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1

NSDL National Science Digital Library

Determine the fractal dimensions of several line-deformation fractals. Input the scale factor and number of similar copies, and the dimension will be calculated. Fractal Dimensions is one of the Interactivate assessment explorers.

2

Surface Fractal Dimension of Bentonite and its Application in Calculation of Swelling Deformation

NASA Astrophysics Data System (ADS)

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.

Xiang, G. S.; Xu, Y. F.; Jiang, H.

2014-09-01

3

The Calculation of Fractal Dimension in the Presence of Non-Fractal Clutter

NASA Technical Reports Server (NTRS)

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.

Herren, Kenneth A.; Gregory, Don A.

1999-01-01

4

Effect of Image Processing of a Leaf Photograph on the Calculated Fractal Dimension of Leaf Veins

Digital photography is a promised method for estimating the fractal characteristics of leaf veins. In this study, the effects\\u000a of different threshold levels and image processing methods using Adobe Photoshop software on the fractal dimension values\\u000a were examined from a digital photo of nectarine leaf. The results showed that the nectarine leaf vein has typical fractal\\u000a characteristics and its fractal

Yun Kong; Shaohui Wang; Chengwei Ma; Baoming Li; Yuncong Yao

2007-01-01

5

FERImage: an interactive program for fractal dimension, d(per) and d(min) calculation.

A computer program has been written for the determination of the D fractal dimension at low scale, of the d(per) representative parameter of the periodical region at high scale, and the d(min), representative parameter of the minimum elemental cell which is repeated in the periodical structure from the variogram. It carries out the simultaneous obtention of the three previous parameters developed by Bonetto and Ladaga. The program also allows to obtain fractal dimension values from the Fourier power spectrum. FERImage has been developed so that the users could choose the rank where the behavior is fractal, not only in the variogram method but also in the Fourier spectrum method. PMID:11405304

Bianchi, F D; Bonetto, R D

2001-01-01

6

Fractal dimension of color fractal images.

Fractal dimension is a very useful metric for the analysis of the images with self-similar content, such as textures. For its computation there exist several approaches, the probabilistic algorithm being accepted as the most elegant approach. However, all the existing methods are defined for 1-D signals or binary images, with extension to grayscale images. Our purpose is to propose a color version of the probabilistic algorithm for the computation of the fractal dimension. To validate this new approach, we also propose an extension of the existing algorithm for the generation of probabilistic fractals, in order to obtain color fractal images. Then we show the results of our experiments and conclude this paper. PMID:20643608

Ivanovici, Mihai; Richard, Noël

2011-01-01

7

Dimension of Fractal Basin Boundaries.

NASA Astrophysics Data System (ADS)

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. We look for universal scalings of the dimension of fractal basin boundaries near type I and type III intermittency transitions to chaos. Type I intermittency can occur as the system experiences a saddle-node (tangent) bifurcation and type III intermittency can occur as the system experiences an inverted period doubling bifurcation. At these bifurcations, multiple attractors with fractal basin boundaries can be created. It is found the dimension scales, with the parameter, according to the power law d = d_{o } - k| p - p_{c}| ^{beta} with beta = 1/2, where p is the system parameter, p _{c} is the bifurcation value, k is a scaling constant, and d_{o} is the dimension of the basin boundaries at p _{c}. For type I intermittency d_{o} < D and d _{o} = D for type III intermittency. This scaling was confirmed in numerical experiments near the type I and type III intermittency creating bifurcations values for the forced damped pendulum and the type I intermittency creating bifurcation value of period-3 window for the logistic map. (Abstract shortened with permission of author.).

Park, Bae-Sig

8

The fractal dimension of the wind

In this paper the concept of a fractal and its dimension is presented. The fractal dimension of the horizontal component of several wind speed time series is determined by the variation method. It is concluded that the fractal dimensions D=1.60 can be used to describe the wind both at Windsor, MA, and Altamont, CA. The fractal dimension is related to the universal estimate of the decay rate of turbulent kinetic energy in the wind.

Syu, C.Y.; Kirchhoff, R.H. (Univ. of Massachusetts, Amherst, MA (United States). Dept. of Mechanical Engineering)

1993-08-01

9

Fractal Compression Coding Based on Fractal Dimension Feature Blocks

The paper associates fractal dimension with the image disorder degree, which is introduced into pre-classification for fractal compression coding process together with the concept of quartered-tree coding, then a fractal compression coding approach based on fractal dimension feature blocks is brought forward. The algorithm improves coding time, compression ratio, but the peak-peak signal-noise ratio suffers a little. The validity of

Kai Shuang; Ning Xiao; Feng Xu; Dayue Lv; Wang Yu

2008-01-01

10

How to calculate the fractal dimension of a complex network: the box covering algorithm

NASA Astrophysics Data System (ADS)

Covering a network with the minimum possible number of boxes can reveal interesting features for the network structure, especially in terms of self-similar or fractal characteristics. Considerable attention has been recently devoted to this problem, with the finding that many real networks are self-similar fractals. Here we present, compare and study in detail a number of algorithms that we have used in previous papers towards this goal. We show that this problem can be mapped to the well-known graph colouring problem and then we simply can apply well-established algorithms. This seems to be the most efficient method, but we also present two other algorithms based on burning which provide a number of other benefits. We argue that the algorithms presented provide a solution close to optimal and that another algorithm that can significantly improve this result in an efficient way does not exist. We offer to anyone that finds such a method to cover his/her expenses for a one-week trip to our lab in New York (details in http://jamlab.org).

Song, Chaoming; Gallos, Lazaros K.; Havlin, Shlomo; Makse, Hernán A.

2007-03-01

11

Fractal Dimensions of Cranial Sutures and Waveforms

Two quite different shapes of cranial sutures ostensibly yield fractal dimensions. The rare, intricate sutures yield the more valid fractal dimensions because self-similar scaling provides a double-log plot of negative slope. These sutures are fractals over a range of several r values. Some of the highly folded, wavy sutures in humans also fill space except at very tiny values of

C. A. Long; J. E. Long

1992-01-01

12

A Fractal Dimension Survey of Active Region Complexity

NASA Technical Reports Server (NTRS)

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 .

McAteer, R. T. James; Gallagher, Peter; Ireland, Jack

2005-01-01

13

Fractal dimension of bioconvection patterns

NASA Technical Reports Server (NTRS)

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.

Noever, David A.

1990-01-01

14

Fractal dimensions of coals and cokes

Using adsorption data, the authors get formulas for the calculation of fractal dimensions: log[A{sub CO{sub 2}(DP)}/A{sub N{sub 2}(BET)}] = {minus}5.3984(2 {minus} D{sub 1})/2 and log[A{sub CO{sub 2}(BET)}/A{sub N{sub 2}(BET)}] = {minus}4.9569(2 {minus} D{sub 2})/2. The fractal dimensions (D) of 27 coals and 2 cokes have been obtained. The D of coals decreased with the increase of f{sub a} and reached a maximum at H/C equal to 0.66 (or C{sub daf} about 86%). The fractal dimension is relative to ash and volatiles of coal: D = 2.2237 + 0.6249 V{sub daf} + 0.8863 A{sub d}. The relationship between D of coal coke and its conversions (X) obeys the following equation: D = a exp({minus}bX) + c.

Xu, L.; Zhang, D.; Xian, X. [Chongqing Univ. (China). Coll. of Resources and Environment Engineering] [Chongqing Univ. (China). Coll. of Resources and Environment Engineering

1997-06-15

15

Fractal dimension of cerebral surfaces using magnetic resonance images

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.

Majumdar, S.; Prasad, R.R.

1988-11-01

16

Fractal dimensions for diffusion-limited aggregation

A mean-field theory is proposed for fractal dimensions of diffusion-limited aggregates grown on a substrate surface of arbitrary dimensionality. The results are in good agreement with those of the computer simulations for all dimensionalities.

M. Tokuyama; K. Kawasaki

1984-01-01

17

Fractal fractal dimensions of deterministic transport coefficients

If a point particle moves chaotically through a periodic array of scatterers th ea ssociated transport coefficients are typically irregular functions under variation of control parameters. For a piecewise linear two-parameter map we analyse the structure of the associated irregular diffusion coefficient and current by numerically computing dimensions from box-counting and from the autocorrelation function of these graphs. We find

TU Dresden; Fachrichtung Mathematik

18

Surface fractal dimension of single-walled carbon nanotubes

NASA Astrophysics Data System (ADS)

Isolated single-walled carbon nanotubes (SWNTs), SWNT bundles, and ropes (or strands) show a structural self-similar characteristic. By calculating the Hausdorff dimension, it was found that their self-similar organization leads to surface fractality and the value of the surface dimension (Ds) depends on self-similar patterns. Experimentally, Ds obtained by nitrogen adsorption measurements at 77.3 K and by the small-angle x-ray scattering technique in our study proved our calculation that the surface dimension of SWNTs is nonintegral, 2

Sun, Cheng-Hua; Li, Feng; Ying, Zhe; Liu, Chang; Cheng, Hui-Ming

2004-01-01

19

Application of Fractal Dimension on Palsar Data

NASA Astrophysics Data System (ADS)

Study of land cover is the primal task of remote sensing where microwave imaging plays an important role. As an alternate of optical imaging, microwave, in particular, Synthetic Aperture Radar (SAR) imaging is very popular. With the advancement of technology, multi-polarized images are now available, e.g., ALOS-PALSAR (Phased Array type L-band SAR), which are beneficial because each of the polarization channel shows different sensitivity to various land features. Further, using the textural features, various land classes can be classified on the basis of the textural measures. One of the textural measure is fractal dimension. It is noteworthy that fractal dimension is a measure of roughness and thus various land classes can be distinguished on the basis of their roughness. The value of fractal dimension for the surfaces lies between 2.0 and 3.0 where 2.0 represents a smooth surface while 3.0 represents drastically rough surface. The study area covers subset images lying between 2956'53"N, 7750'32"E and 2950'40"N, 7757'19"E. The PALSAR images of the year 2007 and 2009 are considered for the study. In present paper a fractal based classification of PALSAR images has been performed for identification of Water, Urban and Agricultural Area. Since fractals represent the image texture, hence the present study attempts to find the fractal properties of land covers to distinguish them from one another. For the purpose a context has been defined on the basis of a moving window, which is used to estimate the local fractal dimension and then moved over the whole image. The size of the window is an important issue for estimation of textural measures which is considered to be 55 in present study. This procedure, in response, produces a textural map called fractal map. The fractal map is constituted with the help of local fractal dimension values and can be used for contextual classification. In order to study the fractal properties of PALSAR images, the three polarization images viz. HH (Horizontal-Horizontal Polarization), VV (Vertical-Vertical Polarization) and HV (Horizontal-Vertical Polarization) are considered individually. First of all each polarized image is classified in an unsupervised way and various clusters, i.e., four clusters are identified with the help of reference data as Water, Urban and Agricultural Area. For each cluster, the fractal dimension is obtained from the fractal map. Based on the study the ranges of fractal dimension for three classes are Water: 2.0-2.17, Agricultural Area: 2.24-2.72, Urban Area: 2.63-2.92 for HH polarized image; Water: 2.0-2.21, Agricultural Area: 2.20-2.64, Urban; 2.58-2.94 for VV polarized image and Water: 2.0-2.14, Agricultural Area: 2.18-2.58, Urban: 2.46-2.94 for HV polarized image. Since the class Others represents a mixture of various classes, an explicit range of D for this class can not be determined. A closer look at the ranges of fractal dimension indicates that there is an overlapping of the values for different classes, despite of which the classes can be distinguished. Also, the class Water having low value of fractal dimension can be treated as smooth and Urban Area having higher values of fractal dimension can be considered rough in structure while the class Agricultural Area shows an intermediate roughness.

Singh, Dharmendra; Pant, Triloki

20

Trabecular Bone Mechanical Properties and Fractal Dimension

NASA Technical Reports Server (NTRS)

Countermeasures for reducing bone loss and muscle atrophy due to extended exposure to the microgravity environment of space are continuing to be developed and improved. An important component of this effort is finite element modeling of the lower extremity and spinal column. These models will permit analysis and evaluation specific to each individual and thereby provide more efficient and effective exercise protocols. Inflight countermeasures and post-flight rehabilitation can then be customized and targeted on a case-by-case basis. Recent Summer Faculty Fellowship participants have focused upon finite element mesh generation, muscle force estimation, and fractal calculations of trabecular bone microstructure. Methods have been developed for generating the three-dimensional geometry of the femur from serial section magnetic resonance images (MRI). The use of MRI as an imaging modality avoids excessive exposure to radiation associated with X-ray based methods. These images can also detect trabecular bone microstructure and architecture. The goal of the current research is to determine the degree to which the fractal dimension of trabecular architecture can be used to predict the mechanical properties of trabecular bone tissue. The elastic modulus and the ultimate strength (or strain) can then be estimated from non-invasive, non-radiating imaging and incorporated into the finite element models to more accurately represent the bone tissue of each individual of interest. Trabecular bone specimens from the proximal tibia are being studied in this first phase of the work. Detailed protocols and procedures have been developed for carrying test specimens through all of the steps of a multi-faceted test program. The test program begins with MRI and X-ray imaging of the whole bones before excising a smaller workpiece from the proximal tibia region. High resolution MRI scans are then made and the piece further cut into slabs (roughly 1 cm thick). The slabs are X-rayed again and also scanned using dual-energy X-ray absorptiometry (DEXA). Cube specimens are then cut from the slabs and tested mechanically in compression. Correlations between mechanical properties and fractal dimension will then be examined to assess and quantify the predictive capability of the fractal calculations.

Hogan, Harry A.

1996-01-01

21

ERIC Educational Resources Information Center

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)

Esbenshade, Donald H., Jr.

1991-01-01

22

Fractal Dimension in Eeg Signals during Muscle Fatigue

NASA Astrophysics Data System (ADS)

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.

Huang, Haibin; Yao, Bin; Yue, Guang; Brown, Robert; Jing, Liu

2003-10-01

23

Fractal dimension of microbead assemblies used for protein detection.

We use fractal analysis to calculate the protein concentration in a rotating magnetic assembly of microbeads of size 1 ?m, which has optimized parameters of sedimentation, binding sites and magnetic volume. We utilize the original Forrest-Witten method, but due to the relatively small number of bead particles, which is of the order of 500, we use a large number of origins and also a large number of algorithm iterations. We find a value of the fractal dimension in the range 1.70-1.90, as a function of the thrombin concentration, which plays the role of binding the microbeads together. This is in good agreement with previous results from magnetorotation studies. The calculation of the fractal dimension using multiple points of reference can be used for any assembly with a relatively small number of particles. PMID:25195559

Hecht, Ariel; Commiskey, Patrick; Lazaridis, Filippos; Argyrakis, Panos; Kopelman, Raoul

2014-11-10

24

Fractal Dimensions and Entropies of Meragi Songs

NASA Astrophysics Data System (ADS)

Melodies can be treated as time series

Aydemir, Adnan; Gündüz, Güngör

25

Fractal dimension measurement of engineering surfaces

Many types of engineered surfaces have been observed to exhibit a fractal geometry. In some cases, modeling the generation of the surfaces predicts this and provides correlation between the dimension and history and properties of the surfaces. The complex process of machining precision surfaces does not yet lend itself to such detailed modeling, but observations of the correlations between production

John C. Russ

1998-01-01

26

The Correlation Fractal Dimension of Complex Networks

NASA Astrophysics Data System (ADS)

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.

Wang, Xingyuan; Liu, Zhenzhen; Wang, Mogei

2013-05-01

27

Fractal dimension of alumina aggregates grown in two dimensions

NASA Technical Reports Server (NTRS)

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.

Larosa, Judith L.; Cawley, James D.

1992-01-01

28

Estimation of fractal dimensions from transect data

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.

Loehle, C. [Argonne National Lab., IL (United States)

1994-04-01

29

The fractality of lung sounds: A comparison of three waveform fractal dimension algorithms

The fractality of lung sounds: A comparison of three waveform fractal dimension algorithms January Gnitecki *, Zahra Moussavi Faculty of Engineering, Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB Canada R3T 5V6 Abstract Three waveform fractal dimension (FD) analyses

Moussavi, Zahra M. K.

