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.
Low Fractal Dimension Cluster-Dilute Soot Aggregates from a Premixed Flame
NASA Astrophysics Data System (ADS)
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.
2009-06-01
Using a novel morphology segregation technique, we observed minority populations (≈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 ≈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.
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...
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%.
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.
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 aggregates in Titan's atmosphere
NASA Astrophysics Data System (ADS)
Cabane, M.; Rannou, P.; Chassefiere, E.; Israel, G.
1993-04-01
The cluster structure of Titan's atmosphere was modeled by using an Eulerian microphysical model with the specific formulation of microphysical laws applying to fractal particles. The growth of aggregates in the settling phase was treated by introducing the fractal dimension as a parameter of the model. The model was used to obtain a vertical distribution of size and number density of the aggregates for different production altitudes. Results confirm previous estimates of the formation altitude of photochemical aerosols. The vertical profile of the effective radius of aggregates was calculated as a function of the visible optical depth.
Molecular dynamics simulation of fractal aggregate diffusion
NASA Astrophysics Data System (ADS)
Pranami, Gaurav; Lamm, Monica H.; Vigil, R. Dennis
2010-11-01
The diffusion of fractal aggregates constructed with the method by Thouy and Jullien [J. Phys. A 27, 2953 (1994)10.1088/0305-4470/27/9/012] comprised of Np spherical primary particles was studied as a function of the aggregate mass and fractal dimension using molecular dynamics simulations. It is shown that finite-size effects have a strong impact on the apparent value of the diffusion coefficient (D) , but these can be corrected by carrying out simulations using different simulation box sizes. Specifically, the diffusion coefficient is inversely proportional to the length of a cubic simulation box, and the constant of proportionality appears to be independent of the aggregate mass and fractal dimension. Using this result, it is possible to compute infinite dilution diffusion coefficients (Do) for aggregates of arbitrary size and fractal dimension, and it was found that Do∝Np-1/df , as is often assumed by investigators simulating Brownian aggregation of fractal aggregates. The ratio of hydrodynamic radius to radius of gyration is computed and shown to be independent of mass for aggregates of fixed fractal dimension, thus enabling an estimate of the diffusion coefficient for a fractal aggregate based on its radius of gyration.
The fractal aggregation of asphaltenes.
Hoepfner, Michael P; Fávero, Cláudio Vilas Bôas; Haji-Akbari, Nasim; Fogler, H Scott
2013-07-16
This paper discusses time-resolved small-angle neutron scattering results that were used to investigate asphaltene structure and stability with and without a precipitant added in both crude oil and model oil. A novel approach was used to isolate the scattering from asphaltenes that are insoluble and in the process of aggregating from those that are soluble. It was found that both soluble and insoluble asphaltenes form fractal clusters in crude oil and the fractal dimension of the insoluble asphaltene clusters is higher than that of the soluble clusters. Adding heptane also increases the size of soluble asphaltene clusters without modifying the fractal dimension. Understanding the process of insoluble asphaltenes forming fractals with higher fractal dimensions will potentially reveal the microscopic asphaltene destabilization mechanism (i.e., how a precipitant modifies asphaltene-asphaltene interactions). It was concluded that because of the polydisperse nature of asphaltenes, no well-defined asphaltene phase stability envelope exists and small amounts of asphaltenes precipitated even at dilute precipitant concentrations. Asphaltenes that are stable in a crude oil-precipitant mixture are dispersed on the nanometer length scale. An asphaltene precipitation mechanism is proposed that is consistent with the experimental findings. Additionally, it was found that the heptane-insoluble asphaltene fraction is the dominant source of small-angle scattering in crude oil and the previously unobtainable asphaltene solubility at low heptane concentrations was measured.
FRACTAL DIMENSION OF GALAXY ISOPHOTES
Thanki, Sandip; Rhee, George; Lepp, Stephen E-mail: grhee@physics.unlv.edu
2009-09-15
In this paper we investigate the use of the fractal dimension of galaxy isophotes in galaxy classification. We have applied two different methods for determining fractal dimensions to the isophotes of elliptical and spiral galaxies derived from CCD images. We conclude that fractal dimension alone is not a reliable tool but that combined with other parameters in a neural net algorithm the fractal dimension could be of use. In particular, we have used three parameters to segregate the ellipticals and lenticulars from the spiral galaxies in our sample. These three parameters are the correlation fractal dimension D {sub corr}, the difference between the correlation fractal dimension and the capacity fractal dimension D {sub corr} - D {sub cap}, and, thirdly, the B - V color of the galaxy.
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.
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 dimensions of soy protein nanoparticle aggregates determined by dynamic mechanical method
Technology Transfer Automated Retrieval System (TEKTRAN)
Soy protein isolate (SPI) is obtained from soybeans by removing soybean oil and soy carbohydrates. Soy protein nanoparticles were prepared by alkaline hydrolysis of SPI and centrifugal separation process. Structurally, SPI is a globular protein and its aggregates in water consist of sphere-like pr...
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.
Hydrodynamic behavior of fractal aggregates
NASA Astrophysics Data System (ADS)
Wiltzius, Pierre
1987-02-01
Measurements of the radius of gyration RG and the hydrodynamic radius RH of colloidal silica aggregates are reported. These aggregates have fractal geometry and RH is proportional to RG for 500 Å<=RH<=7000 Å, with a ratio RH/RG=0.72+/-0.02. The results are compared with predictions for macromolecules of various shapes. The proportionality of the two radii can be understood with use of the pair correlation function of fractal objects and hydrodynamic interactions on the Oseen level. The value of the ratio remains to be explained.
Fractality à la carte: a general particle aggregation model.
Nicolás-Carlock, J R; Carrillo-Estrada, J L; Dossetti, V
2016-01-19
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.
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.
Surface fractals in liposome aggregation.
Roldán-Vargas, Sándalo; Barnadas-Rodríguez, Ramon; Quesada-Pérez, Manuel; Estelrich, Joan; Callejas-Fernández, José
2009-01-01
In this work, the aggregation of charged liposomes induced by magnesium is investigated. Static and dynamic light scattering, Fourier-transform infrared spectroscopy, and cryotransmission electron microscopy are used as experimental techniques. In particular, multiple intracluster scattering is reduced to a negligible amount using a cross-correlation light scattering scheme. The analysis of the cluster structure, probed by means of static light scattering, reveals an evolution from surface fractals to mass fractals with increasing magnesium concentration. Cryotransmission electron microscopy micrographs of the aggregates are consistent with this interpretation. In addition, a comparative analysis of these results with those previously reported in the presence of calcium suggests that the different hydration energy between lipid vesicles when these divalent cations are present plays a fundamental role in the cluster morphology. This suggestion is also supported by infrared spectroscopy data. The kinetics of the aggregation processes is also analyzed through the time evolution of the mean diffusion coefficient of the aggregates.
Image Segmentation via Fractal Dimension
1987-12-01
statistical expectation K = a proportionality constant H = the Hurst exponent , in interval [0,1] (14:249) Eq (4) is a mathematical generalization of...ease, negatively correlated (24:16). The Hurst exponent is directly related to the fractal diment.ion of the process being modelled by the relation (24...24) DzE.I -H (5) where D = the fractal dimension E m the Euclidean dimension H = the Hurst exponent The effect of N1 on a typical trace can be seen
Backbone fractal dimension and fractal hybrid orbital of protein structure
NASA Astrophysics Data System (ADS)
Peng, Xin; Qi, Wei; Wang, Mengfan; Su, Rongxin; He, Zhimin
2013-12-01
Fractal geometry analysis provides a useful and desirable tool to characterize the configuration and structure of proteins. In this paper we examined the fractal properties of 750 folded proteins from four different structural classes, namely (1) the α-class (dominated by α-helices), (2) the β-class (dominated by β-pleated sheets), (3) the (α/β)-class (α-helices and β-sheets alternately mixed) and (4) the (α + β)-class (α-helices and β-sheets largely segregated) by using two fractal dimension methods, i.e. "the local fractal dimension" and "the backbone fractal dimension" (a new and useful quantitative parameter). The results showed that the protein molecules exhibit a fractal behavior in the range of 1 ⩽ N ⩽ 15 (N is the number of the interval between two adjacent amino acid residues), and the value of backbone fractal dimension is distinctly greater than that of local fractal dimension for the same protein. The average value of two fractal dimensions decreased in order of α > α/β > α + β > β. Moreover, the mathematical formula for the hybrid orbital model of protein based on the concept of backbone fractal dimension is in good coincidence with that of the similarity dimension. So it is a very accurate and simple method to analyze the hybrid orbital model of protein by using the backbone fractal dimension.
Reinforcement of rubber by fractal aggregates
NASA Astrophysics Data System (ADS)
Witten, T. A.; Rubinstein, M.; Colby, R. H.
1993-03-01
Rubber is commonly reinforced with colloidal aggregates of carbon or silica, whose structure has the scale invariance of a fractal object. Reinforced rubbers support large stresses, which often grow faster than linearly with the strain. We argue that under strong elongation the stress arises through lateral compression of the aggregates, driven by the large bulk modulus of the rubber. We derive a power-law relationship between stress and elongation λ when λgg 1. The predicted power p depends on the fractal dimension D and a second structural scaling exponent C. For diffusion-controlled aggregates this power p should lie beween 0.9 and 1.1 ; for reaction-controlled aggregates p should lie between 1.8 and 2.4. For uniaxial compression the analogous powers lie near 4. Practical rubbers filled with fractal aggregates should approach the conditions of validity for these scaling laws. On renforce souvent le caoutchouc avec des agrégats de carbone ou de silice dont la structure a l'invariance par dilatation d'un objet fractal. Les caoutchoucs ainsi renforcés supportent de grandes contraintes qui croissent souvent plus vite que l'élongation. Nous prétendons que, sous élongation forte, cette contrainte apparaît à cause d'une compression latérale des agrégats induite par le module volumique important du caoutchouc. Nous établissons une loi de puissance reliant la contrainte et l'élongation λ quand λgg 1. Cet exposant p dépend de la dimension fractale D et d'un deuxième exposant structural C. Pour des agrégats dont la cinétique de formation est limitée par diffusion, p vaut entre 0,9 et 1,1. Si la cinétique est limitée par le soudage local des particules, p vaut entre 1,8 et 2,4. Sous compression uniaxiale, les puissances homologues valent environ 4. Des caoutchoucs pratiques chargés de tels agrégats devraient approcher des conditions où ces lois d'échelle sont valables.
Emergence of fractals in aggregation with stochastic self-replication.
Hassan, Md Kamrul; Hassan, Md Zahedul; Islam, Nabila
2013-10-01
We propose and investigate a simple model which describes the kinetics of aggregation of Brownian particles with stochastic self-replication. An exact solution and the scaling theory are presented alongside numerical simulation which fully support all theoretical findings. In particular, we show analytically that the particle size distribution function exhibits dynamic scaling and we verify it numerically using the idea of data collapse. Furthermore, the conditions under which the resulting system emerges as a fractal are found, the fractal dimension of the system is given, and the relationship between this fractal dimension and a conserved quantity is pointed out.
Light scattering of fractal aerosol aggregates using T-matrix method
NASA Astrophysics Data System (ADS)
Ding, Kung-Hau
1992-08-01
The aerosols in atmosphere are often random cluster of small primary particles with disordered appearance. They are considered to be fractal aggregates with a noninteger dimensions and have the important property of invariance under scale transformation. The essential fractal morphology are included in this study of scattering properties for aggregated aerosol particles. The fractal aggregates are simulated by using the growth model of hierarchical cluster-cluster aggregation. The T-matrix approach together with the translation addition theorem for vector spherical waves is employed to solve the aggregate scattering problem. Numerical results for scattering cross sections are illustrated for fractal clusters.
Applications of Variance Fractal Dimension: a Survey
NASA Astrophysics Data System (ADS)
Phinyomark, Angkoon; Phukpattaranont, Pornchai; Limsakul, Chusak
2012-04-01
Chaotic dynamical systems are pervasive in nature and can be shown to be deterministic through fractal analysis. There are numerous methods that can be used to estimate the fractal dimension. Among the usual fractal estimation methods, variance fractal dimension (VFD) is one of the most significant fractal analysis methods that can be implemented for real-time systems. The basic concept and theory of VFD are presented. Recent research and the development of several applications based on VFD are reviewed and explained in detail, such as biomedical signal processing and pattern recognition, speech communication, geophysical signal analysis, power systems and communication systems. The important parameters that need to be considered in computing the VFD are discussed, including the window size and the window increment of the feature, and the step size of the VFD. Directions for future research of VFD are also briefly outlined.
Fractal dimension and architecture of trabecular bone.
Fazzalari, N L; Parkinson, I H
1996-01-01
The fractal dimension of trabecular bone was determined for biopsies from the proximal femur of 25 subjects undergoing hip arthroplasty. The average age was 67.7 years. A binary profile of the trabecular bone in the biopsy was obtained from a digitized image. A program written for the Quantimet 520 performed the fractal analysis. The fractal dimension was calculated for each specimen, using boxes whose sides ranged from 65 to 1000 microns in length. The mean fractal dimension for the 25 subjects was 1.195 +/- 0.064 and shows that in Euclidean terms the surface extent of trabecular bone is indeterminate. The Quantimet 520 was also used to perform bone histomorphometric measurements. These were bone volume/total volume (BV/TV) (per cent) = 11.05 +/- 4.38, bone surface/total volume (BS/TV) (mm2/mm3) = 1.90 +/- 0.51, trabecular thickness (Tb.Th) (mm) = 0.12 +/- 0.03, trabecular spacing (Tb.Sp) (mm) = 1.03 +/- 0.36, and trabecular number (Tb.N) (number/mm) = 0.95 +/- 0.25. Pearsons' correlation coefficients showed a statistically significant relationship between the fractal dimension and all the histomorphometric parameters, with BV/TV (r = 0.85, P < 0.0001), BS/TV (r = 0.74, P < 0.0001), Tb.Th (r = 0.50, P < 0.02), Tb.Sp (r = -0.81, P < 0.0001), and Tb.N (r = 0.76, P < 0.0001). This method for calculating fractal dimension shows that trabecular bone exhibits fractal properties over a defined box size, which is within the dimensions of a structural unit for trabecular bone. Therefore, the fractal dimension of trabecular bone provides a measure which does not rely on Euclidean descriptors in order to describe a complex geometry.
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
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
Pre-Service Teachers' Concept Images on Fractal Dimension
ERIC Educational Resources Information Center
Karakus, Fatih
2016-01-01
The analysis of pre-service teachers' concept images can provide information about their mental schema of fractal dimension. There is limited research on students' understanding of fractal and fractal dimension. Therefore, this study aimed to investigate the pre-service teachers' understandings of fractal dimension based on concept image. The…
1987-05-07
CLASSIFICATION Of THIS PAGIL(Whan Data Entered) It is found that the reaction rates of OLA agigregates are comnatible with * that of circular aggregates with...NOSC Wayne State University Code 521 Detroit, Michigan 49207 San Diego, California 91232 Or. W.E. Moerner I.B.M. Corporation Almaden Research Center 650
Aggregates and Superaggregates of Soot with Four Distinct Fractal Morphologies
NASA Technical Reports Server (NTRS)
Sorensen, C. M.; Kim, W.; Fry, D.; Chakrabarti, A.
2004-01-01
Soot formed in laminar diffusion flames of heavily sooting fuels evolves through four distinct growth stages which give rise to four distinct aggregate fractal morphologies. These results were inferred from large and small angle static light scattering from the flames, microphotography of the flames, and analysis of soot sampled from the flames. The growth stages occur approximately over four successive orders of magnitude in aggregate size. Comparison to computer simulations suggests that these four growth stages involve either diffusion limited cluster aggregation or percolation in either three or two dimensions.
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.
NASA Astrophysics Data System (ADS)
Wei, Wei; Cai, Jianchao; Hu, Xiangyun; Han, Qi; Liu, Shuang; Zhou, Yingfang
2016-08-01
A theoretical effective thermal conductivity model for nanofluids is derived based on fractal distribution characteristics of nanoparticle aggregation. Considering two different mechanisms of heat conduction including particle aggregation and convention, the model is expressed as a function of the fractal dimension and concentration. In the model, the change of fractal dimension is related to the variation of aggregation shape. The theoretical computations of the developed model provide a good agreement with the experimental results, which may serve as an effective approach for quantitatively estimating the effective thermal conductivity of nanofluids.
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.
Vahedi, Arman; Gorczyca, Beata
2012-09-01
Here we introduce a distribution of floc fractal dimensions as opposed to a single fractal dimension value into the floc settling velocity model developed in earlier studies. The distribution of fractal dimensions for a single floc size was assumed to cover a range from 1.9 to 3.0. This range was selected based on the theoretically determined fractal dimensions for diffusion-limited and cluster-cluster aggregation. These two aggregation mechanisms are involved in the formation of the lime softening flocs analyzed in this study. Fractal dimensions were generated under the assumption that a floc can have any value of normally distributed fractal dimensions ranging from 1.9-3.0. A range of settling velocities for a single floc size was calculated based on the distribution of fractal dimensions. The assumption of multiple fractal dimensions for a single floc size resulted in a non-unique relationship between the floc size and the floc settling velocity, i.e., several different settling velocities were calculated for one floc size. The settling velocities calculated according to the model ranged from 0 to 10 mm/s (average 2.22 mm/s) for the majority of flocs in the size range of 1-250 μm (average 125 μm). The experimentally measured settling velocities of flocs ranged from 0.1 to 7.1 mm/s (average 2.37 mm/s) for the flocs with equivalent diameters from 10 μm to 260 μm (average 124 μm). Experimentally determined floc settling velocities were predicted well by the floc settling model incorporating distributions of floc fractal dimensions calculated based on the knowledge of the mechanisms of aggregation, i.e., cluster-cluster aggregation and diffusion-limited aggregation.
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.
Dependence of fractal dimension of DLCA clusters on size of primary particles.
Wu, Hua; Lattuada, Marco; Morbidelli, Massimo
2013-07-01
It is well known that clusters generated from colloidal aggregation driven by Brownian motion are typical fractal objects with the fractal dimension in the range of 1.75-1.85 under the diffusion-limited cluster aggregation (DLCA) conditions. In this work, we review and analyze the values of the fractal dimension for DLCA clusters experimentally determined in the literature. It is found that the value of the fractal dimension decreases significantly as the primary particle radius increases. Then, we have properly designed the DLCA experiments, using different radii of the primary particles, and determined the fractal dimensions of the generated clusters. Our results have well confirmed that the fractal dimension indeed decreases as the particle radius increases. To explore the mechanism leading to such dependence, we have performed intense computations through the full T-Matrix theory, and we conclude that this is not related to the effect of the intra-cluster multiple scattering on the slope of the scattering structure factor. The large fractal dimensions of the clusters generated by very small nanoparticles could be explained by thermal restructuring due to their low bonding energies, but no clear explanation can be given for the small fractal dimensions of the clusters made of large particles.
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.
Laser light scattering as a probe of fractal colloid aggregates
NASA Technical Reports Server (NTRS)
Weitz, David A.; Lin, M. Y.
1989-01-01
The extensive use of laser light scattering is reviewed, both static and dynamic, in the study of colloid aggregation. Static light scattering enables the study of the fractal structure of the aggregates, while dynamic light scattering enables the study of aggregation kinetics. In addition, both techniques can be combined to demonstrate the universality of the aggregation process. Colloidal aggregates are now well understood and therefore represent an excellent experimental system to use in the study of the physical properties of fractal objects. However, the ultimate size of fractal aggregates is fundamentally limited by gravitational acceleration which will destroy the fractal structure as the size of the aggregates increases. This represents a great opportunity for spaceborne experimentation, where the reduced g will enable the growth of fractal structures of sufficient size for many interesting studies of their physical properties.
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.
Permeability of collapsed cakes formed by deposition of fractal aggregates upon membrane filtration.
Park, Pyung-Kyu; Lee, Chung-Hak; Lee, Sangho
2006-04-15
We have investigated, theoretically, the physical properties of cake layers formed from aggregates to obtain a better understanding of membrane systems used in conjunction with coagulation/flocculation pretreatment. We developed a model based on fractal theory and incorporated a cake collapse effect to predict the porosity and permeability of the cake layers. The floc size, fractal dimension, and transmembrane pressure were main parameters that we used in these model calculations. We performed experiments using a batch cell device and a confocal laser-scanning microscope to verify the predicted specific cake resistances and porosities under various conditions. Based on the results of the model, the reduction in inter-aggregate porosity is more important than that in intra-aggregate porosity during the cake collapsing process. The specific cake resistance decreases upon increasing the aggregate size and decreasing the fractal dimensions. The modeled porosities and specific cake resistances of the collapsed cake layer agreed reasonably well with those obtained experimentally.
Solar Flare Geometries. I. The Area Fractal Dimension
NASA Astrophysics Data System (ADS)
Aschwanden, Markus J.; Aschwanden, Pascal D.
2008-02-01
In this study we investigate for the first time the fractal dimension of solar flares and find that the flare area observed in EUV wavelengths exhibits a fractal scaling. We measure the area fractal dimension D2, also called the Hausdorff dimension, with a box-counting method, which describes the fractal area as A(L) ~ LD2. We apply the fractal analysis to a statistical sample of 20 GOES X- and M-class flares, including the Bastille Day 2000 July 14 flare, one of the largest flares ever recorded. We find that the fractal area (normalized by the time-integrated flare area Af) varies from near zero at the beginning of the flare to a maximum of A(t)/Af = 0.65 +/- 0.12 after the peak time of the flare, which corresponds to an area fractal dimension in the range of 1.0lesssim D2(t) lesssim 1.89 +/- 0.05. We find that the total EUV flux Ftot(t) is linearly correlated with the fractal area A(t) . From the area fractal dimension D2, the volume fractal dimension D3 can be inferred (subject of Paper II), which is crucial to inferring a realistic volume filling factor, which affects the derived electron densities, thermal energies, and cooling times of solar and stellar flares.
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
Pyramidal fractal dimension for high resolution images.
Mayrhofer-Reinhartshuber, Michael; Ahammer, Helmut
2016-07-01
Fractal analysis (FA) should be able to yield reliable and fast results for high-resolution digital images to be applicable in fields that require immediate outcomes. Triggered by an efficient implementation of FA for binary images, we present three new approaches for fractal dimension (D) estimation of images that utilize image pyramids, namely, the pyramid triangular prism, the pyramid gradient, and the pyramid differences method (PTPM, PGM, PDM). We evaluated the performance of the three new and five standard techniques when applied to images with sizes up to 8192 × 8192 pixels. By using artificial fractal images created by three different generator models as ground truth, we determined the scale ranges with minimum deviations between estimation and theory. All pyramidal methods (PM) resulted in reasonable D values for images of all generator models. Especially, for images with sizes ≥1024×1024 pixels, the PMs are superior to the investigated standard approaches in terms of accuracy and computation time. A measure for the possibility to differentiate images with different intrinsic D values did show not only that the PMs are well suited for all investigated image sizes, and preferable to standard methods especially for larger images, but also that results of standard D estimation techniques are strongly influenced by the image size. Fastest results were obtained with the PDM and PGM, followed by the PTPM. In terms of absolute D values best performing standard methods were magnitudes slower than the PMs. Concluding, the new PMs yield high quality results in short computation times and are therefore eligible methods for fast FA of high-resolution images.
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
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 .
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.
NASA Astrophysics Data System (ADS)
Sui, Jize; Zhao, Peng; Bin-Mohsin, Bandar; Zheng, Liancun; Zhang, Xinxin; Cheng, Zhengdong; Chen, Ying; Chen, Goong
2016-12-01
Nano-suspensions (NS) exhibit unusual thermophysical behaviors once interparticle aggregations and the shear flows are imposed, which occur ubiquitously in applications but remain poorly understood, because existing theories have not paid these attentions but focused mainly on stationary NS. Here we report the critical role of time-dependent fractal aggregation in the unsteady thermal convection of NS systematically. Interestingly, a time ratio λ = tp/tm (tp is the aggregate characteristic time, tm the mean convection time) is introduced to characterize the slow and fast aggregations, which affect distinctly the thermal convection process over time. The increase of fractal dimension reduces both momentum and thermal boundary layers, meanwhile extends the time duration for the full development of thermal convection. We find a nonlinear growth relation of the momentum layer, but a linear one of the thermal layer, with the increase of primary volume fraction of nanoparticles for different fractal dimensions. We present two global fractal scaling formulas to describe these two distinct relations properly, respectively. Our theories and methods in this study provide new evidence for understanding shear-flow and anomalous heat transfer of NS associated non-equilibrium aggregation processes by fractal laws, moreover, applications in modern micro-flow technology in nanodevices.
Sui, Jize; Zhao, Peng; Bin-Mohsin, Bandar; Zheng, Liancun; Zhang, Xinxin; Cheng, Zhengdong; Chen, Ying; Chen, Goong
2016-01-01
Nano-suspensions (NS) exhibit unusual thermophysical behaviors once interparticle aggregations and the shear flows are imposed, which occur ubiquitously in applications but remain poorly understood, because existing theories have not paid these attentions but focused mainly on stationary NS. Here we report the critical role of time-dependent fractal aggregation in the unsteady thermal convection of NS systematically. Interestingly, a time ratio λ = tp/tm (tp is the aggregate characteristic time, tm the mean convection time) is introduced to characterize the slow and fast aggregations, which affect distinctly the thermal convection process over time. The increase of fractal dimension reduces both momentum and thermal boundary layers, meanwhile extends the time duration for the full development of thermal convection. We find a nonlinear growth relation of the momentum layer, but a linear one of the thermal layer, with the increase of primary volume fraction of nanoparticles for different fractal dimensions. We present two global fractal scaling formulas to describe these two distinct relations properly, respectively. Our theories and methods in this study provide new evidence for understanding shear-flow and anomalous heat transfer of NS associated non-equilibrium aggregation processes by fractal laws, moreover, applications in modern micro-flow technology in nanodevices. PMID:27995980
Sui, Jize; Zhao, Peng; Bin-Mohsin, Bandar; Zheng, Liancun; Zhang, Xinxin; Cheng, Zhengdong; Chen, Ying; Chen, Goong
2016-12-20
Nano-suspensions (NS) exhibit unusual thermophysical behaviors once interparticle aggregations and the shear flows are imposed, which occur ubiquitously in applications but remain poorly understood, because existing theories have not paid these attentions but focused mainly on stationary NS. Here we report the critical role of time-dependent fractal aggregation in the unsteady thermal convection of NS systematically. Interestingly, a time ratio λ = tp/tm (tp is the aggregate characteristic time, tm the mean convection time) is introduced to characterize the slow and fast aggregations, which affect distinctly the thermal convection process over time. The increase of fractal dimension reduces both momentum and thermal boundary layers, meanwhile extends the time duration for the full development of thermal convection. We find a nonlinear growth relation of the momentum layer, but a linear one of the thermal layer, with the increase of primary volume fraction of nanoparticles for different fractal dimensions. We present two global fractal scaling formulas to describe these two distinct relations properly, respectively. Our theories and methods in this study provide new evidence for understanding shear-flow and anomalous heat transfer of NS associated non-equilibrium aggregation processes by fractal laws, moreover, applications in modern micro-flow technology in nanodevices.
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.
FRACTAL DIMENSION RESULTS FOR CONTINUOUS TIME RANDOM WALKS
Meerschaert, Mark M.; Nane, Erkan; Xiao, Yimin
2013-01-01
Continuous time random walks impose random waiting times between particle jumps. This paper computes the fractal dimensions of their process limits, which represent particle traces in anomalous diffusion. PMID:23482421
Measuring the Fractal Dimensions of Empirical Cartographic Curves,
1982-01-01
ahhf, by Week smnber) Fractal dimension, Chord length, Line length, Linear regression U.ASTRACF (CO1 o M 6 #d itw 4000"s If Rea.5 I.R OF Wleek amber...The fractal dimension of a curve Is a measure of Its geometric complexity and - can be any men-integer value between 1 and 2 depend ing upon the...curve’s level pair of dividers along a curve, used to calculate the fractal diummlons ofS3 cuvs. It also discusses the *hole of chord length and the wabor
Voronoi cells, fractal dimensions and fibre composites.
Summerscales, J.; Guild, F. J.; Pearce, N. R. L.; Russell, P. M.
2001-02-01
The use of fibre-reinforced polymer matrix composite materials is growing at a faster rate than the gross domestic product (GDP) in many countries. An improved understanding of their processing and mechanical behaviour would extend the potential applications of these materials. For unidirectional composites, it is predicted that localized absence of fibres is related to longitudinal compression failure. The use of woven reinforcements permits more effective manufacture than for unidirectional fibres. It has been demonstrated experimentally that compression strengths of woven composites are reduced when fibres are clustered. Summerscales predicted that clustering of fibres would increase the permeability of the reinforcement and hence expedite the processing of these materials. Commercial fabrics are available which employ this concept using flow-enhancing bound tows. The net effect of clustering fibres is to enhance processability whilst reducing the mechanical properties. The effects reported above were qualitative correlations. To improve the design tools for reinforcement fabrics we have sought to quantify the changes in the micro/meso-structure of woven reinforcement fabrics. Gross differences in the appearance of laminate sections are apparent for different weave styles. The use of automated image analysis is essential for the quantification of subtle changes in fabric architecture. This paper considers Voronoi tessellation and fractal dimensions for the quantification of the microstructures of woven fibre-reinforced composites. It reviews our studies in the last decade of the process-property-structure relationships for commercial and experimental fabric reinforcements in an attempt to resolve the processing vs. properties dilemma. A new flow-enhancement concept has been developed which has a reduced impact on laminate mechanical properties.
Fractal structure and the dynamics of aggregation of synthetic melanin in low pH aqueous solutions
Huang, J.S.; Sung, J.; Eisner, M.; Moss, S.C.; Gallas, J.
1989-01-01
We have used static and dynamic light scattering to study the dynamics of aggregation of synthetic melanin, an amorphous biopolymeric substance, in low pH aqueous solution. We have found that, depending on the final pH value of the solutions, there existed two regimes of the aggregation kinetics, one corresponding to diffusion limited aggregation (DLA), and the other corresponding to reaction limited aggregation (RLA). The precipitates formed in these two regimes can be characterized by fractal structures. We have found fractal dimensions of d/sub f/ = 1.8 for the DLA clusters and d/sub f/ = 2.2 for the RLA clusters. These results agree well with the proposed limits of the fractal dimensions of the gold aggregates formed in aqueous solutions by Weitz et al.
NASA Astrophysics Data System (ADS)
Huang, J. S.; Sung, J.; Eisner, M.; Moss, S. C.; Gallas, J.
1989-01-01
We have used static and dynamic light scattering to study the dynamics of aggregation of synthetic melanin, an amorphous biopolymeric substance, in low pH aqueous solution. We have found that, depending on the final pH value of the solutions, there existed two regimes of the aggregation kinetics, one corresponding to diffusion limited aggregation (DLA), and the other corresponding to reaction limited aggregation (RLA). The precipitates formed in these two regimes can be characterized by fractal structures. We have found fractal dimensions of df =1.8 for the DLA clusters and df =2.2 for the RLA clusters. These results agree well with the proposed limits of the fractal dimensions of the gold aggregates formed in aqueous solutions by Weitz et al.
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.
Fractal aggregate analogues for near solar dust properties
NASA Astrophysics Data System (ADS)
Mann, I.; Okamoto, H.; Mukai, T.; Kimura, H.; Kitada, Y.
1994-11-01
The present study compares properties of near solar dust, deduced from inversion of F-corona brightness data, with calculations of fluffy aggregate particles. It is shown that silicate particles containing a slight amount of absorbing material have temperatures below the blackbody temperature if the impurity amounts to less than 0.1% in volume. This effect is especially significant for porous particles and points to the existence of such a component, possibly cometary dust, in the solar vicinity. In particular the silicate particles with a large impurity, which show a higher temperature than the blackbody at the same solar distance, the sublimation starts closer to the sun and the pure silicate, if it would exist, would survive even about 2 solar radii from the sun. This result which is based on calculations that apply model materials, may possibly explain the wide extended zone of sublimation derived from F-corona brightness data. Another finding of our calculations is an unexpected enhancement of temperature of the two-component aggregates. Namely the silicate aggregate with absorbing impurities attains higher temperature even than the pure carbon. This happens when the volume fraction of absorbing material exceeds 1% and the aggregate with a fractal dimension of 2.98 is located below about 100 solar radii from the sun; this critical solar distance depends on the volume fraction of absorbing material. A similar trend was also seen in the compact particle consisting of two components. This comes from the complex dependence of the energy loss from the particle on the temperature.
Fractal Aggregates in Tennis Ball Systems
ERIC Educational Resources Information Center
Sabin, J.; Bandin, M.; Prieto, G.; Sarmiento, F.
2009-01-01
We present a new practical exercise to explain the mechanisms of aggregation of some colloids which are otherwise not easy to understand. We have used tennis balls to simulate, in a visual way, the aggregation of colloids under reaction-limited colloid aggregation (RLCA) and diffusion-limited colloid aggregation (DLCA) regimes. We have used the…
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.
Smitha, K A; Gupta, A K; Jayasree, R S
2015-09-07
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.
Impact of morphology on the radiative properties of fractal soot aggregates
NASA Astrophysics Data System (ADS)
Doner, Nimeti; Liu, Fengshan
2017-01-01
The impact of morphology on the radiative properties of fractal soot aggregates was investigated using the discrete dipole approximation (DDA). The optical properties of four different types of aggregates of freshly emitted soot with a fractal dimension Df=1.65 and a fractal pre-factor kf=1.76 were calculated. The four types of aggregates investigated are formed by uniform primary particles in point-touch, by uniform but overlapping primary particles, by uniform but enlarged primary particles in point-touch, and formed by point-touch and polydisperse primary particles. The radiative properties of aggregates consisting of N=20, 56 and 103 primary particles were numerically evaluated for a given refractive index at 0.532 and 1.064 μm. The radiative properties of soot aggregates vary strongly with the volume equivalent radius aeff and wavelength. The accuracy of DDA was evaluated in the first and fourth cases against the generalized multi-sphere Mie (GMM) solution in terms of the vertical-vertical differential scattering cross section (Cvv). The model predicted the average relative deviations from the base case to be within 15-25% for Cvv, depending on the number of particles for the aggregate. The scattering cross sections are only slightly affected by the overlapping but more significantly influenced by primary particle polydispersity. It was also found that the enlargement of primary particles by 20% has a strong effect on soot aggregate radiative properties.
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.
Fractal Dimension of Certain Continuous Functions of Unbounded Variation
NASA Astrophysics Data System (ADS)
Liang, Y. S.; Su, W. Y.
Continuous functions on closed intervals are composed of bounded variation functions and unbounded variation functions. Fractal dimension of continuous functions with bounded variation must be one-dimensional (1D). While fractal dimension of continuous functions with unbounded variation may be 1 or not. Certain continuous functions of unbounded variation whose fractal dimensions are 1 have been mainly investigated in the paper. A continuous function on a closed interval with finite unbounded variation points has been proved to be 1D. Furthermore, we deal with continuous functions which have infinite unbounded variation points and part of them have been proved to be 1D. Certain examples of 1D continuous functions which have uncountable unbounded variation points have been given in the present paper.
Relationship between Fractal Dimension and Agreeability of Facial Imagery
NASA Astrophysics Data System (ADS)
Oyama-Higa, Mayumi; Miao, Tiejun; Ito, Tasuo
2007-11-01
Why do people feel happy and good or equivalently empathize more, with smiling face imageries than with ones of expressionless face? To understand what the essential factors are underlying imageries in relating to the feelings, we conducted an experiment by 84 subjects asked to estimate the degree of agreeability about expressionless and smiling facial images taken from 23 young persons to whom the subjects were no any pre-acquired knowledge. Images were presented one at a time to each subject who was asked to rank agreeability on a scale from 1 to 10. Fractal dimensions of facial images were obtained in order to characterize the complexity of the imageries by using of two types of fractal analysis methods, i.e., planar and cubic analysis methods, respectively. The results show a significant difference in the fractal dimension values between expressionless faces and smiling ones. Furthermore, we found a well correlation between the degree of agreeability and fractal dimensions, implying that the fractal dimension optically obtained in relation to complexity in imagery information is useful to characterize the psychological processes of cognition and awareness.
EEG signal features extraction based on fractal dimension.
Finotello, Francesca; Scarpa, Fabio; Zanon, Mattia
2015-01-01
The spread of electroencephalography (EEG) in countless applications has fostered the development of new techniques for extracting synthetic and informative features from EEG signals. However, the definition of an effective feature set depends on the specific problem to be addressed and is currently an active field of research. In this work, we investigated the application of features based on fractal dimension to a problem of sleep identification from EEG data. We demonstrated that features based on fractal dimension, including two novel indices defined in this work, add valuable information to standard EEG features and significantly improve sleep identification performance.
Effect of Na+ on surface fractal dimension of compacted bentonite
NASA Astrophysics Data System (ADS)
Xiang, G. S.; Xu, Y. F.; Jiang, H.
2015-05-01
Compacted Tsukinuno bentonite was immersed into NaCl solutions of different concentrations in oedometers, and the surface fractal dimension of bentonite-saline association was measured by nitrogen adsorption isotherms. The application of the Frenkel-Halsey-Hill equation and the Neimark thermodynamic method to nitrogen adsorption isotherms indicated that the surface roughness was greater for the bentonite-saline association. The surface fractal dimension of bentonite increased in the NaCl solution with low Na+ concentration, but decreased at high Na+ concentration. This process was accompanied by the same tendency in specific surface area and microporosity with the presence of Na+ coating in the clay particles.
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.
NASA Astrophysics Data System (ADS)
Celardo, G. L.; Archetti, D.; Ferrini, G.; Gavioli, L.; Pingue, P.; Cavaliere, E.
2017-01-01
The specific mechanisms which lead to the formation of fractal nanostructures by pulsed laser deposition remain elusive despite intense research efforts, motivated mainly by the technological interest in obtaining tailored nanostructures with simple and scalable production methods. Here we focus on fractal nanostructures of titanium dioxide, TiO2, a strategic material for many applications, obtained by femtosecond laser ablation at ambient conditions. We compare a theoretical model of fractal formation with experimental data. The comparison of theory and experiment confirms that fractal aggregates are formed after landing of the ablated material on the substrate surface by a simple diffusive mechanism. We model the fractal formation through extensive Monte Carlo simulations based on a set of minimal assumptions: TiO2 nanoparticles arrive already formed on the substrate, then they diffuse in a size/mass independent way and stick irreversibly upon touching, thus forming fractal clusters. Despite its simplicity, our model explains the main features of the fractal structures arising from the complex interaction of large TiO2 nanoparticles with different substrates. Indeed our model is able to reproduce both the fractal dimensions and the area distributions of the nanostructures for different densities of the ablated material. Finally we discuss the role of the thermal conductivity of the substrate and the laser fluence on the properties of the fractal nanostructures. Our results represent an advancement towards controlling the production of fractal nanostructures by pulsed laser deposition.
