr, John M.; Deary, Ian J.
2015-01-01
Purpose Cerebral microvascular disease is associated with dementia. Differences in the topography of the retinal vascular network may be a marker for cerebrovascular disease. The association between cerebral microvascular state and non-pathological cognitive ageing is less clear, particularly because studies are rarely able to adjust for pre-morbid cognitive ability level. We measured retinal vascular fractal dimension (Df) as a potential marker of cerebral microvascular disease. We examined the extent to which it contributes to differences in non-pathological cognitive ability in old age, after adjusting for childhood mental ability. Methods Participants from the Lothian Birth Cohort 1936 Study (LBC1936) had cognitive ability assessments and retinal photographs taken of both eyes aged around 73 years (n = 648). IQ scores were available from childhood. Retinal vascular Df was calculated with monofractal and multifractal analysis, performed on custom-written software. Multiple regression models were applied to determine associations between retinal vascular Df and general cognitive ability (g), processing speed, and memory. Results Only three out of 24 comparisons (two eyes × four Df parameters × three cognitive measures) were found to be significant. This is little more than would be expected by chance. No single association was verified by an equivalent association in the contralateral eye. Conclusions The results show little evidence that fractal measures of retinal vascular differences are associated with non-pathological cognitive ageing. PMID:25816017
NASA Astrophysics Data System (ADS)
Bazell, David; Dwek, Eli
1990-09-01
Mathis and Whiffen (1989) have recently suggested that interstellar dust particles are fluffy aggregates of submicron-size particles composed of various astronomical minerals. These dust particles should exhibit optical properties that are quite different from standard dust, characterized by spherical particles of various homogeneous mineral composition. In this paper, the discrete dipole approximation (DDA) method is used to examine the effects of chemical inhomogeneities and spatial structure on the optical properties of interstellar Mathis-Whiffen-type dust particles. The spatial structure of the dust is represented by its fractal dimension, and the chemical inhomogeneities are simulated by randomly assigning the composition of the occupied sites in the structure to be either carbon or silicate. It is found that compositional inhomogeneities are the dominant parameter affecting the shape of the 9.7 and 18 micron silicate bands. Some bands-shape variations can be attributed to the fractal dimension of the dust. The results derived here can be used to explain or constrain variations in these parameters among various astronomical objects.
N'Diaye, Mambaye; Degeratu, Cristinel; Bouler, Jean-Michel; Chappard, Daniel
2013-05-01
Porous structures are becoming more and more important in biology and material science because they help in reducing the density of the grafted material. For biomaterials, porosity also increases the accessibility of cells and vessels inside the grafted area. However, descriptors of porosity are scanty. We have used a series of biomaterials with different types of porosity (created by various porogens: fibers, beads …). Blocks were studied by microcomputed tomography for the measurement of 3D porosity. 2D sections were re-sliced to analyze the microarchitecture of the pores and were transferred to image analysis programs: star volumes, interconnectivity index, Minkowski-Bouligand and Kolmogorov fractal dimensions were determined. Lacunarity and succolarity, two recently described fractal dimensions, were also computed. These parameters provided a precise description of porosity and pores' characteristics. Non-linear relationships were found between several descriptors e.g. succolarity and star volume of the material. A linear correlation was found between lacunarity and succolarity. These techniques appear suitable in the study of biomaterials usable as bone substitutes. PMID:23498228
NASA Technical Reports Server (NTRS)
Bazell, David; Dwek, Eli
1990-01-01
Mathis and Whiffen (1989) have recently suggested that interstellar dust particles are fluffy aggregates of submicron-size particles composed of various astronomical minerals. These dust particles should exhibit optical properties that are quite different from standard dust, characterized by spherical particles of various homogeneous mineral composition. In this paper, the discrete dipole approximation (DDA) method is used to examine the effects of chemical inhomogeneities and spatial structure on the optical properties of interstellar Mathis-Whiffen-type dust particles. The spatial structure of the dust is represented by its fractal dimension, and the chemical inhomogeneities are simulated by randomly assigning the composition of the occupied sites in the structure to be either carbon or silicate. It is found that compositional inhomogeneities are the dominant parameter affecting the shape of the 9.7 and 18 micron silicate bands. Some bands-shape variations can be attributed to the fractal dimension of the dust. The results derived here can be used to explain or constrain variations in these parameters among various astronomical objects.
Objective Auscultation of TCM Based on Wavelet Packet Fractal Dimension and Support Vector Machine
Yan, Jian-Jun; Wang, Yi-Qin; Liu, Guo-Ping; Yan, Hai-Xia; Xia, Chun-Ming; Shen, Xiaojing
2014-01-01
This study was conducted to illustrate that auscultation features based on the fractal dimension combined with wavelet packet transform (WPT) were conducive to the identification the pattern of syndromes of Traditional Chinese Medicine (TCM). The WPT and the fractal dimension were employed to extract features of auscultation signals of 137 patients with lung Qi-deficient pattern, 49 patients with lung Yin-deficient pattern, and 43 healthy subjects. With these features, the classification model was constructed based on multiclass support vector machine (SVM). When all auscultation signals were trained by SVM to decide the patterns of TCM syndromes, the overall recognition rate of model was 79.49%; when male and female auscultation signals were trained, respectively, to decide the patterns, the overall recognition rate of model reached 86.05%. The results showed that the methods proposed in this paper were effective to analyze auscultation signals, and the performance of model can be greatly improved when the distinction of gender was considered. PMID:24883068
NASA Astrophysics Data System (ADS)
Guo, Long; Cai, XU
2009-08-01
It is shown that many real complex networks share distinctive features, such as the small-world effect and the heterogeneous property of connectivity of vertices, which are different from random networks and regular lattices. Although these features capture the important characteristics of complex networks, their applicability depends on the style of networks. To unravel the universal characteristics many complex networks have in common, we study the fractal dimensions of complex networks using the method introduced by Shanker. We find that the average 'density' (ρ(r)) of complex networks follows a better power-law function as a function of distance r with the exponent df, which is defined as the fractal dimension, in some real complex networks. Furthermore, we study the relation between df and the shortcuts Nadd in small-world networks and the size N in regular lattices. Our present work provides a new perspective to understand the dependence of the fractal dimension df on the complex network structure.
A new approach in the BCI research based on fractal dimension as feature and Adaboost as classifier.
Boostani, Reza; Moradi, Mohammad Hassan
2004-12-01
High rate classification of imagery tasks is still one of the hot topics among the brain computer interface (BCI) groups. In order to improve this rate, a new approach based on fractal dimension as feature and Adaboost as classifier is presented for five subjects in this paper. To have a comparison, features such as band power, Hjorth parameters along with LDA classifier have been taken into account. Fractal dimension as a feature with Adaboost and LDA can be considered as alternative combinations for BCI applications. PMID:15876641
A new version of Visual tool for estimating the fractal dimension of images
NASA Astrophysics Data System (ADS)
Grossu, I. V.; Felea, D.; Besliu, C.; Jipa, Al.; Bordeianu, C. C.; Stan, E.; Esanu, T.
2010-04-01
This work presents a new version of a Visual Basic 6.0 application for estimating the fractal dimension of images (Grossu et al., 2009 [1]). The earlier version was limited to bi-dimensional sets of points, stored in bitmap files. The application was extended for working also with comma separated values files and three-dimensional images. New version program summaryProgram title: Fractal Analysis v02 Catalogue identifier: AEEG_v2_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEG_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 9999 No. of bytes in distributed program, including test data, etc.: 4 366 783 Distribution format: tar.gz Programming language: MS Visual Basic 6.0 Computer: PC Operating system: MS Windows 98 or later RAM: 30 M Classification: 14 Catalogue identifier of previous version: AEEG_v1_0 Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 1999 Does the new version supersede the previous version?: Yes Nature of problem: Estimating the fractal dimension of 2D and 3D images. Solution method: Optimized implementation of the box-counting algorithm. Reasons for new version:The previous version was limited to bitmap image files. The new application was extended in order to work with objects stored in comma separated values (csv) files. The main advantages are: Easier integration with other applications (csv is a widely used, simple text file format); Less resources consumed and improved performance (only the information of interest, the "black points", are stored); Higher resolution (the points coordinates are loaded into Visual Basic double variables [2]); Possibility of storing three-dimensional objects (e.g. the 3D Sierpinski gasket). In this version the optimized box-counting algorithm [1] was extended to the three-dimensional case. Summary of revisions:The application interface was changed from SDI (single document interface) to MDI (multi-document interface). One form was added in order to provide a graphical user interface for the new functionalities (fractal analysis of 2D and 3D images stored in csv files). Additional comments: User friendly graphical interface; Easy deployment mechanism. Running time: In the first approximation, the algorithm is linear. References:[1] I.V. Grossu, C. Besliu, M.V. Rusu, Al. Jipa, C.C. Bordeianu, D. Felea, Comput. Phys. Comm. 180 (2009) 1999-2001.[2] F. Balena, Programming Microsoft Visual Basic 6.0, Microsoft Press, US, 1999.
Jitaree, Sirinapa; Phinyomark, Angkoon; Boonyaphiphat, Pleumjit; Phukpattaranont, Pornchai
2015-01-01
Having a classifier of cell types in a breast cancer microscopic image (BCMI), obtained with immunohistochemical staining, is required as part of a computer-aided system that counts the cancer cells in such BCMI. Such quantitation by cell counting is very useful in supporting decisions and planning of the medical treatment of breast cancer. This study proposes and evaluates features based on texture analysis by fractal dimension (FD), for the classification of histological structures in a BCMI into either cancer cells or non-cancer cells. The cancer cells include positive cells (PC) and negative cells (NC), while the normal cells comprise stromal cells (SC) and lymphocyte cells (LC). The FD feature values were calculated with the box-counting method from binarized images, obtained by automatic thresholding with Otsu's method of the grayscale images for various color channels. A total of 12 color channels from four color spaces (RGB, CIE-L*a*b*, HSV, and YCbCr) were investigated, and the FD feature values from them were used with decision tree classifiers. The BCMI data consisted of 1,400, 1,200, and 800 images with pixel resolutions 128 × 128, 192 × 192, and 256 × 256, respectively. The best cross-validated classification accuracy was 93.87%, for distinguishing between cancer and non-cancer cells, obtained using the Cr color channel with window size 256. The results indicate that the proposed algorithm, based on fractal dimension features extracted from a color channel, performs well in the automatic classification of the histology in a BCMI. This might support accurate automatic cell counting in a computer-assisted system for breast cancer diagnosis. PMID:25689353
NASA Astrophysics Data System (ADS)
Bayrak, Yusuf; Bayrak, Erdem
2012-09-01
We investigated the regional variations of Gutenberg-Richter (G-R) parameters (a and b) and fractal (correlation) dimension (DC) and relations among these parameters for the different regions in Western Anatolia (WA). The whole examined area (26-33E, 33-40.5N) is divided into 15 different seismogenic regions based on their tectonic and seismotectonic regimes. We used database including 69,182 earthquakes for the instrumental period from 1900 to 2011. We calculated b value, which is the slope of the frequency-magnitude Gutenberg-Richter relationship, from maximum likelihood method (ML) and DC value, which is the slope of log10C(r) versus log10r, from correlation integral using the least-squares (LS) method. Computed values for 15 different seismogenic regions are mapped using different color scale for different range of values of b and DC. Regional distributions of these parameters reveal information about regional variation of stress level and geological complexity. We concluded Aegean arc and Aegean islands, Alia?a fault and Byk Menderes Graben the most vulnerable regions for occurrence of the large earthquakes in WA considering the computed lowest b-values and the highest DC-values in these regions. Since DC/b values are the highest in these regions, this ratio may be used as an indicator of earthquake hazard levels of different seismogenic zones in a studied region. An effort is made to find relationships between the G-R parameters and fractal dimension. We observed negative correlation between DC and b values and positive correlation between DC and a/b values for different regions of WA. We observed that the relationship between a/b and DC can be used for seismicity, earthquake risk and hazard studies because of the computed high correlation coefficient and fewer scattering of these parameters.
NASA Astrophysics Data System (ADS)
Mihranyan, Albert; Muhel, Mortadha; Strømme, Maria
2009-02-01
The dissolution process of sparingly soluble CaCO3 microparticles and how the fractal surface dimension of the particles changes during dissolution is analyzed. The particles and the dissolution process are studied using scanning electron microscopy, X-ray diffraction, nitrogen adsorption, laser diffraction and conductance measurements. Ball milling of the particles is shown to maintain the particle crystallinity, and to introduce an increased fractal surface dimension in the 1-10 μm size range. Dissolution is found to increase the surface dimension of initially smooth particles and to maintain the fractal surface roughness of milled particles. The dissolution process increases the relative number of small particles (50 nm-1 μm) whereas the larger ones decrease in size. The solubility of the milled fractal particles was ˜1.8 times higher than that for the initially smooth ones. The presented findings show that developing methods for increasing the fractal surface roughness of particles should be of interest for improving the solubility of poorly soluble drug candidates.
Esteban, Francisco J; Padilla, Nelly; Sanz-Cortés, Magdalena; de Miras, Juan Ruiz; Bargalló, Núria; Villoslada, Pablo; Gratacós, Eduard
2010-12-01
In the search for a useful parameter to detect and quantify subtle brain abnormalities in infants with intrauterine growth restriction (IUGR), we hypothesised that the analysis of the structural complexity of grey matter (GM) and white matter (WM) using the fractal dimension (FD), a measurement of the topological complexity of an object, could be established as a useful tool for quantitative studies of infant brain morphology. We studied a sample of 18 singleton IUGR premature infants, (12.72 months corrected age (CA), range: 12 months-14 months), 15 preterm infants matched one-to-one for gestational age (GA) at delivery (12.6 months; range: 12 months-14 months), and 15 neonates born at term (12.4 months; range: 11 months-14 months). The neurodevelopmental outcome was assessed in all subjects at 18 months CA according to the Bayley Scale for Infant and Toddler Development - Third edition (BSID-III). For MRI acquisition and processing, the infants were scanned at 12 months CA, in a TIM TRIO 3T scanner, sleeping naturally. Images were pre-processed using the SPM5 toolbox, the GM and WM segmented under the VBM5 toolbox, and the box-counting method was applied for FD calculation of normal and skeletonized segmented images. The results showed a significant decrease of the FD of the brain GM and WM in the IUGR group when compared to the preterm or at-term controls. We also identified a significant linear tendency of both GM and WM FD from IUGR to preterm and term groups. Finally, multiple linear analyses between the FD of the GM or WM and the neurodevelopmental scales showed a significant regression of the language and motor scales with the FD of the GM. In conclusion, a decreased FD of the GM and WM in IUGR infants could be a sensitive indicator for the investigation of structural brain abnormalities in the IUGR population at 12 months of age, which can also be related to functional disorders. PMID:20633658
Fractal Structure in Human Cerebellum Measured by MRI
NASA Astrophysics Data System (ADS)
Zhang, Luduan; Yue, Guang; Brown, Robert; Liu, Jingzhi
2003-10-01
Fractal dimension has been used to quantify the structures of a wide range of objects in biology and medicine. We measured fractal dimension of human cerebellum (CB) in magnetic resonance images of 24 healthy young subjects (12 men, 12 women). CB images were resampled to a series of image sets with different three-dimensional resolutions. At each resolution, the skeleton of the CB white matter was obtained and the number of pixels belonging to the skeleton was determined. Fractal dimension of the CB skeleton was calculated using the box-counting method. The results indicated that the CB skeleton is a highly fractal structure, with a fractal dimension of 2.57+/-0.01. No significant difference in the CB fractal dimension was observed between men and women. Fractal dimension may serve as a quantitative index for structural complexity of the CB at its developmental, degenerative, or evolutionary stages.
Pore size distribution in porous glass: fractal dimension obtained by calorimetry
NASA Astrophysics Data System (ADS)
Neffati, R.; Rault, J.
2001-05-01
By differential Scanning Calorimetry (DSC), at low heating rate and using a technique of fractionation, we have measured the equilibrium DSC signal (heat flow) J q 0 of two families of porous glass saturated with water. The shape of the DSC peak obtained by these techniques is dependent on the sizes distribution of the pores. For porous glass with large pore size distribution, obtained by sol-gel technology, we show that in the domain of ice melting, the heat flow Jq is related to the melting temperature depression of the solvent, Δ T m , by the scaling law: J q 0˜Δ T m - (1 + D). We suggest that the exponent D is of the order of the fractal dimension of the backbone of the pore network and we discuss the influence of the variation of the melting enthalpy with the temperature on the value of this exponent. Similar D values were obtained from small angle neutron scattering and electronic energy transfer measurements on similar porous glass. The proposed scaling law is explained if one assumes that the pore size distribution is self similar. In porous glass obtained from mesomorphic copolymers, the pore size distribution is very sharp and therefore this law is not observed. One concludes that DSC, at low heating rate ( q? 2°C/min) is the most rapid and less expensive method for determining the pore distribution and the fractal exponent of a porous material.
Entanglement entropy on fractals
NASA Astrophysics Data System (ADS)
Faraji Astaneh, Amin
2016-03-01
We use the heat kernel method to calculate the entanglement entropy for a given entangling region on a fractal. The leading divergent term of the entropy is obtained as a function of the fractal dimension as well as the walk dimension. The power of the UV cutoff parameter is (generally) a fractional number, which, indeed, is a certain combination of these two indices. This exponent is known as the spectral dimension. We show that there is a novel log-periodic oscillatory behavior in the expression of entropy which has root in the complex dimension of the fractal. We finally indicate that the holographic calculation in a certain hyperscaling-violating bulk geometry yields the same leading term for the entanglement entropy, if one identifies the effective dimension of the hyperscaling-violating theory with the spectral dimension of the fractal. We provide additional support by comparing the behavior of the thermal entropy in terms of the temperature, computed for two geometries, the fractal geometry and the hyperscaling-violating background.
Xue, Hong-Xi; He, Jiang; Fan, Qing-Yun; Lü, Chang-Wei; Wang, Xia; Liang, Ying; Sun, Ying; Shen, Li-Li; Sa, Ru-Li
2008-01-01
The expression of surface fractal dimension (SFD) for size fractions of the Yellow River sediment was deduced. Based on the expression, the SFD value of different size fractions of the sediment was calculated. The SFD value of the sediment in the Baotou section of the Yellow River is 1.91, and the SFD value of the sediment smaller than 63 microm is 1.36, indicating strong ablation and separating ability of the Yellow River water. Using the modified fractal model, Freundlich model and Langmuir model to fit the data of heavy metal (Cu, Pb, Zn and Cd) adsorption, it is found that the modified fractal model is more available. And the adsorptive thermodynamics is better described by combining the modified fractal model and metastable equilibrium adsorption (MEA) theory. The variation extents of equilibrium adsorption capacity influenced by different grain size are ranked as Cu > Pb > Zn approximately equal to Cd. For each selected heavy metal, the higher initial concentration is, the stronger variation of adsorption capacity will be. The adsorptions of Cu and Pb are mainly associated with mineral composition of the sediment, while the adsorptions of Zn and Cd are mainly associated with physical characteristics of the sediment surface. PMID:18441918
Topological Vulnerability Evaluation Model Based on Fractal Dimension of Complex Networks
Gou, Li; Wei, Bo; Sadiq, Rehan; Sadiq, Yong; Deng, Yong
2016-01-01
With an increasing emphasis on network security, much more attentions have been attracted to the vulnerability of complex networks. In this paper, the fractal dimension, which can reflect space-filling capacity of networks, is redefined as the origin moment of the edge betweenness to obtain a more reasonable evaluation of vulnerability. The proposed model combining multiple evaluation indexes not only overcomes the shortage of average edge betweenness’s failing to evaluate vulnerability of some special networks, but also characterizes the topological structure and highlights the space-filling capacity of networks. The applications to six US airline networks illustrate the practicality and effectiveness of our proposed method, and the comparisons with three other commonly used methods further validate the superiority of our proposed method. PMID:26751371
NASA Astrophysics Data System (ADS)
Braga, F. L.; Mattos, O. A.; Amorin, V. S.; Souza, A. B.
2015-07-01
Clusters formation models have been extensively studied in literature, and one of the main task of this research area is the analysis of the particle aggregation processes. Some work support that the main characteristics of this processes are strictly correlated to the cluster morphology, for example in DLA. It is expected that in the DLA clusters formation with particles containing different sizes the modification of the aggregation processes can be responsible for changes in the DLA morphology. The present article is going to analyze the formation of DLA clusters of particles with different sizes and show that the aggregates obtained by this approach generate an angle selection mechanism on dendritic growth that influences the shielding effect of the DLA edge and affect the fractal dimension of the clusters.
Topological Vulnerability Evaluation Model Based on Fractal Dimension of Complex Networks.
Gou, Li; Wei, Bo; Sadiq, Rehan; Sadiq, Yong; Deng, Yong
2016-01-01
With an increasing emphasis on network security, much more attentions have been attracted to the vulnerability of complex networks. In this paper, the fractal dimension, which can reflect space-filling capacity of networks, is redefined as the origin moment of the edge betweenness to obtain a more reasonable evaluation of vulnerability. The proposed model combining multiple evaluation indexes not only overcomes the shortage of average edge betweenness's failing to evaluate vulnerability of some special networks, but also characterizes the topological structure and highlights the space-filling capacity of networks. The applications to six US airline networks illustrate the practicality and effectiveness of our proposed method, and the comparisons with three other commonly used methods further validate the superiority of our proposed method. PMID:26751371
Rigoldi, C; Galli, M; Mainardi, L; Albertini, G
2014-04-01
The aim of this study was to explore new techniques in analysing postural control using nonlinear time-series analysis and to relate these results with the clinical knowledge on the postural system in Down syndrome (DS) subjects. In order to achieve the goal, we analysed the time domain and the frequency domain behaviour, the fractal dimension and the entropy of the centre of pressure signal in both directions during quiet standing in 35 participants with DS, comparing the results with a control population. DS patients evidenced a lack in postural control in anterior-posterior direction due to the impairment both in the high organisation and synergies and in the impairments due to ligament laxity and hypotonia. Maintaining posture is a task achieved by the integration of visual, vestibular and somatosensory receptors and the dynamical nature of this signal gives fundamental data about the lack of postural control in specific pathological condition. PMID:22657255
Imaging of the parafoveal capillary network and its integrity analysis using fractal dimension.
Schmoll, Tilman; Singh, Amardeep S G; Blatter, Cedric; Schriefl, Sabine; Ahlers, Christian; Schmidt-Erfurth, Ursula; Leitgeb, Rainer A
2011-01-01
Using a spectral domain OCT system, equipped with a broadband Ti:sapphire laser, we imaged the human retina with 5 µm x 1.3 µm transverse and axial resolution at acquisition rate of 100 kHz. Such imaging speed significantly reduces motion artifacts. Combined with the ultra-high resolution, this allows observing microscopic retinal details with high axial definition without the help of adaptive optics. In this work we apply our system to image the parafoveal capillary network. We demonstrate how already on the intensity level the parafoveal capillaries can be segmented by a simple structural high pass filtering algorithm. This data is then used to quantitatively characterize the capillary network of healthy and diseased eyes. We propose to use the fractal dimension as index for capillary integrity of pathologic disorders. PMID:21559128
Imaging of the parafoveal capillary network and its integrity analysis using fractal dimension
Schmoll, Tilman; Singh, Amardeep S. G.; Blatter, Cedric; Schriefl, Sabine; Ahlers, Christian; Schmidt-Erfurth, Ursula; Leitgeb, Rainer A.
2011-01-01
Using a spectral domain OCT system, equipped with a broadband Ti:sapphire laser, we imaged the human retina with 5 µm x 1.3 µm transverse and axial resolution at acquisition rate of 100 kHz. Such imaging speed significantly reduces motion artifacts. Combined with the ultra-high resolution, this allows observing microscopic retinal details with high axial definition without the help of adaptive optics. In this work we apply our system to image the parafoveal capillary network. We demonstrate how already on the intensity level the parafoveal capillaries can be segmented by a simple structural high pass filtering algorithm. This data is then used to quantitatively characterize the capillary network of healthy and diseased eyes. We propose to use the fractal dimension as index for capillary integrity of pathologic disorders. PMID:21559128
Simplifying the calculation of light scattering properties for black carbon fractal aggregates
NASA Astrophysics Data System (ADS)
Smith, A. J. A.; Grainger, R. G.
2014-02-01
Black carbon fractal aggregates have complicated shapes that make the calculation of their optical properties particularly computationally expensive. Here, a method is presented to estimate fractal aggregate light scattering properties by optimising simplified models to full light scattering calculations. It is found that there are no possible spherical models (at any size or refractive index) that well represent the light scattering in the visible, or near-thermal infrared. As such, parameterisations of the light scattering as a function of the number of aggregate particles is presented as the most pragmatic choice for modelling distributions of black carbon when the large computational overheads of rigorous scattering calculations cannot be justified. This parameterisation can be analytically integrated to provide light scattering properties for log-normal distributions of black carbon fractal aggregates and return extinction cross-sections with 0.1% accuracy for typical black carbon size distributions. Scattering cross-sections and the asymmetry parameter can be obtained to within 3%.
Fuseler, John W; Robichaux, Jacqulyne P; Atiyah, Huda I; Ramsdell, Ann F
2014-03-01
Fractal dimension has emerged as a clinically useful tool in the diagnosis and management of breast cancer. The aim of the present study was to determine if fractal dimension can be applied for the analysis of a pre-clinical breast cancer mouse model, MMTV-cNeu. Using fractal dimension in conjunction with conventional morphometric measurements, the ductal epithelial networks of pubertal-stage MMTV-cNeu mice were quantitatively compared with those of wild-type mice. Significant alterations in ductal epithelial network growth and organization were detected during early neoplasia in MMTV-cNeu mice. Moreover, the left-side networks were significantly more affected relative to their wild-type counterparts than were the right-side networks, a finding that is consistent with elevated left-side tumor incidence reported for breast cancer patients. Taken together these results demonstrate that combined fractal dimension and morphometric analysis is an objective and sensitive approach to quantitatively identify ductal epithelial aberrancies that precede overt mammary carcinoma formation. PMID:24596356
2013-01-01
Background Prostate cancer is a serious public health problem that affects quality of life and has a significant mortality rate. The aim of the present study was to quantify the fractal dimension and Shannon’s entropy in the histological diagnosis of prostate cancer. Methods Thirty-four patients with prostate cancer aged 50 to 75 years having been submitted to radical prostatectomy participated in the study. Histological slides of normal (N), hyperplastic (H) and tumor (T) areas of the prostate were digitally photographed with three different magnifications (40x, 100x and 400x) and analyzed. The fractal dimension (FD), Shannon’s entropy (SE) and number of cell nuclei (NCN) in these areas were compared. Results FD analysis demonstrated the following significant differences between groups: T vs. N and H vs. N groups (p < 0.05) at a magnification of 40x; T vs. N (p < 0.01) at 100x and H vs. N (p < 0.01) at 400x. SE analysis revealed the following significant differences groups: T vs. H and T vs. N (p < 0.05) at 100x; and T vs. H and T vs. N (p < 0.001) at 400x. NCN analysis demonstrated the following significant differences between groups: T vs. H and T vs. N (p < 0.05) at 40x; T vs. H and T vs. N (p < 0.0001) at 100x; and T vs. H and T vs. N (p < 0.01) at 400x. Conclusions The quantification of the FD and SE, together with the number of cell nuclei, has potential clinical applications in the histological diagnosis of prostate cancer. PMID:23414368
Analysis of fractal dimensions in the express diagnostics of bacterial colonies
NASA Astrophysics Data System (ADS)
Ul'Yanov, A. S.
2009-12-01
The features of the formation of speckle structures under irradiation of a model fractal (Sierpinski carpet) have been investigated. The relationship between the fractal properties of the diffraction pattern and the scattering structure parameters (model fractal geometrical sizes, fractal depth) has been analyzed for the irradiation by a focused light beam, whose size is comparable with that of the irradiated object. The results of the computer simulation of the Gaussian beam scattering in bacterial colonies are compared with the experimental data.
NASA Technical Reports Server (NTRS)
Garneau, S.; Plaut, J. J.
2000-01-01
The surface roughness of the Vastitas Borealis Formation on Mars was analyzed with fractal statistics. Root mean square slopes and fractal dimensions were calculated for 74 topographic profiles. Results have implications for radar scattering models.
Determination of the fractal dimension for the epitaxial n-GaAs surface in the local limit
Torkhov, N. A. Bozhkova, V. G.; Ivonin, I. V.; Novikov, V. A.
2009-01-15
Atomic-force microscopy studies of epitaxial n-GaAs surfaces prepared to deposit barrier contacts showed that major relief for such surfaces is characterized by a roughness within 3-15 nm, although 'surges' up to 30-70 nm are observed. Using three independent methods for determining the spatial dimension of the surface, based on the fractal analysis for the surface (triangulation method), its section contours in the horizontal plane, and the vertical section (surface profile), it was shown that the active surface for epitaxial n-GaAs obeys all main features of behavior for fractal Brownian surfaces and, in the local approximation, can be characterized by the fractal dimension D{sub f} slightly differing for various measuring scales. The most accurate triangulation method showed that the fractal dimensions for the studied surface of epitaxial n-GaAs for measurement scales from 0.692 to 0.0186 {mu}m are in the range D{sub f} = 2.490-2.664. The real surface area S{sub real} for n-GaAs epitaxial layers was estimated using a graphical method in the approximation {delta} {sup {yields}} 0 {delta} is the measurement scale parameter). It was shown that the real surface area for epitaxial n-GaAs can significantly (ten times and more) exceed the area of the visible contact window.
Complexity analysis of EEG in patients with schizophrenia using fractal dimension.
Raghavendra, B S; Dutt, D Narayana; Halahalli, Harsha N; John, John P
2009-08-01
We computed Higuchi's fractal dimension (FD) of resting, eyes closed EEG recorded from 30 scalp locations in 18 male neuroleptic-naïve, recent-onset schizophrenia (NRS) subjects and 15 male healthy control (HC) subjects, who were group-matched for age. Schizophrenia patients showed a diffuse reduction of FD except in the bilateral temporal and occipital regions, with the reduction being most prominent bifrontally. The positive symptom (PS) schizophrenia subjects showed FD values similar to or even higher than HC in the bilateral temporo-occipital regions, along with a co-existent bifrontal FD reduction as noted in the overall sample of NRS. In contrast, this increase in FD values in the bilateral temporo-occipital region was absent in the negative symptom (NS) subgroup. The regional differences in complexity suggested by these findings may reflect the aberrant brain dynamics underlying the pathophysiology of schizophrenia and its symptom dimensions. Higuchi's method of measuring FD directly in the time domain provides an alternative for the more computationally intensive nonlinear methods of estimating EEG complexity. PMID:19550026
Small-angle scattering from fat fractals
NASA Astrophysics Data System (ADS)
Anitas, Eugen M.
2014-06-01
A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural information about the underlying fractal structure. We calculate analytically the monodisperse and polydisperse SAS intensity (fractal form factor and structure factor) of a newly introduced model of fat fractals and study its properties in momentum space. The system is a 3D deterministic mass fractal built on an extension of the well-known Cantor fractal. The model allows us to explain a succession of power-law decays and respectively, of generalized power-law decays (GPLD; superposition of maxima and minima on a power-law decay) with arbitrarily decreasing scattering exponents in the range from zero to three. We show that within the model, the present analysis allows us to obtain the edges of all the fractal regions in the momentum space, the number of fractal iteration and the fractal dimensions and scaling factors at each structural level in the fractal. We applied our model to calculate an analytical expression for the radius of gyration of the fractal. The obtained quantities characterizing the fat fractal are correlated to variation of scaling factor with the iteration number.
Jiménez, J; López, A M; Cruz, J; Esteban, F J; Navas, J; Villoslada, P; Ruiz de Miras, J
2014-10-01
This study presents a Web platform (http://3dfd.ujaen.es) for computing and analyzing the 3D fractal dimension (3DFD) from volumetric data in an efficient, visual and interactive way. The Web platform is specially designed for working with magnetic resonance images (MRIs) of the brain. The program estimates the 3DFD by calculating the 3D box-counting of the entire volume of the brain, and also of its 3D skeleton. All of this is done in a graphical, fast and optimized way by using novel technologies like CUDA and WebGL. The usefulness of the Web platform presented is demonstrated by its application in a case study where an analysis and characterization of groups of 3D MR images is performed for three neurodegenerative diseases: Multiple Sclerosis, Intrauterine Growth Restriction and Alzheimer's disease. To the best of our knowledge, this is the first Web platform that allows the users to calculate, visualize, analyze and compare the 3DFD from MRI images in the cloud. PMID:24909817
NASA Astrophysics Data System (ADS)
Karemore, Gopal; Nielsen, Mads
2009-02-01
Structural texture measures are used to address the aspect of breast cancer risk assessment in screening mammograms. The current study investigates whether texture properties characterized by local Fractal Dimension (FD) and Lacunarity contribute to asses breast cancer risk. FD represents the complexity while the Lacunarity characterize the gappiness of a fractal. Our cross-sectional case-control study includes mammograms of 50 patients diagnosed with breast cancer in the subsequent 2-4 years and 50 matched controls. The longitudinal double blind placebo controlled HRT study includes 39 placebo and 36 HRT treated volunteers for two years. ROIs with same dimension (250*150 pixels) were created behind the nipple region on these radiographs. Box counting method was used to calculate the fractal dimension (FD) and the Lacunarity. Paired t-test and Pearson correlation coefficient were calculated. It was found that there were no differences between cancer and control group for FD (P=0.8) and Lacunarity (P=0.8) in crosssectional study whereas earlier published heterogeneity examination of radiographs (BC-HER) breast cancer risk score separated groups (p=0.002). In the longitudinal study, FD decreased significantly (P<0.05) in the HRT treated population while Lacunarity remained insignificant (P=0.2). FD is negatively correlated to Lacunarity (-0.74, P<0.001), BIRADS (-0.34, P<0.001) and Percentage Density (-0.41, P<0.001). FD is invariant to the mammographic texture change from control to cancer population but marginally varying in HRT treated population. This study yields no evidence that lacunarity or FD are suitable surrogate markers of mammographic heterogeneity as they neither pick up breast cancer risk, nor show good sensitivity to HRT.
Li, Qiaowei; Yuan, Yin; Gao, Zhonghai; Chen, Falin
2014-01-01
Background This study aimed to investigate the correlation between quantitative retinal vascular parameters such as central retinal arteriolar equivalent (CRAE) and retinal vascular fractal dimension (D(f)), and cardiovascular risk factors in the Chinese Han population residing in the in islands of southeast China. Methodology/Principle Findings In this cross-sectional study, fundus photographs were collected and semi-automated analysis software was used to analyze retinal vessel diameters and fractal dimensions. Cardiovascular risk factors such as relevant medical history, blood pressure (BP), lipids, and blood glucose data were collected. Subjects had a mean age of 51.9±12.0 years and included 812 (37.4%) males and 1,357 (62.6%) females. Of the subjects, 726 (33.5%) were overweight, 226 (10.4%) were obese, 272 (12.5%) had diabetes, 738 (34.0%) had hypertension, and 1,156 (53.3%) had metabolic syndrome. After controlling for the effects of potential confounders, multivariate analyses found that age (β = 0.06, P = 0.008), sex (β = 1.33, P = 0.015), mean arterial blood pressure (β = −0.12, P<0.001), high-sensitivity C-reactive protein (β = −0.22, P = 0.008), and CRVE (β = 0.23, P<0.001) were significantly associated with CRAE. Age (β = −0.0012, P<0.001), BP classification (prehypertension: β = −0.0075, P = 0.014; hypertension: β = −0.0131, P = 0.002), and hypertension history (β = −0.0007, P = 0.009) were significantly associated with D(f). Conclusions/Significance D(f) exhibits a stronger association with BP than CRAE. Thus, D(f) may become a useful indicator of cardiovascular risk. PMID:25188273
ERIC Educational Resources Information Center
House, Garvey; Zelhart, Paul F.
The complexity (fractal dimension value) of responses to the Rey-Osterrieth Complex Figure Test (ROCFT) between 10 undergraduate students with learning disabilities and a comparison group of 10 students without learning disabilities were compared. The fractal value of responses was assessed under three conditions (copy, immediate, and delay) by…
NASA Astrophysics Data System (ADS)
Rodkin, M. V.; Shatakhtsyan, A. R.
2015-05-01
The method for calculating the fractal correlation dimension is applied for analyzing the data on the locations of large and extralarge ore deposits. The approach implemented in this study differs by a few of important points from that commonly used, e.g., in the calculations of the correlation dimension for a set of the epicenters (hypocenters) of the earthquakes. Firstly, we demonstrate the possibility and advisability of obtaining different dimension estimates for different spatial scales. Such a separation turned out to be useful in distinguishing between the regularities in the location of ore deposits on the scale of an ore cluster, ore province, and entire continent. Secondly, we introduce a new notion, a mixed correlation dimension, and use it for different types of the objects (e.g., Au and Ag). The standard formula for calculating the correlation dimension is trivially generalized on this case. It is shown that the values of the correlation dimension can be lower and higher than the dimension of the hosting medium. The cases when the correlation dimension is higher than that of the hosting medium are interpreted as a "mutual repulsion" of the deposits of the two mentioned types. In contrast, the small correlation dimensions indicate that the deposits of the corresponding types tend to have spatially close locations. The calculations are conducted for the spherical Earth. The method is applied to the data on the large and extralarge world-class ore deposits from the Largest Mineral Deposits of the World (LMDs) geoinformation system (GIS). Different patterns of the studied behavior are illustrated by the model examples.
NASA Astrophysics Data System (ADS)
Phothisonothai, Montri; Nakagawa, Masahiro
In this study, we propose a method of classifying a spontaneous electroencephalogram (EEG) approach to a brain-computer interface. Ten subjects, aged 21-32 years, volunteered to imagine left-and right- hand movements. An independent component analysis based on a fixed-point algorithm is used to eliminate the activities found in the EEG signals. We use a fractal dimension value to reveal the embedded potential responses in the human brain. The different fractal dimension values between the relaxing and imaging periods are computed. Featured data is classified by a three-layer feed-forward neural network based on a simple backpropagation algorithm. Two conventional methods, namely, the use of the autoregressive (AR) model and the band power estimation (BPE) as features, and the linear discriminant analysis (LDA) as a classifier, are selected for comparison in this study. Experimental results show that the proposed method is more effective than the conventional methods.
NASA Astrophysics Data System (ADS)
Bianciardi, G.
2002-11-01
To examine the general properties of ancient genetic codes, we have evaluated the Manhattan and Euclidean fractal dimensions (Dm, De) of tRNAs in Archaea comparing to the values from mRNA in eukaria and of computer-generated random sequences. Here, fractal dimension was used as a tool to measure complexity, where the most complex sequence results to be the random sequences (with D>>1). Dm and De values of ancient informational polymers resulted statistically higher than the ones from bacteria, mitochondria and metazoa and lower than the ones of randomly generated sequences, suggesting that in the primitive Earth informational polymers originated from slightly edited random strings and that during biologic evolution the distance from randomness increased.
Hinrikus, Hiie; Bachmann, Maie; Karai, Deniss; Klonowski, Włodzimierz; Lass, Jaanus; Stepien, Pavel; Stepien, Robert; Tuulik, Viiu
2011-05-01
This study addresses application of Higuchi's fractal dimension (FD) as a measure to evaluate the effect of external periodic stressor on electrical oscillations in the brain. Modulated microwave radiation was applied as a weak periodic stressor with strongly inhomogeneous distribution inside the brain. Experiments were performed on a group of 14 volunteers. Ten cycles (1 min on, 1 min off) of 450-MHz microwave radiation modulated at 40 Hz were applied. Higuchi's FD was calculated in eight symmetric electroencephalographic (EEG) channels located in frontal, temporal, parietal, and occipital areas. FD values averaged over a group detected a small (1-2%) but statistically significant increase with exposure in all EEG channels. FD increased for 12, decreased for one, and was constant for one subject. FD showed the most remarkable effect in temporal and parietal regions of the left hemisphere where the microwave field was maximal. Changes of FD in these regions of the right hemisphere were much higher than expected in accordance with the field distribution. Correlation of FD between different EEG channels was high and retained its value in exposed conditions. Spreading of disturbance between different brain areas is supposed to be crucial for the effect of exposure on the electrical oscillations in the brain. PMID:21465274
Complexity of routes to chaos and global regularity of fractal dimensions in bimodal maps.
Cao, K F; Peng, S L
1999-09-01
The dual-star composition rule of doubly superstable (DSS) sequences presents a complete renormalizable algebraic structure for studying Feigenbaum's metric universality and self-similar classification of DSS sequences in symbolic dynamics of bimodal maps of the interval. Here an important feature is that the complete combinations of up- and down-star products create all the generalized Feigenbaum's routes of transitions to chaos. These routes can be classified into two types: one consists of countably infinitely many regular routes which preserve Feigenbaum's metric universality; another consists of uncountably infinitely many universal nonscaling routes described by the irregularly mixed dual-star products, which break Feigenbaum's asymptotically convergent metric universality although they are structurally universal. The combinatorial complexity of dual-star products may increase the grammatical complexity of languages of symbolic dynamics. Moreover, it is found that there exists a global regularity between the fractal dimensions d and the scaling factors [alpha(C),alpha(D)] for Feigenbaum-type attractors: d(Z)log(/Z/)/alpha(C)(Z)alpha(D)(Z)/=beta((2)), where beta((2)) is independent of the concrete DSS sequences Z. PMID:11970079
Assessing severity of obstructive sleep apnea by fractal dimension sequence analysis of sleep EEG
NASA Astrophysics Data System (ADS)
Zhang, J.; Yang, X. C.; Luo, L.; Shao, J.; Zhang, C.; Ma, J.; Wang, G. F.; Liu, Y.; Peng, C.-K.; Fang, J.