30

Use of Multiple Fractal Dimensions to Quantify Airborne Particle Shape

Fractal dimension has been considered as a quite useful index for quantifying irregular or self-similar shapes. However, shapes of airborne particles may not be fully self-similar and may not be well characterized by a single fractal dimension. Alternatively, they may this study. The results are compared with the result of single fractal dimension.be self-similar at different levels of scale and

Ying Xie; Philip K. Hopke; Gary Casuccio; Brad Henderson

1994-01-01

31

Fractal dimension in nonhyperbolic chaotic scattering

In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic (e.g., there are Kolmogorov-Arnol'd-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic case, although the singular set has zero total length, we present strong evidence that its fractal dimension is 1.

Lau, Y.; Finn, J.M.; Ott, E. (Laboratory for Plasma Research, University of Maryland, College Park, Maryland 20742-3511 (US))

1991-02-25

32

Fractal dimension in nonhyperbolic chaotic scattering

NASA Technical Reports Server (NTRS)

In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic (e.g., there are Kolmogorov-Arnol'd-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic case, although the singular set has zero total length, strong evidence is presented to show that its fractal dimension is 1.

Lau, Yun-Tung; Finn, John M.; Ott, Edward

1991-01-01

33

Single cell correlation fractal dimension of chromatin

Chromatin is a major nuclear component, and it is an active matter of debate to understand its different levels of spatial organization, as well as its implication in gene regulation. Measurements of nuclear chromatin compaction were recently used to understand how DNA is folded inside the nucleus and to detect cellular dysfunctions such as cancer. Super-resolution imaging opens new possibilities to measure chromatin organization in situ. Here, we performed a direct measure of chromatin compaction at the single cell level. We used histone H2B, one of the 4 core histone proteins forming the nucleosome, as a chromatin density marker. Using photoactivation localization microscopy (PALM) and adaptive optics, we measured the three-dimensional distribution of H2B with nanometric resolution. We computed the distribution of distances between every two points of the chromatin structure, namely the Ripley K(r) distribution. We found that the K(r) distribution of H2B followed a power law, leading to a precise measurement of the correlation fractal dimension of chromatin of 2.7. Moreover, using photoactivable GFP fused to H2B, we observed dynamic evolution of chromatin sub-regions compaction. As a result, the correlation fractal dimension of chromatin reported here can be interpreted as a dynamically maintained non-equilibrium state. PMID:24637833

Récamier, Vincent; Izeddin, Ignacio; Bosanac, Lana; Dahan, Maxime; Proux, Florence; Darzacq, Xavier

2014-01-01

34

Spatial-temporal variability of coastline in Bohai Rim based on fractal dimension

NASA Astrophysics Data System (ADS)

This paper extracted the spatial distribution of the continental coastline of Bohai Rim utilizing Remote Sensing and GIS spatial analysis techniques, and calculated the fractal dimension of the coastline by boxcounting method, with a time from 1990 to 2010. Moreover, we analyzed the characteristics of spatialtemporal variability of the coastline's length and fractal dimension, the relationship between the large scales length change and fractal dimension change. During the research period, the coastline length of the study area increased progressively and the most significant change in coastline length was found in Tianjin Municipality. Especially after 2000, the coastline length entered a period of rapid growth. In addition, the fractal dimension of the overall coastline of the study area was between the fractal dimensions of the regional coastlines and was close to the maximum fractal dimensions of these regional coastlines. The fractal dimension of the coastline in Bohai Rim was increasing during the research period, large scale project such as ports construction, reduced tortuous degree of the coastline.

Xu, Ning; Gao, Zhiqiang; Ning, Jicai; Liu, Xiangyang

2014-10-01

35

Maximum likelihood estimator for fractal dimension of fractal signal plus noise

A maximum likelihood estimator (MLE) is developed for estimating the fractal dimension of single variable fraction Brownian motion (fBm) that was corrupted by additive noise. The procedure involve the construction of signal plus noise model and from the model we derive the likelihood function of the signal plus noise. We then estimate the fractal dimension by maximizing the likelihood function.

Adil S. Balghonaim; J. M. Keller

1997-01-01

36

Fractal dimensions and dimensional crossover in superconducting wire networks

We have fabricated Al wire networks in the form of submicron scale fractals including the Sierpinski Gasket (SG), arrays of SG's, and percolation clusters. Measurements have been made of the superconducting phase boundary, Tc(H), and the magnetoconductance above Tc. From analysis of the SG data we have determined the fracton dimension of that fractal. The gasket arrays show a crossover

James M. Gordon; Allen M. Goldman

1988-01-01

37

Local complex dimensions of a fractal string Jacques Levy Vehel

UniversitÂ´e, 4 rue J. Monod, 91893 Orsay Cedex France, jacques.levy-vehel@inria.fr Department of MathematicsLocal complex dimensions of a fractal string Jacques LÂ´evy VÂ´ehel , Franklin Mendivil August 13

Paris-Sud XI, UniversitÃ© de

38

In this paper, a pilot study regarding carotid atherosclerotic plaque instability using B-mode ultrasound (US) images was carried out. The fractal dimension of plaques obtained from ten symptomatic subjects (i.e., subjects having experienced neurological symptoms) as well as from nine asymptomatic subjects, was estimated using a novel method, called the kth nearest neighbour (KNN) method. The results indicated a significant difference, as per the fractal dimension, between the two groups, providing a significantly lower value for the asymptomatic group. Moreover, the phase of the cardiac cycle (systole/diastole) during which the fractal dimension was estimated had no systematic effect on the calculations. The fractal dimension of the plaques was also estimated using a well-known method, namely the box-counting (BC) method. No significant differences between the two groups, as per the fractal dimension, were observed using the BC method. The presented pilot study suggests that the fractal dimension, estimated by the proposed method, could be used as a single determinant for the discrimination of symptomatic and asymptomatic subjects. PMID:12401382

Asvestas, Pantelis; Golemati, Spyretta; Matsopoulos, George K; Nikita, Konstantina S; Nicolaides, Andrew N

2002-09-01

39

Time evolution of the fractal dimension of a mixing front

NASA Astrophysics Data System (ADS)

We present a description of an experimental study of an array of turbulent plumes (from one to nine plumes), investigating the time evolution of the fractal dimension of the plumes and also the spatial evolution of the fractal dimension from one plume to other. We also investigate the effects of bouyancy (different Atwood numbers), the number of plumes and the height of the bouyancy source on the fractal dimension. The plumes are formed by injecting a dense fluid from a small source (from one to nine orifices) into a stationary body of lighter brime (saline solution) contained in a tank. The source fluid was dyed with fluorescein and we use the LIF technique. The plumes were fully turbulent and we have both momentum and bouyancy regimes. The fractal dimensions of contours of concentration were measured. The fractal analysis of the turbulent convective plumes was performed with the box counting algorithm for different intensities of evolving plume images using the special software Ima_Calc. Fractal dimensions between 1.3 and 1.35 are obtained from box counting methods for free convection and neutral boundary layers. Other results have been published which use the box counting method to analyze images of jet sections -produced from LIF techniques. The regions where most of the mixing takes place are also compared with Reactive flow experiments using phenolphthalein and acid-base interfaces performed by Redondo(1994) IMA 43. Eds M. Farge, JC Hunt and C. Vassilicos.

Lopez Gonzalez-Nieto, P.; Grau, J.

2009-04-01

40

The spectrum of fractal dimensions of passively convected scalar gradients

The passive convection of scalar fields by an incompressible fluid flow in two dimensions is investigated numerically. The prescribed flow is chaotic meaning that nearby fluid elements diverge exponentially with time. The gradient of the convected scalar field is of primary interest, and a measure is defined, reflecting the spatial distribution of the regions having large gradient. The dimension spectrum for this measure is computed by the standard box counting technique, and it is found to be fractal. A recent theory proposes that the fractal structure of the scalar gradient can be related to the nonuniform stretching properties of the flow. Using this theory, the fractal dimension spectrum is computed from the distribution of finite time Lyapunov exponents of the flow, and it is found to be in reasonable agreement with the dimension spectrum computed directly by means of box counting.

Varosi, F.; Antonsen, T.M. Jr.; Ott, E. (Univ. of Maryland, College Park, MD (United States))

1991-05-01

41

Describing soil surface microrelief by crossover length and fractal dimension

NASA Astrophysics Data System (ADS)

Accurate description of soil surface topography is essential because different tillage tools produce different soil surface roughness conditions, which in turn affects many processes across the soil surface boundary. Advantages of fractal analysis in soil microrelief assessment have been recognised but the use of fractal indices in practice remains challenging. There is also little information on how soil surface roughness decays under natural rainfall conditions. The objectives of this work were to investigate the decay of initial surface roughness induced by natural rainfall under different soil tillage systems and to compare the performances of a classical statistical index and fractal microrelief indices. Field experiments were performed on an Oxisol at Campinas, São Paulo State (Brazil). Six tillage treatments, namely, disc harrow, disc plow, chisel plow, disc harrow + disc level, disc plow + disc level and chisel plow + disc level were tested. Measurements were made four times, firstly just after tillage and subsequently with increasing amounts of natural rainfall. Duplicated measurements were taken per treatment and date, yielding a total of 48 experimental surfaces. The sampling scheme was a square grid with 25×25 mm point spacing and the plot size was 1350×1350 mm, so that each data set consisted of 3025 individual elevation points. Statistical and fractal indices were calculated both for oriented and random roughness conditions, i.e. after height reading have been corrected for slope and for slope and tillage tool marks. The main drawback of the standard statistical index random roughness, RR, lies in its no spatial nature. The fractal approach requires two indices, fractal dimension, D, which describes how roughness changes with scale, and crossover length, l, specifying the variance of surface microrelief at a reference scale. Fractal parameters D and l, were estimated by two independent self-affine models, semivariogram (SMV) and local root mean square (RMS). Both algorithms, SMV and RMS, gave equivalent results for D and l indices, irrespective of trend removal procedure, even if some bias was present which is in accordance with previous work. Treatments with two tillage operations had the greatest D values, irrespective of evolution stage under rainfall and trend removal procedure. Primary tillage had the greatest initial values of RR and l. Differences in D values between treatments with primary tillage and those with two successive tillage operations were significant for oriented but not for random conditions. The statistical index RR and the fractal indices l and D decreased with increasing cumulative rainfall following different patterns. The l and D decay from initial value was very sharp after the first 24.4 mm cumulative rainfall. For five out of six tillage treatments a significant relationship between D and l was found for the random microrelief conditions allowing a covariance analysis. It was concluded that using RR or l together with D best allow joint description of vertical and horizontal soil roughness variations.

Vidal Vázquez, E.; Miranda, J. G. V.; Paz González, A.

2007-05-01

42

The use of fractal dimension in engineering geology

The method of fractal geometry allows the simulation as well as description of data of many different natural states of orientation,\\u000a distribution and consistency. This affords to the engineering geologist various new possibilities of identifying rock mass\\u000a and rock. We have tried to use fractal dimension (FD) for characterising roughness profiles of shear faces as well as fracture\\u000a trace maps

F. J. Brosch; P. Pölsler; G. Riedmüller

1992-01-01

43

Daily variation of the fractal dimension of the velocity components in the turbulent surface layer

NASA Astrophysics Data System (ADS)

The turbulence is a dominant property within the Planetary Boundary Layer (PBL). It is the main characteristic of the mixing in the lower atmosphere since the atmospheric turbulent fluxes are more efficient than the molecular diffusion. Turbulence can be observed in time series of meteorological variables (wind velocity for example). The sampling rate of observation in that time series has to be high in order to detect the turbulent regime. The analysis of these series presents a self-similarity structure, so the wind velocity can be considered as a fractal magnitude. This work shows a study of the fractal dimension of the wind perturbation series u'and w'components of the wind speed. Fractal dimension of velocity components can be related to others turbulent characteristics of the fluxes close to the ground. Fluctuation of longitudinal and, specially, vertical components depend on stability and, therefore, on the solar cycle. In consequence, the behaviour of fractal dimension should be in agreement with that cycle also. These series have been obtained once it has carried out the necessary transformation to get the mean wind series in short intervals, namely 5 minutes, to ensure the consistent properties of turbulence. The original records available were taken every thirty minutes by sonic anemometers (20 Hz sampling rate) during a week of a field campaign. The data analysed was recorded in the experimental campaign SABLES-98 at the Research Centre for the Lower Atmosphere (CIBA), located in Valladolid province (Spain). It has been calculated the fractal dimension (Komolgorov capacity or box- counting dimension) of the time series of fluctuations of the velocity component along of the mean wind direction and the vertical component (u' = u-U, w' = w -W), both in the physical spaces (velocity-time). It has been studied the time evolution of the fractal dimension during several days and at three levels above the ground (5.8 m, 13.5 m, 32 m). The fractal dimension of theu' and w' components of wind velocity series have been studied, as well as the influence of different turbulent parameters depending on daily cycle: turbulent kinetic energy, friction velocity, difference of temperature between the extreme of the layer studied close of the surface (?T50-0.22m),etc. It has been observed that there is a possible correlation between the fractal dimension and some of these turbulent parameters. Finally, it has been analysed the variation of the fractal dimension versus stability obtained from the Richardson number along of the day.

Tijera, M.; Maqueda, G.; Yagüe, C.; Cano, J. L.

2012-04-01

44

A rapid and accurate method of measuring fractal dimension of a fracture surface is described. The method uses a stereo pair of photomicrographs taken in a scanning electron microscope and digitized for computer analysis. The computer can automatically compare the offset at many places in the two images and calculate their height on the basis of the parallax angle. The

J. J. Friel; C. S. Pande

1993-01-01

45

Fractal dimension and turbulence in Giant HII Regions

We have measured the fractal dimensions of the Giant HII Regions Hubble X and Hubble V in NGC6822 using images obtained with the Hubble's Wide Field Planetary Camera 2 (WFPC2). These measures are associated with the turbulence observed in these regions, which is quantified through the velocity dispersion of emission lines in the visible. Our results suggest low turbulence behaviour.

H. E. Caicedo-Ortiz; E. Santiago-Cortés; J. López-Bonilla; H. O. Castañeda

2015-01-20

46

Fractal dimension and turbulence in Giant HII Regions

We have measured the fractal dimensions of the Giant HII Regions Hubble X and Hubble V in NGC6822 using images obtained with the Hubble's Wide Field Planetary Camera 2 (WFPC2). These measures are associated with the turbulence observed in these regions, which is quantified through the velocity dispersion of emission lines in the visible. Our results suggest low turbulence behaviour.

Caicedo-Ortiz, H E; López-Bonilla, J; Castañeda, H O

2015-01-01

47

Original article Structure and fractal dimensions of root systems

Original article Structure and fractal dimensions of root systems of four co-occurring fruit tree, Bangor, Gwynedd LL57 2UW, UK (Received 1 February 1999; accepted 29 October 1999) Abstract Â Coarse root-auto- matically digitized. Spatial distributions of root length were determined from the digitally

Paris-Sud XI, UniversitÃ© de

48

Fractal dimensions and ƒ;(?) spectrum of the Hénon attractor

NASA Astrophysics Data System (ADS)

We measure the generalized fractal dimensions Dq( q?0) of the Hénon attractor by the box counting and spatial correlation methods. The technique of virtual memory is exploited to handle the extremely large numbers of iterates needed for the convergence of the algorithms. We study quantitatively the oscillations which appear in the usual linear regressions of the log-log plot and which are inherent in lacunar fractal sets. These oscillations are the cause of previous underestimates of the Renyi dimensions and in fact make accurate dimension estimates an elusive goal. The Legendre transform of the D q yields the ƒ(?) spectrum which characterizes the multifractal structure of the attractor. We point out that this spectrum of singularities can be extracted directly from the computed invariant measure, avoiding the log-log regression procedure.

Arneodo, A.; Grasseau, G.; Kostelich, Eric J.

1987-10-01

49

Estimation of Fractal Dimension in Differential Diagnosis of Pigmented Skin Lesions

NASA Astrophysics Data System (ADS)

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.