The Fractal Dimension of the ρ Ophiucus Molecular Cloud Complex
NASA Astrophysics Data System (ADS)
Lee, Yongung; Yi, Di; Kim, Y. S.; Jung, J. H.; Kang, H. W.; Lee, C. H.; Yim, I. S.; Kim, H. G.
2016-12-01
We estimate the fractal dimension of the ρ Ophiuchus Molecular Cloud Complex, associated with star forming regions. We selected a cube (v, l, b) database, obtained with J=1-0 transition lines of \\coand tco at a resolution of 22'' using a multibeam receiver system on the 14-m telescope of the Five College Radio Astronomy Observatory. Using a code developed within IRAF, we identified slice-clouds with two threshold temperatures to estimate the fractal dimension. With threshold temperatures of 2.25 K (3σ) and 3.75 K (5σ), the fractal dimension of the target cloud is estimated to be D = 1.52-1.54, where P ∝ A^{D/2} , which is larger than previous results. We suggest that the sampling rate (spatial resolution) of observed data must be an important parameter when estimating the fractal dimension, and that narrower or wider dispersion around an arbitrary fit line and the intercepts at NP = 100 should be checked whether they relate to rms noise level or characteristic structure of the target cloud. This issue could be investigated by analysing several high resolution databases with different quality (low or moderate sensitivity).
Liver ultrasound image classification by using fractal dimension of edge
NASA Astrophysics Data System (ADS)
Moldovanu, Simona; Bibicu, Dorin; Moraru, Luminita
2012-08-01
Medical ultrasound image edge detection is an important component in increasing the number of application of segmentation, and hence it has been subject of many studies in the literature. In this study, we have classified the liver ultrasound images (US) combining Canny and Sobel edge detectors with fractal analysis in order to provide an indicator about of the US images roughness. We intend to provide a classification rule of the focal liver lesions as: cirrhotic liver, liver hemangioma and healthy liver. For edges detection the Canny and Sobel operators were used. Fractal analyses have been applied for texture analysis and classification of focal liver lesions according to fractal dimension (FD) determined by using the Box Counting method. To assess the performance and accuracy rate of the proposed method the contrast-to-noise (CNR) is analyzed.
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.
The formalism of fractal aggregation phenomena of colloidal drug delivery systems.
Pippa, Natassa; Demetzos, Costas; Danezis, Emmanuel
2012-03-01
Classical Newtonian Physics and Euclidean Geometry are currently used to describe biological phenomena and the processes of drug formulation, which are characterized by homogeneity and linearity. On the other hand, at the mesoscopic level, the principles and the laws of physics are quite different from the Classical Newtonian Physics and Euclidean approach especially at nanoscale dimension. The investigation of the aggregation process of liposomes is of paramount importance due to their applications in pharmaceutical nanotechnology as drug delivery systems and as membrane models, in biosciences. The physical stability and the aggregation process of colloidal systems as well as the surface phenomena are described using the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. The elucidation of the dimensionality of liposome aggregates obeys the fractal approach because the aggregation phenomena are irreversible. This approach can be correlated with the extended DLVO theory, which includes the hydration energy, too.
NASA Astrophysics Data System (ADS)
Besselink, R.; Stawski, T. M.; Van Driessche, A. E. S.; Benning, L. G.
2016-12-01
Densely packed surface fractal aggregates form in systems with high local volume fractions of particles with very short diffusion lengths, which effectively means that particles have little space to move. However, there are no prior mathematical models, which would describe scattering from such surface fractal aggregates and which would allow the subdivision between inter- and intraparticle interferences of such aggregates. Here, we show that by including a form factor function of the primary particles building the aggregate, a finite size of the surface fractal interfacial sub-surfaces can be derived from a structure factor term. This formalism allows us to define both a finite specific surface area for fractal aggregates and the fraction of particle interfacial sub-surfaces at the perimeter of an aggregate. The derived surface fractal model is validated by comparing it with an ab initio approach that involves the generation of a "brick-in-a-wall" von Koch type contour fractals. Moreover, we show that this approach explains observed scattering intensities from in situ experiments that followed gypsum (CaSO4 ṡ 2H2O) precipitation from highly supersaturated solutions. Our model of densely packed "brick-in-a-wall" surface fractal aggregates may well be the key precursor step in the formation of several types of mosaic- and meso-crystals.
Predicting beauty: fractal dimension and visual complexity in art.
Forsythe, A; Nadal, M; Sheehy, N; Cela-Conde, C J; Sawey, M
2011-02-01
Visual complexity has been known to be a significant predictor of preference for artistic works for some time. The first study reported here examines the extent to which perceived visual complexity in art can be successfully predicted using automated measures of complexity. Contrary to previous findings the most successful predictor of visual complexity was Gif compression. The second study examined the extent to which fractal dimension could account for judgments of perceived beauty. The fractal dimension measure accounts for more of the variance in judgments of perceived beauty in visual art than measures of visual complexity alone, particularly for abstract and natural images. Results also suggest that when colour is removed from an artistic image observers are unable to make meaningful judgments as to its beauty.
Fractal dimension of critical clusters in the Φ44 model
NASA Astrophysics Data System (ADS)
Jansen, K.; Lang, C. B.
1991-06-01
We study the d=4 O(4) symmetric nonlinear sigma model at the pseudocritical points for 84-284 lattices. The Fortuin-Kasteleyn-Coniglio-Klein clusters are shown to have fractal dimension df~=3-in accordance with the conjectured scaling relation involving the odd critical exponent δ. For the one cluster algorithm introduced recently by Wolff the dynamical critical exponent z comes out to be compatible with zero in this model.
Fractal Dimension in Eeg Signals during Muscle Fatigue
NASA Astrophysics Data System (ADS)
Huang, Haibin; Yao, Bin; Yue, Guang; Brown, Robert; Jing, Liu
2003-10-01
Fractal dimension (FD) has been successfully used to characterize signals in the format of time series. In this study, we calculated FD of EEG signals recorded during human muscle fatigue as a measure of changes in the EEG signal complexity along fatigue. Subjects performed 200 intermittent handgrip contractions at 100contraction level. Each contraction lasted 2 s, followed by a 5-s rest. EEG data were recorded from the scalp along with handgrip force and muscle EMG signals. The FD computation was based on measurements of the length (Lk) of the signal at 6 different temporal resolutions (k = 1, 2, ¡, 6). FD was determined from the relationship between Lk and k using the least square fit. The results showed that: (1) EEG fractal dimension associated with the motor performance was significantly higher than that during the rest period; (2) changes in the fractal dimension along the process of fatigue showed a significant correlation with the decline in force and EMG signals.
Takehara, Takuma; Ochiai, Fumio; Watanabe, Hiroshi; Suzuki, Naoto
2013-01-01
There is currently substantial literature to suggest that facial emotion recognition is impaired when other-race or inverted faces are presented. This study examined circumplex structures for recognising facial emotions under these conditions, directly measured those structures using a fractal dimension, and examined the difference between fractal dimensions. Results established that emotion ratings for the emotion prototypes used were sufficiently accurate under all conditions. Fractal analyses showed that the fractal dimensions of the circumplexes were significantly higher for recognition of facial emotions in other races than in one's own when the faces were presented upright; the fractal dimensions of the circumplexes were also higher for recognition of emotions in inverted faces than in upright faces in the own-race condition. The results suggest that a lower level of facial emotion recognition is associated with higher fractal dimension and that an increase of fractal dimension may be characterised by lack of facial familiarity.
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).
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
NASA Astrophysics Data System (ADS)
Li, Jian-Hua; Yu, Bo-Ming; Zou, Ming-Qing
2009-11-01
We report a model for the fractal dimension Ds of rough surfaces based on the fractal distribution of roughness elements on surfaces and the fractal character of surface profiles. The proposed model for the fractal dimension Ds is expressed as a function of the fractal dimensions D for conic roughness diameter/height and Dp for surface profile, maximum roughness base diameter λmax, the ratio β of conic roughness height to its base radius as well as the ratio λminλmax of the minimum to the maximal base diameter.
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.
Cosmology in one dimension: fractal geometry, power spectra and correlation
NASA Astrophysics Data System (ADS)
Miller, Bruce N.; Rouet, Jean-Louis
2010-12-01
Concentrations of matter, such as galaxies and galactic clusters, originated as very small density fluctuations in the early universe. The existence of galaxy clusters and super-clusters suggests that a natural scale for the matter distribution may not exist. A point of controversy is whether the distribution is fractal and, if so, over what range of scales. One-dimensional models demonstrate that the important dynamics for cluster formation occur in the position-velocity plane. Here the development of scaling behavior and multifractal geometry is investigated for a family of one-dimensional models for three different, scale-free, initial conditions. The methodology employed includes: (1) the derivation of explicit solutions for the gravitational potential and field for a one-dimensional system with periodic boundary conditions (Ewald sums for one dimension); (2) the development of a procedure for obtaining scale-free initial conditions for the growing mode in phase space for an arbitrary power-law index; (3) the evaluation of power spectra, correlation functions, and generalized fractal dimensions at different stages of the system evolution. It is shown that a simple analytic representation of the power spectra captures the main features of the evolution, including the correct time dependence of the crossover from the linear to nonlinear regime and the transition from regular to fractal geometry. A possible physical mechanism for understanding the self-similar evolution is introduced. It is shown that hierarchical cluster formation depends both on the model and on the initial power spectrum. Under special circumstances a simple relation between the power spectrum, correlation function, and correlation dimension in the highly nonlinear regime is confirmed.
Structure and fractal dimension of protein-detergent complexes
NASA Astrophysics Data System (ADS)
Chen, Sow-Hsin; Teixeira, José
1986-11-01
Small-angle neutron-scattering experiments were made on bovine serum albumin (BSA)-lithium dodecyl sulfate (LDS) complexes in buffer solutions. As increasing amounts of LDS are added, the scattering data indicate that BSA molecules are successively transformed into random coil conformations with LDS forming globular micelles randomly decorating the polypeptide backbones. A cross-section formula is developed which successfully fits small-angle neutron-scattering spectra over the entire Q range. The fractal dimension, the micellar size, and the extent of the denatured protein are simultaneously extracted.
Fractal dimension of electroencephalographic time series and underlying brain processes.
Lutzenberger, W; Preissl, H; Pulvermüller, F
1995-10-01
Fractal dimension has been proposed as a useful measure for the characterization of electrophysiological time series. This paper investigates what the pointwise dimension of electroencephalographic (EEG) time series can reveal about underlying neuronal generators. The following theoretical assumptions concerning brain function were made (i) within the cortex, strongly coupled neural assemblies exist which oscillate at certain frequencies when they are active, (ii) several such assemblies can oscillate at a time, and (iii) activity flow between assemblies is minimal. If these assumptions are made, cortical activity can be considered as the weighted sum of a finite number of oscillations (plus noise). It is shown that the correlation dimension of finite time series generated by multiple oscillators increases monotonically with the number of oscillators. Furthermore, it is shown that a reliable estimate of the pointwise dimension of the raw EEG signal can be calculated from a time series as short as a few seconds. These results indicate that (i) The pointwise dimension of the EEG allows conclusions regarding the number of independently oscillating networks in the cortex, and (ii) a reliable estimate of the pointwise dimension of the EEG is possible on the basis of short raw signals.
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
Spectral asymmetry and Higuchi's fractal dimension measures of depression electroencephalogram.
Bachmann, Maie; Lass, Jaanus; Suhhova, Anna; Hinrikus, Hiie
2013-01-01
This study was aimed to compare two electroencephalogram (EEG) analysis methods, spectral asymmetry index (SASI) and Higuchi's fractal dimension (HFD), for detection of depression. Linear SASI method is based on evaluation of the balance of powers in two EEG frequency bands in one channel selected higher and lower than the alpha band spectrum maximum. Nonlinear HFD method calculates fractal dimension directly in the time domain. The resting EEG signals of 17 depressive patients and 17 control subjects were used as a database for calculations. SASI values were positive for depressive and negative for control group (P < 0.05). SASI provided the true detection rate of 88% in the depressive and 82% in the control group. The calculated HFD values detected a small (3%) increase with depression (P < 0.05). HFD provided the true detection rate of 94% in the depressive group and 76% in the control group. The rate of correct indication in the both groups was 85% using SASI or HFD. Statistically significant variations were not revealed between hemispheres (P > 0.05). The results indicated that the linear EEG analysis method SASI and the nonlinear HFD method both demonstrated a good sensitivity for detection of characteristic features of depression in a single-channel EEG.
Spectral Asymmetry and Higuchi's Fractal Dimension Measures of Depression Electroencephalogram
Bachmann, Maie; Lass, Jaanus; Suhhova, Anna; Hinrikus, Hiie
2013-01-01
This study was aimed to compare two electroencephalogram (EEG) analysis methods, spectral asymmetry index (SASI) and Higuchi's fractal dimension (HFD), for detection of depression. Linear SASI method is based on evaluation of the balance of powers in two EEG frequency bands in one channel selected higher and lower than the alpha band spectrum maximum. Nonlinear HFD method calculates fractal dimension directly in the time domain. The resting EEG signals of 17 depressive patients and 17 control subjects were used as a database for calculations. SASI values were positive for depressive and negative for control group (P < 0.05). SASI provided the true detection rate of 88% in the depressive and 82% in the control group. The calculated HFD values detected a small (3%) increase with depression (P < 0.05). HFD provided the true detection rate of 94% in the depressive group and 76% in the control group. The rate of correct indication in the both groups was 85% using SASI or HFD. Statistically significant variations were not revealed between hemispheres (P > 0.05). The results indicated that the linear EEG analysis method SASI and the nonlinear HFD method both demonstrated a good sensitivity for detection of characteristic features of depression in a single-channel EEG. PMID:24232245
NASA Astrophysics Data System (ADS)
Maggi, Federico
2015-09-01
A comprehensive set of experiments was carried out to investigate the effect of the fractal architecture of granular aggregates on the free-fall acceleration through a still water column. Test aggregates were first generated numerically with a method that allowed to control the fractal dimension d and, next, three stochastic replicates were lithographically fabricated for each of six values of d ranging between 1.9 and 2.7. The recorded position, velocity and acceleration served to analyze their dynamics in the Reynolds and Galilei number space, and to calculate the momentum rate of change and the intensity of drag (viscous and impact) and inertial forces (added mass and Basset-Bousinnesq). Analysis of these forces highlighted a strong dependence on d; additionally, integration of these forces in the particle momentum equation allowed to identify an additional resistance Rx that showed a strong correlation with d. A correlation analysis of Rx with various scaling laws combining velocity and acceleration suggested that Rx could be described by a nonlinear drag force and a force intermediate between drag and inertia. It was therefore concluded that irregular granular fractal aggregates accelerating in water are subject to highly complex and nonlinear hydrodynamic effects caused by surface roughness and volume porosity, and that these effects have tight connection with the internal and external fractal characteristics of the aggregates.
Fractally Fourier decimated homogeneous turbulent shear flow in noninteger dimensions.
Fathali, Mani; Khoei, Saber
2017-02-01
Time evolution of the fully resolved incompressible homogeneous turbulent shear flow in noninteger Fourier dimensions is numerically investigated. The Fourier dimension of the flow field is extended from the integer value 3 to the noninteger values by projecting the Navier-Stokes equation on the fractal set of the active Fourier modes with dimensions 2.7≤d≤3.0. The results of this study revealed that the dynamics of both large and small scale structures are nontrivially influenced by changing the Fourier dimension d. While both turbulent production and dissipation are significantly hampered as d decreases, the evolution of their ratio is almost independent of the Fourier dimension. The mechanism of the energy distribution among different spatial directions is also impeded by decreasing d. Due to this deficient energy distribution, turbulent field shows a higher level of the large-scale anisotropy in lower Fourier dimensions. In addition, the persistence of the vortex stretching mechanism and the forward spectral energy transfer, which are three-dimensional turbulence characteristics, are examined at changing d, from the standard case d=3.0 to the strongly decimated flow field for d=2.7. As the Fourier dimension decreases, these forward energy transfer mechanisms are strongly suppressed, which in turn reduces both the small-scale intermittency and the deviation from Gaussianity. Besides the energy exchange intensity, the variations of d considerably modify the relative weights of local to nonlocal triadic interactions. It is found that the contribution of the nonlocal triads to the total turbulent kinetic energy exchange increases as the Fourier dimension increases.
Fractally Fourier decimated homogeneous turbulent shear flow in noninteger dimensions
NASA Astrophysics Data System (ADS)
Fathali, Mani; Khoei, Saber
2017-02-01
Time evolution of the fully resolved incompressible homogeneous turbulent shear flow in noninteger Fourier dimensions is numerically investigated. The Fourier dimension of the flow field is extended from the integer value 3 to the noninteger values by projecting the Navier-Stokes equation on the fractal set of the active Fourier modes with dimensions 2.7 ≤d ≤3.0 . The results of this study revealed that the dynamics of both large and small scale structures are nontrivially influenced by changing the Fourier dimension d . While both turbulent production and dissipation are significantly hampered as d decreases, the evolution of their ratio is almost independent of the Fourier dimension. The mechanism of the energy distribution among different spatial directions is also impeded by decreasing d . Due to this deficient energy distribution, turbulent field shows a higher level of the large-scale anisotropy in lower Fourier dimensions. In addition, the persistence of the vortex stretching mechanism and the forward spectral energy transfer, which are three-dimensional turbulence characteristics, are examined at changing d , from the standard case d =3.0 to the strongly decimated flow field for d =2.7 . As the Fourier dimension decreases, these forward energy transfer mechanisms are strongly suppressed, which in turn reduces both the small-scale intermittency and the deviation from Gaussianity. Besides the energy exchange intensity, the variations of d considerably modify the relative weights of local to nonlocal triadic interactions. It is found that the contribution of the nonlocal triads to the total turbulent kinetic energy exchange increases as the Fourier dimension increases.
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 use of generalized information dimension in measuring fractal dimension of time series
NASA Astrophysics Data System (ADS)
Ashkenazy, Y.
1999-09-01
An algorithm for calculating generalized fractal dimension of a time series using the general information function is presented. The algorithm is based on a strings sort technique and requires O(N log2 N) computations. A rough estimate for the number of points needed for the fractal dimension calculation is given. The algorithm was tested on analytic example as well as well-known examples, such as, the Lorenz attractor, the Rossler attractor, the van der Pol oscillator, and the Mackey-Glass equation, and compared, successfully, with previous results published in the literature. The computation time for the algorithm suggested in this paper is much less than the computation time according to other methods.
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.
Fractal Dimensions of In Vitro Tumor Cell Proliferation
Lambrou, George I.
2015-01-01
Biological systems are characterized by their potential for dynamic adaptation. One of the challenges for systems biology approaches is their contribution towards the understanding of the dynamics of a growing cell population. Conceptualizing these dynamics in tumor models could help us understand the steps leading to the initiation of the disease and its progression. In vitro models are useful in answering this question by providing information over the spatiotemporal nature of such dynamics. In the present work, we used physical quantities such as growth rate, velocity, and acceleration for the cellular proliferation and identified the fractal structures in tumor cell proliferation dynamics. We provide evidence that the rate of cellular proliferation is of nonlinear nature and exhibits oscillatory behavior. We also calculated the fractal dimensions of our cellular system. Our results show that the temporal transitions from one state to the other also follow nonlinear dynamics. Furthermore, we calculated self-similarity in cellular proliferation, providing the basis for further investigation in this topic. Such systems biology approaches are very useful in understanding the nature of cellular proliferation and growth. From a clinical point of view, our results may be applicable not only to primary tumors but also to tumor metastases. PMID:25883653
Cosmology in One Dimension: Fractal Geometry, Power Spectra and Correlation
NASA Astrophysics Data System (ADS)
Miller, Bruce; Rouet, Jean-Louis
2011-03-01
Concentrations of matter, such as galaxies and galactic clusters, originated as very small density fluctuations in the early universe. The existence of galaxy clusters and super-clusters suggests that a natural scale for the matter distribution may not exist. A point of controversy is whether the distribution is fractal and, if so,over what range of scales. One-dimensional models demonstrate that the important dynamics for cluster formation occur in the position-velocity plane. Here the development of scaling behavior and multifractal geometry is investigated for a family of one-dimensional models for three different, scale-free, initial conditions. A possible physical mechanism for understanding the self-similar evolution is introduced. It is shown that hierarchical cluster formation depends both on the model and the initial power spectrum. Under special circumstances a simple relation between the power spectrum, correlation function, and correlation dimension in the highly nonlinear regime is confirmed.
NASA Astrophysics Data System (ADS)
Lasue, Jeremie; Levasseur-Regourd, Anny-Chantal; Hadamcik, Edith; Botet, Robert; Renard, Jean-Baptiste
In situ missions have shown that cometary dust particles have low densities and are easily fragmenting aggregates [1]. The linear polarization of the solar light scattered by cometary dust corresponds to bell-shaped (with a small negative branch and a maximum below 30%) phase curves with a quasi-linear increase with the wavelength between 30° and 50° phase angle [2]. Such physical properties of the cometary material are reconciled by a fractal model of cometary dust and comet nuclei as formed by aggregation in reduced gravity as studied by laboratory experiments and numerical simulations. Reduced gravity light scattering experiments: The CODAG-LSU experiment (1999) gave the first indication of the light scattering properties transition between single particles and low dimensions fractal aggregates (D 1.3) [3, 4]. Such studies will be pursued on board the ISS with the ICAPS precursor experiment. The PROGRA2 experiment is designed to study the light scattering properties of particles levitated during dedicated microgravity flights or with ground-based configurations [5]. The material properties are chosen so as to be relevant in the context of cosmic dust from cometary and asteroidal origins. It is especially useful to disentangle the effects of varying albedos of constitutive materials [6], shape and size of constitutive grains [7]. Some of the results are interpreted in terms of fractal aggregates growth. Light scattering numerical simulations Based on numerical simulations and in coherence with the experimental results, a model of cometary coma by a mixture of fractal aggregates of up to 256 sub-micron sized spheroidal grains and compact spheroidal particles is shown to reproduce the polarimetric observations of comets such as 1P/Halley or C/1995 O1 Hale-Bopp [8]. Physical parameters of the size distribution of particles (minimum and maximum size, shape of the size distribution and quantity and location of absorbing and non-absorbing particles) can be retrieved
The Three-Point Sinuosity Method for Calculating the Fractal Dimension of Machined Surface Profile
NASA Astrophysics Data System (ADS)
Zhou, Yuankai; Li, Yan; Zhu, Hua; Zuo, Xue; Yang, Jianhua
2015-04-01
The three-point sinuosity (TPS) method is proposed to calculate the fractal dimension of surface profile accurately. In this method, a new measure, TPS is defined to present the structural complexity of fractal curves, and has been proved to follow the power law. Thus, the fractal dimension can be calculated through the slope of the fitted line in the log-log plot. The Weierstrass-Mandelbrot (W-M) fractal curves, as well as the real surface profiles obtained by grinding, sand blasting and turning, are used to validate the effectiveness of the proposed method. The calculation values are compared to those obtained from root-mean-square (RMS) method, box-counting (BC) method and variation method. The results show that the TPS method has the widest scaling region, the least fit error and the highest accuracy among the methods examined, which demonstrates that the fractal characteristics of the fractal curves can be well revealed by the proposed method.
Fractal Dimension Analysis of Transient Visual Evoked Potentials: Optimisation and Applications
Boon, Mei Ying; Henry, Bruce Ian; Chu, Byoung Sun; Basahi, Nour; Suttle, Catherine May; Luu, Chi; Leung, Harry; Hing, Stephen
2016-01-01
Purpose The visual evoked potential (VEP) provides a time series signal response to an external visual stimulus at the location of the visual cortex. The major VEP signal components, peak latency and amplitude, may be affected by disease processes. Additionally, the VEP contains fine detailed and non-periodic structure, of presently unclear relevance to normal function, which may be quantified using the fractal dimension. The purpose of this study is to provide a systematic investigation of the key parameters in the measurement of the fractal dimension of VEPs, to develop an optimal analysis protocol for application. Methods VEP time series were mathematically transformed using delay time, τ, and embedding dimension, m, parameters. The fractal dimension of the transformed data was obtained from a scaling analysis based on straight line fits to the numbers of pairs of points with separation less than r versus log(r) in the transformed space. Optimal τ, m, and scaling analysis were obtained by comparing the consistency of results using different sampling frequencies. The optimised method was then piloted on samples of normal and abnormal VEPs. Results Consistent fractal dimension estimates were obtained using τ = 4 ms, designating the fractal dimension = D2 of the time series based on embedding dimension m = 7 (for 3606 Hz and 5000 Hz), m = 6 (for 1803 Hz) and m = 5 (for 1000Hz), and estimating D2 for each embedding dimension as the steepest slope of the linear scaling region in the plot of log(C(r)) vs log(r) provided the scaling region occurred within the middle third of the plot. Piloting revealed that fractal dimensions were higher from the sampled abnormal than normal achromatic VEPs in adults (p = 0.02). Variances of fractal dimension were higher from the abnormal than normal chromatic VEPs in children (p = 0.01). Conclusions A useful analysis protocol to assess the fractal dimension of transformed VEPs has been developed. PMID:27598422
Imre, Attila R; Rocchini, Duccio
2009-09-01
Different summarized shape indices, like mean shape index (MSI) and area weighted mean shape index (AWMSI) can change over multiple size scales. This variation is important to describe scale heterogeneity of landscapes, but the exact mathematical form of the dependence is rarely known. In this paper, the use of fractal geometry (by the perimeter and area Hausdorff dimensions) made us able to describe the scale dependence of these indices. Moreover, we showed how fractal dimensions can be deducted from existing MSI and AWMSI data. In this way, the equality of a multiscale tabulated MSI and AWMSI dataset and two scale-invariant fractal dimensions has been demonstrated.
NASA Astrophysics Data System (ADS)
Brinkhoff, L. A.; von Savigny, C.; Randall, C. E.; Burrows, J. P.
2015-05-01
The fractal perimeter dimension is a fundamental property of clouds. It describes the cloud shape and is used to improve the understanding of atmospheric processes responsible for cloud shapes. von Savigny et al. (2011) determined the fractal perimeter dimension of noctilucent clouds (or polar mesospheric clouds) for the first time based on a limited data set of cloud images observed with the CIPS (Cloud Imaging and Particle Size) instrument on board the AIM (Aeronomy of Ice in the Mesosphere) satellite. This paper builds on von Savigny et al. (2011) by first presenting a sensitivity analysis of the determination of the fractal perimeter dimension, and secondly presenting results on the seasonal and interhemispheric differences of the perimeter dimension of noctilucent clouds (NLCs). The same method as in the earlier study is applied to an extended data set of satellite images of noctilucent cloud fields taken with the CIPS experiment. The sensitivity studies reveal that cloud holes play an important role for the area-perimeter method, since excluding clouds with holes reduces the dimension value by up to 3%. The results on the fractal perimeter dimension over six NLC seasons from 2007 to 2009 demonstrate that the dimension values of the NLCs neither show significant differences between the seasons nor between the hemispheres.
A Fractal Model for the Capacitance of Lunar Dust and Lunar Dust Aggregates
NASA Technical Reports Server (NTRS)
Collier, Michael R.; Stubbs, Timothy J.; Keller, John W.; Farrell, William M.; Marshall, John; Richard, Denis Thomas
2011-01-01
Lunar dust grains and dust aggregates exhibit clumping, with an uneven mass distribution, as well as features that span many spatial scales. It has been observed that these aggregates display an almost fractal repetition of geometry with scale. Furthermore, lunar dust grains typically have sharp protrusions and jagged features that result from the lack of aeolian weathering (as opposed to space weathering) on the Moon. A perfectly spherical geometry, frequently used as a model for lunar dust grains, has none of these characteristics (although a sphere may be a reasonable proxy for the very smallest grains and some glasses). We present a fractal model for a lunar dust grain or aggregate of grains that reproduces (1) the irregular clumpy nature of lunar dust, (2) the presence of sharp points, and (3) dust features that span multiple scale lengths. We calculate the capacitance of the fractal lunar dust analytically assuming fixed dust mass (i.e. volume) for an arbitrary number of fractal levels and compare the capacitance to that of a non-fractal object with the same volume, surface area, and characteristic width. The fractal capacitance is larger than that of the equivalent non-fractal object suggesting that for a given potential, electrostatic forces on lunar dust grains and aggregates are greater than one might infer from assuming dust grains are sphericaL Consequently, electrostatic transport of lunar dust grains, for example lofting, appears more plausible than might be inferred by calculations based on less realistic assumptions about dust shape and associated capacitance.
NASA Astrophysics Data System (ADS)
Park, Hwangseo; Kim, Hojing; Lee, Sangyoub
1997-05-01
We present a method for generating fractal surfaces of dimension between two and three. By using the method, five fractal surfaces with dimension 2.262, 2.402, 2.524, 2.631, and 2.771 are created. For each of these surfaces, the reaction of carbon monoxide and oxygen is simulated by using a Monte Carlo method based on the ZGB model [Phys. Rev. Lett. 24 (1986) 2553]. The results show that the catalytic CO oxidation proceeds more efficiently on a surface with higher fractal dimension. It is also found that as the fractal dimension of the surface becomes higher, the first-order kinetic phase transition point (y 2) is shifted to a higher partial pressure of CO. This implies that poisoning of the catalyst surface due to CO segregation sets in at a higher CO partial pressure for surfaces with more complexity.
Heterogeneities Analysis Using the Generalized Fractal Dimension and Continuous Wavelet Transform
NASA Astrophysics Data System (ADS)
Ouadfeul, S.; Aliouane, L.; Boudella, A.
2012-04-01
The main goal of this work is analyze heterogeneities from well-logs data using the wavelet transform modulus maxima lines (WTMM). Firstly, the continuous wavelet transform (CWT) with sliding window is calculated. The next step consists to calculate the maxima of the modulus of the CWT and estimate the spectrum of exponents. The three generalized fractal dimensions D0, D1 and D2 are then estimated. Application of the proposed idea at the synthetic and real well-logs data of a borehole located in the Algerian Sahara shows that the fractal dimensions are very sensitive to lithological variations. The generalized fractal dimensions are a very robust tool than can be used for petroleum reservoir characterization. Keywrods: reservoir, Heterogeneities, WTMM, fractal dimension.
NASA Astrophysics Data System (ADS)
Arefiev, K.; Nesterenko, V.; Daneykina, N.
2016-06-01
The results of communication research between the wear resistance of the K applicability hard-alloy cutting tools and the fractal dimension of the wear surface, which is formed on a back side of the cutting edge when processing the materials showing high adhesive activity are presented in the paper. It has been established that the wear resistance of tested cutting tools samples increases according to a fractal dimension increase of their wear surface.
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
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.
Studying neutral hydrogen structures during the epoch of reionization using fractal dimensions
NASA Astrophysics Data System (ADS)
Bandyopadhyay, Bidisha; Choudhury, T. Roy; Seshadri, T. R.
2017-04-01
Fractal dimensions can be used to characterize the clustering and lacunarities in density distributions. We use generalized fractal dimensions to study the neutral hydrogen distribution (H I) during the epoch of reionization. Using a semi-numeric model of ionized bubbles to generate the H I field, we calculate the fractal dimensions for length-scales ∼10 h-1cMpc. We find that the H I field displays significant multifractal behaviour and is not consistent with homogeneity at these scales when the mass-averaged neutral fraction bar{x}_{H I}^M ≳ 0.5. This multifractal nature is driven entirely by the shapes and distribution of the ionized regions. The sensitivity of the fractal dimension to the neutral fraction implies that it can be used for constraining reionization history. We find that the fractal dimension is relatively less sensitive to the value of the minimum mass of ionizing haloes when it is in the range ∼109-1010h-1M⊙. Interestingly, the fractal dimension is very different when the reionization proceeds inside-out compared to when it is outside-in. Thus, the multifractal nature of H I density field at high redshifts can be used to study the nature of reionization.
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.
NASA Astrophysics Data System (ADS)
Chakrabarty, R. K.; Moosmuller, H.; Garro, M. A.; Garro, B. A.; Chancellor, S.; Herald, C. M.
2010-12-01
The three dimensional (3-d) fractal dimension D is considered the key morphological parameter for black carbon (BC) aggregates as it influences BC optical, physical, and chemical properties. Using electron microscopy and image processing techniques, there are several approaches for extracting D and other structural parameters of BC aggregates, which are commonly unknown, from their two dimensional microscopy images. The nested squares method (NSM), the perimeter grid method (PGM), and the ensemble method (EM) have found wide use as image analysis techniques for determination of D for individual and ensemble BC aggregates. However, so far no study has quantified the errors involved in the measured D by these three analysis techniques. In this talk, we highlight the errors associated with using these three fractal analysis techniques for the determination of D and put forth a recommendation on the most reliable analysis technique for use in future research. The talk will conclude by discussing the results obtained by (i) applying the recommended image analysis technique to real-world, flame-generated BC aggregates for calculating their D and other morphological properties; (ii) further making use of the calculated aggregate morphological parameters to theoretically determine BC optical properties using three well-known theories, namely Rayleigh-Debye-Gans (RDG) approximation, volume-equivalent Mie theory, and integral equation formulation for scattering (IEFS); and (iii) comparison of the theoretically determined BC optical properties with experimental data.
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.
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.
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.
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)
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.
Navigation performance in virtual environments varies with fractal dimension of landscape.
Juliani, Arthur W; Bies, Alexander J; Boydston, Cooper R; Taylor, Richard P; Sereno, Margaret E
2016-09-01
Fractal geometry has been used to describe natural and built environments, but has yet to be studied in navigational research. In order to establish a relationship between the fractal dimension (D) of a natural environment and humans' ability to navigate such spaces, we conducted two experiments using virtual environments that simulate the fractal properties of nature. In Experiment 1, participants completed a goal-driven search task either with or without a map in landscapes that varied in D. In Experiment 2, participants completed a map-reading and location-judgment task in separate sets of fractal landscapes. In both experiments, task performance was highest at the low-to-mid range of D, which was previously reported as most preferred and discriminable in studies of fractal aesthetics and discrimination, respectively, supporting a theory of visual fluency. The applicability of these findings to architecture, urban planning, and the general design of constructed spaces is discussed.
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.
Evidence for fractal dimension in asphaltene polymers from electron-spin-relaxation measurements
NASA Astrophysics Data System (ADS)
Raghunathan, P.
1991-08-01
Measurements of low-temperature electron spin—lattice relaxation of VO 2+ centres are reported in five gel-permeation-chromatographic fractions of Athabasca asphaltene polymer. Best fits of the experimental relaxation rate data to a Tn power law lead to Hausdorff fractal dimensions ( df) in the range 1.65-2.0. These fractal values are interpreted in terms of plausible models of polymer structure.
Crack detection in beams in noisy conditions using scale fractal dimension analysis of mode shapes
NASA Astrophysics Data System (ADS)
Bai, R. B.; Ostachowicz, W.; Cao, M. S.; Su, Z.
2014-06-01
Fractal dimension analysis of mode shapes has been actively studied in the area of structural damage detection. The most prominent features of fractal dimension analysis are high sensitivity to damage and instant determination of damage location. However, an intrinsic deficiency is its susceptibility to measurement noise, likely obscuring the features of damage. To address this deficiency, this study develops a novel damage detection method, scale fractal dimension (SFD) analysis of mode shapes, based on combining the complementary merits of a stationary wavelet transform (SWT) and Katz’s fractal dimension in damage characterization. With this method, the SWT is used to decompose a mode shape into a set of scale mode shapes at scale levels, with damage information and noise separated into distinct scale mode shapes because of their dissimilar scale characteristics; the Katz’s fractal dimension individually runs on every scale mode shape in the noise-adaptive condition provided by the SWT to canvass damage. Proof of concept for the SFD analysis is performed on cracked beams simulated by the spectral finite element method; the reliability of the method is assessed using Monte Carlo simulation to mimic the operational variability in realistic damage diagnosis. The proposed method is further experimentally validated on a cracked aluminum beam with mode shapes acquired by a scanning laser vibrometer. The results show that the SFD analysis of mode shapes provides a new strategy for damage identification in noisy conditions.
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.
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)
Huber, Franz J. T.; Will, Stefan; Daun, Kyle J.
2016-11-01
Inferring the size distribution of aerosolized fractal aggregates from the angular distribution of elastically scattered light is a mathematically ill-posed problem. This paper presents a procedure for analyzing Wide-Angle Light Scattering (WALS) data using Bayesian inference. The outcome is probability densities for the recovered size distribution and aggregate morphology parameters. This technique is applied to both synthetic data and experimental data collected on soot-laden aerosols, using a measurement equation derived from Rayleigh-Debye-Gans fractal aggregate (RDG-FA) theory. In the case of experimental data, the recovered aggregate size distribution parameters are generally consistent with TEM-derived values, but the accuracy is impaired by the well-known limited accuracy of RDG-FA theory. Finally, we show how this bias could potentially be avoided using the approximation error technique.
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
Fractal dimension in butterflies' wings: a novel approach to understanding wing patterns?
Castrejón-Pita, A A; Sarmiento-Galán, A; Castrejón-Pita, J R; Castrejón-García, R
2005-05-01
The geometrical complexity in the wings of several, taxonomically different butterflies, is analyzed in terms of their fractal dimension. Preliminary results provide some evidence on important questions about the (dis)similarity of the wing patterns in terms of their fractal dimension. The analysis is restricted to two groups which are widely used in the literature as typical examples of mimicry, and a small number of unrelated species, thus implying the consideration of only a fraction of the wing pattern diversity. The members of the first mimicry ring, composed by the species Danaus plexippus (better known as the monarch butterfly), and the two subspecies Basilarchia archippus obsoleta (or northern viceroy) and Basilarchia archippus hoffmanni (or tropical viceroy), are found to have a very similar value for the fractal dimension of their wing patterns, even though they do not look very similar at first sight. It is also found that the female of another species (Neophasia terlootii), which looks similar to the members of the previous group, does not share the same feature, while the Lycorea ilione albescens does share it. For the members of the second group of mimicry related butterflies, the Greta nero nero and the Hypoleria cassotis, it is shown that they also have very close values for the fractal dimension of their wing patterns. Finally, it is shown that other species, which apparently have very similar wing patterns, do not have the same fractal dimension. A possible, not completely tested hypothesis is then conjectured: the formation of groups by individuals whose wing patterns have an almost equal fractal dimension may be due to the fact that they do share the same developmental raw material, and that this common feature is posteriorly modified by natural selection, possibly through predation.
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.
Electrochemical Growth of Ag Junctions and Diffusion Limited Aggregate (DLA) Fractal Simulation
NASA Astrophysics Data System (ADS)
Olson, Zak; Tuppan, Sam; Kim, Woo-Joong; Seattle University Team
2015-03-01
We attempt construction of a single atom connection between two copper wires. By applying a DC voltage across the wires when immersed in a silver nitrate solution, we deposit silver until a junction is formed. The deposited silver forms a fractal structure that can be simulated with a diffusion limited aggregation model.