2009-10-01
Different sleep stages are associated with distinct dynamical patterns in EEG signals. In this article, we explored the relationship between the sleep architecture and fractal dimension (FD) of sleep EEG. In particular, we applied the FD analysis to the sleep EEG of patients with obstructive sleep apnea-hypopnea syndrome (OSAHS), which is characterized by recurrent oxyhemoglobin desaturation and arousals from sleep, a disease which received increasing public attention due to its significant potential impact on health. We showed that the variation of FD reflects the macrostructure of sleep. Furthermore, the fast fluctuation of FD, as measured by the zero-crossing rate of detrended FD (zDFD), is a useful indicator of sleep disturbance, and therefore, correlates with apnea-hypopnea index (AHI), and hourly number of blood oxygen saturation (SpO 2) decreases greater than 4%, as obstructive apnea/hypopnea disturbs sleep architecture. For practical purpose, a modified index combining zDFD of EEG and body mass index (BMI) may be useful for evaluating the severity of OSAHS symptoms.
Gene Entropy-Fractal Dimension Informatics with Application to Mouse-Human Translational Medicine
Holden, T.; Cheung, E.; Dehipawala, S.; Ye, J.; Tremberger, G.; Lieberman, D.; Cheung, T.
2013-01-01
DNA informatics represented by Shannon entropy and fractal dimension have been used to form 2D maps of related genes in various mammals. The distance between points on these maps for corresponding mRNA sequences in different species is used to study evolution. By quantifying the similarity of genes between species, this distance might be indicated when studies on one species (mouse) would tend to be valid in the other (human). The hypothesis that a small distance from mouse to human could facilitate mouse to human translational medicine success is supported by the studied ESR-1, LMNA, Myc, and RNF4 sequences. ID1 and PLCZ1 have larger separation. The collinearity of displacement vectors is further analyzed with a regression model, and the ID1 result suggests a mouse-chimp-human translational medicine approach. Further inference was found in the tumor suppression gene, p53, with a new hypothesis of including the bovine PKM2 pathways for targeting the glycolysis preference in many types of cancerous cells, consistent with quantum metabolism models. The distance between mRNA and protein coding CDS is proposed as a measure of the pressure associated with noncoding processes. The Y-chromosome DYS14 in fetal micro chimerism that could offer protection from Alzheimer's disease is given as an example. PMID:23586047
NASA Astrophysics Data System (ADS)
Fernandes, Maurício Anderson; Ribeiro Rosa, Edvaldo Antônio; Johann, Aline Cristina Batista Rodrigues; Grégio, Ana Maria Trindade; Trevilatto, Paula Cristina; Azevedo-Alanis, Luciana Reis
2016-01-01
Objectives: To test the capacity of the digital tool, fractal dimension (FD) analysis, in identifying subtle differences in bone pattern in patients with renal osteodystrophy (RO), correlated with the time of hemodialysis, in different regions of interest, delineated on panoramic and periapical radiographs. Study design: A total of 34 patients with chronic renal disease undergoing hemodialysis were submitted to panoramic and periapical radiographs. Different regions of interest were delineated on the mandibular body and ramus. FD was analyzed by means of the software program ImageJ and correlated with the time of hemodialysis. Results: The sample consisted of 34 subjects. The time of hemodialysis varied from 1 to 286 months. There was significant correlation between the time of hemodialysis and the FD values in the region delineated in the mandibular angle (r = 0.498; p = 0.003) and this was shown in the periapical radiographs as well (r = -0.349; p = 0.043). Conclusions: FD analysis was a useful tool in detecting alterations caused by RO in bone pattern, in panoramic and periapical radiographs.
Medical image retrieval and analysis by Markov random fields and multi-scale fractal dimension
NASA Astrophysics Data System (ADS)
Backes, André Ricardo; Cavaleri Gerhardinger, Leandro; do Espírito Santo Batista Neto, João; Martinez Bruno, Odemir
2015-02-01
Many Content-based Image Retrieval (CBIR) systems and image analysis tools employ color, shape and texture (in a combined fashion or not) as attributes, or signatures, to retrieve images from databases or to perform image analysis in general. Among these attributes, texture has turned out to be the most relevant, as it allows the identification of a larger number of images of a different nature. This paper introduces a novel signature which can be used for image analysis and retrieval. It combines texture with complexity extracted from objects within the images. The approach consists of a texture segmentation step, modeled as a Markov Random Field process, followed by the estimation of the complexity of each computed region. The complexity is given by a Multi-scale Fractal Dimension. Experiments have been conducted using an MRI database in both pattern recognition and image retrieval contexts. The results show the accuracy of the proposed method in comparison with other traditional texture descriptors and also indicate how the performance changes as the level of complexity is altered.
Medical image retrieval and analysis by Markov random fields and multi-scale fractal dimension.
Backes, André Ricardo; Gerhardinger, Leandro Cavaleri; Batista Neto, João do Espírito Santo; Bruno, Odemir Martinez
2015-02-01
Many Content-based Image Retrieval (CBIR) systems and image analysis tools employ color, shape and texture (in a combined fashion or not) as attributes, or signatures, to retrieve images from databases or to perform image analysis in general. Among these attributes, texture has turned out to be the most relevant, as it allows the identification of a larger number of images of a different nature. This paper introduces a novel signature which can be used for image analysis and retrieval. It combines texture with complexity extracted from objects within the images. The approach consists of a texture segmentation step, modeled as a Markov Random Field process, followed by the estimation of the complexity of each computed region. The complexity is given by a Multi-scale Fractal Dimension. Experiments have been conducted using an MRI database in both pattern recognition and image retrieval contexts. The results show the accuracy of the proposed method in comparison with other traditional texture descriptors and also indicate how the performance changes as the level of complexity is altered. PMID:25586375
Gómez, Carlos; Mediavilla, Angela; Hornero, Roberto; Abásolo, Daniel; Fernández, Alberto
2009-04-01
Alzheimer's disease (AD) is an irreversible brain disorder of unknown aetiology that gradually destroys brain cells and represents the most prevalent form of dementia in western countries. The main aim of this study was to analyse the magnetoencephalogram (MEG) background activity from 20 AD patients and 21 elderly control subjects using Higuchi's fractal dimension (HFD). This non-linear measure can be used to estimate the dimensional complexity of biomedical time series. Before the analysis with HFD, the stationarity and the non-linear structure of the signals were proved. Our results showed that MEG signals from AD patients had lower HFD values than control subjects' recordings. We found significant differences between both groups at 71 of the 148 MEG channels (p<0.01; Student's t-test with Bonferroni's correction). Additionally, five brain regions (anterior, central, left lateral, posterior and right lateral) were analysed by means of receiver operating characteristic curves, using a leave-one-out cross-validation procedure. The highest accuracy (87.8%) was achieved when the mean HFD over all channels was analysed. To sum up, our results suggest that spontaneous MEG rhythms are less complex in AD patients than in healthy control subjects, hence indicating an abnormal type of dynamics in AD. PMID:18676171
Shimizu, Wataru; Sato, Takaaki; Matsumoto, Taki; Murakami, Yasushi
2012-05-01
Titanium oxide polymers having a low-fractal dimension (d(f) < 2) were rapidly synthesized from titanium tetra-n-butoxide via a catalytic sol-gel process with a hydrazine monohydrochloride catalyst. Different from conventional sol-gel processes aimed at producing low-fractal dimension titanium oxide polymers, the present synthetic strategy needed neither organic ligand to enhance the stability of titanium alkoxides nor an extremely long reaction time in a strongly acidic solution condition, thanks to a drastically accelerated polycondensation reaction. We pursued the structure evolution of the titanium oxide polymers by means of time-resolved small-angle X-ray scattering (Tr-SAXS). The SAXS data unambiguously demonstrate the generation of the expanded polymer-like structure characterized by the fractal dimension of d(f) approximately equal 5/3. The results offer an efficient route to the synthesis of the weakly-branched titanium oxide polymers, which are expected to be used to create a wide range of optical materials having a high refractive index, such as anti-glare coating. PMID:22852301
Higher-order fractal geometry; application to multiple light scattering
Seeley, G.; Keyes, T.; Ohtsuki, T.
1988-01-25
A hierarchy of fractal geometrical exponents D(l), based upon l-rank orientational fluctuations, is proposed; D(0) = D is the usual fractal dimension. The first three D(l) are calculated via computer simulation for a growth model with a tunable fractal dimension for several values in the range 3>D>1, and for bond percolation. The new exponents are used to discuss fractal structure. The second-order light-scattering intensity is evaluated for the growing fractal clusters, and is shown to be sensitive to the higher order D(l).
Fuss, Franz Konstantin
2013-01-01
Standard methods for computing the fractal dimensions of time series are usually tested with continuous nowhere differentiable functions, but not benchmarked with actual signals. Therefore they can produce opposite results in extreme signals. These methods also use different scaling methods, that is, different amplitude multipliers, which makes it difficult to compare fractal dimensions obtained from different methods. The purpose of this research was to develop an optimisation method that computes the fractal dimension of a normalised (dimensionless) and modified time series signal with a robust algorithm and a running average method, and that maximises the difference between two fractal dimensions, for example, a minimum and a maximum one. The signal is modified by transforming its amplitude by a multiplier, which has a non-linear effect on the signal's time derivative. The optimisation method identifies the optimal multiplier of the normalised amplitude for targeted decision making based on fractal dimensions. The optimisation method provides an additional filter effect and makes the fractal dimensions less noisy. The method is exemplified by, and explained with, different signals, such as human movement, EEG, and acoustic signals. PMID:24151522
2013-01-01
Standard methods for computing the fractal dimensions of time series are usually tested with continuous nowhere differentiable functions, but not benchmarked with actual signals. Therefore they can produce opposite results in extreme signals. These methods also use different scaling methods, that is, different amplitude multipliers, which makes it difficult to compare fractal dimensions obtained from different methods. The purpose of this research was to develop an optimisation method that computes the fractal dimension of a normalised (dimensionless) and modified time series signal with a robust algorithm and a running average method, and that maximises the difference between two fractal dimensions, for example, a minimum and a maximum one. The signal is modified by transforming its amplitude by a multiplier, which has a non-linear effect on the signal's time derivative. The optimisation method identifies the optimal multiplier of the normalised amplitude for targeted decision making based on fractal dimensions. The optimisation method provides an additional filter effect and makes the fractal dimensions less noisy. The method is exemplified by, and explained with, different signals, such as human movement, EEG, and acoustic signals. PMID:24151522
Mossotti, Victor G.; Eldeeb, A. Raouf
2000-01-01
Turcotte, 1997, and Barton and La Pointe, 1995, have identified many potential uses for the fractal dimension in physicochemical models of surface properties. The image-analysis program described in this report is an extension of the program set MORPH-I (Mossotti and others, 1998), which provided the fractal analysis of electron-microscope images of pore profiles (Mossotti and Eldeeb, 1992). MORPH-II, an integration of the modified kernel of the program MORPH-I with image calibration and editing facilities, was designed to measure the fractal dimension of the exposed surfaces of stone specimens as imaged in cross section in an electron microscope.
NASA Astrophysics Data System (ADS)
Boness, D. A.; Terrell-Martinez, B.
2010-12-01
As part of an ongoing undergraduate research project of light scattering calculations involving fractal carbonaceous soot aggregates relevant to current anthropogenic and natural sources in Earth's atmosphere, we have read with interest a recent paper [E.T. Wolf and O.B Toon,Science 328, 1266 (2010)] claiming that the Faint Young Sun paradox discussed four decades ago by Carl Sagan and others can be resolved without invoking heavy CO2 concentrations as a greenhouse gas warming the early Earth enough to sustain liquid water and hence allow the origin of life. Wolf and Toon report that a Titan-like Archean Earth haze, with a fractal haze aggregate nature due to nitrogen-methane photochemistry at high altitudes, should block enough UV light to protect the warming greenhouse gas NH3 while allowing enough visible light to reach the surface of the Earth. To test this hypothesis, we have employed a rigorous T-Matrix arbitrary-particle light scattering technique, to avoid the simplifications inherent in Mie-sphere scattering, on haze fractal aggregates at UV and visible wavelenths of incident light. We generate these model aggregates using diffusion-limited cluster aggregation (DLCA) algorithms, which much more closely fit actual haze fractal aggregates than do diffusion-limited aggregation (DLA) algorithms.
2010-01-01
Background Fractal geometry is employ to characterize the irregular objects and had been used in experimental and clinic applications. Starting from a previous work, here we made a theoretical research based on a geometric generalization of the experimental results, to develop a theoretical generalization of the stenotic and restenotic process, based on fractal geometry and Intrinsic Mathematical Harmony. Methods Starting from all the possibilities of space occupation in box-counting space, all arterial prototypes differentiating normality and disease were obtained with a computational simulation. Measures from 2 normal and 3 re-stenosed arteries were used as spatial limits of the generalization. Results A new methodology in animal experimentation was developed, based on fractal geometric generalization. With this methodology, it was founded that the occupation space possibilities in the stenotic process are finite and that 69,249 arterial prototypes are obtained as a total. Conclusions The Intrinsic Mathematical Harmony reveals a supra-molecular geometric self-organization, where the finite and discrete fractal dimensions of arterial layers evaluate objectively the arterial stenosis and restenosis process. PMID:20846449
Meng, Zhiyong; Hashmi, Sara M; Elimelech, Menachem
2013-02-15
The time-evolutions of nanoparticle hydrodynamic radius and aggregate fractal dimension during the aggregation of fullerene (C(60)) nanoparticles (FNPs) were measured via simultaneous multiangle static and dynamic light scattering. The FNP aggregation behavior was determined as a function of monovalent (NaCl) and divalent (CaCl(2)) electrolyte concentration, and the impact of addition of dissolved natural organic matter (humic acid) to the solution was also investigated. In the absence of humic acid, the fractal dimension decreased over time with monovalent and divalent salts, suggesting that aggregates become slightly more open and less compact as they grow. Although the aggregates become slightly more open, the magnitude of the fractal dimension suggests intermediate aggregation between the diffusion- and reaction-limited regimes. We observed different aggregation behavior with monovalent and divalent salts upon the addition of humic acid to the solution. For NaCl-induced aggregation, the introduction of humic acid significantly suppressed the aggregation rate of FNPs at NaCl concentrations lower than 150mM. In this case, the aggregation was intermediate or reaction-limited even at NaCl concentrations as high as 500mM, giving rise to aggregates with a fractal dimension of 2.0. For CaCl(2)-induced aggregation, the introduction of humic acid enhanced the aggregation of FNPs at CaCl(2) concentrations greater than about 5mM due to calcium complexation and bridging effects. Humic acid also had an impact on the FNP aggregate structure in the presence of CaCl(2), resulting in a fractal dimension of 1.6 for the diffusion-limited aggregation regime. Our results with CaCl(2) indicate that in the presence of humic acid, FNP aggregates have a more open and loose structure than in the absence of humic acid. The aggregation results presented in this paper have important implications for the transport, chemical reactivity, and toxicity of engineered nanoparticles in aquatic environments. PMID:23211871
Local connected fractal dimension analysis in gill of fish experimentally exposed to toxicants.
Manera, Maurizio; Giari, Luisa; De Pasquale, Joseph A; Sayyaf Dezfuli, Bahram
2016-06-01
An operator-neutral method was implemented to objectively assess European seabass, Dicentrarchus labrax (Linnaeus, 1758) gill pathology after experimental exposure to cadmium (Cd) and terbuthylazine (TBA) for 24 and 48h. An algorithm-derived local connected fractal dimension (LCFD) frequency measure was used in this comparative analysis. Canonical variates (CVA) and linear discriminant analysis (LDA) were used to evaluate the discrimination power of the method among exposure classes (unexposed, Cd exposed, TBA exposed). Misclassification, sensitivity and specificity, both with original and cross-validated cases, were determined. LCFDs frequencies enhanced the differences among classes which were visually selected after their means, respective variances and the differences between Cd and TBA exposed means, with respect to unexposed mean, were analyzed by scatter plots. Selected frequencies were then scanned by means of LDA, stepwise analysis, and Mahalanobis distance to detect the most discriminative frequencies out of ten originally selected. Discrimination resulted in 91.7% of cross-validated cases correctly classified (22 out of 24 total cases), with sensitivity and specificity, respectively, of 95.5% (1 false negative with respect to 21 really positive cases) and 75% (1 false positive with respect to 3 really negative cases). CVA with convex hull polygons ensured prompt, visually intuitive discrimination among exposure classes and graphically supported the false positive case. The combined use of semithin sections, which enhanced the visual evaluation of the overall lamellar structure; of LCFD analysis, which objectively detected local variation in complexity, without the possible bias connected to human personnel; and of CVA/LDA, could be an objective, sensitive and specific approach to study fish gill lamellar pathology. Furthermore this approach enabled discrimination with sufficient confidence between exposure classes or pathological states and avoided misdiagnosis. PMID:26991750
Comparison of different fractal dimension measuring algorithms for RE-TM M-O films
NASA Technical Reports Server (NTRS)
Bernacki, Bruce E.; Mansuripur, M.
1991-01-01
Noise in magneto-optical recording devices is discussed. In general, it appears that either the divider technique or amplitude spectrum technique may be used interchangeably to measure the fractal dimension (D) in the domain wall structure of ideal images. However, some caveats must be observed for best results. The divider technique is attractive for its simplicity and relatively modest computation requirements. However, it is sensitive to noise, in that noise pixels that touch the domain boundary are interpreted as being part of the boundary, skewing the measurement. Also, it is not useful in measuring nucleation-dominated films or domains that have significant amounts of structure within the interior of the domain wall. The amplitude spectrum method is more complex, and less intuitive than the divider method, and somewhat more expensive to implement computationally. However, since the camera noise tends to be white, the noise can be avoided in the measurement of D by avoiding that portion of the curve that is flat (due to the white noise) when the least squares line is fit to the plot. Also, many image processing software packages include a Fast Fourier Transformation (FFT) facility, while the user will most likely have to write his own edge extraction routine for the divider method. The amplitude spectrum method is a true two dimensional technique that probes the interior of the domain wall, and in fact, can measure arbitrary clusters of domains. It can also be used to measure grey-level images, further reducing processing steps needed to threshold the image.
Bornas, Xavier; Tortella-Feliu, Miquel; Balle, Maria; Llabrés, Jordi
2013-01-01
The cognitive regulation of emotions is important for human adaptation. Self-focused emotion regulation (ER) strategies have been linked to the development and persistence of anxiety and depression. A vast array of research has provided valuable knowledge about the neural correlates of the use of specific self-focused ER strategies; however, the resting neural correlates of cognitive ER styles, which reflect an individual's disposition to engage in different forms of ER in order to manage distress, are largely unknown. In this study, associations between theoretically negative ER style (self-focused or not) and the complexity (fractal dimension, FD) of the resting EEG at frontal, central, parietal, and occipital regions were investigated in 58 healthy volunteers. The Cognitive Emotion Regulation Questionnaire was used as the self-report measure of ER style. Results showed that a diminished FD over the scalp significantly correlated with self-focused ER style scores, even after controlling for negative affect, which has been also considered to influence the use of ER strategies. The lower the EEG FD, the higher were the self-focused ER style scores. Correlational analyses of specific self-focused ER strategies showed that self-blaming and rumination were negatively associated with diminished FD of the EEG, but catastrophizing and blaming others were not. No significant correlations were found for ER strategies more focused on situation or others. Results are discussed within the self-organized criticality theory of brain dynamics: The diminished FD of the EEG may reflect a disposition to engage in self-focused ER strategies as people prone to ruminate and self-blame show a less complex resting EEG activity, which may make it more difficult for them to exit their negative emotional state. PMID:22519470
Comparison of fractal dimension estimation algorithms for epileptic seizure onset detection
NASA Astrophysics Data System (ADS)
Polychronaki, G. E.; Ktonas, P. Y.; Gatzonis, S.; Siatouni, A.; Asvestas, P. A.; Tsekou, H.; Sakas, D.; Nikita, K. S.
2010-08-01
Fractal dimension (FD) is a natural measure of the irregularity of a curve. In this study the performances of three waveform FD estimation algorithms (i.e. Katz's, Higuchi's and the k-nearest neighbour (k-NN) algorithm) were compared in terms of their ability to detect the onset of epileptic seizures in scalp electroencephalogram (EEG). The selection of parameters involved in FD estimation, evaluation of the accuracy of the different algorithms and assessment of their robustness in the presence of noise were performed based on synthetic signals of known FD. When applied to scalp EEG data, Katz's and Higuchi's algorithms were found to be incapable of producing consistent changes of a single type (either a drop or an increase) during seizures. On the other hand, the k-NN algorithm produced a drop, starting close to the seizure onset, in most seizures of all patients. The k-NN algorithm outperformed both Katz's and Higuchi's algorithms in terms of robustness in the presence of noise and seizure onset detection ability. The seizure detection methodology, based on the k-NN algorithm, yielded in the training data set a sensitivity of 100% with 10.10 s mean detection delay and a false positive rate of 0.27 h-1, while the corresponding values in the testing data set were 100%, 8.82 s and 0.42 h-1, respectively. The above detection results compare favourably to those of other seizure onset detection methodologies applied to scalp EEG in the literature. The methodology described, based on the k-NN algorithm, appears to be promising for the detection of the onset of epileptic seizures based on scalp EEG.
Fractal geometrical properties of nuclei
NASA Astrophysics Data System (ADS)
Ma, Wei-Hu; Wang, Jian-Song; Wang, Qi; Mukherjee, S.; Yang, Lei; Yang, Yan-Yun; Huang, Mei-Rong
2015-10-01
We present a new idea to understand the structure of nuclei and compare it to the liquid drop model. After discussing the probability that the nuclear system may be a fractal object with the characteristic of self-similarity, the irregular nuclear structure properties and the self-similarity characteristic are considered to be an intrinsic aspect of the nuclear structure properties. For the description of nuclear geometric properties, the nuclear fractal dimension is an irreplaceable variable similar to the nuclear radius. In order to determine these two variables, a new nuclear potential energy formula which is related to the fractal dimension is put forward and the phenomenological semiempirical Bethe-Weizscker binding energy formula is modified using the fractal geometric theory. One important equation set with two equations is obtained, which is related to the concept that the fractal dimension should be a dynamic parameter in the process of nuclear synthesis. The fractal dimensions of the light nuclei are calculated and their physical meanings are discussed. We compare the nuclear fractal mean density radii with the radii calculated by the liquid drop model for the light stable and unstable nuclei using rational nuclear fractal structure types. In the present model of fractal nuclear structure there is an obvious additional feature compared to the liquid drop model, since the present model can reflect the geometric information of the nuclear structure, especially for nuclei with clusters, such as the ?-cluster nuclei and halo nuclei. Supported by National Basic Research Program of China (973 Program) (2014CB845405, 2013CB8344x), National Natural Science Foundation of China (U1432247, 11205209, 11205221)
Zhang, Lihui; Duan, Feng; Huang, Yaji; Chyang, Chiensong
2015-12-01
The changes in pore structure characteristics of sewage sludge particles under effect of calcium magnesium acetate (CMA) during combustion were investigated, the samples were characterized by N2 isothermal absorption method, and the data were used to analyze the fractal properties of the obtained samples. Results show that reaction time and the mole ratio of calcium to sulfur (Ca/S ratio) have notable impact on the pore structure and morphology of solid sample. The Brunauer-Emmett-Teller (BET) specific surface area (SBET) of sample increases with Ca/S ratio, while significant decreases with reaction time. The fractal dimension D has the similar trend with that of SBET, indicating that the surface roughness of sludge increases under the effect of CMA adding, resulting in improved the sludge combustion and the desulfurization process. PMID:26342334
Crystal, Howard A.; Holman, Susan; Lui, Yvonne W.; Baird, Alison E.; Yu, Hua; Klein, Ronald; Rojas-Soto, Diana Marcella; Gustafson, Deborah R.; Stebbins, Glenn T.
2016-01-01
Objective The fractal dimension of retinal arteries and veins is a measure of the complexity of the vascular tree. We hypothesized that retinal fractal dimension would be associated with brain volume and white matter integrity in HIV-infected women. Design Nested case-control within longitudinal cohort study. Methods Women were recruited from the Brooklyn site of the Women’s Interagency HIV study (WIHS); 34 HIV-infected and 21 HIV-uninfected women with analyzable MRIs and retinal photographs were included. Fractal dimension was determined using the SIVA software program on skeletonized retinal images. The relationship between predictors (retinal vascular measures) and outcomes (quantitative MRI measures) were analyzed with linear regression models. All models included age, intracranial volume, and both arterial and venous fractal dimension. Some models were adjusted for blood pressure, race/ethnicity, and HIV-infection. Results The women were 45.6 ± 7.3 years of age. Higher arterial dimension was associated with larger cortical volumes, but higher venous dimension was associated with smaller cortical volumes. In fully adjusted models, venous dimension was significantly associated with fractional anisotropy (standardized β = -0.41, p = 0.009) and total gray matter volume (β = -0.24, p = 0.03), and arterial dimension with mean diffusivity (β = -0.33,.p = 0.04) and fractional anisotropy (β = 0.34, p = 0.03). HIV-infection was not associated with any retinal or MRI measure. Conclusions Higher venous fractal dimension was associated with smaller cortical volumes and lower fractional anisotropy, whereas higher arterial fractal dimension was associated with the opposite patterns. Longitudinal studies are needed to validate this finding. PMID:27158911
NASA Astrophysics Data System (ADS)
Suzuki, Hiroki; Nagata, Kouji; Sakai, Yasuhiko; Hasegawa, Yutaka
2013-07-01
The fractal geometry of turbulent mixing of high-Schmidt-number scalars in multiscale, fractal-generated turbulence (FGT) is experimentally investigated. The difference between the fractal geometry in FGT and that in classical grid turbulence (CGT) generated by a biplane, single-scale grid is also investigated. Nondimensional concentration fields are measured by a planar laser-induced fluorescence technique whose accuracy has recently been improved by our research group, and the fractal dimensions are calculated by using the box-counting method. The mesh Reynolds number is 2500 for both CGT and FGT. The Schmidt number is about 2100. It is found that the threshold width ΔCth, when applying the box-counting method, does not affect the evaluation of the fractal dimension at large scales; therefore, the fractal dimensions at large scales have been investigated in this study. The results show that the fractal dimension in FGT is larger than that in CGT. In addition, the fractal dimension in FGT monotonically increases with the onset of time (or with the downstream direction), whereas that in CGT is almost constant with time. The investigation of the number of counted boxes in a unit area, together with the above results, suggests that turbulent mixing is more enhanced in FGT from the viewpoints of fractal geometry and expansion of the mixing interface.
Building Fractal Models with Manipulatives.
ERIC Educational Resources Information Center
Coes, Loring
1993-01-01
Uses manipulative materials to build and examine geometric models that simulate the self-similarity properties of fractals. Examples are discussed in two dimensions, three dimensions, and the fractal dimension. Discusses how models can be misleading. (Contains 10 references.) (MDH)
NASA Astrophysics Data System (ADS)
Pepe, S.; Di Martino, G.; Iodice, A.; Manzo, M.; Pepe, A.; Riccio, D.; Ruello, G.; Sansosti, E.; Tizzani, P.; Zinno, I.
2012-04-01
In the last two decades several aspects relevant to volcanic activity have been analyzed in terms of fractal parameters that effectively describe natural objects geometry. More specifically, these researches have been aimed at the identification of (1) the power laws that governed the magma fragmentation processes, (2) the energy of explosive eruptions, and (3) the distribution of the associated earthquakes. In this paper, the study of volcano morphology via satellite images is dealt with; in particular, we use the complete forward model developed by some of the authors (Di Martino et al., 2012) that links the stochastic characterization of amplitude Synthetic Aperture Radar (SAR) images to the fractal dimension of the imaged surfaces, modelled via fractional Brownian motion (fBm) processes. Based on the inversion of such a model, a SAR image post-processing has been implemented (Di Martino et al., 2010), that allows retrieving the fractal dimension of the observed surfaces, dictating the distribution of the roughness over different spatial scales. The fractal dimension of volcanic structures has been related to the specific nature of materials and to the effects of active geodynamic processes. Hence, the possibility to estimate the fractal dimension from a single amplitude-only SAR image is of fundamental importance for the characterization of volcano structures and, moreover, can be very helpful for monitoring and crisis management activities in case of eruptions and other similar natural hazards. The implemented SAR image processing performs the extraction of the point-by-point fractal dimension of the scene observed by the sensor, providing - as an output product - the map of the fractal dimension of the area of interest. In this work, such an analysis is performed on Cosmo-SkyMed, ERS-1/2 and ENVISAT images relevant to active stratovolcanoes in different geodynamic contexts, such as Mt. Somma-Vesuvio, Mt. Etna, Vulcano and Stromboli in Southern Italy, Shinmoe in Japan, Merapi in Indonesia. Preliminary results reveal that the fractal dimension of natural areas, being related only to the roughness of the observed surface, is very stable as the radar illumination geometry, the resolution and the wavelength change, thus holding a very unique property in SAR data inversion. Such a behavior is not verified in case of non-natural objects. As a matter of fact, when the fractal estimation is performed in the presence of either man-made objects or SAR image features depending on geometrical distortions due to the SAR system acquisition (i.e. layover, shadowing), fractal dimension (D) values outside the range of fractality of natural surfaces (2 < D < 3) are retrieved. These non-fractal characteristics show to be heavily dependent on sensor acquisition parameters (e.g. view angle, resolution). In this work, the behaviour of the maps generated starting from the C- and X- band SAR data, relevant to all the considered volcanoes, is analyzed: the distribution of the obtained fractal dimension values is investigated on different zones of the maps. In particular, it is verified that the fore-slope and back-slope areas of the image share a very similar fractal dimension distribution that is placed around the mean value of D=2.3. We conclude that, in this context, the fractal dimension could be considered as a signature of the identification of the volcano growth as a natural process. The COSMO-SkyMed data used in this study have been processed at IREA-CNR within the SAR4Volcanoes project under Italian Space Agency agreement n. I/034/11/0.
NASA Astrophysics Data System (ADS)
Albert, Helena; Perugini, Diego; Martí, Joan
2014-05-01
The volcanic unit of Montaña Reventada is an example of magma mixing in Tenerife (Canary Islands, Spain). The eruptive process has been detonated by a basanite intruding into a phonolite magma chamber. This eruption started with a basanite followed by a phonolite. Montaña Reventada phonolite is characterized by the presence of mafic enclaves. These enclaves represent about the 2% of the outcrop and have been classified like basanites, phono-tephrite and tephri-phonolite. The enclaves have different morphologies, from rounded to complex fingers-like structures, and usually exhibit cuspate terminations. This study aims to provide a new perspective on the 1100 AD Montaña Reventada eruption quantifying the textural heterogeneities related to the enclaves generated by the mixing process. The textural study was carried out using a fractal geometry approach, and its results were used to calculate some parameters related to magma chamber dynamics. Photographs of 67 samples were taken normal to the surface of the enclaves with the aim of delineating the contact between the enclaves and the host rocks. The resulted pictures were processed with the NIH (National Institutes of Health) image analysis software to generate binary images in which enclaves and host rock were replaced by black and white pixels, respectively. The fractal dimension (Dbox) has been computed by using the box-counting method in order to quantify the complexity of the enclaves morphology. Viscosity ratio (μR) between the phonolite and the enclaves has been calculated as follows: log(μR) = 0.013e3.34Dbox PIC The viscosity of the enclaves has been calculated according to the μRvalue with the higher frequency and to the calculated viscosity of the phonolite between 900° and 1200° . We hypothesized that this value corresponds to the amount of mafic magma present in the system, while the other values represent different degrees of mingling and chemical diffusion. Viscosity of the basanite can be computed like: μenclave = (%phonolite *μphonolite)+ (%basanite *μbasanite) PIC μenclaves--(%phonolite *μphonolite) μbasanite = %basanite PIC The minimum percentages which satisfy the relation are 69.5% of basanite and 30.5% of phonolite. Although the amount of mafic magma reaches the 69.5%, the presence of enclaves in the phonolite is just the ≡1% and the amount of basanite erupted before could correspond to the 15% of the phonolite (estimated from stratigraphic sections). Probably a magma body of basanite was still stored in the magma chamber. The volume of basanite still stored during this time may have evolved to a more explosive magma and hence increases the volcanic risk in the area.
NASA Astrophysics Data System (ADS)
Roth, E. J.; Mays, D. C.
2013-12-01
Clogging is an important limitation to essentially any technology or environmental process involving flow in porous media. Examples include (1) groundwater remediation, (2) managed or natural aquifer recharge, (3) hydrocarbon reservoir damage, (4) head loss in water treatment filters, (5) fouling in porous media reactors, and (6) nutrient flow for plants or bacteria. Clogging, that is, a detrimental reduction in permeability, is a common theme in each of these examples. Clogging results from a number of mechanisms, including deposition of colloidal particles (such as clay minerals), which is the focus of this research. Colloid deposits reduce porosity, which is recognized to play an important role in clogging, as expressed in the Kozeny-Carman equation. However, recent research has demonstrated that colloid deposit morphology is also a crucial variable in the clogging process. Accordingly, this presentation reports an ongoing series of laboratory experiments whose goal is to quantify deposit morphology as a fractal dimension, using an innovative technique based on static light scattering (SLS) in refractive index matched (RIM) porous media. For experiments conducted at constant flow, with constant influent suspension concentration, and initially clean porous media, results indicate that clogging is associated with colloid deposits having smaller fractal dimensions, that is, more dendritic and space-filling deposits. This result is consistent with previous research that quantified colloid deposit morphology using an empirical parameter. Clogging by colloid deposits also provides insight into the more complex clogging mechanisms of bioclogging, mineralization, and biomineralization. Although this line of work was originally motivated by problems of clogging in groundwater remediation, the methods used and the insight gained by correlating clogging with fractal dimension are expected to have relevance to other areas where flow in porous media overlaps with colloid science: Hydrogeology, petrology, water treatment, and chemical engineering.
Kerrigan, Jason R; Sanchez-Molina, David; Neggers, Jan; Arregui-Dalmases, Carlos; Velazquez-Ameijide, Juan; Crandall, Jeff R
2014-05-01
The goal of this study was to determine material properties for the anterior cortex and subcortical regions of human patellae and relate those properties to mineral density and fractal dimension of the bone. Ten human patellae were obtained from eight fresh frozen human cadavers and subjected to anteriorly-directed spherical indentation-relaxation experiments using two different sized indenters to two different indentation depths. Response data were fit to a three-mode viscoelastic model obtained through elastic-viscoelastic correspondence of the Hertzian contact relation for spherical indentation. A location-specific effective bone density measurement that more heavily weighted bone material close to the indentation site (by von Mises stress distribution) was determined from micro-computed tomography (38µm resolution) data captured for each specimen. The same imagery data were used to compute location specific fractal dimension estimates for each indentation site. Individual and averaged patella material models verified the hypothesis that when the larger indenter and greater indentation depth is used to engage the surface and deeper (trabecular) bone, the bone exhibits a more compliant response than when only the surface (cortical) bone was engaged (instantaneous elastic modulus was 325MPa vs. 207MPa, p<0.05). Effective bone mineral density was shown to be a significant predictor of the elastic modulus for both small and large indentation types (p<0.05) despite relatively low correlations. Exponential regressions of fractal dimension on elastic modulus showed significant relationships with high correlation for both the small (R(2)=0.93) and large (R(2)=0.97) indentations. PMID:23972564
Guo, Jing; Posnansky, Oleg; Hirsch, Sebastian; Scheel, Michael; Taupitz, Matthias; Braun, Juergen; Sack, Ingolf
2012-06-21
The dynamics of the complex shear modulus, G*, of soft biological tissue is governed by the rigidity and topology of multiscale mechanical networks. Multifrequency elastography can measure the frequency dependence of G* in soft biological tissue, providing information about the structure of tissue networks at multiple scales. In this study, the viscoelastic properties of structure-mimicking phantoms containing tangled paper stripes embedded in agarose gel are investigated by multifrequency magnetic resonance elastography within the dynamic range of 40–120 Hz. The effective media viscoelastic properties are analyzed in terms of the storage modulus (the real part of G*), the loss modulus (the imaginary part of G*) and the viscoelastic powerlaw given by the two-parameter springpot model. Furthermore, diffusion tensor imaging is used for investigating the effect of network structures on water mobility. The following observations were made: the random paper networks with fractal dimensions between 2.481 and 2.755 had no or minor effects on the storage modulus, whereas the loss modulus was significantly increased about 2.2 kPa per fractal dimension unit (R = 0.962, P < 0.01). This structural sensitivity of the loss modulus was significantly correlated with the springpot powerlaw exponent (0.965, P < 0.01), while for the springpot elasticity modulus, a trend was discernable (0.895, P < 0.05). No effect of the paper network on water diffusion was observed. The gel phantoms with embedded paper stripes presented here are a feasible way for experimentally studying the effect of network topology on soft-tissue viscoelastic parameters. In the dynamic range of in vivo elastography, the fractal network dimension primarily correlates to the loss behavior of soft tissue as can be seen from the loss modulus or the powerlaw exponent of the springpot model. These findings represent the experimental underpinning of structure-sensitive elastography for an improved characterization of various soft-tissue diseases. PMID:22674199
NASA Astrophysics Data System (ADS)
Guo, Jing; Posnansky, Oleg; Hirsch, Sebastian; Scheel, Michael; Taupitz, Matthias; Braun, Juergen; Sack, Ingolf
2012-06-01
The dynamics of the complex shear modulus, G*, of soft biological tissue is governed by the rigidity and topology of multiscale mechanical networks. Multifrequency elastography can measure the frequency dependence of G* in soft biological tissue, providing information about the structure of tissue networks at multiple scales. In this study, the viscoelastic properties of structure-mimicking phantoms containing tangled paper stripes embedded in agarose gel are investigated by multifrequency magnetic resonance elastography within the dynamic range of 40-120 Hz. The effective media viscoelastic properties are analyzed in terms of the storage modulus (the real part of G*), the loss modulus (the imaginary part of G*) and the viscoelastic powerlaw given by the two-parameter springpot model. Furthermore, diffusion tensor imaging is used for investigating the effect of network structures on water mobility. The following observations were made: the random paper networks with fractal dimensions between 2.481 and 2.755 had no or minor effects on the storage modulus, whereas the loss modulus was significantly increased about 2.2 kPa per fractal dimension unit (R = 0.962, P < 0.01). This structural sensitivity of the loss modulus was significantly correlated with the springpot powerlaw exponent (0.965, P < 0.01), while for the springpot elasticity modulus, a trend was discernable (0.895, P < 0.05). No effect of the paper network on water diffusion was observed. The gel phantoms with embedded paper stripes presented here are a feasible way for experimentally studying the effect of network topology on soft-tissue viscoelastic parameters. In the dynamic range of in vivo elastography, the fractal network dimension primarily correlates to the loss behavior of soft tissue as can be seen from the loss modulus or the powerlaw exponent of the springpot model. These findings represent the experimental underpinning of structure-sensitive elastography for an improved characterization of various soft-tissue diseases.
Lung cancer-a fractal viewpoint.
Lennon, Frances E; Cianci, Gianguido C; Cipriani, Nicole A; Hensing, Thomas A; Zhang, Hannah J; Chen, Chin-Tu; Murgu, Septimiu D; Vokes, Everett E; Vannier, Michael W; Salgia, Ravi
2015-11-01
Fractals are mathematical constructs that show self-similarity over a range of scales and non-integer (fractal) dimensions. Owing to these properties, fractal geometry can be used to efficiently estimate the geometrical complexity, and the irregularity of shapes and patterns observed in lung tumour growth (over space or time), whereas the use of traditional Euclidean geometry in such calculations is more challenging. The application of fractal analysis in biomedical imaging and time series has shown considerable promise for measuring processes as varied as heart and respiratory rates, neuronal cell characterization, and vascular development. Despite the advantages of fractal mathematics and numerous studies demonstrating its applicability to lung cancer research, many researchers and clinicians remain unaware of its potential. Therefore, this Review aims to introduce the fundamental basis of fractals and to illustrate how analysis of fractal dimension (FD) and associated measurements, such as lacunarity (texture) can be performed. We describe the fractal nature of the lung and explain why this organ is particularly suited to fractal analysis. Studies that have used fractal analyses to quantify changes in nuclear and chromatin FD in primary and metastatic tumour cells, and clinical imaging studies that correlated changes in the FD of tumours on CT and/or PET images with tumour growth and treatment responses are reviewed. Moreover, the potential use of these techniques in the diagnosis and therapeutic management of lung cancer are discussed. PMID:26169924
NASA Astrophysics Data System (ADS)
Saji, Ryoya; Konno, Hidetoshi
2000-02-01
We have studied local irregularity of brain waves using “local fractal dimensions (LFDs)” for two groups of elderly people, one healthy and the other affected by senile dementia. It is determined that (a) the probability distribution of the LFDs for both groups is subject to the universal law of the beta distribution; (b) the stochastic processes of LFDs of the two groups show a marked difference. We have demonstrated the applicability of the present statistical method based on the LFD for estimating the degree of progression of dementia.
NASA Astrophysics Data System (ADS)
Wu, Yongfeng; Batuski, D. J.; Khalil, A.