Aralica, Gorana; Miloševi?, Danko; Konjevoda, Paško; Seiwerth, Sven; Štambuk, Nikola

50

On the fractal dimension of sea surface backscattered signal at low grazing angle

Fractal analysis of sea surface backscattering signal (sea clutter in radar terminology) represents a novel technique for the study of sea surface roughness. When Kirchhoff's assumption is satisfied, the fractal dimension of the signal is linearly related to the fractal dimension of the sea surface. Moreover, such a relationship is independent of transmitted frequency, polarization, time, space, sea wave propagation

Marco Martorella; Fabrizio Berizzi; Enzo Dalle Mese

2004-01-01

51

Construction of Fractal Surfaces by Recurrent Fractal Interpolation Curves

A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system(RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal surfaces by fractal curves and Lipschitz functions and calculate the box-counting dimension of the constructed surfaces. Finally, we combine both methods to have more flexible constructions of fractal surfaces.

Chol-hui Yun; Hyong-chol O.; Hui-chol Choi

2013-03-04

52

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as $\\lambda \\to \\infty$, $\\dim (\\sigma(H_\\lambda)) \\cdot \\log \\lambda$ converges to an explicit constant ($\\approx 0.88137$). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schr\\"odinger dynamics generated by the Fibonacci Hamiltonian.

David Damanik; Mark Embree; Anton Gorodetski; Serguei Tcheremchantsev

2007-05-02

53

Fractal dimension analysis of malignant and benign endobronchial ultrasound nodes

Background Endobronchial ultrasonography (EBUS) has been applied as a routine procedure for the diagnostic of hiliar and mediastinal nodes. The authors assessed the relationship between the echographic appearance of mediastinal nodes, based on endobronchial ultrasound images, and the likelihood of malignancy. Methods The images of twelve malignant and eleven benign nodes were evaluated. A previous processing method was applied to improve the quality of the images and to enhance the details. Texture and morphology parameters analyzed were: the image texture of the echographies and a fractal dimension that expressed the relationship between area and perimeter of the structures that appear in the image, and characterizes the convoluted inner structure of the hiliar and mediastinal nodes. Results Processed images showed that relationship between log perimeter and log area of hilar nodes was lineal (i.e. perimeter vs. area follow a power law). Fractal dimension was lower in the malignant nodes compared with non-malignant nodes (1.47(0.09), 1.53(0.10) mean(SD), Mann–Whitney U test p?Fractal dimension of ultrasonographic images of mediastinal nodes obtained through endobronchial ultrasound differ in malignant nodes from non-malignant. This parameter could differentiate malignat and non-malignat mediastinic and hiliar nodes. PMID:24920158

2014-01-01

54

NASA Astrophysics Data System (ADS)

A rapid and accurate method of measuring fractal dimension of a fracture surface is described. The method uses a stereo pair of photomicrographs taken in a scanning electron microscope and digitized for computer analysis. The computer can automatically compare the offset at many places in the two images and calculate their height on the basis of the parallax angle. The sum of the area of all planes formed by an array of three-dimensional points is an approximation of the true surface area. The scale can be varied both in the microscope and in the computer; therefore, a fractal dimension can be calculated. Unlike previous methods, this one is direct and gave results intermediate between two prior indirect measurements.

Friel, J. J.; Pande, C. S.

1993-01-01

55

A new method of diesel engine fault diagnosis that uses image recognition technology based on fractal dimension is proposed. The Wigner-Ville distributions of six kinds of vibration acceleration signals which are acquired from diesel engine cylinder head are calculated by time-frequency analysis, and a series of time-frequency gray images can be obtained from above distributions by image processing. According to

Yanping Cai; Shu Cheng; Yanping He; Ping Xu

2008-01-01

56

Mapping soil fractal dimension in agricultural fields with GPR

NASA Astrophysics Data System (ADS)

We documented that the mapping of the fractal dimension of the backscattered Ground Penetrating Radar traces (Fractal Dimension Mapping, FDM) accomplished over heterogeneous agricultural fields gives statistically sound combined information about the spatial distribution of Andosol' dielectric permittivity, volumetric and gravimetric water content, bulk density, and mechanical resistance under seven different management systems. The roughness of the recorded traces was measured in terms of a single number H, the Hurst exponent, which integrates the competitive effects of volumetric water content, pore topology and mechanical resistance in space and time. We showed the suitability to combine the GPR traces fractal analysis with routine geostatistics (kriging) in order to map the spatial variation of soil properties by nondestructive techniques and to quantify precisely the differences under contrasting tillage systems. Three experimental plots with zero tillage and 33, 66 and 100% of crop residues imprinted the highest roughness to GPR wiggle traces (mean HR/S=0.15), significantly different to Andosol under conventional tillage (HR/S=0.47).

Oleschko, K.; Korvin, G.; Muñoz, A.; Velazquez, J.; Miranda, M. E.; Carreon, D.; Flores, L.; Martínez, M.; Velásquez-Valle, M.; Brambila, F.; Parrot, J.-F.; Ronquillo, G.

2008-09-01

57

Surface Evaluation by Estimation of Fractal Dimension and Statistical Tools

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

Salac, Petr

2014-01-01

58

NASA Astrophysics Data System (ADS)

By applying fractal geometry analysis to the drainage network of three large watercourses in America and Europe, we have calculated for the first time their fractal dimension. The aim is to interpret the geomorphologic characteristics to better understand the morphoevolutionary processes of these fluvial morphotypes; to identify and discriminate geomorphic phenomena responsible for any difference or convergence of a fractal dimension; to classify hydrographic patterns, and finally to compare the fractal degree with some geomorphic-quantitative indexes. The analyzed catchment of Russian (California, USA), Ebro (Spain), and Volturno (Italy) rivers are situated in Mediterranean-climate regions sensu Köppen, but with different geologic context and tectonic styles. Results show fractal dimensions ranging from 1.08 to 1.50. According to the geological setting and geomorphic indexes of these basins, the lower fractal degree indicates a prevailing tectonics, active or not, while the higher degree indicates the stronger erosion processes on inherited landscapes.

Donadio, Carlo; Magdaleno, Fernando; Mazzarella, Adriano; Mathias Kondolf, G.

2014-08-01

59

Magnetic ordering of spin systems having fractal dimensions Experimental study

NASA Astrophysics Data System (ADS)

It is well-known that cooperative properties such as magnetic ordering can depend on the samples’ dimensions ( Ds) in a qualitative way. However, there have been no samples with well-defined non-integer Ds. The dimension of a given sample has been always discussed on the anisotropy of the electronic/crystal/magnetic structures, which has no definition suitable for quantitative discussion on dimensions vs. properties. On the other hand a particular type of porous samples, i.e. fractal bodies, can have well-defined non-integer Ds dependent exclusively on the geometrical feature of structures, and physical properties of such materials remains unexplored. This paper reports on magnetic ordering in samples covering 2.5 ? D ? 3, in addition to a way of precise control of the fractal dimensions of given samples simply by wax (alkylketene dimer). The results show that the magnetic ordering temperatures, i.e. Néel temperatures ( T N s), of CoO depend on D, and rapidly enhance immediately below D = 3. This means that one can control or enhance the critical temperature simply by tuning D with keeping the remaining magnetic properties unchanged.

Naito, T.; Yamamoto, H.; Okuda, K.; Konishi, K.; Mayama, H.; Yamaguchi, D.; Koizumi, S.; Kubo, K.; Nakamura, T.

2013-10-01

60

Fractal Dimension of Geologically Constrained Crater Populations of Mercury

NASA Astrophysics Data System (ADS)

Data gathered during the Mariner10 and MESSENGER missions are collated in this paper to classify craters into four geo-chronological units constrained to the geological map produced after MESSENGER's flybys. From the global catalogue, we classify craters, constraining them to the geological information derived from the map. We produce a size frequency distribution (SFD) finding that all crater classes show fractal behaviour: with the number of craters inversely proportional to their diameter, the exponent of the SFD (i.e., the fractal dimension of each class) shows a variation among classes. We discuss this observation as possibly being caused by endogenic and/or exogenic phenomena. Finally, we produce an interpretative scenario where, assuming a constant flux of impactors, the slope variation could be representative of rheological changes in the target materials.

Mancinelli, Paolo; Pauselli, Cristina; Perugini, Diego; Lupattelli, Andrea; Federico, Costanzo

2014-08-01

61

Purpose To assess bony trabecular changes potentially caused by loading stress around dental implants using fractal dimension analysis. Methods Fractal dimensions were measured in 48 subjects by comparing radiographs taken immediately after prosthesis delivery with those taken 1 year after functional loading. Regions of interest were isolated, and fractal analysis was performed using the box-counting method with Image J 1.42 software. Wilcoxon signed-rank test was used to analyze the difference in fractal dimension before and after implant loading. Results The mean fractal dimension before loading (1.4213±0.0525) increased significantly to 1.4329±0.0479 at 12 months after loading (P<0.05). Conclusions Fractal dimension analysis might be helpful in detecting changes in peri-implant alveolar trabecular bone patterns in clinical situations. PMID:24236242

Mu, Teh-Jing; Lee, Dong-Won; Park, Kwang-Ho

2013-01-01

62

Hyper-Fractal Analysis: A visual tool for estimating the fractal dimension of 4D objects

NASA Astrophysics Data System (ADS)

This work presents a new version of a Visual Basic 6.0 application for estimating the fractal dimension of images and 3D objects (Grossu et al. (2010) [1]). The program was extended for working with four-dimensional objects stored in comma separated values files. This might be of interest in biomedicine, for analyzing the evolution in time of three-dimensional images. New version program summaryProgram title: Hyper-Fractal Analysis (Fractal Analysis v03) Catalogue identifier: AEEG_v3_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEG_v3_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 745761 No. of bytes in distributed program, including test data, etc.: 12544491 Distribution format: tar.gz Programming language: MS Visual Basic 6.0 Computer: PC Operating system: MS Windows 98 or later RAM: 100M Classification: 14 Catalogue identifier of previous version: AEEG_v2_0 Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 831-832 Does the new version supersede the previous version? Yes Nature of problem: Estimating the fractal dimension of 4D images. Solution method: Optimized implementation of the 4D box-counting algorithm. Reasons for new version: Inspired by existing applications of 3D fractals in biomedicine [3], we extended the optimized version of the box-counting algorithm [1, 2] to the four-dimensional case. This might be of interest in analyzing the evolution in time of 3D images. The box-counting algorithm was extended in order to support 4D objects, stored in comma separated values files. A new form was added for generating 2D, 3D, and 4D test data. The application was tested on 4D objects with known dimension, e.g. the Sierpinski hypertetrahedron gasket, Df=ln(5)/ln(2) (Fig. 1). The algorithm could be extended, with minimum effort, to higher number of dimensions. Easy integration with other applications by using the very simple comma separated values file format for storing multi-dimensional images. Implementation of ?2 test as a criterion for deciding whether an object is fractal or not. User friendly graphical interface. Hyper-Fractal Analysis-Test on the Sierpinski hypertetrahedron 4D gasket (Df=ln(5)/ln(2)?2.32). Running time: In a first approximation, the algorithm is linear [2]. References: [1] V. Grossu, D. Felea, C. Besliu, Al. Jipa, C.C. Bordeianu, E. Stan, T. Esanu, Computer Physics Communications, 181 (2010) 831-832. [2] I.V. Grossu, C. Besliu, M.V. Rusu, Al. Jipa, C. C. Bordeianu, D. Felea, Computer Physics Communications, 180 (2009) 1999-2001. [3] J. Ruiz de Miras, J. Navas, P. Villoslada, F.J. Esteban, Computer Methods and Programs in Biomedicine, 104 Issue 3 (2011) 452-460.

Grossu, I. V.; Grossu, I.; Felea, D.; Besliu, C.; Jipa, Al.; Esanu, T.; Bordeianu, C. C.; Stan, E.

2013-04-01

63

A Power Differentiation Method of Fractal Dimension Estimation for 2-D Signals

Fractal dimension has been used for texture analysis as it is highly correlated with the human perception of surface roughness. Several methods have been proposed for the estimation of the fractal dimension of an image. One of the most popular is via its power spectrum density, provided that it is modeled as a fractional Brownian function. In this paper, a

P. Asvestas; G. K. Matsopoulos; K. S. Nikita

1998-01-01

64

DETERMINING THE FRACTAL DIMENSION OF A TIME SERIES WITH A NEURAL NET

DETERMINING THE FRACTAL DIMENSION OF A TIME SERIES WITH A NEURAL NET MARK J. EMBRECHTS AND YARON DANON Department of Nuclear Engineering & Engineering Physics Rensselaer Polytechnic Institute Troy.NY 12180 1. INTRODUCTION There are several methods for estimating the fractal dimension of a time series

Danon, Yaron

65

Analysis of fractal dimensions of rat bones from film and digital images

NASA Technical Reports Server (NTRS)

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.

Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Yamauchi, M.

2001-01-01

66

COMPARISON OF FRACTAL DIMENSION ALGORITHMS FOR THE COMPUTATION OF EEG BIOMARKERS FOR DEMENTIA

COMPARISON OF FRACTAL DIMENSION ALGORITHMS FOR THE COMPUTATION OF EEG BIOMARKERS FOR DEMENTIA C Goh of the Fractal Dimension of the EEG appears to be a good approach for the computation of biomarkers for dementia for computing reliable biomarkers, specifically, for the assessment of dementia. To achieve this, some

Paris-Sud XI, UniversitÃ© de

67

Numerical Analysis of Dynamical Systems and the Fractal Dimension of Boundaries

A set of MapleV R.4/5 software routines for calculating the numerical evolution of dynamical systems and flexibly plotting the results is presented. The package consists of an initial condition generator (on which the user can impose quite general constraints), a numerical solving manager, plotting commands that allow the user to locate and focus in on regions of possible interest and, finally, a set of routines that calculate the fractal dimension of the boundaries of those regions. A special feature of the software routines presented here is an optional interface in C, permitting fast numerical integration using standard Runge-Kutta methods, or variations, for high precision numerical integration

L. G. S. Duarte; L. A. C. P. da Mota; H. P. de Oliveira; R. O. Ramos; J. E. F. Skea

1998-12-22

68

Scaling exponents for a monkey on a tree - fractal dimensions of randomly branched polymers

We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to 2-loop order and, where available, compare them to numerical results.

Janssen, Hans-Karl

2012-01-01

69

Scaling exponents for a monkey on a tree - fractal dimensions of randomly branched polymers

We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to 2-loop order and, where available, compare them to numerical results.

Hans-Karl Janssen; Olaf Stenull

2012-03-13

70

Scaling exponents for a monkey on a tree: Fractal dimensions of randomly branched polymers

NASA Astrophysics Data System (ADS)

We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large, randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to two-loop order and, where available, compare them to numerical results.

Janssen, Hans-Karl; Stenull, Olaf

2012-05-01

71

Shortest-path fractal dimension for percolation in two and three dimensions.

We carry out a high-precision Monte Carlo study of the shortest-path fractal dimension d(min) for percolation in two and three dimensions, using the Leath-Alexandrowicz method which grows a cluster from an active seed site. A variety of quantities are sampled as a function of the chemical distance, including the number of activated sites, a measure of the radius, and the survival probability. By finite-size scaling, we determine d(min)=1.13077(2) and 1.3756(6) in two and three dimensions, respectively. The result in two dimensions rules out the recently conjectured value d(min)=217/192 [Deng et al., Phys. Rev. E 81, 020102(R) (2010)]. PMID:23367887

Zhou, Zongzheng; Yang, Ji; Deng, Youjin; Ziff, Robert M

2012-12-01

72

A new fuel-cell electrocatalyst based on highly porous carbonized polyacrylonitrile (PAN) microcellular foam with platinum particles was prepared recently in this laboratory. Its surface morphology, one of the most important aspects of a practical electrocatalyst, has been examined in terms of fractal theory and methods. The fractal dimension of the platinum particles dispersed in porous carbonized PAN foam was determined by using chronometric and rotating-disk-electrode methods in oxygen-saturated solutions. A fractal dimension smaller than 2 was obtained, which was attributed to the partially active nature of the surface of this electrocatalytic material. This value of fractal dimension is also proposed to be considered as a reaction dimension. A reaction dimension smaller than 2 may indicate that not all of the platinum particle surface is accessible to the incoming oxygen molecules.