Lévy dusts, Mittag-Leffler statistics, mass fractal lacunarity, and perceived dimension
NASA Astrophysics Data System (ADS)
Blumenfeld, Raphael; Mandelbrot, Benoit B.
1997-07-01
We study the Lévy dusts on the line on two accounts: the fluctuations around the average power law that characterizes the mass-radius relation for self-similar fractals, and the statistics of the intervals between strides along the logarithmic axis (their tail distribution is related to the dust's fractal dimension). The Lévy dusts are suggested as a yardstick of neutral lacunarity, against which non-neutral lacunarity can be measured objectively. A notion of perceived dimension is introduced. We conclude with an application of the Mittag-Leffler statistics to a nonlinear electrical network.
Salinas-Nolasco, Manlio Favio; Méndez-Vivar, Juan
2010-03-16
Among several analysis techniques applied to the study of surface passivation using dicarboxylic acids, small angle X-ray scattering (SAXS) has proved to be relevant in the physicochemical interpretation of the surface association resulting between calcium carbonate and the molecular structure of malonic acid. It is possible to establish chemical affinity principles through bidimensional geometric analysis in terms of the fractal dimension obtained experimentally by SAXS. In this Article, we present results about the adsorption of malonic acid on calcite, using theoretical and mathematical principles of the fractal dimension.
Fractal dimensions of pacing and grip force in drawing and handwriting production.
Fernandes, David N; Chau, Tom
2008-01-01
We performed a repeated measures experiment to show that the pacing of a cyclic, ballistic drawing task has a fractal dimension. We also estimated the dimensionality of the force used to grip the drawing implement. Finally, we present an analysis of pediatric data to show that grip force has a fractal dimension in an actual handwriting task. In our experiment, subjects drew circles of varying sizes and at varying rates on a digitizing tablet, using a pen instrumented to measure radial force applied to its barrel. Subjects also drew circles in synchrony with a metronome. We found strong evidence for fractal scaling of both drawing period and grip force in the circle-drawing study. The dimensionality ranged from fractal Gaussian noise (fGn) to fractal Brownian motion, with Hurst coefficients clustering around the value for 1/f noise. When the subjects were required to synchronize their drawing with a metronome, the Hurst coefficient for the drawing period decreased, while the coefficient for grip force did not. This result indicates that independent processes control the variations in pacing and grip force. Grip force in the handwriting study also displayed fractal properties, with Hurst coefficients in the range of correlated fGn. We draw parallels between our handwriting measurements and studies of human gait.
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.
Complex Patterns in Financial Time Series Through HIGUCHI’S Fractal Dimension
NASA Astrophysics Data System (ADS)
Grace Elizabeth Rani, T. G.; Jayalalitha, G.
2016-11-01
This paper analyzes the complexity of stock exchanges through fractal theory. Closing price indices of four stock exchanges with different industry sectors are selected. Degree of complexity is assessed through Higuchi’s fractal dimension. Various window sizes are considered in evaluating the fractal dimension. It is inferred that the data considered as a whole represents random walk for all the four indices. Analysis of financial data through windowing procedure exhibits multi-fractality. Attempts to apply moving averages to reduce noise in the data revealed lower estimates of fractal dimension, which was verified using fractional Brownian motion. A change in the normalization factor in Higuchi’s algorithm did improve the results. It is quintessential to focus on rural development to realize a standard and steady growth of economy. Tools must be devised to settle the issues in this regard. Micro level institutions are necessary for the economic growth of a country like India, which would induce a sporadic development in the present global economical scenario.
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 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.
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.
Kalauzi, Aleksandar; Bojić, Tijana; Vuckovic, Aleksandra
2012-07-01
The exact mathematical relationship between FFT spectrum and fractal dimension (FD) of an experimentally recorded signal is not known. In this work, we tried to calculate signal FD directly from its Fourier amplitudes. First, dependence of Higuchi's FD of mathematical sinusoids on their individual frequencies was modeled with a two-parameter exponential function. Next, FD of a finite sum of sinusoids was found to be a weighted average of their FDs, weighting factors being their Fourier amplitudes raised to a fractal degree. Exponent dependence on frequency was modeled with exponential, power and logarithmic functions. A set of 280 EEG signals and Weierstrass functions were analyzed. Cross-validation was done within EEG signals and between them and Weierstrass functions. Exponential dependence of fractal exponents on frequency was found to be the most accurate. In this work, signal FD was for the first time expressed as a fractal weighted average of FD values of its Fourier components, also allowing researchers to perform direct estimation of signal fractal dimension from its FFT spectrum.
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.
Pacagnelli, Francis Lopes; Sabela, Ana Karênina Dias de Almeida; Mariano, Thaoan Bruno; Ozaki, Guilherme Akio Tamura; Castoldi, Robson Chacon; do Carmo, Edna Maria; Carvalho, Robson Francisco; Tomasi, Loreta Casquel; Okoshi, Katashi; Vanderlei, Luiz Carlos Marques
2016-01-01
Background Right-sided heart failure has high morbidity and mortality, and may be caused by pulmonary arterial hypertension. Fractal dimension is a differentiated and innovative method used in histological evaluations that allows the characterization of irregular and complex structures and the quantification of structural tissue changes. Objective To assess the use of fractal dimension in cardiomyocytes of rats with monocrotaline-induced pulmonary arterial hypertension, in addition to providing histological and functional analysis. Methods Male Wistar rats were divided into 2 groups: control (C; n = 8) and monocrotaline-induced pulmonary arterial hypertension (M; n = 8). Five weeks after pulmonary arterial hypertension induction with monocrotaline, echocardiography was performed and the animals were euthanized. The heart was dissected, the ventricles weighed to assess anatomical parameters, and histological slides were prepared and stained with hematoxylin/eosin for fractal dimension analysis, performed using box-counting method. Data normality was tested (Shapiro-Wilk test), and the groups were compared with non-paired Student t test or Mann Whitney test (p < 0.05). Results Higher fractal dimension values were observed in group M as compared to group C (1.39 ± 0.05 vs. 1.37 ± 0.04; p < 0.05). Echocardiography showed lower pulmonary artery flow velocity, pulmonary acceleration time and ejection time values in group M, suggesting function worsening in those animals. Conclusion The changes observed confirm pulmonary-arterial-hypertension-induced cardiac dysfunction, and point to fractal dimension as an effective method to evaluate cardiac morphological changes induced by ventricular dysfunction. PMID:27223643
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.
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
Local fuzzy fractal dimension and its application in medical image processing.
Zhuang, Xiaodong; Meng, Qingchun
2004-09-01
The local fuzzy fractal dimension (LFFD) is proposed to extract local fractal feature of medical images. The definition of LFFD is an extension of the pixel-covering method by incorporating the fuzzy set. Multi-feature edge detection is implemented with the LFFD and the Sobel operator. The LFFD can also serve as a characteristic of motion in medical image sequences. The experimental results show that the LFFD is an important feature of edge areas in medical images and can provide information for segmentation of echocardiogram image sequences.
Neuronal differentiation and synapse formation occur in space and time with fractal dimension.
Waliszewski, Przemyslaw; Konarski, Jerzy
2002-03-15
The analysis of a set of experimental data obtained by an independent team of researchers confirms that neuronal differentiation or synapse formation do occur in time and space with fractal dimension. The interacting cells create first a dynamic system with its own attractor, (i.e., a fragment of time and space where the dynamic processes occur and where no further evolution of the system is possible at all owing to the action of the intrasystemic forces unless some extrasystemic forces act upon it). This attractor is then modified in the active manner by the differentiating cells until the system attains a degenerated stationary state and differentiation ends. The fractal structure of the system is also lost in the course of tumor progression. Our data indicate that the cellular system can attain the degenerated stationary state, leaving the attractor with a fractal dimension directly or undergoing diversification into many attractors and going through the areas of deterministic chaos. Since evolution of the cellular system is driven by the cooperative dynamic processes, as reflected by the changes of the mean fractal dimension between the intervals of the Gompertzian curve, it is likely that cells differentiate into neurons and create synapses with a conjugated probability and non-Gaussian distribution rather than with the classical probability and the Gaussian distribution. These findings can help to optimize features of artificial neural networks. They also define a simple in vitro biological model for biophysical and biochemical studies on natural neural networks.
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.
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
Structural five-fold symmetry in the fractal morphology of diffusion-limited aggregates
NASA Astrophysics Data System (ADS)
Arneodo, A.; Argoul, F.; Muzy, J. F.; Tabard, M.
1992-09-01
The statistical self-similarity of the geometry of diffusion-limited aggregates and the multifractal nature of the growth probability distribution on the surface of the growing clusters are investigated using the wavelet transform. This study reveals the existence of a predominant structural five-fold symmetry in the internal frozen region as well as in the active outer region of the interface. This observation is corroborated by a statistical analysis of the screening effects that govern diffusion-limited aggregation (DLA) growth in linear and sector-shaped cells. The existence of this symmetry is likely to be a clue to a hierarchichal fractal ordering. We report on the discovery of Fibonacci sequences in the inner extinct region of large mass off-lattice DLA clusters, with a branching ratio which converges asymptotically to the golden mean. We suggest an interpretation of the DLA morphology as a “quasifractal” counterpart of the well-ordered snowflake fractal architecture.
Transition of fractal dimension in a latticed dynamical system
Duong-van, M.
1986-03-01
We study a recursion relation that manifests two distinct routes to turbulence, both of which reproduce commonly observed phenomena: the Feigenbaum route, with period-doubling frequencies; and a much more general route with noncommensurate frequencies and frequency entrainment, and locking. Intermittency and large-scale aperiodic spatial patterns are reproduced in this new route. In the oscillatory instability regime the fracal dimension saturates at D/sub F/ approx. = 2.6 with imbedding dimensions while in the turbulent regime D/sub F/ saturates at 6.0. 19 refs., 3 figs.
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.
Effect of Fractal Dimension on the Strain Behavior of Particulate Media
NASA Astrophysics Data System (ADS)
Altun, Selim; Sezer, Alper; Goktepe, A. Burak
2016-12-01
In this study, the influence of several fractal identifiers of granular materials on dynamic behavior of a flexible pavement structure as a particulate stratum is considered. Using experimental results and numerical methods as well, 15 different grain-shaped sands obtained from 5 different sources were analyzed as pavement base course materials. Image analyses were carried out by use of a stereomicroscope on 15 different samples to obtain quantitative particle shape information. Furthermore, triaxial compression tests were conducted to determine stress-strain and shear strength parameters of sands. Additionally, the dynamic response of the particulate media to standard traffic loads was computed using finite element modeling (FEM) technique. Using area-perimeter, line divider and box counting methods, over a hundred grains for each sand type were subjected to fractal analysis. Relationships among fractal dimension descriptors and dynamic strain levels were established for assessment of importance of shape descriptors of sands at various scales on the dynamic behavior. In this context, the advantage of fractal geometry concept to describe irregular and fractured shapes was used to characterize the sands used as base course materials. Results indicated that fractal identifiers can be preferred to analyze the effect of shape properties of sands on dynamic behavior of pavement base layers.
Di Ieva, Antonio; Grizzi, Fabio; Tschabitscher, Manfred; Colombo, Piergiuseppe; Casali, Massimiliano; Simonelli, Matteo; Widhalm, Georg; Muzzio, Pier Carlo; Matula, Christian; Chiti, Arturo; Rodriguez y Baena, Riccardo
2010-09-01
Neuroradiological and metabolic imaging is a fundamental diagnostic procedure in the assessment of patients with primary and metastatic brain tumors. The correlation between objective parameters capable of quantifying the neoplastic angioarchitecture and imaging data may improve our understanding of the underlying physiopathology and make it possible to evaluate treatment efficacy in brain tumors. Only a few studies have so far correlated the quantitative parameters measuring the neovascularity of brain tumors with the metabolic profiles measured by means of amino acid uptake in positron emission tomography (PET) scans. Fractal geometry offers new mathematical tools for the description and quantification of complex anatomical systems, including microvascularity. In this study, we evaluated the microvascular network complexity of six cases of human glioblastoma multiforme quantifying the surface fractal dimension on CD34 immunostained specimens. The microvascular fractal dimension was estimated by applying the box-counting algorithm. As the fractal dimension depends on the density, size and shape of the vessels, and their distribution pattern, we defined it as an index of the whole complexity of microvascular architecture and compared it with the uptake of (11)C-methionine (MET) assessed by PET. The different fractal dimension values observed showed that the same histological category of brain tumor had different microvascular network architectures. Fractal dimension ranged between 1.19 and 1.77 (mean: 1.415+/-0.225), and the uptake of (11)C-methionine ranged between 1.30 and 5.30. A statistically significant direct correlation between the microvascular fractal dimension and the uptake of (11)C-methionine (p=0.02) was found. Our preliminary findings indicate that that vascularity (estimated on the histologic specimens by means of the fractal dimension) and (11)C-methionine uptake (assessed by PET) closely correlate in glioblastoma multiforme and that microvascular
Nuclear fractal dimension as a prognostic factor in oral squamous cell carcinoma.
Goutzanis, L; Papadogeorgakis, N; Pavlopoulos, P M; Katti, K; Petsinis, V; Plochoras, I; Pantelidaki, C; Kavantzas, N; Patsouris, E; Alexandridis, C
2008-04-01
Strong theoretical reasons exist for using fractal geometry in measurements of natural objects, including most objects studied in pathology. Indeed, fractal dimension provides a more precise and theoretically more appropriate approximation of their structure properties and especially their shape complexity. The aim of our study was to evaluate the nuclear fractal dimension (FD) in tissue specimens from patients with oral cavity carcinomas in order to assess its potential value as prognostic factor. Relationships between FD and other factors including clinicopathologic characteristics were also investigated. Histological sections from 48 oral squamous cell carcinomas as well as from 17 non-malignant mucosa specimens were stained with Hematoxylin-Eosin for pathological examination and with Feulgen for nuclear complexity evaluation. The sections were evaluated by image analysis using fractal analysis software to quantify nuclear FD by the box-counting method. Carcinomas presented higher mean values of FD compared to normal mucosa. Well differentiated neoplasms had lower FD values than poorly differentiated ones. FD was significantly correlated with the nuclear size. Patients with FD lower than the median value of the sample had statistically significant higher survival rates. Within the sample of patients studied, FD was proved to be an independent prognostic factor of survival in oral cancer patients. In addition this study provides evidence that there are several statistically significant correlations between FD and other morphometric characteristics or clinicopathologic factors in oral squamous cell carcinomas.
NASA Astrophysics Data System (ADS)
Monceau, Pascal
2006-09-01
The extension of the phase diagram of the q -state Potts model to noninteger dimension is investigated by means of Monte Carlo simulations on Sierpinski and Menger fractal structures. Both multicanonical and canonical simulations have been carried out with the help of the Wang-Landau and the Wolff cluster algorithms. Lower bounds are provided for the critical values qc of q where a first-order transition is expected in the cases of two structures whose fractal dimension is smaller than 2: The transitions associated with the seven-state and ten-state Potts models on Sierpinski carpets with fractal dimensions df≃1.8928 and df≃1.7925 , respectively, are shown to be second-order ones, the renormalization eigenvalue exponents yh are calculated, and bounds are provided for the renormalization eigenvalue exponents yt and the critical temperatures. Moreover, the results suggest that second-order transitions are expected to occur for very large values of q when the fractal dimension is lowered below 2—that is, in the case of hierarchically weakly connected systems with an infinite ramification order. At last, the transition associated with the four-state Potts model on a fractal structure with a dimension df≃2.631 is shown to be a weakly first-order one.
Radial distribution function for hard spheres in fractal dimensions: A heuristic approximation.
Santos, Andrés; de Haro, Mariano López
2016-06-01
Analytic approximations for the radial distribution function, the structure factor, and the equation of state of hard-core fluids in fractal dimension d (1≤d≤3) are developed as heuristic interpolations from the knowledge of the exact and Percus-Yevick results for the hard-rod and hard-sphere fluids, respectively. In order to assess their value, such approximate results are compared with those of recent Monte Carlo simulations and numerical solutions of the Percus-Yevick equation for a fractal dimension [M. Heinen et al., Phys. Rev. Lett. 115, 097801 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.097801], a good agreement being observed.
NASA Astrophysics Data System (ADS)
Zhang, Chen; Ni, Zhiwei; Ni, Liping; Tang, Na
2016-10-01
Feature selection is an important method of data preprocessing in data mining. In this paper, a novel feature selection method based on multi-fractal dimension and harmony search algorithm is proposed. Multi-fractal dimension is adopted as the evaluation criterion of feature subset, which can determine the number of selected features. An improved harmony search algorithm is used as the search strategy to improve the efficiency of feature selection. The performance of the proposed method is compared with that of other feature selection algorithms on UCI data-sets. Besides, the proposed method is also used to predict the daily average concentration of PM2.5 in China. Experimental results show that the proposed method can obtain competitive results in terms of both prediction accuracy and the number of selected features.
NASA Astrophysics Data System (ADS)
Tijera, Manuel; Maqueda, Gregorio; Yagüe, Carlos
2016-11-01
In this work the relation between integral scale and fractal dimension and the type of stratification in fully developed turbulence is analyzed. The integral scale corresponds to that in which energy from larger scales is incoming into a turbulent regime. One of the aims of this study is the understanding of the relation between the integral scale and the bulk Richardson number, which is one of the most widely used indicators of stability close to the ground in atmospheric studies. This parameter will allow us to verify the influence of the degree of stratification over the integral scale of the turbulent flows in the atmospheric boundary layer (ABL). The influence of the diurnal and night cycles on the relationship between the fractal dimension and integral scale is also analyzed. The fractal dimension of wind components is a turbulent flow characteristic, as has been shown in previous works, where its relation to stability was highlighted. Fractal dimension and integral scale of the horizontal (u') and vertical (w') velocity fluctuations have been calculated using the mean wind direction as a framework. The scales are obtained using sonic anemometer data from three elevations 5.8, 13 and 32 m above the ground measured during the SABLES 98 field campaign (Cuxart et al., 2000). In order to estimate the integral scales, a method that combines the normalized autocorrelation function and the best Gaussian fit (R2 ≥ 0.70) has been developed. Finally, by comparing, at the same height, the scales of u' and w' velocity components, it is found that the turbulent flows are almost always anisotropic.
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.
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.
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
NASA Astrophysics Data System (ADS)
Dellino, P.; Liotino, G.
2002-03-01
Image processing analysis is used to check the ability of the fractal dimension for quantitatively describing the shape of volcanic ash particles. Digitized scanning electron microscopy images of fine pyroclasts from the eruptions of Monte Pilato-Rocche Rosse (Lipari, Italy) are investigated to test the efficiency of the fractal dimension to discriminate between particles of different eruptive processes. Multivariate analysis of multiple fractal components allows distinction between magmatic particles and phreatomagmatic particles, which however is less significant than the discrimination obtained in previous studies by the use of simple 'adimensional' shape parameters. Approximation of the actual particle boundary and the not rotation invariant nature of the fractal data frequently result in a significant scatter of data points in the Mandelbrot-Richardson plot. Such behavior obscures in some cases the actual information of particle shape and renders the discriminating power of fractal analysis less effective than classical shape descriptors. Data less affected by scatter reveal that phreatomagmatic particles of the Monte Pilato-Rocche Rosse eruptions are true (mono) fractals, whereas magmatic particles are multifractals. The textural (small-scale) fractal of magmatic particles is similar to the fractal dimension value of phreatomagmatic particles, and is attributed to the rheological behavior of melt upon brittle fragmentation. The structural (large-scale) fractal of magmatic particles refers to the walls of ruptured vesicles that lay on the particle outline. The high difference between the values of the textural and structural fractals of magmatic particles of the Monte Pilato-Rocche Rosse eruptions suggests two distinct and independent processes in the formation of such pyroclasts. At the scales corresponding to the textural fractal, the fragmentation process is independent of vesicles. Magmatic fragmentation is not simply related to growth, expansion
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)
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.
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
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
Rate of diffusion-limited reactions for a fractal aggregate of reactive spheres
NASA Astrophysics Data System (ADS)
Tseng, Chin-Yao; Tsao, Heng-Kwong
2002-08-01
We study the reaction rate for a fractal cluster of perfectly absorbing, stationary spherical sinks in a medium containing a mobile reactant. The effectiveness factor eta, which is defined as the ratio of the total reaction rate of the cluster to that without diffusional interactions, is calculated. The scaling behavior of eta is derived for arbitrary fractal dimension based on the Kirkwood-Riseman approximation. The asymptotic as well as the finite size scaling of eta are confirmed numerically by the method of multipole expansion, which has been proven to be an excellent approximation. The fractal assembly is made of N spheres with its dimension varying from D<1 to D=3. The number of sinks can be as high as NapproxO(104). The asymptotic scaling behavior of the effectiveness factor is eta][approxN1/D-1 for D>1, eta][approx(ln N)-1 for D=1, and eta][approxN0 for D<1. The crossover behavior indicates that while in the regime of D>1 the screening effect of diffusive interactions grows with the size, for D<1 it is limited in a finite range and decays with decreasing D. The conclusion is also applicable to transport phenomena like dissolution, heat conduction, and sedimentation.
Lacunarity and Fractal Dimension: An Approach to the Evolutionary Characterization of AGNs
NASA Astrophysics Data System (ADS)
Peña, V. J.; Hernández, J. A.; Plata, A.
2006-06-01
The Active Galaxy Nuclei AGN are systems whose behavior shows high complexity and multidependency of evolutionary parameters. With this consideration, it is proposed the fractal characterization of an AGN using the analysis of Lacunarity and fractal dimension as a methodology to quantify dynamic space patterns. In general, the fractal set of complex dynamic systems only show invariance at big scales. In order to study the processes that define the morphologic characteristics of a set, it is necessary to establish parameters that show the distribution of dimension and scales. This measurement can be obtained by considering the variations of both, multyfractal spectrum and Lacunarity of the same set. For an AGN, the filling factor is a distribution measurement of the densely grouped electrons in the Broad Line Region BLR. This region plays an important role in the object morphology. we in this paper carry out the analogy between the filling factor and Lacunarity, with this, we give a new approach for both the dynamic and morphologic study of the AGN.
The area-to-mass ratio and fractal dimension of marine flocs
NASA Astrophysics Data System (ADS)
Bowers, D. G.; McKee, D.; Jago, C. F.; Nimmo-Smith, W. A. M.
2017-04-01
Optical instruments have proved invaluable in the study of suspended matter in the sea but the measurements they provide are more closely related to the cross-sectional area of the particles in suspension than the mass measured by filtration or predicted by theory. In this paper, we examine the factors controlling the relationship between particle area and mass, using the fractal model of particle structure as a theoretical framework. Both theory and observation agree that the area-to-mass ratio of particles (symbol A*) decreases with increasing fractal dimension (symbol Nf) as particles hide behind each other in compact flocs. The equation A* = 0.253-0.081Nf, in which A* is in m2 g-1 explains 81% of the variance in the area:mass ratio at 151 stations in coastal waters. In contrast, the effect of floc size on A* is small. Three optical parameters - beam attenuation, diffuse attenuation and remote sensing reflectance, expressed per unit mass of suspended material, all decrease with increasing Nf. As our understanding of the flocculation process grows and it becomes possible to predict the fractal dimension of particles as a function of the environmental conditions in which the flocs form, these results will lead to improved calibration of optical instruments in terms of the mass concentration of suspended materials and to better models of sediment suspension and transport.
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
NASA Astrophysics Data System (ADS)
Jing, Juntao; Feng, Pingfa; Wei, Shiliang; Zhao, Hong; Liu, Yunfeng
2016-11-01
Rotary ultrasonic grinding machining (RUGM) has been employed in Si3N4 ceramics parts machining widely, and the surface morphology is related with surface friction and wear properties directly. It is necessary to investigation on the surface morphology characterization to improve surface quality. Surface morphology of Si3N4 ceramics for rotary ultrasonic grinding machining was investigated based on fractal theory in the paper. The fractal features of surface morphology have been proved with qualitative and quantitative investigation. Differential box-counting method and peleg-blanket method was applied to calculate fractal dimension, but low calculation accuracy was found. So a new fractal dimension calculation method called perimeter-volume method has been proposed. The results show that the calculation deviation rate of morphology fractal dimension is only 2.5%. Meanwhile, the influence of spindle speed, cutting depth, feed rate and cutting force on fractal dimension has also been investigated. The investigation results provide the support for surface morphology optimization.
NASA Astrophysics Data System (ADS)
Billiones, R. G.; Tackx, M. L.; Daro, M. H.
1999-03-01
Water samples from the Scheldt estuary were collected in three fractions: (a) unfiltered water, (b) water filtered through a 50 μ m net and (c) water filtered through a 300 μm net. Particles easily recognisable from the majority of the amorphous particles were isolated and their geometric dimensions measured. From the measurements, shape factors were calculated. Measurement of fractal dimensions was attempted. From the first fraction, the particles isolated and measured were circular and chained diatoms. In the second fraction, zooplankters were easily distinguishable and representatives of the three dominant groups (cladocerans, cyclopoids and calanoids) were measured. In the third fraction, detrital pieces from monocotyledon and dicotyledon plants were recognised, isolated and measured. Fractal dimensions were only measurable in particles from fraction 3. The geometric features, shape factors and fractal dimensions of the particles were tested and proven to be effective ' fingerprints ' to distinguish these particles from the majority of the unidentifiable amorphous particles in the samples.
Zheng, Xiuqing; Liao, Zhiwu; Hu, Shaoxiang; Li, Ming; Zhou, Jiliu
2013-01-01
NLMs is a state-of-art image denoising method; however, it sometimes oversmoothes anatomical features in low-dose CT (LDCT) imaging. In this paper, we propose a simple way to improve the spatial adaptivity (SA) of NLMs using pointwise fractal dimension (PWFD). Unlike existing fractal image dimensions that are computed on the whole images or blocks of images, the new PWFD, named pointwise box-counting dimension (PWBCD), is computed for each image pixel. PWBCD uses a fixed size local window centered at the considered image pixel to fit the different local structures of images. Then based on PWBCD, a new method that uses PWBCD to improve SA of NLMs directly is proposed. That is, PWBCD is combined with the weight of the difference between local comparison windows for NLMs. Smoothing results for test images and real sinograms show that PWBCD-NLMs with well-chosen parameters can preserve anatomical features better while suppressing the noises efficiently. In addition, PWBCD-NLMs also has better performance both in visual quality and peak signal to noise ratio (PSNR) than NLMs in LDCT imaging.
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.
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.)
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.
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.
The fourth dimension of life: fractal geometry and allometric scaling of organisms.
West, G B; Brown, J H; Enquist, B J
1999-06-04
Fractal-like networks effectively endow life with an additional fourth spatial dimension. This is the origin of quarter-power scaling that is so pervasive in biology. Organisms have evolved hierarchical branching networks that terminate in size-invariant units, such as capillaries, leaves, mitochondria, and oxidase molecules. Natural selection has tended to maximize both metabolic capacity, by maximizing the scaling of exchange surface areas, and internal efficiency, by minimizing the scaling of transport distances and times. These design principles are independent of detailed dynamics and explicit models and should apply to virtually all organisms.
Network-Growth Rule Dependence of Fractal Dimension of Percolation Cluster on Square Lattice
NASA Astrophysics Data System (ADS)
Tanaka, Shu; Tamura, Ryo
2013-05-01
To investigate the network-growth rule dependence of certain geometric aspects of percolation clusters, we propose a generalized network-growth rule introducing a generalized parameter q and we study the time evolution of the network. The rule we propose includes a rule in which elements are randomly connected step by step and the rule recently proposed by Achlioptas et al. [Science 323 (2009) 1453]. We consider the q-dependence of the dynamics of the number of elements in the largest cluster. As q increases, the percolation step is delayed. Moreover, we also study the q-dependence of the roughness and the fractal dimension of the percolation cluster.
Power spectrum and fractal dimension of laser backscattering from the ocean.
Churnside, James H; Wilson, James J
2006-11-01
We flew an airborne lidar perpendicular to the coastline along straight-line transects that varied in length between 230 and 280 km. The sample spacing was approximately 3 m, so we sampled almost five decades of spatial scales. Except for the return from right at the surface, the power spectra of backscattered power had a power-law dependence on spatial frequency, with a slope of approximately 1.49. This corresponds to a fractal dimension of 1.76. This implies that the distribution is not as patchy as that of a purely turbulent process.
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).
Cichański, Artur; Nowicki, Krzysztof; Mazurkiewicz, Adam; Topoliński, Tomasz
2010-01-01
The paper presents linear, logarithmic and exponential regression tabecular bone indices, fractal dimensions and strength. The analysis of the above parameters was supported by determining non-parametric correlation coefficients: Spearman's ρ, gamma and Kendall's τ. The principal components' analysis (PCA) was also performed in order to reduce the number of indices describing the variance in the data set. The analysis showed the most independent indices: lacunarity (λm, λmin, λmax), BMD, Conn.D., SMI, DA, ρA and age.
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…
Some results on the behavior and estimation of the fractal dimensions of distributions on attractors
NASA Astrophysics Data System (ADS)
Cutler, C. D.
1991-02-01
The strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation. In this paper we first provide a rigorous mathematical framework for the study of dimension, focusing on pointwise dimension σ( x) and the generalized Renyi dimensions D(q), and give a rigorous proof of inequalities first derived by Grassberger and Procaccia and Hentschel and Procaccia. We then specialize to the problem of statistical estimation of the correlation dimension ν and information dimension σ. It has been recognized for some time that the error estimates accompanying the usual procedures (which generally involve least squares methods and nearest neighbor calculations) grossly underestimate the true statistical error involved. In least squares analyses of ν and σ we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates. We then develop an estimation procedure for σ which corrects for an important bias term (the local measure density) and provides confidence intervals for σ. The general applicability of this method is illustrated with various numerical examples.
NASA Astrophysics Data System (ADS)
Chakrabarty, Rajan K.; Moosmüller, Hans; Arnott, W. Patrick; Garro, Mark A.; Slowik, Jay G.; Cross, Eben S.; Han, Jeong-Ho; Davidovits, Paul; Onasch, Timothy B.; Worsnop, Douglas R.
2007-10-01
This study compares the optical coefficients of size-selected soot particles measured at a wavelength of 870 nm with those predicted by three theories, namely, Rayleigh-Debye-Gans (RDG) approximation, volume-equivalent Mie theory, and integral equation formulation for scattering (IEFS). Soot particles, produced by a premixed ethene flame, were size-selected using two differential mobility analyzers in series, and their scattering and absorption coefficients were measured with nephelometry and photoacoustic spectroscopy. Scanning electron microscopy and image processing techniques were used for the parameterization of the structural properties of the fractal-like soot aggregates. The aggregate structural parameters were used to evaluate the predictions of the optical coefficients based on the three light-scattering and absorption theories. Our results show that the RDG approximation agrees within 10% with the experimental results and the exact electromagnetic calculations of the IEFS theory. Volume-equivalent Mie theory overpredicts the experimental scattering coefficient by a factor of ˜3.2. The optical coefficients predicted by the RDG approximation showed pronounced sensitivity to changes in monomer mean diameter, the count median diameter of the aggregates, and the geometric standard deviation of the aggregate number size distribution.
Friction factor for aerosol fractal aggregates over the entire Knudsen range
NASA Astrophysics Data System (ADS)
Corson, James; Mulholland, George W.; Zachariah, Michael R.
2017-01-01
We develop an approach for computing the hydrodynamic friction tensor and scalar friction coefficient for an aerosol fractal aggregate in the transition regime. Our approach involves solving the Bhatnagar-Gross-Krook equation for the velocity field around a sphere and using the velocity field to calculate the force on each primary sphere in the aggregate due to the presence of the other spheres. It is essentially an extension of Kirkwood-Riseman theory from the continuum flow regime to the entire Knudsen range (Knudsen number from 0.01 to 100 based on the primary sphere radius). Our results compare well to published direct simulation Monte Carlo results, and they converge to the correct continuum and free molecule limits. Our calculations for clusters with up to 100 spheres support the theory that aggregate slip correction factors collapse to a single curve when plotted as a function of an appropriate aggregate Knudsen number. This self-consistent-field approach calculates the friction coefficient very quickly, so the approach is well-suited for testing existing scaling laws in the field of aerosol science and technology, as we demonstrate for the adjusted sphere scaling method.
Chappard, D; Legrand, E; Haettich, B; Chalès, G; Auvinet, B; Eschard, J P; Hamelin, J P; Baslé, M F; Audran, M
2001-11-01
Trabecular bone has been reported as having two-dimensional (2-D) fractal characteristics at the histological level, a finding correlated with biomechanical properties. However, several fractal dimensions (D) are known and computational ways to obtain them vary considerably. This study compared three algorithms on the same series of bone biopsies, to obtain the Kolmogorov, Minkowski-Bouligand, and mass-radius fractal dimensions. The relationships with histomorphometric descriptors of the 2-D trabecular architecture were investigated. Bone biopsies were obtained from 148 osteoporotic male patients. Bone volume (BV/TV), trabecular characteristics (Tb.N, Tb.Sp, Tb.Th), strut analysis, star volumes (marrow spaces and trabeculae), inter-connectivity index, and Euler-Poincaré number were computed. The box-counting method was used to obtain the Kolmogorov dimension (D(k)), the dilatation method for the Minkowski-Bouligand dimension (D(MB)), and the sandbox for the mass-radius dimension (D(MR)) and lacunarity (L). Logarithmic relationships were observed between BV/TV and the fractal dimensions. The best correlation was obtained with D(MR) and the lowest with D(MB). Lacunarity was correlated with descriptors of the marrow cavities (ICI, star volume, Tb.Sp). Linear relationships were observed among the three fractal techniques which appeared highly correlated. A cluster analysis of all histomorphometric parameters provided a tree with three groups of descriptors: for trabeculae (Tb.Th, strut); for marrow cavities (Euler, ICI, Tb.Sp, star volume, L); and for the complexity of the network (Tb.N and the three D's). A sole fractal dimension cannot be used instead of the classic 2-D descriptors of architecture; D rather reflects the complexity of branching trabeculae. Computation time is also an important determinant when choosing one of these methods.
Fractal Dimension of Tumor Microvasculature by DCE-US: Preliminary Study in Mice.
Saidov, Tamerlan; Heneweer, Carola; Kuenen, Maarten; von Broich-Oppert, Julian; Wijkstra, Hessel; Rosette, Jean de la; Mischi, Massimo
2016-12-01
Neoangiogenesis, which results in the formation of an irregular network of microvessels, plays a fundamental role in the growth of several types of cancer. Characterization of microvascular architecture has therefore gained increasing attention for cancer diagnosis, treatment monitoring and evaluation of new drugs. However, this characterization requires immunohistologic analysis of the resected tumors. Currently, dynamic contrast-enhanced ultrasound imaging (DCE-US) provides new options for minimally invasive investigation of the microvasculature by analysis of ultrasound contrast agent (UCA) transport kinetics. In this article, we propose a different method of analyzing UCA concentration that is based on the spatial distribution of blood flow. The well-known concept of Mandelbrot allows vascular networks to be interpreted as fractal objects related to the regional blood flow distribution and characterized by their fractal dimension (FD). To test this hypothesis, the fractal dimension of parametric maps reflecting blood flow, such as UCA wash-in rate and peak enhancement, was derived for areas representing different microvascular architectures. To this end, subcutaneous xenograft models of DU-145 and PC-3 prostate-cancer lines in mice, which show marked differences in microvessel density spatial distribution inside the tumor, were employed to test the ability of DCE-US FD analysis to differentiate between the two models. For validation purposes, the method was compared with immunohistologic results and UCA dispersion maps, which reflect the geometric properties of microvascular architecture. The results showed good agreement with the immunohistologic analysis, and the FD analysis of UCA wash-in rate and peak enhancement maps was able to differentiate between the two xenograft models (p < 0.05).
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.
Automatic prediction of tumour malignancy in breast cancer with fractal dimension
Chan, Alan
2016-01-01
Breast cancer is one of the most prevalent types of cancer today in women. The main avenue of diagnosis is through manual examination of histopathology tissue slides. Such a process is often subjective and error-ridden, suffering from both inter- and intraobserver variability. Our objective is to develop an automatic algorithm for analysing histopathology slides free of human subjectivity. Here, we calculate the fractal dimension of images of numerous breast cancer slides, at magnifications of 40×, 100×, 200× and 400×. Using machine learning, specifically, the support vector machine (SVM) method, the F1 score for classification accuracy of the 40× slides was found to be 0.979. Multiclass classification on the 40× slides yielded an accuracy of 0.556. A reduction of the size and scope of the SVM training set gave an average F1 score of 0.964. Taken together, these results show great promise in the use of fractal dimension to predict tumour malignancy. PMID:28083100
Lattice Dynamics of the Binary Aperiodic Chains of Atoms I:. Fractal Dimension of Phonon Spectra
NASA Astrophysics Data System (ADS)
Salejda, Włodzimierz
The microscopic harmonic model of lattice dynamics of the binary chains of atoms is formulated and studied numerically. The dependence of spring constants of the nearest-neighbor (NN) interactions on the average distance between atoms are taken into account. The covering fractal dimensions Df{( c ; )} of the Cantor-set-like phonon spec-tra (PS) of generalized Fibonacci and non-Fibonaccian aperiodic chains containing of 16384≤N≤33461 atoms are determined numerically. The dependence of Df{( c ; )} on the strength Q of NN interactions and on R=mH/mL, where mH and mL denotes the mass of heavy and light atoms, respectively, are calculated for a wide range of Q and R. In particular we found: (1) The fractal dimension Df{( c ; )} of the PS for the so-called goldenmean, silver-mean, bronze-mean, dodecagonal and Severin chain shows a local maximum at increasing magnitude of Q and R>1 (2) At sufficiently large Q we observe power-like diminishing of Df{( c ; )} , i.e. Df{( c ; )} ( {R > 1, Q} ; ) = a ḑot Qα , where α=-0.14±0.02 and α=-0.10±0.02 for the above specified chains and so-called octagonal, copper-mean, nickel-mean, Thue-Morse, Rudin-Shapiro chain, respectively.
Angle closure glaucoma detection using fractal dimension index on SS-OCT images.
Ni, Soe Ni; Marzilianol, Pina; Wong, Hon-Tym
2014-01-01
Optical coherence tomography (OCT) is a high resolution, rapid and non-invasive screening tool for angle closure glaucoma. In this paper, we propose a new strategy for automatic and landmark invariant quantification of the anterior chamber angle of the eye using swept source optical coherence tomography (SS-OCT) images. Seven hundred and eight swept source optical coherence tomography SS-OCT images from 148 patients with average age of (59.48 ± 8.97) were analyzed in this study. The angle structure is measured by fractal dimension (FD) analysis to quantify the complexity or changes of angle recess. We evaluated the FD index with biometric parameters for classification of open angle and angle closure glaucoma. The proposed fractal dimension index gives a better representation of the angle configuration for capturing the nature of the angle dynamics involved in different forms of open and closed angle glaucoma (average FD (standard deviation): 1.944 (0.045) for open and 1.894 (0.043) for closed angle). It showed that the proposed approach has promising potential to become a computer aided diagnostic tool for angle closure glaucoma (ACG) disease.