2007-12-01
The fractal dimension of the spatial distribution of galaxies can be characterized by various statistical and topological methods, such as the box counting and the two-point correlation function. Here we develop a new way to get fractal information, that is the Metric Space Technique (MST). It allows multiple measures to be simultaneously applied for quantitative analysis of any type of structure distribution. All such distributions are considered to be elements of multi-parameter space, and the analysis is based on considering a sample's output functions, which characterize the distributions in multi-parameter space. We use a dozen slices of a volume of space containing many newly measured galaxies from Sloan Digital Sky Survey Data Release 5. We compare results with that of mock samples of galaxies from N-body simulation with current best estimates of cosmological parameters and nested-pairs simulations, and random catalogs. By systematically studying those slices including hundreds of thousands of galaxies, we demonstrated that in the local universe there exists a fractal structure from MST. We also apply the method to 2MASS and WMAP surveys and get interesting results.
Lin, Fan; Zhu, Pengli; Huang, Feng; Li, Qiaowei; Yuan, Yin; Gao, Zhonghai; Yu, Peng; Lin, Jing; Chen, Falin
2015-05-01
The objective of this study was to evaluate the association of the central retinal arteriolar equivalent (CRAE) and the retinal vascular fractal dimension, two quantitative parameters that reflect microcirculation, with aortic stiffness. In this cross-sectional study, we identified the cardiovascular risk factors in 2169 subjects using a health questionnaire, physical examinations and laboratory examinations. We evaluated the aortic stiffness using noninvasive brachial-ankle pulse wave velocity (baPWV) and assessed the microcirculatory alterations with CRAE and retinal vascular fractal dimension, which were measured using fundus photography and semiautomatic quantitative software, respectively. The increase in baPWV (Q1-Q4) correlated with an increased likelihood of the central retinal artery narrowing and a reduction in the retinal vascular fractal dimension. Further adjustment of the cardiovascular risk factors diminished the association between baPWV and CRAE, but increased the association between baPWV and retinal vascular fractal dimension. Elevated baPWV correlates with reduced CRAE and retinal vascular fractal dimension. Such a finding supports macrocirculation- and microcirculation-associated hypotheses. PMID:25716651
Sadana, A
1998-01-01
The diffusion-limited binding kinetics of antigen (analyte), in solution with antibody (receptor) immobilized on a biosensor surface, is analyzed within a fractal framework. Most of the data presented is adequately described by a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot. A single example of a dual-fractal analysis is also presented. It is of interest to note that the binding-rate coefficient (k) and the fractal dimension (Df) both exhibit changes in the same and in the reverse direction for the antigen-antibody systems analyzed. Binding-rate coefficient expressions, as a function of the Df developed for the antigen-antibody binding systems, indicate the high sensitivity of the k on the Df when both a single- and a dual-fractal analysis are used. For example, for a single-fractal analysis, and for the binding of antibody Mab 0.5 beta in solution to gp120 peptide immobilized on a BIAcore biosensor, the order of dependence on the Df was 4.0926. For a dual-fractal analysis, and for the binding of 25-100 ng/mL TRITC-LPS (lipopolysaccharide) in solution with polymyxin B immobilized on a fiberoptic biosensor, the order of dependence of the binding-rate coefficients, k1 and k2, on the fractal dimensions, Df1 and Df2, were 7.6335 and -11.55, respectively. The fractional order of dependence of the k(s) on the Df(s) further reinforces the fractal nature of the system. The k(s) expressions developed as a function of the Df(s) are of particular value, since they provide a means to better control biosensor performance, by linking it to the heterogeneity on the surface, and further emphasize, in a quantitative sense, the importance of the nature of the surface in biosensor performance. PMID:9779572
NASA Astrophysics Data System (ADS)
Cámara, Joaquín; Gómez-Miguel, Vicente; Martín, Miguel Ángel
2016-03-01
Geologists know that drainage networks can exhibit different drainage patterns depending on the hydrogeological properties of the underlying materials. Geographic Information System (GIS) technologies and the increasing availability and resolution of digital elevation data have greatly facilitated the delineation, quantification, and study of drainage networks. This study investigates the possibility of inferring geological information of the underlying material from fractal and linear parameters describing drainage networks automatically extracted from 5-m-resolution LiDAR digital terrain model (DTM) data. According to the lithological information (scale 1:25,000), the study area is comprised of 30 homogeneous bedrock lithologies, the lithological map units (LMUs). These are mostly igneous and metamorphic rocks, but also include some sedimentary rocks. A statistical classification model of the LMUs by rock type has been proposed based on both the fractal dimension and drainage density of the overlying drainage networks. The classification model has been built using 16 LMUs, and it has correctly classified 13 of the 14 LMUs used for its validation. Results for the study area show that LMUs, with areas ranging from 177.83 ± 0.01 to 3.16 ± 0.01 km2, can be successfully classified by rock type using the fractal dimension and the drainage density of the drainage networks derived from medium resolution LiDAR DTM data with different flow support areas. These results imply that the information included in a 5-m-resolution LiDAR DTM and the appropriate techniques employed to manage it are the only inputs required to identify the underlying geological materials.
NASA Astrophysics Data System (ADS)
Cmara, Joaqun; Gmez-Miguel, Vicente; Martn, Miguel ngel
2015-07-01
Geologists know that drainage networks can exhibit different drainage patterns depending on the hydrogeological properties of the underlying materials. Geographic Information System (GIS) technologies and the increasing availability and resolution of digital elevation data have greatly facilitated the delineation, quantification, and study of drainage networks. This study investigates the possibility of inferring geological information of the underlying material from fractal and linear parameters describing drainage networks automatically extracted from 5-m-resolution LiDAR digital terrain model (DTM) data. According to the lithological information (scale 1:25,000), the study area is comprised of 30 homogeneous bedrock lithologies, the lithological map units (LMUs). These are mostly igneous and metamorphic rocks, but also include some sedimentary rocks. A statistical classification model of the LMUs by rock type has been proposed based on both the fractal dimension and drainage density of the overlying drainage networks. The classification model has been built using 16 LMUs, and it has correctly classified 13 of the 14 LMUs used for its validation. Results for the study area show that LMUs, with areas ranging from 177.83 0.01 to 3.16 0.01 km2, can be successfully classified by rock type using the fractal dimension and the drainage density of the drainage networks derived from medium resolution LiDAR DTM data with different flow support areas. These results imply that the information included in a 5-m-resolution LiDAR DTM and the appropriate techniques employed to manage it are the only inputs required to identify the underlying geological materials.
Récamier, Vincent; Izeddin, Ignacio; Bosanac, Lana; Dahan, Maxime; Proux, Florence; Darzacq, Xavier
2014-01-01
Chromatin is a major nuclear component, and it is an active matter of debate to understand its different levels of spatial organization, as well as its implication in gene regulation. Measurements of nuclear chromatin compaction were recently used to understand how DNA is folded inside the nucleus and to detect cellular dysfunctions such as cancer. Super-resolution imaging opens new possibilities to measure chromatin organization in situ. Here, we performed a direct measure of chromatin compaction at the single cell level. We used histone H2B, one of the 4 core histone proteins forming the nucleosome, as a chromatin density marker. Using photoactivation localization microscopy (PALM) and adaptive optics, we measured the three-dimensional distribution of H2B with nanometric resolution. We computed the distribution of distances between every two points of the chromatin structure, namely the Ripley K(r) distribution. We found that the K(r) distribution of H2B followed a power law, leading to a precise measurement of the correlation fractal dimension of chromatin of 2.7. Moreover, using photoactivable GFP fused to H2B, we observed dynamic evolution of chromatin sub-regions compaction. As a result, the correlation fractal dimension of chromatin reported here can be interpreted as a dynamically maintained non-equilibrium state. PMID:24637833
Zouein, Fouad A; Kurdi, Mazen; Booz, George W; Fuseler, John W
2014-08-01
Hearts of mice with reduction of function mutation in STAT3 (SA/SA) develop fibrotic collagen foci and reduced systolic function with hypertension. This model was used to determine if fractal dimension and image analysis can provide a quantitative description of myocardial fibrosis using routinely prepared trichome-stained material. Collagen was characterized by relative density [integrated optical density/area (IOD/A)] and fractal dimension (D), an index of complexity. IOD/A of collagen in wild type mice increased with hypertension while D decreased, suggesting tighter collagen packing that could eventually stiffen the myocardium as in diastolic heart failure. Reduced STAT3 function caused modest collagen fibrosis with increased IOD/A and D, indicating more tightly packed, but more disorganized collagen than normotensive and hypertensive controls. Hypertension in SA/SA mice resulted in large regions where myocytes were lost and replaced by fibrotic collagen characterized by decreased density and increased disorder. This indicates that collagen associated with reparative fibrosis in SA/SA hearts experiencing hypertension was highly disorganized and more space filling. Loss of myocytes and their replacement by disordered collagen fibers may further weaken the myocardium leading to systolic heart failure. Our findings highlight the utility of image analysis in revealing importance of a cellular protein for normal and reparative extracellular matrix deposition. PMID:25410603
Comprehensive Fractal Description of Porosity of Coal of Different Ranks
Ren, Jiangang; Zhang, Guocheng; Song, Zhimin; Liu, Gaofeng; Li, Bing
2014-01-01
We selected, as the objects of our research, lignite from the Beizao Mine, gas coal from the Caiyuan Mine, coking coal from the Xiqu Mine, and anthracite from the Guhanshan Mine. We used the mercury intrusion method and the low-temperature liquid nitrogen adsorption method to analyze the structure and shape of the coal pores and calculated the fractal dimensions of different aperture segments in the coal. The experimental results show that the fractal dimension of the aperture segment of lignite, gas coal, and coking coal with an aperture of greater than or equal to 10 nm, as well as the fractal dimension of the aperture segment of anthracite with an aperture of greater than or equal to 100 nm, can be calculated using the mercury intrusion method; the fractal dimension of the coal pore, with an aperture range between 2.03 nm and 361.14 nm, can be calculated using the liquid nitrogen adsorption method, of which the fractal dimensions bounded by apertures of 10 nm and 100 nm are different. Based on these findings, we defined and calculated the comprehensive fractal dimensions of the coal pores and achieved the unity of fractal dimensions for full apertures of coal pores, thereby facilitating, overall characterization for the heterogeneity of the coal pore structure. PMID:24955407
Biometric feature extraction using local fractal auto-correlation
NASA Astrophysics Data System (ADS)
Chen, Xi; Zhang, Jia-Shu
2014-09-01
Image texture feature extraction is a classical means for biometric recognition. To extract effective texture feature for matching, we utilize local fractal auto-correlation to construct an effective image texture descriptor. Three main steps are involved in the proposed scheme: (i) using two-dimensional Gabor filter to extract the texture features of biometric images; (ii) calculating the local fractal dimension of Gabor feature under different orientations and scales using fractal auto-correlation algorithm; and (iii) linking the local fractal dimension of Gabor feature under different orientations and scales into a big vector for matching. Experiments and analyses show our proposed scheme is an efficient biometric feature extraction approach.
A fractal model for optical properties of tissue
NASA Astrophysics Data System (ADS)
Li, Zhi-fang; Li, Hui
2011-06-01
An analytical theory integrating the Rayleigh-Debye-Gans theory with fractal aggregate theory (RDG-FA) for a better understanding of the optical properties of fractal-like tissue at a high-resolution scale is present. The calculated optical properties of tissue are consistent with the reported values. And the optical properties are related to the fractal dimension and the correlation length. In addition, the exponents of the inverse power law spectral dependence of scattering coefficient and reduced scattering coefficient are piecewise linearly depending on the fractal dimension, indicating that the exponents can serve as a tools for distinguishing the different tissue.
Wagenseil, R.
1991-01-01
There are persistent difficulties in monitoring nonpoint source pollution and in the related field of hydrology. The problems stem from variations in spatial distribution which are poorly understood and difficult to model with established methods. Two recent developments may offer a solution, if they are combined with care. The first development is the increasing capability of computer mapping, called geographic information systems (GIS). These systems can store, retrieve, and manipulate data with an explicit spatial structure. The second development is the field of fractal mathematics. Fractal mathematics includes geometric sets which have simple descriptions, despite complex appearances. One family of such fractal sets are the Brownian surfaces, which capture many of the qualities of natural land surfaces in a simple statistical model. Up until now, the Brownian models have been constrained by the assumption that the same statistical relationship holds over the entire surface. This is called the constraint of stationarity. The need to study how the landscape differs by location leads to relaxing the constraint of stationarity. This, in turn, causes some profound changes in the model. A special computer program applies the new model to a set of three-dimensional digital maps of natural terrain (DEMs). The model performs well, and highlights differences in landforms. This suggests several new approaches to spatial variation.
Preliminary calculation of cylinder dimensions for aircraft engines
NASA Technical Reports Server (NTRS)
Schwager, Otto
1921-01-01
It is extremely important in building aircraft engines to determine the requisite cylinder dimensions as accurately as possible, in order that the weight required for a given power shall not be excessive. This report presents a calculation method that depends on the air requirement of the fuel.
Verifying the Dependence of Fractal Coefficients on Different Spatial Distributions
Gospodinov, Dragomir; Marekova, Elisaveta; Marinov, Alexander
2010-01-21
A fractal distribution requires that the number of objects larger than a specific size r has a power-law dependence on the size N(r) = C/r{sup D}propor tor{sup -D} where D is the fractal dimension. Usually the correlation integral is calculated to estimate the correlation fractal dimension of epicentres. A 'box-counting' procedure could also be applied giving the 'capacity' fractal dimension. The fractal dimension can be an integer and then it is equivalent to a Euclidean dimension (it is zero of a point, one of a segment, of a square is two and of a cube is three). In general the fractal dimension is not an integer but a fractional dimension and there comes the origin of the term 'fractal'. The use of a power-law to statistically describe a set of events or phenomena reveals the lack of a characteristic length scale, that is fractal objects are scale invariant. Scaling invariance and chaotic behavior constitute the base of a lot of natural hazards phenomena. Many studies of earthquakes reveal that their occurrence exhibits scale-invariant properties, so the fractal dimension can characterize them. It has first been confirmed that both aftershock rate decay in time and earthquake size distribution follow a power law. Recently many other earthquake distributions have been found to be scale-invariant. The spatial distribution of both regional seismicity and aftershocks show some fractal features. Earthquake spatial distributions are considered fractal, but indirectly. There are two possible models, which result in fractal earthquake distributions. The first model considers that a fractal distribution of faults leads to a fractal distribution of earthquakes, because each earthquake is characteristic of the fault on which it occurs. The second assumes that each fault has a fractal distribution of earthquakes. Observations strongly favour the first hypothesis.The fractal coefficients analysis provides some important advantages in examining earthquake spatial distribution, which are: - Simple way to quantify scale-invariant distributions of complex objects or phenomena by a small number of parameters. - It is becoming evident that the applicability of fractal distributions to geological problems could have a more fundamental basis. Chaotic behaviour could underlay the geotectonic processes and the applicable statistics could often be fractal.The application of fractal distribution analysis has, however, some specific aspects. It is usually difficult to present an adequate interpretation of the obtained values of fractal coefficients for earthquake epicenter or hypocenter distributions. That is why in this paper we aimed at other goals - to verify how a fractal coefficient depends on different spatial distributions. We simulated earthquake spatial data by generating randomly points first in a 3D space - cube, then in a parallelepiped, diminishing one of its sides. We then continued this procedure in 2D and 1D space. For each simulated data set we calculated the points' fractal coefficient (correlation fractal dimension of epicentres) and then checked for correlation between the coefficients values and the type of spatial distribution.In that way one can obtain a set of standard fractal coefficients' values for varying spatial distributions. These then can be used when real earthquake data is analyzed by comparing the real data coefficients values to the standard fractal coefficients. Such an approach can help in interpreting the fractal analysis results through different types of spatial distributions.
Mincione, Gabriella; Di Nicola, Marta; Di Marcantonio, Maria Carmela; Muraro, Raffaella; Piattelli, Adriano; Rubini, Corrado; Penitente, Enrico; Piccirilli, Marcello; Aprile, Giuseppe; Perrotti, Vittoria; Artese, Luciano
2015-10-01
Fractal dimension (FD) in tissue specimens from patients with oral squamous cell carcinoma (OSCC) was evaluated. FD values in different stages of OSCC, and the correlations with clinicopathological variables and patient survival were investigated. Histological sections from OSCC and control non-neoplastic mucosa specimens were stained with hematoxylin-eosin for pathological analysis and with Feulgen for nuclear evaluation. FD in OSCC groups vs. controls revealed statistically significant differences (P<0.001). In addition, a progressive increase of FD from stage I and II lesions and stage III and IV lesions was observed, with statistically significant differences (P=0.003). Moreover, different degrees of tumor differentiation showed a significant difference in the average nuclear FD values (P=0.001). A relationship between FD and patients' survival was also detected with lower FD values associated to longer survival time and higher FD values with shorter survival time (P=0.034). These data showed that FD significantly increased during OSCC progression. Thus, FD could represent a novel prognostic tool for OSCC, as FD values significantly correlated with patient survival. Fractal geometry could give insights into tumor morphology and could become an useful tool for analyzing irregular tumor growth patterns. PMID:25367085
Fractal analysis of motor imagery recognition in the BCI research
NASA Astrophysics Data System (ADS)
Chang, Chia-Tzu; Huang, Han-Pang; Huang, Tzu-Hao
2011-12-01
A fractal approach is employed for the brain motor imagery recognition and applied to brain computer interface (BCI). The fractal dimension is used as feature extraction and SVM (Support Vector Machine) as feature classifier for on-line BCI applications. The modified Inverse Random Midpoint Displacement (mIRMD) is adopted to calculate the fractal dimensions of EEG signals. The fractal dimensions can effectively reflect the complexity of EEG signals, and are related to the motor imagery tasks. Further, the SVM is employed as the classifier to combine with fractal dimension for motor-imagery recognition and use mutual information to show the difference between two classes. The results are compared with those in the BCI 2003 competition and it shows that our method has better classification accuracy and mutual information (MI).
Fractal characteristics of fracture morphology of steels irradiated with high-energy ions
NASA Astrophysics Data System (ADS)
Xian, Yongqiang; Liu, Juan; Zhang, Chonghong; Chen, Jiachao; Yang, Yitao; Zhang, Liqing; Song, Yin
2015-06-01
A fractal analysis of fracture surfaces of steels (a ferritic/martensitic steel and an oxide-dispersion-strengthened ferritic steel) before and after the irradiation with high-energy ions is presented. Fracture surfaces were acquired from a tensile test and a small-ball punch test (SP). Digital images of the fracture surfaces obtained from scanning electron microscopy (SEM) were used to calculate the fractal dimension (FD) by using the pixel covering method. Boundary of binary image and fractal dimension were determined with a MATLAB program. The results indicate that fractal dimension can be an effective parameter to describe the characteristics of fracture surfaces before and after irradiation. The rougher the fracture surface, the larger the fractal dimension. Correlation of the change of fractal dimension with the embrittlement of the irradiated steels is discussed.
Sorensen, C M; Cai, J; Lu, N
1992-10-20
A new method for the in situ optical determination of the soot-cluster monomer particle radius a, the number of monomers per cluster N, and the fractal dimension D is presented. The method makes use of a comparison of the volume-equivalent sphere radius determined from scattering-extinction measurements RSe and the radius of gyration Rg, which is determined from the optical structure factor. The combination of these data with the measured turbidity permits for a novel measurement of D. The parameters a and N are obtained from a graphical network-analysis scheme that compares R(se) and Rg. Corrections for cluster polydispersity are presented. The effects of uncertainty in various input parameters and assumptions are discussed. The method is illustrated by an application to data obtained from a premixed methane-oxygen flame, and reasonable values of a, N, andD are obtained. PMID:20733873
Cake porosity analysis using 1D-3D fractal dimensions in coagulation-microfiltration of NOM.
Raspati, G S; Leiknes, T O
2015-01-01
Fouling during coagulation-ceramic microfiltration of natural organic matter was investigated. Two process configurations (inline coagulation (IC) and tank coagulation (TC)) and two process conditions (types of coagulants-aluminum-based PAX and iron-based PIX-and G-values) were studied. The rate of irreversible fouling corresponding to the increase of initial transmembrane pressure after backwash of IC-PAX was lowest followed by TC-PAX and TC-PIX, while the performance of IC-PIX was found worst. The 1D and 2D fractal analysis revealed that flocs from IC were morphologically different from those of TC, leading to different filtration characteristics. The 3D fractal analysis revealed two groups of morphologically similar flocs: one led to successful filtration experiments, whereas the other led to unsuccessful ones. Cake porosity was found dependent on the floc morphology. Thus, such an approach was found complementary with fouling analysis by means of a membrane fouling model and minimization of fouling phenomenon was achieved by combining the two approaches. PMID:25768221
Berry, Hugues
2002-01-01
Conventional equations for enzyme kinetics are based on mass-action laws, that may fail in low-dimensional and disordered media such as biological membranes. We present Monte Carlo simulations of an isolated Michaelis-Menten enzyme reaction on two-dimensional lattices with varying obstacle densities, as models of biological membranes. The model predicts that, as a result of anomalous diffusion on these low-dimensional media, the kinetics are of the fractal type. Consequently, the conventional equations for enzyme kinetics fail to describe the reaction. In particular, we show that the quasi-stationary-state assumption can hardly be retained in these conditions. Moreover, the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial substrate concentration increase. The simulations indicate that these two influences are mainly additive. Finally, the simulations show pronounced S-P segregation over the lattice at obstacle densities compatible with in vivo conditions. This phenomenon could be a source of spatial self organization in biological membranes. PMID:12324410
A fractal analysis of quaternary, Cenozoic-Mesozoic, and Late Pennsylvanian sea level changes
NASA Technical Reports Server (NTRS)
Hsui, Albert T.; Rust, Kelly A.; Klein, George D.
1993-01-01
Sea level changes are related to both climatic variations and tectonic movements. The fractal dimensions of several sea level curves were compared to a modern climatic fractal dimension of 1.26 established for annual precipitation records. A similar fractal dimension (1.22) based on delta(O-18/O-16) in deep-sea sediments has been suggested to characterize climatic change during the past 2 m.y. Our analysis indicates that sea level changes over the past 150,000 to 250,000 years also exhibit comparable fractal dimensions. Sea level changes for periods longer than about 30 m.y. are found to produce fractal dimensions closer to unity and Missourian (Late Pennsylvanian) sea level changes yield a fractal dimension of 1.41. The fact that these sea level curves all possess fractal dimensions less than 1.5 indicates that sea level changes exhibit nonperiodic, long-run persistence. The different fractal dimensions calculated for the various time periods could be the result of a characteristic overprinting of the sediment recored by prevailing processes during deposition. For example, during the Quaternary, glacio-eustatic sea level changes correlate well with the present climatic signature. During the Missourian, however, mechanisms such as plate reorganization may have dominated, resulting in a significantly different fractal dimension.
Fractal analysis of surface topography in ground monocrystal sapphire
NASA Astrophysics Data System (ADS)
Wang, Qiuyan; Liang, Zhiqiang; Wang, Xibin; Zhao, Wenxiang; Wu, Yongbo; Zhou, Tianfeng
2015-02-01
The surface characterization of ground monocrystal sapphire is investigated by fractal analysis method. A serial of ground sapphire surfaces in ductile removal and brittle removal mode are obtained by grinding experiments, and their three dimensional (3D) and two dimensional (2D) fractal dimensions are calculated and analyzed by box-counting methods. The 3D surface fractal dimension Ds shows a negative correlation with the surface roughness parameter Ra and is sensitive to the ground surface defects. For the ground surface with larger fractal dimension Ds, its micro-topography is more exquisite with minor defects. Once the fractal dimension Ds become smaller, deep cracks and pronounced defects are exhibited in ground surface. Moreover, the material removal mode can be implied from the distribution of 2D cross-sectional profile fractal dimension DL. The workpiece surface generated in ductile removal mode has high surface quality with high 2D and 3D fractal dimensions. This study indicates that the box-counting fractal analysis is an effective method to evaluate ground sapphire surface comprehensively.
NASA Astrophysics Data System (ADS)
Carletti, Timoteo; Righi, Simone
2010-05-01
In this paper we define a new class of weighted complex networks sharing several properties with fractal sets, and whose topology can be completely analytically characterized in terms of the involved parameters and of the fractal dimension. General networks with fractal or hierarchical structures can be set in the proposed framework that moreover could be used to provide some answers to the widespread emergence of fractal structures in nature.
Local Box-Counting to Determine Fractal Dimension of High-Order Chaos
NASA Astrophysics Data System (ADS)
Osaka, Motohisa; Ito, Nobuyasu
To determine the attractor dimension of chaotic dynamics, the box-counting method has the difficulty in getting accurate estimates because the boxes are not weighted by their relative probabilities. We present a new method to minimize this difficulty. The local box-counting method can be quite effective in determining the attractor dimension of high-order chaos as well as low-order chaos.
Wei, Mao-Hong; Lin, Hui-Long
2014-03-01
The alpine meadow in the source region of the Yangtze and Yellow River is suffering serious deterioration. Though great efforts have been put into, the restoration for the degraded grassland is far from being effective, mainly due to poor understanding of the degradation mechanism of alpine meadow in this region. In order to clarify the formation mechanism of degradation grassland and provide the new ideas for restoration, degradation sequences of the alpine meadow in the source region of the Yangtze and Yellow River were taken as target systems to analyze the soil particle size distribution, the fractal dimension of the soil particle size, and the relationship between soil erosion modulus and fractal dimension. The results showed that, with increasing grassland degradation, the percentage contents of clay increased while the percentage contents of silt sand and very fine sand showed a decreasing trend. The fractal dimension presented a positive correlation with clay among the degradation sequences while negative correlations were found with very fine sand and silt sand. The curvilinear regression of fractal dimension and erosion modulus fitted a quadratic function. Judged by the function, fractal dimension 2.81 was the threshold value of soil erosion. The threshold value has an indicative meaning on predicting the breakout of grazing-induced erosion and on restoration of the degraded grassland. Taking fractal dimension of 2.81 as the restoration indicator, adoption of corresponding measures to make fractal dimension less than 2.81, would an effective way to restore the degradation grassland. PMID:24984483
Fractal feature analysis and classification in medical imaging
Chen, C.C.; Fox, M.D. . Dept. of Computer Science); DaPonte, J.S. )
1989-06-01
Following Mandelbrot's fractal theory, it is found that the fractal dimension could be obtained in medical images by the concept of fractional Brownian motion. An estimation concept for determination of the fractal dimension based upon the concept of fractional Brownian motion is discussed. Two applications are found: classification and edge enhancement and detection. For the purpose of classification, a normalized fractional Brownian motion feature vector is defined from this estimation concept. For edge enhancement and detection application, a transformed image is obtained by calculating the fractal dimension of each pixel over the whole medical image. Preliminary results using projection radiographs suggest that the fractal based image transformation appears to hold promise as an edge enhancement and preprocessing algorithm that does not increase noise in the way that gradient operators do.
Fractal analysis of high-resolution CT images as a tool for quantification of lung diseases
Uppaluri, R.; Mitsa, T.; Galvin, J.R.
1995-12-31
Fractal geometry is increasingly being used to model complex naturally occurring phenomena. There are two types of fractals in nature-geometric fractals and stochastic fractals. The pulmonary branching structure is a geometric fractal and the intensity of its grey scale image is a stochastic fractal. In this paper, the authors attempt to quantify the texture of CT lung images using properties of both types of fractals. A simple algorithm for detecting of abnormality in human lungs, based on 2-D and 3-D fractal dimensions, is presented. This method involves calculating the local fractal dimensions, based on intensities, in the 2-D slice to air edge enhancement. Following this, grey level thresholding is performed and a global fractal dimension, based on structure, for the entire data is estimated in 2-D and 3-D. High Resolution CT images of normal and abnormal lungs were analyzed. Preliminary results showed that classification of normal and abnormal images could be obtained based on the differences between their global fractal dimensions.
NASA Astrophysics Data System (ADS)
Pepe, S.; Solaro, G.; Ricciardi, G. P.; Tizzani, P.
2008-10-01
We investigated the existence of a fractal law (power law) distribution of size pyroclastic fragments erupted during the fallout phase of the 79 A.D. Plinian eruption at Mt. Vesuvius. In particular, we performed a particle size distribution analysis on 18 white and grey pumice samples collected in six sites distributed in the SW sector of Mt. Vesuvius. Our measurements show that the fragmentation of samples in the investigated range (from 32 mm to 850 μm) follows a power law, guaranteeing the scale invariance of the process. The relationship frequency-size distribution of the fragments is verified independently from the nature (i.e., pumices and lithics) and stratigraphic height of the considered samples in the pyroclastic deposit. Therefore, the fractal fragmentation theory can be indicated for evaluating the relationship between the intensity of fragmentation (fractal dimension D) and eruption energy. In this way the apparent chaotic distribution of the particles in the fallout deposits hides a self-organized complexity revealed by the retrieved power law distribution. We further remark that a key aspect of our analysis is the founded evidence that the fractal dimension of the lithics is systematically greater than that of the pumices.
Beretta-Piccoli, Matteo; D’Antona, Giuseppe; Barbero, Marco; Fisher, Beth; Dieli-Conwright, Christina M.; Clijsen, Ron; Cescon, Corrado
2015-01-01
Purpose Over the past decade, linear and non-linear surface electromyography descriptors for central and peripheral components of fatigue have been developed. In the current study, we tested fractal dimension (FD) and conduction velocity (CV) as myoelectric descriptors of central and peripheral fatigue, respectively. To this aim, we analyzed FD and CV slopes during sustained fatiguing contractions of the quadriceps femoris in healthy humans. Methods A total of 29 recreationally active women (mean age±standard deviation: 24±4 years) and two female elite athletes (one power athlete, age 24 and one endurance athlete, age 30 years) performed two knee extensions: (1) at 20% maximal voluntary contraction (MVC) for 30 s, and (2) at 60% MVC held until exhaustion. Surface EMG signals were detected from the vastus lateralis and vastus medialis using bidimensional arrays. Results Central and peripheral fatigue were described as decreases in FD and CV, respectively. A positive correlation between FD and CV (R=0.51, p<0.01) was found during the sustained 60% MVC, probably as a result of simultaneous motor unit synchronization and a decrease in muscle fiber CV during the fatiguing task. Conclusions Central and peripheral fatigue can be described as changes in FD and CV, at least in young, healthy women. The significant correlation between FD and CV observed at 60% MVC suggests that a mutual interaction between central and peripheral fatigue can arise during submaximal isometric contractions. PMID:25880369
Wu, Yu-Te; Shyu, Kuo-Kai; Jao, Chii-Wen; Wang, Zun-Yun; Soong, Bing-Wen; Wu, Hsiu-Mei; Wang, Po-Shan
2010-01-01
Multiple system atrophy of the cerebellar type (MSA-C) is a degenerative neurological disease of the central nervous system. This study aims to demonstrate that the morphological changes of cerebellar structure, specifically, the cerebellum white matter (CBWM) and cerebellum gray matter (CBGM) from T1-weighted magnetic resonance (MR) images, can be quantified by three-dimensional (3D) fractal dimension (FD) analysis, which is a measure of complexity. Twenty-three MSA-C patients and twenty-one normal subjects participated in this study. The results of this study show that MSA-C patients presented significantly lower FD values compared to the control group, and that morphological change in the CBWM dominates the cerebellar degeneration. In addition, the FD analysis method is superior to conventional volumetric methods in quantifying the structural changes of WM and GM because it exhibits smaller variances and less gender effect. Since a decrease of cerebellar FD value indicates degeneration of the cerebellar structure, this study further suggests that the morphological changes of cerebellar structures (CBGM and CBWM) can be characterized by FD analysis. PMID:19635573
de Oliveira, Marcos Aurélio Barboza; Brandi, Antônio Carlos; dos Santos, Carlos Alberto; Botelho, Paulo Henrique Husseni; Cortez, José Luís Lasso; de Godoy, Moacir Fernandes; Braile, Domingo Marcolino
2014-01-01
Introduction Solutions that cause elective cardiac arrest are constantly evolving, but the ideal compound has not yet been found. The authors compare a new cardioplegic solution with histidine-tryptophan-glutamate (Group 2) and other one with histidine-tryptophan-cetoglutarate (Group 1) in a model of isolated rat heart. Objective To quantify the fractal dimension and Shannon entropy in rat myocytes subjected to cardioplegia solution using histidine-tryptophan with glutamate in an experimental model, considering the caspase markers, IL-8 and KI-67. Methods Twenty male Wistar rats were anesthetized and heparinized. The chest was opened, the heart was withdrawn and 40 ml/kg of cardioplegia (with histidine-tryptophan-cetoglutarate or histidine-tryptophan-glutamate solution) was infused. The hearts were kept for 2 hours at 4ºC in the same solution, and thereafter placed in the Langendorff apparatus for 30 min with Ringer-Locke solution. Analyzes were performed for immunohistochemical caspase, IL-8 and KI-67. Results The fractal dimension and Shannon entropy were not different between groups histidine-tryptophan-glutamate and histidine-tryptophan-acetoglutarate. Conclusion The amount of information measured by Shannon entropy and the distribution thereof (given by fractal dimension) of the slices treated with histidine-tryptophan-cetoglutarate and histidine-tryptophan-glutamate were not different, showing that the histidine-tryptophan-glutamate solution is as good as histidine-tryptophan-acetoglutarate to preserve myocytes in isolated rat heart. PMID:25140464
Jurczyszyn, Kamil; Osiecka, Beata J; Ziółkowski, Piotr
2012-01-01
Fractal dimension analysis (FDA) is modern mathematical method widely used to describing of complex and chaotic shapes when classic methods fail. The main aim of this study was evaluating the influence of photodynamic therapy (PDT) with cystein proteases inhibitors (CPI) on the number and morphology of blood vessels inside tumor and on increase of effectiveness of combined therapy in contrast to PDT and CPI used separately. Animals were divided into four groups: control, treated using only PDT, treated using only CPI and treated using combined therapy, PDT and CPI. Results showed that time of animal survival and depth of necrosis inside tumor were significantly higher in CPI+PDT group in contrast to other groups. The higher value of fractal dimension (FD) was observed in control group, while the lowest value was found in the group which was treated by cystein protease inhibitors. The differences between FD were observed in CPI group and PDT+CPI group in comparison to control group. Our results revealed that fractal dimension analysis is a very useful tool in estimating differences between irregular shapes like blood vessels in PDT treated tumors. Thus, the implementation of FDA algorithms could be useful method in evaluating the efficacy of PDT. PMID:22991578
Jurczyszyn, Kamil; Osiecka, Beata J.; Ziółkowski, Piotr
2012-01-01
Fractal dimension analysis (FDA) is modern mathematical method widely used to describing of complex and chaotic shapes when classic methods fail. The main aim of this study was evaluating the influence of photodynamic therapy (PDT) with cystein proteases inhibitors (CPI) on the number and morphology of blood vessels inside tumor and on increase of effectiveness of combined therapy in contrast to PDT and CPI used separately. Animals were divided into four groups: control, treated using only PDT, treated using only CPI and treated using combined therapy, PDT and CPI. Results showed that time of animal survival and depth of necrosis inside tumor were significantly higher in CPI+PDT group in contrast to other groups. The higher value of fractal dimension (FD) was observed in control group, while the lowest value was found in the group which was treated by cystein protease inhibitors. The differences between FD were observed in CPI group and PDT+CPI group in comparison to control group. Our results revealed that fractal dimension analysis is a very useful tool in estimating differences between irregular shapes like blood vessels in PDT treated tumors. Thus, the implementation of FDA algorithms could be useful method in evaluating the efficacy of PDT. PMID:22991578
Fractal properties of aggregates of metal nanoclusters on solid surface
NASA Astrophysics Data System (ADS)
Samsonov, V. M.; Kuznetsova, Yu. V.; D'yakova, E. V.
2016-02-01
AFM images are used to determine and analyze fractal characteristics (cluster fraction dimension and lacunarity) of aggregates of Au and Ag nanoclusters on metal films of the same metal produced with the aid of thermal vacuum deposition on mica surface. A fractal dimension of 1.6 that corresponds to typical samples with relatively uniform distribution of nanoclusters on the film surface is in agreement with the mean value calculated from experimental data of Belko et al., who studied the fractal dimension of Au nanoclusters on a different dielectric (quartz) surface. When a compact single aggregate of Au nanoclusters is formed on a certain active center or defect, the fractal cluster dimension decreases to 1.4. The experimental data are compared with the results of existing theoretical models of association of nanoclusters in 2D systems.
Pereira, Luis M
2010-06-01
Pharmacokinetics (PK) has been traditionally dealt with under the homogeneity assumption. However, biological systems are nowadays comprehensively understood as being inherently fractal. Specifically, the microenvironments where drug molecules interact with membrane interfaces, metabolic enzymes or pharmacological receptors, are unanimously recognized as unstirred, space-restricted, heterogeneous and geometrically fractal. Therefore, classical Fickean diffusion and the notion of the compartment as a homogeneous kinetic space must be revisited. Diffusion in fractal spaces has been studied for a long time making use of fractional calculus and expanding on the notion of dimension. Combining this new paradigm with the need to describe and explain experimental data results in defining time-dependent rate constants with a characteristic fractal exponent. Under the one-compartment simplification this strategy is straightforward. However, precisely due to the heterogeneity of the underlying biology, often at least a two-compartment model is required to address macroscopic data such as drug concentrations. This simple modelling step-up implies significant analytical and numerical complications. However, a few methods are available that make possible the original desideratum. In fact, exploring the full range of parametric possibilities and looking at different drugs and respective biological concentrations, it may be concluded that all PK modelling approaches are indeed particular cases of the fractal PK theory. PMID:20461596
P-adic coverage method in fractal analysis of showers
NASA Astrophysics Data System (ADS)
Dedovich, T. G.; Tokarev, M. V.
2011-11-01
Self-similarity in multiple processes at high energies is considered. It is assumed that a parton cascade transforms into a hadron shower with a fractal structure. The box counting (BC) method used to calculate the fractal dimension is analyzed. The parton shower with permissible 1/3 parts of pseudorapidity space, which corresponds to a triadic Cantor set, was used as a test fractal. It was found that there is an optimal set of bins (a parameter of the BC method) that allows one to find the fractal dimension with maximal accuracy. The optimal set of bins is shown to depend on the fractal generation law. The P-adic coverage (PaC) method is proposed and used in the fractal analysis. This method makes it possible to determine the fractal dimension of a shower as accurately as possible, the number of fractal levels and partons at each branching point during the parton shower evolution, the type of cascade (either random or regular), and its structure. It is shown to be applicable to an analysis of the regular and random N-ary cascades with permissible 1/ k parts of the space studied.
NASA Astrophysics Data System (ADS)
Posnansky, Oleg; Guo, Jing; Hirsch, Sebastian; Papazoglou, Sebastian; Braun, Jürgen; Sack, Ingolf
2012-06-01
Recent advances in dynamic elastography and biorheology have revealed that the complex shear modulus, G*, of various biological soft tissues obeys a frequency-dependent powerlaw. This viscoelastic powerlaw behavior implies that mechanical properties are communicated in tissue across the continuum of scales from microscopic to macroscopic. For deriving constitutive constants from the dispersion of G* in a biological tissue, a hierarchical fractal model is introduced that accounts for multiscale networks. Effective-media powerlaw constants are derived by a constitutive law based on cross-linked viscoelastic clusters embedded in a rigid environment. The spatial variation of G* is considered at each level of hierarchy by an iterative coarse-graining procedure. The establishment of cross-links in this model network is associated with an increasing fractal dimension and an increasing viscoelastic powerlaw exponent. This fundamental relationship between shear modulus dynamics and fractal dimension of the mechanical network in tissue is experimentally reproduced in phantoms by applying shear oscillatory rheometry to layers of tangled paper strips embedded in agarose gel. Both model and experiments demonstrate the sensitivity of G* to the density of the mechanical network in tissue, corroborating disease-related alterations of the viscoelastic powerlaw exponent in human parenchyma demonstrated by in vivo elastography.