Ye, S. [Inst. de Recherche d`Hydro-Quebec, Varennes, Quebec (Canada)]|[Inst. National de la Recherche Scientifique, Varennes, Quebec (Canada). Lab. de Recherche sur les Materiaux Avances; Vijh, A.K. [Inst. de Recherche d`Hydro-Quebec, Varennes, Quebec (Canada); Dao, L.H. [Inst. National de la Recherche Scientifique, Varennes, Quebec (Canada). Lab. de Recherche sur les Materiaux Avances

1997-05-01

73

Purpose. This study describes how to identify the coincidence of desired planning isodose curves with film experimental results by using a mathematical fractal dimension characteristic method to avoid the errors caused by visual inspection in the intensity modulation radiation therapy (IMRT). Methods and Materials. The isodose curves of the films delivered by linear accelerator according to Plato treatment planning system were acquired using Osiris software to aim directly at a single interested dose curve for fractal characteristic analysis. The results were compared with the corresponding planning desired isodose curves for fractal dimension analysis in order to determine the acceptable confidence level between the planning and the measurement. Results. The film measured isodose curves and computer planning curves were deemed identical in dose distribution if their fractal dimensions are within some criteria which suggested that the fractal dimension is a unique fingerprint of a curve in checking the planning and film measurement results. The dose measured results of the film were presumed to be the same if their fractal dimension was within 1%. Conclusions. This quantitative rather than qualitative comparison done by fractal dimension numerical analysis helps to decrease the quality assurance errors in IMRT dosimetry verification. PMID:23956976

Wu, Jia-Ming; Kuo, Chung-Ming; Chen, Ching-Jiang

2013-01-01

74

Estimation of fractal dimension of images using a fixed mass approach

A method from the field of chaotic dynamics is applied for the estimation of fractal dimension of images. The method is compared with other well-known algorithms on a set of computer generated images of known fractal di- mension. The results confirm the superiority of the method in terms of accuracy, dynamic range and computational time. \\

Pantelis Asvestas; George K. Matsopoulos; Konstantina S. Nikita

1999-01-01

75

Theory of Fractals and Fractional Dimension in Physics and Engineering of Wave Processes

Presented in this paper is the new synergetic approach on the basis of fractional measuring, fractal dimension, scaling, fractals and deterministic chaos, which has been developed in IRE RAS as applied to the problems of modern nonlinear radio physics and radio engineering since the eighties of the 20th century. Researches have been traditionally performed in the framework of modern fundamental

A. A. Potapov

2007-01-01

76

NASA Astrophysics Data System (ADS)

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.

Park, Hwangseo; Kim, Hojing; Lee, Sangyoub

1997-05-01

77

Properties relating to porosity of solids (fractal dimensions, surface roughness parameters) were evaluated from atomic force\\u000a microscopy (AFM) and nitrogen adsorption-desorption isotherms measured at 77 K for selected high-temperature [(RE) Ba2Cu3O7?x, RE=Y, Sm] superconductors. Adsorption capacity, specific surface area, fractal dimensions were determined from adsorption-desorption\\u000a isotherms. The adsorption isotherms of all samples were S-shaped and belong to type II according

G. W. Ch?dzy?ski; P. Staszczuk; D. Sternik; M. B?achnio

2008-01-01

78

Assessment of disintegrant efficacy with fractal dimensions from real-time MRI.

An efficient disintegrant is capable of breaking up a tablet in the smallest possible particles in the shortest time. Until now, comparative data on the efficacy of different disintegrants is based on dissolution studies or the disintegration time. Extending these approaches, this study introduces a method, which defines the evolution of fractal dimensions of tablets as surrogate parameter for the available surface area. Fractal dimensions are a measure for the tortuosity of a line, in this case the upper surface of a disintegrating tablet. High-resolution real-time MRI was used to record videos of disintegrating tablets. The acquired video images were processed to depict the upper surface of the tablets and a box-counting algorithm was used to estimate the fractal dimensions. The influence of six different disintegrants, of different relative tablet density, and increasing disintegrant concentration was investigated to evaluate the performance of the novel method. Changing relative densities hardly affect the progression of fractal dimensions, whereas an increase in disintegrant concentration causes increasing fractal dimensions during disintegration, which are also reached quicker. Different disintegrants display only minor differences in the maximal fractal dimension, yet the kinetic in which the maximum is reached allows a differentiation and classification of disintegrants. PMID:25234864

Quodbach, Julian; Moussavi, Amir; Tammer, Roland; Frahm, Jens; Kleinebudde, Peter

2014-11-20

79

Evaluation of the fractal dimension as a pattern recognition feature using neural networks

NASA Astrophysics Data System (ADS)

In the past fractal dimension has often been computed using a stochastic approach based on a random walk process, which has been found to be very time consuming. More recently, mathematical morphology has been used to compute the fractal dimension in a more timely fashion. This paper describes how the fractal dimension computed using mathematical morphology can be used in the texture analysis of ultrasonic imagery. The discriminatory ability of the fractal dimension as a pattern recognition feature is evaluated and compared to more traditional parameters. This analysis includes comparisons with statistical features in which each parameter is treated as an independent variable and in which interactions between those variables are evaluated. Pattern recognition techniques include Stepwise Discriminant Analysis, Linear Discriminant Analysis, and Nearest Neighbor Analysis in addition to Backpropagation Neural Network Classifiers. Our results identify the fractal dimension as one of the most important parameters for distinguishing between normal and abnormal livers. In this study, consisting of 186 images, a significant statistical difference was found for both the mean and standard deviation of the fractal dimension between the normal and abnormal groups using parametric and nonparametric statistical techniques.

DaPonte, John S.; Parikh, Jo Ann; Decker, James; Vitale, Joseph N.

1993-09-01

80

The purposes of our studies are to examine whether or not fractal-feature distance deduced from virtual volume method can\\u000a simulate observer performance indices and to investigate the physical meaning of pseudo fractal dimension and complexity.\\u000a Contrast-detail (C-D) phantom radiographs were obtained at various mAs values (0.5-4.0 mAs) and 140 kVp with a computed radiography\\u000a system, and the reference image was

K. Imai; M. Ikeda; Y. Enchi; T. Niimi

2009-01-01

81

Fractal dimensions of silica gels generated using reactive molecular dynamics simulations

We have used molecular dynamics simulations based on a three-body potential with charge transfer to generate nanoporous silica aerogels. Care was taken to reproduce the sol-gel condensation reaction that forms the gel backbone as realistically as possible and to thereby produce credible gel structures. The self-similarity of aerogel structures was investigated by evaluating their fractal dimension from geometric correlations. For comparison, we have also generated porous silica glasses by rupturing dense silica and computed their fractal dimension. The fractal dimension of the porous silica structures was found to be process dependent. Finally, we have determined that the effect of supercritical drying on the fractal nature of condensed silica gels is not appreciable.

Bhattacharya, Sudin; Kieffer, John [Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2158 (United States); Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136 (United States)

2005-03-01

82

Fractal Dimension Characterization of in-vivo Laser Doppler Flowmetry signals

NASA Astrophysics Data System (ADS)

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.

Srinivasan, Gayathri; Sujatha, N.

83

NASA Astrophysics Data System (ADS)

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.

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

84

Fractal dimension analysis of weight-bearing bones of rats during skeletal unloading

NASA Technical Reports Server (NTRS)

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.

Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Sanhueza, A. I.; Yamauchi, M.

2001-01-01

85

Fractal dimension of chromatin is an independent prognostic factor for survival in melanoma

Background Prognostic factors in malignant melanoma are currently based on clinical data and morphologic examination. Other prognostic features, however, which are not yet used in daily practice, might add important information and thus improve prognosis, treatment, and survival. Therefore a search for new markers is desirable. Previous studies have demonstrated that fractal characteristics of nuclear chromatin are of prognostic importance in neoplasias. We have therefore investigated whether the fractal dimension of nuclear chromatin measured in routine histological preparations of malignant melanomas could be a prognostic factor for survival. Methods We examined 71 primary superficial spreading cutaneous melanoma specimens (thickness ? 1 mm) from patients with a minimum follow up of 5 years. Nuclear area, form factor and fractal dimension of chromatin texture were obtained from digitalized images of hematoxylin-eosin stained tissue micro array sections. Clark's level, tumor thickness and mitotic rate were also determined. Results The median follow-up was 104 months. Tumor thickness, Clark's level, mitotic rate, nuclear area and fractal dimension were significant risk factors in univariate Cox regressions. In the multivariate Cox regression, stratified for the presence or absence of metastases at diagnosis, only the Clark level and fractal dimension of the nuclear chromatin were included as independent prognostic factors in the final regression model. Conclusion In general, a more aggressive behaviour is usually found in genetically unstable neoplasias with a higher number of genetic or epigenetic changes, which on the other hand, provoke a more complex chromatin rearrangement. The increased nuclear fractal dimension found in the more aggressive melanomas is the mathematical equivalent of a higher complexity of the chromatin architecture. So, there is strong evidence that the fractal dimension of the nuclear chromatin texture is a new and promising variable in prognostic models of malignant melanomas. PMID:20525386

2010-01-01

86

Radial viscous fingers and diffusion-limited aggregation: Fractal dimension and growth sites

We show that fractal viscous fingers can be formed in a Hele-Shaw cell with radial symmetry, thereby permitting their study-for the first time-without the complicating effects of boundary conditions such as those present in the conventional linear cell. We find-for a wide range of shear-thinning fluids, flow rates, and plate separations-that radial viscous fingers have a fractal dimension df=1.70+\\/-0.05, the

Gérard Daccord; Johann Nittmann; H. Eugene Stanley

1986-01-01

87

Concepts of scaling and fractal dimension in the design of a fractal detector of radio signals

Concepts of hardware implementation of fractal nonparametric detectors of radar signals (FNDRSs) are analyzed. The structure\\u000a of the first FNDRS prototype is determined on the basis of concepts of sample topology and fractal signatures. Experiments\\u000a in which frequency and time scalings are used for detection of sinusoidal and pulsed signals in the presence of additive noises\\u000a and interferences on the

Yu. V. Gulyaev; S. A. Nikitov; A. A. Potapov; V. A. German

2006-01-01

88

Electroconvulsive therapy (ECT), in which electrical current is used to induce seizures, is an effective treatment in psychiatry. Different methods of analyzing the electroencephalogram (EEG) changes during ECT have been studied for predicting clinical outcome. Analysis of the fractal dimension (FD) is one such method. Mid-seizure and post-seizure FD has been shown to correlate with antidepressant effect. In this study, we examined whether the highest fractal dimension achieved during each ECT session changed over the course of 6 ECTs. The sample for this study came from a randomized controlled trial, comparing the efficacy of bifrontal and bitemporal electrode placements in schizophrenia. EEG was recorded using bilateral frontal pole leads during all ECT sessions. In 40 of the 114 randomized patients, we could obtain artifact-free EEGs for the first 6 ECT sessions. FD was calculated using standardized algorithms. For each session, the average of 5 highest FDs was calculated. The change in this value over a course of 6 ECTs was analyzed using repeated-measures analysis of variance. The average highest FD remained virtually unchanged across the 6 ECT sessions. Means (standard deviations) average maximum FDs over the 6 sessions were 1.57 (0.075), 1.57 (0.064), 1.56 (0.064), 1.57 (0.062), 1.55 (0.07), and 1.56 (0.067); occasion effect, F = 0.5, P = .75. Group effect (F = 0.01, P = .92) and group × occasion interaction effect (F = 1.88, P = .1) were not significant, suggesting no influence of electrode placement on maximum FD. Seizure duration, however, showed significant decline over the course of ECT. Maximum FD of ECT-induced EEG seizure remains fairly constant over frontal poles across the first 6 ECT sessions, which is true irrespective of ECT electrode placements. PMID:23760035

Rakesh, Gopalkumar; Abhishekh, Hulegar A; Thirthalli, Jagadisha; Phutane, Vivek H; Muralidharan, Kesavan; Candade, Vittal S; Gangadhar, Bangalore N

2014-04-01

89

NASA Technical Reports Server (NTRS)

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.

Emerson, Charles W.; Sig-NganLam, Nina; Quattrochi, Dale A.

2004-01-01

90

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.

Smith, R.L., E-mail: firefan@ufl.edu; Mecholsky, J.J., E-mail: jmech@ufl.edu

2011-05-15

91

Fractal dimensions of aggregates formed in different fluid mechanical environments

The fractal properties of aggregates formed under two different fluid mechanical environments, a paddle mixer and a rolling cylinder, were measured using three different techniques: a non-steady state method requiring both volume and length size distributions, a steady state size distribution method, and an aggregate property scaling method. Based on cumulative size distributions and the non-steady state method, aggregates produced

Bruce E. Logan; John R. Kilps

1995-01-01

92

A robust marker to describe mass, hydrophobicity and polarizability distribution holds the key to deciphering structural and folding constraints within proteins. Since each of these distributions is inhomogeneous in nature, the construct should be sensitive in describing the patterns therein. We show, for the first time, that the hydrophobicity and polarizability distributions in protein interior follow fractal scaling. It is

Anirban Banerji; Indira Ghosh; Sotirios Koutsopoulos

2009-01-01

93

Unilateral contact problems with fractal geometry and fractal friction laws: methods of calculation

The present paper deals with two interrelated subjects: the fractal geometry and the fractal behaviour in unilateral contact\\u000a problems. More specifically, throughout this paper both the interfaces and the friction laws holding on these interfaces are\\u000a modelled by means of the fractal geometry. It is important to notice here that the fractality of the induced friction laws\\u000a takes into account

E. S. Mistakidis; O. K. Panagouli; P. D. Panagiotopoulos

1998-01-01

94

NASA Technical Reports Server (NTRS)

The use of renormalization group techniques on fragmentation problems is examined. The equations which represent fractals and the size-frequency distributions of fragments are presented. Method for calculating the size distributions of asteriods and meteorites are described; the frequency-mass distribution for these interplanetary objects are due to fragmentation. The application of two renormalization group models to fragmentation is analyzed. It is observed that the models yield a fractal behavior for fragmentation; however, different values for the fractal dimension are produced . It is concluded that fragmentation is a scale invariant process and that the fractal dimension is a measure of the fragility of the fragmented material.

Turcotte, D. L.

1986-01-01

95

Unilateral contact problems with fractal geometry and fractal friction laws: methods of calculation

NASA Astrophysics Data System (ADS)

The present paper deals with two interrelated subjects: the fractal geometry and the fractal behaviour in unilateral contact problems. More specifically, throughout this paper both the interfaces and the friction laws holding on these interfaces are modelled by means of the fractal geometry. It is important to notice here that the fractality of the induced friction laws takes into account the randomness of the interface asperities causing the friction forces. According to the fractal model introduced in this paper, both the fractal law and the fractal interface are considered to be graphs of two different fractal interpolation functions which are the ``fixed points'' of two contractive operators. Using this method, the fractal friction law is approximated by a sequence of nonmonotone possibly multivalued classical C0-curves. The numerical treatment of each arizing nonmonotone problem is accomplished by an advanced solution method which approximates the nonmonotone problem by a sequence of monotone subproblems. Numerical applications from the static analysis of cracked structures with a prescribed fractal geometry and fractal interface laws are included in order to illustrate the theory.

Mistakidis, E. S.; Panagouli, O. K.; Panagiotopoulos, P. D.

96

Comparison of fractal dimension estimation algorithms for epileptic seizure onset detection

The fractal dimension (FD) is a natural measure of the irregularity of a curve. In this study the performances of two FD-based methodologies are compared in terms of their ability to detect the onset of epileptic seizures in scalp EEG. The FD algorithms used is Katzpsilas, which has been broadly utilized in the EEG analysis literature, and the k-nearest neighbor

Georgia E. Polychronaki; Periklis Y. Ktonas; Stylianos Gatzonis; Pantelis A. Asvestas; Eirini Spanou; Anna Siatouni; Hara Tsekou; Damianos Sakas; Konstantina S. Nikita

2008-01-01

97

In this paper, a pilot study regarding carotid atherosclerotic plaque instability using B-mode ultrasound (US) images was carried out. The fractal dimension of plaques obtained from ten symptomatic subjects (i.e., subjects having experienced neurological symptoms) as well as from nine asymptomatic subjects, was estimated using a novel method, called the kth nearest neighbour (KNN) method. The results indicated a significant

Pantelis Asvestas; Spyretta Golemati; George K. Matsopoulos; Konstantina S. Nikita; Andrew N. Nicolaides

2002-01-01

98

In this paper, a pilot study regarding carotid atherosclerotic plaque instability using B-mode ultra- sound (US) images was carried out. The fractal dimension of plaques obtained from ten symptomatic subjects (i.e., subjects having experienced neurological symptoms) as well as from nine asymptomatic subjects, was estimated using a novel method, called the kth nearest neighbour (KNN) method. The results indicated a

PANTELIS ASVESTAS; SPYRETTA GOLEMATI; GEORGE K. MATSOPOULOS; KONSTANTINA S. NIKITA; ANDREW N. NICOLAIDES

2002-01-01

99

We analyze large diffusion-limited aggregates and uncover a discrete scaling invariance in their inner structure, which can be quantified by the introduction of a set of complex fractal dimensions. We provide a theoretical framework and prediction of their values based on renormalization group theory and a previous wavelet analysis.