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.
NASA Astrophysics Data System (ADS)
Inclan, Rosa Maria
2016-04-01
Knowledge on three dimensional soil pore architecture is important to improve our understanding of the factors that control a number of critical soil processes as it controls biological, chemical and physical processes at various scales. Computed Tomography (CT) images provide increasingly reliable information about the geometry of pores and solids in soils at very small scale with the benefit that is a non-invasive technique. Fractal formalism has revealed as a useful tool in these cases where highly complex and heterogeneous meda are studied. One of these quantifications is mass dimension (Dm) and spectral dimension (d) applied to describe the water and gas diffusion coefficients in soils (Tarquis et al., 2012). In this work, intact soil samples were collected from the first three horizons of La Herreria soil. This station is located in the lowland mountain area of Sierra de Guadarrama (Santolaria et al., 2015) and it represents a highly degraded type of site as a result of the livestock keeping. The 3D images, of 45.1 micro-m resolution (256x256x256 voxels), were obtained and then binarized following the singularity-CA method (Martín-Sotoca et al. 2016). Based on these images Dm and d were estimated. The results showed an statistical difference in porosity, Dm and d for each horizon. This fact has a direct implication in diffusion parameters for a pore network modeling based on both fractal dimensions. These soil parameters will constitute a basis for site characterization for further studies regarding soil degradation; determining the interaction between soil, plant and atmosphere with respect to human induced activities as well as the basis for several nitrogen and carbon cycles modeling. References Martin Sotoca; J.J. Ana M. Tarquis, Antonio Saa Requejo, and Juan B. Grau (2016). Pore detection in Computed Tomography (CT) soil 3D images using singularity map analysis. Geophysical Research Abstracts, 18, EGU2016-829. Santolaria-Canales, Edmundo and the Gu
Irreversible aggregation of interacting particles in one dimension.
Sidi Ammi, Hachem; Chame, Anna; Touzani, M'hammed; Benyoussef, Abdelilah; Pierre-Louis, Olivier; Misbah, Chaouqi
2005-04-01
We present a study of the aggregation of interacting particles in one dimension. This situation, for example, applies to atoms trapped along linear defects at the surface of a crystal. Simulations are performed with two lattice models. In the first model, the borders of atoms and islands interact in a vectorial manner via force monopoles. In the second model, each atom carries a dipole. These two models lead to qualitatively similar but quantitatively different behaviors. In both cases, the final average island size S(f) does not depend on the interactions in the limits of very low and very high coverages. For intermediate coverages, S(f) exhibits an asymmetric behavior as a function of the interaction strength: while it saturates for attractive interactions, it decreases for repulsive interactions. A class of mean-field models is designed, which allows one to retrieve the interaction dependence on the coverage dependence of the average island size with a good accuracy.
Phulari, Rashmi G S; Rathore, Rajendrasinh S; Talegaon, Trupti Pramod
2016-01-01
Background: Various clinical and histological factors have helped in predicting the survival of patients with oral squamous cell carcinoma (OSCC). However, there has been a need for more specialized diagnostic and prognostic factors to avoid subjective variation among opinion. Thus, fractal dimension (FD) can be used as an index of the morphological changes that the epithelial cells undergo during their transformation into neoplastic cell. In oral cancer study, nuclear FD (NFD) can be used as a quantitative index to discriminate between normal, dysplastic and neoplastic oral mucosa. Aim: To use nuclear fractal geometry to compare the morphometric complexity in the normal, epithelial dysplasia and OSCC cases and to verify the difference among the various histological grades of dysplasia and OSCC. It was fulfilled by estimating the FDs of the nuclear surface. Materials and Methods: Histopathologically diagnosed cases of epithelial dysplasia and OSCC were taken from the archives. Photomicrographs were captured with the help of Lawrence and Mayo research microscope. The images were then subjected to image analysis using the Image J software with FracLac plugin java 1.6 to obtain FDs. FD of ten selected nuclei was calculated using the box-counting algorithm. Statistical Analysis: was done using descriptive analysis, ANOVA and Tukey's honest significant difference post hoc tests with STATAIC-13 software. Results and Conclusion: NFD can provide valuable information to discriminate between normal mucosa, dysplasia and carcinoma objectively without subjective discrimination. PMID:27721604
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)
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.
Zhang, Jiong; Bekkers, Erik; Abbasi-Sureshjani, Samaneh
2016-01-01
The retinal fractal dimension (FD) is a measure of vasculature branching pattern complexity. FD has been considered as a potential biomarker for the detection of several diseases like diabetes and hypertension. However, conflicting findings were found in the reported literature regarding the association between this biomarker and diseases. In this paper, we examine the stability of the FD measurement with respect to (1) different vessel annotations obtained from human observers, (2) automatic segmentation methods, (3) various regions of interest, (4) accuracy of vessel segmentation methods, and (5) different imaging modalities. Our results demonstrate that the relative errors for the measurement of FD are significant and FD varies considerably according to the image quality, modality, and the technique used for measuring it. Automated and semiautomated methods for the measurement of FD are not stable enough, which makes FD a deceptive biomarker in quantitative clinical applications. PMID:27703803
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.
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)
Lipping, T.; Olejarczyk, E.; Parts, M.
2004-07-01
The microwave radiation effects on EEG-signal have been studied by comparison with photo-stimulaton. The study of photos-stimulation effects at 16 Hz frequency and microwave radiation stimulation effects at 450 MHz modulated with 7 Hz frequency show fractal dimension increase.
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
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
NASA Astrophysics Data System (ADS)
Pérez-López, R.; Paredes, C.; Muñoz-Martín, A.
2005-04-01
The spatial distribution of faults is usually described as a fractal set characterised by the fractal dimension. In this work, we have filtered fault patterns interpreted from digital elevation models, aerial photographs and field maps, by using structural geological parameters of the stress ellipsoid (stress tensor direction and stress ratio R') and age of deformation. From these filtered structural maps, we have obtained the fractal dimension associated with the fracture patterns developed during Permo-Triassic and Alpine tectonic events on a Variscan granitic massif located in the Spanish Central System. Oriented fractal dimensions were calculated on several transects crossing the fault-filtered maps. The fractal dimension ( D), calculated by 1-D box-counting, describes an ellipse on a polar plot with the short axis as the minimum value ( DHmin) and the long axis as the maximum value ( DHmax) of the fractal dimensions measured. From these analyses, we have defined the F-parameter as a function of the maximum value, minimum value and vertical value of fractal dimension ( Dz), F=( Dz- DHmin)/( DHmax- DHmin). Finally we have established, from a local scale analysis, a perpendicular relationship between the principal axes of the ellipse of the fractal spatial anisotropy of fractures and the principal axes of the stress tensor ( σHmax, σHmin and σz) that generates this dynamic pattern of fractures. Furthermore, the F-parameter and the stress ratio R' are equivalents and, applied in this area, both show a triaxial extension.
NASA Astrophysics Data System (ADS)
Monceau, Pascal; Hsiao, Pai-Yi
2002-09-01
We study the Wolff cluster size distributions obtained from Monte Carlo simulations of the Ising phase transition on Sierpinski fractals with Hausdorff dimensions Df between 2 and 3. These distributions are shown to be invariant when going from an iteration step of the fractal to the next under a scaling of the cluster sizes involving the exponent (β/ν)+(γ/ν). Moreover, the decay of the autocorrelation functions at the critical points enables us to calculate the Wolff dynamical critical exponents z for three different values of Df. The Wolff algorithm is more efficient in reducing the critical slowing down when Df is lowered.
Puškaš, Nela; Zaletel, Ivan; Stefanović, Bratislav D; Ristanović, Dušan
2015-03-04
Pyramidal neurons of the mammalian cerebral cortex have specific structure and pattern of organization that involves the presence of apical dendrite. Morphology of the apical dendrite is well-known, but quantification of its complexity still remains open. Fractal analysis has proved to be a valuable method for analyzing the complexity of dendrite morphology. The aim of this study was to establish the fractal dimension of apical dendrite arborization of pyramidal neurons in distinct neocortical laminae by using the modified box-counting method. A total of thirty, Golgi impregnated neurons from the rat brain were analyzed: 15 superficial (cell bodies located within lamina II-III), and 15 deep pyramidal neurons (cell bodies situated within lamina V-VI). Analysis of topological parameters of apical dendrite arborization showed no statistical differences except in total dendritic length (p=0.02), indicating considerable homogeneity between the two groups of neurons. On the other hand, average fractal dimension of apical dendrite was 1.33±0.06 for the superficial and 1.24±0.04 for the deep cortical neurons, showing statistically significant difference between these two groups (p<0.001). In conclusion, according to the fractal dimension values, apical dendrites of the superficial pyramidal neurons tend to show higher structural complexity compared to the deep ones.
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.
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
Fractal dimension of cohesive sediment flocs at steady state under seven shear flow conditions
Zhu, Zhongfan; Yu, Jingshan; Wang, Hongrui; ...
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
From static micrographs to particle aggregation dynamics in three dimensions.
Häbel, H; Särkkä, A; Rudemo, M; Hamngren Blomqvist, C; Olsson, E; Abrahamsson, C; Nordin, M
2016-04-01
Studies on colloidal aggregation have brought forth theories on stability of colloidal gels and models for aggregation dynamics. Still, a complete link between developed frameworks and obtained laboratory observations has to be found. In this work, aggregates of silica nanoparticles (20 nm) are studied using diffusion limited cluster aggregation (DLCA) and reaction limited cluster aggregation (RLCA) models. These processes are driven by the probability of particles to aggregate upon collision. This probability of aggregation is one in the DLCA and close to zero in the RLCA process. We show how to study the probability of aggregation from static micrographs on the example of a silica nanoparticle gel at 9 wt%. The analysis includes common summary functions from spatial statistics, namely the empty space function and Ripley's K-function, as well as two newly developed summary functions for cluster analysis based on graph theory. One of the new cluster analysis functions is related to the clustering coefficient in communication networks and the other to the size of a cluster. All four topological summary statistics are used to quantitatively compare in plots and in a least-square approach experimental data to cluster aggregation simulations with decreasing probabilities of aggregation. We study scanning transmission electron micrographs and utilize the intensity-mass thickness relation present in such images to create comparable micrographs from three-dimensional simulations. Finally, a characterization of colloidal silica aggregates and simulated structures is obtained, which allows for an evaluation of the cluster aggregation process for different aggregation scenarios. As a result, we find that the RLCA process fits the experimental data better than the DLCA process.
Fractal dimension of debris-avalanche deposits in the Hawaiian submarine landslide deposits
NASA Astrophysics Data System (ADS)
Yokose, H.; Yamato, S.
2005-12-01
17 landslide deposits on the flanks of the southern Hawaiian Ridge have been classified into two major types: SLUMPS, which moved slowly as a coherent mass, and DEBRIS AVALANCHES, which moved quickly.The debris-avalanche deposits are predominant on submarine flanks of volcanic ocean islands elsewhere in the world. Such huge landslides are considered to produce giant tsunamis and megaturbidites covering large areas of abyssal plains. Based on the small scale topographic elements, we reinvestigated the distribution areas and emplacement styles of the debris-avalanche deposits, which differ from those previously proposed from GLORIA images without benefit of detailed bathymetric data or direct seafloor observations. There are several types of small scale topographic elements in the debris-avalanche deposits previously proposed: source amphitheater, toppled blocks, marginal levee, slide-emplaced blocks, chute, mud wave, hummocky terrain. They are very similar to those appeared in subaerial volcanic debris-avalanche fields. However, no correlation between the collapse height and runout distance are observed in the submarine debris-avalanche deposits. The hummocky terrains can be classified into two types: FLAT-TYPE, which is distributed in the nearly flat abyssal plain, less than 0.5 degree, and SLOPE-TYPE, which located on the lower part of the submarine flanks, greater than 1 degree. The size of hummocks in a slope-type hummocky terrain have an unimodal distribution pattern with a broad peak in the number of hummocks versus height category diagram. On the contrary, the size of hummocks in flat-type hummocky terrains have a power law distribution pattern in the same diagram. The fractal dimensions calculated from these diagrams are 1.19 (Nuuanu landslide), 2.32 (Ka Lae landslide) and 2.96 (Alika 2 debris-avalanche), respectively. They are expected to reflect the processes and degree of fragmentation. Therefore, among the debris_]avalanche deposits proposed previously
Losa, Gabriele A; Castelli, Christian
2005-11-01
An analytical strategy combining fractal geometry and grey-level co-occurrence matrix (GLCM) statistics was devised to investigate ultrastructural changes in oestrogen-insensitive SK-BR3 human breast cancer cells undergoing apoptosis in vitro. Apoptosis was induced by 1 microM calcimycin (A23187 Ca(2+) ionophore) and assessed by measuring conventional cellular parameters during the culture period. SK-BR3 cells entered the early stage of apoptosis within 24 h of treatment with calcimycin, which induced detectable changes in nuclear components, as documented by increased values of most GLCM parameters and by the general reduction of the fractal dimensions. In these affected cells, morphonuclear traits were accompanied by the reduction of distinct gangliosides and loss of unidentifiable glycolipid molecules at the cell surface. All these changes were shown to be involved in apoptosis before the detection of conventional markers, which were only measurable during the active phases of apoptotic cell death. In overtly apoptotic cells treated with 1 microM calcimycin for 72 h, most nuclear components underwent dramatic ultrastructural changes, including marginalisation and condensation of chromatin, as reflected in a significant reduction of their fractal dimensions. Hence, both fractal and GLCM analyses confirm that the morphological reorganisation of nuclei, attributable to a loss of structural complexity, occurs early in apoptosis.
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)
Verbovšek, T.
2009-04-01
Fractal dimensions of fracture networks (D) are usually determined from 2-D objects, like the digitized fracture traces in outcrops. Sometimes, extrapolations to higher dimensions are required if the measurements (for example fracture traces in the boreholes or in the scanlines) are performed in 1-D environment (D1-D) and are later upscaled to higher dimensions (D2-D). For isotropic fractals this relation should be straight-forward according to the theory: D2-D = D1-D +1, as the intersection of a 2-D fractal with a plane results in a fractal with D1-D equal to D2-D minus one. Some authors have questioned this relation and proposed different empirical relationships. Still, there exist very few field studies of natural fracture networks to support or test such a relationship. The study was therefore focused on the analysis of 23 natural fracture networks in Triassic dolomites in Slovenia. The traces of these fractures were analyzed separately in both 1-D and 2-D environments, and relationships between the obtained fractal dimensions were determined. For 2-D data, the digitized images of fracture traces in 2048x2048 pixel resolution were analyzed by the box-counting method, considering truncation and censoring effects (the 'cut-off' method, using only the valid data right of the cut-off points) and also by considering the complete data range interval (the 'full' method). These values were consequently compared to 1-D values. Those were obtained by dissecting images in both x- and y-directions into 2048 smaller linear images of 1-pixel width, simulating the intersection with a plane. Such line images were then examined by the fracture line-counting method, a 1-D equivalent of the box-counting technique. Results show that the values of all fractal dimensions, regardless of the different fracture networks or the method used, lie in a very narrow data range, and the standard deviations are very small (up to 0.03). The small range can be attributed to a similar fracturing
NASA Astrophysics Data System (ADS)
Alonso, C.; Benito, R. M.; Tarquis, A. M.
2012-04-01
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
Aggregating Political Dimensions: Of the Feasibility of Political Indicators
ERIC Educational Resources Information Center
Sanin, Francisco Gutierrez; Buitrago, Diana; Gonzalez, Andrea
2013-01-01
Political indicators are widely used in academic writing and decision making, but remain controversial. This paper discusses the problems related to the aggregation functions they use. Almost always, political indicators are aggregated by weighted averages or summations. The use of such functions is based on untenable assumptions (existence of…
NASA Astrophysics Data System (ADS)
Kou, Jian-Long; Lu, Hang-Jun; Wu, Feng-Min; Xu, You-Sheng
2008-05-01
A tumour vascular network, characterized as an irregularly stochastic growth, is different from the normal vascular network. We systematically analyse the dependence of the branching. It is found that anastomosis of tumour on time is according to a number of tumour images, and both the fractal dimensions and multifractal spectra of the tumours are obtained. In the cases studied, the fractal dimensions of the tumour vascular network increase with time and the multifractal spectrum not only rises entirely but also shifts right. In addition, the best drug delivery stage is discussed according to the difference of the singularity exponent δα(δα = αmax — αmin), which shows some change in the growth process of the tumour vascular network. A common underlying principle is obtained from our analysis along with previous results.
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.
Manera, M; Dezfuli, B S; Borreca, C; Giari, L
2014-11-01
Fractal analysis is a reliable method for describing, summarizing object complexity and heterogeneity and has been widely used in biology and medicine to deal with scale, size and shape management problems. The aim of present survey was to use fractal analysis as a complexity measure to characterize mast cells (MCs) degranulation in a rainbow trout ex vivo model (isolated organ bath). Compound 48/80, a condensation product of N-methyl-p-methoxyphenethylamine with formaldehyde, was adopted as MCs degranulation agent in trout intestinal strips. Fractal dimension (D), as a measure of complexity, 'roughness' and lacunarity (λ), as a measure of rotational and translational invariance, heterogeneity, in other words, of the texture, were compared in MCs images taken from intestinal strips before and after compound 48/80 addition to evaluate if and how they were affected by degranulation. Such measures were also adopted to evaluate their discrimination efficacy between compound 48/80 degranulated group and not degranulated group and the results were compared with previously reported data obtained with conventional texture analysis (image histogram, run-length matrix, co-occurrence matrix, autoregressive model, wavelet transform) on the same experimental material. Outlines, skeletons and original greyscale images were fractal analysed to evaluate possible significant differences in the measures values according to the analysed feature. In particular, and considering outline and skeleton as analysed features, fractal dimensions from compound 48/80 treated intestinal strips were significantly higher than the corresponding untreated ones (paired t and Wilcoxon test, p < 0.05), whereas corresponding lacunarity values were significantly lower (paired Wilcoxon test, p < 0.05) but only for outline as analysed feature. Outlines roughness increase is consistent with an increased granular mediators interface, favourable for their biological action; while lacunarity (image
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.
NASA Astrophysics Data System (ADS)
Zuo, Xue; Zhu, Hua; Zhou, Yuankai; Ding, Cong; Sun, Guodong
2016-08-01
Relationships between material hardness, turning parameters (spindle speed and feed rate) and surface parameters (surface roughness Ra, fractal dimension D and characteristic roughness τ∗) are studied and modeled using response surface methodology (RSM). The experiments are carried out on a CNC lathe for six carbon steel material AISI 1010, AISI 1020, AISI 1030, AISI 1045, AISI 1050 and AISI 1060. The profile of turned surface and the surface roughness value are measured by a JB-5C profilometer. Based on the profile data, D and τ∗ are computed through the root-mean-square method. The analysis of variance (ANOVA) reveals that spindle speed is the most significant factors affecting Ra, while material hardness is the most dominant parameter affecting τ∗. Material hardness and spindle speed have the same influence on D. Feed rate has less effect on three surface parameters than spindle speed and material hardness. The second-order models of RSM are established for estimating Ra, D and τ∗. The validity of the developed models is approximately 80%. The response surfaces show that a surface with small Ra and large D and τ∗ can be obtained by selecting a high speed and a large hardness material. According to the established models, Ra, D and τ∗ of six carbon steels surfaces can be predicted under cutting conditions studied in this paper. The results have an instructive meaning to estimate the surface quality before turning.
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.
Linear correlation between fractal dimension of EEG signal and handgrip force.
Liu, J Z; Yang, Q; Yao, B; Brown, R W; Yue, G H
2005-08-01
Fractal dimension (FD) has been proved useful in quantifying the complexity of dynamical signals in biology and medicine. In this study, we measured FDs of human electroencephalographic (EEG) signals at different levels of handgrip forces. EEG signals were recorded from five major motor-related cortical areas in eight normal healthy subjects. FDs were calculated using three different methods. The three physiological periods of handgrip (command preparation, movement and holding periods) were analyzed and compared. The results showed that FDs of the EEG signals during the movement and holding periods increased linearly with handgrip force, whereas FD during the preparation period had no correlation with force. The results also demonstrated that one method (Katz's) gave greater changes in FD, and thus, had more power in capturing the dynamic changes in the signal. The linear increase of FD, together with results from other EEG and neuroimaging studies, suggest that under normal conditions the brain recruits motor neurons at a linear progress when increasing the force.
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)
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.
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
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.
Technology Transfer Automated Retrieval System (TEKTRAN)
The aggregate structure of phthalic anhydride (PA) modified soy protein isolate (SPI) was investigated by estimating its fractal dimension from the equilibrated dynamic strain sweep experiments. The estimated fractal dimensions of the filler aggregates were less than 2, indicating that these partic...
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.
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
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.
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.
Comparison of fractal dimension estimation algorithms for epileptic seizure onset detection.
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.
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.
Observation of two different fractal structures in nanoparticle, protein and surfactant complexes
Mehan, Sumit Kumar, Sugam Aswal, V. K.
2014-04-24
Small angle neutron scattering has been carried out from a complex of nanoparticle, protein and surfactant. Although all the components are similarly (anionic) charged, we have observed strong interactions in their complex formation. It is characterized by the coexistence of two different mass fractal structures. The first fractal structure is originated from the protein and surfactant interaction and second from the depletion effect of first fractal structure leading the nanoparticle aggregation. The fractal structure of protein-surfactant complex represents to bead necklace structure of micelle-like clusters of surfactant formed along the unfolded protein chain. Its fractal dimension depends on the surfactant to protein ratio (r) and decreases with the increase in r. However, fractal dimension of nanoparticle aggregates in nanoparticle-protein complex is found to be independent of protein concentration and governed by the diffusion limited aggregation like morphology.
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.
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)
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
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)
Modeling EEG fractal dimension changes in wake and drowsy states in humans--a preliminary study.
Bojić, Tijana; Vuckovic, Aleksandra; Kalauzi, Aleksandar
2010-01-21
Aim of this preliminary study was to examine and compare topographic distribution of Higuchi's fractal dimension (FD, measure of signal complexity) of EEG signals between states of relaxed wakefulness and drowsiness, as well as their FD differences. The experiments were performed on 10 healthy individuals using a fourteen-channel montage. An explanation is offered on the causes of the detected FD changes. FD values of 60s records belonging to wake (Hori's stage 1) and drowsy (Hori's stages 2-4) states were calculated for each channel and each subject. In 136 out of 140 epochs an increase in FD was obtained. Relationship between signal FD and its relative alpha amplitude was mathematically modeled and we quantitatively demonstrated that the increase in FD was predominantly due to a reduction in alpha activity. The model was generalized to include other EEG oscillations. By averaging FD values for each channel across 10 subjects, four clusters (O2O1; T6P4T5P3; C3F3F4C4F8F7; T4T3) for the wake and two clusters (O2O1P3T6P4T5; C3C4F4F3F8T4T3F7) for the drowsy state were statistically verified. Topographic distribution of FD values in wakefulness showed a lateral symmetry and a partial fronto-occipital gradient. In drowsiness, a reduction in the number of clusters was detected, due to regrouping of channels T3, T4, O1 and O2. Topographic distribution of absolute FD differences revealed largest values at F7, O1 and F3. Reorganization of channel clusters showed that regionalized brain activity, specific for wakefulness, became more global by entering into drowsiness. Since the global increase in FD during wake-to-drowsy transition correlated with the decrease of alpha power, we inferred that increase of EEG complexity may not necessarily be an index of brain activation.
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.
Fractal signatures in analogs of interplanetary dust particles
NASA Astrophysics Data System (ADS)
Katyal, Nisha; Banerjee, Varsha; Puri, Sanjay
2014-10-01
Interplanetary dust particles (IDPs) are an important constituent of the earths stratosphere, interstellar and interplanetary medium, cometary comae and tails, etc. Their physical and optical characteristics are significantly influenced by the morphology of silicate aggregates which form the core in IDPs. In this paper we reinterpret scattering data from laboratory analogs of cosmic silicate aggregates created by Volten et al. (2007) [1] to extract their morphological features. By evaluating the structure factor, we find that the aggregates are mass fractals with a mass fractal dimension dm≃1.75. The same fractal dimension also characterizes clusters obtained from diffusion limited aggregation (DLA). This suggests that the analogs are formed by an irreversible aggregation of stochastically transported silicate particles.
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)
Wu, Yu; Cheng, Tianhai; Zheng, Lijuan; Chen, Hao
2016-10-01
Light absorption enhancement of aged soot aerosols is highly sensitive to the morphologies and mixing states of soot aggregates and their non-absorbing coatings, such as organic materials. The quantification of these effects on the optical properties of thinly coated soot aerosols is simulated using an effective model with fixed volume fractions. Fractal aggregated soot was simulated using the diffusion limited aggregation (DLA) algorithm and discretized into soot dipoles. The dipoles of non-absorbing aerosols, whose number was fixed by the volume fraction, were further generated from the neighboring random edge dipoles. Their optical properties were calculated using the discrete dipole approximation (DDA) method and were compared with other commonly used models. The optical properties of thinly coated soot calculated using the fixed volume fraction model are close to (less than ~10% difference) the results of the fixed coating thickness model, except their asymmetry parameters (up to ~25% difference). In the optical simulations of thinly coated soot aerosols, this relative difference of asymmetry parameters and phase functions between these realistic models may be notable. The realizations of the fixed volume fraction model may introduce smaller variation of optical results than those of the fixed coating thickness model. Moreover, the core-shell monomers model and homogeneous aggregated spheres model with the Maxwell-Garnett (MG) theory may underestimate (up to ~20%) the cross sections of thinly coated soot aggregates. The single core-shell sphere model may largely overestimate (up to ~150%) the cross sections and single scattering albedo of thinly coated soot aggregates, and it underestimated (up to ~60%) their asymmetry parameters. It is suggested that the widely used single core-shell sphere approximation may not be suitable for the single scattering calculations of thinly coated soot aerosols.
NASA Astrophysics Data System (ADS)
Monceau, P.; Hsiao, P.-Y.
2003-02-01
We study the cluster size distributions generated by the Wolff algorithm in the framework of the Ising model on Sierpinski fractals with Hausdorff dimension Df between 1 and 2. We show that these distributions exhibit a scaling property involving the magnetic exponent yh associated with one of the eigen-direction of the renormalization flows. We suggest that a single cluster tends to invade the whole lattice as Df tends towards the lower critical dimension of the Ising model, namely 1. The autocorrelation times associated with the Wolff and Swendsen-Wang algorithms enable us to calculate dynamical exponents; the cluster algorithms are shown to be more efficient in reducing the critical slowing down when Df is lowered.
NASA Astrophysics Data System (ADS)
He, Qi; Yang, Zhao; Gong, Bin; Wang, Jingjing; Xiao, Kaijun; Yang, Shang-Tian
2017-02-01
This work aimed to establish an effective approach to evaluate the quality of frozen fish, focusing on changes in fish tissue structure and chemical composition during storage. Fresh tilapia samples were treated by coating with tangerine peel (TP) extract and then stored at ‑4, ‑8 and ‑18 °C, respectively, for 40 days. The frozen fish tissues were analyzed for structural and chemical changes. Fractal dimension, which quantifies the porous structure formed in the tissue samples, texture properties including hardness and springiness, and moisture content and water activity all decreased during the storage, while the extents of lipid oxidation, measured as peroxide value and thiobarbituric acid concentration, and protein degradation, monitored with total volatile basic nitrogen and trichloroacetic acid soluble peptides, increased. The change rates of these parameters decreased with decreasing the storage temperature and by applying TP extract. A model was developed for predicting fractal dimension, which indicated the quality of preserved tilapia and thus can be used to predict the shelf life under different storage temperatures. The results demonstrated that TP extract could extend the shelf life of frozen tilapia by 35–45% by inhibiting changes in tissue structure, moisture loss, lipid oxidation and protein degradation during frozen storage.
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
He, Qi; Yang, Zhao; Gong, Bin; Wang, Jingjing; Xiao, Kaijun; Yang, Shang-Tian
2017-01-01
This work aimed to establish an effective approach to evaluate the quality of frozen fish, focusing on changes in fish tissue structure and chemical composition during storage. Fresh tilapia samples were treated by coating with tangerine peel (TP) extract and then stored at −4, −8 and −18 °C, respectively, for 40 days. The frozen fish tissues were analyzed for structural and chemical changes. Fractal dimension, which quantifies the porous structure formed in the tissue samples, texture properties including hardness and springiness, and moisture content and water activity all decreased during the storage, while the extents of lipid oxidation, measured as peroxide value and thiobarbituric acid concentration, and protein degradation, monitored with total volatile basic nitrogen and trichloroacetic acid soluble peptides, increased. The change rates of these parameters decreased with decreasing the storage temperature and by applying TP extract. A model was developed for predicting fractal dimension, which indicated the quality of preserved tilapia and thus can be used to predict the shelf life under different storage temperatures. The results demonstrated that TP extract could extend the shelf life of frozen tilapia by 35–45% by inhibiting changes in tissue structure, moisture loss, lipid oxidation and protein degradation during frozen storage. PMID:28169365
He, Qi; Yang, Zhao; Gong, Bin; Wang, Jingjing; Xiao, Kaijun; Yang, Shang-Tian
2017-02-07
This work aimed to establish an effective approach to evaluate the quality of frozen fish, focusing on changes in fish tissue structure and chemical composition during storage. Fresh tilapia samples were treated by coating with tangerine peel (TP) extract and then stored at -4, -8 and -18 °C, respectively, for 40 days. The frozen fish tissues were analyzed for structural and chemical changes. Fractal dimension, which quantifies the porous structure formed in the tissue samples, texture properties including hardness and springiness, and moisture content and water activity all decreased during the storage, while the extents of lipid oxidation, measured as peroxide value and thiobarbituric acid concentration, and protein degradation, monitored with total volatile basic nitrogen and trichloroacetic acid soluble peptides, increased. The change rates of these parameters decreased with decreasing the storage temperature and by applying TP extract. A model was developed for predicting fractal dimension, which indicated the quality of preserved tilapia and thus can be used to predict the shelf life under different storage temperatures. The results demonstrated that TP extract could extend the shelf life of frozen tilapia by 35-45% by inhibiting changes in tissue structure, moisture loss, lipid oxidation and protein degradation during frozen storage.
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.
NASA Astrophysics Data System (ADS)
Cymberknop, L.; Legnani, W.; Pessana, F.; Bia, D.; Zócalo, Y.; Armentano, R. L.
2011-12-01
The advent of vascular diseases, such as hypertension and atherosclerosis, is associated to significant alterations in the physical properties of arterial vessels. Evaluation of arterial biomechanical behaviour is related to the assessment of three representative indices: arterial compliance, arterial distensibility and arterial stiffness index. Elasticity is the most important mechanical property of the arterial wall, whose natures is strictly non-linear. Intervention of elastin and collagen fibres, passive constituent elements of the arterial wall, is related to the applied wall stress level. Concerning this, appropriate tools are required to analyse the temporal dynamics of the signals involved, in order to characterize the whole phenomenon. Fractal geometry can be mentioned as one of those techniques. The aim of this study consisted on arterial pressure and diameter signals processing, by means of nonlinear techniques based on fractal geometry. Time series morphology was related to different arterial stiffness states, generated by means of blood flow variations, during experiences performed in vitro.
Dual Fractal Dimension and Long-Range Correlation of Chinese Stock Prices
NASA Astrophysics Data System (ADS)
Chen, Chaoshi; Wang, Lei
2012-03-01
The recently developed modified inverse random midpoint displacement (mIRMD) and conventional detrended fluctuation analysis (DFA) algorithms are used to analyze the tick-by-tick high-frequency time series of Chinese A-share stock prices and indexes. A dual-fractal structure with a crossover at about 10 min is observed. The majority of the selected time series show visible persistence within this time threshold, but approach a random walk on a longer time scale. The phenomenon is found to be industry-dependent, i.e., the crossover is much more prominent for stocks belonging to cyclical industries than for those belonging to noncyclical (defensive) industries. We have also shown that the sign series show a similar dual-fractal structure, while like generally found, the magnitude series show a much longer time persistence.
NASA Astrophysics Data System (ADS)
Shimada, Hirohiko; Hikami, Shinobu
2016-12-01
The fractal dimensions of polymer chains and high-temperature graphs in the Ising model both in three dimension are determined using the conformal bootstrap applied for the continuation of the O( N) models from N=1 (Ising model) to N=0 (polymer). Even for non-integer N, the O( N) sum rule allows one to study the unitarity bound formally defined from the positivity, which may be violated in a non-unitary CFT. This unitarity bound of the scaling dimension for the O( N)-symmetric-tensor develops a kink as a function of the fundamental field as in the case of the energy operator dimension in the Z_2 (Ising) sum rule. Although this kink structure becomes less pronounced as N tends to zero, we found instead an emerging asymmetric minimum in the current central charge C_J. Despite the non-unitarity of the O( N) model at non-integer N, we find the C_J-kink along the unitarity bound lies very close to the location of the infrared (IR) O( N) CFT estimated by other methods. It is pointed out that certain level degeneracies at the IR CFT should induce these singular shapes of the unitarity bounds. As an application to the quantum and classical spin systems, we also predict critical exponents associated with the N=1 supersymmetry, which could be relevant for locating the corresponding fixed point in the phase diagram.
NASA Astrophysics Data System (ADS)
Koshiro, Yoko; Watanabe, Manabu; Takai, Rikuo; Hagiwara, Tomoaki; Suzuki, Toru
Size and shape of ice crystals in frozen food materials are very important because they affect not only quality of foods but also the viability of industrial processing such as freeze-drying of concentration. In this study, 30%wt sucrose solution is used as test samples. For examining the effect of stabilizerspectine and xantan gum is added to the sucrose solution. They are frozen on the cold stage of microscope to be observed their growing ice crystals under the circumstance of -10°C. Their size and shape are measured and quantitatively evaluated by applying fractal analysis. lce crystal of complicated shape has large fractal dimension, and vice versa. It successflly categorized the ice crystals into two groups; one is a group of large size and complicated shape, and the other is a group of small size and plain shape. The critical crystal size between the two groups is found to become larger with increasing holding time. It suggests a phenomenological model for metamorphoses process of ice crystals. Further, it is indicated that xantan gum is able to suppress the smoothing of ice crystals.
Acedo, L; Yuste, S B
2001-01-01
We address the problem of evaluating the number S(N)(t) of distinct sites visited up to time t by N noninteracting random walkers all starting from the same origin in fractal media. For a wide class of fractals (of which the percolation cluster at criticality and the Sierpinski gasket are typical examples) we propose, for large N and after the short-time compact regime, an asymptotic series for S(N)(t) analogous to that found for Euclidean media: S(N)(t) approximately S(N)(t)(1-Delta). Here S(N)(t) is the number of sites (volume) inside a hypersphere of radius L[ln(N)/c]1/v where L is the root-mean-square chemical displacement of a single random walker, and v and c determine how fast 1-Gamma(t)(l) (the probability that a given site at chemical distance l from the origin is visited by a single random walker by time t) decays for large values of l/L: 1-Gamma(t)(l) approximately exp[-c(l/L)(v)]. For the fractals considered in this paper, v=d(l)w/((d(l)w)-1), d(l)w being the chemical-diffusion exponent. The corrective term Delta is expressed as a series in ln(-n)(N)ln(m) ln(N) (with n> or =1 and 0< or =m< or =n), which is given explicitly up to n=2. This corrective term contributes substantially to the final value of S(N)(t) even for relatively large values of N.
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
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.
Frozen Fractals All Around: Aggregate Particles in the Plumes of Enceladus
NASA Astrophysics Data System (ADS)
Gao, P.; Kopparla, P.; Zhang, X.; Ingersoll, A. P.
2015-12-01
Estimates of the total particulate mass of the plumes of Enceladus are important to constrain theories of particle formation and transport at the surface and interior of the satellite. We revisit the calculations of Ingersoll & Ewald (2011), who estimated the particulate mass of the Enceladus plumes from strongly forward scattered light in Cassini ISS images. We model the plume as a combination of spherical particles and irregular aggregates resulting from the coagulation of spherical monomers, the latter of which allows for plumes of lower particulate mass. Though a continuum of solutions are permitted by the model, the best fits to the ISS data consist either of low mass plumes composed entirely of small aggregates or high mass plumes composed of large aggregates and spheres. The high mass plumes can be divided into a population of large aggregates with total particulate mass of 116 ± 12 × 103 kg, and a mixed population of spheres and aggregates consisting of a few large monomers that has a total plume particulate mass of 166 ± 42 × 103 kg, consistent with the results of Ingersoll & Ewald (2011). Meanwhile, the low particulate mass aggregate plumes have masses of 25 ± 4 × 103 kg, leading to a solid to vapor mass ratio of 0.07 ± 0.01 for the plume. If indeed the plumes are made of such aggregates, then a vapor-based origin for the plume particles is possible. The process of aggregate formation by the coagulation of monomers, which depends on the bulk monomer number density inside the plume vents, requires a total plume vent cross sectional area of at most 1800 m2 to allow for the aggregates to form before the monomers are ejected into space. Differentiation between the high mass and low mass solutions may be possible if forward scattering observations are taken at scattering angles <2°, or else an independent plume particulate mass measurement becomes available.
Alados, Concepcion L.; Escos, Juan; Emlen, John M.
1995-01-01
The effects of inbreeding on the developmental instability of skulls of dorcas (Gazella dorcas) and dama (G. dama) gazelles were investigated. In total, 132 dorcas gazelle skulls and 74 dama gazelle skulls from the Estación Experimental de Zonas Aridas in Almera, Spain, were measured. The fluctuating asymmetry of 9 meristic characters, consisting of the numbers of foramina on the two sides of the skull and mandible, was calculated. Although only the foramen infraorbitalis showed a significant increase in asymmetry with inbreeding in dorcas gazelles, the sum of the foramina in 5 of the skull regions clearly indicates an increase in asymmetry with inbreeding in both dorcas and dama gazelles. The fractal dimension of the sagittal suture was calculated by means of the coastline method. A greater effect of inbreeding on the sagittal suture in dama than in dorcas gazelle was observed, in concordance with the more evident deleterious effects of inbreeding depression in dama than in dorcas gazelles.
NASA Astrophysics Data System (ADS)
Raupov, Dmitry S.; Myakinin, Oleg O.; Bratchenko, Ivan A.; Kornilin, Dmitry V.; Zakharov, Valery P.; Khramov, Alexander G.