2014-01-01
Background The evaluation of intestinal trophism, mainly the mucosal layer, is an important issue in various conditions associated with injury, atrophy, recovery, and healing of the gut. The aim of the present study was to evaluate the kinetics of the proliferation and apoptosis of enterocytes by immunohistochemistry and to assess the complexity of intestinal mucosa by fractal dimension (FD) analysis in Solea solea fed different experimental diets. Results Histomorphological evaluation of all intestinal segments did not show signs of degeneration or inflammation. Cell proliferation index and FD were significantly reduced with a diet high in mussel meal (MM; p = 0.0034 and p = 0.01063, respectively), while apoptotic index did not show any significant difference for the same comparison (p = 0.3859). Linear regression analysis between apoptotic index (independent variable) and FD (dependent variable) showed a statistically significant inverse relationship (p = 0.002528). Linear regression analysis between cell proliferation index (independent variable) and FD (dependent variable) did not show any significant correlation (p = 0.131582). Conclusions The results demonstrated that diets containing increasing levels of mussel meal in substitution of fishmeal did not incite a hyperplastic response of the intestinal mucosa. The mussel meal, which is derived from molluscs, could mimic the characteristics of the sole’s natural prey, being readily digestible, even without increasing the absorptive surface of intestinal mucosa. Interestingly, from this study emerged that FD could be used as a numeric indicator complementary to in situ quantification methods to measure intestinal trophism, in conjunction with functional parameters. PMID:24997003
Ali, Zulfiqar; Elamvazuthi, Irraivan; Alsulaiman, Mansour; Muhammad, Ghulam
2016-01-01
Voice disorders are associated with irregular vibrations of vocal folds. Based on the source filter theory of speech production, these irregular vibrations can be detected in a non-invasive way by analyzing the speech signal. In this paper we present a multiband approach for the detection of voice disorders given that the voice source generally interacts with the vocal tract in a non-linear way. In normal phonation, and assuming sustained phonation of a vowel, the lower frequencies of speech are heavily source dependent due to the low frequency glottal formant, while the higher frequencies are less dependent on the source signal. During abnormal phonation, this is still a valid, but turbulent noise of source, because of the irregular vibration, affects also higher frequencies. Motivated by such a model, we suggest a multiband approach based on a three-level discrete wavelet transformation (DWT) and in each band the fractal dimension (FD) of the estimated power spectrum is estimated. The experiments suggest that frequency band 1-1562 Hz, lower frequencies after level 3, exhibits a significant difference in the spectrum of a normal and pathological subject. With this band, a detection rate of 91.28 % is obtained with one feature, and the obtained result is higher than all other frequency bands. Moreover, an accuracy of 92.45 % and an area under receiver operating characteristic curve (AUC) of 95.06 % is acquired when the FD of all levels is fused. Likewise, when the FD of all levels is combined with 22 Multi-Dimensional Voice Program (MDVP) parameters, an improvement of 2.26 % in accuracy and 1.45 % in AUC is observed. PMID:26531753
Kim, Songkil; Lee, Kwang-Sung; Zachariah, Michael R; Lee, Donggeun
2010-04-15
It has been a big challenge to explore a direct relation of experimental parameters such as pH, electrolyte concentration, particle size, and temperature with the final structures of aggregates, because Monte Carlo simulations have been performed on the basis of arbitrarily chosen sticking probability. We attempted to incorporate colloidal theory to Monte Carlo simulations for two model systems of CuO- and SiO(2)-water systems, so as to resolve this difficulty. Conducting three-dimensional off-lattice MC simulations at various pHs for both systems, we investigated effects of pH on fractal structures of aggregates, encompassing the whole aggregation regime from diffusion-limited cluster-cluster aggregation to reaction-limited cluster-cluster aggregation. Moreover, developing a functional analysis, we found an explicit correlation between experimental parameters, sticking probability, and the fractal dimension of aggregates for both systems. PMID:20132942
[Dimensional fractal of post-paddy wheat root architecture].
Chen, Xin-xin; Ding, Qi-shuo; Li, Yi-nian; Xue, Jin-lin; Lu, Ming-zhou; Qiu, Wei
2015-06-01
To evaluate whether crop rooting system was directionally dependent, a field digitizer was used to measure post-paddy wheat root architectures. The acquired data was transferred to Pro-E, in which virtual root architecture was reconstructed and projected to a series of planes each separated in 10° apart. Fractal dimension and fractal abundance of root projections in all the 18 planes were calculated, revealing a distinctive architectural distribution of wheat root in each direction. This strongly proved that post-paddy wheat root architecture was directionally dependent. From seedling to turning green stage, fractal dimension of the 18 projections fluctuated significantly, illustrating a dynamical root developing process in the period. At the jointing stage, however, fractal indices of wheat root architecture resumed its regularity in each dimension. This wheat root architecture recovered its dimensional distinctness. The proposed method was applicable for precision modeling field state root distribution in soil. PMID:26572023
Johansen, Daniel; Trewhella, Jill; Goldenberg, David P
2011-01-01
Small-angle X-ray scattering (SAXS) was used to characterize the bacteriophage λ N protein, a 107 residue intrinsically disordered protein (IDP) that functions as a transcriptional antitermination factor. The SAXS data were used to estimate both the average radius of gyration and the fractal dimension, a measure of the protein's internal scaling properties, under a variety of solution conditions. In the absence of denaturants, the radius of gyration was 38 ± 3.5 Å and the fractal dimension was 1.76 ± 0.05, slightly larger than the value predicted for a well-solvated polymer with excluded volume (1.7). Neither the radius of gyration nor the fractal dimension changed significantly on the addition of urea, further indicating that the protein is extensively unfolded and well solvated in the absence of denaturant. The addition of NaCl or D2O was found to promote aggregation, but did not appear to affect the properties of the monomeric form. The experimental SAXS profiles were also compared with those predicted by a computational model for a random-coil polypeptide, with an adjustable solvation energy term. The experimental data were well fit to the model with the solvation energy close to zero. These results indicate that the λ N protein is among the more expanded members of the broad class of IDPs, most likely because of its high content of charged residues and a large net charge (+15 at neutral pH). The expanded nature of the conformational ensemble may play a role in facilitating the interactions of the protein with other components of the dynamic transcriptional complex. PMID:21936008
NASA Technical Reports Server (NTRS)
Wiscombe, W.
1999-01-01
The purpose of this paper is discuss the concept of fractal dimension; multifractal statistics as an extension of this; the use of simple multifractal statistics (power spectrum, structure function) to characterize cloud liquid water data; and to understand the use of multifractal cloud liquid water models based on real data as input to Monte Carlo radiation models of shortwave radiation transfer in 3D clouds, and the consequences of this in two areas: the design of aircraft field programs to measure cloud absorptance; and the explanation of the famous "Landsat scale break" in measured radiance.
NASA Astrophysics Data System (ADS)
Wurm, Gerhard; Schnaiter, Martin
Individual cosmic dust particles come in very complex shapes but there are a variety of environments where dust particles are aggregates of smaller grains. These aggregates often can be characterized by a fractal particle structure. Within this fractal framework, morphologically averaged particle properties depend only on a few parameters. Despite the complexity of the individual particles, fractal dimension, aggregate size, and building-block size are sufficient parameters for many applications. This paper reviews the process of aggregation to generate fractal aggregates with focus on some general features of those aggregates (mass to surface ratios, aspect ratios). Some implications for aerodynamical and optical properties (gas- grain coupling times, polarization, extinction) are discussed.
NASA Astrophysics Data System (ADS)
Davarpanah, A.; Babaie, H. A.
2012-12-01
The interaction of the thermally induced stress field of the Yellowstone hotspot (YHS) with existing Basin and Range (BR) fault blocks, over the past 17 m.y., has produced a new, spatially and temporally variable system of normal faults around the Snake River Plain (SRP) in Idaho and Wyoming-Montana area. Data about the trace of these new cross faults (CF) and older BR normal faults were acquired from a combination of satellite imageries, DEM, and USGS geological maps and databases at scales of 1:24,000, 1:100,000, 1:250,000, 1:1000, 000, and 1:2,500, 000, and classified based on their azimuth in ArcGIS 10. The box-counting fractal dimension (Db) of the BR fault traces, determined applying the Benoit software, and the anisotropy intensity (ellipticity) of the fractal dimensions, measured with the modified Cantor dust method applying the AMOCADO software, were measured in two large spatial domains (I and II). The Db and anisotropy of the cross faults were studied in five temporal domains (T1-T5) classified based on the geologic age of successive eruptive centers (12 Ma to recent) of the YHS along the eastern SRP. The fractal anisotropy of the CF system in each temporal domain was also spatially determined in the southern part (domain S1), central part (domain S2), and northern part (domain S3) of the SRP. Line (fault trace) density maps for the BR and CF polylines reveal a higher linear density (trace length per unit area) for the BR traces in the spatial domain I, and a higher linear density of the CF traces around the present Yellowstone National Park (S1T5) where most of the seismically active faults are located. Our spatio-temporal analysis reveals that the fractal dimension of the BR system in domain I (Db=1.423) is greater than that in domain II (Db=1.307). It also shows that the anisotropy of the fractal dimension in domain I is less eccentric (axial ratio: 1.242) than that in domain II (1.355), probably reflecting the greater variation in the trend of the BR system in domain I. The CF system in the S1T5 domain has the highest fractal dimension (Db=1.37) and the lowest anisotropy eccentricity (1.23) among the five temporal domains. These values positively correlate with the observed maxima on the fault trace density maps. The major axis of the anisotropy ellipses is consistently perpendicular to the average trend of the normal fault system in each domain, and therefore approximates the orientation of extension for normal faulting in each domain. This fact gives a NE-SW and NW-SE extension direction for the BR system in domains I and II, respectively. The observed NE-SW orientation of the major axes of the anisotropy ellipses in the youngest T4 and T5 temporal domains, oriented perpendicular to the mean trend of the normal faults in the these domains, suggests extension along the NE-SW direction for cross faulting in these areas. The spatial trajectories (form lines) of the minor axes of the anisotropy ellipses, and the mean trend of fault traces in the T4 and T5 temporal domains, define a large parabolic pattern about the axis of the eastern SRP, with its apex at the Yellowstone plateau.
Extended fractal analysis method and its application for linear rivers
NASA Astrophysics Data System (ADS)
Wang, Liqin; Long, Yi; Cui, Shilin
2008-10-01
Extended fractal analysis method can analyze the fractal character (i.e. self-similarity) objectively, especially the difference and change of the shape and the structure in different observation scale intervals. As one of the common fractal objects, river on the map can be surveyed its length and quantified the complexity of its shape and structure as well as its partial details with Extended Fractal Dimension Analysis method (abbreviated as EFDA). Compared to the traditional method, EFDA has unparalleled advantages. Considering the extended fractal character with scaling variance, and based on its simulating function adopting the Inverse Logistic Model, the paper gained the extended fractal function for quantifying the length of the river depending on the different observing scales. Furthermore, based on the mathematical derivation of its simulating function and fractal analysis, the paper obtained the relevant parameter for establishing Meta Fractal Dimension (abbreviated as MFD) Model to quantify the local complexity of the river on the map. Several experiments based on the China's seven major rivers done indicate that this method is easy to operate and has a relatively high calculation precision and a logical result of spatial analysis.
Persistence intervals of fractals
NASA Astrophysics Data System (ADS)
Máté, Gabriell; Heermann, Dieter W.
2014-07-01
Objects and structures presenting fractal like behavior are abundant in the world surrounding us. Fractal theory provides a great deal of tools for the analysis of the scaling properties of these objects. We would like to contribute to the field by analyzing and applying a particular case of the theory behind the P.H. dimension, a concept introduced by MacPherson and Schweinhart, to seek an intuitive explanation for the relation of this dimension and the fractality of certain objects. The approach is based on recently elaborated computational topology methods and it proves to be very useful for investigating scaling hidden in dimensions lower than the “native” dimension in which the investigated object is embedded. We demonstrate the applicability of the method with two examples: the Sierpinski gasket-a traditional fractal-and a two dimensional object composed of short segments arranged according to a circular structure.
Fractal characteristics of ozonometric network
NASA Technical Reports Server (NTRS)
Gruzdev, Alexander N.
1994-01-01
The fractal (correlation) dimensions are calculated which characterize the distribution of stations in the ground-based total ozone measuring network and the distribution of nodes in a latitude-longitude grid. The dimension of the ground-based ozonometric network equals 1.67 +/- 0.1 with an appropriate scaling in the 60 to 400 km range. For the latitude-longitude grid two scaling regimes are revealed. One regime, with the dimension somewhat greater than one, is peculiar to smaller scales and limited from a larger scale by the latitudinal resolution of the grid. Another scaling regime, with the dimension equal 1.84, ranges up to 15,000 km scale. The fact that the dimension of a measuring network is less than two possesses problems in observing sparse phenomena. This has to have important consequences for ozone statistics.
Waliszewski, Przemyslaw
2016-01-01
Background: Tumor grading, PSA concentration, and stage determine a risk of prostate cancer patients with accuracy of about 70%. An approach based on the fractal geometrical model was proposed to eliminate subjectivity from the evaluation of tumor aggressiveness and to improve the prediction. This study was undertaken to validate classes of equivalence for the spatial distribution of cancer cell nuclei in a larger, independent set of prostate carcinomas. Methods: The global fractal capacity D0, information D1 and correlation D2 dimension, the local fractal dimension (LFD) and the local connected fractal dimension (LCFD), Shannon entropy H and lacunarity λ were measured using computer algorithms in digitalized images of both the reference set (n = 60) and the test set (n = 208) of prostate carcinomas. Results: Prostate carcinomas were re-stratified into seven classes of equivalence. The cut-off D0-values 1.5450, 1.5820, 1.6270, 1.6490, 1.6980, 1.7640 defined the classes from C1 to C7, respectively. The other measures but the D1 failed to define the same classes of equivalence. The pairs (D0, LFD), (D0, H), (D0, λ), (D1, LFD), (D1, H), (D1, λ) characterized the spatial distribution of cancer cell nuclei in each class. The co-application of those measures enabled the subordination of prostate carcinomas to one out of three clusters associated with different tumor aggressiveness. For D0 < 1.5820, LFD < 1.3, LCFD > 1.5, H < 0.7, and λ > 0.8, the class C1 or C2 contains low complexity low aggressive carcinomas exclusively. For D0 > 1.6980, LFD > 1.7644, LCFD > 1.7051, H > 0.9, and λ < 0.7, the class C6 or C7 contains high complexity high aggressive carcinomas. Conclusions: The cut-off D0-values defining the classes of equivalence were validated in this study. The cluster analysis suggested that the number of the subjective Gleason grades and the number of the objective classes of equivalence could be decreased from seven to three without a loss of clinically relevant information. Two novel quantitative criteria based on the complexity and the diversity measures enabled the identification of low or high aggressive prostate carcinomas and should be verified in the future multicenter, randomized studies. PMID:26903883
Chappard, Daniel; Stancu, Izabela-Cristina
2015-04-01
Porosity is an important factor to consider in a large variety of materials. Porosity can be visualized in bone or 3D synthetic biomaterials by microcomputed tomography (microCT). Blocks of porous poly(2-hydroxyethyl methacrylate) were prepared with polystyrene beads of different diameter (500, 850, 1160 and 1560 ?m) and analysed by microCT. On each 2D binarized microCT section, pixels of the pores which belong to the same image column received the same pseudo-colour according to a look up table. The same colour was applied on the same column of a frontal plane image which was constructed line by line from all images of the microCT stack. The fractal dimension Df of the frontal plane image was measured as well as the descriptors of the 3D models (porosity, 3D fractal dimension D3D, thickness, density and separation of material walls. Porosity, thickness Df and D3D increased with the size of the porogen beads. A linear correlation was observed between Df and D3D. This method provides quantitative and qualitative analysis of porosity on a single frontal plane image of a porous object. PMID:25556606
NASA Astrophysics Data System (ADS)
Yang, Yang; Wang, Ya Ping; Li, Chunyan; Gao, Shu; Shi, Benwei; Zhou, Liang; Wang, Dandan; Li, Gaocong; Dai, Chen
2016-01-01
Interactions between turbulence, suspended sediment concentration (SSC), settling velocity, effective density, fractal dimension, and floc size were studied on the tide-dominated, muddy coastal shelf of the southwestern Yellow Sea, China. The measurements were carried out in July 2013 at two sites located in water depths of 21.2 and 22.1 m. Negative correlations were observed between shear rate, SSC, effective density, and mean floc size, which supports the results of previous numerical, experimental, and field studies. A significant positive correlation was observed between near-bed SSC and shear rate, an indication that SSC variations are controlled by turbulence and re-suspension. In addition, significant linear relationships were found between settling velocity and other parameters (floc size, turbulence, SSC, effective density, and fractal dimension) at the two sites, indicating that the controlling factors on settling velocity are spatially variable. Principal component analysis was applied to determine the relative importance of turbulence, flocculation ability, and SSC as controls on floc size in situ. The relative contributions of turbulence, flocculation ability, and SSC to floc size (at both sites) were ~33.0%, 30.3%, and 29.7%, respectively, this being a new field-based quantitative analysis of the controls on floc size. The findings demonstrate that, in nature, flocculation ability affects floc size to the same degree as turbulence and SSC. Therefore, predictions of floc size in coastal marine environments require constraints not only on turbulence and SSC, but also on flocculation ability.
NASA Astrophysics Data System (ADS)
Yang, Yang; Wang, Ya Ping; Li, Chunyan; Gao, Shu; Shi, Benwei; Zhou, Liang; Wang, Dandan; Li, Gaocong; Dai, Chen
2016-04-01
Interactions between turbulence, suspended sediment concentration (SSC), settling velocity, effective density, fractal dimension, and floc size were studied on the tide-dominated, muddy coastal shelf of the southwestern Yellow Sea, China. The measurements were carried out in July 2013 at two sites located in water depths of 21.2 and 22.1 m. Negative correlations were observed between shear rate, SSC, effective density, and mean floc size, which supports the results of previous numerical, experimental, and field studies. A significant positive correlation was observed between near-bed SSC and shear rate, an indication that SSC variations are controlled by turbulence and re-suspension. In addition, significant linear relationships were found between settling velocity and other parameters (floc size, turbulence, SSC, effective density, and fractal dimension) at the two sites, indicating that the controlling factors on settling velocity are spatially variable. Principal component analysis was applied to determine the relative importance of turbulence, flocculation ability, and SSC as controls on floc size in situ. The relative contributions of turbulence, flocculation ability, and SSC to floc size (at both sites) were ~33.0%, 30.3%, and 29.7%, respectively, this being a new field-based quantitative analysis of the controls on floc size. The findings demonstrate that, in nature, flocculation ability affects floc size to the same degree as turbulence and SSC. Therefore, predictions of floc size in coastal marine environments require constraints not only on turbulence and SSC, but also on flocculation ability.
Gravitation theory in a fractal space-time
Agop, M.; Gottlieb, I.
2006-05-15
Assimilating the physical space-time with a fractal, a general theory is built. For a fractal dimension D=2, the virtual geodesics of this space-time implies a generalized Schroedinger type equation. Subsequently, a geometric formulation of the gravitation theory on a fractal space-time is given. Then, a connection is introduced on a tangent bundle, the connection coefficients, the Riemann curvature tensor and the Einstein field equation are calculated. It results, by means of a dilation operator, the equivalence of this model with quantum Einstein gravity.
Cluster-Cluster Aggregation Calculations of Fractal Haze Particles: Titan and the Early Earth
NASA Astrophysics Data System (ADS)
Terrell-Martinez, Bernice; Boness, David
2010-10-01
The atmosphere of the Archean Earth (3.8 to 2.5 billion years ago) is thought to have been dominated by a thick hydrocarbon haze similar to that of Titan's current atmosphere. To understand radiative transport in the atmospheres of the early Earth and of Titan, it is necessary to compute light scattering in UV, visible, and IR wavelength ranges for realistic fractal aggregate hydrocarbon aerosol particles. We report preliminary work on MATLAB, True BASIC, and Fortran programs to simulate the growth of fractal aggregate aerosols through diffusion limited aggregation (DLA) and cluster-cluster aggregation (CCA) physical processes. The results of these computations are being used with a T-Matrix light scattering program to test recently published, widely-reported conclusions about the early Earth and the faint young Sun paradox [E. T. Wolf and O. B. Toon, Science 328, 1266 (2010)]. This modeling is also relevant to understanding atmospheric carbonaceous soot aerosol anthropogenic and natural effects on climate change of Earth today.
Fractal accretion of cosmic grains.
NASA Astrophysics Data System (ADS)
Tullet, P.
1999-05-01
This study is concerned with the formations that might arise in space if cosmic dust grains with a range of sizes collide and coalesce. Simulations with circular particles in two dimensions show that the resulting structures are fractal objects with a fractal dimension of 1.37. In the low gravity of space these fractal growths might snowball and form large, low-density growths with a fluffy texture.
Dimension of chaotic attractors
Farmer, J.D.; Ott, E.; Yorke, J.A.
1982-09-01
Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on probabilistic properties (that is, they depend on the frequency with which a typical trajectory visits different regions of the attractor). Both our example and the previous work that we review support the conclusion that all of the probabilistic dimensions take on the same value, which we call the dimension of the natural measure, and all of the metric dimensions take on a common value, which we call the fractal dimension. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.
Mossotti, Victor G.; Eldeeb, A. Raouf; Oscarson, Robert
1998-01-01
MORPH-I is a set of C-language computer programs for the IBM PC and compatible minicomputers. The programs in MORPH-I are used for the fractal analysis of scanning electron microscope and electron microprobe images of pore profiles exposed in cross-section. The program isolates and traces the cross-sectional profiles of exposed pores and computes the Richardson fractal dimension for each pore. Other programs in the set provide for image calibration, display, and statistical analysis of the computed dimensions for highly complex porous materials. Requirements: IBM PC or compatible; minimum 640 K RAM; mathcoprocessor; SVGA graphics board providing mode 103 display.
Fractal dynamics in chaotic quantum transport.
Kotimäki, V; Räsänen, E; Hennig, H; Heller, E J
2013-08-01
Despite several experiments on chaotic quantum transport in two-dimensional systems such as semiconductor quantum dots, corresponding quantum simulations within a real-space model have been out of reach so far. Here we carry out quantum transport calculations in real space and real time for a two-dimensional stadium cavity that shows chaotic dynamics. By applying a large set of magnetic fields we obtain a complete picture of magnetoconductance that indicates fractal scaling. In the calculations of the fractality we use detrended fluctuation analysis-a widely used method in time-series analysis-and show its usefulness in the interpretation of the conductance curves. Comparison with a standard method to extract the fractal dimension leads to consistent results that in turn qualitatively agree with the previous experimental data. PMID:24032907
Fractal analysis of time varying data
Vo-Dinh, Tuan; Sadana, Ajit
2002-01-01
Characteristics of time varying data, such as an electrical signal, are analyzed by converting the data from a temporal domain into a spatial domain pattern. Fractal analysis is performed on the spatial domain pattern, thereby producing a fractal dimension D.sub.F. The fractal dimension indicates the regularity of the time varying data.
Fractal characterization of fracture surfaces in concrete
Saouma, V.E.; Barton, C.C.; Gamaleldin, N.A.
1990-01-01
Fractal geometry is used to characterize the roughness of cracked concrete surfaces through a specially built profilometer, and the fractal dimension is subsequently correlated to the fracture toughness and direction of crack propagation. Preliminary results indicate that the fracture surface is indeed fractal over two orders of magnitudes with a dimension of approximately 1.20. ?? 1990.
NASA Astrophysics Data System (ADS)
Dewey, T. G.
1990-06-01
Quantum time-evolution propagators are derived for a particle on a fractal lattice. It is shown that the path of such a particle represents a self-affine fractal. A relationship is obtained between the fractal dimension of the path (as determined by three different operational definitions) and the fractal dimension of the lattice. From these relationships it is seen that a quantum-mechanical particle in a one-dimensional Euclidean space also has a path which is a self-affine fractal. A strong analogy exists between these observations and the difference between Brownian motion and fractional Brownian motion. The uncertainty principle for a particle on a fractal lattice is derived and the phenomena of persistence and antipersistence is related to the uncertainty in momentum. The fractal quantum propagators are then used to develop a quantum theory of fractal rate constants. The theory of Miller and co-workers [J. Chem. Phys. 79, 4889 (1983); 90, 904 (1989)] is used to derive expressions for a bimolecular rate constant. Using the position-flux cross correlation function, scaling laws for the time dependence of the ``rate constant'' is obtained. These results are discussed in relation to experimental and theoretical results for time-dependent rate constants in percolation clusters.
NASA Astrophysics Data System (ADS)
Maria, Anton; Carey, Steven
2007-03-01
The morphology of volcanic particles can yield insight into magma fragmentation, transport processes, and style of eruption. However, the complexity and variability of volcanic particle shapes make quantitative characterization difficult. The technique applied in this study is based on fractal geometry, which has been successfully used to characterize a wide variety of particles and shapes. Here, fractal data is produced by dilation of the 2-D particle boundary to produce a full spectrum of fractal dimensions over a range of scales for each particle. Multiple fractal dimensions, which can be described as a fractal spectrum curve, are calculated by taking the first derivative of data points on a standard Richardson plot. Use of multiple fractal dimensions results in more effective discrimination than expressions of shape based on one or two fractal dimensions. Quantitative comparisons are carried out using multivariate statistical techniques such as cluster and principal components analysis. Applications to samples from well-documented eruptions (e.g. Mt. St. Helens 1980, Tambora 1815, Surtsey 1963-64) indicate that the fractal spectrum technique provides a useful means of characterizing volcanic particles and can be helpful for identifying the products of specific fragmentation processes (volatile exsolution, phreatomagmatic, quench granulation) and modes of volcanic transport/deposition (tephra fall, pyroclastic flow, blast/surge).
Rheological and fractal hydrodynamics of aerobic granules.
Tijani, H I; Abdullah, N; Yuzir, A; Ujang, Zaini
2015-06-01
The structural and hydrodynamic features for granules were characterized using settling experiments, predefined mathematical simulations and ImageJ-particle analyses. This study describes the rheological characterization of these biologically immobilized aggregates under non-Newtonian flows. The second order dimensional analysis defined as D2=1.795 for native clusters and D2=1.099 for dewatered clusters and a characteristic three-dimensional fractal dimension of 2.46 depicts that these relatively porous and differentially permeable fractals had a structural configuration in close proximity with that described for a compact sphere formed via cluster-cluster aggregation. The three-dimensional fractal dimension calculated via settling-fractal correlation, U∝l(D) to characterize immobilized granules validates the quantitative measurements used for describing its structural integrity and aggregate complexity. These results suggest that scaling relationships based on fractal geometry are vital for quantifying the effects of different laminar conditions on the aggregates' morphology and characteristics such as density, porosity, and projected surface area. PMID:25836036
Fractal analysis: A new remote sensing tool for lava flows
NASA Technical Reports Server (NTRS)
Bruno, B. C.; Taylor, G. J.; Rowland, S. K.; Lucey, P. G.; Self, S.
1992-01-01
Many important quantitative parameters have been developed that relate to the rheology and eruption and emplacement mechanics of lavas. This research centers on developing additional, unique parameters, namely the fractal properties of lava flows, to add to this matrix of properties. There are several methods of calculating the fractal dimension of a lava flow margin. We use the 'structured walk' or 'divider' method. In this method, we measure the length of a given lava flow margin by walking rods of different lengths along the margin. Since smaller rod lengths transverse more smaller-scaled features in the flow margin, the apparent length of the flow outline will increase as the length of the measuring rod decreases. By plotting the apparent length of the flow outline as a function of the length of the measuring rod on a log-log plot, fractal behavior can be determined. A linear trend on a log-log plot indicates that the data are fractal. The fractal dimension can then be calculated from the slope of the linear least squares fit line to the data. We use this 'structured walk' method to calculate the fractal dimension of many lava flows using a wide range of rod lengths, from 1/8 to 16 meters, in field studies of the Hawaiian islands. We also use this method to calculate fractal dimensions from aerial photographs of lava flows, using lengths ranging from 20 meters to over 2 kilometers. Finally, we applied this method to orbital images of extraterrestrial lava flows on Venus, Mars, and the Moon, using rod lengths up to 60 kilometers.
Fractal structures and processes
Bassingthwaighte, J.B.; Beard, D.A.; Percival, D.B.; Raymond, G.M.
1996-06-01
Fractals and chaos are closely related. Many chaotic systems have fractal features. Fractals are self-similar or self-affine structures, which means that they look much of the same when magnified or reduced in scale over a reasonably large range of scales, at least two orders of magnitude and preferably more (Mandelbrot, 1983). The methods for estimating their fractal dimensions or their Hurst coefficients, which summarize the scaling relationships and their correlation structures, are going through a rapid evolutionary phase. Fractal measures can be regarded as providing a useful statistical measure of correlated random processes. They also provide a basis for analyzing recursive processes in biology such as the growth of arborizing networks in the circulatory system, airways, or glandular ducts. {copyright} {ital 1996 American Institute of Physics.}
Naik, Ganesh R; Arjunan, Sridhar; Kumar, Dinesh
2011-06-01
The surface electromyography (sEMG) signal separation and decphompositions has always been an interesting research topic in the field of rehabilitation and medical research. Subtle myoelectric control is an advanced technique concerned with the detection, processing, classification, and application of myoelectric signals to control human-assisting robots or rehabilitation devices. This paper reviews recent research and development in independent component analysis and Fractal dimensional analysis for sEMG pattern recognition, and presents state-of-the-art achievements in terms of their type, structure, and potential application. Directions for future research are also briefly outlined. PMID:21416388
Fractal model of anomalous diffusion.
Gmachowski, Lech
2015-12-01
An equation of motion is derived from fractal analysis of the Brownian particle trajectory in which the asymptotic fractal dimension of the trajectory has a required value. The formula makes it possible to calculate the time dependence of the mean square displacement for both short and long periods when the molecule diffuses anomalously. The anomalous diffusion which occurs after long periods is characterized by two variables, the transport coefficient and the anomalous diffusion exponent. An explicit formula is derived for the transport coefficient, which is related to the diffusion constant, as dependent on the Brownian step time, and the anomalous diffusion exponent. The model makes it possible to deduce anomalous diffusion properties from experimental data obtained even for short time periods and to estimate the transport coefficient in systems for which the diffusion behavior has been investigated. The results were confirmed for both sub and super-diffusion. PMID:26129728
Jartti, Tuomas T; Kuusela, Tom A; Kaila, Timo J; Tahvanainen, Kari U O; Välimäki, Ilkka A T
1998-01-01
Aims To study the dose-response effects of intravenous terbutaline on the cardiovascular and respiratory autonomic nervous regulation. Methods The study followed a randomized, placebo-controlled crossover design in six healthy adult volunteers. The terbutaline dose ranged from 10 to 30 μg min−1. We continuously measured electrocardiogram, finger systolic arterial pressure (SAP) and flow-volume spirometry in supine and upright positions at baseline and during 3 h drug infusion. The periodic variability components of R-R intervals (time between successive heart beats) and SAP in relation to respiration were assessed using spectral analysis techniques. The regularity of the time series was assessed by approximate entropy (ApEn) and the convolutedness by fractal dimension (FD). Results Terbutaline dose-dependently decreased total variability of R-R intervals, low frequency (LF) variability of R-R intervals (10 s waves), high frequency (HF) variability of R-R intervals (respiratory variability), total variability of SAP, HF variability of SAP, baroreflex sensitivity, plasma potassium concentration, approximate entropy of R-R interval and of SAP as well as fractal dimension of R-R interval. Terbutaline dose-dependently increased heart rate, LF/HF ratios of R-R intervals and of SAP, LF variability of SAP, minute ventilation and plasma terbutaline concentration. Conclusions Terbutaline infusion decreases parasympathetic cardiovascular reactivity, baroreflex sensitivity, dimensionality of heart rate and plasma potassium concentration; it increases sympathetic dominance in cardiovascular autonomic balance, minute ventilation, and the regularity of heart rate and blood pressure time series. PMID:9517372
Testing Fractal Methods on Observed and Simulated Solar Magnetograms
NASA Technical Reports Server (NTRS)
Adams, M.; Falconer, D. A.; Lee, J. K.; Jones, C.
2003-01-01
The term "magnetic complexity" has not been sufficiently quantified. To accomplish this, we must understand the relationship between the observed magnetic field of solar active regions and fractal dimension measurements. Using data from the Marshall Space Flight Center's vector magnetograph ranging from December 1991 to July 2001, we compare the results of several methods of calculating a fractal dimension, e.g., Hurst coefficient, the Higuchi method, power spectrum, and 2-D Wavelet Packet Analysis. In addition, we apply these methods to synthetic data, beginning with representations of very simple dipole regions, ending with regions that are magnetically complex.
Routes to fractality and entropy in Liesegang systems.
Kalash, Leen; Sultan, Rabih
2014-06-01
Liesegang bands are formed when solutions of co-precipitate ions interdiffuse in a 1D gel matrix. In a recent study [R. F. Sultan, Acta. Mech. Sin. 27, 119 (2011)], Liesegang patterns have been characterized as fractal structures. In addition to experimentally obtained patterns, geometric Liesegang patterns were constructed in conformity with the well-known empirical laws. Both mathematical fractal dimensions and box count dimensions for images of PbF2 and PbI2 Liesegang patterns have been calculated. Liesegang patterns can also be described by the entropy state function, and categorized as more or less ordered structures. We revisit the relation between entropy and fractal dimension, and apply it to simulated geometrical Liesegang patterns. We have resort to three different routes for the estimation of the entropy of a Liesegang pattern. The HarFA software enabled the calculation of the Hausdorff dimension and the topological entropy, then the information dimension and the Shannon entropy. In a third pathway, analytical calculations were carried out by estimating the probability of occurrence of a fractal element or coverage. The product of Shannon entropy and Boltzmann constant yields the thermodynamic entropy. The values for PbF2 and PbI2 Liesegang patterns attained the order of magnitude of the reported Third Law entropies, but yet remained lower, in conformity with the more ordered Liesegang structures. PMID:24985435
Routes to fractality and entropy in Liesegang systems
Kalash, Leen; Sultan, Rabih
2014-06-01
Liesegang bands are formed when solutions of co-precipitate ions interdiffuse in a 1D gel matrix. In a recent study [R. F. Sultan, Acta. Mech. Sin. 27, 119 (2011)], Liesegang patterns have been characterized as fractal structures. In addition to experimentally obtained patterns, geometric Liesegang patterns were constructed in conformity with the well-known empirical laws. Both mathematical fractal dimensions and box count dimensions for images of PbF{sub 2} and PbI{sub 2} Liesegang patterns have been calculated. Liesegang patterns can also be described by the entropy state function, and categorized as more or less ordered structures. We revisit the relation between entropy and fractal dimension, and apply it to simulated geometrical Liesegang patterns. We have resort to three different routes for the estimation of the entropy of a Liesegang pattern. The HarFA software enabled the calculation of the Hausdorff dimension and the topological entropy, then the information dimension and the Shannon entropy. In a third pathway, analytical calculations were carried out by estimating the probability of occurrence of a fractal element or coverage. The product of Shannon entropy and Boltzmann constant yields the thermodynamic entropy. The values for PbF{sub 2} and PbI{sub 2} Liesegang patterns attained the order of magnitude of the reported Third Law entropies, but yet remained lower, in conformity with the more ordered Liesegang structures.
Radlinski, A.P.; Radlinska, E.Z.; Agamalian, M.; Wignall, G.D.; Lindner, P.; Randl, O.G.
1999-04-01
The analysis of small- and ultra-small-angle neutron scattering data for sedimentary rocks shows that the pore-rock fabric interface is a surface fractal (D{sub s}=2.82) over 3 orders of magnitude of the length scale and 10 orders of magnitude in intensity. The fractal dimension and scatterer size obtained from scanning electron microscopy image processing are consistent with neutron scattering data. {copyright} {ital 1999} {ital The American Physical Society}
Roughness Perception of Haptically Displayed Fractal Surfaces
NASA Technical Reports Server (NTRS)
Costa, Michael A.; Cutkosky, Mark R.; Lau, Sonie (Technical Monitor)
2000-01-01
Surface profiles were generated by a fractal algorithm and haptically rendered on a force feedback joystick, Subjects were asked to use the joystick to explore pairs of surfaces and report to the experimenter which of the surfaces they felt was rougher. Surfaces were characterized by their root mean square (RMS) amplitude and their fractal dimension. The most important factor affecting the perceived roughness of the fractal surfaces was the RMS amplitude of the surface. When comparing surfaces of fractal dimension 1.2-1.35 it was found that the fractal dimension was negatively correlated with perceived roughness.
Barton, C.C.; Troussov, G.L. )
1996-01-01
A new method based on fractal geometry has been developed and computerized for assessing the size, number, and total volume of undiscovered, conventionally recoverable hydrocarbon accumulations based on fitting a truncated fractal (power-law) distribution to a log-log plot of the cumulative size-frequency distribution of discovered accumulations in a play or other geologically or geographically defined region.
Barton, C.C.; Troussov, G.L.
1996-12-31
A new method based on fractal geometry has been developed and computerized for assessing the size, number, and total volume of undiscovered, conventionally recoverable hydrocarbon accumulations based on fitting a truncated fractal (power-law) distribution to a log-log plot of the cumulative size-frequency distribution of discovered accumulations in a play or other geologically or geographically defined region.
Juergens, H.; Peitgen, H.O.; Saupe, D. )
1990-08-01
The pathological structures conjured up by 19th-century mathematicians have, in recent years, taken the form of fractals, mathematical figures that have fractional dimension rather than the integral dimensions of familiar geometric figures (such as one-dimensional lines or two-dimensional planes). Fractals are much more than a mathematical curiosity. They offer an extremely compact method for describing objects and formations. Many structures have an underlying geometric regularity, known as scale invariance or self-similarity. If one examines these objects at different size scales, one repeatedly encounters the same fundamental elements. The repetitive pattern defines the fractional, or fractal, dimension of the structure. Fractal geometry seems to describe natural shapes and forms more gracefully and succinctly than does Euclidean geometry. Scale invariance has a noteworthy parallel in contemporary chaos theory, which reveals that many phenomena, even though they follow strict deterministic rules, are in principle unpredictable. Chaotic events, such as turbulence in the atmosphere or the beating of a human heart, show similar patterns of variation on different time scales, much as scale-invariant objects show similar structural patterns on different spatial scales. The correspondence between fractals and chaos is no accident. Rather it is a symptom of a deep-rooted relation: fractal geometry is the geometry of chaos.
Fractal processes in soil water retention
Tyler, S.W.; Wheatcraft, S.W. )
1990-05-01
The authors propose a physical conceptual model for soil texture and pore structure that is based on the concept of fractal geometry. The motivation for a fractal model of soil texture is that some particle size distributions in granular soils have already been shown to display self-similar scaling that is typical of fractal objects. Hence it is reasonable to expect that pore size distributions may also display fractal scaling properties. The paradigm that they used for the soil pore size distribution is the Sierpinski carpet, which is a fractal that contains self similar holes (or pores) over a wide range of scales. The authors evaluate the water retention properties of regular and random Sierpinski carpets and relate these properties directly to the Brooks and Corey (or Campbell) empirical water retention model. They relate the water retention curves directly to the fractal dimension of the Sierpinski carpet and show that the fractal dimension strongly controls the water retention properties of the Sierpinski carpet soil. Higher fractal dimensions are shown to mimic clay-type soils, with very slow dewatering characteristics and relatively low fractal dimensions are shown to mimic a sandy soil with relatively rapid dewatering characteristics. Their fractal model of soil water retention removes the empirical fitting parameters from the soil water retention models and provides paramters which are intrinsic to the nature of the fractal porous structure. The relative permeability functions of Burdine and Mualem are also shown to be fractal directly from fractal water retention results.
Fractal analysis of narwhal space use patterns.
Laidre, Kristin L; Heide-Jørgensen, Mads P; Logsdon, Miles L; Hobbs, Roderick C; Dietz, Rune; VanBlaricom, Glenn R
2004-01-01
Quantifying animal movement in response to a spatially and temporally heterogeneous environment is critical to understanding the structural and functional landscape influences on population viability. Generalities of landscape structure can easily be extended to the marine environment, as marine predators inhabit a patchy, dynamic system, which influences animal choice and behavior. An innovative use of the fractal measure of complexity, indexing the linearity of movement paths over replicate temporal scales, was applied to satellite tracking data collected from narwhals (Monodon monoceros) (n = 20) in West Greenland and the eastern Canadian high Arctic. Daily movements of individuals were obtained using polar orbiting satellites via the ARGOS data location and collection system. Geographic positions were filtered to obtain a daily good quality position for each whale. The length of total pathway was measured over seven different temporal length scales (step lengths), ranging from one day to one week, and a seasonal mean was calculated. Fractal dimension (D) was significantly different between seasons, highest during summer (D = 1.61, SE 0.04) and winter (D = 1.69, SE 0.06) when whales made convoluted movements in focal areas. Fractal dimension was lowest during fall (D = 1.34, SE 0.03) when whales were migrating south ahead of the forming sea ice. There were no significant effects of size category or sex on fractal dimension by season. The greater linearity of movement during the migration period suggests individuals do not intensively forage on patchy resources until they arrive at summer or winter sites. The highly convoluted movements observed during summer and winter suggest foraging or searching efforts in localized areas. Significant differences between the fractal dimensions on two separate wintering grounds in Baffin Bay suggest differential movement patterns in response to the dynamics of sea ice. PMID:16351924
Bies, Alexander; Taylor, Richard; Sereno, Margaret
2015-01-01
Edges are significant, ubiquitous features of natural scenes. Basic properties of visual stimuli such as edges should be controlled for in experiments and reported in the literature. Currently, no commonly reported image statistics describe natural scenes' edges. An edge's fractal dimension (Df) could serve as a statistic that quantifies edge roughness in an image across scales. Researchers have often relied on hand tracing to isolate edges in natural scenes for box-counting, a Df measurement technique. For a typical experiment's stimulus set, this would be unfeasibly time consuming. To expedite the process, we developed an algorithm to isolate selected edges of a natural scene for fractal analysis. Our algorithm consists of a three-step manual component (select specific color channels and average their intensity maps, apply an intensity-based threshold, and choose a set of binary objects to retain) followed by a two-step automated component (draw the edges and perform a box-count). We implemented our algorithm in Matlab and applied it to 89 images of clouds. We found that clouds as viewed from the ground have mean Df=1.34 (SD=0.11). We also computed the slope (β) of the radially averaged power spectrum for each image to test for a relationship between Df and β. We found no significant correlation between Df and β (r(89)=0.145, p=0.175). This implies that an image's textures may be independent from the Df of the textures' borders. This distinction is important because β can be computed with full automation. While computing Df for natural image's objects' edges has been time-intensive, our algorithm allows for quick determination of this critical scene statistic. Df could be used characterize the roughness of edges in visually presented natural scene stimuli. Studying how multi-scale contours affect visual processing would complement the literature on the visual processing of texture. Meeting abstract presented at VSS 2015. PMID:26326457
Fractal structure of asphaltene aggregates.