D. Sornette; A. Johansen; A. Arneodo; J. F. Muzy; H. Saleur

1996-01-01

100

Fractal dimension based sand ripple suppression for mine hunting with sidescan sonar

1 Fractal dimension based sand ripple suppression for mine hunting with sidescan sonar J. D. B. Nelson and N. G. Kingsbury Abstract--Sand ripples present a difficult challenge to current mine hunting approaches. We propose a robust and adaptive method that suppresses sand ripples prior to the detection stage

Nelson, James

101

Detection of explosive lung and bowel sounds by means of fractal dimension

An efficient technique for detecting explosive lung sounds (LS) (fine\\/coarse crackles and squawks) or bowel sounds (BS) in clinical auscultative recordings is presented. The technique is based on a fractal-dimension (FD) analysis of the recorded LS and BS obtained from controls and patients with pulmonary and bowel pathology, respectively. Experimental results demonstrate the efficiency of the proposed method, since it

Leontios J. Hadjileontiadis; Ioannis T. Rekanos

2003-01-01

102

Cusp-scaling behavior in fractal dimension of chaotic scattering Adilson E. Motter1

Cusp-scaling behavior in fractal dimension of chaotic scattering Adilson E. Motter1 and Ying-Cheng Lai1,2 1 Department of Mathematics, Center for Systems Science and Engineering Research, Arizona State University, Tempe, Arizona 85287 2 Departments of Electrical Engineering and Physics, Arizona State

Lai, Ying-Cheng

103

Capillary pressure in a porous medium with distinct pore surface and pore volume fractal dimensions

Capillary pressure in a porous medium with distinct pore surface and pore volume fractal dimensions M. R. Deinert Department of Mechanical Engineering, University of Texas at Austin, Austin, Texas 78758, USA A. Dathe Biological and Environmental Engineering, Cornell University, Ithaca, New York 14850

Deinert, Mark

104

It is well known that angiogenesis is a complex process that accompanies neoplastic growth, but pituitary tumours are less vascularized than normal pituitary glands. Several analytical methods aimed at quantifying the vascular system in two-dimensional histological sections have been proposed, with very discordant results. In this study we investigated the non-Euclidean geometrical complexity of the two-dimensional microvasculature of normal pituitary glands and pituitary adenomas by quantifying the surface fractal dimension that measures its space-filling property. We found a statistical significant difference between the mean vascular surface fractal dimension estimated in normal versus adenomatous tissues (P = 0.01), normal versus secreting adenomatous tissues (P = 0.0003), and normal versus non-secreting adenomatous tissues (P = 0.047), whereas the difference between the secreting and non-secreting adenomatous tissues was not statistically significant. This study provides the first demonstration that fractal dimension is an objective and valid quantitator of the two-dimensional geometrical complexity of the pituitary gland microvascular network in physiological and pathological states. Further studies are needed to compare the vascular surface fractal dimension estimates in different subtypes of pituitary tumours and correlate them with clinical parameters in order to evaluate whether the distribution pattern of vascular growth is related to a particular state of the pituitary gland. PMID:17784937

Di Ieva, Antonio; Grizzi, Fabio; Ceva-Grimaldi, Giorgia; Russo, Carlo; Gaetani, Paolo; Aimar, Enrico; Levi, Daniel; Pisano, Patrizia; Tancioni, Flavio; Nicola, Giancarlo; Tschabitscher, Manfred; Dioguardi, Nicola; Baena, Riccardo Rodriguez y

2007-01-01

105

Fractal spatial distribution of pancreatic islets in three dimensions: a self-avoiding growth model

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, have 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, 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 fractal dimension 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. PMID:23629025

Jo, Junghyo; Hörnblad, Andreas; Kilimnik, German; Hara, Manami; Ahlgren, Ulf; Periwal, Vipul

2013-01-01

106

NASA Astrophysics Data System (ADS)

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.

Jo, Junghyo; Hörnblad, Andreas; Kilimnik, German; Hara, Manami; Ahlgren, Ulf; Periwal, Vipul

2013-06-01

107

Are fractal dimensions of the spatial distribution of mineral deposits meaningful?

It has been proposed that the spatial distribution of mineral deposits is bifractal. An implication of this property is that the number of deposits in a permissive area is a function of the shape of the area. This is because the fractal density functions of deposits are dependent on the distance from known deposits. A long thin permissive area with most of the deposits in one end, such as the Alaskan porphyry permissive area, has a major portion of the area far from known deposits and consequently a low density of deposits associated with most of the permissive area. On the other hand, a more equi-dimensioned permissive area, such as the Arizona porphyry permissive area, has a more uniform density of deposits. Another implication of the fractal distribution is that the Poisson assumption typically used for estimating deposit numbers is invalid. Based on datasets of mineral deposits classified by type as inputs, the distributions of many different deposit types are found to have characteristically two fractal dimensions over separate non-overlapping spatial scales in the range of 5-1000 km. In particular, one typically observes a local dimension at spatial scales less than 30-60 km, and a regional dimension at larger spatial scales. The deposit type, geologic setting, and sample size influence the fractal dimensions. The consequence of the geologic setting can be diminished by using deposits classified by type. The crossover point between the two fractal domains is proportional to the median size of the deposit type. A plot of the crossover points for porphyry copper deposits from different geologic domains against median deposit sizes defines linear relationships and identifies regions that are significantly underexplored. Plots of the fractal dimension can also be used to define density functions from which the number of undiscovered deposits can be estimated. This density function is only dependent on the distribution of deposits and is independent of the definition of the permissive area. Density functions for porphyry copper deposits appear to be significantly different for regions in the Andes, Mexico, United States, and western Canada. Consequently, depending on which regional density function is used, quite different estimates of numbers of undiscovered deposits can be obtained. These fractal properties suggest that geologic studies based on mapping at scales of 1:24,000 to 1:100,000 may not recognize processes that are important in the formation of mineral deposits at scales larger than the crossover points at 30-60 km. ?? 2008 International Association for Mathematical Geology.

Raines, G.L.

2008-01-01

108

Fractal Dimension of EEG Activity Senses Neuronal Impairment in Acute Stroke

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

Zappasodi, Filippo; Olejarczyk, Elzbieta; Marzetti, Laura; Assenza, Giovanni; Pizzella, Vittorio; Tecchio, Franca

2014-01-01

109

NASA Astrophysics Data System (ADS)

A set of Maple routines is presented, fully compatible with the new releases of Maple (14 and higher). The package deals with the numerical evolution of dynamical systems and provide flexible plotting of the results. The package also brings an initial conditions generator, a numerical solver manager, and a focusing set of routines that allow for better analysis of the graphical display of the results. The novelty that the package presents an optional C interface is maintained. This allows for fast numerical integration, even for the totally inexperienced Maple user, without any C expertise being required. Finally, the package provides the routines to calculate the fractal dimension of boundaries (via box counting). New version program summary Program Title: Ndynamics Catalogue identifier: %Leave blank, supplied by Elsevier. Licensing provisions: no. Programming language: Maple, C. Computer: Intel(R) Core(TM) i3 CPU M330 @ 2.13 GHz. Operating system: Windows 7. RAM: 3.0 GB Keywords: Dynamical systems, Box counting, Fractal dimension, Symbolic computation, Differential equations, Maple. Classification: 4.3. Catalogue identifier of previous version: ADKH_v1_0. Journal reference of previous version: Comput. Phys. Commun. 119 (1999) 256. Does the new version supersede the previous version?: Yes. Nature of problem Computation and plotting of numerical solutions of dynamical systems and the determination of the fractal dimension of the boundaries. Solution method The default method of integration is a fifth-order Runge-Kutta scheme, but any method of integration present on the Maple system is available via an argument when calling the routine. A box counting [1] method is used to calculate the fractal dimension [2] of the boundaries. Reasons for the new version The Ndynamics package met a demand of our research community for a flexible and friendly environment for analyzing dynamical systems. All the user has to do is create his/her own Maple session, with the system to be studied, and use the commands on the package to (for instance) calculate the fractal dimension of a certain boundary, without knowing or worrying about a single line of C programming. So the package combines the flexibility and friendly aspect of Maple with the fast and robust numerical integration of the compiled (for example C) basin. The package is old, but the problems it was designed to dealt with are still there. Since Maple evolved, the package stopped working, and we felt compelled to produce this version, fully compatible with the latest version of Maple, to make it again available to the Maple user. Summary of revisions Deprecated Maple Packages and Commands: Paraphrasing the Maple in-built help files, "Some Maple commands and packages are deprecated. A command (or package) is deprecated when its functionality has been replaced by an improved implementation. The newer command is said to supersede the older one, and use of the newer command is strongly recommended". So, we have examined our code to see if some of these occurrences could be dangerous for it. For example, the "readlib" command is unnecessary, and we have removed its occurrences from our code. We have checked and changed all the necessary commands in order for us to be safe in respect to danger from this source. Another change we had to make was related to the tools we have implemented in order to use the interface for performing the numerical integration in C, externally, via the use of the Maple command "ssystem". In the past, we had used, for the external C integration, the DJGPP system. But now we present the package with (free) Borland distribution. The compilation and compiling commands are now slightly changed. For example, to compile only, we had used "gcc-c"; now, we use "bcc32-c", etc. All this installation (Borland) is explained on a "README" file we are submitting here to help the potential user. Restrictions Besides the inherent restrictions of numerical integration methods, this version of the package only deals w

Avellar, J.; Duarte, L. G. S.; da Mota, L. A. C. P.; de Melo, N.; Skea, J. E. F.

2012-09-01

110

The use of the fractal description to characterize engineering surfaces and wear particles

Fractals can be extremely useful when applied to tribology. Obtaining fractal descriptions of engineering surfaces and wear particles requires surface topography information to be measured, digitized and processed. Such procedures can be rigorous. This article compares various methods to calculate profile and surface fractal dimension. Profile fractal dimension is computed using three available methods, corresponding to the yard-stick, the power

C. Q Yuan; J Li; X. P Yan; Z Peng

2003-01-01

111

An in situ laser light scattering method has been developed for line measurement of aggregate size and morphology. Planar multiangular light scattering measurements were interpreted by using Rayleigh–Debye–Gans scattering theory for fractal aggregates to simultaneously obtain the mean radius of gyration and the fractal dimension along the flame axis, which are the parameters characterizing size and morphology of aggregates. The

H. W. Kim; M. Choi

2003-01-01

112

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.

Huang, F. [College of Science, China Agricultural University, Beijing 100083 (China); Peng, R. D. [State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Beijing 100083 (China); Liu, Y. H. [Institute of Complexity Science, Qingdao University, Qingdao 266071 (China); Chen, Z. Y. [Department of Physics, Beijing University of Chemical Technology, Beijing 100029 (China); Ye, M. F.; Wang, L. [Institute of Physics, Chinese Academy of Science, Beijing 100190 (China)

2012-09-15

113

Zone specific fractal dimension of retinal images as predictor of stroke incidence.

Fractal dimensions (FDs) are frequently used for summarizing the complexity of retinal vascular. However, previous techniques on this topic were not zone specific. A new methodology to measure FD of a specific zone in retinal images has been developed and tested as a marker for stroke prediction. Higuchi's fractal dimension was measured in circumferential direction (FDC) with respect to optic disk (OD), in three concentric regions between OD boundary and 1.5 OD diameter from its margin. The significance of its association with future episode of stroke event was tested using the Blue Mountain Eye Study (BMES) database and compared against spectrum fractal dimension (SFD) and box-counting (BC) dimension. Kruskal-Wallis analysis revealed FDC as a better predictor of stroke (H = 5.80, P = 0.016, ? = 0.05) compared with SFD (H = 0.51, P = 0.475, ? = 0.05) and BC (H = 0.41, P = 0.520, ? = 0.05) with overall lower median value for the cases compared to the control group. This work has shown that there is a significant association between zone specific FDC of eye fundus images with future episode of stroke while this difference is not significant when other FD methods are employed. PMID:25485298

Aliahmad, Behzad; Kumar, Dinesh Kant; Hao, Hao; Unnikrishnan, Premith; Che Azemin, Mohd Zulfaezal; Kawasaki, Ryo; Mitchell, Paul

2014-01-01

114

Fractal Dimension and Vessel Complexity in Patients with Cerebral Arteriovenous Malformations

The fractal dimension (FD) can be used as a measure for morphological complexity in biological systems. The aim of this study was to test the usefulness of this quantitative parameter in the context of cerebral vascular complexity. Fractal analysis was applied on ten patients with cerebral arteriovenous malformations (AVM) and ten healthy controls. Maximum intensity projections from Time-of-Flight MRI scans were analyzed using different measurements of FD, the Box-counting dimension, the Minkowski dimension and generalized dimensions evaluated by means of multifractal analysis. The physiological significance of this parameter was investigated by comparing values of FD first, with the maximum slope of contrast media transit obtained from dynamic contrast-enhanced MRI data and second, with the nidus size obtained from X-ray angiography data. We found that for all methods, the Box-counting dimension, the Minkowski dimension and the generalized dimensions FD was significantly higher in the hemisphere with AVM compared to the hemisphere without AVM indicating that FD is a sensitive parameter to capture vascular complexity. Furthermore we found a high correlation between FD and the maximum slope of contrast media transit and between FD and the size of the central nidus pointing out the physiological relevance of FD. The proposed method may therefore serve as an additional objective parameter, which can be assessed automatically and might assist in the complex workup of AVMs. PMID:22815946

Reishofer, Gernot; Koschutnig, Karl; Enzinger, Christian; Ebner, Franz; Ahammer, Helmut

2012-01-01

115

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

Gheonea, Dan Ionu?; Streba, Costin Teodor; Vere, Cristin Constantin; ?erb?nescu, Mircea; Pirici, Daniel; Com?nescu, Maria; Streba, Leti?ia Adela Maria; Ciurea, Marius Eugen; Mogoant?, Stelian; Rogoveanu, Ion

2014-01-01

116

Variations in the fractal dimension of fluctuations of the ionospheric F2 layer critical frequency

NASA Astrophysics Data System (ADS)

The 40-year period of observations of short-term variations (with characteristic times of up to 1-2 days) in the critical frequency of the ionospheric F2 layer ( foF2) is analyzed. The continuous (with a step of 1 h) series of fluctuations ( F) of the foF2 critical frequency (with eliminated daily variations) has been calculated using the hourly variations in foF2 at Moscow stations. The fractal dimension (FRH) of the fluctuations, characterizing short-term variations in foF2, has been determined and analyzed on a 30-day interval, using the Higuchi method. It has been established that FRH estimates substantially change in time. The 11-year cycle, which is in antiphase with the solar cycle, and the total annual and semiannual variations, similar to the variations observed in the normalized critical frequency of the E region and in the electron density of the D region, are clearly defined in these changes. Thus, the parameters of fast variations in the ionospheric F2 layer are affected by the phase of the 11-year solar cycle and by the position of the Earth in the orbit or seasonal variations in the atmosphere.

Chertoprud, V. E.; Leshchenko, L. N.; Givishvili, G. V.; Goncharova, E. E.; Ivanov-Kholodny, G. S.

2009-08-01

117

A new way of describing meiosis that uses fractal dimension to predict metaphase I.