2016-04-01
Optical coherence tomography (OCT) is usually employed for the measurement of tumor topology, which reflects structural changes of a tissue. We investigated the possibility of OCT in detecting changes using a computer texture analysis method based on Haralick texture features, fractal dimension and the complex directional field method from different tissues. These features were used to identify special spatial characteristics, which differ healthy tissue from various skin cancers in cross-section OCT images (B-scans). Speckle reduction is an important pre-processing stage for OCT image processing. In this paper, an interval type-II fuzzy anisotropic diffusion algorithm for speckle noise reduction in OCT images was used. The Haralick texture feature set includes contrast, correlation, energy, and homogeneity evaluated in different directions. A box-counting method is applied to compute fractal dimension of investigated tissues. Additionally, we used the complex directional field calculated by the local gradient methodology to increase of the assessment quality of the diagnosis method. The complex directional field (as well as the "classical" directional field) can help describe an image as set of directions. Considering to a fact that malignant tissue grows anisotropically, some principal grooves may be observed on dermoscopic images, which mean possible existence of principal directions on OCT images. Our results suggest that described texture features may provide useful information to differentiate pathological from healthy patients. The problem of recognition melanoma from nevi is decided in this work due to the big quantity of experimental data (143 OCT-images include tumors as Basal Cell Carcinoma (BCC), Malignant Melanoma (MM) and Nevi). We have sensitivity about 90% and specificity about 85%. Further research is warranted to determine how this approach may be used to select the regions of interest automatically.
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.
Fractal dimension of sparkles in automotive metallic coatings by multispectral imaging measurements.
Medina, José M; Díaz, José A; Vignolo, Carlos
2014-07-23
Sparkle in surface coatings is a property of mirror-like pigment particles that consists of remarkable bright spots over a darker surround under unidirectional illumination. We developed a novel nondestructive method to characterize sparkles based on the multispectral imaging technique, and we focused on automotive metallic coatings containing aluminum flake pigments. Multispectral imaging was done in the visible spectrum at different illumination angles around the test sample. Reflectance spectra at different spatial positions were mapped to color coordinates and visualized in different color spaces. Spectral analysis shows that sparkles exhibit higher reflectance spectra and narrower bandwidths. Colorimetric analysis indicates that sparkles present higher lightness values and are far apart from the bulk of color coordinates spanned by the surround. A box-counting procedure was applied to examine the fractal organization of color coordinates in the CIE 1976 L*a*b* color space. A characteristic noninteger exponent was found at each illumination position. The exponent was independent of the illuminant spectra. Together, these results demonstrate that sparkles are extreme deviations relative to the surround and that their spectral properties can be described as fractal patterns within the color space. Multispectral reflectance imaging provides a powerful, noninvasive method for spectral identification and classification of sparkles from metal flake pigments on the micron scale.
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
NASA Astrophysics Data System (ADS)
Chadee, X. T.
2007-05-01
The fractal dimension, Lyapunov-exponent spectrum, and predictability are analyzed for chaotic attractors in the atmosphere by analyzing the time series of daily wind speeds over the Caribbean region. It can be shown that this dimension is greater than 8. However, the number of data points may be too small to obtain a reliable estimate of the Grassberger-Procaccia (1983a) correlation dimension because of the limitations discussed by Ruelle (1990). These results lead us to claim that there probably exist no low-dimensional strange attractors in the atmosphere. Because the fractal dimension has not yet been saturated, the Kolmogorov entropy and the error-doubling time obtained by the method of Grassberger and Procaccia (1983b) are sensitive to the selection of the time delay and are thus unreliable. A practical and more reliable method for estimating the Kolmogorov entropy and error-doubling time involves the computation of the Lyapunov-exponent spectrum using the algorithm of Zeng et al. (1991). Using this method, it is found that the error-doubling time is 2-3 days for time series over the Caribbean region. This is comparable to the predictability time found by Waelbrock (1995) for a single station in Mexico. The predictability time over land is slightly less than that over ocean which tends to have higher climatic signal-to-noise ratio. This analysis impacts on the selection of prediction tools (deterministic chaotic linear and non-linear maps or linear stochastic modeling) for wind speeds in the short term for wind energy farm resource planning and management. We conclude that short term wind predictions in the Caribbean region, for a few days ahead, may be best done with a stochastic model instead of a deterministic chaotic model. References Grassberger, P., and I. Procaccia. 1983a. Measuring the strangeness of attractors. Physica D 9: 189-208. Grassberger, P., and I. Procaccia. 1983b. Estimating the Kolmogorov entropy from a chaotic signal. Phys. Rev. A. 28
A study on solar dust ring formation based on fractal dust models.
NASA Astrophysics Data System (ADS)
Kimura, H.; Ishimoto, H.; Mukai, T.
1997-10-01
Using the fractal aggregate model for circumsolar dust grains, the nature of the circumsolar dust clouds is examined. As a fractal dimension of the aggregate decreases, the porosity of the aggregate increases. Consequently, its temperature becomes independent of its size, and approaches that of its constituent particles. This evidence suggests that the fractal aggregates with different sizes and made of the same chemical components sublimate at the same solar distance. This implies that the distance of the sublimation zone depends on the chemical composition alone. We have found that the aggregates consisting of silicate material, as well as carbon material, sublimate in the solar F-corona. On the other hand, a ratio of radiation pressure force to solar gravity on the fractal aggregate scarcely increases with decreasing size due to sublimation, in contrast with a strong dependence of its ratio on its size for a compact sphere. Our computer simulation for dynamical evolution of fractal aggregates suggests that they produce a narrow ring structure in the circumsolar dust cloud, compared with that expected for spherical dust grains. When the aggregates have more fluffy structure with a small fractal dimension, however, it is found that the circumsolar dust clouds would make no remarkable ring structure.
Fractal dimensions of wave functions and local spectral measures on the Fibonacci chain
NASA Astrophysics Data System (ADS)
Macé, Nicolas; Jagannathan, Anuradha; Piéchon, Frédéric
2016-05-01
We present a theoretical framework for understanding the wave functions and spectrum of an extensively studied paradigm for quasiperiodic systems, namely the Fibonacci chain. Our analytical results, which are obtained in the limit of strong modulation of the hopping amplitudes, are in good agreement with published numerical data. In the perturbative limit, we show a symmetry of wave functions under permutation of site and energy indices. We compute the wave-function renormalization factors and from them deduce analytical expressions for the fractal exponents corresponding to individual wave functions, as well as their global averages. The multifractality of wave functions is seen to appear at next-to-leading order in ρ . Exponents for the local spectral density are given, in extremely good accord with numerical calculations. Interestingly, our analytical results for exponents are observed to describe the system rather well even for values of ρ well outside the domain of applicability of perturbation theory.
Qian, A R; Li, D; Han, J; Gao, X; Di, S M; Zhang, W; Hu, L F; Shang, Peng
2012-05-01
Osteoblasts, the bone-forming cells, respond to various mechanical forces, such as stretch and fluid shear force in essentially similar ways. The cytoskeleton, as the load-bearing architecture of the cell, is sensitive to altered inertial forces. Disruption of the cytoskeleton will result in alteration of cellular structure and function. However, it is difficult to quantitatively illustrate cytoskeletal rearrangement because of the complexity of cytoskeletal structure. Usually, the morphological changes in actin organization caused by external stimulus are basically descriptive. In this study, fractal dimensions (D) analysis was used to quantify the morphological changes in the actin cytoskeleton of osteoblast-like cells (MC3T3-E1) under simulated microgravity using 3-D/2-D clinostats. The ImageJ software was used to count the fractal dimension of actin cytoskeleton by box-counting methods. Real-time PCR and immunofluroscent assays were used to further confirm the results obtained by fractal dimension analysis. The results showed significant decreases in D value of actin cytoskeleton, β-actin mRNA expression, and the mean fluorescence intensity of F-actin in osteoblast-like cells after 24 or 48 h of incubation under 3-D/2-D clinorotation condition compared with control. The findings indicate that 3-D/2-D clinorotation affects both actin cytoskeleton architecture and mRNA expression, and fractal may be a promising approach for quantitative analysis of the changes in cytoskeleton in different environments.
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.
NASA Astrophysics Data System (ADS)
Orbach, R.
1986-02-01
Random structures often exhibit fractal geometry, defined in terms of the mass scaling exponent, D, the fractal dimension. The vibrational dynamics of fractal networks are expressed in terms of the exponent d double bar, the fracton dimensionality. The eigenstates on a fractal network are spatially localized for d double bar less than or equal to 2. The implications of fractal geometry are discussed for thermal transport on fractal networks. The electron-fracton interaction is developed, with a brief outline given for the time dependence of the electronic relaxation on fractal networks. It is suggested that amorphous or glassy materials may exhibit fractal properties at short length scales or, equivalently, at high energies. The calculations of physical properties can be used to test the fractal character of the vibrational excitations in these materials.
D'Agata, Roberta; Palladino, Pasquale
2017-01-01
Gold nanoparticles (AuNPs) exhibit unique properties that can be modulated through a tailored surface functionalization, enabling their targeted use in biochemical sensing and medical diagnostics. In particular, streptavidin-modified AuNPs are increasingly used for biosensing purposes. We report here a study of AuNPs surface-functionalized with streptavidin-biotinylated oligonucleotide, focussing on the role played by the oligonucleotide probes in the stabilization/destabilization of the functionalized nanoparticle dispersion. The behaviour of the modified AuNP dispersion as a consequence of the competitive displacement of the biotinylated oligonucleotide has been investigated and the critical role of displaced oligonucletides in triggering the quasi one-dimensional aggregation of nanoparticles is demonstrated for the first time. The thorough understanding of the fundamental properties of bioconjugated AuNPs is of great importance for the design of highly sensitive and reliable functionalized AuNP-based assays. PMID:28144559
Spasic, Sladjana; Kalauzi, Aleksandar; Kesic, Srdjan; Obradovic, Milica; Saponjic, Jasna
2011-11-21
We used spectral analysis and Higuchi fractal dimension (FD) to correlate the EEG spectral characteristics of the sensorimotor cortex, hippocampus, and pons with their corresponding EEG signal complexities in anesthetized rats. We have explored the quantitative relationship between the mean FDs and EEG wide range high frequency (8-50 Hz) activity during ketamine/xylazine versus nembutal anesthesia at surgical plane. Using FD we detected distinct inter-structure complexity pattern and uncovered for the first time that the polygraphically and behaviorally defined anesthetized state at surgical plane as equal during experiment in two anesthetic regimens, is not the same with respect to the degree of neuronal activity (degree of generalized neuronal inhibition achieved) at different brain levels. Using the correlation of certain brain structure EEG spectral characteristics with their corresponding FDs, and the surrogate data modeling, we determined what particular frequency band contributes to EEG complexities in ketamine/xylazine versus nembutal anesthesia. In this study we have shown that the quantitative relationship between higher frequency EEG amplitude and EEG complexity is the best-modeled by surrogate data as a 3rd order polynomial. On the base of our EEG amplitude/EEG complexity relationship model, and the evidenced spectral differences in ketamine versus nembutal anesthesia we have proved that higher amplitudes of sigma, beta, and gamma frequency in ketamine anesthesia yields to higher FDs.
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.
NASA Astrophysics Data System (ADS)
Han, H. H.; Wang, Y. L.; Ren, G. L.; LI, J. Q.; Gao, T.; Yang, M.; Yang, J. L.
2016-11-01
Remote sensing plays an important role in mineral exploration of “One Belt One Road” plan. One of its applications is extracting and locating hydrothermal alteration zones that are related to mines. At present, the extracting method for alteration anomalies from principal component image mainly relies on the data's normal distribution, without considering the nonlinear characteristics of geological anomaly. In this study, a Fractal Dimension Change Point Model (FDCPM), calculated by the self-similarity and mutability of alteration anomalies, is employed to quantitatively acquire the critical threshold of alteration anomalies. The realization theory and access mechanism of the model are elaborated by an experiment with ASTER data in Beishan mineralization belt, also the results are compared with traditional method (De-Interfered Anomalous Principal Component Thresholding Technique, DIAPCTT). The results show that the findings produced by FDCPM are agree with well with a mounting body of evidence from different perspectives, with the extracting accuracy over 80%, indicating that FDCPM is an effective extracting method for remote sensing alteration anomalies, and could be used as an useful tool for mineral exploration in similar areas in Silk Road Economic Belt.
Fractal radar scattering from soil.
Oleschko, Klaudia; Korvin, Gabor; Figueroa, Benjamin; Vuelvas, Marco Antonio; Balankin, Alexander S; Flores, Lourdes; Carreón, Dora
2003-04-01
A general technique is developed to retrieve the fractal dimension of self-similar soils through microwave (radar) scattering. The technique is based on a mathematical model relating the fractal dimensions of the georadargram to that of the scattering structure. Clear and different fractal signatures have been observed over four geosystems (soils and sediments) compared in this work.
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
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
Observation of a crossover in kinetic aggregation of Palladium colloids
NASA Astrophysics Data System (ADS)
Ghafari, M.; Ranjbar, M.; Rouhani, S.
2015-10-01
We use field emission scanning electron microscope (FE-SEM) to investigate the growth of palladium colloids over the surface of thin films of WO3/glass. The film is prepared by Pulsed Laser Deposition (PLD) at different temperatures. A PdCl2 (aq) droplet is injected on the surface and in the presence of steam hydrogen the droplet is dried through a reduction reaction process. Two distinct aggregation regimes of palladium colloids are observed over the substrates. We argue that the change in aggregation dynamics emerges when the measured water drop Contact Angel (CA) for the WO3/glass thin films passes a certain threshold value, namely CA ≈ 46°, where a crossover in kinetic aggregation of palladium colloids occurs. Our results suggest that the mass fractal dimension of palladium aggregates follows a power-law behavior. The fractal dimension (Df) in the fast aggregation regime, where the measured CA values vary from 27° up to 46° according to different substrate deposition temperatures, is Df = 1.75(± 0.02) - the value of Df is in excellent agreement with kinetic aggregation of other colloidal systems in fast aggregation regime. Whereas for the slow aggregation regime, with CA = 58°, the fractal dimension changes abruptly to Df = 1.92(± 0.03). We have also used a modified Box-Counting method to calculate fractal dimension of gray-level images and observe that the crossover at around CA ≈ 46° remains unchanged.
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.
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.
NASA Astrophysics Data System (ADS)
Ng, Hoi Dick; Abderrahmane, Hamid Ait; Bates, Kevin R.; Nikiforakis, Nikos
2011-11-01
The interface between air and a rectangular block of sulphur hexafluoride (SF6), impulsively accelerated by the passage of a planar shock wave, undergoes Richtmyer-Meshkov instability and the flow becomes turbulent. The evolution of the interface was previously simulated using a multi-component model based on a thermodynamically consistent and fully conservative formulation and results were validated against available experimental data (Bates et al. Richtmyer-Meshkov instability induced by the interaction of a shock wave with a rectangular block of SF6, Phys Fluids, 2007; 19:036101). In this study, the CFD results are analyzed using the fractal theory approach and the evolution of fractal dimension of the interface during the transition of the flow into fully developed turbulence is measured using the standard box-counting method. It is shown that as the Richtmyer-Meshkov instability on the interface develops and the flow becomes turbulent, the fractal dimension of the interface increases asymptotically toward a value close to 1.39, which agrees well to those measured for classical shear and fully developed turbulences.
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.
NASA Astrophysics Data System (ADS)
Rodríguez Pascua, M. A.; De Vicente, G.; Calvo, J. P.; Pérez-López, R.
2003-05-01
A paleoseismic data set derived from the relationship between the thickness of seismites, 'mixed layers' in lacustrine Miocene deposits and the magnitude of the earthquakes is presented. The relationship between both parameters was calibrated by the threshold of fluidification limits in the interval of magnitude 5 and 5.5. The mixed layers (deformational sediment structures due to seismic activity) were observed in varved sediments from three Neogene lacustrine basins near Hellı´n (Albacete, Spain), El Cenajo, Elche de la Sierra and Hı´jar, and are interpreted as liquefaction features due to seismic phenomena. These paleoseismic structures were dated (relative values) by measurements of cyclic annual sedimentation in the varved sediments. From these observations, we are able to establish a recurrence interval of 130 years with events for magnitude bigger than or equal to four. Both paleoseismicity and instrumental seismicity data sets obey the Gutenberg-Richter law and the 'b' value is close to 0.86. The fractal dimension (dimension of capacity) of spatial distribution of potentially active faults (faults oriented according to the stress tensor regime in the area) was measured by the box-counting technique ( D0=1.73). According to the Aki empirical relation ( D0=2 b) for the instrumental seismicity and paleoseismic data sets in the area, the fractal dimension is close to 1.72. The similar value of the fractal dimension obtained by both techniques shows homogeneous seismic dynamics during the studied time interval. Moreover, the better established 'b' value of the paleoseismic data sets (0.86) compared with the 'b' value for the incomplete historic seismicity (<0.5) in the area increases the seismic series beyond the historic seismic record.
Experimental Evidence of Dynamical Scaling in a Two-Dimensional Fractal Growth
NASA Astrophysics Data System (ADS)
Miyashita, Satoru; Saito, Yukio; Uwaha, Makio
1997-04-01
A dynamical scaling law of fractal aggregation is testedusing electrochemical deposition without an external electric field.Silver metal leaves grow on the edge of a Cu plate placed in a thin cell containing an AgNO3-water solution due to the difference in ionization tendency between Ag and Cu. We find that the tip height h(t) satisfies the dynamical scaling relationh(t)= c-1/(d-D_f) \\tilde{g}(tc2/(d-D_f)) with respect to the solute concentration cin the space dimension d=2 with the fractal dimension Df=1.71 of the diffusion-limited aggregation.
NASA Astrophysics Data System (ADS)
Matsushita, Mitsugu; Family, Fereydoon; Honda, Katsuya
1987-10-01
A scaling description of the crossover from isotropic to anisotropic cluster growth for ordinary diffusion-limited aggregation (DLA) in two dimensions developed recently by Family and Hentschel is extended to the generalized DLA or η model. The dependence of various exponents necessary to characterize the anisotropic growth of the local-growth probability exponent η of the generalized DLA is obtained explicitly. The η dependence of the exponent β describing the variation of the crossover mass Nc on the degree of symmetry m,Nc~mβ, is derived. The results indicate that the anisotropic star-shaped clusters can be easily observed for η>1, while their appearance is much more difficult for η<1. All our results are consistent with those of computer simulations reported so far.
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
NASA Astrophysics Data System (ADS)
Knutson, Paul; Dahlberg, E. Dan
2003-10-01
In examples of fractals such as moon craters, rivers,2 cauliflower,3 and bread,4 the actual growth process of the fractal object is missed. In the simple experiment described here, one can observe and record the growth of calcium carbonate crystals — a ubiquitous material found in marble and seashells — in real time. The video frames can be digitized and analyzed to determine the fractal dimension.
Fractal titanium oxide under inverse 10-ns laser deposition in air and water
NASA Astrophysics Data System (ADS)
Pan, Aifei; Wang, Wenjun; Mei, Xuesong; Lin, Qijing; Cui, Jianlei; Wang, Kedian; Zhai, Zhaoyang
2017-04-01
This paper presents the preparation of different kinds of titanium oxide fractal structures on the surface of titanium by inverse pulsed laser deposition (IPLD) in air and water. In air, two-dimensional fractal structures are obtained with a low pulse energy. However, their branches units are aggregated and nanoscale branches disappear due to the high substrate temperature, causing the low fractal dimension of structure. When a higher laser energy is applied, the preformed deposited material forms a porous film, which reduces heat transfer from substrate. Therefore, three-dimensional and one-dimensional fractal structures with nanoscale branches on the topside of the film can be obtained. Then the desired two-dimensional fractal structures with nano-branches are obtained in water due to the water-induced rapid cooling of substrate temperature and plasma shock wave-induced particle's expansion along the surface of substrate. Meanwhile, the asymmetry of fractal structure units analyzed by diffusion limited aggregation (DLA) model is caused by the difference of the distance between the initial deposited particles. In addition, when the pulse energy goes up to 111 mJ, the branches of two-dimensional fractal structure units are also aggregated and form isolated particles. The idea about modification of substrate temperature and water can guide the preparation of the desired titanium oxide fractal structures in pulsed laser deposition (PLD), which is also applicable to other materials.
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.
NASA Astrophysics Data System (ADS)
Skorupski, Krzysztof
2015-05-01
Black carbon (BC) particles are a product of incomplete combustion of carbon-based fuels. One of the possibilities of studying the optical properties of BC structures is to use the DDA (Discrete Dipole Approximation) method. The main goal of this work was to investigate its accuracy and to approximate the most reliable simulation parameters. For the light scattering simulations the ADDA code was used and for the reference program the superposition T-Matrix code by Mackowski was selected. The study was divided into three parts. First, DDA simulations for a single particle (sphere) were performed. The results proved that the meshing algorithm can significantly affect the particle shape, and therefore, the extinction diagrams. The volume correction procedure is recommended for sparse or asymmetrical meshes. In the next step large fractal-like aggregates were investigated. When sparse meshes are used, the impact of the volume correction procedure cannot be easily predicted. In some cases it can even lead to more erroneous results. Finally, the optical properties of fractal-like aggregates composed of spheres in point contact were compared to much more realistic structures made up of connected, non-spherical primary particles.
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.
Landscape roughness analysis of Mt. Etna volcanic complex detected via fractal geometry
NASA Astrophysics Data System (ADS)
De Luca, Claudio; Bonfante, Antonello; Di Martino, Gerardo; Iodice, Antonio; Manzo, Mariarosaria; Pepe, Antonio; Pepe, Susi; Riccio, Daniele; Sansosti, Eugenio; Tizzani, Pietro; Zinno, Ivana
2013-04-01
fractal dimension map was quantized in order to identify and aggregate areas that are homogeneous from a fractal viewpoint, i.e., areas having a similar surface roughness. The comparison with the geological map of Mt. Etna volcanic complex reveals that the level of roughness is strictly connected with the lava flow age. More specifically, the roughness of lava flows, which is measured by the fractal dimension, decreases with the increasing of time. In other words, our analysis demonstrates that the volcanic depositional processes of the lava flows produce surfaces with high fractal dimension (roughness), whereas the exogenous phenomena, as the erosion processes, significantly reduce the fractal dimension associated with eruptive mechanisms. The COSMO-SkyMed and ALOS data used in this study have been processed at REA-CNR within the SAR4Volcanoes project under Italian Space Agency agreement n. I/034/11/0. G. Di Martino, D. Riccio, I. Zinno, "SAR Imaging of Fractal Surfaces," IEEE Trans. Geosci. Remote Sens., vol.50, no.2, pp. 630-644, Feb. 2012. S. Pepe, G. Di Martino, A. Iodice, M. Manzo, A. Pepe, D. Riccio, G. Ruello, E. Sansosti, P. Tizzani, I. Zinno; "Analysis of the fractal dimension of volcano geomorphology through Synthetic Aperture Radar (SAR) amplitude images acquired in C and X band", Geophysical Research Abstracts Vol. 14, EGU2012-6573-1, 2012, EGU General Assembly 2012.
NASA Astrophysics Data System (ADS)
Warchalowski, Wiktor; Krawczyk, Malgorzata J.
2017-03-01
We found the Lindenmayer systems for line graphs built on selected fractals. We show that the fractal dimension of such obtained graphs in all analysed cases is the same as for their original graphs. Both for the original graphs and for their line graphs we identified classes of nodes which reflect symmetry of the graph.
Applications of fractal geometry to dynamical evolution of sunspots
Milovanov, A.V.; Zelenyi, L.M. )
1993-07-01
A fractal model for sunspot dynamics is presented. Formation of a sunspot in the solar photosphere is considered from the viewpoint of aggregation of magnetic flux tubes on a fractal geometry. Fine structure of the magnetic flux tubes is analyzed for a broad class of non-Maxwellian plasma distribution functions. The sunspot fractal dimension is proved to depend on the parameters of the plasma distribution function, enabling one to investigate intrinsic properties of the solar plasma by means of powerful geometrical methods. Magnetic field dissipation in the tubes is shown to result in effective sunspot decay. Sunspot formation and decay times as well as the diffusion constant [ital K] deduced by using the fractal model, are in a good agreement with observational data. Disappearance of umbras in decaying sunspots is interpreted as a second-order phase transition reminiscent of the transition through the Curie point in ferromagnetics.
Effect of Dust Coagulation Dynamics on the Geometry of Aggregates
NASA Technical Reports Server (NTRS)
Nakamura, R.
1996-01-01
Master equation gives a more fundamental description of stochastic coagulation processes rather than popular Smoluchowski's equation. In order to examine the effect of the dynamics on the geometry of resulting aggregates, we study Master equation with a rigorous Monte Carlo algorithm. It is found that Cluster-Cluster aggregation model is a good approximation of orderly growth and the aggregates have fluffy structures with a fractal dimension approx. 2. A scaling analysis of Smoluchowski's equation also supports this conclusion.
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
Synthesis, Analysis, and Processing of Fractal Signals
1991-10-01
fractal dimension of the underlying signal , when defined. Robust estimation of the fractal dimension of 1/f processes is important in a number of...modeling errors. The resulting parameter estimation algorithms, which compute both fractal dimension parameters and the accompanying signal and noise...Synthesis, Analysis, and Processing of Fractal Signals RLE Technical Report No. 566 Gregory W. Wornell October 1991 Research Laboratory of
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.
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.
Tissue as a self-organizing system with fractal dynamics.
Waliszewski, P; Konarski, J
2001-01-01
Cell is a supramolecular dynamic network. Screening of tissue-specific cDNA library and results of Relative RT-PCR indicate that the relationship between genotype, (i.e., dynamic network of genes and their protein regulatory elements) and phenotype is non-bijective, and mendelian inheritance is a special case only. This implies non-linearity, complexity, and quasi-determinism, (i.e., co-existence of deterministic and non-deterministic events) of dynamic cellular network; prerequisite conditions for the existence of fractal structure. Indeed, the box counting method reveals that morphological patterns of the higher order, such as gland-like structures or populations of differentiating cancer cells possess fractal dimension and self-similarity. Since fractal space is not filled out randomly, a variety of morphological patterns of functional states arises. The expansion coefficient characterizes evolution of fractal dynamics. The coefficient indicates what kind of interactions occurs between cells, and how far from the limiting integer dimension of the Euclidean space the expanding population of cells is. We conclude that cellular phenomena occur in the fractal space; aggregation of cells is a supracollective phenomenon (expansion coefficient > 0), and differentiation is a collective one (expansion coefficient < 0). Fractal dimension or self-similarity are lost during tumor progression. The existence of fractal structure in a complex tissue system denotes that dynamic cellular phenomena generate an attractor with the appropriate organization of space-time. And vice versa, this attractor sets up physical limits for cellular phenomena during their interactions with various fields. This relationship can help to understand the emergence of extraterrestial forms of life. Although those forms can be composed of non-carbon molecules, fractal structure appears to be the common feature of all interactive biosystems.
Fractal structures and their computer simulation in rapidly quenched Ai-Mn alloys
NASA Astrophysics Data System (ADS)
Zhang, Meng; Hong, Chunyong; Dai, Hong; Yang, Pinsheng; Tan, Yuxi
1999-02-01
The microstructure of rapidly quenched A1-Mn alloys were studied by TEM and SEM. The icosahedral phase in AI-Mn alloys was observed to show various types of fractal morphologies, which may be classified into four kinds: 1) dendritic shape, 2) flower-like shape, 3) granular shape and 4) grain-oscillation shape. After being digitized by a computer, the fractal dimensions (D) of these morphologies were calculated. Based on the traditional diffusion limited aggregation (DLA) model computer simulations were made with two-seeded and many-seeded clusters, which reflect the growing mechanism of some fractal structures in A1-Mn alloys. It is suggested that these fractal structures are formed by many icosahedral particles about 20 nm in size aggregating during the rapid quenching process.
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.
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 Geometry of Architecture
NASA Astrophysics Data System (ADS)
Lorenz, Wolfgang E.
In Fractals smaller parts and the whole are linked together. Fractals are self-similar, as those parts are, at least approximately, scaled-down copies of the rough whole. In architecture, such a concept has also been known for a long time. Not only architects of the twentieth century called for an overall idea that is mirrored in every single detail, but also Gothic cathedrals and Indian temples offer self-similarity. This study mainly focuses upon the question whether this concept of self-similarity makes architecture with fractal properties more diverse and interesting than Euclidean Modern architecture. The first part gives an introduction and explains Fractal properties in various natural and architectural objects, presenting the underlying structure by computer programmed renderings. In this connection, differences between the fractal, architectural concept and true, mathematical Fractals are worked out to become aware of limits. This is the basis for dealing with the problem whether fractal-like architecture, particularly facades, can be measured so that different designs can be compared with each other under the aspect of fractal properties. Finally the usability of the Box-Counting Method, an easy-to-use measurement method of Fractal Dimension is analyzed with regard to architecture.
Multilayer adsorption on fractal surfaces.
Vajda, Péter; Felinger, Attila
2014-01-10
Multilayer adsorption is often observed in liquid chromatography. The most frequently employed model for multilayer adsorption is the BET isotherm equation. In this study we introduce an interpretation of multilayer adsorption measured on liquid chromatographic stationary phases based on the fractal theory. The fractal BET isotherm model was successfully used to determine the apparent fractal dimension of the adsorbent surface. The nonlinear fitting of the fractal BET equation gives us the estimation of the adsorption equilibrium constants and the monolayer saturation capacity of the adsorbent as well. In our experiments, aniline and proline were used as test molecules on reversed phase and normal phase columns, respectively. Our results suggest an apparent fractal dimension 2.88-2.99 in the case of reversed phase adsorbents, in the contrast with a bare silica column with a fractal dimension of 2.54.
NASA Astrophysics Data System (ADS)
Lei, L.; McCormick, M. P.; Su, J.
2015-12-01
Detection of the PBL heights and the PBL structure is very important for understanding the dynamic of the PBL since heat, water vapor and pollutions which come from the surface must transport through the PBL before they can affect the upper atmosphere. Fractal dimension (FD) retrieved from the three wavelengths lidar signals and the range- corrected signal (RCS) of 1064nm were used to analyses the PBL height and structure in Hampton University (HU, 37.02° N, 76.33° W). And the result shows that the new method has the potential to determine the top of different layer at same time. Combination of the FD and RCS signal also can be used to derive the structure of the PBL. Also the PBL evolution and the long time variety of the PBL in Hampton were analyzed. Wavelet covariance transform (WCT) was used to objectively determine the top and structure of the PBL from the FD signal and RCS signal.
NASA Astrophysics Data System (ADS)
Raupov, Dmitry S.; Myakinin, Oleg O.; Bratchenko, Ivan A.; Zakharov, Valery P.; Khramov, Alexander G.
2016-10-01
In this paper, we propose a report about our examining of the validity of OCT in identifying changes using a skin cancer texture analysis compiled from Haralick texture features, fractal dimension, Markov random field method and the complex directional features from different tissues. Described features have been used to detect specific spatial characteristics, which can differentiate healthy tissue from diverse skin cancers in cross-section OCT images (B- and/or C-scans). In this work, we used an interval type-II fuzzy anisotropic diffusion algorithm for speckle noise reduction in OCT images. The Haralick texture features as contrast, correlation, energy, and homogeneity have been calculated in various directions. A box-counting method is performed to evaluate fractal dimension of skin probes. Markov random field have been used for the quality enhancing of the classifying. Additionally, we used the complex directional field calculated by the local gradient methodology to increase of the assessment quality of the diagnosis method. Our results demonstrate that these texture features may present helpful information to discriminate tumor from healthy tissue. The experimental data set contains 488 OCT-images with normal skin and tumors as Basal Cell Carcinoma (BCC), Malignant Melanoma (MM) and Nevus. All images were acquired from our laboratory SD-OCT setup based on broadband light source, delivering an output power of 20 mW at the central wavelength of 840 nm with a bandwidth of 25 nm. We obtained sensitivity about 97% and specificity about 73% for a task of discrimination between MM and Nevus.
Structure and kinetics of shear aggregation in turbulent flows. I. Early stage of aggregation.
Bäbler, Matthäus U; Moussa, Amgad S; Soos, Miroslav; Morbidelli, Massimo
2010-08-17
Aggregation of rigid colloidal particles leads to fractal-like structures that are characterized by a fractal dimension d(f) which is a key parameter for describing aggregation processes. This is particularly true in shear aggregation where d(f) strongly influences aggregation kinetics. Direct measurement of d(f) in the early stages of shear aggregation is however difficult, as the aggregates are small and few in number. An alternative method for determining d(f) is to use an aggregation model that when fitted to the time evolution of the cluster mass distribution allows for estimating d(f). Here, we explore three such models, two of which are based on an effective collision sphere and one which directly incorporates the permeable structure of the aggregates, and we apply them for interpreting the initial aggregate growth measured experimentally in a turbulent stirred tank reactor. For the latter, three polystyrene latexes were used that differed only in the size of the primary particles (d(p) = 420, 600, and 810 nm). It was found that all three models describe initial aggregation kinetics reasonably well using, however, substantially different values for d(f). To discriminate among the models, we therefore also studied the regrowth of preformed aggregates where d(f) was experimentally accessible. It was found that only the model that directly incorporates the permeable structure of the aggregates is able to predict correctly this second type of experiments. Applying this model to the initial aggregation kinetics, we conclude that the actual initial fractal dimension is d(f) = 2.07 +/- 0.04 as found from this model.
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.}
NASA Astrophysics Data System (ADS)
Mezzenga, Raffaele; Fischer, Peter
2013-04-01
The aggregation of proteins is of fundamental relevance in a number of daily phenomena, as important and diverse as blood coagulation, medical diseases, or cooking an egg in the kitchen. Colloidal food systems, in particular, are examples that have great significance for protein aggregation, not only for their importance and implications, which touches on everyday life, but also because they allow the limits of the colloidal science analogy to be tested in a much broader window of conditions, such as pH, ionic strength, concentration and temperature. Thus, studying the aggregation and self-assembly of proteins in foods challenges our understanding of these complex systems from both the molecular and statistical physics perspectives. Last but not least, food offers a unique playground to study the aggregation of proteins in three, two and one dimensions, that is to say, in the bulk, at air/water and oil/water interfaces and in protein fibrillation phenomena. In this review we will tackle this very ambitious task in order to discuss the current understanding of protein aggregation in the framework of foods, which is possibly one of the broadest contexts, yet is of tremendous daily relevance.
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.
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}
Fractal Physiology and the Fractional Calculus: A Perspective
West, Bruce J.
2010-01-01
This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a
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.
Turbulent wakes of fractal objects.
Staicu, Adrian; Mazzi, Biagio; Vassilicos, J C; van de Water, Willem
2003-06-01
Turbulence of a windtunnel flow is stirred using objects that have a fractal structure. The strong turbulent wakes resulting from three such objects which have different fractal dimensions are probed using multiprobe hot-wire anemometry in various configurations. Statistical turbulent quantities are studied within inertial and dissipative range scales in an attempt to relate changes in their self-similar behavior to the scaling of the fractal objects.
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
Multi-scale interactions in Dictyostelium discoideum aggregation
NASA Astrophysics Data System (ADS)
Dixon, James A.; Kelty-Stephen, Damian G.
2012-12-01
Cellular aggregation is essential for a wide range of phenomena in developmental biology, and a crucial event in the life-cycle of Dictyostelium discoideum. The current manuscript presents an analysis of multi-scale interactions involved in D. discoideum aggregation and non-aggregation events. The multi-scale fractal dimensions of a sequence of microscope images were used to estimate changing structure at different spatial scales. Three regions showing aggregation and three showing non-aggregation were considered. The results showed that both aggregation and non-aggregation regions were strongly multi-fractal. Analyses of the over-time relationships among nine scales of the generalized dimension, D(q), were conducted using vector autoregression and vector error-correction models. Both types of regions showed evidence that across-scale interactions serve to maintain the equilibrium of the system. Aggregation and non-aggregation regions also showed different patterns of effects of individual scales on other scales. Specifically, aggregation regions showed greater effects of both the smallest and largest scales on the smaller scale structures. The results suggest that multi-scale interactions are responsible for maintaining and altering the cellular structures during aggregation.
Characterization and modeling of thermal diffusion and aggregation in nanofluids.
Gharagozloo, Patricia E.; Goodson, Kenneth E.
2010-05-01
Fluids with higher thermal conductivities are sought for fluidic cooling systems in applications including microprocessors and high-power lasers. By adding high thermal conductivity nanoscale metal and metal oxide particles to a fluid the thermal conductivity of the fluid is enhanced. While particle aggregates play a central role in recent models for the thermal conductivity of nanofluids, the effect of particle diffusion in a temperature field on the aggregation and transport has yet to be studied in depth. The present work separates the effects of particle aggregation and diffusion using parallel plate experiments, infrared microscopy, light scattering, Monte Carlo simulations, and rate equations for particle and heat transport in a well dispersed nanofluid. Experimental data show non-uniform temporal increases in thermal conductivity above effective medium theory and can be well described through simulation of the combination of particle aggregation and diffusion. The simulation shows large concentration distributions due to thermal diffusion causing variations in aggregation, thermal conductivity and viscosity. Static light scattering shows aggregates form more quickly at higher concentrations and temperatures, which explains the increased enhancement with temperature reported by other research groups. The permanent aggregates in the nanofluid are found to have a fractal dimension of 2.4 and the aggregate formations that grow over time are found to have a fractal dimension of 1.8, which is consistent with diffusion limited aggregation. Calculations show as aggregates grow the viscosity increases at a faster rate than thermal conductivity making the highly aggregated nanofluids unfavorable, especially at the low fractal dimension of 1.8. An optimum nanoparticle diameter for these particular fluid properties is calculated to be 130 nm to optimize the fluid stability by reducing settling, thermal diffusion and aggregation.
NASA Astrophysics Data System (ADS)
Olivier, B. J.; Sorensen, C. M.
1990-02-01
Dynamic light scattering is used to study the dependence of the aggregation kernel homogeneity λ on the aggregant concentration [HCl] for aqueous gold sols. We find the cluster growth kinetics are well described by a powerlaw, Rapp~tz/D, where Rapp is the measured apparent radius, D the cluster fractal dimension, and z=1/(1-λ) for all aggregant concentrations. The values for the dynamic exponent z, and hence the homogeneity λ, are functions of HCl concentration. We find the larger HCl concentrations yield a fast-aggregation regime characterized by λ~=-0.6. Smaller HCl concentrations yield a continuum of aggregation regimes characterized by homogeneities evolving from λ~=-0.6 towards 1.0. Our results do not support the view that aggregation in gold colloids is based on two limiting regimes, diffusion-limited and reaction-limited aggregation.
Applications of Fractal Analytical Techniques in the Estimation of Operational Scale
NASA Technical Reports Server (NTRS)
Emerson, Charles W.; Quattrochi, Dale A.