Rahmani, Nazmul H G; Dabros, Tadeusz; Masliyah, Jacob H
2005-05-15
A photographic technique coupled with image analysis was used to measure the size and fractal dimension of asphaltene aggregates formed in toluene-heptane solvent mixtures. First, asphaltene aggregates were examined in a Couette device and the fractal-like aggregate structures were quantified using boundary fractal dimension. The evolution of the floc structure with time was monitored. The relative rates of shear-induced aggregation and fragmentation/restructuring determine the steady-state floc structure. The average floc structure became more compact or more organized as the floc size distribution attained steady state. Moreover, the higher the shear rate is, the more compact the floc structure is at steady state. Second, the fractal dimensions of asphaltene aggregates were also determined in a free-settling test. The experimentally determined terminal settling velocities and characteristic lengths of the aggregates were utilized to estimate the 2D and 3D fractal dimensions. The size-density fractal dimension (D(3)) of the asphaltene aggregates was estimated to be in the range from 1.06 to 1.41. This relatively low fractal dimension suggests that the asphaltene aggregates are highly porous and very tenuous. The aggregates have a structure with extremely low space-filling capacity. PMID:15837477
Guthold, M.; Liu, W.; Stephens, B.; Lord, S. T.; Hantgan, R. R.; Erie, D. A.; Taylor, R. M.; Superfine, R.
2004-01-01
We report protocols and techniques to image and mechanically manipulate individual fibrin fibers, which are key structural components of blood clots. Using atomic force microscopy-based lateral force manipulations we determined the rupture force, FR, of fibrin fibers as a function of their diameter, D, in ambient conditions. As expected, the rupture force increases with increasing diameter; however, somewhat unexpectedly, it increases as FR ∼ D1.30±0.06. Moreover, using a combined atomic force microscopy-fluorescence microscopy instrument, we determined the light intensity, I, of single fibers, that were formed with fluorescently labeled fibrinogen, as a function of their diameter, D. Similar to the force data, we found that the light intensity, and thus the number of molecules per cross section, increases as I ∼ D1.25±0.11. Based on these findings we propose that fibrin fibers are fractals for which the number of molecules per cross section increases as about D1.3. This implies that the molecule density varies as ρ(D) ∼ D−0.7, i.e., thinner fibers are denser than thicker fibers. Such a model would be consistent with the observation that fibrin fibers consist of 70–80% water and only 20–30% protein, which also suggests that fibrin fibers are very porous. PMID:15465869
Flocculation control study based on fractal theory*
Chang, Ying; Liu, Qian-jun; Zhang, Jin-song
2005-01-01
A study on flocculation control based on fractal theory was carried out. Optimization test of chemical coagulant dosage confirmed that the fractal dimension could reflect the flocculation degree and settling characteristics of aggregates and the good correlation with the turbidity of settled effluent. So that the fractal dimension can be used as the major parameter for flocculation system control and achieve self-acting adjustment of chemical coagulant dosage. The fractal dimension flocculation control system was used for further study carried out on the effects of various flocculation parameters, among which are the dependency relationship among aggregates fractal dimension, chemical coagulant dosage, and turbidity of settled effluent under the conditions of variable water quality and quantity. And basic experimental data were obtained for establishing the chemical coagulant dosage control model mainly based on aggregates fractal dimension. PMID:16187420
NASA Astrophysics Data System (ADS)
Kayal, J. R.; Das, Vishal; Ghosh, Uma
2012-12-01
We examined seismic characteristics, b value and fractal dimension of the aftershock sequence of the January 26, 2001 Bhuj earthquake (Mw 7.7) that occurred in the Kutch failed rift basin, western margin of the Stable Continental Region (SCR) of India. A total of about 2,000 events (M ≥ 2.0) were recorded within two and a half months, immediately after the main shock. Some 795 events were precisely relocated by simultaneous inversion. These relocated events are used for mapping the frequency-magnitude relation ( b value) and fractal correlation dimension (Dc) to understand the seismic characteristics of the aftershocks and the source zone of the main shock. The surface maps of the b value and Dc reveal two distinct tectonic arms or zones of the V-shaped aftershock area, western zone and eastern zone. The b value is relatively higher (~1.6) in the western zone compared to a lower value (~1.4) in the eastern zone. The Dc map also shows a higher value (1.2-1.35) in the western zone compared to a lower Dc (0.80-1.15) in the eastern zone; this implies a positive correlation between Dc and b value. Two cross sections, E-W and N-S, are examined. The E-W sections show similar characteristics, higher b value and higher Dc in the western zone and lower in the eastern zone with depth. The N-S sections across the fault zones, however, show unique features; it imaged both the b and Dc characteristics convincingly to identify two known faults, the Kutch Mainland fault and the South Wagad fault (SWF), one stepping over the other with a seismogenic source zone at depth (20-35 km). The source zone at depth is imaged with a relatively lower b and higher Dc at the `fault end' of the SWF showing a negative correlation. These observations, corroborated with the seismic tomography as well as with the proposed geological/tectonic model, shed a new light to our understanding on seismogenesis of the largest SCR earthquake in India in the recent years.
Texture Analysis In Cytology Using Fractals
NASA Astrophysics Data System (ADS)
Basu, Santanu; Barba, Joseph; Chan, K. S.
1990-01-01
We present some preliminary results of a study aimed to assess the actual effectiveness of fractal theory in the area of medical image analysis for texture description. The specific goal of this research is to utilize "fractal dimension" to discriminate between normal and cancerous human cells. In particular, we have considered four types of cells namely, breast, bronchial, ovarian and uterine. A method based on fractal Brownian motion theory is employed to compute the "fractal dimension" of cells. Experiments with real images reveal that the range of scales over which the cells exhibit fractal property can be used as the discriminatory feature to identify cancerous cells.
Target Detection Using Fractal Geometry
NASA Technical Reports Server (NTRS)
Fuller, J. Joseph
1991-01-01
The concepts and theory of fractal geometry were applied to the problem of segmenting a 256 x 256 pixel image so that manmade objects could be extracted from natural backgrounds. The two most important measurements necessary to extract these manmade objects were fractal dimension and lacunarity. Provision was made to pass the manmade portion to a lookup table for subsequent identification. A computer program was written to construct cloud backgrounds of fractal dimensions which were allowed to vary between 2.2 and 2.8. Images of three model space targets were combined with these backgrounds to provide a data set for testing the validity of the approach. Once the data set was constructed, computer programs were written to extract estimates of the fractal dimension and lacunarity on 4 x 4 pixel subsets of the image. It was shown that for clouds of fractal dimension 2.7 or less, appropriate thresholding on fractal dimension and lacunarity yielded a 64 x 64 edge-detected image with all or most of the cloud background removed. These images were enhanced by an erosion and dilation to provide the final image passed to the lookup table. While the ultimate goal was to pass the final image to a neural network for identification, this work shows the applicability of fractal geometry to the problems of image segmentation, edge detection and separating a target of interest from a natural background.
Fractal texture analysis of the healing process after bone loss.
Borowska, Marta; Szarmach, Janusz; Oczeretko, Edward
2015-12-01
Radiological assessment of treatment effectiveness of guided bone regeneration (GBR) method in postresectal and postcystal bone loss cases, observed for one year. Group of 25 patients (17 females and 8 males) who underwent root resection with cystectomy were evaluated. The following combination therapy of intraosseous deficits was used, consisting of bone augmentation with xenogenic material together with covering regenerative membranes and tight wound closure. The bone regeneration process was estimated, comparing the images taken on the day of the surgery and 12 months later, by means of Kodak RVG 6100 digital radiography set. The interpretation of the radiovisiographic image depends on the evaluation ability of the eye looking at it, which leaves a large margin of uncertainty. So, several texture analysis techniques were developed and used sequentially on the radiographic image. For each method, the results were the mean from the 25 images. These methods compute the fractal dimension (D), each one having its own theoretic basis. We used five techniques for calculating fractal dimension: power spectral density method, triangular prism surface area method, blanket method, intensity difference scaling method and variogram analysis. Our study showed a decrease of fractal dimension during the healing process after bone loss. We also found evidence that various methods of calculating fractal dimension give different results. During the healing process after bone loss, the surfaces of radiographic images became smooth. The result obtained show that our findings may be of great importance for diagnostic purpose. PMID:26362075
Kesić, Srdjan; Nikolić, Ljiljana; Savić, Aleksandar G; Petković, Branka; Spasić, Sladjana Z
2014-01-01
Aim of this study was to investigate the application of normalized mean of the empirical Higuchi fractal dimension (FD) distributions, as a new approach to analyze the spontaneous bioelectrical activity of garden snail (Helix pomatia) Br neuron. The effect of ouabain on modulation of Br neuron bursting activity before and after the exposure to 10 mT static magnetic field (SMF) was observed by analyzing the following parameters: action potential (AP), interspike interval (ISI) and interbursting interval (IBI) components. Normalized mean of the empirical FD distributions were formed for the following experimental conditions: Control 1, Ouabain 1, Control 2, SMF 2, ASMF 2, Control 3, SMF 3 and Ouabain ASMF 3. Our main results have shown that ouabain without SMF induced increase in participation of AP and a decrease in participation of IBI components compared to the first control condition. However, in the presence of 10 mT SMF, ouabain-induced changes of measured parameters of Br neuron activity were less pronounced compared to the third control condition. We have shown that normalized mean of the empirical FD distributions is a useful method for detecting the changes in AP, ISI, and IBI components of complex bursting activity in altered physiological states. PMID:24968407
Haitao, Sun; Ning, Li; Lijun, Guo; Fei, Gao
2011-01-01
Objective The aim of this study was to use fractal dimension (FD) analysis on multidetector CT (MDCT) images for quantifying the morphological changes of the pulmonary artery tree in patients with pulmonary hypertension (PH). Materials and Methods Fourteen patients with PH and 17 patients without PH as controls were studied. All of the patients underwent contrast-enhanced helical CT and transthoracic echocardiography. The pulmonary artery trees were generated using post-processing software, and the FD and projected image area of the pulmonary artery trees were determined with ImageJ software in a personal computer. The FD, the projected image area and the pulmonary artery pressure (PAP) were statistically evaluated in the two groups. Results The FD, the projected image area and the PAP of the patients with PH were higher than those values of the patients without PH (p < 0.05, t-test). There was a high correlation of FD with the PAP (r = 0.82, p < 0.05, partial correlation analysis). There was a moderate correlation of FD with the projected image area (r = 0.49, p < 0.05, partial correlation analysis). There was a correlation of the PAP with the projected image area (r = 0.65, p < 0.05, Pearson correlation analysis). Conclusion The FD of the pulmonary arteries in the PH patients was significantly higher than that of the controls. There is a high correlation of FD with the PAP. PMID:21603288
Frankel, A.
1991-01-01
The high-frequency falloff ??-y of earthquake displacement spectra and the b value of aftershock sequences are attributed to the character of spatially varying strength along fault zones. I assume that the high frequency energy of a main shock is produced by a self-similar distribution of subevents, where the number of subevents with radii greater than R is proportional to R-D, D being the fractal dimension. In the model, an earthquake is composed of a hierarchical set of smaller earthquakes. The static stress drop is parameterized to be proportional to R??, and strength is assumed to be proportional to static stress drop. I find that a distribution of subevents with D = 2 and stress drop independent of seismic moment (?? = 0) produces a main shock with an ??-2 falloff, if the subevent areas fill the rupture area of the main shock. By equating subevents to "islands' of high stress of a random, self-similar stress field on a fault, I relate D to the scaling of strength on a fault, such that D = 2 - ??. Thus D = 2 corresponds to constant stress drop scaling (?? = 0) and scale-invariant fault strength. A self-similar model of aftershock rupture zones on a fault is used to determine the relationship between the b value, the size distribution of aftershock rupture zones, and the scaling of strength on a fault. -from Author
Milovanovic, Petar; Djuric, Marija; Rakocevic, Zlatko
2012-01-01
There is an increasing interest in bone nano-structure, the ultimate goal being to reveal the basis of age-related bone fragility. In this study, power spectral density (PSD) data and fractal dimensions of the mineralized bone matrix were extracted from atomic force microscope topography images of the femoral neck trabeculae. The aim was to evaluate age-dependent differences in the mineralized matrix of human bone and to consider whether these advanced nano-descriptors might be linked to decreased bone remodeling observed by some authors and age-related decline in bone mechanical competence. The investigated bone specimens belonged to a group of young adult women (n = 5, age: 20–40 years) and a group of elderly women (n = 5, age: 70–95 years) without bone diseases. PSD graphs showed the roughness density distribution in relation to spatial frequency. In all cases, there was a fairly linear decrease in magnitude of the power spectra with increasing spatial frequencies. The PSD slope was steeper in elderly individuals (−2.374 vs. −2.066), suggesting the dominance of larger surface morphological features. Fractal dimension of the mineralized bone matrix showed a significant negative trend with advanced age, declining from 2.467 in young individuals to 2.313 in the elderly (r = 0.65, P = 0.04). Higher fractal dimension in young women reflects domination of smaller mineral grains, which is compatible with the more freshly remodeled structure. In contrast, the surface patterns in elderly individuals were indicative of older tissue age. Lower roughness and reduced structural complexity (decreased fractal dimension) of the interfibrillar bone matrix in the elderly suggest a decline in bone toughness, which explains why aged bone is more brittle and prone to fractures. PMID:22946475
Fractal geometry of collision cascades
Rossi, F.; Parkin, D.M.; Nastasi, M.
1989-01-01
The fractal nature of self-ion collision cascades is first described using an inverse power potential and then by the more realistic potential of Biersack--Ziegler. Based on the model of Cheng et al. and TRIM Monte Carlo simulations, the average cascade fractal dimension is a function of both atomic mass and initial energy. The instantaneous fractal dimension increases as the cascade evolves. A critical energy E/sub c/ for producing a dense subcascade is derived and it is shown that E/sub c/ agrees well with the onset energy for constant damage efficiency.
ERIC Educational Resources Information Center
Camp, Dane R.
1991-01-01
After introducing the two-dimensional Koch curve, which is generated by simple recursions on an equilateral triangle, the process is extended to three dimensions with simple recursions on a regular tetrahedron. Included, for both fractal sequences, are iterative formulae, illustrations of the first several iterations, and a sample PASCAL program.…
Application of the fractal theory on the study of filter cake constructure
Xu, X.; Xu, J.; Deng, C.; Qian, L.; Yan, K.
1995-12-31
Cake filtration is a complex process and the cake constructure is very difficult to describe in theory. Cake constructure parameters, such as the cake porosity, pore size shape and even its distribution, are main factors influencing the filtration results but have not been thoroughly understood yet. In this paper the fractal theory, an effective mathematical method in describing the self-similar phenomenon is used to investigate the filter cake constructure, and the scanning electron microscope and automatic image analyzer are used to measure the cake constructure. Cakes which formed in different conditions are examined and the fractal dimension of the cake are calculated. The study shows that the constructure of the filter cake can be approximated by Sierpinski fractal geometry and that the fractal dimension of filter cake, related to the particle characteristics, slurry concentration and filtration pressure is a good parameter to describe the pore size distribution and the cake penetrability.
Squarcina, Letizia; De Luca, Alberto; Bellani, Marcella; Brambilla, Paolo; Turkheimer, Federico E; Bertoldo, Alessandra
2015-02-21
Fractal geometry can be used to analyze shape and patterns in brain images. With this study we use fractals to analyze T1 data of patients affected by schizophrenia or bipolar disorder, with the aim of distinguishing between healthy and pathological brains using the complexity of brain structure, in particular of grey matter, as a marker of disease. 39 healthy volunteers, 25 subjects affected by schizophrenia and 11 patients affected by bipolar disorder underwent an MRI session. We evaluated fractal dimension of the brain cortex and its substructures, calculated with an algorithm based on the box-count algorithm. We modified this algorithm, with the aim of avoiding the segmentation processing step and using all the information stored in the image grey levels. Moreover, to increase sensitivity to local structural changes, we computed a value of fractal dimension for each slice of the brain or of the particular structure. To have reference values in comparing healthy subjects with patients, we built a template by averaging fractal dimension values of the healthy volunteers data. Standard deviation was evaluated and used to create a confidence interval. We also performed a slice by slice t-test to assess the difference at slice level between the three groups. Consistent average fractal dimension values were found across all the structures in healthy controls, while in the pathological groups we found consistent differences, indicating a change in brain and structures complexity induced by these disorders. PMID:25633275
Fractality in the neuron axonal topography of the human brain based on 3-D diffusion MRI
NASA Astrophysics Data System (ADS)
Katsaloulis, P.; Ghosh, A.; Philippe, A. C.; Provata, A.; Deriche, R.
2012-05-01
In this work the fractal architecture of the neuron axonal topography of the human brain is evaluated, as derived from 3-D diffusion MRI (dMRI) acquisitions. This is a 3D extension of work performed previously in 2D regions of interest (ROIs), where the fractal dimension of the neuron axonal topography was computed from dMRI data. A group study with 18 subjects is here conducted and the fractal dimensions D f of the entire 3-D volume of the brains is estimated via the box counting, the correlation dimension and the fractal mass dimension methods. The neuron axon data is obtained using tractography algorithms on diffusion tensor imaging of the brain. We find that all three calculations of D f give consistent results across subjects, namely, they demonstrate fractal characteristics in the short and medium length scales: different fractal exponents prevail at different length scales, an indication of multifractality. We surmise that this complexity stems as a collective property emerging when many local brain units, performing different functional tasks and having different local topologies, are recorded together.
NASA Astrophysics Data System (ADS)
Squarcina, Letizia; De Luca, Alberto; Bellani, Marcella; Brambilla, Paolo; Turkheimer, Federico E.; Bertoldo, Alessandra
2015-02-01
Fractal geometry can be used to analyze shape and patterns in brain images. With this study we use fractals to analyze T1 data of patients affected by schizophrenia or bipolar disorder, with the aim of distinguishing between healthy and pathological brains using the complexity of brain structure, in particular of grey matter, as a marker of disease. 39 healthy volunteers, 25 subjects affected by schizophrenia and 11 patients affected by bipolar disorder underwent an MRI session. We evaluated fractal dimension of the brain cortex and its substructures, calculated with an algorithm based on the box-count algorithm. We modified this algorithm, with the aim of avoiding the segmentation processing step and using all the information stored in the image grey levels. Moreover, to increase sensitivity to local structural changes, we computed a value of fractal dimension for each slice of the brain or of the particular structure. To have reference values in comparing healthy subjects with patients, we built a template by averaging fractal dimension values of the healthy volunteers data. Standard deviation was evaluated and used to create a confidence interval. We also performed a slice by slice t-test to assess the difference at slice level between the three groups. Consistent average fractal dimension values were found across all the structures in healthy controls, while in the pathological groups we found consistent differences, indicating a change in brain and structures complexity induced by these disorders.
Time profiles of short and middle GRB fractal analysis and fireball model.
NASA Astrophysics Data System (ADS)
Arkhangelskaja, I.
In present work results of GRB time profile fractal analysis are presented. The fractal index is a time profile characteristic which is sensitive to change of shape and indentness of an event time profile. Many experimental signals are fractal in nature, and, importantly, fractal dimensions must be different for time profiles of events which are caused by different physical processes.There are two important featuries of the fractal index for our proposed project: 1) the fractal indexes of Poisson statisticsdominated sets are equal to 1.5; 2) invariance of fractal index to event duration: if there are two bursts with the same form of time profile but different duration, the fractal indexes of these bursts will be the same. The combined data processing for two different types of BATSE data is presented as an example. We try to separate some additional types of Gamma-Ray Bursts(GRB) using fractal analysis of GRB time profiles (now 3 classes of GRB are known : short, middle, long with mean durations t90 0.7s, 3s and 25s correspondingly). We study time profiles of short and middle GRB using LAD discriminator (DISCLA) data with 64 ms time resolution for bursts with t90 > 2s from the 5B BATSE catalog and time-tagged event (TTE) data (in this data, time of registration of first 32767 photons recorded with 2 mks time resolution) for bursts with t90 < 2s. TTE and DISCLA are very different types of data and we can use both types in one dataset because these data have the same background fractal indexes - 1.5 for both dataset. We analyse time profiles of 2000 GRB from 5B BATSE catalog by fractal analysis and obtain some new additional subclasses of GRB. There are 4 subclasses in fractal index distribution of short GRB with mean fractal indexies 1.05 +/- 0.03, 1.31 +/- 0.05,1.51 +/- 0.03,1.90 +/- 0.003 and there are 3 subclasses in fractal index distribution of middle GRB with mean fractal indexies 1.25 +/- 0.03, 1.47 +/- 0.03,1.87 +/- 0.03. We calculate time profiles using fireball model and study fractal index distribution of these time profiles. We obtain that fractal indexies of such time profiles are in region 1.213D1.400 and these events are correspond to subclass of short and middle GRB with maximum at D=1.31. and D=1.25 correspondly.
Quantitative Characterization Of Basaltic Tephra Using The Fractal Spectrum Technique
NASA Astrophysics Data System (ADS)
Maria, A.
2006-12-01
Geologists have studied volcanic eruptions on Hawaii more closely than anywhere else. Even so, processes of magma fragmentation during Hawaiian style eruptions (e.g. lava fountaining) are not well understood. Furthermore, the products of these eruptions have not been fully characterized. Analysis of tephra shape is particularly useful for understanding the nature of eruptions, as particle morphology reflects numerous volcanic parameters (e.g. magma viscosity, volatile content, interaction with water, transport processes). The technique applied in this study, based on fractal geometry, uses data produced by dilation of the 2-D particle boundary to produce a full spectrum of fractal dimensions over a range of scales for each particle. Multiple fractal dimensions, which can be described as a fractal spectrum curve, are calculated by taking the first derivative of data points on a standard Richardson plot. Previous applications of this technique have proven helpful for characterizing basaltic to rhyolitic products of specific fragmentation processes, and modes of volcanic transport/deposition. In this study, all the samples are basaltic, eliminating the variable of composition, and include material from two Hawaiian lava-fountaining events (Mauna Ulu, 1969; Kilauea Iki, 1959), as well as material from Masaya, Nicaragua (San Judas Formation) that is thought to have been unusually explosive, for comparison. One of our goals is to identify characteristic particle shapes formed during the lava-fountain events. Use of multiple fractal dimensions results in more effective discrimination than expressions of shape based on one or two fractal dimensions, and also allows use of multivariate statistical techniques. Cluster analysis provides a visual display of the similarities of particles in a sample, and facilitates identification of the types of shapes that are most characteristic of a given deposit. Use of principal components analysis to summarize the data as accurately as possible using a few components, facilitates comparison between samples.
Analysis of fractals with combined partition
NASA Astrophysics Data System (ADS)
Dedovich, T. G.; Tokarev, M. V.
2016-03-01
The space—time properties in the general theory of relativity, as well as the discreteness and non-Archimedean property of space in the quantum theory of gravitation, are discussed. It is emphasized that the properties of bodies in non-Archimedean spaces coincide with the properties of the field of P-adic numbers and fractals. It is suggested that parton showers, used for describing interactions between particles and nuclei at high energies, have a fractal structure. A mechanism of fractal formation with combined partition is considered. The modified SePaC method is offered for the analysis of such fractals. The BC, PaC, and SePaC methods for determining a fractal dimension and other fractal characteristics (numbers of levels and values of a base of forming a fractal) are considered. It is found that the SePaC method has advantages for the analysis of fractals with combined partition.
NASA Astrophysics Data System (ADS)
Shi, Juanjuan; Liang, Ming; Guan, Yunpeng
2016-02-01
The conventional way for bearing fault diagnosis under variable rotational speed generally includes prefiltering, resampling based on shaft rotating frequency and order spectrum analysis. However, its application is confined by three major obstacles: a) knowledge-demanding parameter determination required by prefiltering, b) unavailable shaft rotating frequency for resampling as it is coupled with instantaneous fault characteristic frequency (IFCF) by a fault characteristic coefficient (FCC) which cannot be decided without knowing what fault actually exists, and c) complicated and error-prone resampling process. As such, we propose a new method to address these problems. The proposed method free from prefiltering and resampling mainly contains the following steps: a) extracting envelope by windowed fractal dimension (FD) transform, requiring no prefiltering, b) with the envelope signal, performing short time Fourier transform (STFT) to get a clear time frequency representation (TFR), from which the IFCF and the basic demodulator for generalized demodulation (GD) can be obtained, c) applying the generalized demodulation to the envelope signal with the current demodulator, converting the trajectory of the current time-frequency component into a linear path parallel to the time axis, d) frequency analyzing the demodulated signal, followed by searching the amplitude of the constant frequency where the linear path is situated. Updating demodulator via multiplying the basic demodulator by different real numbers (i.e., coefficient λ) and repeating the steps (c)-(d), the resampling-free order spectrum is then obtained. Based on the resulting spectrum, the final diagnosis decision can be made. The proposed method for its implementation on the example of simulated data is presented. Finally, experimental data are employed to validate the effectiveness of the proposed technique.
Doherty, Cailbhe; Bleakley, Chris; Hertel, Jay; Caulfield, Brian; Ryan, John; Delahunt, Eamonn
2014-01-01
Instrumented postural control analysis plays an important role in evaluating the effects of injury on dynamic stability during balance tasks, and is often conveyed with measures based on the displacement of the center-of-pressure (COP) assessed with a force platform. However, the desired outcome of the task is frequently characterized by a loss of dynamic stability, secondary to injury. Typically, these failed trials are discarded during research investigations, with the potential loss of informative data pertaining to task success. The novelty of the present study is that COP characteristics of failed trials in injured participants are compared to successful trial data in another injured group, and a control group of participants, using the fractal dimension (FD) method. Three groups of participants attempted a task of eyes closed single limb stance (SLS): twenty-nine participants with acute ankle sprain successfully completed the task on their non-injured limb (successful injury group); twenty eight participants with acute ankle sprain failed their attempt on their injured limb (failed injury group); sixteen participants with no current injury successfully completed the task on their non-dominant limb (successful non-injured group). Between trial analyses of these groups revealed significant differences in COP trajectory FD (successful injury group: 1.58±0.06; failed injury group: 1.54±0.07; successful non-injured group: 1.64±0.06) with a large effect size (0.27). These findings demonstrate that successful eyes-closed SLS is characterized by a larger FD of the COP path when compared to failed trials, and that injury causes a decrease in COP path FD. PMID:24746034
Fractal analysis of deformation-induced dislocation patterns
Zaiser, M. ); Bay, K. . Inst. fuer Theoretische und Angewandte Physik); Haehner, P. . Joint Research Centre TU Braunschweig . Inst. fuer Metallphysik und Nukleare Festkoerperphysik)
1999-06-22
The paper reports extensive analyses of the fractal geometry of cellular dislocation structures observed in Cu deformed in multiple-slip orientation. Several methods presented for the determination of fractal dimensions are shown to give consistent results. Criteria are formulated which allow the distinguishing of fractal from non-fractal patterns, and implications of fractal dislocation patterning for quantitative metallography are discussed in detail. For an interpretation of the findings a theoretical model is outlined according to which dislocation cell formation is associated to a noise-induced structural transition far from equilibrium. This allows relating the observed fractal dimensions to the stochastic properties of deformation by collective dislocation glide.
Fractal characteristics and microstructure evolution of magnetron sputtering Cu thin films
NASA Astrophysics Data System (ADS)
Du, Shiwen; Li, Yongtang
2013-01-01
How to describe surface morphology characteristic and microstructure evolution are the hottest researches of current thin film researches. But in traditional characterization of surface morphology, the roughness parameters are scale related. And the microstructure evolution of thin film during post-treatment is usually not considered in detail. To give a better understanding of the roughness of thin films topography, fractal method is carried out. In addition, microstructure evolution of thin films is analyzed based on the crystallography and energy theory. Cu thin films are deposited on Si (100) substrates by magnetron sputtering, and then annealed at different temperatures. Surface topography is characterized by atomic force microscope (AFM). Triangular prism surface area (TPSA) algorithm is used to calculate the fractal dimension of the AFM images. Apparent scale effect exists between the surface morphology roughness and film thickness. Relationship between the fractal dimension and roughness is analyzed by linear regression method and linear relationship exists between fractal dimension and surface roughness root mean square (RMS). Fractal dimension can be characterized as a scale independence parameter to represent the complex degree and roughness level of surface. With the increase of annealing temperature, surface roughness and fractal dimension decrease. But when the annealing temperature exceeds the recrystallization temperature, due to the agglomeration and coalescence of Cu grain, surface roughness and fractal dimension increase. Scale effect and changing regularity of grain growth and shape evolution for different film thickness under different annealing temperatures are analyzed. Based on minimum total free energy, regularity of grain growth and changing is proposed. The proposed research has some theory significance and applicative value of Cu interconnect process and development of MEMS.
NASA Astrophysics Data System (ADS)
Kong, Xiangguo; Wang, Enyuan; Hu, Shaobin; Shen, Rongxi; Li, Xuelong; Zhan, Tangqi
2016-01-01
Aimed at exploring the influence of methane to coal and studying fractal characteristics and acoustic emission (AE) features in the damage evolution, the triaxial compression experiments of coal containing methane were conducted, and acoustic emission response was collected simultaneously in the loading process. Based on the method for calculating the correlation dimension, the fractal dimension was calculated with regard to time series of acoustic emission. Our experimental results indicate that AE response and fractal dimension can reflect the evolution and propagation of cracks in the loading process. Corresponding to the load-time, acoustic emission experiences active, linearly increasing, rapidly augmenting and decreasing stage. However, the fractal dimension of AE develops from chaos to orderly state. Late loading, a continued slowdown in fractal dimension, can be used as a precursory signal of coal sample destruction. In addition, the amount of gas in the coal sample will influence the evolution of pore and fracture, which causes a variation in the acoustic emission signals and fractal dimension. The maximum bearing load reduces 18.85% and 49.18% within pore pressure of 0.75 and 1.5 MPa, compared with it (24.4 kN) of the coal sample (without gas). What's more, the increase of pore pressure will cause the growth of AE count and energy, but the correlation dimension of AE parameters drops. This study is helpful for us to understand the effects of methane to coal and the evolution mechanism of cracks, and it can be applied to the research on occurrence mechanism and early warning of coal and gas outburst.
NASA Technical Reports Server (NTRS)
Bruno, B. C.; Taylor, G. J.; Rowland, S. K.; Lucey, P. G.; Self, S.
1992-01-01
Results are presented of a preliminary investigation of the fractal nature of the plan-view shapes of lava flows in Hawaii (based on field measurements and aerial photographs), as well as in Idaho and the Galapagos Islands (using aerial photographs only). The shapes of the lava flow margins are found to be fractals: lava flow shape is scale-invariant. This observation suggests that nonlinear forces are operating in them because nonlinear systems frequently produce fractals. A'a and pahoehoe flows can be distinguished by their fractal dimensions (D). The majority of the a'a flows measured have D between 1.05 and 1.09, whereas the pahoehoe flows generally have higher D (1.14-1.23). The analysis is extended to other planetary bodies by measuring flows from orbital images of Venus, Mars, and the moon. All are fractal and have D consistent with the range of terrestrial a'a and have D consistent with the range of terrestrial a'a and pahoehoe values.
ERIC Educational Resources Information Center
Gray, Shirley B.
1992-01-01
This article traces the historical development of fractal geometry from early in the twentieth century and offers an explanation of the mathematics behind the recursion formulas and their representations within computer graphics. Also included are the fundamentals behind programing for fractal graphics in the C Language with appropriate
ERIC Educational Resources Information Center
Osler, Thomas J.
1999-01-01
Because fractal images are by nature very complex, it can be inspiring and instructive to create the code in the classroom and watch the fractal image evolve as the user slowly changes some important parameter or zooms in and out of the image. Uses programming language that permits the user to store and retrieve a graphics image as a disk file.…
ERIC Educational Resources Information Center
Dewdney, A. K.
1991-01-01
Explores the subject of fractal geometry focusing on the occurrence of fractal-like shapes in the natural world. Topics include iterated functions, chaos theory, the Lorenz attractor, logistic maps, the Mandelbrot set, and mini-Mandelbrot sets. Provides appropriate computer algorithms, as well as further sources of information. (JJK)
Fractal analysis of Mesoamerican pyramids.
Burkle-Elizondo, Gerardo; Valdez-Cepeda, Ricardo David
2006-01-01
A myth of ancient cultural roots was integrated into Mesoamerican cult, and the reference to architecture denoted a depth religious symbolism. The pyramids form a functional part of this cosmovision that is centered on sacralization. The space architecture works was an expression of the ideological necessities into their conception of harmony. The symbolism of the temple structures seems to reflect the mathematical order of the Universe. We contemplate two models of fractal analysis. The first one includes 16 pyramids. We studied a data set that was treated as a fractal profile to estimate the Df through variography (Dv). The estimated Fractal Dimension Dv = 1.383 +/- 0.211. In the second one we studied a data set to estimate the Dv of 19 pyramids and the estimated Fractal Dimension Dv = 1.229 +/- 0.165. PMID:16393505
Thermodynamics of Photons on Fractals
Akkermans, Eric; Dunne, Gerald V.; Teplyaev, Alexander
2010-12-03
A thermodynamical treatment of a massless scalar field (a photon) confined to a fractal spatial manifold leads to an equation of state relating pressure to internal energy, PV{sub s}=U/d{sub s}, where d{sub s} is the spectral dimension and V{sub s} defines the 'spectral volume'. For regular manifolds, V{sub s} coincides with the usual geometric spatial volume, but on a fractal this is not necessarily the case. This is further evidence that on a fractal, momentum space can have a different dimension than position space. Our analysis also provides a natural definition of the vacuum (Casimir) energy of a fractal. We suggest ways that these unusual properties might be probed experimentally.
Constraints on Titan's Topography Through Fractal Analysis of Shorelines
NASA Astrophysics Data System (ADS)
Sharma, P.; Byrne, S.
2008-12-01
The recent discovery of hydrocarbon lakes at Titan's North Pole by the Radio Detection and Ranging (RADAR) instrument onboard the Cassini spacecraft is one of the most exciting discoveries of the Cassini-Huygens mission. Previous analyses of terrestrial coastlines have revealed them to be closely approximated by self-similar fractals. Coastline length increases as the measuring scale decreases because smaller measuring scales are sensitive to smaller features of the coastline. The measured perimeter can be related to the measuring scale by a power law whose exponent is 1-D, where D is the fractal dimension. The value of D provides a means to quantify the complexity (ruggedness) of a coastline with higher values indicating higher complexity. As pooled liquids form equipotential surfaces, coastlines are equivalent to topographic contour lines. The complexity of a coastline can therefore be related to the complexity of the surface it is embedded in through fractal theory. Thus, a statistical characterization of Titan's topography can be extracted through analysis of these shorelines. We have carried out this analysis for coastlines on Titan and have related the coastline roughness parameters to topography parameters for Titan's landscape. In this study, we used projected Cassini Radar observations (resolution of about 350m/pixel near the centre of the swath). The shorelines of 290 of these North Polar Titanian lakes have been manually outlined at the full resolution of the dataset. Their fractal dimensions were calculated via two methods: the ruler method and the box-counting method. Our results show Titan's coastlines do exhibit fractal properties with fractal dimensions comparable to published estimates of the terrestrial coastlines of Britain and Germany. Such high values of this roughness parameter show that Titanian coastlines are intricate by terrestrial standards, which implies a rugged landscape. We will report on this statistical characterization of Titan's topography and spatial variations in landscape roughness.
Fractal Analysis of Optical Coherence Tomography of Normal and Malignant Breast Tissue
NASA Astrophysics Data System (ADS)
Sullivan, Amanda C.; Hunt, John P.; Oldenburg, Amy L.
2011-03-01
Optical coherence tomography (OCT) provides real-time imaging of tissue several mean free photon paths into tissue by heterodyne detection of backscattered light. OCT can potentially be used to rapidly assess tumor margins during breast cancer resection, however, currently it is difficult to differentiate between normal and malignant tissues with OCT. Because cancer is characterized morphologically by increasing disorder, we investigated the fractal dimension of OCT images of normal and cancerous breast tissue. 3D OCT images of 44 specimens were collected, then tissues were histologically processed to independently determine distinct regions of adipose, stroma and cancer. The fractal dimension of each tissue type was then calculated with a one-dimensional box-counting algorithm applied to the OCT axial scans. We found that the fractal dimensions of stromal tissues were significantly higher than those of cancer (P 10 -6) , whilethoseofadiposetissueweresignificantlylowerthanthoseofcancer (P 10 - 4) .
Fractal dynamics of bioconvective patterns
NASA Technical Reports Server (NTRS)
Noever, David A.
1991-01-01
Biologically generated cellular patterns, sometimes called bioconvective patterns, are found to cluster into aggregates which follow fractal growth dynamics akin to diffusion-limited aggregation (DLA) models. The pattern formed is self-similar with fractal dimension of 1.66 +/-0.038. Bioconvective DLA branching results from thermal roughening which shifts the balance between ordering viscous forces and disordering cell motility and random diffusion. The phase diagram for pattern morphology includes DLA, boundary spokes, random clusters, and reverse clusters.
Fractal Electronic Circuits Assembled From Nanoclusters
NASA Astrophysics Data System (ADS)
Fairbanks, M. S.; McCarthy, D.; Taylor, R. P.; Brown, S. A.
2009-07-01
Many patterns in nature can be described using fractal geometry. The effect of this fractal character is an array of properties that can include high internal connectivity, high dispersivity, and enhanced surface area to volume ratios. These properties are often desirable in applications and, consequently, fractal geometry is increasingly employed in technologies ranging from antenna to storm barriers. In this paper, we explore the application of fractal geometry to electrical circuits, inspired by the pervasive fractal structure of neurons in the brain. We show that, under appropriate growth conditions, nanoclusters of Sb form into islands on atomically flat substrates via a process close to diffusion-limited aggregation (DLA), establishing fractal islands that will form the basis of our fractal circuits. We perform fractal analysis of the islands to determine the spatial scaling properties (characterized by the fractal dimension, D) of the proposed circuits and demonstrate how varying growth conditions can affect D. We discuss fabrication approaches for establishing electrical contact to the fractal islands. Finally, we present fractal circuit simulations, which show that the fractal character of the circuit translates into novel, non-linear conduction properties determined by the circuit's D value.
Fractal nature of hydrocarbon deposits. 2. Spatial distribution
Barton, C.C.; Schutter, T.A; Herring, P.R.; Thomas, W.J. ); Scholz, C.H. )
1991-03-01
Hydrocarbons are unevenly distributed within reservoirs and are found in patches whose size distribution is a fractal over a wide range of scales. The spatial distribution of the patches is also fractal and this can be used to constrain the design of drilling strategies also defined by a fractal dimension. Fractal distributions are scale independent and are characterized by a power-law scaling exponent termed the fractal dimension. The authors have performed fractal analyses on the spatial distribution of producing and showing wells combined and of dry wells in 1,600-mi{sup 2} portions of the Denver and Powder River basins that were nearly completely drilled on quarter-mile square-grid spacings. They have limited their analyses to wells drilled to single stratigraphic intervals so that the map pattern revealed by drilling is representative of the spatial patchiness of hydrocarbons at depth. The fractal dimensions for the spatial patchiness of hydrocarbons in the two basins are 1.5 and 1.4, respectively. The fractal dimension for the pattern of all wells drilled is 1.8 for both basins, which suggests a drilling strategy with a fractal dimension significantly higher than the dimensions 1.5 and 1.4 sufficient to efficiently and economically explore these reservoirs. In fact, the fractal analysis reveals that the drilling strategy used in these basins approaches a fractal dimension of 2.0, which is equivalent to random drilling with no geologic input. Knowledge of the fractal dimension of a reservoir prior to drilling would provide a basis for selecting and a criterion for halting a drilling strategy for exploration whose fractal dimension closely matches that of the spatial fractal dimension of the reservoir, such a strategy should prove more efficient and economical than current practice.