Meiosis, the reductive nuclear division, is a continuum, but for purposes of communication, is described in stages. In sexually-reproducing organisms, including the dwarf mistletoe Arceuthobium americanum, prophase I of meiosis is prolonged (8 months for female A. americanum). Conversely, metaphase I, where chromosome pairs line up along a dividing cell's "equator", is relatively brief, difficult to predict, but critical regarding the random distribution of the paternal and maternal chromosomes in sexual organisms. However, descriptions of meiosis as either a continuum or stages are limited to qualitative observations. A quantification of meiosis can provide mathematical descriptors and allow for the prediction of when chromosomes reach the equator; this will not only be useful to researchers of cell division, but also to those requiring a large sample of metaphase I materials. Here, the probability-density function was used to calculate the fractal dimension of A. americanum nuclei undergoing early meiosis, and it predicted the onset of metaphase I by 2 days. PMID:16094465

Ross, Cynthia M

2005-01-01

118

A new way of describing meiosis that uses fractal dimension to predict metaphase I

Meiosis, the reductive nuclear division, is a continuum, but for purposes of communication, is described in stages. In sexually-reproducing organisms, including the dwarf mistletoe Arceuthobium americanum, prophase I of meiosis is prolonged (8 months for female A. americanum). Conversely, metaphase I, where chromosome pairs line up along a dividing cell's "equator", is relatively brief, difficult to predict, but critical regarding the random distribution of the paternal and maternal chromosomes in sexual organisms. However, descriptions of meiosis as either a continuum or stages are limited to qualitative observations. A quantification of meiosis can provide mathematical descriptors and allow for the prediction of when chromosomes reach the equator; this will not only be useful to researchers of cell division, but also to those requiring a large sample of metaphase I materials. Here, the probability-density function was used to calculate the fractal dimension of A. americanum nuclei undergoing early meiosis, and it predicted the onset of metaphase I by 2 days. PMID:16094465

2005-01-01

119

Fractal dimension of non-network space of a catchment basin

NASA Astrophysics Data System (ADS)

Topographically convex regions within a catchment basin represent varied degrees of hill-slopes. The non-network space (M), the characterization of which we address in this letter, is akin to the space that is achieved by subtracting channelized portions contributed due to concave regions from the watershed space (X). This non-network space is similar to non-channelized convex region within a catchment basin. We propose an alternative shape-dependent quantity like fractal dimension to characterize this non-network space. Towards this goal, we decompose the non-network space in two-dimensional discrete space into simple non-overlapping disks (NODs) of various sizes by employing mathematical morphological transformations and certain logical operations. Furthermore, we plot the number of NODs of less than threshold radius against the radius, and compute the shape-dependent fractal dimension of non-network space.

Sagar, B. S. Daya; Chockalingam, L.

2004-06-01

120

Comparison of fractal dimension estimation algorithms for epileptic seizure onset detection

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

G. E. Polychronaki; P. Y. Ktonas; S. Gatzonis; A. Siatouni; P. A. Asvestas; H. Tsekou; D. Sakas; K. S. Nikita

2010-01-01

121

Characterization of anomalies by applying methods of fractal analysis

Fractal analysis is applied in a variety of research fields to characterize nonstationary data. Here, fractal analysis is used as a tool of characterization in time series. The fractal dimension is calculated by Higuchi`s method, and the effect of small data size on accuracy is studied in detail. Three types of fractal-based anomaly indicators are adopted: (a) the fractal dimension, (b) the error of the fractal dimension, and (c) the chi-square value of the linear fitting of the fractal curve in the wave number domain. Fractal features of time series can be characterized by introducing these three measures. The proposed method is applied to various simulated fractal time series with ramp, random, and periodic noise anomalies and also to neutron detector signals acquired in a nuclear reactor. Fractal characterization can successfully supplement conventional signal analysis methods especially if nonstationary and non-Gaussian features of the signal become important.

Sakuma, M.; Kozma, R.; Kitamura, M. [Tohoku Univ., Sendai (Japan). Dept. of Nuclear Engineering

1996-01-01

122

NASA Astrophysics Data System (ADS)

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.

Kikuchi, Tsuneo; Nakazawa, Toshihiro; Furukawa, Tetsuo; Higuchi, Toshiyuki; Maruyama, Yukio; Sato, Sojun

1995-05-01

123

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)

Ul'yanov, A S; Lyapina, A M; Ulianova, O V; Fedorova, V A; Uianov, S S

2011-04-30

124

Fractal dimension of chromatin: potential molecular diagnostic applications for cancer prognosis

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

Metze, Konradin

2013-01-01

125

A Brief Historical Introduction to Fractals and Fractal Geometry

ERIC Educational Resources Information Center

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…

Debnath, Lokenath

2006-01-01

126

Fractal structure of interstellar cirrus

NASA Technical Reports Server (NTRS)

The paper investigates the fractal structure of some of the infrared cirrus discovered by IRAS. The clouds studied are characterized by a perimeter with fractal dimension 1.26 + or - 0.04 using the method of Lovejoy. This dimension was found to be essentially constant from region to region. Methods of calculating this dimension are compared and the relation of this quantity to some other observable cloud properties is discussed.

Bazell, D.; Desert, F. X.

1988-01-01

127

ERIC Educational Resources Information Center

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.)

McCartney, M.; Myers, D.; Sun, Y.

2008-01-01

128

NASA Astrophysics Data System (ADS)

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

129

THE FRACTAL DIMENSION OF STAR-FORMING REGIONS AT DIFFERENT SPATIAL SCALES IN M33

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.

Sanchez, Nestor; Alfaro, Emilio J. [Instituto de Astrofisica de AndalucIa, CSIC, Apdo. 3004, E-18080, Granada (Spain); Anez, Neyda [Departamento de Fisica, Facultad Experimental de Ciencias, Universidad del Zulia, Maracaibo (Venezuela, Bolivarian Republic of); Odekon, Mary Crone, E-mail: nestor@iaa.e [Department of Physics, Skidmore College, Saratoga Springs, NY 12866 (United States)

2010-09-01

130

Measuring capital market efficiency: long-term memory, fractal dimension and approximate entropy

NASA Astrophysics Data System (ADS)

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).

Kristoufek, Ladislav; Vosvrda, Miloslav

2014-07-01

131

Several studies have investigated the fractal and multifractal nature of magnetic features in the solar photosphere and its variation with the solar magnetic activity cycle. Here we extend those studies by examining the fractal geometry of bright magnetic features at higher atmospheric levels, specifically in the solar chromosphere. We analyze structures identified in CaIIK images obtained with the Precision Solar Photometric Telescopes (PSPTs) at Osservatorio Astronomico di Roma (OAR) and Mauna Loa Solar Observatory (MLSO). Fractal dimension estimates depend on the estimator employed, the quality of the images, and the structure identification techniques used. We examine both real and simulated data and employ two different perimeter-area estimators in order to understand the sensitivity of the deduced fractal properties to pixelization and image quality. The fractal dimension of bright 'magnetic' features in CaIIK images ranges between values of 1.2 and 1.7 for small and large structures respectively. This size dependency largely reflects the importance of image pixelization in the measurement of small objects. The fractal dimension of chromospheric features does not show any clear systematic variation with time over the period examined, the descending phase of solar cycle 23. These conclusions, and the analysis of both real and synthetic images on which they are based, are important in the interpretation of previously reported results.

Serena Criscuoli; Mark Rast; Ilaria Ermolli; Mauro Centrone

2006-09-27

132

NASA Astrophysics Data System (ADS)

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' (?(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.

Guo, Long; Cai, XU

2009-08-01

133

Objective Auscultation of TCM Based on Wavelet Packet Fractal Dimension and Support Vector Machine

This study was conducted to illustrate that auscultation features based on the fractal dimension combined with wavelet packet transform (WPT) were conducive to the identification the pattern of syndromes of Traditional Chinese Medicine (TCM). The WPT and the fractal dimension were employed to extract features of auscultation signals of 137 patients with lung Qi-deficient pattern, 49 patients with lung Yin-deficient pattern, and 43 healthy subjects. With these features, the classification model was constructed based on multiclass support vector machine (SVM). When all auscultation signals were trained by SVM to decide the patterns of TCM syndromes, the overall recognition rate of model was 79.49%; when male and female auscultation signals were trained, respectively, to decide the patterns, the overall recognition rate of model reached 86.05%. The results showed that the methods proposed in this paper were effective to analyze auscultation signals, and the performance of model can be greatly improved when the distinction of gender was considered. PMID:24883068

Yan, Jian-Jun; Wang, Yi-Qin; Liu, Guo-Ping; Yan, Hai-Xia; Xia, Chun-Ming; Shen, Xiaojing

2014-01-01

134

We consider a construction of recurrent fractal interpolation surfaces with function vertical scaling factors and estimation of their box-counting dimension. A recurrent fractal interpolation surface (RFIS) is an attractor of a recurrent iterated function system (RIFS) which is a graph of bivariate interpolation function. For any given data set on rectangular grids, we construct general recurrent iterated function systems with function vertical scaling factors and prove the existence of bivariate functions whose graph are attractors of the above constructed RIFSs. Finally, we estimate lower and upper bounds for the box-counting dimension of the constructed RFISs.

Chol-Hui Yun; Hui-Chol Choi; Hyong-Chol O

2013-07-10

135

NASA Technical Reports Server (NTRS)

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.

Garneau, S.; Plaut, J. J.

2000-01-01

136

Fractal approach to measuring roughness of geomembranes

Fractal analysis was used to evaluate the roughness of four commercially available geomembranes used in waste-containment systems. Fractal analysis uses the concept of the fractal dimension D as a way to calculate the roughness of simple and complex profiles. In the present study, it was determined that the fractal dimension D of the geomembrane profiles increased as the roughness of their profiles increased. For example, the smoothest of the four geomembrane profiles had a fractal dimension D equal to 1.001; the roughest of the geomembrane profiles had a fractal dimension D equal to 1.1345. The difference in the fractal dimension D for the four geomembrane profiles was found to be relatively small. Thus, when the fractal dimension D is used to evaluate the roughness of geomembranes, at least four-digit precision after the decimal point is advisable. The sensitivity of the fractal dimension to inputs such as the direction of measurement and cut as well as the size of the segment length used to cover the geomembrane profiles were also analyzed.

Vallejo, L.E.; Zhou, Y. [Univ. of Pittsburgh, PA (United States). Dept. of Civil Engineering] [Univ. of Pittsburgh, PA (United States). Dept. of Civil Engineering

1995-05-01

137

NASA Astrophysics Data System (ADS)

Using our standard pore-level model, we have extended our earlier study of the crossover from fractal viscous fingering to compact /linear flow at a characteristic crossover time, ? , in three dimensions to systems with as many as a 106 pore bodies. These larger systems enable us to investigate the flows in the fully compact/well-past-crossover regime. The center of mass of the injected fluid exhibits basically the same behavior as found earlier but with an improved characteristic time. However, our earlier study of much smaller systems was unable to study the interfacial width in the important well-past-crossover regime, t?? . Now, we can study both the time evolution and roughness of the interfacial width. The interfacial width exhibits the same fractal-to-compact crossover as the center of mass, with the same characteristic time. In the fully compact regime, t?? , the interfacial width grows approximately linearly with time so that the standard growth exponent is approximately unity, ?=1.0±0.1 . We find that neither is the interface self-affine nor is the roughness of the interface in the compact regime consistent with an effective long-range surface tension as assumed by various theories. In fact, similar to Lévy flights, the height variations across the interface appear to be random with occasional large height variations.

Ferer, M.; Bromhal, Grant S.; Smith, Duane H.

2009-07-01

138

Nonlinear Geoscience: Fractals

NSDL National Science Digital Library

This site, from the Earth Monitoring Station at the University of North Carolina, provides an examination of self-similarity and fractals in geoscience. The generation of artificial fractals, fractal dimensions and definitions, and fractal geometry are explained. The site also contains examples such as river basins, coastlines, and lightening that exhibit fractal characteristics.

2007-03-30

139

The aim of this study was to explore new techniques in analysing postural control using nonlinear time-series analysis and to relate these results with the clinical knowledge on the postural system in Down syndrome (DS) subjects. In order to achieve the goal, we analysed the time domain and the frequency domain behaviour, the fractal dimension and the entropy of the centre of pressure signal in both directions during quiet standing in 35 participants with DS, comparing the results with a control population. DS patients evidenced a lack in postural control in anterior-posterior direction due to the impairment both in the high organisation and synergies and in the impairments due to ligament laxity and hypotonia. Maintaining posture is a task achieved by the integration of visual, vestibular and somatosensory receptors and the dynamical nature of this signal gives fundamental data about the lack of postural control in specific pathological condition. PMID:22657255

Rigoldi, C; Galli, M; Mainardi, L; Albertini, G

2014-04-01

140

NASA Astrophysics Data System (ADS)

Using special thermogravimetry and sorptometry methods physicochemical properties of carbon nanotube surfaces were investigated. A numerical and analytical procedure for the evaluation of total heterogeneous properties on the basis of liquid thermodesorption from the sample surfaces under the quasi-equilibrium conditions are presented. The desorption energy distribution was derived from the mass loss Q-TG and the differential mass loss Q-DTG curves of thermodesorption of pre-adsorbed polar and apolar liquid films. It is shown that the samples are highly sensitive to benzene vapor because the mechanism of liquid adsorption depends largely on the active surface centers and molecular interactions. For the first time, the evaluation of the fractal dimensions of nanotubes using the sorptometry and thermogravimetry data is presented.

Staszczuk, Piotr; Matyjewicz, Magdalena; Kowalska, Ewa; Radomska, Joanna; Byszewski, Przemyslaw; Kozlowski, M.

2003-04-01

141

Background Prostate cancer is a serious public health problem that affects quality of life and has a significant mortality rate. The aim of the present study was to quantify the fractal dimension and Shannon’s entropy in the histological diagnosis of prostate cancer. Methods Thirty-four patients with prostate cancer aged 50 to 75 years having been submitted to radical prostatectomy participated in the study. Histological slides of normal (N), hyperplastic (H) and tumor (T) areas of the prostate were digitally photographed with three different magnifications (40x, 100x and 400x) and analyzed. The fractal dimension (FD), Shannon’s entropy (SE) and number of cell nuclei (NCN) in these areas were compared. Results FD analysis demonstrated the following significant differences between groups: T vs. N and H vs. N groups (p?

2013-01-01

142

Fractal dimension of landscape silhouette outlines as a predictor of landscape preference

The aim of this study was to explore the suggestion that fractal characteristics may play a role in aesthetic experiences by providing possible empirical evidence for connections between landscape preference and fractal properties. This approach was motivated by the knowledge that many natural forms are fractal and that, in preference research, naturalness has been found an important predictor. For reasons

Caroline M. Hagerhall; Terry Purcell; Richard Taylor

2004-01-01

143

Journal of Coastal Research 22 5 1300Â1304 West Palm Beach, Florida September 2006 Fractal Analysis(5), 1300Â1304. West Palm Beach (Florida), ISSN 0749-0208. Average fractal dimensions (D) are calculated

Perfect, Ed

144

NASA Astrophysics Data System (ADS)

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.

Karemore, Gopal; Nielsen, Mads

2009-02-01

145

This study presents a Web platform (http://3dfd.ujaen.es) for computing and analyzing the 3D fractal dimension (3DFD) from volumetric data in an efficient, visual and interactive way. The Web platform is specially designed for working with magnetic resonance images (MRIs) of the brain. The program estimates the 3DFD by calculating the 3D box-counting of the entire volume of the brain, and also of its 3D skeleton. All of this is done in a graphical, fast and optimized way by using novel technologies like CUDA and WebGL. The usefulness of the Web platform presented is demonstrated by its application in a case study where an analysis and characterization of groups of 3D MR images is performed for three neurodegenerative diseases: Multiple Sclerosis, Intrauterine Growth Restriction and Alzheimer's disease. To the best of our knowledge, this is the first Web platform that allows the users to calculate, visualize, analyze and compare the 3DFD from MRI images in the cloud. PMID:24909817

Jiménez, J; López, A M; Cruz, J; Esteban, F J; Navas, J; Villoslada, P; Ruiz de Miras, J

2014-10-01

146

The fault diagnosis technology based on fractal geometry for logging truck engine

The paper discusses the fundamental conceptions and properties of fractal geometry. The definitions of fractal dimension are\\u000a described and the methods of calculating fractal dimension are introduced. The paper researches the peculiarities of fault\\u000a diagnosis for logging truck engine and puts forward the technical way of diagnosing the faults with the help of the fractal\\u000a geometry.