2000-01-01
resolutions to represent the varying form of a phenomenon as the pixel size is increased in a convolution process. We have observed that for images of homogeneous land covers, the fractal dimension varies linearly with changes in resolution or pixel size over the range of past, current, and planned space-borne sensors. This relationship differs significantly in images of agricultural, urban, and forest land covers, with urban areas retaining the same level of complexity, forested areas growing smoother, and agricultural areas growing more complex as small pixels are aggregated into larger, mixed pixels. Images of scenes having a mixture of land covers have fractal dimensions that exhibit a non-linear, complex relationship to pixel size. Measuring the fractal dimension of a difference image derived from two images of the same area obtained on different dates showed that the fractal dimension increased steadily, then exhibited a sharp decrease at increasing levels of pixel aggregation. This breakpoint of the fractal dimension/resolution plot is related to the spatial domain or operational scale of the phenomenon exhibiting the predominant visible difference between the two images (in this case, mountain snow cover). The degree to which an image departs from a theoretical ideal fractal surface provides clues as to how much information is altered or lost in the processes of rescaling and rectification. The measured fractal dimension of complex, composite land covers such as urban areas also provides a useful textural index that can assist image classification of complex scenes.
Population balance modeling of antibodies aggregation kinetics.
Arosio, Paolo; Rima, Simonetta; Lattuada, Marco; Morbidelli, Massimo
2012-06-21
The aggregates morphology and the aggregation kinetics of a model monoclonal antibody under acidic conditions have been investigated. Growth occurs via irreversible cluster-cluster coagulation forming compact, fractal aggregates with fractal dimension of 2.6. We measured the time evolution of the average radius of gyration,
Numerical Simulations of Single and Multiple Scattering by Fractal Ice Clusters
NASA Technical Reports Server (NTRS)
Dlugach, Janna M.; Mishchenko, Michael I.; Mackowski, Daniel W.
2011-01-01
We consider the scattering model in the form of a vertically and horizontally homogeneous particulate slab of an arbitrary optical thickness composed of widely separated fractal aggregates built of small spherical ice monomers. The aggregates are generated by applying three different approaches, including simulated cluster-cluster aggregation (CCA) and diffusion-limited aggregation (DLA) procedures. Having in mind radar remote-sensing applications, we report and analyze the results of computations of the backscattering circular polarization ratio obtained using efficient superposition T-matrix and vector radiative-transfer codes. The computations have been performed at a wavelength of 12.6 cm for fractal aggregates with the following characteristics: monomer refractive index m=1.78+i0.003, monomer radius r=1 cm, monomer packing density p=0.2, overall aggregate radii R in the range 4<=R<=10 cm and fractal dimensions D(sub f) 2.5 and 3. We show that for aggregates generated with simulated CCA and DLA procedures, the respective values of the backscattering circular polarization ratio differ weakly for D(sub f) 2.5, but the differences can increase somewhat for D(sub f)3, especially in case of an optically semi-infinite medium. For aggregates with a spheroidal overall shape, the dependence of the circular polarization ratio on the cluster morphology can be quite significant and increases with increasing the aspect ratio of the circumscribing spheroid.
A single parameter description of aggregate morphologies in two-dimensions
NASA Astrophysics Data System (ADS)
Das, Tamoghna; Bandi, Mahesh
2015-03-01
A morphological hierarchy of two-dimensional aggregates has been studied using molecular dynamics. Particulate aggregates resulting from the competition between short-range attraction and long-range repulsion show a transition from non-compact to compact to percolated `gel' structures as the competition varies at a constant temperature and density. A three-dimensional (3D) parameter space controlling the competition is mapped to a single dimensionless parameter Λ. A unique relation found between the reduced second virial coefficient B2*, computed for a large set of points in the 3D parameter space, and Λ provides strong support for the proposed description. The observed morphologies were further quantified using an entropic measure S2 of positional information. A simple scaling relation between S2 and Λ shows the promise of describing the static structures of aggregates in terms of geometry alone. This work was supported by the OIST Graduate University with subsidy funding from the Cabinet Office, Government of Japan.
Molecular Dynamics Simulation Of The Energetic Reaction Between Ni And Al Nanoparticle Aggregates
NASA Astrophysics Data System (ADS)
Sparks, Jacob; Hawa, Takumi
2013-03-01
Molecular Dynamics simulations are used to simulate the energetic reaction of Ni and Al particles at the nanometer scale. The effect of particle size and structures on reaction time and temperature for Ni and Al separate nanoparticles has been considered. The differences in melting temperature and phase change behavior between Al and Ni are expected to produce differing results for the nanoparticle aggregates systems. Simulation results show that the sintering time increases with increasing mass of the aggregates and with decreasing the fractal dimension of the aggregate. The final temperature of the systems increases with decreasing the primary particle sizes when mass of the aggregates remains unchanged. The phenomenological model is a power law including a dependence on the number of particles in an aggregates and fractal dimension is also developed.
Young, G C; Dey, S; Rogers, A D; Exton, D
2017-01-01
We present a method to construct and analyse 3D models of underwater scenes using a single cost-effective camera on a standard laptop with (a) free or low-cost software, (b) no computer programming ability, and (c) minimal man hours for both filming and analysis. This study focuses on four key structural complexity metrics: point-to-point distances, linear rugosity (R), fractal dimension (D), and vector dispersion (1/k). We present the first assessment of accuracy and precision of structure-from-motion (SfM) 3D models from an uncalibrated GoPro™ camera at a small scale (4 m2) and show that they can provide meaningful, ecologically relevant results. Models had root mean square errors of 1.48 cm in X-Y and 1.35 in Z, and accuracies of 86.8% (R), 99.6% (D at scales 30-60 cm), 93.6% (D at scales 1-5 cm), and 86.9 (1/k). Values of R were compared to in-situ chain-and-tape measurements, while values of D and 1/k were compared with ground truths from 3D printed objects modelled underwater. All metrics varied less than 3% between independently rendered models. We thereby improve and rigorously validate a tool for ecologists to non-invasively quantify coral reef structural complexity with a variety of multi-scale metrics.
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
In situ observation of fractal growth during a-Si crystallization in a Cu3Si matrix
NASA Astrophysics Data System (ADS)
Russell, S. W.; Li, Jian; Mayer, J. W.
1991-11-01
We observe that the crystallization of amorphous Si thin films in contact with a copper silicide layer occurs at a temperature of around 485 °C in the form of dendrites with a fractal dimension of 1.7. The in situ observation of both the silicidation reaction, forming Cu3Si, and the subsequent crystallization of the remaining amorphous silicon in the silicide matrix, were observed during annealing in a transmission electron microscope. We estimate the radial growth rate of these crystallites at 5 nm/s at this temperature. The fractal dimension of the dendrites indicates a growth process similar to one known as diffusion-limited aggregation.
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
Estimation of Surface Soil Moisture Using Fractal
NASA Astrophysics Data System (ADS)
Chen, Yen Chang; He, Chun Hsuan
2016-04-01
This study establishes the relationship between surface soil moisture and fractal dimension. The surface soil moisture is one of important factors in the hydrological cycle of surface evaporation. It could be used in many fields, such as reservoir management, early drought warning systems, irrigation scheduling and management, and crop yield estimations. Soil surface cracks due to dryness can be used to describe drought conditions. Soil cracking phenomenon and moisture have a certain relationship, thus this study makes used the fractal theory to interpret the soil moisture represented by soil cracks. The fractal dimension of surface soil cracking is a measure of the surface soil moisture. Therefore fractal dimensions can also be used to indicate how dry of the surface soil is. This study used the sediment in the Shimen Reservoir to establish the fractal dimension and soil moisture relation. The soil cracking is created under the control of temperature and thickness of surface soil layers. The results show the increase in fractal dimensions is accompanied by a decreases in surface soil moisture. However the fractal dimensions will approach a constant even the soil moisture continually decreases. The sigmoid function is used to fit the relation of fractal dimensions and surface soil moistures. The proposed method can be successfully applied to estimate surface soil moisture. Only a photo taken from the field is needed and is sufficient to provide the fractal dimension. Consequently, the surface soil moisture can be estimated quickly and accurately.
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.
Robustness of the fractal regime for the multiple-scattering structure factor
NASA Astrophysics Data System (ADS)
Katyal, Nisha; Botet, Robert; Puri, Sanjay
2016-08-01
In the single-scattering theory of electromagnetic radiation, the fractal regime is a definite range in the photon momentum-transfer q, which is characterized by the scaling-law behavior of the structure factor: S(q) ∝ 1 /q df. This allows a straightforward estimation of the fractal dimension df of aggregates in Small-Angle X-ray Scattering (SAXS) experiments. However, this behavior is not commonly studied in optical scattering experiments because of the lack of information on its domain of validity. In the present work, we propose a definition of the multiple-scattering structure factor, which naturally generalizes the single-scattering function S(q). We show that the mean-field theory of electromagnetic scattering provides an explicit condition to interpret the significance of multiple scattering. In this paper, we investigate and discuss electromagnetic scattering by three classes of fractal aggregates. The results obtained from the TMatrix method show that the fractal scaling range is divided into two domains: (1) a genuine fractal regime, which is robust; (2) a possible anomalous scaling regime, S(q) ∝ 1 /qδ, with exponent δ independent of df, and related to the way the scattering mechanism uses the local morphology of the scatterer. The recognition, and an analysis, of the latter domain is of importance because it may result in significant reduction of the fractal regime, and brings into question the proper mechanism in the build-up of multiple-scattering.
Fractal applications to complex crustal problems
NASA Technical Reports Server (NTRS)
Turcotte, Donald L.
1989-01-01
Complex scale-invariant problems obey fractal statistics. The basic definition of a fractal distribution is that the number of objects with a characteristic linear dimension greater than r satisfies the relation N = about r exp -D where D is the fractal dimension. Fragmentation often satisfies this relation. The distribution of earthquakes satisfies this relation. The classic relationship between the length of a rocky coast line and the step length can be derived from this relation. Power law relations for spectra can also be related to fractal dimensions. Topography and gravity are examples. Spectral techniques can be used to obtain maps of fractal dimension and roughness amplitude. These provide a quantitative measure of texture analysis. It is argued that the distribution of stress and strength in a complex crustal region, such as the Alps, is fractal. Based on this assumption, the observed frequency-magnitude relation for the seismicity in the region can be derived.
NASA Astrophysics Data System (ADS)
Turcotte, Donald L.
Tectonic processes build landforms that are subsequently destroyed by erosional processes. Landforms exhibit fractal statistics in a variety of ways; examples include (1) lengths of coast lines; (2) number-size statistics of lakes and islands; (3) spectral behavior of topography and bathymetry both globally and locally; and (4) branching statistics of drainage networks. Erosional processes are dominant in the development of many landforms on this planet, but similar fractal statistics are also applicable to the surface of Venus where minimal erosion has occurred. A number of dynamical systems models for landforms have been proposed, including (1) cellular automata; (2) diffusion limited aggregation; (3) self-avoiding percolation; and (4) advective-diffusion equations. The fractal statistics and validity of these models will be discussed. Earthquakes also exhibit fractal statistics. The frequency-magnitude statistics of earthquakes satisfy the fractal Gutenberg-Richter relation both globally and locally. Earthquakes are believed to be a classic example of self-organized criticality. One model for earthquakes utilizes interacting slider-blocks. These slider block models have been shown to behave chaotically and to exhibit self-organized criticality. The applicability of these models will be discussed and alternative approaches will be presented. Fragmentation has been demonstrated to produce fractal statistics in many cases. Comminution is one model for fragmentation that yields fractal statistics. It has been proposed that comminution is also responsible for much of the deformation in the earth's crust. The brittle disruption of the crust and the resulting earthquakes present an integrated problem with many fractal aspects.
Order-fractal transitions in abstract paintings
Calleja, E.M. de la; Cervantes, F.; Calleja, J. de la
2016-08-15
In this study, we determined the degree of order for 22 Jackson Pollock paintings using the Hausdorff–Besicovitch fractal dimension. Based on the maximum value of each multi-fractal spectrum, the artworks were classified according to the year in which they were painted. It has been reported that Pollock’s paintings are fractal and that this feature was more evident in his later works. However, our results show that the fractal dimension of these paintings ranges among values close to two. We characterize this behavior as a fractal-order transition. Based on the study of disorder-order transition in physical systems, we interpreted the fractal-order transition via the dark paint strokes in Pollock’s paintings as structured lines that follow a power law measured by the fractal dimension. We determined self-similarity in specific paintings, thereby demonstrating an important dependence on the scale of observations. We also characterized the fractal spectrum for the painting entitled Teri’s Find. We obtained similar spectra for Teri’s Find and Number 5, thereby suggesting that the fractal dimension cannot be rejected completely as a quantitative parameter for authenticating these artworks. -- Highlights: •We determined the degree of order in Jackson Pollock paintings using the Hausdorff–Besicovitch dimension. •We detected a fractal-order transition from Pollock’s paintings between 1947 and 1951. •We suggest that Jackson Pollock could have painted Teri’s Find.
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.…
Fractal scattering of microwaves from soils.
Oleschko, K; Korvin, G; Balankin, A S; Khachaturov, R V; Flores, L; Figueroa, B; Urrutia, J; Brambila, F
2002-10-28
Using a combination of laboratory experiments and computer simulation we show that microwaves reflected from and transmitted through soil have a fractal dimension correlated to that of the soil's hierarchic permittivity network. The mathematical model relating the ground-penetrating radar record to the mass fractal dimension of soil structure is also developed. The fractal signature of the scattered microwaves correlates well with some physical and mechanical properties of soils.
NASA Astrophysics Data System (ADS)
Marks-Tarlow, Terry
Linear concepts of time plus the modern capacity to track history emerged out of circular conceptions characteristic of ancient and traditional cultures. A fractal concept of time lies implicitly within the analog clock, where each moment is treated as unique. With fractal geometry the best descriptor of nature, qualities of self-similarity and scale invariance easily model her endless variety and recursive patterning, both in time and across space. To better manage temporal aspects of our lives, a fractal concept of time is non-reductive, based more on the fullness of being than on fragments of doing. By using a fractal concept of time, each activity or dimension of life is multiply and vertically nested. Each nested cycle remains simultaneously present, operating according to intrinsic dynamics and time scales. By adding the vertical axis of simultaneity to the horizontal axis of length, time is already full and never needs to be filled. To attend to time's vertical dimension is to tap into the imaginary potential for infinite depth. To switch from linear to fractal time allows us to relax into each moment while keeping in mind the whole.
Segmentation of histological structures for fractal analysis
NASA Astrophysics Data System (ADS)
Dixon, Vanessa; Kouznetsov, Alexei; Tambasco, Mauro
2009-02-01
Pathologists examine histology sections to make diagnostic and prognostic assessments regarding cancer based on deviations in cellular and/or glandular structures. However, these assessments are subjective and exhibit some degree of observer variability. Recent studies have shown that fractal dimension (a quantitative measure of structural complexity) has proven useful for characterizing structural deviations and exhibits great potential for automated cancer diagnosis and prognosis. Computing fractal dimension relies on accurate image segmentation to capture the architectural complexity of the histology specimen. For this purpose, previous studies have used techniques such as intensity histogram analysis and edge detection algorithms. However, care must be taken when segmenting pathologically relevant structures since improper edge detection can result in an inaccurate estimation of fractal dimension. In this study, we established a reliable method for segmenting edges from grayscale images. We used a Koch snowflake, an object of known fractal dimension, to investigate the accuracy of various edge detection algorithms and selected the most appropriate algorithm to extract the outline structures. Next, we created validation objects ranging in fractal dimension from 1.3 to 1.9 imitating the size, structural complexity, and spatial pixel intensity distribution of stained histology section images. We applied increasing intensity thresholds to the validation objects to extract the outline structures and observe the effects on the corresponding segmentation and fractal dimension. The intensity threshold yielding the maximum fractal dimension provided the most accurate fractal dimension and segmentation, indicating that this quantitative method could be used in an automated classification system for histology specimens.
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.
Fractal analysis of Xylella fastidiosa biofilm formation
NASA Astrophysics Data System (ADS)
Moreau, A. L. D.; Lorite, G. S.; Rodrigues, C. M.; Souza, A. A.; Cotta, M. A.
2009-07-01
We have investigated the growth process of Xylella fastidiosa biofilms inoculated on a glass. The size and the distance between biofilms were analyzed by optical images; a fractal analysis was carried out using scaling concepts and atomic force microscopy images. We observed that different biofilms show similar fractal characteristics, although morphological variations can be identified for different biofilm stages. Two types of structural patterns are suggested from the observed fractal dimensions Df. In the initial and final stages of biofilm formation, Df is 2.73±0.06 and 2.68±0.06, respectively, while in the maturation stage, Df=2.57±0.08. These values suggest that the biofilm growth can be understood as an Eden model in the former case, while diffusion-limited aggregation (DLA) seems to dominate the maturation stage. Changes in the correlation length parallel to the surface were also observed; these results were correlated with the biofilm matrix formation, which can hinder nutrient diffusion and thus create conditions to drive DLA growth.
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.
Transient shear viscosity of weakly aggregating polystyrene latex dispersions
NASA Astrophysics Data System (ADS)
de Rooij, R.; Potanin, A. A.; van den Ende, D.; Mellema, J.
1994-04-01
The transient behavior of the viscosity (stress growth) of a weakly aggregating polystyrene latex dispersion after a step from a high shear rate to a lower shear rate has been measured and modeled. Single particles cluster together into spherical fractal aggregates. The steady state size of these aggregates is determined by the shear stresses exerted on the latter by the flow field. The restructuring process taking place when going from a starting situation with monodisperse spherical aggregates to larger monodisperse spherical aggregates is described by the capture of primary fractal aggregates by growing aggregates until a new steady state is reached. It is assumed that the aggregation mechanism is diffusion limited. The model is valid if the radii of primary aggregates Rprim are much smaller than the radii of the growing aggregates. Fitting the model to experimental data at two volume fractions and a number of step sizes in shear rate yielded physically reasonable values of Rprim at fractal dimensions 2.1≤df≤2.2. The latter range is in good agreement with the range 2.0≤df≤2.3 obtained from steady shear results. The experimental data have also been fitted to a numerical solution of the diffusion equation for primary aggregates for a cell model with moving boundary, also yielding 2.1≤df≤2.2. The range for df found from both approaches agrees well with the range df≊2.1-2.2 determined from computer simulations on diffusion-limited aggregation including restructuring or thermal breakup after formation of bonds. Thus a simple model has been put forward which may capture the basic features of the aggregating model dispersion on a microstructural level and leads to physically acceptable parameter values.
Aggregate Morphology Evolution by Sintering: Number & Diameter of Primary Particles
Eggersdorfer, Max L.; Kadau, Dirk; Herrmann, Hans J.; Pratsinis, Sotiris E.
2013-01-01
The structure of fractal-like agglomerates (physically-bonded) and aggregates (chemically- or sinter-bonded) is important in aerosol synthesis of nanoparticles, and in monitoring combustion emissions and atmospheric particles. It influences also particle mobility, scattering, and eventually performance of nanocomposites, suspensions and devices made with such particles. Here, aggregate sintering by viscous flow of amorphous materials (silica, polymers) and grain boundary diffusion of crystalline ceramics (titania, alumina) or metals (Ni, Fe, Ag etc.) is investigated. A scaling law is found between average aggregate projected area and equivalent number of constituent primary particles during sintering: from fractal-like agglomerates to aggregates and eventually compact particles (e.g. spheres). This is essentially a relation independent of time, material properties and sintering mechanisms. It is used to estimate the equivalent primary particle diameter and number in aggregates. The evolution of aggregate morphology or structure is quantified by the effective fractal dimension (Df) and mass-mobility exponent (Dfm) and the corresponding prefactors. The Dfm increases monotonically during sintering converging to 3 for a compact particle. Therefore Dfm and its prefactor could be used to gauge the degree or extent of sintering of agglomerates made by a known collision mechanism. This analysis is exemplified by comparison to experiments of silver nanoparticle aggregates sintered at different temperatures in an electric tube furnace. PMID:23658467
Aggregate Morphology Evolution by Sintering: Number & Diameter of Primary Particles.
Eggersdorfer, Max L; Kadau, Dirk; Herrmann, Hans J; Pratsinis, Sotiris E
2012-04-01
The structure of fractal-like agglomerates (physically-bonded) and aggregates (chemically- or sinter-bonded) is important in aerosol synthesis of nanoparticles, and in monitoring combustion emissions and atmospheric particles. It influences also particle mobility, scattering, and eventually performance of nanocomposites, suspensions and devices made with such particles. Here, aggregate sintering by viscous flow of amorphous materials (silica, polymers) and grain boundary diffusion of crystalline ceramics (titania, alumina) or metals (Ni, Fe, Ag etc.) is investigated. A scaling law is found between average aggregate projected area and equivalent number of constituent primary particles during sintering: from fractal-like agglomerates to aggregates and eventually compact particles (e.g. spheres). This is essentially a relation independent of time, material properties and sintering mechanisms. It is used to estimate the equivalent primary particle diameter and number in aggregates. The evolution of aggregate morphology or structure is quantified by the effective fractal dimension (Df ) and mass-mobility exponent (Dfm ) and the corresponding prefactors. The Dfm increases monotonically during sintering converging to 3 for a compact particle. Therefore Dfm and its prefactor could be used to gauge the degree or extent of sintering of agglomerates made by a known collision mechanism. This analysis is exemplified by comparison to experiments of silver nanoparticle aggregates sintered at different temperatures in an electric tube furnace.
Fractal Weyl law for Linux Kernel architecture
NASA Astrophysics Data System (ADS)
Ermann, L.; Chepelianskii, A. D.; Shepelyansky, D. L.
2011-01-01
We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be ν ≈ 0.65 that corresponds to the fractal dimension of the network d ≈ 1.3. An independent computation of the fractal dimension by the cluster growing method, generalized for directed networks, gives a close value d ≈ 1.4. The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension d < 2.
Aggregation kinetics and structure of cryoimmunoglobulins clusters
NASA Astrophysics Data System (ADS)
Spirito, M. De; Chiappini, R.; Bassi, F. Andreasi; Stasio, E. Di; Giardina, B.; Arcovito, G.
2002-02-01
Cryoimmunoglobulins are pathological antibodies characterized by a temperature-dependent reversible insolubility. Rheumatoid factors (RF) are immunoglobulins possessing anti-immunoglobulin activity and usually consist of an IgM antibody that recognizes IgG as antigen. These proteins are present in sera of patients affected by a large variety of different pathologies, such as HCV infection, neoplastic and autoimmune diseases. Aggregation and precipitation of cryoimmunoglobulins, leading to vasculiti, are physical phenomena behind such pathologies. A deep knowledge of the physico-chemical mechanisms regulating such phenomena plays a fundamental role in biological and clinical applications. In this work, a preliminary investigation of the aggregation kinetics and of the final macromolecular structure of the aggregates is presented. Through static light scattering techniques, the gyration radius Rg and the fractal dimension Dm of the growing clusters have been determined. However, while the initial aggregation mechanism could be described using the universal reaction-limited cluster-cluster aggregation (RLCCA) theory, at longest times from the beginning of the process, the RLCCA theory fails and a restructuring of clusters is observed together with an increase of the cluster fractal dimension Dm up to a value Dm∼3. The time tn, at which the restructuring takes place, and the final cluster size can be modulated by varying the quenching temperature.
Phosphorus recovery from wastewater by struvite crystallization: property of aggregates.
Ye, Zhilong; Shen, Yin; Ye, Xin; Zhang, Zhaoji; Chen, Shaohua; Shi, Jianwen
2014-05-01
Struvite crystallization is a promising method to remove and recover phosphorus from wastewater to ease both the scarcity of phosphorus rock resources and water eutrophication worldwide. To date, although various kinds of reactor systems have been developed, supporting methods are required to control the struvite fines flushing out of the reactors. As an intrinsic property, aggregation is normally disregarded in the struvite crystallization process, although it is the key factor in final particle size and therefore guarantees phosphorus recovery efficiency. The present study developed a method to analyze the characteristics of struvite aggregates using fractal geometry, and the influence of operational parameters on struvite aggregation was evaluated. Due to its typical orthorhombic molecular structure, struvite particles are prone to crystallize into needle or rod shapes, and aggregate at the corners or edges of crystals. The determined fractal dimension (Dpf) of struvite aggregates was 1.52-1.31, with the corresponding range of equivalent diameter (d0.5) at 295.9-85.4 μm. Aggregates formed in relatively low phosphorus concentrations (3.0-5.0 mmol/L) and mildly alkaline conditions (pH 9.0-9.5) displayed relatively compact structures, large aggregate sizes and high aggregation strength. Increasing pH values led to continuous decrease of aggregate sizes, while the variation of Dpf was insignificant. As to the aggregate evolution, fast growth in a short time followed by a long steady stage was observed.
NASA Astrophysics Data System (ADS)
Beech, M.
1989-02-01
The author discusses some of the more recent research on fractal astronomy and results presented in several astronomical studies. First, the large-scale structure of the universe is considered, while in another section one drops in scale to examine some of the smallest bodies in our solar system; the comets and meteoroids. The final section presents some thoughts on what influence the fractal ideology might have on astronomy, focusing particularly on the question recently raised by Kadanoff, "Fractals: where's the physics?"
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.
Generalized Mandelbrot rule for fractal sections
NASA Astrophysics Data System (ADS)
Meisel, L. V.
1992-01-01
Mandelbrot's rule for sections is generalized to apply to the Hentschel and Procaccia fractal dimension at arbitrary q and on arbitrary sections. It is shown that for almost all (n-m)-dimensional sections, Dn(q)=Dn-m(q)+m, where the Dr(q) are box-counting, Hentschel, and Procaccia generalized fractal dimensions of r-dimensional sections of homogeneous fractal point sets in Rn and Dn-m(q)>0. The rule applies for finite ``thickness'' sections as well as ``true'' sections and can be interpreted for inhomogenous fractal sets.
Fractal dynamics of earthquakes
Bak, P.; Chen, K.
1995-05-01
Many objects in nature, from mountain landscapes to electrical breakdown and turbulence, have a self-similar fractal spatial structure. It seems obvious that to understand the origin of self-similar structures, one must understand the nature of the dynamical processes that created them: temporal and spatial properties must necessarily be completely interwoven. This is particularly true for earthquakes, which have a variety of fractal aspects. The distribution of energy released during earthquakes is given by the Gutenberg-Richter power law. The distribution of epicenters appears to be fractal with dimension D {approx} 1--1.3. The number of after shocks decay as a function of time according to the Omori power law. There have been several attempts to explain the Gutenberg-Richter law by starting from a fractal distribution of faults or stresses. But this is a hen-and-egg approach: to explain the Gutenberg-Richter law, one assumes the existence of another power-law--the fractal distribution. The authors present results of a simple stick slip model of earthquakes, which evolves to a self-organized critical state. Emphasis is on demonstrating that empirical power laws for earthquakes indicate that the Earth`s crust is at the critical state, with no typical time, space, or energy scale. Of course the model is tremendously oversimplified; however in analogy with equilibrium phenomena they do not expect criticality to depend on details of the model (universality).
Morphological properties of atmospheric aerosol aggregates
Xiong, C.; Friedlander, S. K.
2001-01-01
Ultrafine particles (smaller than about 0.1 μm) are often emitted from combustion and other high-temperature processes in the form of fractal-like aggregates composed of solid nanoparticles. Results of a study of atmospheric aggregates are reported. Particles were collected on transmission electron microscope grids fitted on the last two stages of a single-jet eight-stage low-pressure impactor for periods of a few minutes. Photomicrographs of transmission electron microscope grids from the impactor stages were analyzed to obtain the fractal dimension (Df) and prefactor (A) for aggregates. Df increased from near 1 to above 2 as the number of primary particles making up the aggregates increased from 10 to 180. Total particle concentrations in size ranges roughly equivalent to the low-pressure impactor stages were measured with a mobility analyzer and condensation particle counter. In one set of measurements, the fraction of the particles present as aggregates was about 60% for particles with aerodynamic diameters between 50 and 75 nm and 34% for the range 75 to 120 nm. The total aggregate concentration in the 50- to 120-nm size range was about 400 ml−1. The primary particles that make up atmospheric aggregates are more polydisperse than soot aggregates generated from a single laboratory source, an ethane/oxygen flame. Most measurements were made in the Los Angeles area, where the aggregates may represent a signature for diesel emissions. Rural aggregate concentrations in the size range 50 to 120 nm were less than 1% of the concentrations at urban sites. The data will permit better estimates of atmospheric aggregate residence times, transport, and deposition in the lung, optical extinction, and heterogenous nucleation. PMID:11592995
Morphological properties of atmospheric aerosol aggregates.
Xiong, C; Friedlander, S K
2001-10-09
Ultrafine particles (smaller than about 0.1 microm) are often emitted from combustion and other high-temperature processes in the form of fractal-like aggregates composed of solid nanoparticles. Results of a study of atmospheric aggregates are reported. Particles were collected on transmission electron microscope grids fitted on the last two stages of a single-jet eight-stage low-pressure impactor for periods of a few minutes. Photomicrographs of transmission electron microscope grids from the impactor stages were analyzed to obtain the fractal dimension (D(f)) and prefactor (A) for aggregates. D(f) increased from near 1 to above 2 as the number of primary particles making up the aggregates increased from 10 to 180. Total particle concentrations in size ranges roughly equivalent to the low-pressure impactor stages were measured with a mobility analyzer and condensation particle counter. In one set of measurements, the fraction of the particles present as aggregates was about 60% for particles with aerodynamic diameters between 50 and 75 nm and 34% for the range 75 to 120 nm. The total aggregate concentration in the 50- to 120-nm size range was about 400 ml(-1). The primary particles that make up atmospheric aggregates are more polydisperse than soot aggregates generated from a single laboratory source, an ethane/oxygen flame. Most measurements were made in the Los Angeles area, where the aggregates may represent a signature for diesel emissions. Rural aggregate concentrations in the size range 50 to 120 nm were less than 1% of the concentrations at urban sites. The data will permit better estimates of atmospheric aggregate residence times, transport, and deposition in the lung, optical extinction, and heterogeneous nucleation.
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.
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.
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.…
Stadnitski, Tatjana
2012-01-01
WHEN INVESTIGATING FRACTAL PHENOMENA, THE FOLLOWING QUESTIONS ARE FUNDAMENTAL FOR THE APPLIED RESEARCHER: (1) What are essential statistical properties of 1/f noise? (2) Which estimators are available for measuring fractality? (3) Which measurement instruments are appropriate and how are they applied? The purpose of this article is to give clear and comprehensible answers to these questions. First, theoretical characteristics of a fractal pattern (self-similarity, long memory, power law) and the related fractal parameters (the Hurst coefficient, the scaling exponent α, the fractional differencing parameter d of the autoregressive fractionally integrated moving average methodology, the power exponent β of the spectral analysis) are discussed. Then, estimators of fractal parameters from different software packages commonly used by applied researchers (R, SAS, SPSS) are introduced and evaluated. Advantages, disadvantages, and constrains of the popular estimators ([Formula: see text] power spectral density, detrended fluctuation analysis, signal summation conversion) are illustrated by elaborate examples. Finally, crucial steps of fractal analysis (plotting time series data, autocorrelation, and spectral functions; performing stationarity tests; choosing an adequate estimator; estimating fractal parameters; distinguishing fractal processes from short-memory patterns) are demonstrated with empirical time series.
Stadnitski, Tatjana
2012-01-01
When investigating fractal phenomena, the following questions are fundamental for the applied researcher: (1) What are essential statistical properties of 1/f noise? (2) Which estimators are available for measuring fractality? (3) Which measurement instruments are appropriate and how are they applied? The purpose of this article is to give clear and comprehensible answers to these questions. First, theoretical characteristics of a fractal pattern (self-similarity, long memory, power law) and the related fractal parameters (the Hurst coefficient, the scaling exponent α, the fractional differencing parameter d of the autoregressive fractionally integrated moving average methodology, the power exponent β of the spectral analysis) are discussed. Then, estimators of fractal parameters from different software packages commonly used by applied researchers (R, SAS, SPSS) are introduced and evaluated. Advantages, disadvantages, and constrains of the popular estimators (d^ML, power spectral density, detrended fluctuation analysis, signal summation conversion) are illustrated by elaborate examples. Finally, crucial steps of fractal analysis (plotting time series data, autocorrelation, and spectral functions; performing stationarity tests; choosing an adequate estimator; estimating fractal parameters; distinguishing fractal processes from short-memory patterns) are demonstrated with empirical time series. PMID:22586408
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
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)
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.
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.
Aggregation in charged nanoparticles solutions induced by different interactions
NASA Astrophysics Data System (ADS)
Abbas, S.; Kumar, Sugam; Aswal, V. K.; Kohlbrecher, J.
2016-05-01
Small-angle neutron scattering (SANS) has been used to study the aggregation of anionic silica nanoparticles as induced through different interactions. The nanoparticle aggregation is induced by addition of salt (NaCl), cationic protein (lysozyme) and non-ionic surfactant (C12E10) employing different kind of interactions. The results show that the interaction in presence of salt can be explained using DLVO theory whereas non-DLVO forces play important role for interaction of nanoparticles with protein and surfactant. The presence of salt screens the repulsion between charged nanoparticles giving rise to a net attraction in the DLVO potential. On the other hand, strong electrostatic attraction between nanoparticle and oppositely charged protein leads to protein-mediated nanoparticle aggregation. In case of non-ionic surfactant, the relatively long-range attractive depletion interaction is found to be responsible for the particle aggregation. Interestingly, the completely different interactions lead to similar kind of aggregate morphology. The nanoparticle aggregates formed are found to have mass fractal nature having a fractal dimension (~2.5) consistent with diffusion limited type of fractal morphology in all three cases.
Specimen and aggregate size effect on the fracture behavior of concrete
NASA Astrophysics Data System (ADS)
Islam, Md. Shahidul
2000-12-01
Fracture in concrete has been a subject of extensive research for the past thirty years. Although much has been accomplished, concrete still remains the least known of the engineering materials with respect to its fracture behavior. The objective of this research was to study the influence of micro-structural parameters, such as aggregate size, and macroscopic parameters, such as specimen size, on the fracture toughness of concrete. In addition, the effect of notch-depth ratio on the fracture behavior of concrete was studied. Over 300 concrete specimens of various sizes were prepared with various maximum aggregate sizes. There is a well pronounced trend observed in all groups of specimens tested. Critical energy release rate of concrete in the presence of compressive force increases with crack length, i.e., exhibits a typical R-curve behavior. For fixed specimen size, toughness (energy release rate or fracture energy) increases with an increase in maximum aggregate size. Fracture surface interaction, i.e., interlocking, bridging, etc. is more pronounced in the specimens with larger maximum size aggregates. For specimens with fixed maximum aggregate size, toughness increases with an increase in specimen size. For geometrically similar specimens, i.e., the shape and all dimensionless parameters are the same, the toughness evaluated for larger specimens is noticeably higher than that for smaller specimens. Energy release rates for long-notch specimens were relatively lower than those for short-notch specimens. Fracture energy for short-notch specimen is higher than that for long notch specimen. Concrete surfaces exhibit fractal characteristics over the range of scales studied. Fractal dimension increases with an increase in maximum aggregate size and specimen size as well. The fractal dimension correlates very well with surface roughness. The rougher the surface, the higher the fractal dimension. A good correlation also exits between the fractal dimension and fracture
Simulation of red blood cell aggregation in shear flow.
Lim, B; Bascom, P A; Cobbold, R S
1997-01-01
A simulation model has been developed for red blood cell (RBC) aggregation in shear flow. It is based on a description of the collision rates of RBC, the probability of particles sticking together, and the breakage of aggregates by shear forces. The influence of shear rate, hematocrit, aggregate fractal dimension, and binding strength on aggregation kinetics were investigated and compared to other theoretical and experimental results. The model was used to simulate blood flow in a long large diameter tube under steady flow conditions at low Reynolds numbers. The time and spatial distribution of the state of aggregation are shown to be in qualitative agreement with previous B-mode ultrasound studies in which a central region of low echogenicity was noted. It is suggested that the model can provide a basis for interpreting prior measurements of ultrasound echogenicity and may help relate them to the local state of aggregation.
Shape characteristics of equilibrium and non-equilibrium fractal clusters.