Multispectral image fusion based on fractal features
NASA Astrophysics Data System (ADS)
Tian, Jie; Chen, Jie; Zhang, Chunhua
2004-01-01
Imagery sensors have been one indispensable part of the detection and recognition systems. They are widely used to the field of surveillance, navigation, control and guide, et. However, different imagery sensors depend on diverse imaging mechanisms, and work within diverse range of spectrum. They also perform diverse functions and have diverse circumstance requires. So it is unpractical to accomplish the task of detection or recognition with a single imagery sensor under the conditions of different circumstances, different backgrounds and different targets. Fortunately, the multi-sensor image fusion technique emerged as important route to solve this problem. So image fusion has been one of the main technical routines used to detect and recognize objects from images. While, loss of information is unavoidable during fusion process, so it is always a very important content of image fusion how to preserve the useful information to the utmost. That is to say, it should be taken into account before designing the fusion schemes how to avoid the loss of useful information or how to preserve the features helpful to the detection. In consideration of these issues and the fact that most detection problems are actually to distinguish man-made objects from natural background, a fractal-based multi-spectral fusion algorithm has been proposed in this paper aiming at the recognition of battlefield targets in the complicated backgrounds. According to this algorithm, source images are firstly orthogonally decomposed according to wavelet transform theories, and then fractal-based detection is held to each decomposed image. At this step, natural background and man-made targets are distinguished by use of fractal models that can well imitate natural objects. Special fusion operators are employed during the fusion of area that contains man-made targets so that useful information could be preserved and features of targets could be extruded. The final fused image is reconstructed from the composition of source pyramid images. So this fusion scheme is a multi-resolution analysis. The wavelet decomposition of image can be actually considered as special pyramid decomposition. According to wavelet decomposition theories, the approximation of image (formula available in paper) at resolution 2j+1 equal to its orthogonal projection in space , that is, where Ajf is the low-frequency approximation of image f(x, y) at resolution 2j and , , represent the vertical, horizontal and diagonal wavelet coefficients respectively at resolution 2j. These coefficients describe the high-frequency information of image at direction of vertical, horizontal and diagonal respectively. Ajf, , and are independent and can be considered as images. In this paper J is set to be 1, so the source image is decomposed to produce the son-images Af, D1f, D2f and D3f. To solve the problem of detecting artifacts, the concepts of vertical fractal dimension FD1, horizontal fractal dimension FD2 and diagonal fractal dimension FD3 are proposed in this paper. The vertical fractal dimension FD1 corresponds to the vertical wavelet coefficients image after the wavelet decomposition of source image, the horizontal fractal dimension FD2 corresponds to the horizontal wavelet coefficients and the diagonal fractal dimension FD3 the diagonal one. These definitions enrich the illustration of source images. Therefore they are helpful to classify the targets. Then the detection of artifacts in the decomposed images is a problem of pattern recognition in 4-D space. The combination of FD0, FD1, FD2 and FD3 make a vector of (FD0, FD1, FD2, FD3), which can be considered as a united feature vector of the studied image. All the parts of the images are classified in the 4-D pattern space created by the vector of (FD0, FD1, FD2, FD3) so that the area that contains man-made objects could be detected. This detection can be considered as a coarse recognition, and then the significant areas in each son-images are signed so that they can be dealt with special rules. There has been various fusion rules developed with each one aiming at a special problem. These rules have different performance, so it is very important to select an appropriate rule during the design of an image fusion system. Recent research denotes that the rule should be adjustable so that it is always suitable to extrude the features of targets and to preserve the pixels of useful information. In this paper, owing to the consideration that fractal dimension is one of the main features to distinguish man-made targets from natural objects, the fusion rule was defined that if the studied region of image contains man-made target, the pixels of the source image whose fractal dimension is minimal are saved to be the pixels of the fused image, otherwise, a weighted average operator is adopted to avoid loss of information. The main idea of this rule is to store the pixels with low fractal dimensions, so it can be named Minimal Fractal dimensions (MFD) fusion rule. This fractal-based algorithm is compared with a common weighted average fusion algorithm. An objective assessment is taken to the two fusion results. The criteria of Entropy, Cross-Entropy, Peak Signal-to-Noise Ratio (PSNR) and Standard Gray Scale Difference are defined in this paper. Reversely to the idea of constructing an ideal image as the assessing reference, the source images are selected to be the reference in this paper. It can be deemed that this assessment is to calculate how much the image quality has been enhanced and the quantity of information has been increased when the fused image is compared with the source images. The experimental results imply that the fractal-based multi-spectral fusion algorithm can effectively preserve the information of man-made objects with a high contrast. It is proved that this algorithm could well preserve features of military targets because that battlefield targets are most man-made objects and in common their images differ from fractal models obviously. Furthermore, the fractal features are not sensitive to the imaging conditions and the movement of targets, so this fractal-based algorithm may be very practical.
Edge detection and image segmentation of space scenes using fractal analyses
NASA Technical Reports Server (NTRS)
Cleghorn, Timothy F.; Fuller, J. J.
1992-01-01
A method was developed for segmenting images of space scenes into manmade and natural components, using fractal dimensions and lacunarities. Calculations of these parameters are presented. Results are presented for a variety of aerospace images, showing that it is possible to perform edge detections of manmade objects against natural background such as those seen in an aerospace environment.
Large-dimension configuration-interaction calculations of positron binding to the group-II atoms
Bromley, M. W. J.; Mitroy, J.
2006-03-15
The configuration-interaction (CI) method is applied to the calculation of the structures of a number of positron binding systems, including e{sup +}Be, e{sup +}Mg, e{sup +}Ca, and e{sup +}Sr. These calculations were carried out in orbital spaces containing about 200 electron and 200 positron orbitals up to l=12. Despite the very large dimensions, the binding energy and annihilation rate converge slowly with l, and the final values do contain an appreciable correction obtained by extrapolating the calculation to the l{yields}{infinity} limit. The binding energies were 0.00317 hartree for e{sup +}Be, 0.0170 hartree for e{sup +}Mg, 0.0189 hartree for e{sup +}Ca, and 0.0131 hartree for e{sup +}Sr.
Kmetyk, L.N.; Yarrington, P.
1989-05-01
Calculations were performed with the CTH and HULL finite difference wavecodes to evaluate computational capabilities for predicting depth and diameter of target cavities produced in high velocity penetration events. The calculations simulated selected tests in a set of armor penetration experiments conducted by the US Army Ballistic Research Laboratory and reported earlier in the literature. The tests and simulations involved penetration of semi-infinite targets by long rod projectiles over a range of impact velocities from 1.3 to 4.5 km/sec. Comparisons are made between the calculated and measured dimensions of the target cavities, and the sensitivity of the predicted results to target property variations is investigated. 9 refs., 18 figs., 3 tabs.
Fractal-based characterization of structural changes in biomedical images
NASA Astrophysics Data System (ADS)
Swarnakar, Vivek; Acharya, Raj S.; Sibata, Claudio H.; Shin, Kyu H.
1996-04-01
In the present work, distinct structures appearing in biomedical images are modeled as fractals. Within an image, the relevant structures are associated to a fractal dimension. Changes in the dimension values, as a function of time, reflect alterations of structural properties. Accurate and robust estimation of this dimension, leads to a precise characterization of changes undergone by the structure. The Continuous Pyramidal Alternating Sequential Filter method is proposed as a robust and accurate fractal dimension estimator. A study on bedrest data of human subjects was conducted. Bedrest is an accepted model for the study of osteoporosis. Here the spine is modeled as a fractal structure. Fractal model were also applied towards analysis of breast cancer and brain tumors. Results from these different studies confirm that fractals can suitably model a variety of biological structures. These studies also suggest that fractal models can be effectively utilized to detect temporal changes undergone by the structures.
Fractal Distribution of Experimentally Generated Pyroclasts
NASA Astrophysics Data System (ADS)
Kueppers, U.; Perugini, D.; Dingwell, D. B.
2005-12-01
Despite recent advances by means of experiments and high-resolution surveys, volcanic eruptions remain highly unpredictable in terms of the type of activity and the duration an imminent eruption will probably exhibit. This uncertainty hinders hazard assessment tremendously. In an effort to counter this problem, a comparison of natural deposits and pyroclasts from laboratory experiments has been undertaken in order to enable estimation of the physical conditions during volcanic eruptions. Three sample sets of Unzen volcano, Japan, have been investigated in order to evaluate the influence of open porosity in combination with applied gas overpressure on the fragmentation behaviour and on the pyroclast generation (fragmentation efficiency). All experiments have been performed at 850 °C and at initial pressure values above the respective fragmentation threshold. The set-up allowed for accurate simulation of explosive volcanic fragmentation whilst investigating the resulting pyroclast generation. The generated pyroclasts have been analysed for their grain-size distribution and the fractality of that distribution. The grain-size distribution was analysed by dry sieving for particles bigger than 250 μm and laser refraction of the suspended particles smaller than 250 μm. Laser refraction was found to be applicable to the size analysis of pyroclasts from natural samples. The grain-size analysis exhibits a clear dependence of applied pressure and open porosity on the resulting pyroclasts: i.e. the fragmentation efficiency was found to have increased with increasing potential energy for fragmentation (gas fraction × applied pressure). The fractal fragmentation theory was applied to the achieved grain-size distribution. The fractal dimension of fragmentation (Df) was calculated for all experiments for samples with different open porosity. Results show a general linear increase of Df, i.e. intensity of fragmentation, as the pressure increases. An additional important point is the variation of intercept of linear fitting of data. In particular, the intercept increases with the open porosity of the samples indicating that the intensity of the fragmentation process increases with the open porosity of the samples. These results indicate that fractal fragmentation theory may allow for quantifying fragmentation processes during explosive volcanic eruptions, a feature that is difficult to study by using classical statistical methods. The results may help in evaluating volcanic risk by estimating the explosivity (e.g. pressure in the conduit and possibly other parameters) from the value of fractal dimension of grain-size distribution of natural deposits. This may give the opportunity to draw iso-Df or iso-explosivity contour maps based on fractal statistics.
ERIC Educational Resources Information Center
Jurgens, Hartmut; And Others
1990-01-01
The production and application of images based on fractal geometry are described. Discussed are fractal language groups, fractal image coding, and fractal dialects. Implications for these applications of geometry to mathematics education are suggested. (CW)
Fractal universe and quantum gravity.
Calcagni, Gianluca
2010-06-25
We propose a field theory which lives in fractal spacetime and is argued to be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and causal. The system flows from an ultraviolet fixed point, where spacetime has Hausdorff dimension 2, to an infrared limit coinciding with a standard four-dimensional field theory. Classically, the fractal world where fields live exchanges energy momentum with the bulk with integer topological dimension. However, the total energy momentum is conserved. We consider the dynamics and the propagator of a scalar field. Implications for quantum gravity, cosmology, and the cosmological constant are discussed. PMID:20867360
Fractal Universe and Quantum Gravity
Calcagni, Gianluca
2010-06-25
We propose a field theory which lives in fractal spacetime and is argued to be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and causal. The system flows from an ultraviolet fixed point, where spacetime has Hausdorff dimension 2, to an infrared limit coinciding with a standard four-dimensional field theory. Classically, the fractal world where fields live exchanges energy momentum with the bulk with integer topological dimension. However, the total energy momentum is conserved. We consider the dynamics and the propagator of a scalar field. Implications for quantum gravity, cosmology, and the cosmological constant are discussed.
Fractal analysis of complex microstructure in castings
Lu, S.Z.; Lipp, D.C.; Hellawell, A.
1995-12-31
Complex microstructures in castings are usually characterized descriptively which often raises ambiguity and makes it difficult to relate the microstructure to the growth kinetics or mechanical properties in processing modeling. Combining the principle of fractal geometry and computer image processing techniques, it is feasible to characterize the complex microstructures numerically by the parameters of fractal dimension, D, and shape factor, a, without ambiguity. Procedures of fractal measurement and analysis are described, and a test case of its application to cast irons is provided. The results show that the irregular cast structures may all be characterized numerically by fractal analysis.
NASA Astrophysics Data System (ADS)
Gao, Guang-Lei; Ding, Guo-Dong; Zhao, Yuan-Yuan; Wu, Bin; Zhang, Yu-Qing; Guo, Jian-Bin; Qin, Shu-Gao; Bao, Yan-Feng; Yu, Ming-Han; Liu, Yun-Dong
2016-02-01
We constructed an aeolian soil database across arid, semi-arid, and dry sub-humid regions, China. Soil particle size distribution was measured with a laser diffraction technique, and fractal dimensions were calculated. The results showed that: (i) the predominant soil particle size distributed in fine and medium sand classifications, and fractal dimensions covered a wide range from 2.0810 to 2.6351; (ii) through logarithmic transformations, fractal dimensions were significantly positive correlated with clay and silt contents (R2 = 0.81 and 0.59, P < 0.01), and significantly negative correlated with sand content (R2 = 0.50, P < 0.01); (3) hierarchical cluster analysis divided the plots into three types which were similar to sand dune types indicating desertification degree. In a large spatial scale, fractal dimensions are still sensitive to wind-induced desertification. Therefore, we highly recommend that fractal dimension be used as a reliable and quantitative parameter to monitor soil environment changes in desertified regions. This improved information provides a firm basis for better understanding of desertification processes.
Fractal structure of lunar topography: An interpretation of topographic characteristics
NASA Astrophysics Data System (ADS)
Cao, Wei; Cai, Zhanchuan; Tang, Zesheng
2015-06-01
Over the years, fractal geometry has been applied extensively in many fields of geoscience. Based on the global gridded data generated from the Lunar Reconnaissance Orbiter, we carry out our fractal measure to interpret lunar fractures by using qualitative (similar ratio) and quantitative (fractal dimension) approaches of fractal geometry. We find that most of the lunar surface exhibits fractal behavior over the given scales ranging from 1 to 256 m. Lunar maria have higher fractal dimensions than other geological units, while those of volcanic areas and highlands are lower than their surroundings. Simple and flat surfaces have low values of similar ratios and these areas indicate low surface roughness and young ages. Older-aged areas, such as the Hertzsprung basin, have low fractal dimensions and high similar ratios by their complicated topography.
NASA Astrophysics Data System (ADS)
Burdzy, Krzysztof; Hołyst, Robert; Pruski, Łukasz
2013-05-01
We investigate a process of random walks of a point particle on a two-dimensional square lattice of size n×n with periodic boundary conditions. A fraction p⩽20% of the lattice is occupied by holes (p represents macroporosity). A site not occupied by a hole is occupied by an obstacle. Upon a random step of the walker, a number of obstacles, M, can be pushed aside. The system approaches equilibrium in (nlnn)2 steps. We determine the distribution of M pushed in a single move at equilibrium. The distribution F(M) is given by Mγ where γ=-1.18 for p=0.1, decreasing to γ=-1.28 for p=0.01. Irrespective of the initial distribution of holes on the lattice, the final equilibrium distribution of holes forms a fractal with fractal dimension changing from a=1.56 for p=0.20 to a=1.42 for p=0.001 (for n=4,000). The trace of a random walker forms a distribution with expected fractal dimension 2.
Fractal geometry of aggregate snowflakes revealed by triple-wavelength radar measurements
NASA Astrophysics Data System (ADS)
Stein, T. H. M.; Westbrook, C. D.; Nicol, J. C.
2015-01-01
Radar reflectivity measurements from three different wavelengths are used to retrieve information about the shape of aggregate snowflakes in deep stratiform ice clouds. Dual-wavelength ratios are calculated for different shape models and compared to observations at 3, 35, and 94 GHz. It is demonstrated that many scattering models, including spherical and spheroidal models, do not adequately describe the aggregate snowflakes that are observed. The observations are consistent with fractal aggregate geometries generated by a physically based aggregation model. It is demonstrated that the fractal dimension of large aggregates can be inferred directly from the radar data. Fractal dimensions close to 2 are retrieved, consistent with previous theoretical models and in situ observations.
Fractal Geometry in the High School Classroom.
ERIC Educational Resources Information Center
Camp, Dane R.
1995-01-01
Discusses classroom activities that involve applications of fractal geometry. Includes an activity sheet that explores Pascal's triangle, Sierpinsky's gasket, and modular arithmetic in two and three dimensions. (Author/MKR)
The Fractal Structure of Basic Particles
NASA Astrophysics Data System (ADS)
Li, Shuming; Li, Lihua; Li, Shuyun; Li, Shuwei
2000-03-01
A single photon with very high energy can form a tiny black hole bound by its gravitational force, whose state is referred as Space-Time Quantum of Action (STQA). The Schwarzschild radius, energy and duration of STQA can be calculated. It is very intriguing to find out that the product of space interval, energy and time interval of STQA is a constant (STQAC). In addition, STQAC is proven to be the minimum of all particles, which means that the product of space interval, energy and time interval for other particles is integer times as much as STQAC. Thus it is reasonable to hypothesize that the STQA is the basic unit of all kinds of particles. We deduced that STQAs construct all the particles in various fractal dimensions. The dimension of the universe is calculated to be three. Since the product of energy and time interval of the basic unit STQA is the Plank constant, the real quantum of action can be found, which leads to a new explanation to the in-determinant principle of Quantum theory. We can foresee many practical applications of this finding. One example is the novel design of fractal antennas that might lead to revolution in wireless communications.
NASA Astrophysics Data System (ADS)
Heck, Andre
The application of fractal mathematics to problems in astrophysics is discussed in reviews and reports. Topics addressed include fractal structures and the angular correlation function of galaxies, the approach to homogeneity of a fractal cell universe, a fractal cascading model for the large-scale galaxy distribution, and fractals and multifractals in the description of the cosmic structure. Consideration is given to a one-dimensional simulation with implications for the homogeneity of the expanding universe, self-gravitational fractal configurations, the fractal dimensions of nebulae, chaotic dynamics in pulsating stars, and the fractal dimension of the solar granulation.
Fractals and cosmological large-scale structure
NASA Technical Reports Server (NTRS)
Luo, Xiaochun; Schramm, David N.
1992-01-01
Observations of galaxy-galaxy and cluster-cluster correlations as well as other large-scale structure can be fit with a 'limited' fractal with dimension D of about 1.2. This is not a 'pure' fractal out to the horizon: the distribution shifts from power law to random behavior at some large scale. If the observed patterns and structures are formed through an aggregation growth process, the fractal dimension D can serve as an interesting constraint on the properties of the stochastic motion responsible for limiting the fractal structure. In particular, it is found that the observed fractal should have grown from two-dimensional sheetlike objects such as pancakes, domain walls, or string wakes. This result is generic and does not depend on the details of the growth process.
Fractal nature of fat crystal networks
NASA Astrophysics Data System (ADS)
Narine, Suresh S.; Marangoni, Alejandro G.
1999-02-01
The quantification of microstructure in fat crystal networks is studied using the relationship of the shear elastic modulus (G') to the volume fraction of solid fat (Φ) via the mass fractal dimension (D) of the network. Results from application of a scaling theory (weak-link regime theory), developed for colloidal gels, to the microstructure of fat crystal networks are presented and discussed. A method to measure mass fractal dimensions and chemical length exponents or backbone fractal dimensions (x) from in situ polarized light microscope (PLM) images of the microstructural network of fat crystals is developed and applied to the fat systems studied. Fractal dimensions measured from in situ PLM images of the various fat systems are in good agreement with fractal dimensions measured using rheological measurements and the weak-link regime theory (percent deviations range from 0.40% to 2.50%). The crystallization behavior of the various fat systems is studied using differential scanning calorimetry, and the potential for altering G' by changing crystallization conditions using the fractal dimension of the network as an indicator is discussed.
Fractals in DNA sequence analysis
NASA Astrophysics Data System (ADS)
Yu, Zu-Guo; Vo, Anh; Gong, Zhi-Min; Long, Shun-Chao
2002-12-01
Fractal methods have been successfully used to study many problems in physics, mathematics, engineering, finance and even in biology. There has been an increasing interest in unravelling the mysteries of DNA; for example, how can we distinguish coding and noncoding sequences and the problems of classification and evolution relationship of organisms are key problems in bioinformatics. Although much research has been carried out by taking into consideration the long-range correlations in DNA sequences and the global fractal dimension has been used in these works by other people, the models and methods are somewhat rough and the results are not satisfactory. In recent years, our group has introduced a time series model (statistical point of view) and a visual representation (geometrical point of view) to DNA sequence analysis. We have also used fractal dimension, correlation dimension, the Hurst exponent and the dimension spectrum (multifractal analysis) to discuss problems in this field. In this paper, we introduce these fractal models and methods and the results of DNA sequence analysis.
Fractal Characterization of Hyperspectral Imagery
NASA Technical Reports Server (NTRS)
Qiu, Hon-Iie; Lam, Nina Siu-Ngan; Quattrochi, Dale A.; Gamon, John A.
1999-01-01
Two Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) hyperspectral images selected from the Los Angeles area, one representing urban and the other, rural, were used to examine their spatial complexity across their entire spectrum of the remote sensing data. Using the ICAMS (Image Characterization And Modeling System) software, we computed the fractal dimension values via the isarithm and triangular prism methods for all 224 bands in the two AVIRIS scenes. The resultant fractal dimensions reflect changes in image complexity across the spectral range of the hyperspectral images. Both the isarithm and triangular prism methods detect unusually high D values on the spectral bands that fall within the atmospheric absorption and scattering zones where signature to noise ratios are low. Fractal dimensions for the urban area resulted in higher values than for the rural landscape, and the differences between the resulting D values are more distinct in the visible bands. The triangular prism method is sensitive to a few random speckles in the images, leading to a lower dimensionality. On the contrary, the isarithm method will ignore the speckles and focus on the major variation dominating the surface, thus resulting in a higher dimension. It is seen where the fractal curves plotted for the entire bandwidth range of the hyperspectral images could be used to distinguish landscape types as well as for screening noisy bands.
Fractal analysis of the 3D microstructure of porous materials
NASA Astrophysics Data System (ADS)
Khlyupin, A. N.; Dinariev, O. Yu.
2015-06-01
Statistical and geometrical characteristics of the microstructure of the porous space of rock samples are investigated on the basis of 3D images obtained by X-ray microtomography. It is shown that the surface of the porous space exhibits fractal properties. As a result of fractal analysis of 3D micromodels, the leading fractal dimension, the multifractal spectra of generalized dimensions, and other structural and geometrical parameters are obtained.
Fractal characterization of brain lesions in CT images
Jauhari, Rajnish K.; Trivedi, Rashmi; Munshi, Prabhat; Sahni, Kamal
2005-12-15
Fractal Dimension (FD) is a parameter used widely for classification, analysis, and pattern recognition of images. In this work we explore the quantification of CT (computed tomography) lesions of the brain by using fractal theory. Five brain lesions, which are portions of CT images of diseased brains, are used for the study. These lesions exhibit self-similarity over a chosen range of scales, and are broadly characterized by their fractal dimensions.
Fractal Theory for Permeability Prediction, Venezuelan and USA Wells
NASA Astrophysics Data System (ADS)
Aldana, Milagrosa; Altamiranda, Dignorah; Cabrera, Ana
2014-05-01
Inferring petrophysical parameters such as permeability, porosity, water saturation, capillary pressure, etc, from the analysis of well logs or other available core data has always been of critical importance in the oil industry. Permeability in particular, which is considered to be a complex parameter, has been inferred using both empirical and theoretical techniques. The main goal of this work is to predict permeability values on different wells using Fractal Theory, based on a method proposed by Pape et al. (1999). This approach uses the relationship between permeability and the geometric form of the pore space of the rock. This method is based on the modified equation of Kozeny-Carman and a fractal pattern, which allows determining permeability as a function of the cementation exponent, porosity and the fractal dimension. Data from wells located in Venezuela and the United States of America are analyzed. Employing data of porosity and permeability obtained from core samples, and applying the Fractal Theory method, we calculated the prediction equations for each well. At the beginning, this was achieved by training with 50% of the data available for each well. Afterwards, these equations were tested inferring over 100% of the data to analyze possible trends in their distribution. This procedure gave excellent results in all the wells in spite of their geographic distance, generating permeability models with the potential to accurately predict permeability logs in the remaining parts of the well for which there are no core samples, using even porority logs. Additionally, empirical models were used to determine permeability and the results were compared with those obtained by applying the fractal method. The results indicated that, although there are empirical equations that give a proper adjustment, the prediction results obtained using fractal theory give a better fit to the core reference data.
Riemann zeros, prime numbers, and fractal potentials.
van Zyl, Brandon P; Hutchinson, David A W
2003-06-01
Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be D=1.5 for the Riemann zeros and D=1.8 for the prime numbers. This result is somewhat surprising since the nearest-neighbor spacings of the Riemann zeros are known to be chaotically distributed, whereas the primes obey almost Poissonlike statistics. Our findings show that the fractal dimension is dependent on both level statistics and spectral rigidity, Delta(3), of the energy levels. PMID:16241330
Determination of fish gender using fractal analysis of ultrasound images.
McEvoy, Fintan J; Tomkiewicz, Jonna; Støttrup, Josianne G; Overton, Julie L; McEvoy, Conni; Svalastoga, Eiliv
2009-01-01
The gender of cod Gadus morhua can be determined by considering the complexity in their gonadal ultrasonographic appearance. The fractal dimension (D(B)) can be used to describe this feature in images. B-mode gonadal ultrasound images in 32 cod, where gender was known, were collected. Fractal analysis was performed on these images and D(B) was determined using the box counting method. A receiver-operating curve (ROC) was drawn for D(B) as a test for male fish. Using a range of D(B) values, the maximum accuracy for this test was calculated and compared with the accuracy for identifying male fish by subjective analysis alone. The mean (and standard deviation) of the fractal dimension D(B) for male fish was 1.554 (0.073) while for female fish it was 1.468 (0.061); the difference was statistically significant (P = 0.001). The area under the ROC curve was 0.84 indicating the value of fractal analysis in gender determination in cod. Maximum accuracy (0.84) for D(B) as a test for male fish was obtained using the threshold value D(B) = 1.5058 compared with an accuracy of 0.78 for subjective image evaluation. The use of two thresholds, D(B) < 1.4475 (females) and D(B) > 1.5054 (males) gives an 80% certainty in the classification result. Fractal analysis is useful for gender determination in cod. This or a similar form of analysis may have wide application in veterinary imaging as a tool for quantification of complexity in images. PMID:19788038
Manera, M; Giari, L; Depasquale, J A; Dezfuli, B S
2016-03-01
The objective of this study was to compare expert versus fractal analysis as new methods to evaluate branchial lamellar pathology in European sea bass Dicentrarchus labrax (Linnaeus, 1758) experimentally exposed to cadmium and to terbuthylazine. In particular, guided expert quantitative and fractal analysis were performed on selected images from semithin sections to test possible differences according to exposure class (unexposed, cadmium exposed, or terbuthylazine exposed) and the discrimination power of the two methods. With respect to guided expert quantitative analysis, the following elementary pathological features were assessed according to pre-determined cover classes: 'epithelial lifting', 'epithelial shrinkage', 'epithelial swelling', 'pillar cells coarctation', 'pillar cells detachment', 'channels fusion', 'chloride cells swelling' and 'chloride cells invasion'. Considering fractal analysis, DB (box dimension), DM (mass dimension), Dx¯ (mean fractal dimension) as fractal dimensions and lacunarity from DM and Dx¯ scan types were calculated both from the outlined and skeletonized (one pixel wide lines) images. Despite significant differences among experimental classes, only expert analysis provided good discrimination with correct classification of 91.7 % of the original cases, and of 87.5 % of the cross-validated cases, with a sensitivity of 95.45 % and 91.3 %, respectively, and a specificity of 75 % in both cases. Guided expert quantitative analysis appears to be a reliable method to objectively characterize fish gill pathology and may represent a powerful tool in environmental biomonitoring to ensure proper standardization and reproducibility. Though fractal analysis did not equal the discrimination power of the expert method, it certainly warrants further study to evaluate local variations in complexity or possible multiple scaling rules. PMID:26469527
Spontaneous optical fractal pattern formation.
Huang, J G; McDonald, G S
2005-05-01
We report, for the first time, spontaneous nonlinear optical spatial fractals. The proposed generic mechanism employs intrinsic nonlinear dynamics both to generate an initial pattern seed and to fill out structure across decades of spatial scale. We demonstrate this in one of the simplest of nonlinear optical systems, composed of a Kerr slice and a single-feedback mirror. In this case, the smallest pattern scales are limited by either the optical wavelength or the diffusion length of the medium photoexcitation. The dimension characteristics of these particular fractals are also derived. PMID:15904294
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Lam, Nina Siu-Ngan; Quattrochi, Dale A.
1999-01-01
Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images is the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimension-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.
Nucleation of squat cracks in rail, calculation of crack initiation angles in three dimensions
NASA Astrophysics Data System (ADS)
Naeimi, Meysam; Li, Zili; Dollevoet, Rolf
2015-07-01
A numerical model of wheel-track system is developed for nucleation of squat-type fatigue cracks in rail material. The model is used for estimating the angles of squat cracks in three dimensions. Contact mechanics and multi-axial fatigue analysis are combined to study the crack initiation mechanism in rails. Nonlinear material properties, actual wheel-rail geometries and realistic loading conditions are considered in the modelling process. Using a 3D explicit finite element analysis the transient rolling contact behaviour of wheel on rail is simulated. Employing the critical plane concept, the material points with the largest possibility of crack initiation are determined; based on which, the 3D orientations/angles of the possible squat cracks are estimated. Numerical estimations are compared with sample results of experimental observations on a rail specimen with squat from the site. The findings suggest a proper agreement between results of modelling and experiment. It is observed that squat cracks initiate at an in-plane angle around 13°-22° relative to the rail surface. The initiation angle seen on surface plane is calculated around 29°-48°, while the crack tend to initiate in angles around 25°-31° in the rail cross-section.
Koo, Imhoi; Zhao, Yaping; Zhang, Jun; Kim, Seongho; Zhang, Xiang
2012-01-01
A method of calculating the second dimension hold-up time for comprehensive two-dimensional gas chromatographic (GC×GC) data was developed by incorporating the temperature information of the second dimension column into the calculation model. The model was developed by investigating the relationship between the coefficients in each of six literature reported nonlinear models and the relationship between each coefficient and the second dimension column temperature. The most robust nonlinear function was selected and further used to construct the new model for calculation of the second dimension retention time, in which the coefficients that have significant correlation with the column temperature are replaced with expressions of column temperature. An advantage of the proposed equation is that eight parameters could explain the second dimension hold-up time as well as retention time corresponding to n-alkanes and column temperature in the entire chromatographic region, including the chromatographic region not bounded by the retention times of n-alkanes. To optimize the experimental design for collecting the isothermal data of n-alkanes to create the second dimension hold-up time model, the column temperature difference and the number of isothermal experiments should be considered simultaneously. It was concluded that a total of 5 or 6 isothermal experiments with temperature difference of 40 or 50 °C are enough to generate an accurate model. The test mean squared error (MSE) of those conditions ranges from 0.0428 to 0.0532 for calculation of the second dimension hold-up time for GC×GC data. PMID:22964052
On the ubiquitous presence of fractals and fractal concepts in pharmaceutical sciences: a review.
Pippa, Natassa; Dokoumetzidis, Aristides; Demetzos, Costas; Macheras, Panos
2013-11-18
Fractals have been very successful in quantifying nature's geometrical complexity, and have captured the imagination of scientific community. The development of fractal dimension and its applications have produced significant results across a wide variety of biomedical applications. This review deals with the application of fractals in pharmaceutical sciences and attempts to account the most important developments in the fields of pharmaceutical technology, especially of advanced Drug Delivery nano Systems and of biopharmaceutics and pharmacokinetics. Additionally, fractal kinetics, which has been applied to enzyme kinetics, drug metabolism and absorption, pharmacokinetics and pharmacodynamics are presented. This review also considers the potential benefits of using fractal analysis along with considerations of nonlinearity, scaling, and chaos as calibration tools to obtain information and more realistic description on different parts of pharmaceutical sciences. As a conclusion, the purpose of the present work is to highlight the presence of fractal geometry in almost all fields of pharmaceutical research. PMID:24025993
NASA Astrophysics Data System (ADS)
Wuorinen, Charles
2015-03-01
Any of the arts may produce exemplars that have fractal characteristics. There may be fractal painting, fractal poetry, and the like. But these will always be specific instances, not necessarily displaying intrinsic properties of the art-medium itself. Only music, I believe, of all the arts possesses an intrinsically fractal character, so that its very nature is fractally determined. Thus, it is reasonable to assert that any instance of music is fractal...
[Fractal analysis of liver fibrosis].
Soda, G; Nardoni, S; Bosco, D; Grizzi, F; Dioguardi, N; Melis, M
2003-04-01
This study realized by two different study groups use of Fractal geometry to quantify the complex collagen deposition during chronic liver disease. Thirty standard needle liver biopsy specimens were obtained from patients with chronic HCV-related disease. Three mu-thick sections were cut and stained by means of Picrosirius stain, in order to visualise collagen matrix. The degree of fibrosis was measured using a quantitative scoring system based on the computer-assisted evaluation of the fractal dimension of the deposited collagen surface. The obtained results by both study groups, show that the proposed method is reproducible, rapid and inexpensive. The complex distribution of its collagenous components can be quantified using a single numerical score. This study demonstrated that it is possible to quantify the collagen's irregularity in an objective manner, and that the study of the fractal properties of the collagen shapes is likely to reveal more about its structure and the complex behaviour of its development. PMID:12768879
Characterizing Hyperspectral Imagery (AVIRIS) Using Fractal Technique
NASA Technical Reports Server (NTRS)
Qiu, Hong-Lie; Lam, Nina Siu-Ngan; Quattrochi, Dale
1997-01-01
With the rapid increase in hyperspectral data acquired by various experimental hyperspectral imaging sensors, it is necessary to develop efficient and innovative tools to handle and analyze these data. The objective of this study is to seek effective spatial analytical tools for summarizing the spatial patterns of hyperspectral imaging data. In this paper, we (1) examine how fractal dimension D changes across spectral bands of hyperspectral imaging data and (2) determine the relationships between fractal dimension and image content. It has been documented that fractal dimension changes across spectral bands for the Landsat-TM data and its value [(D)] is largely a function of the complexity of the landscape under study. The newly available hyperspectral imaging data such as that from the Airborne Visible Infrared Imaging Spectrometer (AVIRIS) which has 224 bands, covers a wider spectral range with a much finer spectral resolution. Our preliminary result shows that fractal dimension values of AVIRIS scenes from the Santa Monica Mountains in California vary between 2.25 and 2.99. However, high fractal dimension values (D > 2.8) are found only from spectral bands with high noise level and bands with good image quality have a fairly stable dimension value (D = 2.5 - 2.6). This suggests that D can also be used as a summary statistics to represent the image quality or content of spectral bands.
Dimensionally frustrated diffusion towards fractal adsorbers.
Nair, Pradeep R; Alam, Muhammad A
2007-12-21
Diffusion towards a fractal adsorber is a well-researched problem with many applications. While the steady-state flux towards such adsorbers is known to be characterized by the fractal dimension (D{F}) of the surface, the more general problem of time-dependent adsorption kinetics of fractal surfaces remains poorly understood. In this Letter, we show that the time-dependent flux to fractal adsorbers (1
Random sequential adsorption on fractals
NASA Astrophysics Data System (ADS)
Ciesla, Michal; Barbasz, Jakub
2012-07-01
Irreversible adsorption of spheres on flat collectors having dimension d < 2 is studied. Molecules are adsorbed on Sierpinski's triangle and carpet-like fractals (1 < d < 2), and on general Cantor set (d < 1). Adsorption process is modeled numerically using random sequential adsorption (RSA) algorithm. The paper concentrates on measurement of fundamental properties of coverages, i.e., maximal random coverage ratio and density autocorrelation function, as well as RSA kinetics. Obtained results allow to improve phenomenological relation between maximal random coverage ratio and collector dimension. Moreover, simulations show that, in general, most of known dimensional properties of adsorbed monolayers are valid for non-integer dimensions.
Random sequential adsorption on fractals.
Ciesla, Michal; Barbasz, Jakub
2012-07-28
Irreversible adsorption of spheres on flat collectors having dimension d < 2 is studied. Molecules are adsorbed on Sierpinski's triangle and carpet-like fractals (1 < d < 2), and on general Cantor set (d < 1). Adsorption process is modeled numerically using random sequential adsorption (RSA) algorithm. The paper concentrates on measurement of fundamental properties of coverages, i.e., maximal random coverage ratio and density autocorrelation function, as well as RSA kinetics. Obtained results allow to improve phenomenological relation between maximal random coverage ratio and collector dimension. Moreover, simulations show that, in general, most of known dimensional properties of adsorbed monolayers are valid for non-integer dimensions. PMID:22852643
A fractal-like resistive network
NASA Astrophysics Data System (ADS)
Saggese, A.; De Luca, R.
2014-11-01
The equivalent resistance of a fractal-like network is calculated by means of approaches similar to those employed in defining the equivalent resistance of an infinite ladder. Starting from an elementary triangular circuit, a fractal-like network, named after Saggese, is developed. The equivalent resistance of finite approximations of this network is measured, and the didactical implications of the model are highlighted.
Aesthetic Responses to Exact Fractals Driven by Physical Complexity
Bies, Alexander J.; Blanc-Goldhammer, Daryn R.; Boydston, Cooper R.; Taylor, Richard P.; Sereno, Margaret E.
2016-01-01
Fractals are physically complex due to their repetition of patterns at multiple size scales. Whereas the statistical characteristics of the patterns repeat for fractals found in natural objects, computers can generate patterns that repeat exactly. Are these exact fractals processed differently, visually and aesthetically, than their statistical counterparts? We investigated the human aesthetic response to the complexity of exact fractals by manipulating fractal dimensionality, symmetry, recursion, and the number of segments in the generator. Across two studies, a variety of fractal patterns were visually presented to human participants to determine the typical response to exact fractals. In the first study, we found that preference ratings for exact midpoint displacement fractals can be described by a linear trend with preference increasing as fractal dimension increases. For the majority of individuals, preference increased with dimension. We replicated these results for other exact fractal patterns in a second study. In the second study, we also tested the effects of symmetry and recursion by presenting asymmetric dragon fractals, symmetric dragon fractals, and Sierpinski carpets and Koch snowflakes, which have radial and mirror symmetry. We found a strong interaction among recursion, symmetry and fractal dimension. Specifically, at low levels of recursion, the presence of symmetry was enough to drive high preference ratings for patterns with moderate to high levels of fractal dimension. Most individuals required a much higher level of recursion to recover this level of preference in a pattern that lacked mirror or radial symmetry, while others were less discriminating. This suggests that exact fractals are processed differently than their statistical counterparts. We propose a set of four factors that influence complexity and preference judgments in fractals that may extend to other patterns: fractal dimension, recursion, symmetry and the number of segments in a pattern. Conceptualizations such as Berlyne’s and Redies’ theories of aesthetics also provide a suitable framework for interpretation of our data with respect to the individual differences that we detect. Future studies that incorporate physiological methods to measure the human aesthetic response to exact fractal patterns would further elucidate our responses to such timeless patterns.
Computerized analysis of mammographic parenchymal patterns using fractal analysis
NASA Astrophysics Data System (ADS)
Li, Hui; Giger, Maryellen L.; Huo, Zhimin; Olopade, Olufunmilayo I.; Chinander, Michael R.; Lan, Li; Bonta, Ioana R.
2003-05-01
Mammographic parenchymal patterns have been shown to be associated with breast cancer risk. Fractal-based texture analyses, including box-counting methods and Minkowski dimension, were performed within parenchymal regions of normal mammograms of BRCA1/BRCA2 gene mutation carriers and within those of women at low risk for developing breast cancer. Receiver Operating Characteristic (ROC) analysis was used to assess the performance of the computerized radiographic markers in the task of distinguishing between high and low-risk subjects. A multifractal phenomenon was observed with the fractal analyses. The high frequency component of fractal dimension from the conventional box-counting technique yielded an Az value of 0.84 in differentiating between two groups, while using the LDA to estimate the fractal dimension yielded an Az value of 0.91 for the high frequency component. An Az value of 0.82 was obtained with fractal dimensions extracted using the Minkowski algorithm.
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Lam, Nina Siu-Ngan; Quattrochi, Dale A.
1999-01-01
Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images of the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimensional-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.
NASA Technical Reports Server (NTRS)
Pandey, Apoorva; Chakrabarty, Rajan K.; Liu, Li; Mishchenko, Michael I.
2015-01-01
Soot aggregates (SAs)-fractal clusters of small, spherical carbonaceous monomers-modulate the incoming visible solar radiation and contribute significantly to climate forcing. Experimentalists and climate modelers typically assume a spherical morphology for SAs when computing their optical properties, causing significant errors. Here, we calculate the optical properties of freshly-generated (fractal dimension Df = 1.8) and aged (Df = 2.6) SAs at 550 nm wavelength using the numericallyexact superposition T-Matrix method. These properties were expressed as functions of equivalent aerosol diameters as measured by contemporary aerosol instruments. This work improves upon previous efforts wherein SA optical properties were computed as a function of monomer number, rendering them unusable in practical applications. Future research will address the sensitivity of variation in refractive index, fractal prefactor, and monomer overlap of SAs on the reported empirical relationships.
Fractal nature of humic materials
Rice, J.A. . Dept. of Chemistry); Lin, J.S. )
1992-01-01
Fractals are geometric representatives of strongly disordered systems whose structure is described by nonintegral dimensions. A fundamental tenet of fractal geometry is that disorder persists at any characterization scale-length used to describe the system. The nonintegral nature of these fractal dimensions is the result of the realization that a disordered system must possess more structural detail than an ordered system with classical dimensions of 1, 2, or 3 in order to accommodate this disorder within disorder.'' Thus from a fractal perspective, disorder is seen as an inherent characteristic of the system rather than as a perturbative phenomena forced upon it. Humic materials are organic substances that are formed by the profound alteration of organic matter in a natural environment. They can be operationally divided into 3 fractions; humic acid (soluble in base), fulvic acid (soluble in acid or base), and humin (insoluble in acid or base). Each of these fraction has been shown to be an extremely heterogeneous mixture. These mixtures have proven so intractable that they may represent the ultimate in molecular disorder. In fact, based on the characteristics that humic materials must possess in order to perform their functions in natural systems, it has been proposed that the fundamental chemical characteristic of a humic material is not a discrete chemical structure but a pronounced lack of order on a molecular level. If the fundamental chemical characteristic of a humic material is a strongly disordered nature, as has been proposed, then humic materials should be amenable to characterization by fractal geometry. The purpose of this paper is to test this hypothesis.
Fractal nature of humic materials
Rice, J.A.; Lin, J.S.
1992-03-01
Fractals are geometric representatives of strongly disordered systems whose structure is described by nonintegral dimensions. A fundamental tenet of fractal geometry is that disorder persists at any characterization scale-length used to describe the system. The nonintegral nature of these fractal dimensions is the result of the realization that a disordered system must possess more structural detail than an ordered system with classical dimensions of 1, 2, or 3 in order to accommodate this ``disorder within disorder.`` Thus from a fractal perspective, disorder is seen as an inherent characteristic of the system rather than as a perturbative phenomena forced upon it. Humic materials are organic substances that are formed by the profound alteration of organic matter in a natural environment. They can be operationally divided into 3 fractions; humic acid (soluble in base), fulvic acid (soluble in acid or base), and humin (insoluble in acid or base). Each of these fraction has been shown to be an extremely heterogeneous mixture. These mixtures have proven so intractable that they may represent the ultimate in molecular disorder. In fact, based on the characteristics that humic materials must possess in order to perform their functions in natural systems, it has been proposed that the fundamental chemical characteristic of a humic material is not a discrete chemical structure but a pronounced lack of order on a molecular level. If the fundamental chemical characteristic of a humic material is a strongly disordered nature, as has been proposed, then humic materials should be amenable to characterization by fractal geometry. The purpose of this paper is to test this hypothesis.