Du Yuanhu; Zhu Jianxin; Wu Yuecheng

1996-01-01

147

NASA Astrophysics Data System (ADS)

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 (?R) between the phonolite and the enclaves has been calculated as follows: log(?R) = 0.013e3.34Dbox PIC The viscosity of the enclaves has been calculated according to the ?Rvalue with the higher frequency and to the calculated viscosity of the phonolite between 900° and 1200° . We hypothesized that this value corresponds to the amount of mafic magma present in the system, while the other values represent different degrees of mingling and chemical diffusion. Viscosity of the basanite can be computed like: ?enclave = (%phonolite *?phonolite)+ (%basanite *?basanite) PIC ?enclaves--(%phonolite *?phonolite) ?basanite = %basanite PIC The minimum percentages which satisfy the relation are 69.5% of basanite and 30.5% of phonolite. Although the amount of mafic magma reaches the 69.5%, the presence of enclaves in the phonolite is just the ?1% and the amount of basanite erupted before could correspond to the 15% of the phonolite (estimated from stratigraphic sections). Probably a magma body of basanite was still stored in the magma chamber. The volume of basanite still stored during this time may have evolved to a more explosive magma and hence increases the volcanic risk in the area.

Albert, Helena; Perugini, Diego; Martí, Joan

2014-05-01

148

Structural investigations of fat fractals using small-angle scattering

NASA Astrophysics Data System (ADS)

Experimental small-angle scattering (SAS) data characterized, on a double logarithmic scale, by a succession of power-law decays with decreasing values of scattering exponents, can be described in terms of fractal structures with positive Lebesgue measure (fat fractals). Here we present a theoretical model for fat fractals and show how one can extract structural information about the underlying fractal using SAS method, for the well known fractals existing in the literature: Vicsek and Menger sponge. We calculate analytically the fractal structure factor and study its properties in momentum space. The models allow us to obtain the fractal dimension at each structural level inside the fractal, the number of particles inside the fractal and about the most common distances between the center of mass of the particles.

Anitas, Eugen M.

2015-01-01

149

NASA Astrophysics Data System (ADS)

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.

Boness, D. A.; Terrell-Martinez, B.

2010-12-01

150

Fractal analysis of engineering ceramics ground surface

NASA Astrophysics Data System (ADS)

Traditional methods of roughness characterization cannot properly reflect the differences between metal and engineering ceramics surfaces. Therefore, fractal method is introduced to characterize the engineering ceramics ground surface in this paper. This article compares various methods to calculate profile dimension. By comparison, the variation method is suitable for extracting fractal dimension of engineering ceramics ground surface. The precision of variation method is further improved by modifying the error. In view of engineering ceramics ground surface, we have analyzed the relationship between fractal dimension and traditional roughness parameters, surface texture, surface function and material property using modified variation method.

Liang, Xiaohu; Lin, Bin; Han, Xuesong; Chen, Shangong

2012-06-01

151

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 (?), 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 heterogeneity) reduction is consistent with the biological informative content decrease, due to granule content depletion. In spite of the significant differences in fractal dimension and lacunarity values registered according to the analysed feature (greyscale obtained values were, on average, lower than those obtained from outlines and skeletons; General Linear Model, p < 0.01), the discrimination power between not degranulated and degranulated MCs was, on average, the same and fully comparable with previously performed texture analysis on the same experimental material (outline and skeleton misclassification error, 20% [two false negative cases]; greyscale misclassification error, 30% [two false negative cases and one false positive case]). Fractal analysis proved to be a reliable and objective method for the characterization of MCs degranulation. PMID:25087582

Manera, M; Dezfuli, B S; Borreca, C; Giari, L

2014-11-01

152

Unbiased estimation of multi-fractal dimensions of finite data sets

We present a novel method for determining multi-fractal properties from experimental data. It is based on maximising the likelihood that the given finite data set comes from a particular set of parameters in a multi-parameter family of well known multi-fractals. By comparing characteristic correlations obtained from the original data with those that occur in artificially generated multi-fractals with the {\\em same} number of data points, we expect that predicted multi-fractal properties are unbiased by the finiteness of the experimental data.

A. J. Roberts; A. Cronin

1996-02-01

153

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

D. A. Boness; B. Terrell-Martinez

2010-01-01

154

Standard methods for computing the fractal dimensions of time series are usually tested with continuous nowhere differentiable functions, but not benchmarked with actual signals. Therefore they can produce opposite results in extreme signals. These methods also use different scaling methods, that is, different amplitude multipliers, which makes it difficult to compare fractal dimensions obtained from different methods. The purpose of this research was to develop an optimisation method that computes the fractal dimension of a normalised (dimensionless) and modified time series signal with a robust algorithm and a running average method, and that maximises the difference between two fractal dimensions, for example, a minimum and a maximum one. The signal is modified by transforming its amplitude by a multiplier, which has a non-linear effect on the signal's time derivative. The optimisation method identifies the optimal multiplier of the normalised amplitude for targeted decision making based on fractal dimensions. The optimisation method provides an additional filter effect and makes the fractal dimensions less noisy. The method is exemplified by, and explained with, different signals, such as human movement, EEG, and acoustic signals. PMID:24151522

2013-01-01

155

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

156

Medical image retrieval and analysis by Markov random fields and multi-scale fractal dimension

NASA Astrophysics Data System (ADS)

Many Content-based Image Retrieval (CBIR) systems and image analysis tools employ color, shape and texture (in a combined fashion or not) as attributes, or signatures, to retrieve images from databases or to perform image analysis in general. Among these attributes, texture has turned out to be the most relevant, as it allows the identification of a larger number of images of a different nature. This paper introduces a novel signature which can be used for image analysis and retrieval. It combines texture with complexity extracted from objects within the images. The approach consists of a texture segmentation step, modeled as a Markov Random Field process, followed by the estimation of the complexity of each computed region. The complexity is given by a Multi-scale Fractal Dimension. Experiments have been conducted using an MRI database in both pattern recognition and image retrieval contexts. The results show the accuracy of the proposed method in comparison with other traditional texture descriptors and also indicate how the performance changes as the level of complexity is altered.

Backes, André Ricardo; Cavaleri Gerhardinger, Leandro; do Espírito Santo Batista Neto, João; Martinez Bruno, Odemir

2015-02-01

157

Gene Entropy-Fractal Dimension Informatics with Application to Mouse-Human Translational Medicine

DNA informatics represented by Shannon entropy and fractal dimension have been used to form 2D maps of related genes in various mammals. The distance between points on these maps for corresponding mRNA sequences in different species is used to study evolution. By quantifying the similarity of genes between species, this distance might be indicated when studies on one species (mouse) would tend to be valid in the other (human). The hypothesis that a small distance from mouse to human could facilitate mouse to human translational medicine success is supported by the studied ESR-1, LMNA, Myc, and RNF4 sequences. ID1 and PLCZ1 have larger separation. The collinearity of displacement vectors is further analyzed with a regression model, and the ID1 result suggests a mouse-chimp-human translational medicine approach. Further inference was found in the tumor suppression gene, p53, with a new hypothesis of including the bovine PKM2 pathways for targeting the glycolysis preference in many types of cancerous cells, consistent with quantum metabolism models. The distance between mRNA and protein coding CDS is proposed as a measure of the pressure associated with noncoding processes. The Y-chromosome DYS14 in fetal micro chimerism that could offer protection from Alzheimer's disease is given as an example. PMID:23586047

Holden, T.; Cheung, E.; Dehipawala, S.; Ye, J.; Tremberger, G.; Lieberman, D.; Cheung, T.

2013-01-01

158

Medical image retrieval and analysis by Markov random fields and multi-scale fractal dimension.

Many Content-based Image Retrieval (CBIR) systems and image analysis tools employ color, shape and texture (in a combined fashion or not) as attributes, or signatures, to retrieve images from databases or to perform image analysis in general. Among these attributes, texture has turned out to be the most relevant, as it allows the identification of a larger number of images of a different nature. This paper introduces a novel signature which can be used for image analysis and retrieval. It combines texture with complexity extracted from objects within the images. The approach consists of a texture segmentation step, modeled as a Markov Random Field process, followed by the estimation of the complexity of each computed region. The complexity is given by a Multi-scale Fractal Dimension. Experiments have been conducted using an MRI database in both pattern recognition and image retrieval contexts. The results show the accuracy of the proposed method in comparison with other traditional texture descriptors and also indicate how the performance changes as the level of complexity is altered. PMID:25586375

Backes, André Ricardo; Gerhardinger, Leandro Cavaleri; Neto, João do Espírito Santo Batista; Bruno, Odemir Martinez

2015-02-01

159

The average roughness and fractal dimension of articular cartilage during drying.

Cartilage is a unique material in part because of it biphasic properties. The structure of cartilage is a porous matrix of collagen fibers, permeated with synovial fluid which creates a gliding and near frictionless motion in articulating joints. However, during in vitro testing or surgery, there exists potential for cartilage to dehydrate, or dry out. The effects of this drying can influence experimental results. It is likely that drying also changes joint performance in vivo. In in vitro testing of equine cartilage explants exposed to open air, the average roughness of cartilage changes from 1.85?±?0.78 to 3.66?±?1.41?µm SD in a 5-h period. Significant changes in roughness in individual samples are seen within 10?min of exposure to open air. However, the multi-scale nature of cartilage, characterized by the fractal dimension, does not significantly change during the same period. The current study attempts to quantify the magnitude of error that is introduced when cartilage is removed from its native environment. PMID:24173958

Smyth, P A; Rifkin, R; Jackson, R L; Hanson, R R

2014-01-01

160

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.

Mossotti, Victor G.; Eldeeb, A. Raouf

2000-01-01

161

Building Fractal Models with Manipulatives.

ERIC Educational Resources Information Center

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)

Coes, Loring

1993-01-01

162

Fractal Geometric Characterization of Functionally Graded Materials

Fractal Geometric Characterization of Functionally Graded Materials A. Saharan1 ; M. Ostoja graded materials (FGM) is studied from the standpoint of fractal geometry. First, upon introducing fractals, and an interfacial fractal dimension is estimated for varying degrees of fineness. Avariation

Ostoja-Starzewski, Martin

163

Comparison of fractal dimension estimation algorithms for epileptic seizure onset detection.

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. PMID:20571184

Polychronaki, G E; Ktonas, P Y; Gatzonis, S; Siatouni, A; Asvestas, P A; Tsekou, H; Sakas, D; Nikita, K S

2010-08-01

164

Comparison of different fractal dimension measuring algorithms for RE-TM M-O films

NASA Technical Reports Server (NTRS)

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.

Bernacki, Bruce E.; Mansuripur, M.

1991-01-01

165

NASA Astrophysics Data System (ADS)

In the last two decades several aspects relevant to volcanic activity have been analyzed in terms of fractal parameters that effectively describe natural objects geometry. More specifically, these researches have been aimed at the identification of (1) the power laws that governed the magma fragmentation processes, (2) the energy of explosive eruptions, and (3) the distribution of the associated earthquakes. In this paper, the study of volcano morphology via satellite images is dealt with; in particular, we use the complete forward model developed by some of the authors (Di Martino et al., 2012) that links the stochastic characterization of amplitude Synthetic Aperture Radar (SAR) images to the fractal dimension of the imaged surfaces, modelled via fractional Brownian motion (fBm) processes. Based on the inversion of such a model, a SAR image post-processing has been implemented (Di Martino et al., 2010), that allows retrieving the fractal dimension of the observed surfaces, dictating the distribution of the roughness over different spatial scales. The fractal dimension of volcanic structures has been related to the specific nature of materials and to the effects of active geodynamic processes. Hence, the possibility to estimate the fractal dimension from a single amplitude-only SAR image is of fundamental importance for the characterization of volcano structures and, moreover, can be very helpful for monitoring and crisis management activities in case of eruptions and other similar natural hazards. The implemented SAR image processing performs the extraction of the point-by-point fractal dimension of the scene observed by the sensor, providing - as an output product - the map of the fractal dimension of the area of interest. In this work, such an analysis is performed on Cosmo-SkyMed, ERS-1/2 and ENVISAT images relevant to active stratovolcanoes in different geodynamic contexts, such as Mt. Somma-Vesuvio, Mt. Etna, Vulcano and Stromboli in Southern Italy, Shinmoe in Japan, Merapi in Indonesia. Preliminary results reveal that the fractal dimension of natural areas, being related only to the roughness of the observed surface, is very stable as the radar illumination geometry, the resolution and the wavelength change, thus holding a very unique property in SAR data inversion. Such a behavior is not verified in case of non-natural objects. As a matter of fact, when the fractal estimation is performed in the presence of either man-made objects or SAR image features depending on geometrical distortions due to the SAR system acquisition (i.e. layover, shadowing), fractal dimension (D) values outside the range of fractality of natural surfaces (2 < D < 3) are retrieved. These non-fractal characteristics show to be heavily dependent on sensor acquisition parameters (e.g. view angle, resolution). In this work, the behaviour of the maps generated starting from the C- and X- band SAR data, relevant to all the considered volcanoes, is analyzed: the distribution of the obtained fractal dimension values is investigated on different zones of the maps. In particular, it is verified that the fore-slope and back-slope areas of the image share a very similar fractal dimension distribution that is placed around the mean value of D=2.3. We conclude that, in this context, the fractal dimension could be considered as a signature of the identification of the volcano growth as a natural process. The COSMO-SkyMed data used in this study have been processed at IREA-CNR within the SAR4Volcanoes project under Italian Space Agency agreement n. I/034/11/0.

Pepe, S.; Di Martino, G.; Iodice, A.; Manzo, M.; Pepe, A.; Riccio, D.; Ruello, G.; Sansosti, E.; Tizzani, P.; Zinno, I.

2012-04-01

166

Biometric feature extraction using local fractal auto-correlation

NASA Astrophysics Data System (ADS)

Image texture feature extraction is a classical means for biometric recognition. To extract effective texture feature for matching, we utilize local fractal auto-correlation to construct an effective image texture descriptor. Three main steps are involved in the proposed scheme: (i) using two-dimensional Gabor filter to extract the texture features of biometric images; (ii) calculating the local fractal dimension of Gabor feature under different orientations and scales using fractal auto-correlation algorithm; and (iii) linking the local fractal dimension of Gabor feature under different orientations and scales into a big vector for matching. Experiments and analyses show our proposed scheme is an efficient biometric feature extraction approach.

Chen, Xi; Zhang, Jia-Shu

2014-09-01

167

The goal of this study was to determine material properties for the anterior cortex and subcortical regions of human patellae and relate those properties to mineral density and fractal dimension of the bone. Ten human patellae were obtained from eight fresh frozen human cadavers and subjected to anteriorly-directed spherical indentation-relaxation experiments using two different sized indenters to two different indentation depths. Response data were fit to a three-mode viscoelastic model obtained through elastic-viscoelastic correspondence of the Hertzian contact relation for spherical indentation. A location-specific effective bone density measurement that more heavily weighted bone material close to the indentation site (by von Mises stress distribution) was determined from micro-computed tomography (38µm resolution) data captured for each specimen. The same imagery data were used to compute location specific fractal dimension estimates for each indentation site. Individual and averaged patella material models verified the hypothesis that when the larger indenter and greater indentation depth is used to engage the surface and deeper (trabecular) bone, the bone exhibits a more compliant response than when only the surface (cortical) bone was engaged (instantaneous elastic modulus was 325MPa vs. 207MPa, p<0.05). Effective bone mineral density was shown to be a significant predictor of the elastic modulus for both small and large indentation types (p<0.05) despite relatively low correlations. Exponential regressions of fractal dimension on elastic modulus showed significant relationships with high correlation for both the small (R(2)=0.93) and large (R(2)=0.97) indentations. PMID:23972564

Kerrigan, Jason R; Sanchez-Molina, David; Neggers, Jan; Arregui-Dalmases, Carlos; Velazquez-Ameijide, Juan; Crandall, Jeff R

2014-05-01

168

NASA Astrophysics Data System (ADS)

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.