Mansfield, Marc L; Douglas, Jack F
2013-07-28
It is often difficult in practice to discriminate between equilibrium and non-equilibrium nanoparticle or colloidal-particle clusters that form through aggregation in gas or solution phases. Scattering studies often permit the determination of an apparent fractal dimension, but both equilibrium and non-equilibrium clusters in three dimensions frequently have fractal dimensions near 2, so that it is often not possible to discriminate on the basis of this geometrical property. A survey of the anisotropy of a wide variety of polymeric structures (linear and ring random and self-avoiding random walks, percolation clusters, lattice animals, diffusion-limited aggregates, and Eden clusters) based on the principal components of both the radius of gyration and electric polarizability tensor indicates, perhaps counter-intuitively, that self-similar equilibrium clusters tend to be intrinsically anisotropic at all sizes, while non-equilibrium processes such as diffusion-limited aggregation or Eden growth tend to be isotropic in the large-mass limit, providing a potential means of discriminating these clusters experimentally if anisotropy could be determined along with the fractal dimension. Equilibrium polymer structures, such as flexible polymer chains, are normally self-similar due to the existence of only a single relevant length scale, and are thus anisotropic at all length scales, while non-equilibrium polymer structures that grow irreversibly in time eventually become isotropic if there is no difference in the average growth rates in different directions. There is apparently no proof of these general trends and little theoretical insight into what controls the universal anisotropy in equilibrium polymer structures of various kinds. This is an obvious topic of theoretical investigation, as well as a matter of practical interest. To address this general problem, we consider two experimentally accessible ratios, one between the hydrodynamic and gyration radii, the other
A Fractal Nature for Polymerized Laminin
Hochman-Mendez, Camila; Cantini, Marco; Moratal, David; Salmeron-Sanchez, Manuel; Coelho-Sampaio, Tatiana
2014-01-01
Polylaminin (polyLM) is a non-covalent acid-induced nano- and micro-structured polymer of the protein laminin displaying distinguished biological properties. Polylaminin stimulates neuritogenesis beyond the levels achieved by ordinary laminin and has been shown to promote axonal regeneration in animal models of spinal cord injury. Here we used confocal fluorescence microscopy (CFM), scanning electron microscopy (SEM) and atomic force microscopy (AFM) to characterize its three-dimensional structure. Renderization of confocal optical slices of immunostained polyLM revealed the aspect of a loose flocculated meshwork, which was homogeneously stained by the antibody. On the other hand, an ordinary matrix obtained upon adsorption of laminin in neutral pH (LM) was constituted of bulky protein aggregates whose interior was not accessible to the same anti-laminin antibody. SEM and AFM analyses revealed that the seed unit of polyLM was a flat polygon formed in solution whereas the seed structure of LM was highly heterogeneous, intercalating rod-like, spherical and thin spread lamellar deposits. As polyLM was visualized at progressively increasing magnifications, we observed that the morphology of the polymer was alike independently of the magnification used for the observation. A search for the Hausdorff dimension in images of the two matrices showed that polyLM, but not LM, presented fractal dimensions of 1.55, 1.62 and 1.70 after 1, 8 and 12 hours of adsorption, respectively. Data in the present work suggest that the intrinsic fractal nature of polymerized laminin can be the structural basis for the fractal-like organization of basement membranes in the neurogenic niches of the central nervous system. PMID:25296244
Quantitative evaluation of midpalatal suture maturation via fractal analysis
Kwak, Kyoung Ho; Kim, Yong-Il; Kim, Yong-Deok
2016-01-01
Objective The purpose of this study was to determine whether the results of fractal analysis can be used as criteria for midpalatal suture maturation evaluation. Methods The study included 131 subjects aged over 18 years of age (range 18.1–53.4 years) who underwent cone-beam computed tomography. Skeletonized images of the midpalatal suture were obtained via image processing software and used to calculate fractal dimensions. Correlations between maturation stage and fractal dimensions were calculated using Spearman's correlation coefficient. Optimal fractal dimension cut-off values were determined using a receiver operating characteristic curve. Results The distribution of maturation stages of the midpalatal suture according to the cervical vertebrae maturation index was highly variable, and there was a strong negative correlation between maturation stage and fractal dimension (−0.623, p < 0.001). Fractal dimension was a statistically significant indicator of dichotomous results with regard to maturation stage (area under curve = 0.794, p < 0.001). A test in which fractal dimension was used to predict the resulting variable that splits maturation stages into ABC and D or E yielded an optimal fractal dimension cut-off value of 1.0235. Conclusions There was a strong negative correlation between fractal dimension and midpalatal suture maturation. Fractal analysis is an objective quantitative method, and therefore we suggest that it may be useful for the evaluation of midpalatal suture maturation. PMID:27668195
Comparison of two fractal interpolation methods
NASA Astrophysics Data System (ADS)
Fu, Yang; Zheng, Zeyu; Xiao, Rui; Shi, Haibo
2017-03-01
As a tool for studying complex shapes and structures in nature, fractal theory plays a critical role in revealing the organizational structure of the complex phenomenon. Numerous fractal interpolation methods have been proposed over the past few decades, but they differ substantially in the form features and statistical properties. In this study, we simulated one- and two-dimensional fractal surfaces by using the midpoint displacement method and the Weierstrass-Mandelbrot fractal function method, and observed great differences between the two methods in the statistical characteristics and autocorrelation features. From the aspect of form features, the simulations of the midpoint displacement method showed a relatively flat surface which appears to have peaks with different height as the fractal dimension increases. While the simulations of the Weierstrass-Mandelbrot fractal function method showed a rough surface which appears to have dense and highly similar peaks as the fractal dimension increases. From the aspect of statistical properties, the peak heights from the Weierstrass-Mandelbrot simulations are greater than those of the middle point displacement method with the same fractal dimension, and the variances are approximately two times larger. When the fractal dimension equals to 1.2, 1.4, 1.6, and 1.8, the skewness is positive with the midpoint displacement method and the peaks are all convex, but for the Weierstrass-Mandelbrot fractal function method the skewness is both positive and negative with values fluctuating in the vicinity of zero. The kurtosis is less than one with the midpoint displacement method, and generally less than that of the Weierstrass-Mandelbrot fractal function method. The autocorrelation analysis indicated that the simulation of the midpoint displacement method is not periodic with prominent randomness, which is suitable for simulating aperiodic surface. While the simulation of the Weierstrass-Mandelbrot fractal function method has
Robinson,, Gilpin R.; Brown, William M.
2002-01-01
The United States uses large quantities of natural aggregate to build and maintain a continuously expanding infrastructure. In recent years, per capita demand for aggregate in the United States has grown to about 9.7 metric tons (10.7 tons) per person per year. Over the next 25 years, the aggregate industry expects to mine quantities equivalent to all aggregate mined in the United States over the past 100 years. The issues surrounding supply and demand for aggregate in the mid-Atlantic states of Maryland, Pennsylvania, Virginia, and West Virginia illustrate competing requirements for industrial minerals and many simultaneous social and environmental objectives.
Lagrangian and Hamiltonian Mechanics on Fractals Subset of Real-Line
NASA Astrophysics Data System (ADS)
Golmankhaneh, Alireza Khalili; Golmankhaneh, Ali Khalili; Baleanu, Dumitru
2013-11-01
A discontinuous media can be described by fractal dimensions. Fractal objects has special geometric properties, which are discrete and discontinuous structure. A fractal-time diffusion equation is a model for subdiffusive. In this work, we have generalized the Hamiltonian and Lagrangian dynamics on fractal using the fractional local derivative, so one can use as a new mathematical model for the motion in the fractal media. More, Poisson bracket on fractal subset of real line is suggested.
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.
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.; W. Vannier, Michael; Salgia, Ravi
2016-01-01
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)
Pan, Xuezai; Shang, Xudong; Wang, Minggang; Zuo-Fei
With the purpose of researching the changing regularities of the Cantor set’s multi-fractal spectrums and generalized fractal dimensions under different probability factors, from statistical physics, the Cantor set is given a mass distribution, when the mass is given with different probability ratios, the different multi-fractal spectrums and the generalized fractal dimensions will be acquired by computer calculation. The following conclusions can be acquired. On one hand, the maximal width of the multi-fractal spectrum and the maximal vertical height of the generalized fractal dimension will become more and more narrow with getting two probability factors closer and closer. On the other hand, when two probability factors are equal to 1/2, both the multi-fractal spectrum and the generalized fractal dimension focus on the value 0.6309, which is not the value of the physical multi-fractal spectrum and the generalized fractal dimension but the mathematical Hausdorff dimension.
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.
Effect of surface properties of elastomer colloids on their coalescence and aggregation kinetics.
Gauer, Cornelius; Wu, Hua; Morbidelli, Massimo
2009-10-20
We study the aggregation kinetics of two elastomer colloids with similar bulk polymer properties but with different surface charge groups in order to understand the role of the surface properties in particle coalescence during aggregation. It is confirmed that clusters of the elastomer particles stabilized purely by ionic surfactants coalesce in both reaction-limited and diffusion-limited aggregation (RLCA and DLCA) regimes and that the coalescence is independent of the coagulant type. On the other hand, clusters formed by elastomer particles stabilized by charged polymer end groups, which are fixed on the particle surface, are fractal objects with a fractal dimension of 1.7 in the DLCA and 2.1 in the RLCA regime. This indicates insignificant cluster coalescence during aggregation, most likely due to a hindrance effect of the fixed charges.
Effects of aggregation on scattering and radiative properties of soot aerosols
NASA Astrophysics Data System (ADS)
Liu, Li; Mishchenko, Michael I.
2005-06-01
The superposition T-matrix method is used to compute the scattering matrix elements and optical cross sections for a wide variety of fractal-like soot aggregates in random orientation at a visible wavelength 0.628 μm. The effects of the fractal dimension and prefactor, the monomer radius, the number of monomers in the aggregate, and the refractive index on light scattering and absorption by aggregated soot particles are analyzed. It is shown that the configuration of the monomers can have a substantial effect and that aggregation can result in a significant enhancement of extinction and scattering relative to those computed from the Lorenz-Mie theory, assuming that there are no electromagnetic interactions between the monomers. Thus one must take the effects of soot agglomeration and cluster morphology into account in radiative transfer modeling and remote sensing applications.
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.
Fractal signatures in the aperiodic Fibonacci grating.
Verma, Rupesh; Banerjee, Varsha; Senthilkumaran, Paramasivam
2014-05-01
The Fibonacci grating (FbG) is an archetypal example of aperiodicity and self-similarity. While aperiodicity distinguishes it from a fractal, self-similarity identifies it with a fractal. Our paper investigates the outcome of these complementary features on the FbG diffraction profile (FbGDP). We find that the FbGDP has unique characteristics (e.g., no reduction in intensity with increasing generations), in addition to fractal signatures (e.g., a non-integer fractal dimension). These make the Fibonacci architecture potentially useful in image forming devices and other emerging technologies.
Black carbon fractal morphology and short-wave radiative impact: a modelling study
NASA Astrophysics Data System (ADS)
Kahnert, M.; Devasthale, A.
2011-08-01
We investigate the impact of the morphological properties of freshly emitted black carbon aerosols on optical properties and on radiative forcing. To this end, we model the optical properties of fractal black carbon aggregates by use of numerically exact solutions to Maxwell's equations within a spectral range from the UVC to the mid-IR. The results are coupled to radiative transfer computations, in which we consider six realistic case studies representing different atmospheric pollution conditions and surface albedos. The spectrally integrated radiative impacts of black carbon are compared for two different fractal morphologies, which brace the range of recently reported experimental observations of black carbon fractal structures. We also gauge our results by performing corresponding calculations based on the homogeneous sphere approximation, which is commonly employed in climate models. We find that at top of atmosphere the aggregate models yield radiative impacts that can be as much as 2 times higher than those based on the homogeneous sphere approximation. An aggregate model with a low fractal dimension can predict a radiative impact that is higher than that obtained with a high fractal dimension by a factor ranging between 1.1-1.6. Although the lower end of this scale seems like a rather small effect, a closer analysis reveals that the single scattering optical properties of more compact and more lacy aggregates differ considerably. In radiative flux computations there can be a partial cancellation due to the opposing effects of differences in the optical cross sections and asymmetry parameters. However, this cancellation effect can strongly depend on atmospheric conditions and is therefore quite unpredictable. We conclude that the fractal morphology of black carbon aerosols and their fractal parameters can have a profound impact on their radiative forcing effect, and that the use of the homogeneous sphere model introduces unacceptably high biases in
Black carbon fractal morphology and short-wave radiative impact: a modelling study
NASA Astrophysics Data System (ADS)
Kahnert, M.; Devasthale, A.
2011-11-01
We investigate the impact of the morphological properties of freshly emitted black carbon aerosols on optical properties and on radiative forcing. To this end, we model the optical properties of fractal black carbon aggregates by use of numerically exact solutions to Maxwell's equations within a spectral range from the UVC to the mid-IR. The results are coupled to radiative transfer computations, in which we consider six realistic case studies representing different atmospheric pollution conditions and surface albedos. The spectrally integrated radiative impacts of black carbon are compared for two different fractal morphologies, which brace the range of recently reported experimental observations of black carbon fractal structures. We also gauge our results by performing corresponding calculations based on the homogeneous sphere approximation, which is commonly employed in climate models. We find that at top of atmosphere the aggregate models yield radiative impacts that can be as much as 2 times higher than those based on the homogeneous sphere approximation. An aggregate model with a low fractal dimension can predict a radiative impact that is higher than that obtained with a high fractal dimension by a factor ranging between 1.1-1.6. Although the lower end of this scale seems like a rather small effect, a closer analysis reveals that the single scattering optical properties of more compact and more lacy aggregates differ considerably. In radiative flux computations there can be a partial cancellation due to the opposing effects of different error sources. However, this cancellation effect can strongly depend on atmospheric conditions and is therefore quite unpredictable. We conclude that the fractal morphology of black carbon aerosols and their fractal parameters can have a profound impact on their radiative forcing effect, and that the use of the homogeneous sphere model introduces unacceptably high biases in radiative impact studies. We emphasise that there
Long-term Differences in Tillage and Land Use Affect Intra-aggregate Pore Heterogeneity
Kravchenko, A.N.; Wang, A.N.W.; Smucker, A.J.M.; Rivers, M.L.
2012-10-25
Recent advances in computed tomography provide measurement tools to study internal structures of soil aggregates at micrometer resolutions and to improve our understanding of specific mechanisms of various soil processes. Fractal analysis is one of the data analysis tools that can be helpful in evaluating heterogeneity of the intra-aggregate internal structures. The goal of this study was to examine how long-term tillage and land use differences affect intra-aggregate pore heterogeneity. The specific objectives were: (i) to develop an approach to enhance utility of box-counting fractal dimension in characterizing intra-aggregate pore heterogeneity; (ii) to examine intra-aggregate pores in macro-aggregates (4-6 mm in size) using the computed tomography scanning and fractal analysis, and (iii) to compare heterogeneity of intra-aggregate pore space in aggregates from loamy Alfisol soil subjected to 20 yr of contrasting management practices, namely, conventional tillage (chisel plow) (CT), no-till (NT), and native succession vegetation (NS). Three-dimensional images of the intact aggregates were obtained with a resolution of 14.6 {micro}m at the Advanced Photon Source, Argonne National Laboratory, Argonne, IL. Proposed box-counting fractal dimension normalization was successfully implemented to estimate heterogeneity of pore voxel distributions without bias associated with different porosities in soil aggregates. The aggregates from all three studied treatments had higher porosity associated with large (>100 {micro}m) pores present in their centers than in their exteriors. Pores 15 to 60 {micro}m were equally abundant throughout entire aggregates but their distributions were more heterogeneous in aggregate interiors. The CT aggregates had greater numbers of pores 15 to 60 {micro}m than NT and NS. Distribution of pore voxels belonging to large pores was most heterogeneous in the aggregates from NS, followed by NT and by CT. This result was consistent with presence of
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)
Losa, Gabriele A
2009-01-01
The extension of the concepts of Fractal Geometry (Mandelbrot [1983]) toward the life sciences has led to significant progress in understanding complex functional properties and architectural / morphological / structural features characterising cells and tissues during ontogenesis and both normal and pathological development processes. It has even been argued that fractal geometry could provide a coherent description of the design principles underlying living organisms (Weibel [1991]). Fractals fulfil a certain number of theoretical and methodological criteria including a high level of organization, shape irregularity, functional and morphological self-similarity, scale invariance, iterative pathways and a peculiar non-integer fractal dimension [FD]. Whereas mathematical objects are deterministic invariant or self-similar over an unlimited range of scales, biological components are statistically self-similar only within a fractal domain defined by upper and lower limits, called scaling window, in which the relationship between the scale of observation and the measured size or length of the object can be established (Losa and Nonnenmacher [1996]). Selected examples will contribute to depict complex biological shapes and structures as fractal entities, and also to show why the application of the fractal principle is valuable for measuring dimensional, geometrical and functional parameters of cells, tissues and organs occurring within the vegetal and animal realms. If the criteria for a strict description of natural fractals are met, then it follows that a Fractal Geometry of Life may be envisaged and all natural objects and biological systems exhibiting self-similar patterns and scaling properties may be considered as belonging to the new subdiscipline of "fractalomics".
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.
Impact of diffusion limited aggregates of impurities on nematic ordering
NASA Astrophysics Data System (ADS)
Harkai, S.; Ambrožič, M.; Kralj, S.
2017-02-01
We study the impact of random bond-type disorder on two-dimensional (2D) orientational ordering of nematic liquid crystal (LC) configurations. The lattice Lebwohl-Lasher pseudospin model is used to model orientational ordering perturbed by frozen-in rod-like impurities of concentration p exhibiting the isotropic orientational probability distribution. The impurities are either (i) randomly spatially distributed or (ii) form diffusion limited aggregation (DLA)-type patterns characterized by the fractal dimensions df, where we consider cases df ∼ 1.7 and df ∼ 1.9. The degree of orientational ordering is quantified in terms of the orientational pair correlation function G(r) . Simulations reveal that the DLA pattern imposed disorder has a significantly weaker impact for a given concentration of impurities. Furthermore, if samples are quenched from the isotropic LC phase, then the fractal dimension is relatively strongly imprinted on quantitative characteristics of G(r) .
Analysis of sludge aggregates produced during electrocoagulation of model wastewater.
Załęska-Chróst, B; Wardzyńska, R
2016-01-01
This paper presents the results of the study of sludge aggregates produced during electrocoagulation of model wastewater of a composition corresponding to the effluents from the cellulose and paper industry. Wastewater was electrocoagulated statically using aluminium electrodes with a current density of 31.25 A m(-2) and 62.50 A m(-2). In subsequent stages of the treatment, sludge flocs were collected, their size was studied and their floc settling velocity (30-520 μm s(-1)) and fractal dimension (D) were determined. The values of D ranged from 1.53 to 1.95 and were directly proportional to the degree of wastewater treatment. Higher values of D were determined for sludge with lower water content (after 24 hours' settling). Fractal dimension can therefore be used as an additional parameter of wastewater treatment control.
Aggregate breakup in a contracting nozzle.
Soos, Miroslav; Ehrl, Lyonel; Bäbler, Matthäus U; Morbidelli, Massimo
2010-01-05
The breakup of dense aggregates in an extensional flow was investigated experimentally. The flow was realized by pumping the suspension containing the aggregates through a contracting nozzle. Variation of the cluster mass distribution during the breakage process was measured by small-angle light scattering. Because of the large size of primary particles and the dense aggregate structure image analysis was used to determine the shape and structure of the produced fragments. It was found, that neither aggregate structure, characterized by a fractal dimension d(f) = 2.7, nor shape, characterized by an average aspect ratio equal to 1.5, was affected by breakage. Several passes through the nozzle were required to reach the steady state. This is explained by the radial variation of the hydrodynamic stresses at the nozzle entrance, characterized through computational fluid dynamics, which implies that only the fraction of aggregates whose strength is smaller than the local hydrodynamic stress is broken during one pass through the nozzle. Scaling of the steady-state aggregate size as a function of the hydrodynamic stress was used to determine the aggregate strength.
NASA Astrophysics Data System (ADS)
Ferreira, Quirina; António Ribeiro, Paulo; Raposo, Maria
2013-06-01
Morphology of poly(allylamine hydrochloride) and poly[1-[4-(3-carboxy-4-hydroxyphenylazo) benzenesulfonamido]-1,2-ethanediyl, sodium salt] (PAZO) layer-by-layer (LBL) films is shown to influence the orientation of PAZO chromophores with respect to solid support surface, which in turn is related with observed red-shifts changes of the chromophore absorbance peak position relative to that of solution spectrum, as the bilayers are being deposited. For the first bilayers, an increase of red shift values is observed, while roughness and grain radius are kept practically constant; after the 5th bilayer, the red-shift values decrease, while grain sizes increase and the number of grains decreases. This behavior is consistent with adsorption of coiled PAZO molecules, treated as pseudo-particles, with the chromophores head-to-head oriented-J aggregates. These aggregates adsorb perpendicularly to the substrate surface for the first layers and, as roughness and grain radius increase, the adsorption of the J aggregates takes place parallel to the solid support surface, which gives rise to a decrease in the red shift value. Moreover, the adsorption of these pseudo-particles follows a fractal growth characterized by a scaling exponent of α = 0.80 ± 0.02 and a temporal growth exponent of β = 0.17 ± 0.02. These values suggest a layer growth according with Villain model, which accounts for the interactions between deposited particles and the surface. This is in accordance with the electrostatic forces driving LbL film formation and accounts for the observed morphology behavior for the different number of layers.
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.
Protein oleogels from heat-set whey protein aggregates.
de Vries, Auke; Wesseling, Anne; van der Linden, Erik; Scholten, Elke
2017-01-15
In this research we use heat-set whey protein aggregates (diameter∼200nm) as novel building blocks for structure formation in liquid oil to form oleogels. To transfer the aggregates to the oil phase, a solvent exchange procedure to sunflower oil was applied using acetone as an intermediate solvent. We found that agglomeration of the aggregates was prevented and the particle size in oil did not change from that in the initial aqueous phase. The small protein aggregates assemble into a space-spanning network, thereby providing solid-like properties to liquid oil. From oscillatory rheology we conclude that the aggregates are highly effective in forming a network. Already at ∼3% we found that G'>G″ and G' scales with protein concentration as G'∼cp(5.3). Applying a fractal gel network theory to the rheological data we deduce that the gels are in the strong link regime with a fractal dimension of 2.2. The results show that protein aggregates, besides their well-known functionality in aqueous solvents, are capable of forming a network in liquid oil. This provides a novel and promising way to design oleogels with tuneable rheological properties, applicable to e.g. foods, pharmaceuticals and/or cosmetics.
[Fractal analysis in the diagnosis of breast tumors].
Crişan, D A; Lesaru, M; Dobrescu, R; Vasilescu, C
2007-01-01
Last years studies made by researchers from over the world show that fractal geometry is a viable alternative for image analysis. Fractal features of natural forms give to fractal analysis new valences in various fields, medical imaging being a very important one. This paper intend to prove that fractal dimension, as a way to characterize the complexity of a form, can be used for diagnosis of mammographic lesions classified BI-RADS 4, further investigations being not necessary. The experiments made on 30 cases classified BI-RADS 4 confirmed that 89% of benign lesions have an average fractal dimension under the threshold 1.4, meanwhile malign lesions are characterized, in a similar percentage, by an average fractal dimension over that threshold.
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.
Mesoscale Simulation of Asphaltene Aggregation.
Wang, Jiang; Ferguson, Andrew L
2016-08-18
Asphaltenes constitute a heavy aromatic crude oil fraction with a propensity to aggregate and precipitate out of solution during petroleum processing. Aggregation is thought to proceed according to the Yen-Mullins hierarchy, but the molecular mechanisms underlying mesoscopic assembly remain poorly understood. By combining coarse-grained molecular models parametrized using all-atom data with high-performance GPU hardware, we have performed molecular dynamics simulations of the aggregation of hundreds of asphaltenes over microsecond time scales. Our simulations reveal a hierarchical self-assembly mechanism consistent with the Yen-Mullins model, but the details are sensitive and depend on asphaltene chemistry and environment. At low concentrations asphaltenes exist predominantly as dispersed monomers. Upon increasing concentration, we first observe parallel stacking into 1D rod-like nanoaggregates, followed by the formation of clusters of nanoaggregates associated by offset, T-shaped, and edge-edge stacking. Asphaltenes possessing long aliphatic side chains cannot form nanoaggregate clusters due to steric repulsions between their aliphatic coronae. At very high concentrations, we observe a porous percolating network of rod-like nanoaggregates suspended in a sea of interpenetrating aliphatic side chains with a fractal dimension of ∼2. The lifetime of the rod-like aggregates is described by an exponential distribution reflecting a dynamic equilibrium between coagulation and fragmentation.
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.
Khan, Iftheker A; Aich, Nirupam; Afrooz, A R M Nabiul; Flora, Joseph R V; Schierz, P Ariette; Ferguson, P Lee; Sabo-Attwood, Tara; Saleh, Navid B
2013-11-01
Aggregate structure of covalently functionalized chiral specific semiconducting single-walled carbon nanotubes (SWNTs) was systematically studied employing static light scattering (SLS). Fractal dimensions (Df) of two specific chirality SWNTs-SG65 and SG76 with (6, 5) and (7, 6) chiral enrichments-were measured under four biological exposure media conditions, namely: Dulbecco's Modified Eagle Medium (DMEM), Minimum Essential Medium (MEM), Roswell Park Memorial Institute (RPMI) 1640 medium, and 0.9% saline solution. The SWNTs exhibited chiral dependence on Df with SG65 showing more fractal or loosely bound aggregate structures, i.e., lower Df values (range of 2.24±0.03 to 2.64±0.05), compared to the SG76 sample (range of 2.58±0.13 to 2.90±0.08). All the Df values reported are highly reproducible, measured from multiple SLS runs and estimated with 'random block-effects' statistical analysis that yielded all p values to be <0.001. The key mechanism for such difference in Df between the SWNT samples was identified as the difference in van der Waals (VDW) interaction energies of these samples, where higher VDW of SG76 resulted in tighter packing density. Effect of medium type showed lower sensitivity; however, presence of di-valent cations (Ca(2+)) in DMEM and MEM media resulted in relatively loose or more fractal aggregates. Moreover, presence of fetal bovine serum (FBS) and bovine serum albumin (BSA), used to mimic the in vitro cell culture condition, reduced the Df values, i.e., created more fractal structures. Steric hindrance to aggregation was identified as the key mechanism for creating the fractal structures. Also, increase in FBS concentration from 1% to 10% resulted in increasingly lower Df values.
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...
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.
Image analysis of sludge aggregates obtained at preliminary treatment of sewage.
Smoczyński, L; Ratnaweera, H; Kosobucka, M; Kvaal, K; Smoczyński, M
2014-01-01
The results of wastewater treatment by Al and Fe salts and by electrocoagulation with aluminum electrodes are discussed and interpreted. Those processes used alone or combined with biological treatment, were analyzed for 50 and 90% removal of phosphates. Scanning electron microscopy (SEM) of the resulting sludge from three coagulation processes defined the perimeter P and area A of 129-142 differently sized objects in each contrast-enhanced image. Plots of lg A against lg P revealed that the analyzed sludge samples were made of self-similar aggregates-flocs with fractal characteristics. The slope of 'log plots' was used to determine surface fractal dimension Da, which was extrapolated to volumetric fractal dimension Dv. Dv was applied in a quantitative description of sludge aggregates-flocs. Aggregates-flocs of sludge obtained by Al ions (pre-polymerized Al and electrocoagulation) were characterized by higher values of Dv in comparison with sludge obtained by iron salts. The structure of {Al(OH)(3)} and {Fe(OH)(3)} aggregate-flocs was graphically simulated to determine the effect of size distribution and Dv on sweep flocculation and sludge separation and dehydration. Phosphate removal efficiency of 50% occurred at low ratios of Al:P and Fe:P. Adsorption-charge neutralization was suggested during coagulation with pre-polymerized coagulants, and sweep flow mechanism during electrocoagulation.
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.
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.
SANS spectra of the fractal supernucleosomal chromatin structure models
NASA Astrophysics Data System (ADS)
Ilatovskiy, Andrey V.; Lebedev, Dmitry V.; Filatov, Michael V.; Petukhov, Michael G.; Isaev-Ivanov, Vladimir V.
2012-03-01
The eukaryotic genome consists of chromatin—a nucleoprotein complex with hierarchical architecture based on nucleosomes, the organization of higher-order chromatin structures still remains unknown. Available experimental data, including SANS spectra we had obtained for whole nuclei, suggested fractal nature of chromatin. Previously we had built random-walk supernucleosomal models (up to 106 nucleosomes) to interpret our SANS spectra. Here we report a new method to build fractal supernucleosomal structure of a given fractal dimension or two different dimensions. Agreement between calculated and experimental SANS spectra was significantly improved, especially for model with two fractal dimensions—3 and 2.
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.
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.
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).
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
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
NASA Astrophysics Data System (ADS)
Coskun, Aycan; Sonmez, Harun; Ercin Kasapoglu, K.; Ozge Dinc, S.; Celal Tunusluoglu, M.
2010-05-01
dimensions. To determine fractal dimensions of more than hundred andesite blocks in cores, a computer program namely FRACRUN were developed. Fractal geometry has been used as practical and popular tool to define particularly irregular shaped bodies in literature since the theory of fractal was developed by Mandelbrot (1967) (Hyslip and Vallejo, 1997; Kruhl and Nega, 1996; Bagde etal., 2002; Gulbin and Evangulova, 2003; Pardini, 2003; Kolay and Kayabali, 2006; Hamdi, 2008; Zorlu, 2009 and Sezer, 2009). Although there are some methods to determine fractal dimensions, square grid-cell count method for 2D and segment count method for 1D were followed in the algorithm of FRACRUN. FRACRUN has capable of determine fractal dimensions of many closed polygons on a single surface. In the study, a database composed of uniaxial compressive strength, volumetric block proportion, fractal dimensions and number of blocks for each core was established. Finally, prediction models were developed by regression analyses and compared with the empirical equations proposed by Sonmez et al. (2006). Acknowledgement This study is a product of ongoing project supported by TUBITAK (The Scientific and Technological Research Council of Turkey - Project No: 108Y002). References Bagde, M.N., Raina, A.K., Chakraborty, A.K., Jethwa, J.L., 2002. Rock mass characterization by fractal dimension. Engineering Geology 63, 141-155. Gokceoglu, C., 2002. A fuzzy triangular chart to predict the uniaxial compressive strength of the Ankara agglomerates from their petrographic composition. Engineering Geology, 66 (1-2), 39-51. Gulbin, Y.L., Evangulova, E.B., 2003. Morphometry of quartz aggregates in granites: fractal images referring to nucleation and growth processes. Mathematical Geology 35 (7), 819-833 Hamdi, E., 2008. A fractal description of simulated 3D discontinuity networks. Rock Mechanics and Rock Engineering 41, 587-599. Hyslip, J.P., Vallejo, L.E., 1997. Fractals analysis of the roughness and size distribution
Fractal and multifractal analysis: a review.
Lopes, R; Betrouni, N
2009-08-01
Over the last years, fractal and multifractal geometries were applied extensively in many medical signal (1D, 2D or 3D) analysis applications like pattern recognition, texture analysis and segmentation. Application of this geometry relies heavily on the estimation of the fractal features. Various methods were proposed to estimate the fractal dimension or multifractal spectral of a signal. This article presents an overview of these algorithms, the way they work, their benefits and their limits. The aim of this review is to explain and to categorize the various algorithms into groups and their application in the field of medical signal analysis.
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.
Fresnel diffraction of fractal grating and self-imaging effect.
Wang, Junhong; Zhang, Wei; Cui, Yuwei; Teng, Shuyun
2014-04-01
Based on the self-similarity property of fractal, two types of fractal gratings are produced according to the production and addition operations of multiple periodic gratings. Fresnel diffractions of fractal grating are analyzed theoretically, and the general mathematic expressions of the diffraction intensity distributions of fractal grating are deduced. The gray-scale patterns of the 2D diffraction distributions of fractal grating are provided through numerical calculations. The diffraction patterns take on the periodicity along the longitude and transverse directions. The 1D diffraction distribution at some certain distances shows the same structure as the fractal grating. This indicates that the self-image of fractal grating is really formed in the Fresnel diffraction region. The experimental measurement of the diffraction intensity distribution of fractal grating with different fractal dimensions and different fractal levels is performed, and the self-images of fractal grating are obtained successfully in experiments. The conclusions of this paper are helpful for the development of the application of fractal grating.
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.
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.
Evaluation of 3D Printer Accuracy in Producing Fractal Structure.
Kikegawa, Kana; Takamatsu, Kyuuichirou; Kawakami, Masaru; Furukawa, Hidemitsu; Mayama, Hiroyuki; Nonomura, Yoshimune
2017-01-01
Hierarchical structures, also known as fractal structures, exhibit advantageous material properties, such as water- and oil-repellency as well as other useful optical characteristics, owing to its self-similarity. Various methods have been developed for producing hierarchical geometrical structures. Recently, fractal structures have been manufactured using a 3D printing technique that involves computer-aided design data. In this study, we confirmed the accuracy of geometrical structures when Koch curve-like fractal structures with zero to three generations were printed using a 3D printer. The fractal dimension was analyzed using a box-counting method. This analysis indicated that the fractal dimension of the third generation hierarchical structure was approximately the same as that of the ideal Koch curve. These findings demonstrate that the design and production of fractal structures can be controlled using a 3D printer. Although the interior angle deviated from the ideal value, the side length could be precisely controlled.
Investigations of human EEG response to viewing fractal patterns.
Hagerhall, Caroline M; Laike, Thorbjörn; Taylor, Richard P; Küller, Marianne; Küller, Rikard; Martin, Theodore P
2008-01-01
Owing to the prevalence of fractal patterns in natural scenery and their growing impact on cultures around the world, fractals constitute a common feature of our daily visual experiences, raising an important question: what responses do fractals induce in the observer? We monitored subjects' EEG while they were viewing fractals with different fractal dimensions, and the results show that significant effects could be found in the EEG even by employing relatively simple silhouette images. Patterns with a fractal dimension of 1.3 elicited the most interesting EEG, with the highest alpha in the frontal lobes but also the highest beta in the parietal area, pointing to a complicated interplay between different parts of the brain when experiencing this pattern.
[Chaos and fractals and their applications in electrocardial signal research].
Jiao, Qing; Guo, Yongxin; Zhang, Zhengguo
2009-06-01
Chaos and fractals are ubiquitous phenomena of nature. A system with fractal structure usually behaves chaos. As a complicated nonlinear dynamics system, heart has fractals structure and behaves as chaos. The deeper inherent mechanism of heart can be opened out when the chaos and fractals theory is utilized in the research of the electrical activity of heart. Generally a time series of a system was used for describing the status of the strange attractor of the system. The indices include Poincare plot, fractals dimension, Lyapunov exponent, entropy, scaling exponent, Hurst index and so on. In this article, the basic concepts and the methods of chaos and fractals were introduced firstly. Then the applications of chaos and fractals theories in the study of electrocardial signal were expounded with example of how they are used for ventricular fibrillation.
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.
Fractal structures and fractal functions as disease indicators
Escos, J.M; Alados, C.L.; Emlen, J.M.
1995-01-01
Developmental instability is an early indicator of stress, and has been used to monitor the impacts of human disturbance on natural ecosystems. Here we investigate the use of different measures of developmental instability on two species, green peppers (Capsicum annuum), a plant, and Spanish ibex (Capra pyrenaica), an animal. For green peppers we compared the variance in allometric relationship between control plants, and a treatment group infected with the tomato spotted wilt virus. The results show that infected plants have a greater variance about the allometric regression line than the control plants. We also observed a reduction in complexity of branch structure in green pepper with a viral infection. Box-counting fractal dimension of branch architecture declined under stress infection. We also tested the reduction in complexity of behavioral patterns under stress situations in Spanish ibex (Capra pyrenaica). Fractal dimension of head-lift frequency distribution measures predator detection efficiency. This dimension decreased under stressful conditions, such as advanced pregnancy and parasitic infection. Feeding distribution activities reflect food searching efficiency. Power spectral analysis proves to be the most powerful tool for character- izing fractal behavior, revealing a reduction in complexity of time distribution activity under parasitic infection.
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 properties of quantum spacetime.
Benedetti, Dario
2009-03-20
We show that, in general, a spacetime having a quantum group symmetry has also a scale-dependent fractal dimension which deviates from its classical value at short scales, a phenomenon that resembles what is observed in some approaches to quantum gravity. In particular, we analyze the cases of a quantum sphere and of kappa-Minkowski spacetime, the latter being relevant in the context of quantum gravity.
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.
NASA Astrophysics Data System (ADS)
Afrooz, A. R. M. Nabiul; Hussain, Saber M.; Saleh, Navid B.
2014-12-01
Most in vitro nanotoxicological assays are performed after 24 h exposure. However, in determining size and shape effect of nanoparticles in toxicity assays, initial characterization data are generally used to describe experimental outcome. The dynamic size and structure of aggregates are typically ignored in these studies. This brief communication reports dynamic evolution of aggregation characteristics of gold nanoparticles. The study finds that gradual increase in aggregate size of gold nanospheres (AuNS) occurs up to 6 h duration; beyond this time period, the aggregation process deviates from gradual to a more abrupt behavior as large networks are formed. Results of the study also show that aggregated clusters possess unique structural conformation depending on nominal diameter of the nanoparticles. The differences in fractal dimensions of the AuNS samples likely occurred due to geometric differences, causing larger packing propensities for smaller sized particles. Both such observations can have profound influence on dosimetry for in vitro nanotoxicity analyses.
Aggregation of liposomes induced by calcium: A structural and kinetic study
NASA Astrophysics Data System (ADS)
Roldán-Vargas, Sándalo; Martín-Molina, Alberto; Quesada-Pérez, Manuel; Barnadas-Rodríguez, Ramon; Estelrich, Joan; Callejas-Fernández, José
2007-02-01
In this work, the calcium-induced aggregation of phosphatidylserine liposomes is probed by means of the analysis of the kinetics of such process as well as the aggregate morphology. This novel characterization of liposome aggregation involves the use of static and dynamic light-scattering techniques to obtain kinetic exponents and fractal dimensions. For salt concentrations larger than 5mM , a diffusion-limited aggregation regime is observed and the Brownian kernel properly describes the time evolution of the diffusion coefficient. For slow kinetics, a slightly modified multiple contact kernel is required. In any case, a time evolution model based on the numerical resolution of Smoluchowski’s equation is proposed in order to establish a theoretical description for the aggregating system. Such a model provides an alternative procedure to determine the dimerization constant, which might supply valuable information about interaction mechanisms between phospholipid vesicles.
A quantitative investigation of the effect of pore morphology on soil aggregate stability
NASA Astrophysics Data System (ADS)
Papadopoulos, A.
2009-04-01
Soil structure determines the operating environment for all physical, chemical and biological processes within the soil. Soil aggregate stability is an important measure for assessing soil structure quality. Non-destructive tomography techniques such as X-ray Computed Tomography (CT) offer great opportunities to quantitatively investigate the soil porous architecture which can provide important information for understanding soil processes and function in a multi-scale manner. For instance, the intra-aggregate pore space is of great importance for microbial activity, the sequestration of organic carbon and water flow. This paper investigates the effect of pore morphology on soil aggregate stability. Apparent porosity, pore size distribution, average pore size and fractal perimeter dimension (pore roughness) were measured from the images of the reconstructed 2-D image stacks. A new theoretical concept of soil aggregate stability is proposed. A strong relationship was observed between soil aggregate stability and pore morphological complexity.
Influence of buoyancy on drainage of a fractal porous medium
NASA Astrophysics Data System (ADS)
Huinink, H. P.; Michels, M. A.
2002-10-01
The influence of stabilizing hydrostatic pressure gradients on the drainage of a fractal porous medium is studied. The invasion process is treated with invasion percolation (IP) in a gradient. Fractality is mimicked by randomly closing bonds of a network. Two length scales govern the problem: the characteristic length of the pore structure ξs and a length scale ξg above which buoyancy determines the structure of the cluster. When ξs<ξg the local structure of the invading cluster is governed by the interplay of capillarity and the fractal properties of the pore space. Only parts of the backbone of the pore structure can be invaded. Therefore, the obtained fractal dimension for small systems L<ξs is much lower (1.40) than the one for ordinary IP (1.82). On larger length scales, ξs
Retinal fractals and acute lacunar stroke.
Cheung, Ning; Liew, Gerald; Lindley, Richard I; Liu, Erica Y; Wang, Jie Jin; Hand, Peter; Baker, Michelle; Mitchell, Paul; Wong, Tien Y
2010-07-01
This study aimed to determine whether retinal fractal dimension, a quantitative measure of microvascular branching complexity and density, is associated with lacunar stroke. A total of 392 patients presenting with acute ischemic stroke had retinal fractal dimension measured from digital photographs, and lacunar infarct ascertained from brain imaging. After adjusting for age, gender, and vascular risk factors, higher retinal fractal dimension (highest vs lowest quartile and per standard deviation increase) was independently and positively associated with lacunar stroke (odds ratio [OR], 4.27; 95% confidence interval [CI], 1.49-12.17 and OR, 1.85; 95% CI, 1.20-2.84, respectively). Increased retinal microvascular complexity and density is associated with lacunar stroke.
Scaling and fractal behaviour underlying meiotic recombination.