Microtopographic Inspection and Fractal Analysis of Skin Neoplasia
NASA Astrophysics Data System (ADS)
Costa, Manuel F. M.; Hipolito, Alberto Valencia; Gutierrez, Gustavo Fidel; Chanona, Jorge; Gallegos, Eva Ramn
2008-04-01
Early detection of skin cancer is fundamental to a successful treatment. Changes in the shape, including the relief, of skin lesions are an indicator of a possible malignity. Optical microtopographic inspection of skin lesions can be used to identify diagnostic patterns of benign and malign skin' lesions. Statistical parameters like the mean roughness (Ra) may allow the discrimination between different types of lesions and degree of malignity. Fractal analysis of bi-dimensional and 3D images of skin lesions can validate or complement that assessment by calculation of its fractal dimensions (FD). On the study herein reported the microtopographic inspection of the skin lesions were performed using the optical triangulation based microtopographer developed at the Physics Department of the University of Minho, MICROTOP.03.MFC. The patients that participated in this research work were men and women older than 15 years with the clinical and histopathology diagnoses of: melanoma, basocellular carcinoma, epidermoide carcinoma, actinic keratosis, keratoacantosis and benign nevus. Latex impressions of the lesions were taken and microtopographically analyzed. Characteristic information for each type of studied lesion was obtained. For melanoma it was observed that on the average these tumors present an increased roughness of around 67 percent compared to the roughness of the healthy skin. This feature allows the distinction from other tumors as basocellular carcinoma (were the roughness increase was in the average of 49 percent) and benign lesions as the epidermoide cyst (37 percent) or the seborrhea keratosis (4 percent). Tumor size and roughness are directly proportional to the grade of malignality. The characterization of the fractal geometry of 2D (histological slides) and 3D images of skin lesions was performed by obtaining its FD evaluated by means of the Box counting method. Results obtained showed that the average fractal dimension of histological slide images (FDh) corresponding to some neoplasia is higher (1.334+/-0.072) than those for healthy skin (1.091+/-0.082). A significant difference between the fractal dimensions of neoplasia and healhty skin (>0.001) was registered. The FD of microtopography maps (FDm) can also distinguish between healthy and malignant tissue in general (2.277+/-0.070 to 2.309+/-0.040), but not discriminate the different types of skin neoplasias. The combination of the rugometric evaluation and fractal geometry characterization provides valuable information about the malignity of skin lesions and type of lesion.
NASA Astrophysics Data System (ADS)
López, Carmen; Martí, Joan; Abella, Rafael; Tarraga, Marta
2014-05-01
The impossibility of observing magma migration inside the crust obliges us to rely on geophysical data and mathematical modelling to interpret precursors and to forecast volcanic eruptions. Of the geophysical signals that may be recorded before and during an eruption, deformation and seismicity are two of the most relevant as they are directly related to its dynamic. The final phase of the unrest episode that preceded the 2011-2012 eruption on El Hierro (Canary Islands) was characterized by local and accelerated deformation and seismic energy release indicating an increasing fracturing and a migration of the magma. Application of time varying fractal analysis to the seismic data and the characterization of the seismicity pattern and the strain and the stress rates allow us to identify different stages in the source mechanism and to infer the geometry of the path used by the magma and associated fluids to reach the Earth's surface. The results obtained illustrate the relevance of such studies to understanding volcanic unrest and the causes that govern the initiation of volcanic eruptions.
NASA Astrophysics Data System (ADS)
López, Carmen; Martí, Joan; Abella, Rafael; Tarraga, Marta
2014-07-01
The impossibility of observing magma migration inside the crust obliges us to rely on geophysical data and mathematical modelling to interpret precursors and to forecast volcanic eruptions. Of the geophysical signals that may be recorded before and during an eruption, deformation and seismicity are two of the most relevant as they are directly related to its dynamic. The final phase of the unrest episode that preceded the 2011-2012 eruption on El Hierro (Canary Islands) was characterized by local and accelerated deformation and seismic energy release indicating an increasing fracturing and a migration of the magma. Application of time varying fractal analysis to the seismic data and the characterization of the seismicity pattern and the strain and the stress rates allow us to identify different stages in the source mechanism and to infer the geometry of the path used by the magma and associated fluids to reach the Earth's surface. The results obtained illustrate the relevance of such studies to understanding volcanic unrest and the causes that govern the initiation of volcanic eruptions.
Fractal Analysis of DNA Sequence Data
NASA Astrophysics Data System (ADS)
Berthelsen, Cheryl Lynn
DNA sequence databases are growing at an almost exponential rate. New analysis methods are needed to extract knowledge about the organization of nucleotides from this vast amount of data. Fractal analysis is a new scientific paradigm that has been used successfully in many domains including the biological and physical sciences. Biological growth is a nonlinear dynamic process and some have suggested that to consider fractal geometry as a biological design principle may be most productive. This research is an exploratory study of the application of fractal analysis to DNA sequence data. A simple random fractal, the random walk, is used to represent DNA sequences. The fractal dimension of these walks is then estimated using the "sandbox method." Analysis of 164 human DNA sequences compared to three types of control sequences (random, base -content matched, and dimer-content matched) reveals that long-range correlations are present in DNA that are not explained by base or dimer frequencies. The study also revealed that the fractal dimension of coding sequences was significantly lower than sequences that were primarily noncoding, indicating the presence of longer-range correlations in functional sequences. The multifractal spectrum is used to analyze fractals that are heterogeneous and have a different fractal dimension for subsets with different scalings. The multifractal spectrum of the random walks of twelve mitochondrial genome sequences was estimated. Eight vertebrate mtDNA sequences had uniformly lower spectra values than did four invertebrate mtDNA sequences. Thus, vertebrate mitochondria show significantly longer-range correlations than do invertebrate mitochondria. The higher multifractal spectra values for invertebrate mitochondria suggest a more random organization of the sequences. This research also includes considerable theoretical work on the effects of finite size, embedding dimension, and scaling ranges.
Fractal analysis of DNA sequence data
Berthelsen, C.L.
1993-01-01
DNA sequence databases are growing at an almost exponential rate. New analysis methods are needed to extract knowledge about the organization of nucleotides from this vast amount of data. Fractal analysis is a new scientific paradigm that has been used successfully in many domains including the biological and physical sciences. Biological growth is a nonlinear dynamic process and some have suggested that to consider fractal geometry as a biological design principle may be most productive. This research is an exploratory study of the application of fractal analysis to DNA sequence data. A simple random fractal, the random walk, is used to represent DNA sequences. The fractal dimension of these walks is then estimated using the [open quote]sandbox method[close quote]. Analysis of 164 human DNA sequences compared to three types of control sequences (random, base-content matched, and dimer-content matched) reveals that long-range correlations are present in DNA that are not explained by base or dimer frequencies. The study also revealed that the fractal dimension of coding sequences was significantly lower than sequences that were primarily noncoding, indicating the presence of longer-range correlations in functional sequences. The multifractal spectrum is used to analyze fractals that are heterogeneous and have a different fractal dimension for subsets with different scalings. The multifractal spectrum of the random walks of twelve mitochondrial genome sequences was estimated. Eight vertebrate mtDNA sequences had uniformly lower spectra values than did four invertebrate mtDNA sequences. Thus, vertebrate mitochondria show significantly longer-range correlations than to invertebrate mitochondria. The higher multifractal spectra values for invertebrate mitochondria suggest a more random organization of the sequences. This research also includes considerable theoretical work on the effects of finite size, embedding dimension, and scaling ranges.
Characterization of branch complexity by fractal analyses
Alados, C.L.; Escos, J.; Emlen, J.M.; Freeman, D.C.
1999-01-01
The comparison between complexity in the sense of space occupancy (box-counting fractal dimension D(c) and information dimension D1) and heterogeneity in the sense of space distribution (average evenness index f and evenness variation coefficient J(cv)) were investigated in mathematical fractal objects and natural branch structures. In general, increased fractal dimension was paired with low heterogeneity. Comparisons between branch architecture in Anthyllis cytisoides under different slope exposure and grazing impact revealed that branches were more complex and more homogeneously distributed for plants on northern exposures than southern, while grazing had no impact during a wet year. Developmental instability was also investigated by the statistical noise of the allometric relation between internode length and node order. In conclusion, our study demonstrated that fractal dimension of branch structure can be used to analyze the structural organization of plants, especially if we consider not only fractal dimension but also shoot distribution within the canopy (lacunarity). These indexes together with developmental instability analyses are good indicators of growth responses to the environment.
NASA Astrophysics Data System (ADS)
Kober, Martin; Koch, Benjamin; Bleicher, Marcus
2007-12-01
We calculate the inclusive cross section of double Z-boson production within large extra dimensions at the Large Hadron Collider (LHC). Using perturbatively quantized gravity in the ADD (Arkani-Hamed, Dvali, Dimopoulos) model we perform a first order calculation of the graviton mediated contribution to the pp→ZZ+x cross section. At low energies (e.g. Tevatron) this additional contribution is very small, making it virtually unobservable, for a fundamental mass scale above 2500 GeV. At LHC energies, however, the calculation indicates that the ZZ-production rate within the ADD model should differ significantly from the standard model if the new fundamental mass scale would be below 15000 GeV. A comparison with the observed production rate at the LHC might therefore provide direct hints on the number and structure of the extra dimensions.
Magnetohydrodynamics of fractal media
Tarasov, Vasily E.
2006-05-15
The fractal distribution of charged particles is considered. An example of this distribution is the charged particles that are distributed over the fractal. The fractional integrals are used to describe fractal distribution. These integrals are considered as approximations of integrals on fractals. Typical turbulent media could be of a fractal structure and the corresponding equations should be changed to include the fractal features of the media. The magnetohydrodynamics equations for fractal media are derived from the fractional generalization of integral Maxwell equations and integral hydrodynamics (balance) equations. Possible equilibrium states for these equations are considered.
Fractality à la carte: a general particle aggregation model.
Nicolás-Carlock, J R; Carrillo-Estrada, J L; Dossetti, V
2016-01-01
In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters' fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension. PMID:26781204
Growth of fractal structures in flames with silicon admixture
NASA Astrophysics Data System (ADS)
Smirnov, B. M.; Dutka, M.; van Essen, V. M.; Gersen, S.; Visser, P.; Vainchtein, D.; De Hosson, J. Th. M.; Levinsky, H. B.; Mokhov, A. V.
2012-06-01
Transmission electron microscopy (TEM) measurements and theoretical analysis are combined to construct the physical picture of formation of SiO2 fractal aggregates in a methane/hexamethyldisiloxane/air atmospheric pressure flame. The formation of SiO2 fractal aggregates is described as a multistage process. The first stage is combustion of fuel in a narrow flame front region with formation of main combustion products including SiO2 molecules. Further downstream SiO2 molecules join in liquid nanoclusters. After cooling combustion products due to heat losses to surroundings, the nanoclusters become solid in a cold flame region and join in fractal aggregates there. Along with growth of fractal aggregates, the restructuring process proceeds in a cold flame region that leads to the decrease of the fractal dimension of fractal aggregates. The measured parameters of fractal aggregates are in accord with those followed from theoretical models.
Fractality à la carte: a general particle aggregation model
Nicolás-Carlock, J. R.; Carrillo-Estrada, J. L.; Dossetti, V.
2016-01-01
In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters’ fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension. PMID:26781204
Fractality à la carte: a general particle aggregation model
NASA Astrophysics Data System (ADS)
Nicolás-Carlock, J. R.; Carrillo-Estrada, J. L.; Dossetti, V.
2016-01-01
In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters’ fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension.
Stability limits for bioconvective fractals - Microgravity prospects
NASA Technical Reports Server (NTRS)
Noever, David A.
1992-01-01
Fractal objects are delicate aggregates which show self-similar behavior and vanishing density for increasing length scales. In practice real fractals in nature however possess only a limited region of verifiable self-similarity. As natural fractal objects increase in size, they become easier to disrupt mechanically. Herein the effects of thermal vibrations and gravity are investigated as deforming forces on fractal aggregation. Example calculations are carried out on a biological fractal formed from the surface aggregation of various cells such as alga and bacteria. For typical cell parameters, the predicted diameter of this so-called 'bioconvective' fractal agrees well with the observed limits of about 5 cm. On earth, this size represents an experimental maximum for finding bioconvective fractal objects. To extend this size range of fractals available for statistical study, a reduced gravity environment offers one way to achieve larger fractals. For these enhanced sizes, the present scaling predicts that microgravity can yield up to a 35-fold improvement in extending statistical resolution.
Shimizu, Wataru; Nakamura, Satoshi; Sato, Takaaki; Murakami, Yasushi
2012-08-21
Amorphous titanium dioxide (TiO(2)) thin films exhibiting high refractive indices (n ≈ 2.1) and high transparency were fabricated by spin-coating titanium oxide liquid precursors having a weakly branched polymeric structure. The precursor solution was prepared from titanium tetra-n-butoxide (TTBO) via the catalytic sol-gel process with hydrazine monohydrochloride used as a salt catalyst, which serves as a conjugate acid-base pair catalyst. Our unique catalytic sol-gel technique accelerated the overall polycondensation reaction of partially hydrolyzed alkoxides, which facilitated the formation of liner polymer-like titanium oxide aggregates having a low fractal dimension of ca. (5)/(3), known as a characteristic of the so-called "expanded polymer chain". Such linear polymeric features are essential to the production of highly dense amorphous TiO(2) thin films; mutual interpenetration of the linear polymeric aggregates avoided the creation of void space that is often generated by the densification of high-fractal-dimension (particle-like) aggregates produced in a conventional sol-gel process. The mesh size of the titanium oxide polymers can be tuned either by water concentration or the reaction time, and the smaller mesh size in the liquid precursor led to a higher n value of the solid thin film, thanks to its higher local electron density. The reaction that required no addition of organic ligand to stabilize titanium alkoxides was advantageous to overcoming issues from organic residues such as coloration. The dense amorphous film structure suppressed light scattering loss owing to its extremely smooth surface and the absence of inhomogeneous grains or particles. Furthermore, the fabrication can be accomplished at a low heating temperature of <80 °C. Indeed, we successfully obtained a transparent film with a high refractive index of n = 2.064 (at λ = 633 nm) on a low-heat-resistance plastic, poly(methyl methacrylate), at 60 °C. The result offers an efficient route to high-refractive-index amorphous TiO(2) films as well as base materials for a wider range of applications. PMID:22817104
Fuzzy fractals, chaos, and noise
Zardecki, A.
1997-05-01
To distinguish between chaotic and noisy processes, the authors analyze one- and two-dimensional chaotic mappings, supplemented by the additive noise terms. The predictive power of a fuzzy rule-based system allows one to distinguish ergodic and chaotic time series: in an ergodic series the likelihood of finding large numbers is small compared to the likelihood of finding them in a chaotic series. In the case of two dimensions, they consider the fractal fuzzy sets whose {alpha}-cuts are fractals, arising in the context of a quadratic mapping in the extended complex plane. In an example provided by the Julia set, the concept of Hausdorff dimension enables one to decide in favor of chaotic or noisy evolution.
Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics
NASA Astrophysics Data System (ADS)
Chernodub, Maxim N.; Ouvry, Stéphane
2015-10-01
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasimomentum space, the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider web in contrast to the Hofstadter butterfly for unbiased motion.
Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics.
Chernodub, Maxim N; Ouvry, Stéphane
2015-10-01
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasimomentum space, the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider web in contrast to the Hofstadter butterfly for unbiased motion. PMID:26565163
Fractal analysis of scatter imaging signatures to distinguish breast pathologies
NASA Astrophysics Data System (ADS)
Eguizabal, Alma; Laughney, Ashley M.; Krishnaswamy, Venkataramanan; Wells, Wendy A.; Paulsen, Keith D.; Pogue, Brian W.; López-Higuera, José M.; Conde, Olga M.
2013-02-01
Fractal analysis combined with a label-free scattering technique is proposed for describing the pathological architecture of tumors. Clinicians and pathologists are conventionally trained to classify abnormal features such as structural irregularities or high indices of mitosis. The potential of fractal analysis lies in the fact of being a morphometric measure of the irregular structures providing a measure of the object's complexity and self-similarity. As cancer is characterized by disorder and irregularity in tissues, this measure could be related to tumor growth. Fractal analysis has been probed in the understanding of the tumor vasculature network. This work addresses the feasibility of applying fractal analysis to the scattering power map (as a physical modeling) and principal components (as a statistical modeling) provided by a localized reflectance spectroscopic system. Disorder, irregularity and cell size variation in tissue samples is translated into the scattering power and principal components magnitude and its fractal dimension is correlated with the pathologist assessment of the samples. The fractal dimension is computed applying the box-counting technique. Results show that fractal analysis of ex-vivo fresh tissue samples exhibits separated ranges of fractal dimension that could help classifier combining the fractal results with other morphological features. This contrast trend would help in the discrimination of tissues in the intraoperative context and may serve as a useful adjunct to surgeons.
Comparison of ictal and interictal EEG signals using fractal features.
Wang, Yu; Zhou, Weidong; Yuan, Qi; Li, Xueli; Meng, Qingfang; Zhao, Xiuhe; Wang, Jiwen
2013-12-01
The feature analysis of epileptic EEG is very significant in diagnosis of epilepsy. This paper introduces two nonlinear features derived from fractal geometry for epileptic EEG analysis. The features of blanket dimension and fractal intercept are extracted to characterize behavior of EEG activities, and then their discriminatory power for ictal and interictal EEGs are compared by means of statistical methods. It is found that there is significant difference of the blanket dimension and fractal intercept between interictal and ictal EEGs, and the difference of the fractal intercept feature between interictal and ictal EEGs is more noticeable than the blanket dimension feature. Furthermore, these two fractal features at multi-scales are combined with support vector machine (SVM) to achieve accuracies of 97.58% for ictal and interictal EEG classification and 97.13% for normal, ictal and interictal EEG classification. PMID:24156671
Fractal analysis of yeast cell optical speckle
NASA Astrophysics Data System (ADS)
Flamholz, A.; Schneider, P. S.; Subramaniam, R.; Wong, P. K.; Lieberman, D. H.; Cheung, T. D.; Burgos, J.; Leon, K.; Romero, J.
2006-02-01
Steady state laser light propagation in diffuse media such as biological cells generally provide bulk parameter information, such as the mean free path and absorption, via the transmission profile. The accompanying optical speckle can be analyzed as a random spatial data series and its fractal dimension can be used to further classify biological media that show similar mean free path and absorption properties, such as those obtained from a single population. A population of yeast cells can be separated into different portions by centrifuge, and microscope analysis can be used to provide the population statistics. Fractal analysis of the speckle suggests that lower fractal dimension is associated with higher cell packing density. The spatial intensity correlation revealed that the higher cell packing gives rise to higher refractive index. A calibration sample system that behaves similar as the yeast samples in fractal dimension, spatial intensity correlation and diffusion was selected. Porous silicate slabs with different refractive index values controlled by water content were used for system calibration. The porous glass as well as the yeast random spatial data series fractal dimension was found to depend on the imaging resolution. The fractal method was also applied to fission yeast single cell fluorescent data as well as aging yeast optical data; and consistency was demonstrated. It is concluded that fractal analysis can be a high sensitivity tool for relative comparison of cell structure but that additional diffusion measurements are necessary for determining the optimal image resolution. Practical application to dental plaque bio-film and cam-pill endoscope images was also demonstrated.
Hexagonal and Pentagonal Fractal Multiband Antennas
NASA Technical Reports Server (NTRS)
Tang, Philip W.; Wahid, Parveen
2005-01-01
Multiband dipole antennas based on hexagonal and pentagonal fractals have been analyzed by computational simulations and functionally demonstrated in experiments on prototypes. These antennas are capable of multiband or wide-band operation because they are subdivided into progressively smaller substructures that resonate at progressively higher frequencies by virtue of their smaller dimensions. The novelty of the present antennas lies in their specific hexagonal and pentagonal fractal configurations and the resonant frequencies associated with them. These antennas are potentially applicable to a variety of multiband and wide-band commercial wireless-communication products operating at different frequencies, including personal digital assistants, cellular telephones, pagers, satellite radios, Global Positioning System receivers, and products that combine two or more of the aforementioned functions. Perhaps the best-known prior multiband antenna based on fractal geometry is the Sierpinski triangle antenna (also known as the Sierpinski gasket), shown in the top part of the figure. In this antenna, the scale length at each iteration of the fractal is half the scale length of the preceding iteration, yielding successive resonant frequencies related by a ratio of about 2. The middle and bottom parts of the figure depict the first three iterations of the hexagonal and pentagonal fractals along with typical dipole-antenna configuration based on the second iteration. Successive resonant frequencies of the hexagonal fractal antenna have been found to be related by a ratio of about 3, and those of the pentagonal fractal antenna by a ratio of about 2.59.
Fractal antenna and fractal resonator primer
NASA Astrophysics Data System (ADS)
Cohen, Nathan
2015-03-01
Self-similarity and fractals have opened new and important avenues for antenna and electronic solutions over the last 25 years. This primer provides an introduction to the benefits provided by fractal geometry in antennas, resonators, and related structures. Such benefits include, among many, wider bandwidths, smaller sizes, part-less electronic components, and better performance. Fractals also provide a new generation of optimized design tools, first used successfully in antennas but applicable in a general fashion.
NASA Astrophysics Data System (ADS)
Chang, Kuo-En; Lin, Tang-Huang; Lien, Wei-Hung
2015-04-01
Anthropogenic pollutants or smoke from biomass burning contribute significantly to global particle aggregation emissions, yet their aggregate formation and resulting ensemble optical properties are poorly understood and parameterized in climate models. Particle aggregation refers to formation of clusters in a colloidal suspension. In clustering algorithms, many parameters, such as fractal dimension, number of monomers, radius of monomer, and refractive index real part and image part, will alter the geometries and characteristics of the fractal aggregation and change ensemble optical properties further. The cluster-cluster aggregation algorithm (CCA) is used to specify the geometries of soot and haze particles. In addition, the Generalized Multi-particle Mie (GMM) method is utilized to compute the Mie solution from a single particle to the multi particle case. This computer code for the calculation of the scattering by an aggregate of spheres in a fixed orientation and the experimental data have been made publicly available. This study for the model inputs of optical determination of the monomer radius, the number of monomers per cluster, and the fractal dimension is presented. The main aim in this study is to analyze and contrast several parameters of cluster aggregation aforementioned which demonstrate significant differences of optical properties using the GMM method finally. Keywords: optical properties, fractal aggregation, GMM, CCA
Fractal simulation of the resistivity and capacitance of arsenic selenide
Balkhanov, V. K. Bashkuev, Yu. B.
2010-03-15
The temperature dependences of the ac resistivity R and ac capacitance C of arsenic selenide were measured more than four decades ago [V. I. Kruglov and L. P. Strakhov, in Problems of Solid State Electronics, Vol. 2 (Leningrad Univ., Leningrad, 1968)]. According to these measurements, the frequency dependences are R {proportional_to} {omega}{sup -0.80{+-}0.01} and {Delta}C {proportional_to} {omega}{sup -0.120{+-}0.006} ({omega} is the circular frequency and {Delta}C is measured from the temperature-independent value C{sub 0}). According to fractal-geometry methods, R {proportional_to} {omega}{sup 1-3/h} and {Delta}C {proportional_to} {omega}{sup -2+3/h}, where h is the walk dimension of the electric current in arsenic selenide. Comparison of the experimental and theoretical results indicates that the walk dimensions calculated from the frequency dependences of resistivity and capacitance are h{sub R} = 1.67 {+-} 0.02 and h{sub C} = 1.60 {+-} 0.08, which are in agreement with each other within the measurement errors. The fractal dimension of the distribution of conducting sections is D = 1/h = 0.6. Since D < 1, the conducting sections are spatially separated and form a Cantor set.
Analysis of transient flow and starting pressure gradient of power-law fluid in fractal porous media
NASA Astrophysics Data System (ADS)
Tan, Xiao-Hua; Li, Xiao-Ping; Zhang, Lie-Hui; Liu, Jian-Yi; Cai, Jianchao
2015-09-01
A transient flow model for power-law fluid in fractal porous media is derived by combining transient flow theory with the fractal properties of tortuous capillaries. Pressure changes of transient flow for power-law fluid in fractal porous media are related to pore fractal dimension, tortuosity fractal dimension and the power-law index. Additionally, the starting pressure gradient model of power-law fluid in fractal porous media is established. Good agreement between the predictions of the present model and that of the traditional empirical model is obtained, the sensitive parameters that influence the starting pressure gradient are specified and their effects on the starting pressure gradient are discussed.
Mechanical test and fractal analysis on anisotropic fracture of cortical bone
NASA Astrophysics Data System (ADS)
Yin, Dagang; Chen, Bin; Ye, Wei; Gou, Jihua; Fan, Jinghong
2015-12-01
The mechanical properties of the cortical bone of fresh bovine femora along three different directions are tested through four-point bending experiments. It is indicated that the fracture energy along the transversal direction of the bone is distinctly larger than those of the longitudinal and radial directions. The fracture surfaces of the three different directions are observed by scanning electron microscope (SEM). It is shown that the roughness of the fracture surface of the transversal direction is obviously larger than those of the fracture surfaces of the longitudinal and radial directions. It is also revealed that the osteons in the bone are perpendicular to the fracture surface of the transversal direction and parallel to the fracture surfaces of the longitudinal and radial directions. Based on these experimental results, the fractal dimensions of the fracture surfaces of different directions are calculated by box-counting method in MATLAB. The calculated results show that the fractal dimension of the fracture surface of the transversal direction is remarkably larger than those of the fracture surfaces of the longitudinal and radial directions. The fracture energies of different directions are also calculated based on their fractal models. It is denoted that the fracture energy of the transversal direction is remarkably larger than those of the longitudinal and radial directions. The calculated results are in good agreement with the tested results.
ERIC Educational Resources Information Center
Barton, Ray
1990-01-01
Presented is an educational game called "The Chaos Game" which produces complicated fractal images. Two basic computer programs are included. The production of fractal images by the Sierpinski gasket and the Chaos Game programs is discussed. (CW)
Chaos, Fractals, and Polynomials.
ERIC Educational Resources Information Center
Tylee, J. Louis; Tylee, Thomas B.
1996-01-01
Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)
Fractal analysis of bone structure with applications to osteoporosis and microgravity effects
Acharya, R.S.; Swarnarkar, V.; Krishnamurthy, R.; Hausman, E.; LeBlanc, A.; Lin, C.; Shackelford, L.
1995-12-31
The authors characterize the trabecular structure with the aid of fractal dimension. The authors use Alternating Sequential filters to generate a nonlinear pyramid for fractal dimension computations. The authors do not make any assumptions of the statistical distributions of the underlying fractal bone structure. The only assumption of the scheme is the rudimentary definition of self similarity. This allows them the freedom of not being constrained by statistical estimation schemes. With mathematical simulations, the authors have shown that the ASF methods outperform other existing methods for fractal dimension estimation. They have shown that the fractal dimension remains the same when computed with both the X-Ray images and the MRI images of the patella. They have shown that the fractal dimension of osteoporotic subjects is lower than that of the normal subjects. In animal models, the authors have shown that the fractal dimension of osteoporotic rats was lower than that of the normal rats. In a 17 week bedrest study, they have shown that the subject`s prebedrest fractal dimension is higher than that of the postbedrest fractal dimension.
Orthogonal polynomial approach to calculate the two-nucleon transition operator in three dimensions
NASA Astrophysics Data System (ADS)
Topolnicki, Kacper; Golak, Jacek; Skibiński, Roman; Witała, Henryk
2016-02-01
We give a short report on the possibility to use orthogonal polynomials (OP) in calculations that involve the two-nucleon (2N) transition operator. The presented work adds another approach to the set of previously developed methods (described in Phys. Rev. C 81, 034006 (2010); Few-Body Syst. 53, 237 (2012); K. Topolnicki, PhD thesis, Jagiellonian University (2014)) and is applied to the transition operator calculated at laboratory kinetic energy 300MeV. The new results for neutron-neutron and neutron-proton scattering observables converge to the results presented in Few-Body Syst. 53, 237 (2012) and to results obtained using the Arnoldi algorithm (Y. Saad, Iterative methods for sparse linear systems (SIAM Philadelphia, PA, USA 2003)). The numerical cost of the calculations performed using the new scheme is large and the new method can serve only as a backup to cross-check the previously used calculation schemes.
NASA Astrophysics Data System (ADS)
Hediger, T.; Passamante, A.; Farrell, Mary Eileen
1990-05-01
An algorithm to estimate the average local intrinsic dimension (
Array Patterns of Fractal Linear Array Antennas Based on Cantor Set
NASA Astrophysics Data System (ADS)
Deepika Rani, N.; Sri Devi, P. V.
2012-03-01
A fractal is a recursively generated object having a fractional dimension. Antennas can be designed using the recursive nature of a fractal. In this paper general expression for array factor of fractal linear array based on cantor set was compared with conventional linear array. The similarity of the radiation patterns and their fractal features are examined for various iterations with the simulated results using MATLAB.
Fractal geometry characterization of geothermal reservoir fracture networks
Watanabe, K.; Takahashi, H.
1995-01-01
As a new procedure for modeling geothermal energy extraction systems, a two-dimensional modeling technique for subsurface fracture networks on the basis of `fractal geometry` is presented. Models of fracture networks are generated by distributing fractures randomly in space and by using the fractal relation between fracture length r and the number of fractures N expressed with a fractal dimension D as N = Cr(exp -D), where C is a constant that signifies the fracture density within the rock mass. This procedure makes it possible to characterize geothermal reservoirs by parameters measured from field data, such as from core sampling. In this characterization the fracture density parameter C of a geothermal reservoir is used as a parameter to model the subsurface fracture network. Using this fracture network model, the connectivities of the water flow paths between wells are calculated by means of a Monte Carlo simulation, and the result is then compared with that derived from a percolation model. We show that many fewer fractures are required to connect two wells for the fracture network model than for the percolation model. The transmissivities between wells for the fracture network model are also obtained as a function of the fracture density parameter C. The results show that the transmissivities in geothermal reservoirs are significantly dependent upon the fracture density of rock mass, and they can be predicted from the fracture density parameter C of the reservoirs.
Ferretti; Zhang; Buffle
1998-12-15
The structure of hematite aggregates in the presence of fairly monodisperse polyacrylic acid (PAA) with two different molecular weights (Mw = 1.36 x 10(6), Mw/Mn = 1.53; Mw = 3.69 x 10(4), Mw/Mn = 1.60) was studied using static light scattering (SLS). The fractal dimensions were calculated from the scattering exponents, after taking into account the finite size of aggregates, using exponential and Gaussian cutoff functions. Three flocculation regimes, namely, pre-DLA, DLA (diffusion-limited aggregation), and post-DLA, were defined based on the polymer concentration. In the DLA regime, fractal dimension values, Df = 1.84 +/- 0.02 and 1.73 +/- 0.02, were obtained using exponential and Gaussian cutoff functions, respectively. A fractal dimension of approximately 2.0 was found, as expected, in the pre-DLA regime (at PAA concentrations lower than the optimal dosage for a DLA regime) where the flocculation rate was reaction limited. In contrast, in the post-DLA regime, the flocculation was slow but the structure of aggregates was as tenuous as in the DLA regime with a fractal dimension Df approximately 1.8. Moreover, for all three regimes, the Df values were independent of the molecular weights of PAA. The lower fractal dimension in post-DLA was probably due to the increased concentration of polymer chains between adjacent particles in aggregates. The steric hindrance favored tip-to-tip aggregation, leading to a more tenuous structure. Copyright 1998 Academic Press. PMID:9845695
Ferretti, R.; Zhang, J.; Buffle, J.
1998-12-15
The structure of hematite aggregates in the presence of fairly monodisperse polyacrylic acid (PAA) with two different molecular weights (M{sub w} = 1.36 {times} 10{sup 6}, M{sub w}/M{sub n} = 1.53; M{sub w} = 3.69 {times} 10{sup 4}, M{sub w}/M{sub n} = 1.60) was studied using static light scattering (SLS). The fractal dimensions were calculated from the scattering exponents, after taking into account the finite size of aggregates, using exponential and Gaussian cutoff functions. Three flocculation regimes, namely, pre-DLA, DLA (diffusion-limited aggregation), and post-DLA, were defined based on the polymer concentration. In the DLA regime, fractal dimension values, D{sub f} = 1.84 {+-} 0.02 and 1.73 {+-} 0.02, were obtained using exponential and Gaussian cutoff functions, respectively. A fractal dimension of approximately 2.0 was found, as expected, in the pre-DLA regime (at PAA concentrations lower than the optimal dosage for a DLA regime) where the flocculation rate was reaction limited. In contrast, in the post-DLA regime, the flocculation was slow but the structure of aggregates was as tenuous as in the DLA regime with a fractal dimension D{sub f} {approx} 1.8. Moreover, for all three regimes, the D{sub f} values were independent of the molecular weights of PAA. The lower fractal dimension in post-DLA was probably due to the increased concentration of polymer chains between adjacent particles in aggregates. The steric hindrance favored tip-to-tip aggregation, leading to a more tenuous structure.
Fractal aggregates evolution of methyl red in liquid crystal.
Ciuchi, F; Sorriso-Valvo, L; Mazzulla, A; Redondo, J M
2009-06-01
The spontaneous formation of dendritic aggregates is observed in a two-dimensional confined layered system consisting of a film composed of liquid crystal, dye and solvent cast above a polymer substrate. The observed aggregates are promoted by phase separation processes induced by dye diffusion and solvent evaporation. The growth properties of the aggregates are studied through the temporal evolution of their topological properties (surface, perimeter, fractal dimension). The fractal dimension of the completely formed structures, when they are coexistent with different types of structures, is consistent with theoretical and experimental values obtained for Diffusion-Limited Aggregates. Under different experimental conditions (temperature and local dye concentration) the structure forms without interactions with other kinds of structures, and its equilibrium fractal dimension is smaller. The fractal dimension is thus not a universal property of the observed structures, but rather depends on the experimental conditions. PMID:19513769
ERIC Educational Resources Information Center
Fraboni, Michael; Moller, Trisha
2008-01-01
Fractal geometry offers teachers great flexibility: It can be adapted to the level of the audience or to time constraints. Although easily explained, fractal geometry leads to rich and interesting mathematical complexities. In this article, the authors describe fractal geometry, explain the process of iteration, and provide a sample exercise.
Fractal interpretation of intermittency
Hwa, R.C.
1991-12-01
Implication of intermittency in high-energy collisions is first discussed. Then follows a description of the fractal interpretation of intermittency. A basic quantity with asymptotic fractal behavior is introduced. It is then shown how the factorial moments and the G moments can be expressed in terms of it. The relationship between the intermittency indices and the fractal indices is made explicit.
ERIC Educational Resources Information Center
Fraboni, Michael; Moller, Trisha
2008-01-01
Fractal geometry offers teachers great flexibility: It can be adapted to the level of the audience or to time constraints. Although easily explained, fractal geometry leads to rich and interesting mathematical complexities. In this article, the authors describe fractal geometry, explain the process of iteration, and provide a sample exercise.…
Fractal structure of the time distribution of microfracturing in rocks
NASA Astrophysics Data System (ADS)
Feng, Xia-Ting; Seto, Masahiro
1999-01-01
Using acoustic emission data obtained from laboratory double torsion tests, we have analysed the fractal nature of a series of 29 granite microfracturing processes in time. The data represent a wide variety of timescales, stress environments (increasing load with a constant displacement rate, relaxation, creep), soaking conditions [air, water, dodecyl trimethyl ammonium bromide (DTAB), polyethelene oxide (PEO)], and material anisotropy. We find that the time distribution of rock microfracturing displays fractal and multifractal properties. In some cases, it has a single fractal or a multifractal structure. In other cases, it changes from a single fractal structure into a multifractal structure as the system evolves dynamically. We suggest that the heterogeneity of the rock, the distribution of joints or weak planes, the stress level, and the nature of the microfracturing mechanism lead to these multifractal properties. Whatever the fractal structure of the system, a lower fractal dimension is generally produced at near-failure of the rock due to an increased clustering. This result concerning the fractal-dimension decrease is consistent with the conclusion drawn from the spatial distribution of rock microfracturing. Therefore, from the vantage point of observation of the time distribution of rock microfracturing, the decrease of the fractal dimension has a potential use as a rock failure predictor.
Fractality in high energy cosmic rays
NASA Astrophysics Data System (ADS)
Martinic, N. J.; Ticona, R.; Poma, I.; Osco, F.; Gutiérrez, R.
2003-07-01
Using EAS data from the EAS-EXC group (EAS plus hadronic calorimeter and nuclear emulsion-X chamber hybrid experiment) on Mount Chacaltaya during the 1990's the fractal properties of the energetic cosmic ray fluxes was investigated. A discrimination in the hadron content of the energetic primaries furnishes two different data samples, the total EAS fluxes and the hadron-less fluxes. The sidereal diurnal variation of the hadron-less EAS time variation shows values of 0.5% with a phase at about 2.29 hr sidereal time. With the help of the Crassberger Procaccia algorithm the fractal dimensions have been investigated of, on the one hand, the differences in arrival times of the hadron-less showers and on the other hand the EAS fluxes time series with an integration time of five min. The obtained fractal dimensions show inconclusive evidence of continuous chaotic component in the data samples investigated.
Fractal structure of the interplanetary magnetic field
NASA Technical Reports Server (NTRS)
Burlaga, L. F.; Klein, L. W.
1985-01-01
Under some conditions, time series of the interplanetary magnetic field strength and components have the properties of fractal curves. Magnetic field measurements made near 8.5 AU by Voyager 2 from June 5 to August 24, 1981 were self-similar over time scales from approximately 20 sec to approximately 3 x 100,000 sec, and the fractal dimension of the time series of the strength and components of the magnetic field was D = 5/3, corresponding to a power spectrum P(f) approximately f sup -5/3. Since the Kolmogorov spectrum for homogeneous, isotropic, stationary turbulence is also f sup -5/3, the Voyager 2 measurements are consistent with the observation of an inertial range of turbulence extending over approximately four decades in frequency. Interaction regions probably contributed most of the power in this interval. As an example, one interaction region is discussed in which the magnetic field had a fractal dimension D = 5/3.
Defocus measurement for random self-affine fractal surfaces.
Wang, Jun; Zhou, Wei; Lim, Lennie E; Asundi, Anand K
2008-03-01
We studied correlation between fractal dimensions and image contrast for metallic surfaces. The study has led to an interesting finding that the maximum fractal dimension of the object surface under imaging gives the best focal plane. The significant finding can be made use of to estimate the best focal plane or measure the focus error with high sensitivity of a few microns, which are well within depth of field of the microscopic imaging system. PMID:18542378
A precise calculation of the fundamental string tension in SU (N) gauge theories in 2 + 1 dimensions
NASA Astrophysics Data System (ADS)
Bringoltz, Barak; Teper, Michael
2007-02-01
We use lattice techniques to calculate the continuum string tensions of SU (N) gauge theories in 2 + 1 dimensions. We attempt to control all systematic errors at a level that allows us to perform a precise test of the analytic prediction of Karabali, Kim and Nair. We find that their prediction is within 3% of our values for all N and that the discrepancy decreases with increasing N. When we extrapolate our results to N = ∞ we find that there remains a discrepancy of ≃ 1 %, which is a convincing ∼ 6 σ effect. Thus, while the Karabali-Nair analysis is remarkably accurate at N = ∞, it is not exact.
Boyd, O.S.
2006-01-01
We have created a second-order finite-difference solution to the anisotropic elastic wave equation in three dimensions and implemented the solution as an efficient Matlab script. This program allows the user to generate synthetic seismograms for three-dimensional anisotropic earth structure. The code was written for teleseismic wave propagation in the 1-0.1 Hz frequency range but is of general utility and can be used at all scales of space and time. This program was created to help distinguish among various types of lithospheric structure given the uneven distribution of sources and receivers commonly utilized in passive source seismology. Several successful implementations have resulted in a better appreciation for subduction zone structure, the fate of a transform fault with depth, lithospheric delamination, and the effects of wavefield focusing and defocusing on attenuation. Companion scripts are provided which help the user prepare input to the finite-difference solution. Boundary conditions including specification of the initial wavefield, absorption and two types of reflection are available. ?? 2005 Elsevier Ltd. All rights reserved.
A fractal model for large eddy simulation of turbulent flow
NASA Astrophysics Data System (ADS)
Scotti, A.; Meneveau, C.
1999-03-01
A new class of subgrid closures for large eddy simulation (LES) of turbulence is developed, based on the construction of synthetic, fractal subgrid-scale fields. The relevant mathematical tool, fractal interpolation, allows to interpolate the resolved velocity with fields that have fluctuations down to much smaller scales and to compute the required stresses explicitly. In one dimension, the approach is used in the context of the coarse-grained Burgers equation. Then, fractal interpolation is extended to three dimensions and is used to formulate a subgrid model for the filtered Navier-Stokes equations. The model is applied to LES of both steady and freely decaying isotropic turbulence. We find that the assumption of fractality per sè is not enough to yield physically meaningful results, and we explore several variants of the model in which the rules to generate the synthetic fields explicitly incorporate the condition that energy dissipation take place. In one dimension, this is accomplished by means of an additional transport equation that allows to dynamically determine the fractal dimension. In three dimensions, good results are obtained only once the fractal dimension is allowed to vary in different eigendirections of the resolved strain-rate tensor so as to (nearly) maximize energy dissipation.