Waxman, D; Stoletzki, N
2010-01-01
In this paper we investigate some of the mathematical properties of meiotic recombination. Working within the framework of a genetic model with n loci, where alpha alleles are possible at each locus, we find that the proportion of all possible diploid parental genotypes that can produce a particular haploid gamete is exp[-n log(alpha(2)/[2alpha-1])]. We show that this proportion connects recombination with a fractal geometry of dimension log(2alpha-1)/log(alpha). The fractal dimension of a geometric object manifests itself when it is measured at increasingly smaller length scales. Decreasing the length scale of a geometric object is found to be directly analogous, in a genetics problem, to specifying a multilocus haplotype at a larger number of loci, and it is here that the fractal dimension reveals itself.
Synthesis of Cobalt Oxides Thin Films Fractal Structures by Laser Chemical Vapor Deposition
Haniam, P.; Kunsombat, C.; Chiangga, S.; Songsasen, A.
2014-01-01
Thin films of cobalt oxides (CoO and Co3O4) fractal structures have been synthesized by using laser chemical vapor deposition at room temperature and atmospheric pressure. Various factors which affect the density and crystallization of cobalt oxides fractal shapes have been examined. We show that the fractal structures can be described by diffusion-limited aggregation model and discuss a new possibility to control the fractal structures. PMID:24672354
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.
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.
Fractal analysis of multiscale spatial autocorrelation among point data
De Cola, L.
1991-01-01
The analysis of spatial autocorrelation among point-data quadrats is a well-developed technique that has made limited but intriguing use of the multiscale aspects of pattern. In this paper are presented theoretical and algorithmic approaches to the analysis of aggregations of quadrats at or above a given density, in which these sets are treated as multifractal regions whose fractal dimension, D, may vary with phenomenon intensity, scale, and location. The technique is illustrated with Matui's quadrat house-count data, which yield measurements consistent with a nonautocorrelated simulated Poisson process but not with an orthogonal unit-step random walk. The paper concludes with a discussion of the implications of such analysis for multiscale geographic analysis systems. -Author
Aggregates, broccoli and cauliflower
NASA Astrophysics Data System (ADS)
Grey, Francois; Kjems, Jørgen K.
1989-09-01
Naturally grown structures with fractal characters like broccoli and cauliflower are discussed and compared with DLA-type aggregates. It is suggested that the branching density can be used to characterize the growth process and an experimental method to determine this parameter is proposed.
Three-dimensional reconstruction and fractal geometric analysis of serrated adenoma.
Iwabuchi, Masahiro; Endoh, Mareyuki; Hiwatashi, Nobuo; Kinouchi, Yoshitaka; Shimosegawa, Tooru; Masuda, Takayuki; Moriya, Takuya; Sasano, Hironobu
2002-03-01
Serrated adenoma (SA) is a relatively newly defined entity of colorectal neoplasm first characterized by Longacre and Fenoglio-Preiser in 1990. This lesion is characterized by a complicated serrated edge of crypts. In this study, we performed three-dimensional (3-D) reconstruction, including 3-D distribution patterns of Ki-67-positive cells and fractal dimension of SA, in order to evaluate the nature of the complicated architecture, including its possible morphogenesis. We studied nine colonoscopic polypectomy specimens including three SAs, three tubular adenomas (TAs), and three hyperplastic polyps (HPs). Sixty serial tissue sections per case were stained alternately with hematoxylin and eosin (H&E) and Ki-67 immunostain. Each serial image was then digitized for 3-D computer analysis and the distribution pattern of Ki-67-positive cells was evaluated. Ki-67-immunostained sections were also subjected to 2-D quantitative morphometric study. In addition, the fractal dimensions of images from H&E-stained sections were examined using a box-counting method. Results of the 3-D reconstruction study demonstrated that glandular budding and branching were more frequent in SA than in TA or HP. These findings were confirmed quantitatively by the results of fractal geometric analysis of these polyps (fractal dimension:1.34 +/- 0.08 for SA, 1.23 +/- 0.07 for TA, and 1.28 +/- 0.12 for HP). Ki-67-positive cells in HP were localized mainly in the bottom of crypts and those in TA were diffusely distributed, while Ki-67-positive cells in SA were mainly aggregated in the depressed sites of serrated epithelia. These findings were also confirmed quantitatively using 2-D morphometry. These distribution patterns of the proliferative zone of SA are considered to contribute to the formation of the characteristic serrated epithelia and the complicated morphological appearance of SA.
The optical properties of hygroscopic soot aggregates with water coating
NASA Astrophysics Data System (ADS)
Wu, Yu; Cheng, Tianhai; Zheng, Lijuan
2014-05-01
Anthropogenic aerosols, such as soot, have modified the Earth's radiation balance by scattering and absorbing solar and long-wave radiative transmission, which have largely influenced the global climate change since the industrial era. Based on transmission electron microscope images (TEM), soot particles are shown as the complex, fractal-like aggregate structures. In humid atmospheric environments, these soot aggregates tend to acquire a water coating, which introduces further complexity to the problem of determining the optical properties of the aggregates. The hygroscopic growth of soot aggregates is important for the aging of these absorbing aerosols, which can significantly influence the optical properties of these kinds of soot particles. In this paper, according to the specific volume fractions of soot core in the water coated soot particle, the monomers of fractal soot aggregates are modeled as semi-external mixtures (physical contact) with constant radius of soot core and variable size of water coating. The single scattering properties of these hygroscopic soot particles, such as scattering matrices, the cross sections of extinction, absorption and scattering, single scattering albedo (SSA), and asymmetry parameter (ASY), are calculated using the numerically exact superposition T-matrix method. The morphological effects are compared with different monomer numbers and fractal dimensions of the soot aggregates, as well as different size of water coating for these concentric spherical monomers. The results have shown that SSA, cross sections of extinction and absorption are increased for soot aggregates with thicker weakly absorbing coating on the monomers. It is found that the SSA of aged soot aggregates with hygroscopic grown are remarkably (~50% for volume fraction of soot aggregates is 0.5, at 0.670μm) larger than fresh soot particles without the consideration of water coating, due to the size of water coating and the morphological features, such as the
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.
Studying fractal geometry on submicron length scales by small-angle scattering
Wong, P.; Lin, J.
1988-08-01
Recent studies have shown that internal surfaces of porous geological materials, such as rocks and lignite coals, can be described by fractals down to atomic length scales. In this paper, the basic properties of self-similar and self-affine fractals are reviewed and how fractal dimensions can be measured by small-angle scattering experiments are discussed.
Fractal characterization of neural correlates of consciousness
NASA Astrophysics Data System (ADS)
Ibañez-Molina, A. J.; Iglesias-Parro, S.
2013-01-01
In this work we present a novel experimental paradigm, based on binocular rivalry, to address the study of internally and externally generated conscious percepts. Assuming the nonlinear nature of the EEG signals, we propose the use of fractal dimension to characterize the complexity of the EEG associated with each percept. Data analysis showed significant differences in complexity between the internally and externally generated percepts. Moreover, EEG complexity of auditory and visual percepts was unequal. These results support fractal dimension analyses as a new tool to characterize conscious perception.
Antenna Miniaturization Using Koch Snowflake Fractal Geometry
NASA Astrophysics Data System (ADS)
Minal, Dhama, Nitin
2010-11-01
The Wireless Industry is witnessing an volatile emergence today in present era. Also requires the performance over several frequency bands or are reconfigurable as the demands on the system changes. This Paper Presents Rectangular, Koch Fractal Patch Antennas on Single and Multilayer Substrate With and Without Air-Gap using Advanced Design System Simulator (ADS). Fractal Antenna provides Miniaturization over conventional microstrip Antennas. The Antennas Have Been Designed on FR4 substrate with ∈ = 4.2, h = 1.53 and the initial Dimension of the simple Rectangular Patch is 36.08 * 29.6 mm. The experimental Resonant Frequencies of the Fractal Patch with 1st, 2nd & 3rd are observed 2.22, 2.14 & 2.02 GHz Respectively in comparison to Rectangular Patch with 2.43 GHz. The reduced Impedance bandwidth of the Fractal Patch has been improved by designing the patch over multilayer substrate with varying Air-gap between two Substrate. As we increase the air- gap between the two substrate layer further enhancement in impedance bandwidth of Fractal antenna has been Obtained. The Radiation pattern of Koch Fractal antenna is as similar to rectangular patch antenna but with better H-plane Cross Polarization for fractal patch. The all simulated Results are in close Agreement with experimental Results.
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.
Kinetic properties of fractal stellar media
NASA Astrophysics Data System (ADS)
Chumak, O. V.; Rastorguev, A. S.
2017-01-01
Kinetic processes in fractal stellar media are analysed in terms of the approach developed in our earlier paper involving a generalization of the nearest neighbour and random force distributions to fractal media. Diffusion is investigated in the approximation of scale-dependent conditional density based on an analysis of the solutions of the corresponding Langevin equations. It is shown that kinetic parameters (time-scales, coefficients of dynamic friction, diffusion, etc.) for fractal stellar media can differ significantly both qualitatively and quantitatively from the corresponding parameters for a quasi-uniform random media with limited fluctuations. The most important difference is that in the fractal case, kinetic parameters depend on spatial scalelength and fractal dimension of the medium studied. A generalized kinetic equation for stellar media (fundamental equation of stellar dynamics) is derived in the Fokker-Planck approximation with the allowance for the fractal properties of the spatial stellar density distribution. Also derived are its limit forms that can be used to describe small departures of fractal gravitating medium from equilibrium.
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.
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.
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.
Catastrophes in the multi-fractal dynamics of social-economic systems
NASA Astrophysics Data System (ADS)
Kudinov, A. N.; Tsvetkov, V. P.; Tsvetkov, I. V.
2011-06-01
In the present paper, the concept of multi-fractal dynamics is developed. The problem concerning catastrophes in this dynamics is studied in detail. In the framework of the concept of fractal curve as a thick curve, it is proved that the cell approach to measuring the fractal dimension D is equivalent to measuring the dependence of the length L of the line on the scope δ. The introduction of a fractal scale of temperatures T f is suggested.
Monitoring the Depth of Anaesthesia Using Fractal Complexity Method
NASA Astrophysics Data System (ADS)
Klonowski, W.; Olejarczyk, E.; Stepien, R.; Jalowiecki, P.; Rudner, R.
We propose a simple and effective method of characterizing complexity of EEG-signals for monitoring the depth of anaesthesia using Higuchi's fractal dimension method. We demonstrate that the proposed method may compete with the widely used BIS monitoring method.
Paradigms of Complexity: Fractals and Structures in the Sciences
NASA Astrophysics Data System (ADS)
Novak, Miroslav M.
The Table of Contents for the book is as follows: * Preface * The Origin of Complexity (invited talk) * On the Existence of Spatially Uniform Scaling Laws in the Climate System * Multispectral Backscattering: A Fractal-Structure Probe * Small-Angle Multiple Scattering on a Fractal System of Point Scatterers * Symmetric Fractals Generated by Cellular Automata * Bispectra and Phase Correlations for Chaotic Dynamical Systems * Self-Organized Criticality Models of Neural Development * Altered Fractal and Irregular Heart Rate Behavior in Sick Fetuses * Extract Multiple Scaling in Long-Term Heart Rate Variability * A Semi-Continous Box Counting Method for Fractal Dimension Measurement of Short Single Dimension Temporal Signals - Preliminary Study * A Fractional Brownian Motion Model of Cracking * Self-Affine Scaling Studies on Fractography * Coarsening of Fractal Interfaces * A Fractal Model of Ocean Surface Superdiffusion * Stochastic Subsurface Flow and Transport in Fractal Fractal Conductivity Fields * Rendering Through Iterated Function Systems * The σ-Hull - The Hull Where Fractals Live - Calculating a Hull Bounded by Log Spirals to Solve the Inverse IFS-Problem by the Detected Orbits * On the Multifractal Properties of Passively Convected Scalar Fields * New Statistical Textural Transforms for Non-Stationary Signals: Application to Generalized Mutlifractal Analysis * Laplacian Growth of Parallel Needles: Their Mullins-Sekerka Instability * Entropy Dynamics Associated with Self-Organization * Fractal Properties in Economics (invited talk) * Fractal Approach to the Regional Seismic Event Discrimination Problem * Fractal and Topological Complexity of Radioactive Contamination * Pattern Selection: Nonsingular Saffman-Taylor Finger and Its Dynamic Evolution with Zero Surface Tension * A Family of Complex Wavelets for the Characterization of Singularities * Stabilization of Chaotic Amplitude Fluctuations in Multimode, Intracavity-Doubled Solid-State Lasers * Chaotic
Hashemi, S. M.; Jagodič, U.; Mozaffari, M. R.; Ejtehadi, M. R.; Muševič, I.; Ravnik, M.
2017-01-01
Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter. PMID:28117325
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)
Thamrin, Cindy; Stern, Georgette; Frey, Urs
2010-06-01
There is increasing interest in the study of fractals in medicine. In this review, we provide an overview of fractals, of techniques available to describe fractals in physiological data, and we propose some reasons why a physician might benefit from an understanding of fractals and fractal analysis, with an emphasis on paediatric respiratory medicine where possible. Among these reasons are the ubiquity of fractal organisation in nature and in the body, and how changes in this organisation over the lifespan provide insight into development and senescence. Fractal properties have also been shown to be altered in disease and even to predict the risk of worsening of disease. Finally, implications of a fractal organisation include robustness to errors during development, ability to adapt to surroundings, and the restoration of such organisation as targets for intervention and treatment.
NASA Astrophysics Data System (ADS)
Hashemi, S. M.; Jagodič, U.; Mozaffari, M. R.; Ejtehadi, M. R.; Muševič, I.; Ravnik, M.
2017-01-01
Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter.
Hashemi, S M; Jagodič, U; Mozaffari, M R; Ejtehadi, M R; Muševič, I; Ravnik, M
2017-01-24
Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter.
a Fractal Network Model for Fractured Porous Media
NASA Astrophysics Data System (ADS)
Xu, Peng; Li, Cuihong; Qiu, Shuxia; Sasmito, Agus Pulung
2016-04-01
The transport properties and mechanisms of fractured porous media are very important for oil and gas reservoir engineering, hydraulics, environmental science, chemical engineering, etc. In this paper, a fractal dual-porosity model is developed to estimate the equivalent hydraulic properties of fractured porous media, where a fractal tree-like network model is used to characterize the fracture system according to its fractal scaling laws and topological structures. The analytical expressions for the effective permeability of fracture system and fractured porous media, tortuosity, fracture density and fraction are derived. The proposed fractal model has been validated by comparisons with available experimental data and numerical simulation. It has been shown that fractal dimensions for fracture length and aperture have significant effect on the equivalent hydraulic properties of fractured porous media. The effective permeability of fracture system can be increased with the increase of fractal dimensions for fracture length and aperture, while it can be remarkably lowered by introducing tortuosity at large branching angle. Also, a scaling law between the fracture density and fractal dimension for fracture length has been found, where the scaling exponent depends on the fracture number. The present fractal dual-porosity model may shed light on the transport physics of fractured porous media and provide theoretical basis for oil and gas exploitation, underground water, nuclear waste disposal and geothermal energy extraction as well as chemical engineering, etc.
Deterministic fractals: extracting additional information from small-angle scattering data.
Cherny, A Yu; Anitas, E M; Osipov, V A; Kuklin, A I
2011-09-01
The small-angle scattering curves of deterministic mass fractals are studied and analyzed in momentum space. In the fractal region, the curve I(q)q(D) is found to be log-periodic with good accuracy, and the period is equal to the scaling factor of the fractal. Here, D and I(q) are the fractal dimension and the scattering intensity, respectively. The number of periods of this curve coincides with the number of fractal iterations. We show that the log-periodicity of I(q)q(D) in the momentum space is related to the log-periodicity of the quantity g(r)r(3-D) in the real space, where g(r) is the pair distribution function. The minima and maxima positions of the scattering intensity are estimated explicitly by relating them to the pair distance distribution in real space. It is shown that the minima and maxima are damped with increasing polydispersity of the fractal sets; however, they remain quite pronounced even at sufficiently large values of polydispersity. A generalized self-similar Vicsek fractal with controllable fractal dimension is introduced, and its scattering properties are studied to illustrate the above findings. In contrast with the usual methods, the present analysis allows us to obtain not only the fractal dimension and the edges of the fractal region, but also the fractal iteration number, the scaling factor, and the number of structural units from which the fractal is composed.
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.
Dynamic structure factor of vibrating fractals.
Reuveni, Shlomi; Klafter, Joseph; Granek, Rony
2012-02-10
Motivated by novel experimental work and the lack of an adequate theory, we study the dynamic structure factor S(k,t) of large vibrating fractal networks at large wave numbers k. We show that the decay of S(k,t) is dominated by the spatially averaged mean square displacement of a network node, which evolves subdiffusively in time, ((u[over →](i)(t)-u[over →](i)(0))(2))∼t(ν), where ν depends on the spectral dimension d(s) and fractal dimension d(f). As a result, S(k,t) decays as a stretched exponential S(k,t)≈S(k)e(-(Γ(k)t)(ν)) with Γ(k)∼k(2/ν). Applications to a variety of fractal-like systems are elucidated.
Menger sponge-like fractal body created by a novel template method.
Mayama, H; Tsujii, K
2006-09-28
We have established experimental strategies on how to create a Menger sponge-like fractal body and how to control its fractal dimension. The essence was to utilize alkylketene dimer (AKD), which spontaneously forms super-water-repellent fractal surface. We prepared "fractal AKD particles" with fractal surface structure as templates of pores in fractal body. The fractal body was synthesized by filling the remained space between the packed template particles with a tetramethyl orthosilicate solution, solidifying it by the sol-gel process, and removing the template by calcinations. We have succeeded in systematically creating fractal bodies of silica with different cross-sectional fractal dimensions D(cs)=1.87, 1.84, and 1.80 using "fractal template particles" compressed under the ratio=1.0, 2.0, and 3.0, respectively. We also discussed the possibilities of their fractal geometries in comparison with mathematical models. We concluded that the created fractal bodies were close to a Menger sponge and its modified one. Our experimental strategy allows us to design fractality of porous materials.
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.
Pulse regime in formation of fractal fibers
NASA Astrophysics Data System (ADS)
Smirnov, B. M.
2016-11-01
The pulse regime of vaporization of a bulk metal located in a buffer gas is analyzed as a method of generation of metal atoms under the action of a plasma torch or a laser beam. Subsequently these atoms are transformed into solid nanoclusters, fractal aggregates and then into fractal fibers if the growth process proceeds in an external electric field. We are guided by metals in which transitions between s and d-electrons of their atoms are possible, since these metals are used as catalysts and filters in interaction with gas flows. The resistance of metal fractal structures to a gas flow is evaluated that allows one to find optimal parameters of a fractal structure for gas flow propagation through it. The thermal regime of interaction between a plasma pulse or a laser beam and a metal surface is analyzed. It is shown that the basic energy from an external source is consumed on a bulk metal heating, and the efficiency of atom evaporation from the metal surface, that is the ratio of energy fluxes for vaporization and heating, is 10-3-10-4 for transient metals under consideration. A typical energy flux ( 106 W/cm2), a typical surface temperature ( 3000 K), and a typical pulse duration ( 1 μs) provide a sufficient amount of evaporated atoms to generate fractal fibers such that each molecule of a gas flow collides with the skeleton of fractal fibers many times.
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.
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.
Fractal structure of films deposited in a tokamak
Budaev, V. P.; Khimchenko, L. N.
2007-04-15
The surface of amorphous films deposited in the T-10 tokamak was studied in a scanning tunnel microscope. The surface relief on a scale from 10 nm to 100 {mu}m showed a stochastic surface topography and revealed a hierarchy of grains. The observed variety of irregular structures of the films was studied within the framework of the concept of scale invariance using the methods of fractal geometry and statistical physics. The experimental probability density distribution functions of the surface height variations are close in shape to the Cauchy distribution. The stochastic surface topography of the films is characterized by a Hurst parameter of H = 0.68-0.85, which is evidence of a nontrivial self-similarity of the film structure. The fractal character and porous structure of deposited irregular films must be considered as an important issue related to the accumulation of tritium in the ITER project. The process of film growth on the surface of tokamak components exposed to plasma has been treated within the framework of the general concept of inhomogeneous surface growth. A strong turbulence of the edge plasma in tokamaks can give rise to fluctuations in the incident flux of particles, which leads to the growth of fractal films with grain dimensions ranging from nano-to micrometer scale. The shape of the surface of some films found in the T-10 tokamak has been interpreted using a model of diffusion-limited aggregation (DLA). The growth of films according to the discrete DLA model was simulated using statistics of fluctuations observed in a turbulent edge plasma of the T-10 tokamak. The modified DLA model reproduces well the main features of the surface of some films deposited in tokamaks.
NASA Astrophysics Data System (ADS)
Yao, Yanbin; Liu, Dameng; Tang, Dazhen; Tang, Shuheng; Huang, Wenhui; Liu, Zhihua; Che, Yao
2009-06-01
To better understand the characteristics of seepage-pores (pore radius larger than 100 nanometers) and their influence on the permeability of coals, we have conducted fractal analyses for 34 fresh coal samples (mean maximum vitrinite reflectance Ro,max from 0.43% to 4.21%) from North, Northwest and Northeast China. Mercury porosimetry data indicate that the coals are fractal, with pore radius ranging from 0.1 to 50 μm. Calculated fractal dimensions of these coals range from 2.61 to 2.98, higher than those from other kinds of rocks such as sandstone, shale, and carbonate. The data suggest that the coals have more complicated and inhomogeneous pore structures than other rocks. The fractal dimension has a negative correlation with the petrologic permeability of coals, particularly for higher rank coals (with 1.47-4.21% Ro,max), from which a strong negative linear correlation ( R2=0.85) between fractal dimension and permeability is observed. A 'U-shaped' trend between fractal dimensions and coal ranks is observed, with the minimum fractal dimensions occurring at 1.1-1.3% Ro,max. The sub-bituminous, high volatile bituminous, semi-anthracite, and anthracite have higher fractal dimensions. The effects of coal rank upon fractal dimensions are mainly due to the variety of micropore contents and aromaticity of coals with coalification.
Fractals properties of EEG during event-related desynchronization of motor imagery.
Nguyen, Ngoc Quang; Truong, Quang Dang Khoa; Kondo, Toshiyuki
2015-01-01
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.
Spatial behavior analysis at the global level using fractal geometry.
Sambrook, Roger C
2008-01-01
Previous work has suggested that an estimate of fractal dimension can provide a useful metric for quantifying settlement patterns. This study uses fractal methods to investigate settlement patterns at a global scale showing that the scaling behavior of the pattern of the world's largest cities corresponds to that typically observed for coastlines and rivers. This serves to validate the use of fractal dimension as a scale-independent measure of settlement patterns which can be correlated with other physical features. Such a measure may be a useful validation criterion for models of human settlement and spatial behavior.
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
Colloidal Aggregate Structure under Shear by USANS
NASA Astrophysics Data System (ADS)
Chatterjee, Tirtha; van Dyk, Antony K.; Ginzburg, Valeriy V.; Nakatani, Alan I.
2015-03-01
Paints are complex formulations of polymeric binders, inorganic pigments, dispersants, surfactants, colorants, rheology modifiers, and other additives. A commercially successful paint exhibits a desired viscosity profile over a wide shear rate range from 10-5 s-1 for settling to >104 s-1 for rolling, and spray applications. Understanding paint formulation structure is critical as it governs the paint viscosity profile. However, probing paint formulation structure under shear is a challenging task due to the formulation complexity containing structures with different hierarchical length scales and their alterations under the influence of an external flow field. In this work mesoscale structures of paint formulations under shear are investigated using Ultra Small-Angle Neutron Scattering (rheo-USANS). Contrast match conditions were utilized to independently probe the structure of latex binder particle aggregates and the TiO2 pigment particle aggregates. Rheo-USANS data revealed that the aggregates are fractal in nature and their self-similarity dimensions and correlations lengths depend on the chemistry of the binder particles, the type of rheology modifier present and the shear stress imposed upon the formulation. These results can be explained in the framework of diffusion and reaction limited transient aggregates structure evolution under simple shear.
NASA Astrophysics Data System (ADS)
Ono, Kiminori; Matsukawa, Yoshiya; Saito, Yasuhiro; Matsushita, Yohsuke; Aoki, Hideyuki; Era, Koki; Aoki, Takayuki; Yamaguchi, Togo
2015-06-01
This study presents the validity and ability of an aggregate mean free path cluster-cluster aggregation (AMP-CCA) model, which is a direct Monte Carlo simulation, to predict the aggregate morphology with diameters form about 15-200 nm by comparing the particle size distributions (PSDs) with the results of the previous stochastic approach. The PSDs calculated by the AMP-CCA model with the calculated aggregate as a coalesced spherical particle are in reasonable agreement with the results of the previous stochastic model regardless of the initial number concentration of particles. The shape analysis using two methods, perimeter fractal dimension and the shape categories, has demonstrated that the aggregate structures become complex with increasing the initial number concentration of particles. The AMP-CCA model provides a useful tool to calculate the aggregate morphology and PSD with reasonable accuracy.
NASA Astrophysics Data System (ADS)
Ingersoll, A. P.; Nakajima, M.; Ewald, S.; Gao, P.
2015-12-01
Postberg et al (2009) argued that the observed plume activity requires large vapor chambers above the evaporating liquid (left figure). Here we argue that large vapor chambers are unnecessary, and that a liquid-filled crack 1 meter wide extending along the 500 km length of the tiger stripes would be an adequate source (right figure). We consider controlled boiling (companion paper by Nakajima and Ingersoll 2015AGU) regulated by friction between the gas and the walls. Postberg et al use formulas from Rayleigh-Benard convection, which we argue does not apply when bubbles are transferring their latent heat across the liquid-gas interface. We show that modest convection currents in the liquid (few cm/s) can supply energy to the boiling zone and prevent it from freezing. Hedman et al (2013) reported brightness variations with orbital phase, but they also reported that their 2005 observations were roughly 50% higher than the 2009 observations. Here we extend the observation period to 2015 (Ingersoll and Ewald 2015). Our analysis relies on ISS images whereas Hedman et al rely on VIMS near-IR images, which have 40 times lower resolution. We successfully separate the brightness of the plume from the E-ring background. Our earlier analysis of the particle size distribution (Ingersoll and Ewald 2011) allows us to correct for differences in scattering angle. We confirm a general decline in activity over the 10-year period, but we find hints of fluctuations on shorter time scales. Kempf (Cassini project science meeting, Jan 22, 2015) reported that the mass of particles in the plumes could be an order of magnitude less than that reported by Ingersoll and Ewald (2011). Kempf used in situ particle measurements by CDA, whereas I&E used brightness observations and the assumption that the particles are solid ice. Here we show (Gao et al 2015AGU) that fractal aggregates fit the brightness data just as well as solid ice, and are consistent with the lower mass reported by Kempf.
Effect of the particle shape on the optical properties of black carbon aggregates
NASA Astrophysics Data System (ADS)
Skorupski, Krzysztof
2016-04-01
Small particles tend to connect to each other and create large geometries, namely aggregates. To simplify the light scattering simulation process, they are usually modelled as assemblies of spheres positioned in point contact. This is a rough approximation because connections between them always exist. In this work we present answers to the three following questions: which optical properties of fractal-like aggregates are strongly dependent on the particle shape, what is the magnitude of the relative extinction error σCext when non-spherical particles are modelled as spheres and whether the relative extinction error σCext is dependent on the aggregate size Np. The paper was aimed at tropospheric black carbon particles and their complex refractive index m was based on the work by Chang and Charalampopoulos. The incident wavelength λ varied from λ = 300nm to λ = 900nm. For the light scattering simulations the ADDA algorithm was used. The polarizability expression was IGT_SO (approximate Integration of Greens Tensor over the dipole) and each particle, regardless of its shape, was composed of ca. Nd ≍ 1000 volume elements (dipoles). In the study, fractal-like aggregates consisted of up to Np = 300 primary particles with the volume equivalent to the volume of a sphere with the radius rp = 15nm. The fractal dimension was Df = 1:8 and the fractal prefactor was kf = 1:3. Geometries were generated with the tunable CC (Cluster-Cluster) algorithm proposed by Filippov et al. The results show that when the extinction cross section σCext is considered, the changes caused by the particle shape, which are especially visible for longer wavelengths λ cannot be neglected. The most significant difference can be observed for the regular tetrahedron. The relative extinction error σCext diminishes slightly along with the number of primary particles Np. However, even when large fractal-like aggregates are studied, it should not be considered as non-existent. On the contrary
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.
Fractal Metrology for biogeosystems analysis
NASA Astrophysics Data System (ADS)
Torres-Argüelles, V.; Oleschko, K.; Tarquis, A. M.; Korvin, G.; Gaona, C.; Parrot, J.-F.; Ventura-Ramos, E.
2010-11-01
The solid-pore distribution pattern plays an important role in soil functioning being related with the main physical, chemical and biological multiscale and multitemporal processes of this complex system. In the present research, we studied the aggregation process as self-organizing and operating near a critical point. The structural pattern is extracted from the digital images of three soils (Chernozem, Solonetz and "Chocolate" Clay) and compared in terms of roughness of the gray-intensity distribution quantified by several measurement techniques. Special attention was paid to the uncertainty of each of them measured in terms of standard deviation. Some of the applied methods are known as classical in the fractal context (box-counting, rescaling-range and wavelets analyses, etc.) while the others have been recently developed by our Group. The combination of these techniques, coming from Fractal Geometry, Metrology, Informatics, Probability Theory and Statistics is termed in this paper Fractal Metrology (FM). We show the usefulness of FM for complex systems analysis through a case study of the soil's physical and chemical degradation applying the selected toolbox to describe and compare the structural attributes of three porous media with contrasting structure but similar clay mineralogy dominated by montmorillonites.
MRI Image Processing Based on Fractal Analysis
Marusina, Mariya Y; Mochalina, Alexandra P; Frolova, Ekaterina P; Satikov, Valentin I; Barchuk, Anton A; Kuznetcov, Vladimir I; Gaidukov, Vadim S; Tarakanov, Segrey A
2017-01-01
Background: Cancer is one of the most common causes of human mortality, with about 14 million new cases and 8.2 million deaths reported in in 2012. Early diagnosis of cancer through screening allows interventions to reduce mortality. Fractal analysis of medical images may be useful for this purpose. Materials and Methods: In this study, we examined magnetic resonance (MR) images of healthy livers and livers containing metastases from colorectal cancer. The fractal dimension and the Hurst exponent were chosen as diagnostic features for tomographic imaging using Image J software package for image processings FracLac for applied for fractal analysis with a 120x150 pixel area. Calculations of the fractal dimensions of pathological and healthy tissue samples were performed using the box-counting method. Results: In pathological cases (foci formation), the Hurst exponent was less than 0.5 (the region of unstable statistical characteristics). For healthy tissue, the Hurst index is greater than 0.5 (the zone of stable characteristics). Conclusions: The study indicated the possibility of employing fractal rapid analysis for the detection of focal lesions of the liver. The Hurst exponent can be used as an important diagnostic characteristic for analysis of medical images.
[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.
Respiratory Onset Detection Using Variance Fractal Dimension
2007-11-02
V.R., 1994. “The synchronization of respiration and swallow sounds with videofluroscopy during swallowing”. Dysphagia 9, 162-167 [3] Tarrant S.C, Ellis...R.E, Flack F.C, and Selley W.G., 1997. “Comparative review of techniques for recording respiratory events at rest and during deglutition”, Dysphagia
He, Chiquan; Zhao, Kuiyi
2003-04-01
By using the principles and methods of fractal geometry theory, the relationship between above ground biomass and plant length or sheath height of Carex lasiocarpa population was studied. The results showed that there was a good static fractal relationship between them, and the resulted fractal dimension was an efficient description of the accumulation of above ground biomass in each organ. The dynamic fractal relationship showed that during the whole growing season, the increase of above ground biomass had a self-similarity, being a fractal growth process, and the pattern of its increase was the fractal dimension D. Based on these results, a fractal growth model of Carex lasiocarpa population was established, which regarded the bigger grass as the result of the amplification of seedling growth.
T-matrix calculations of fractal black carbon atmospheric aerosol particle optical scattering
NASA Astrophysics Data System (ADS)
Smith, Anna; Boness, David
2008-05-01
To better constrain global climate change computer models, and thereby to more fully understand the full extent of anthropogenic climate change, it is necessary to understand the physics of light scattering from those atmospheric aerosol particles that are caused by human activities. The IPCC AR4 report on the physical basis of climate change lists uncertainty in the effects of black carbon aerosol particles, caused by burning fossil fuels and organic matter, as one of the greatest uncertainties in current climate change understanding. This study hopes to increase the knowledge of how aerosols contribute to radiative forcing by using more realistic modeling of scattering properties. We use D. W. Mackowski's T- matrix code on fractal aggregates of uniform spherical monomers and compare this with fractal scattering predicted by the Raleigh-Debye-Gans approximation. The T-matrix code is checked for accuracy with one spherical particle as found with Mie theory. Scattering properties found using the T-matrix method are performed as a function of fractal dimension and number of monomers. Preliminary results will be presented. Future work will involve comparison with soot particle optical scattering measurements made at Seattle University.
NASA Astrophysics Data System (ADS)
Zhang, Xiaoyang; Wu, Caifang; Li, Teng
2017-02-01
The micropore structure of a tight sandstone is the decisive factor in determining its reserve and seepage characteristics. An accurate description of the pore structures and a complete characterization of the gas-water permeability are critical when exploring for tight sandstone gas. One simple and effective way to quantitatively characterize the heterogeneity and complexity of the pore structures in a low permeability reservoir is the fractal dimension. In this study, three different methods, each utilizing mercury intrusion porosimetry (MIP) data, were adopted to analyze the fractal dimensions and the fractal curves of sandstones from the no. 8 layer of the Xiashihezi Formation (He 8 member) in the Linxing block, dated from the Middle Permian. The morphological features of the fractal curves, the characteristics of the fractal dimensions and the theoretical differences between these three methods were also discussed. The results show that the fractal dimensions obtained by method I reflect the characteristics of the remaining pores that are not intruded by mercury, and they show that the involved pore scales are more comprehensive. While in methods II and III, both obtain the fractal dimensions of the pores intruded by mercury, the difference between them is in the selection of a simplified pore shape model, which results in the fractal dimensions differing by a value of 1 between them. No matter which method is adopted, the pore structures of tight sandstone reservoirs in the Linxing block exhibit fractal characteristics. However, the fractal dimensions obtained by method I are more suitable for describing the complexity and petrophysical properties of the tight sandstone pores in the He 8 member of the Linxing block. The fractal curves obtained by different methods are consistent to a certain extent in terms of morphological changes. Small pores (
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.
Discrimination of walking patterns using wavelet-based fractal analysis.
Sekine, Masaki; Tamura, Toshiyo; Akay, Metin; Fujimoto, Toshiro; Togawa, Tatsuo; Fukui, Yasuhiro
2002-09-01
In this paper, we attempted to classify the acceleration signals for walking along a corridor and on stairs by using the wavelet-based fractal analysis method. In addition, the wavelet-based fractal analysis method was used to evaluate the gait of elderly subjects and patients with Parkinson's disease. The triaxial acceleration signals were measured close to the center of gravity of the body while the subject walked along a corridor and up and down stairs continuously. Signal measurements were recorded from 10 healthy young subjects and 11 elderly subjects. For comparison, two patients with Parkinson's disease participated in the level walking. The acceleration signal in each direction was decomposed to seven detailed signals at different wavelet scales by using the discrete wavelet transform. The variances of detailed signals at scales 7 to 1 were calculated. The fractal dimension of the acceleration signal was then estimated from the slope of the variance progression. The fractal dimensions were significantly different among the three types of walking for individual subjects (p < 0.01) and showed a high reproducibility. Our results suggest that the fractal dimensions are effective for classifying the walking types. Moreover, the fractal dimensions were significantly higher for the elderly subjects than for the young subjects (p < 0.01). For the patients with Parkinson's disease, the fractal dimensions tended to be higher than those of healthy subjects. These results suggest that the acceleration signals change into a more complex pattern with aging and with Parkinson's disease, and the fractal dimension can be used to evaluate the gait of elderly subjects and patients with Parkinson's disease.
[Fractal characteristics of soil particles in surface layer of black soil].
Miao, Chi-Yuan; Wang, Ya-Feng; Wei, Xin; Xu, Xia; Shi, Wen
2007-09-01
Based on the second national soil survey of China, the fractal dimension of soil particles in the surface layers of 36 typical profiles of black soil was calculated. The results showed that the fractal dimension was 2.5831-2.8230, being increased with decreasing diameter of soil texture, but the variability was inconspicuous. The fractal dimension was negatively correlated with the contents of sand (2-0.02 mm) and silt (0.02-0.002 mm) (P < 0.05), but positively correlated with clay (< 0.002 mm) content (P < 0.01). No significant correlations were observed between soil particle fractal dimension and soil organic matter, total nitrogen, total phosphorus, total potassium, and pH. The fractal dimension of soil particles could be used as a comprehensive and quantitative index in evaluating the degradation degree of black soil.
Fragmentation of Fractal Random Structures
NASA Astrophysics Data System (ADS)
Elçi, Eren Metin; Weigel, Martin; Fytas, Nikolaos G.
2015-03-01
We analyze the fragmentation behavior of random clusters on the lattice under a process where bonds between neighboring sites are successively broken. Modeling such structures by configurations of a generalized Potts or random-cluster model allows us to discuss a wide range of systems with fractal properties including trees as well as dense clusters. We present exact results for the densities of fragmenting edges and the distribution of fragment sizes for critical clusters in two dimensions. Dynamical fragmentation with a size cutoff leads to broad distributions of fragment sizes. The resulting power laws are shown to encode characteristic fingerprints of the fragmented objects.
Application study of fractal theory in mechanical transmission
NASA Astrophysics Data System (ADS)
Zhao, Han; Wu, Qilin
2016-09-01
Mechanical transmissions are applied widely in various electrical and mechanical products, but some qualities of some high-end products can't meet people's demand, and need to be improved with some new methods or theories. The fractal theory is a new mathematic tool, which provides a new approach for the further study in the area of the mechanical transmission, and helps to solve some problems. The basic contents of the fractal theory are introduced firstly, especially the two important concepts, the self-similar fractal and the fractal dimension. Then, the deferent application of the fractal theory in this area are given to display how to further the study and improve some important characteristics of the mechanical transmission, such as contact surfaces, manufacturing precise, friction and wear, stiffness, strength, dynamics, fault diagnosis, etc. Finally, the problems of the fractal theory and its application are discussed, and some weaknesses, such as the calculation capacity of the fractal theory is not strong, are pointed out. Some new solutions are suggested, such as combining the fractal theory with the fuzzy theory, the chaos theory and so on. The new application fields of the fractal theory in the area of the mechanical transmission are proposed.
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.
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.
NASA Astrophysics Data System (ADS)
Tremberger, George, Jr.; Flamho