Multi-Scale Fractal Analysis of Image Texture and Pattern
NASA Technical Reports Server (NTRS)
Emerson, Charles W.
1998-01-01
Fractals embody important ideas of self-similarity, in which the spatial behavior or appearance of a system is largely independent of scale. Self-similarity is defined as a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. An ideal fractal (or monofractal) curve or surface has a constant dimension over all scales, although it may not be an integer value. This is in contrast to Euclidean or topological dimensions, where discrete one, two, and three dimensions describe curves, planes, and volumes. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution. However, most geographical phenomena are not strictly self-similar at all scales, but they can often be modeled by a stochastic fractal in which the scaling and self-similarity properties of the fractal have inexact patterns that can be described by statistics. Stochastic fractal sets relax the monofractal self-similarity assumption and measure many scales and resolutions in order to represent the varying form of a phenomenon as a function of local variables across space. In image interpretation, pattern is defined as the overall spatial form of related features, and the repetition of certain forms is a characteristic pattern found in many cultural objects and some natural features. Texture is the visual impression of coarseness or smoothness caused by the variability or uniformity of image tone or color. A potential use of fractals concerns the analysis of image texture. In these situations it is commonly observed that the degree of roughness or inexactness in an image or surface is a function of scale and not of experimental technique. The fractal dimension of remote sensing data could yield quantitative insight on the spatial complexity and information content contained within these data. A software package known as the Image Characterization and Modeling System (ICAMS) was used to explore how fractal dimension is related to surface texture and pattern. The ICAMS software was verified using simulated images of ideal fractal surfaces with specified dimensions. The fractal dimension for areas of homogeneous land cover in the vicinity of Huntsville, Alabama was measured to investigate the relationship between texture and resolution for different land covers.
Fractals properties of EEG during event-related desynchronization of motor imagery.
Nguyen, Ngoc Quang; Truong, Quang Dang Khoa; Kondo, Toshiyuki
2015-08-01
Chaos and fractal dimension are emerging modalities for the research of electroencephalogram (EEG) signal processing. The capability of measuring non-linear characteristics of the fractal dimension enables new methodologies to identify distinct brain activities. Recent studies on the topic focus on utilizing various types of fractals as features in order to design better brain state classification system. However, we have little insight about the EEG signals projected in fractal dimension. In this paper, we investigate the relationship between the non-linear characteristics of ongoing EEG signals and event-related desynchronization (ERD) during motor imagery. We observed a considerable synchronization between ERD and fractal dimension. This finding suggests further usage of chaos and fractal theory in investigating brain activities. PMID:26737207
[Fractal characteristics of mature aerobic granular sludge cultivated by different carbon sources].
Gao, Jing-feng; Su, Kai; Zhang, Qian; Chen, Ran-ni; Wang, Jin-hui; Peng, Yong-zhen
2010-08-01
In order to characterize the dense and regular level of aerobic granular sludge (AGS) which was cultivated by different carbon sources, the SEM images of mature AGS were used in the study. The calculation process using Photoshop and Fips2 on fractal characteristics of these granules was built. The lowest box-counting dimension was bulking aerobic granular sludge cultivated by glucose (No. 1), which was 1.794 +/- 0.011; the box-counting dimension of AGS cultivated by peptone (No. 3) and domestic sewage (No. 5) were reached up to 1.866 +/- 0.018 and 1.880 +/- 0.015. The boundary box-counting dimension of the AGS was also calculated and the average value was 1.14. The AGS cultivated by beer (No. 6) was more regular in shape and the boundary box-counting dimension was 1.115 +/- 0.003. The AGS cultivated by landfill leachate (No. 7) was the most irregular in shape. This study indicates that fractal dimension provides an approach for quantification of dense and regular level of AGS, furthermore, it could be used to characterize the status of AGS, such as bulking. PMID:21090307
Fractal structure in one-dimensional sheet model and expansion law
NASA Astrophysics Data System (ADS)
Tatekawa, Takayuki; Maeda, Kei-Ichi
One-dimensional sheet model was handled as an easy model to analyze the nature of the gravity from the old days. For one-dimensional sheet model with cosmic expansion we analyzed evolution of fractal dimension of structure formed from primordial fractal density fluctuation. We found that the fractal dimension of nonlinear structure is independent of fractal dimension of initial pattern. Recently, the scale-free structure formed in one-dimensional sheet model without cosmic expansion. In this paper, we analyze one-dimensional sheet model with other expansion rate.
Fractals in art and nature: why do we like them?
NASA Astrophysics Data System (ADS)
Spehar, Branka; Taylor, Richard P.
2013-03-01
Fractals have experienced considerable success in quantifying the visual complexity exhibited by many natural patterns, and continue to capture the imagination of scientists and artists alike. Fractal patterns have also been noted for their aesthetic appeal, a suggestion further reinforced by the discovery that the poured patterns of the American abstract painter Jackson Pollock are also fractal, together with the findings that many forms of art resemble natural scenes in showing scale-invariant, fractal-like properties. While some have suggested that fractal-like patterns are inherently pleasing because they resemble natural patterns and scenes, the relation between the visual characteristics of fractals and their aesthetic appeal remains unclear. Motivated by our previous findings that humans display a consistent preference for a certain range of fractal dimension across fractal images of various types we turn to scale-specific processing of visual information to understand this relationship. Whereas our previous preference studies focused on fractal images consisting of black shapes on white backgrounds, here we extend our investigations to include grayscale images in which the intensity variations exhibit scale invariance. This scale-invariance is generated using a 1/f frequency distribution and can be tuned by varying the slope of the rotationally averaged Fourier amplitude spectrum. Thresholding the intensity of these images generates black and white fractals with equivalent scaling properties to the original grayscale images, allowing a direct comparison of preferences for grayscale and black and white fractals. We found no significant differences in preferences between the two groups of fractals. For both set of images, the visual preference peaked for images with the amplitude spectrum slopes from 1.25 to 1.5, thus confirming and extending the previously observed relationship between fractal characteristics of images and visual preference.
Aperture correlation of a fractal fracture
Wang, J.S.Y.; Narasimhan, T.N.; Scholz, C.H.
1988-03-10
A rough-walled facture is modeled by fractal geometry. In the fractal fracture model, the rock surfaces are characterized by a fractal dimension D between 2 and 3, with lower D for smoother surfaces and higher D for rougher surfaces. The mismatch due to shear displacement between two mirror-image fractal surfaces determines the fracture aperture distribution. An analytic equation is derived for the varioram ..gamma../sub f/(r) describing the spatial correlation of the aperture of a fractal fracture. The aperture of a smooth fracture with low D is highly correlated over distances much larger than the shear displacement. The aperture of a rough fracture with high D becomes uncorrelated within a range shorter than the shear displacement. Near the origin, r = 0, of the variogram, the variogram is proportional to r/sup 6-2//sup D/. For the special case of Brownian fractal with D = 2.5, the variogram is proportional to r, which is the same r dependence exhibited by the spherical model widely used in geostatistical analyses. copyright American Geophysical Union 1988
Fractal aggregates induced by liposome-liposome interaction in the presence of Ca2+.
Sabín, J; Prieto, G; Ruso, J M; Sarmiento, F
2007-10-01
We present a study of the fractal dimension of clusters of large unilamellar vesicles (LUVs) formed by egg yolk phosphatidylcholine (EYPC), dimyristoylphosphocholine (DMPC) and dipalmitoylphosphocholine (DPPC) induced by Ca2+ . Fractal dimensions were calculated by application of two methods, measuring the angular dependency of the light scattered by the clusters and following the evolution of the cluster size. In all cases, the fractal dimensions fell in the range from 2.1 to 1.8, corresponding to two regimes: diffusion-limited cluster aggregation (DLCA) and reaction-limited cluster aggregation (RLCA). Whereas DMPC clusters showed a typical transition from the RLCA to the DLCA aggregation, EYPC exhibited an unusual behaviour, since the aggregation was limited for a higher concentration than the critical aggregation concentration. The behaviour of DPPC was intermediate, with a transition from the RLCA to the DLCA regimes with cluster sizes depending on Ca2+ concentration. Studies on the reversibility of the aggregates show that EYPC and DPPC clusters can be re-dispersed by dilution with water. DMPC does not present reversibility. Reversibility is evidence of the existence of secondary minima in the DLVO potential between two liposomes. To predict these secondary minima, a correction of the DLVO model was necessary taking into account a repulsive force of hydration. PMID:18000643
Fractal network model for contact conductance
Majumdar, A. ); Tien, C.L. )
1991-08-01
The topography of rough surfaces strongly influences the conduction of heat and electricity between two surfaces in contact. Roughness measurements on a variety of surfaces have shown that their structure follows a fractal geometry whereby similar images of the surface appear under repeated magnification. Such a structure is characterized by the fractal dimension D, which lies between 2 and 3 for a surface and between 1 and 2 for a surface profile. This paper uses the fractal characterization of surface roughness to develop a new network model for analyzing heat conduction between two contacting rough surfaces. The analysis yields the simple result that the contact conductance h and the real area of contact A{sub t} are related as h {approximately} A{sub t}{sup D/2} where D is the fractal dimension of the surface profile. Contact mechanics of fractal surfaces has shown that A{sub t} varies with the load F as A{sub t} {approximately} F{sup {eta}} where {eta} ranges from 1 to 1.33 depending on the value of D. This proves that the ocnductance and load are related as h {approximately} F{sup {eta}D/2} and resolves the anomaly in previous investigations, which theoretically and experimentally obtained different values for the load exponent. The analytical results agreed well with previous experiments although there is a tendency for overprediction.
McConathy, R.K.
1983-03-01
The study describes the gradients of stomatal size and density in the crown of a mature forest-grown tulip-poplar (Liriodendron tulipifera L.) in eastern Tennessee. These data are used to predict leaf resistance to vapor diffusion in relation to stomatal width and boundary layer resistance. Stomatal density on individual leaves did not vary, but density increased with increasing crown height. Stomatal size decreased with increasing height of leaves within the crown. Stomatal size and density variations interacted to result in a constant number of stomata per leaf at all crown heights. Stomatal diffusive resistance values calculated from stomatal measurements and varying environmental parameters indicated that stomatal resistance controlled transpiration water losses only at small apertures (<0.6 ..mu..m). Boundary layer resistance was controlling at large stomatal apertures (>0.6 ..mu..m) and at low wind speeds (approx.100 cm/s). Under normal forest conditions tulip-poplar stomatal resistance exercised more control over transpiration than did boundary layer resistance.
Full dimension Rb2He ground triplet potential energy surface and quantum scattering calculations.
Guillon, Grégoire; Viel, Alexandra; Launay, Jean-Michel
2012-05-01
We have developed a three-dimensional potential energy surface for the lowest triplet state of the Rb(2)He complex. A global analytic fit is provided as in the supplementary material [see supplementary material at http://dx.doi.org/10.1063/1.4709433 for the corresponding Fortran code]. This surface is used to perform quantum scattering calculations of (4)He and (3)He colliding with (87)Rb(2) in the partial wave J = 0 at low and ultralow energies. For the heavier helium isotope, the computed vibrational relaxation probabilities show a broad and strong shape resonance for a collisional energy of 0.15 K and a narrow Feshbach resonance at about 17 K for all initial Rb(2) vibrational states studied. The broad resonance corresponds to an efficient relaxation mechanism that does not occur when (3)He is the colliding partner. The Feshbach resonance observed at higher collisional energy is robust with respect to the isotopic substitution. However, its effect on the vibrational relaxation mechanism is faint for both isotopes. PMID:22583230
The fractal energy measurement and the singularity energy spectrum analysis
NASA Astrophysics Data System (ADS)
Xiong, Gang; Zhang, Shuning; Yang, Xiaoniu
2012-12-01
The singularity exponent (SE) is the characteristic parameter of fractal and multifractal signals. Based on SE, the fractal dimension reflecting the global self-similar character, the instantaneous SE reflecting the local self-similar character, the multifractal spectrum (MFS) reflecting the distribution of SE, and the time-varying MFS reflecting pointwise multifractal spectrum were proposed. However, all the studies were based on the depiction of spatial or differentiability characters of fractal signals. Taking the SE as the independent dimension, this paper investigates the fractal energy measurement (FEM) and the singularity energy spectrum (SES) theory. Firstly, we study the energy measurement and the energy spectrum of a fractal signal in the singularity domain, propose the conception of FEM and SES of multifractal signals, and investigate the Hausdorff measure and the local direction angle of the fractal energy element. Then, we prove the compatibility between FEM and traditional energy, and point out that SES can be measured in the fractal space. Finally, we study the algorithm of SES under the condition of a continuous signal and a discrete signal, and give the approximation algorithm of the latter, and the estimations of FEM and SES of the Gaussian white noise, Fractal Brownian motion and the multifractal Brownian motion show the theoretical significance and application value of FEM and SES.
NASA Astrophysics Data System (ADS)
Usov, V. V.; Gopkalo, E. E.; Shkatulyak, N. M.; Gopkalo, A. P.; Cherneva, T. S.
2015-09-01
Crystallographic texture and fracture features are studied after low-cycle fatigue tests of laboratory specimens cut from the base metal and the characteristic zones of a welded joint in a pipeline after its longterm operation. The fractal dimensions of fracture surfaces are determined. The fractal dimension is shown to increase during the transition from ductile to quasi-brittle fracture, and a relation between the fractal dimension of a fracture surface and the fatigue life of the specimen is found.
Triangular constellations in fractal measures
NASA Astrophysics Data System (ADS)
Wilkinson, Michael; Grant, John
2014-09-01
The local structure of a fractal set is described by its dimension D, which is the exponent of a power-law relating the mass {\\cal N} in a ball to its radius \\varepsilon{:}\\ {\\cal N}\\sim \\varepsilon^D . It is desirable to characterise the shapes of constellations of points sampling a fractal measure, as well as their masses. The simplest example is the distribution of shapes of triangles formed by triplets of points, which we investigate for fractals generated by chaotic dynamical systems. The most significant parameter describing the triangle shape is the ratio z of its area to the radius of gyration squared. We show that the probability density of z has a phase transition: P(z) is independent of ɛ and approximately uniform below a critical flow compressibility \\beta_{\\text{c}} , which we estimate. For \\beta>\\beta_{\\text{c}} the distribution appears to be described by two power laws: P(z)\\sim z^{\\alpha_1} when 1\\gg z\\gg z_{\\text{c}}(\\varepsilon) , and P(z)\\sim z^{\\alpha_2} when z\\ll z_{\\text{c}}(\\varepsilon) .
Fractal characterization of a fractured chalk reservoir - The Laegerdorf case
Stoelum, H.H.; Koestler, A.G.; Feder, J.; Joessang, T.; Aharony, A.
1991-03-01
What is the matrix block size distribution of a fractured reservoir In order to answer this question and assess the potential of fractal geometry as a method of characterization of fracture networks, a pilot study has been done of the fractured chalk quarry in Laegerdorf. The fractures seen on the quarry walls were traced in the field for a total area of {approximately}200 {times} 45 m. The digitized pictures have been analyzed by a standard box-counting method. This analysis gave a fractal dimension of similarity varying from 1.33 for fractured areas between faults, to 1.43 for the fault zone, and 1.53 for the highly deformed fault gouge. The amplitude showed a similar trend. The fractal dimension for the whole system of fractures is {approximately}1.55. In other words, fracture networks in chalk have a nonlinear, fractal geometry, and so matrix block size is a scaling property of chalk reservoirs. In terms of rock mechanics, the authors interpret the variation of the fractal dimension as follows: A small fractal dimension and amplitude are associated with brittle deformation in the elastic regime, while a large fractal dimension and amplitude are associated with predominantly ductile, strain softening deformation in the plastic regime. The interaction between the two regimes of deformation in the rock body is a key element of successful characterization and may be approached by seeing the rock as a non-Newtonian viscoelastic medium. The fractal dimension for the whole is close to a material independent limit that constrains the development of fractures.
FORTRAN programs for calculating nonlinear seismic ground response in two dimensions
Joyner, W.B.
1978-01-01
The programs described here were designed for calculating the nonlinear seismic response of a two-dimensional configuration of soil underlain by a semi-infinite elastic medium representing bedrock. There are two programs. One is for plane strain motions, that is, motions in the plane perpendicular to the long axis of the structure, and the other is for antiplane strain motions, that is motions parallel to the axis. The seismic input is provided by specifying what the motion of the rock-soil boundary would be if the soil were absent and the boundary were a free surface. This may be done by supplying a magnetic tape containing the values of particle velocity for every boundary point at every instant of time. Alternatively, a punch card deck may be supplied giving acceleration values at every instant of time. In the plane strain program it is assumed that the acceleration values apply simultaneously to every point on the boundary; in the antiplane strain program it is assumed that the acceleration values characterize a plane shear wave propagating upward in the underlying elastic medium at a specified angle with the vertical. The nonlinear hysteretic behavior of the soil is represented by a three-dimensional rheological model. A boundary condition is used which takes account of finite rigidity in the elastic substratum. The computations are performed by an explicit finite-difference scheme that proceeds step by step in space and time. Computations are done in terms of stress departures from an unspecified initial state. Source listings are provided here along with instructions for preparing the input. A more detailed discussion of the method is presented elsewhere.
Unifying iteration rule for fractal objects
NASA Astrophysics Data System (ADS)
Kittel, A.; Parisi, J.; Peinke, J.; Baier, G.; Klein, M.; Rössler, O. E.
1997-03-01
We introduce an iteration rule for real numbers capable to generate attractors with dragon-, snowflake-, sponge-, or Swiss-flag-like cross sections. The idea behind it is the mapping of a torus into two (or more) shrunken and twisted tori located inside the previous one. Three distinct parameters define the symmetry, the dimension, and the connectedness or disconnectedness of the fractal object. For some selected triples of parameter values, a couple of well known fractal geometries (e.g. the Cantor set, the Sierpinski gasket, or the Swiss flag) can be gained as special cases.
Fractal boundaries in magnetotail particle dynamics
NASA Technical Reports Server (NTRS)
Chen, J.; Rexford, J. L.; Lee, Y. C.
1990-01-01
It has been recently established that particle dynamics in the magnetotail geometry can be described as a nonintegrable Hamiltonian system with well-defined entry and exit regions through which stochastic orbits can enter and exit the system after repeatedly crossing the equatorial plane. It is shown that the phase space regions occupied by orbits of different numbers of equatorial crossings or different exit modes are separated by fractal boundaries. The fractal boundaries in an entry region for stochastic orbits are examined and the capacity dimension is determined.
Fractal geometry in quantum mechanics, field theory and spin systems
NASA Astrophysics Data System (ADS)
Krger, H.
The goal of this article is to review the role of fractal geometry in quantum physics. There are two aspects: (a) The geometry of underlying space (space-time in relativistic systems) is fractal and one studies the dynamics of the quantum system. Example: percolation. (b) The underlying space-time is regular, and fractal geometry which shows up in particular observables is generated by the dynamics of the quantum system. Example: Brownian motion (imaginary time quantum mechanics), zig-zag paths of propagation in quantum mechanics (Feynman's path integral). Historically, the first example of fractal geometry in quantum mechanics was invoked by Feynman and Hibbs describing the self-similarity (fractal behavior) of paths occurring in the path integral. We discuss the geometry of such paths. We present analytical as well as numerical results, yielding Hausdorff dimension dH=2. Velocity-dependent interactions (propagation in a solid, Brueckner's theory of nuclear matter) allow for dH<2. Next, we consider quantum field theory. We discuss the relation of self-similarity, the renormalization group equation, scaling laws and critical behavior, also violation of scale invariance, like logarithmic scaling corrections in hadron structure functions. We discuss the fractal geometry of paths of the path integral in field theory. We present numerical results for the length of propagation and fractal dimension for the free fermion propagator which is relevant for the geometry of quark propagation in QCD. Then we look at order parameters for the confinement phase in QCD. The fractal dimension of closed monopole current loops is such an order parameter. We discuss properties of a fractal Wilson loop. We look at critical phenomena, in particular at critical exponents and its relation to non-integer dimension of space-time by use of an underlying fractal geometry with the purpose to determine lower or upper critical dimensions. As an example we consider the U(1) model of lattice gauge theory. As another topic we discuss fractal geometry and Hausdorff dimension of quantum gravity and also for gravity coupled to matter, like to the Ising model or to the 3-state Potts model. Finally, we study the role that fractal geometry plays in spin physics, in particular for the purpose to describe critical clusters.
Exploring Fractals in the Classroom.
ERIC Educational Resources Information Center
Naylor, Michael
1999-01-01
Describes an activity involving six investigations. Introduces students to fractals, allows them to study the properties of some famous fractals, and encourages them to create their own fractal artwork. Contains 14 references. (ASK)
Retinal vascular fractals in Behçet’s Disease: A screening method?
Norouzpour, Amir; Mehdizadeh, Alireza
2015-01-01
Objective The branching pattern of retinal vessels may be affected in Behçet’s Disease (BD). Fractal analysis can be used as a new method to quantify the changes of the vascular branching pattern. In this study, we examined, for the first time, the relationship between retinal fractal dimension (Df) and retinal vascular changes seen in patients with BD. Methods We conducted a retrospective study of 10 new cases of BD with clinically ocular involvement. Color fundus images taken from both eyes of the participants have been analyzed, and Df of the whole retinal vasculature was quantified using a novel computer-based program. The resultant Df was compared with that of healthy individuals. Results The mean Df, calculated from 20 fundus images of cases with BD, was 1.59 ± 0.064. It was lower than that of healthy participants (1.65 ± 0.060) significantly (P = 0.013). Conclusion Retinal fractal analysis of cases with BD has been performed for the first time, and the results showed that early retinal vascular changes seen in new cases of BD are associated with lower retinal Df. Retinal fractal analysis in BD can be practically utilized as a potential tool for screening of retinal involvement, evaluating the prognosis and the response to treatment. PMID:26155081
Fractals: To Know, to Do, to Simulate.
ERIC Educational Resources Information Center
Talanquer, Vicente; Irazoque, Glinda
1993-01-01
Discusses the development of fractal theory and suggests fractal aggregates as an attractive alternative for introducing fractal concepts. Describes methods for producing metallic fractals and a computer simulation for drawing fractals. (MVL)
Analytical estimation of the correlation dimension of integer lattices.
Lacasa, Lucas; Gmez-Gardees, Jess
2014-12-01
Recently [L. Lacasa and J. Gmez-Gardees, Phys. Rev. Lett. 110, 168703 (2013)], a fractal dimension has been proposed to characterize the geometric structure of networks. This measure is an extension to graphs of the so called correlation dimension, originally proposed by Grassberger and Procaccia to describe the geometry of strange attractors in dissipative chaotic systems. The calculation of the correlation dimension of a graph is based on the local information retrieved from a random walker navigating the network. In this contribution, we study such quantity for some limiting synthetic spatial networks and obtain analytical results on agreement with the previously reported numerics. In particular, we show that up to first order, the correlation dimension ? of integer lattices ?(d) coincides with the Haussdorf dimension of their coarsely equivalent Euclidean spaces, ??=?d. PMID:25554021
Analytical estimation of the correlation dimension of integer lattices
Lacasa, Lucas; Gómez-Gardeñes, Jesús
2014-12-01
Recently [L. Lacasa and J. Gómez-Gardeñes, Phys. Rev. Lett. 110, 168703 (2013)], a fractal dimension has been proposed to characterize the geometric structure of networks. This measure is an extension to graphs of the so called correlation dimension, originally proposed by Grassberger and Procaccia to describe the geometry of strange attractors in dissipative chaotic systems. The calculation of the correlation dimension of a graph is based on the local information retrieved from a random walker navigating the network. In this contribution, we study such quantity for some limiting synthetic spatial networks and obtain analytical results on agreement with the previously reported numerics. In particular, we show that up to first order, the correlation dimension β of integer lattices ℤ{sup d} coincides with the Haussdorf dimension of their coarsely equivalent Euclidean spaces, β = d.
How Fractal are Coastlines Really? Observation and Theory
NASA Astrophysics Data System (ADS)
Murray, A.; Barton, C. C.
2007-12-01
Rocky coastlines have been held up as a prime example of fractal geometry since Mandelbrot introduced the concept. However, we will present a map of the fractal dimensions measured for the contiguous United States coastline which shows that many open-ocean sand--and even rocky--coastlines have fractal dimensions close to one; i.e. they tend to not be very fractal. The fractal nature of rocky coastlines likely represents an inherited fluvial or glacial signature that tends to be erased by coastal processes. Recent theoretical and numerical-modeling developments indicate that wave-driven coastal processes on sandy shores tend to produce one-dimensional coastlines. Gradients in alongshore sediment flux tend to smooth a shoreline, as long as the local wave climate is dominated by 'low-angle' waves (waves that approach the coastline in deep water from angles, relative to the coastline orientation, that are lower than the sediment-flux- maximizing angle). Even when a regional wave climate is dominated by high-angle waves--which produce an instability in plan-view shoreline shape--on the large scale, coastlines self organize in a way that produces locally low-angle-dominated wave climates almost everywhere. These processes explain why wave-dominated sandy coastlines, such as the Carolina and Texas coasts, exhibit fractal dimensions barely above one; wave- driven alongshore transport is an anti-fractal landsculpting agent over a range of scales greater than 0.2 km. In contrast, fluvial landsculpting produces famously fractal topography. When rapid sea-level rise causes the approximately horizontal plane of sea level to intersect a fractal fluvial topography, a fractal coastline results. Where wave energy is low, relative to rock erodibility, the fluvial fractal signature can persist. However, on the rocky West Coast of the US, fractal dimensions are relatively low (1.1 - 1.2), suggesting modification by wave-driven processes; that the production and rearrangement of sediment into ever-expanding pocket beaches has been reducing the fractality of this high-wave-energy, relatively easily eroded coastline. Glacially carved coastlines, such as that of Maine (and some parts of western Britain and Norway), exhibit high fractal dimensions (approximately 1.5), where erodibility is low enough the self-similarity of the intersection of sea-level with a glacially sculpted topography remains. Although wave-driven coastal processes tend to generate low-fractal-dimension shorelines, on sandy coastlines dominated by tidal currents, coastal processes also etch a fractal dendritic network of channels into the coastline. Tidally dominated coastlines, such as those in the Georgia Bight (Southeastern US), sport highly fractal shapes as a result (fractal dimensions approximately 1.5).
Pantic, Igor; Nesic, Zorica; Paunovic Pantic, Jovana; Radojević-Škodrić, Sanja; Cetkovic, Mila; Basta Jovanovic, Gordana
2016-05-21
Fractal analysis and Gray level co-occurrence matrix method represent two novel mathematical algorithms commonly used in medical sciences as potential parts of computer-aided diagnostic systems. In this study, we tested the ability of these methods to discriminate the kidney medullar tissue suffering from reperfusion injury, from normal tissue. A total of 320 digital micrographs of Periodic acid-Schiff (PAS) - stained kidney medulla from 16 Wistar albino mice (20 per animal), were analyzed using National Institutes of Health ImageJ software (NIH, Bethesda, MD) and its plugins. 160 micrographs were obtained from the experimental group with induced reperfusion injury, and another 160 were obtained from the controls. For each micrograph we calculated the values of fractal dimension, lacunarity, as well as five GLCM features: angular second moment, entropy, inverse difference moment, GLCM contrast, and GLCM correlation. Discriminatory value of the parameters was tested using receiver operating characteristic (ROC) analysis, by measuring the area below ROC curve. The results indicate that certain features of GLCM algorithm have excellent discriminatory ability in evaluation of damaged kidney tissue. Fractal dimension and lacunarity as parameters of fractal analysis also had a relatively good discriminatory value in differentiation of injured from the normal tissue. Both methods have potentially promising application in future design of novel techniques applicable in cell physiology, histology and pathology. PMID:26964774
Edges of Saturn's rings are fractal.
Li, Jun; Ostoja-Starzewski, Martin
2015-01-01
The images recently sent by the Cassini spacecraft mission (on the NASA website http://saturn.jpl.nasa.gov/photos/halloffame/) show the complex and beautiful rings of Saturn. Over the past few decades, various conjectures were advanced that Saturn's rings are Cantor-like sets, although no convincing fractal analysis of actual images has ever appeared. Here we focus on four images sent by the Cassini spacecraft mission (slide #42 "Mapping Clumps in Saturn's Rings", slide #54 "Scattered Sunshine", slide #66 taken two weeks before the planet's Augus't 200'9 equinox, and slide #68 showing edge waves raised by Daphnis on the Keeler Gap) and one image from the Voyager 2' mission in 1981. Using three box-counting methods, we determine the fractal dimension of edges of rings seen here to be consistently about 1.63 ~ 1.78. This clarifies in what sense Saturn's rings are fractal. PMID:25883885
Fractal and Multifractal Analysis of Human Gait
NASA Astrophysics Data System (ADS)
Muñoz-Diosdado, A.; del Río Correa, J. L.; Angulo-Brown, F.
2003-09-01
We carried out a fractal and multifractal analysis of human gait time series of young and old individuals, and adults with three illnesses that affect the march: The Parkinson's and Huntington's diseases and the amyotrophic lateral sclerosis (ALS). We obtained cumulative plots of events, the correlation function, the Hurst exponent and the Higuchi's fractal dimension of these time series and found that these fractal markers could be a factor to characterize the march, since we obtained different values of these quantities for youths and adults and they are different also for healthy and ill persons and the most anomalous values belong to ill persons. In other physiological signals there is complexity lost related with the age and the illness, in the case of the march the opposite occurs. The multifractal analysis could be also a useful tool to understand the dynamics of these and other complex systems.
Fractal analysis of the structural complexity of the connective tissue in human carotid bodies
Guidolin, Diego; Porzionato, Andrea; Tortorella, Cinzia; Macchi, Veronica; De Caro, Raffaele
2014-01-01
The carotid body (CB) may undergo different structural changes during perinatal development, aging, or in response to environmental stimuli. In the previous literature, morphometric approaches to evaluate these changes have considered quantitative first order parameters, such as volumes or densities, while changes in spatial disposition and/or complexity of structural components have not yet been considered. In the present study, different strategies for addressing morphological complexity of CB, apart from the overall amount of each tissue component, were evaluated and compared. In particular, we considered the spatial distribution of connective tissue in the carotid bodies of young control subjects, young opiate-related deaths and aged subjects, through analysis of dispersion (Morisita's index), gray level co-occurrence matrix (entropy, angular second moment, variance, correlation), and fractal analysis (fractal dimension, lacunarity). Opiate-related deaths and aged subjects showed a comparable increase in connective tissue with respect to young controls. However, the Morisita's index (p < 0.05), angular second moment (p < 0.05), fractal dimension (p < 0.01), and lacunarity (p < 0.01) permitted to identify significant differences in the disposition of the connective tissue between these two series. A receiver operating characteristic (ROC) curve was also calculated to evaluate the efficiency of each parameter. The fractal dimension and lacunarity, with areas under the ROC curve of 0.9651 (excellent accuracy) and 0.8835 (good accuracy), respectively, showed the highest discriminatory power. They evidenced higher level of structural complexity in the carotid bodies of opiate-related deaths than old controls, due to more complex branching of intralobular connective tissue. Further analyses will have to consider the suitability of these approaches to address other morphological features of the CB, such as different cell populations, vascularization, and innervation. PMID:25414672
Application of fractal geometry to damage development and brittle fracture in materials
Anderson, T.L.
1989-01-01
The concepts of fractal geometry are applicable to a wide range of problems in materials science. One obvious application is the characterization of irregular surfaces (e.g., fracture surfaces) by means of a fractal dimension. Several papers on this subject have recently appeared. A somewhat less obvious use of the fractal dimension involves characterization of the fragmented (nonuniform) nature of microstructural features such as second phase particles and microcracks. This article utilizes fractal geometry to develop simple models for microcrack growth. Both stable and unstable growth are considered. These results are potentially applicable to a wide range of materials including composites, ceramics and structural steels.
Fractal Structures Driven by Self-gravity: Molecular Clouds and the Universe
NASA Astrophysics Data System (ADS)
Combes, Francoise
1998-09-01
In the interstellar medium, as well as in the Universe, large density fluctuations are observed, that obey power-law density distributions and correlation functions. These structures are hierarchical, chaotic, turbulent, but are also self-organizing. The apparent disorder is not random noise, but can be described by a fractal, with a deterministic fractal dimension. We present a new theory of the self-gravity thermodynamics, that could explain the existence of these fractal structures, and predict their fractal dimension. The media obeying scaling laws can be considered critical, as in second order phase transitions for instance.
Evaluation of Two Fractal Methods for Magnetogram Image Analysis
NASA Technical Reports Server (NTRS)
Stark, B.; Adams, M.; Hathaway, D. H.; Hagyard, M. J.
1997-01-01
Fractal and multifractal techniques have been applied to various types of solar data to study the fractal properties of sunspots as well as the distribution of photospheric magnetic fields and the role of random motions on the solar surface in this distribution. Other research includes the investigation of changes in the fractal dimension as an indicator for solar flares. Here we evaluate the efficacy of two methods for determining the fractal dimension of an image data set: the Differential Box Counting scheme and a new method, the Jaenisch scheme. To determine the sensitivity of the techniques to changes in image complexity, various types of constructed images are analyzed. In addition, we apply this method to solar magnetogram data from Marshall Space Flight Centers vector magnetograph.
Facilitated diffusion of proteins through crumpled fractal DNA globules
NASA Astrophysics Data System (ADS)
Smrek, Jan; Grosberg, Alexander Y.
2015-07-01
We explore how the specific fractal globule conformation, found for the chromatin fiber of higher eukaryotes and topologically constrained dense polymers, affects the facilitated diffusion of proteins in this environment. Using scaling arguments and supporting Monte Carlo simulations, we relate DNA looping probability distribution, fractal dimension, and protein nonspecific affinity for the DNA to the effective diffusion parameters of the proteins. We explicitly consider correlations between subsequent readsorption events of the proteins, and we find that facilitated diffusion is faster for the crumpled globule conformation with high intersegmental surface dimension than in the case of dense fractal conformations with smooth surfaces. As a byproduct, we obtain an expression for the macroscopic conductivity of a hypothetic material consisting of conducting fractal nanowires immersed in a weakly conducting medium.
NASA Astrophysics Data System (ADS)
Batkovich, D. V.; Chetyrkin, K. G.; Kompaniets, M. V.
2016-05-01
We report on a completely analytical calculation of the field anomalous dimension γφ and the critical exponent η for the O (n)-symmetric φ4 model at the record six loop level. We successfully compare our result for γφ with n = 1 with the predictions based on the method of the Borel resummation combined with a conformal mapping (Kazakov et al., 1979 [40]). Predictions for seven loop contribution to the field anomalous dimensions are given.
Correlated Fractal Percolation and the Palis Conjecture
NASA Astrophysics Data System (ADS)
Dekking, Michel; Don, Henk
2010-04-01
Let F 1 and F 2 be independent copies of one-dimensional correlated fractal percolation, with almost sure Hausdorff dimensions dim H( F 1) and dim H( F 2). Consider the following question: does dim H( F 1)+dim H( F 2)>1 imply that their algebraic difference F 1- F 2 will contain an interval? The well known Palis conjecture states that `generically' this should be true. Recent work by Kuijvenhoven and the first author (Dekking and Kuijvenhoven in J. Eur. Math. Soc., to appear) on random Cantor sets cannot answer this question as their condition on the joint survival distributions of the generating process is not satisfied by correlated fractal percolation. We develop a new condition which permits us to solve the problem, and we prove that the condition of Dekking and Kuijvenhoven (J. Eur. Math. Soc., to appear) implies our condition. Independently of this we give a solution to the critical case, yielding that a strong version of the Palis conjecture holds for fractal percolation and correlated fractal percolation: the algebraic difference contains an interval almost surely if and only if the sum of the Hausdorff dimensions of the random Cantor sets exceeds one.
Lattice animals, fractality and criticality in hadronic and partonic systems
NASA Astrophysics Data System (ADS)
Moretto, L. G.; Elliot, J. B.; Lake, P. T.; Phair, L.
2011-01-01
The cluster description of near coexistence phases (e.g. Fisher theory) requires an evaluation of cluster surface entropy. This surface degeneracy can be estimated with lattice models where clusters appear. The maximum probability lies near the maximum cluster surface. At low temperatures, clusters are forced to be nearly spherical by the surface energy and the associated Boltzmann factor. At higher temperatures and near criticality, the fractal dimension of clusters changes so that clusters become fractal. In the MIT bag model, where there is no surface energy, bags are always fractal.
Fractal pharmacokinetics of the drug mibefradil in the liver
NASA Astrophysics Data System (ADS)
Fuite, J.; Marsh, R.; Tuszyński, J.
2002-08-01
We explore the ramifications of the fractal geometry of the key organ for drug elimination, the liver, on pharmacokinetic data analysis. A formalism is developed for the use of a combination of well-stirred Euclidean and fractal compartments in the body. Perturbation analysis is carried out to obtain analytical solutions for the drug concentration time evolution. These results are then fitted to experimental data collected from clinically instrumented dogs [see, A. Skerjanec et al., J. Pharm. Sci. 85, 189 (1995)] using the drug mibefradil. The thus obtained spectral fractal dimension has a range of values that is consistent with the value found in independently performed ultrasound experiments on the liver.
Fractal geometry of some Martian lava flow margins: Alba Patera
NASA Technical Reports Server (NTRS)
Kauhanen, K.
1993-01-01
Fractal dimension for a few lava flow margins on the gently sloping flanks of Alba Patera were measured using the structured walk method. Fractal behavior was observed at scales ranging from 20 to 100 pixels. The upper limit of the linear part of log(margin length) vs. log(scale) profile correlated well to the margin length. The lower limit depended on resolution and flow properties.
Thermodynamics with fractal structure, Tsallis statistics, and hadrons
NASA Astrophysics Data System (ADS)
Deppman, A.
2016-03-01
A system presenting fractal structure in its thermodynamical functions is introduced, and it is shown that Tsallis statistics is the correct framework for describing the thermodynamical aspects of such a fractal. Its Haussdorf dimension and its Lipshitz-Hölder exponent are determined in terms of the entropic index q . The connections with the intermittency in experimental data are discussed. The thermodynamical aspects of the thermofractal is related to the microscopic interaction of its components through the S -matrix.
Hagerhall, C M; Laike, T; Kller, M; Marcheschi, E; Boydston, C; Taylor, R P
2015-01-01
Psychological and physiological benefits of viewing nature have been extensively studied for some time. More recently it has been suggested that some of these positive effects can be explained by nature's fractal properties. Virtually all studies on human responses to fractals have used stimuli that represent the specific form of fractal geometry found in nature, i.e. statistical fractals, as opposed to fractal patterns which repeat exactly at different scales. This raises the question of whether human responses like preference and relaxation are being driven by fractal geometry in general or by the specific form of fractal geometry found in nature. In this study we consider both types of fractals (statistical and exact) and morph one type into the other. Based on the Koch curve, nine visual stimuli were produced in which curves of three different fractal dimensions evolve gradually from an exact to a statistical fractal. The patterns were shown for one minute each to thirty-five subjects while qEEG was continuously recorded. The results showed that the responses to statistical and exact fractals differ, and that the natural form of the fractal is important for inducing alpha responses, an indicator of a wakefully relaxed state and internalized attention. PMID:25575556
Crystallization of space: Space-time fractals from fractal arithmetic
NASA Astrophysics Data System (ADS)
Aerts, Diederik; Czachor, Marek; Kuna, Maciej
2016-02-01
Fractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated with appropriate Lie groups. The construction is illustrated by explicit examples.
Fractal Segmentation and Clustering Analysis for Seismic Time Slices
NASA Astrophysics Data System (ADS)
Ronquillo, G.; Oleschko, K.; Korvin, G.; Arizabalo, R. D.
2002-05-01
Fractal analysis has become part of the standard approach for quantifying texture on gray-tone or colored images. In this research we introduce a multi-stage fractal procedure to segment, classify and measure the clustering patterns on seismic time slices from a 3-D seismic survey. Five fractal classifiers (c1)-(c5) were designed to yield standardized, unbiased and precise measures of the clustering of seismic signals. The classifiers were tested on seismic time slices from the AKAL field, Cantarell Oil Complex, Mexico. The generalized lacunarity (c1), fractal signature (c2), heterogeneity (c3), rugosity of boundaries (c4) and continuity resp. tortuosity (c5) of the clusters are shown to be efficient measures of the time-space variability of seismic signals. The Local Fractal Analysis (LFA) of time slices has proved to be a powerful edge detection filter to detect and enhance linear features, like faults or buried meandering rivers. The local fractal dimensions of the time slices were also compared with the self-affinity dimensions of the corresponding parts of porosity-logs. It is speculated that the spectral dimension of the negative-amplitude parts of the time-slice yields a measure of connectivity between the formation's high-porosity zones, and correlates with overall permeability.
NASA Astrophysics Data System (ADS)
McAteer, R. T. J.
2013-06-01
When Mandelbrot, the father of modern fractal geometry, made this seemingly obvious statement he was trying to show that we should move out of our comfortable Euclidean space and adopt a fractal approach to geometry. The concepts and mathematical tools of fractal geometry provides insight into natural physical systems that Euclidean tools cannot do. The benet from applying fractal geometry to studies of Self-Organized Criticality (SOC) are even greater. SOC and fractal geometry share concepts of dynamic n-body interactions, apparent non-predictability, self-similarity, and an approach to global statistics in space and time that make these two areas into naturally paired research techniques. Further, the iterative generation techniques used in both SOC models and in fractals mean they share common features and common problems. This chapter explores the strong historical connections between fractal geometry and SOC from both a mathematical and conceptual understanding, explores modern day interactions between these two topics, and discusses how this is likely to evolve in