Saji, Ryoya; Konno, Hidetoshi

2000-02-01

169

In this paper, we present the finite cube elements method (FCEM); a novel numerical tool for calculating the gravity anomaly g and structural index SI of solid models with defined boundaries and variable density distributions, tilted or in normal position (e.g. blocks, faulted blocks, cylinders, spheres, hemispheres, triaxial ellipsoids). Extending the calculation to fractal objects, such as Menger sponges of

Mostafa E. Mostafa

2008-01-01

170

Verifying the Dependence of Fractal Coefficients on Different Spatial Distributions

A fractal distribution requires that the number of objects larger than a specific size r has a power-law dependence on the size N(r) = C/r{sup D}propor tor{sup -D} where D is the fractal dimension. Usually the correlation integral is calculated to estimate the correlation fractal dimension of epicentres. A 'box-counting' procedure could also be applied giving the 'capacity' fractal dimension. The fractal dimension can be an integer and then it is equivalent to a Euclidean dimension (it is zero of a point, one of a segment, of a square is two and of a cube is three). In general the fractal dimension is not an integer but a fractional dimension and there comes the origin of the term 'fractal'. The use of a power-law to statistically describe a set of events or phenomena reveals the lack of a characteristic length scale, that is fractal objects are scale invariant. Scaling invariance and chaotic behavior constitute the base of a lot of natural hazards phenomena. Many studies of earthquakes reveal that their occurrence exhibits scale-invariant properties, so the fractal dimension can characterize them. It has first been confirmed that both aftershock rate decay in time and earthquake size distribution follow a power law. Recently many other earthquake distributions have been found to be scale-invariant. The spatial distribution of both regional seismicity and aftershocks show some fractal features. Earthquake spatial distributions are considered fractal, but indirectly. There are two possible models, which result in fractal earthquake distributions. The first model considers that a fractal distribution of faults leads to a fractal distribution of earthquakes, because each earthquake is characteristic of the fault on which it occurs. The second assumes that each fault has a fractal distribution of earthquakes. Observations strongly favour the first hypothesis.The fractal coefficients analysis provides some important advantages in examining earthquake spatial distribution, which are: - Simple way to quantify scale-invariant distributions of complex objects or phenomena by a small number of parameters. - It is becoming evident that the applicability of fractal distributions to geological problems could have a more fundamental basis. Chaotic behaviour could underlay the geotectonic processes and the applicable statistics could often be fractal.The application of fractal distribution analysis has, however, some specific aspects. It is usually difficult to present an adequate interpretation of the obtained values of fractal coefficients for earthquake epicenter or hypocenter distributions. That is why in this paper we aimed at other goals - to verify how a fractal coefficient depends on different spatial distributions. We simulated earthquake spatial data by generating randomly points first in a 3D space - cube, then in a parallelepiped, diminishing one of its sides. We then continued this procedure in 2D and 1D space. For each simulated data set we calculated the points' fractal coefficient (correlation fractal dimension of epicentres) and then checked for correlation between the coefficients values and the type of spatial distribution.In that way one can obtain a set of standard fractal coefficients' values for varying spatial distributions. These then can be used when real earthquake data is analyzed by comparing the real data coefficients values to the standard fractal coefficients. Such an approach can help in interpreting the fractal analysis results through different types of spatial distributions.

Gospodinov, Dragomir [Plovdiv University 'Paisii Hilendarski', 24, Tsar Asen Str., Plovdiv (Bulgaria); Geophysical Institute of Bulgarian Academy of Sciences, Akad. G. Bonchev Str., bl.3, Sofia (Bulgaria); Marekova, Elisaveta; Marinov, Alexander [Plovdiv University 'Paisii Hilendarski', 24, Tsar Asen Str., Plovdiv (Bulgaria)

2010-01-21

171

Fractal Dimension of the 1999 Chamoli Earthquake from Aftershock Studies in Garhwal Himalaya

The Aftershock sequence of Chamoli earthquake ( M w 6.4) of 29 March 1999 is analyzed to study the fractal structure in space, time and magnitude distribution. The b value is found to be 0.63 less than which is usually observed worldwide and in the Himalayas. This indicates that the numbers of smaller earthquakes are relatively less than the larger

Richa Jain; B. K. Rastogi; V. P. Dimri

2003-01-01

172

ACCELERATION OF A PROCEDURE TO GENERATE FRACTAL CURVES OF A GIVEN DIMENSION THROUGH THE

described the use of grammatical evolution to automatically generate L Systems (LS) representing fractal; c) application of a grammar-evolution based genetic algorithm to get a grammar representing, by means of a representation scheme: vector graphics or turtle graphics. In a previous work [Alfonseca

Alfonseca, Manuel

173

NSDL National Science Digital Library

Dr. Mary Ann Connors, a faculty member in the Department of Mathematics and Statistics at University of Massachusetts Amherst has developed this website based on a curriculum project previously funded by the National Science Foundation. Exploring Fractals provides an introduction to fractals, explores concepts such as shape and dimension, provides some classroom investigations, demonstrates how to create simple fractals, and offers some additional information for teachers. Diagrams and pictures are used as part of the explanations. Other Internet resources for further investigation are also provided.

2006-01-19

174

NSDL National Science Digital Library

Dr. Mary Ann Connors, a faculty member in the Department of Mathematics and Statistics at University of Massachusetts Amherst has developed this website based on a curriculum project previously funded by the National Science Foundation. Exploring Fractals provides an introduction to fractals, explores concepts such as shape and dimension, provides some classroom investigations, demonstrates how to create simple fractals, and offers some additional information for teachers. Diagrams and pictures are used as part of the explanations. Other Internet resources for further investigation are also provided.

175

NSDL National Science Digital Library

Dr. Mary Ann Connors, a faculty member in the Department of Mathematics and Statistics at University of Massachusetts Amherst has developed this website based on a curriculum project previously funded by the National Science Foundation. Exploring Fractals provides an introduction to fractals, explores concepts such as shape and dimension, provides some classroom investigations, demonstrates how to create simple fractals, and offers some additional information for teachers. Diagrams and pictures are used as part of the explanations. Other Internet resources for further investigation are also provided.

2007-12-12

176

A fractal analysis of quaternary, Cenozoic-Mesozoic, and Late Pennsylvanian sea level changes

NASA Technical Reports Server (NTRS)

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.

Hsui, Albert T.; Rust, Kelly A.; Klein, George D.

1993-01-01

177

Fractal analysis of surface topography in ground monocrystal sapphire

NASA Astrophysics Data System (ADS)

The surface characterization of ground monocrystal sapphire is investigated by fractal analysis method. A serial of ground sapphire surfaces in ductile removal and brittle removal mode are obtained by grinding experiments, and their three dimensional (3D) and two dimensional (2D) fractal dimensions are calculated and analyzed by box-counting methods. The 3D surface fractal dimension Ds shows a negative correlation with the surface roughness parameter Ra and is sensitive to the ground surface defects. For the ground surface with larger fractal dimension Ds, its micro-topography is more exquisite with minor defects. Once the fractal dimension Ds become smaller, deep cracks and pronounced defects are exhibited in ground surface. Moreover, the material removal mode can be implied from the distribution of 2D cross-sectional profile fractal dimension DL. The workpiece surface generated in ductile removal mode has high surface quality with high 2D and 3D fractal dimensions. This study indicates that the box-counting fractal analysis is an effective method to evaluate ground sapphire surface comprehensively.

Wang, Qiuyan; Liang, Zhiqiang; Wang, Xibin; Zhao, Wenxiang; Wu, Yongbo; Zhou, Tianfeng

2015-02-01

178

NSDL National Science Digital Library

This lesson is designed to develop students' understanding of fractals and fractal dimension. This lesson provides links to discussions and activities related to fractals as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

2010-01-01

179

gasket or the von Koch snowflake or even to a Cantor set. These are examples of fractals (the word is due Let S be von Koch Snowflake, obtained by from the unit interval on the real line by replacing the same construction of each of resulting four line segments, and so on. Covering the snowflake

Bullett, Shaun

180

Fractal dimension (FD) in tissue specimens from patients with oral squamous cell carcinoma (OSCC) was evaluated. FD values in different stages of OSCC, and the correlations with clinicopathological variables and patient survival were investigated. Histological sections from OSCC and control non-neoplastic mucosa specimens were stained with hematoxylin-eosin for pathological analysis and with Feulgen for nuclear evaluation. FD in OSCC groups vs. controls revealed statistically significant differences (P < 0.001). In addition, a progressive increase of FD from stage I and II lesions and stage III and IV lesions was observed, with statistically significant differences (P = 0.003). Moreover, different degrees of tumor differentiation showed a significant difference in the average nuclear FD values (P = 0.001). A relationship between FD and patients' survival was also detected with lower FD values associated to longer survival time and higher FD values with shorter survival time (P = 0.034). These data showed that FD significantly increased during OSCC progression. Thus, FD could represent a novel prognostic tool for OSCC, as FD values significantly correlated with patient survival. Fractal geometry could give insights into tumor morphology and could become an useful tool for analyzing irregular tumor growth patterns. PMID:25367085

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

2014-11-01

181

A generalized volume dimension of complex networks

NASA Astrophysics Data System (ADS)

The fractal and self-similarity properties are investigated in many real complex networks. The volume dimension method is an effective tool to measure the fractal property of complex networks. In this paper, a new volume dimension measure is proposed based on the node degree of complex networks. We apply the proposed method to calculate the fractal dimension of some real networks and Newman–Watts (NW) small-world. The results show that the proposed method is effective when dealing with the fractal dimension problem of complex networks. In addition, we find that the fractal dimension is mainly influenced by the probability of ‘adding edges’ and the average length of the small-world network.

Wei, Daijun; Wei, Bo; Zhang, Haixin; Gao, Cai; Deng, Yong

2014-10-01

182

This study examines the statistical properties of volatility. Fractal dimension, probability distribution and two-point volatility correlation are used to measure and compare volatility among six different markets for the 12-year period from Jan. 1 1990 to Dec. 31 2001. New York market is found to be the strongest among the six in terms of market efficiency. Moreover, the Tokyo and

Hai-Chin YU; Ming-Chang Huang

2003-01-01

183

NASA Astrophysics Data System (ADS)

To better understand the characteristics of coal pores and their influence on coal reservoirs, coal pores and fissures in eight main coalfields of North China were analyzed by using mercury porosimetry, scanning electron microscopy (SEM), electron microscope and photometer. Fractal characteristics of coal pores (size distribution and structure) were researched by using thermodynamics model. This new approach has been developed proposing the relationships between fractal dimensions and coal pore-fissure characteristics. Mercury porosimetry data indicate that the coals are fractal, with pore radius ranging from 0.1 to 50 ?m. 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 coals have more complicated and inhomogeneous pore structures than other rocks. The results show that: 1) By scanning electron microscopy (SEM) imaging analysis and fractal appraisal, the coal reservoirs generally have very high heterogeneity. The coal pore-fissure system was divided into six types (including gas pores, fissures, scrap pores and their combinations). 2) Coal pore structures have fractal characteristics and fractal dimensions reflect the characterization of pore structures that controlled by the composition (e.g., ash, moisture, volatile component) and pore parameters (e.g., pore diameter, micro pores content) of coal. 3) The fractal dimensions of coal pores have good correlations with heterogeneity of coal pore structures. Bigger the fractal dimensions, higher the heterogeneity of pore structures. 4) The fractal dimensions and petrologic permeability of coals have strong negative exponent correlation. However, the fractal dimensions and petrologic permeability of coals have no obvious correlation. In this paper, the thermodynamics model was used to investigate the relationship between heterogeneity, petrologic permeability and coal rank with the fractal dimensions. A horizontal ‘U-shaped' trend between fractal dimensions and coal ranks is observed, with the minimum fractal dimensions occurring at 1.0-2.4% Ro, m. The effects of coal rank upon fractal dimensions are mainly due to the variety of micropore contents and aromaticity of coals during coalification. Pore structure of coals was characterized by using the fractal theory of multiporous material. Based on the fractal theory of geometry, the fractal dimensions which tested by mercury porosimetry of pores in coal reservoirs range from 2.0 to 3.5. The situation that the value exceeds 3.0 may reflect the strong structural movement, high metamorphism and excessive pressure. Moreover, the correlation between pore characteristics and fractal dimension was gained by using SEM images assistant analysis.

Cai, Yidong; Liu, Dameng; Yao, Yanbin

2010-05-01

184

Two imported commercial cheeses (Cheddar and Gouda) were analyzed to characterize their textural changes during storage period from 176 to 362 days at 4°C. Fractal dimension analysis was used to examine the structure of the cheeses, and a scaling model based on stress values was used to calculate fractal dimension (Df) by the Hausdorff dimension count method. It was found

Hong-Hua Xu; Yin-Yuan He; Xin-Huai Zhao; Tie-Jing Li

2011-01-01

185

Fractal analysis of high-resolution CT images as a tool for quantification of lung diseases

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.

Uppaluri, R.; Mitsa, T.; Galvin, J.R. [Univ. of Iowa, Iowa City, IA (United States)

1995-12-31

186

The Quasi-Fractal Structure of Fish Brain Neurons

The box-counting method for calculating the fractal dimension (D) with the ImageJ 1.20s software is used as a tool for quantitative analysis of the neuronal morphology in the fish brain. The fractal dimension was determined for several types of neurons in the brain of two teleost species, Pholidapus dybowskii and Oncorhynchus keta. These results were compared with those obtained for

V. V. Isaeva; E. V. Pushchina; Yu. A. Karetin

2004-01-01

187

Fractal dimension of sparkles in automotive metallic coatings by multispectral imaging measurements.

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. PMID:24945784

Medina, José M; Díaz, José A; Vignolo, Carlos

2014-07-23

188

Fractal geometry of lightning strokes

Fractal dimension is defined, and methods of approximating it are presented. Several pictures of lightning and a picture of an electric arc from a point charge were digitized, and the fractal dimension of the resultant arrays was computed. The use of the diffusion limited aggregation (DLA) fractal growth method to model dielectric breakdown of a point charge is discussed. To

Charles I. Richman

1990-01-01

189

Fault diagnosis of diesel engine based on adaptive wavelet packets and EEMD-fractal dimension

NASA Astrophysics Data System (ADS)

In this paper a novel method for de-noising nonstationary vibration signal and diagnosing diesel engine faults is presented. The method is based on the adaptive wavelet threshold (AWT) de-noising, ensemble empirical mode decomposition (EEMD) and correlation dimension (CD). A new adaptive wavelet packet (WP) thresholding function for vibration signal de-noising is used in this paper. To alleviate the mode mixing problem occurring in EMD, ensemble empirical mode decomposition (EEMD) is presented. With EEMD, the components with truly physical meaning can be extracted from the signal. Utilizing the advantage of EEMD, this paper proposes a new AWT-EEMD-based method for fault diagnosis of diesel engine. A study of correlation dimension in engine condition monitoring is reported also. Some important influencing factors relating directly to the computational precision of correlation dimension are discussed. Industrial engine normal and fault vibration signals measured from different operating conditions are analyzed using the above method.

Wang, Xia; Liu, Changwen; Bi, Fengrong; Bi, Xiaoyang; Shao, Kang

2013-12-01

190

Fractal dimension analysis (FDA) is modern mathematical method widely used to describing of complex and chaotic shapes when classic methods fail. The main aim of this study was evaluating the influence of photodynamic therapy (PDT) with cystein proteases inhibitors (CPI) on the number and morphology of blood vessels inside tumor and on increase of effectiveness of combined therapy in contrast to PDT and CPI used separately. Animals were divided into four groups: control, treated using only PDT, treated using only CPI and treated using combined therapy, PDT and CPI. Results showed that time of animal survival and depth of necrosis inside tumor were significantly higher in CPI+PDT group in contrast to other groups. The higher value of fractal dimension (FD) was observed in control group, while the lowest value was found in the group which was treated by cystein protease inhibitors. The differences between FD were observed in CPI group and PDT+CPI group in comparison to control group. Our results revealed that fractal dimension analysis is a very useful tool in estimating differences between irregular shapes like blood vessels in PDT treated tumors. Thus, the implementation of FDA algorithms could be useful method in evaluating the efficacy of PDT. PMID:22991578

Jurczyszyn, Kamil; Osiecka, Beata J.; Zió?kowski, Piotr

2012-01-01

191

A study on the computerized fractal analysis of architectural distortion in screening mammograms

Architectural distortion (AD) is a sign of malignancy often missed during mammographic interpretation. The purpose of this study was to explore the application of fractal analysis to the investigation of AD in screening mammograms. The study was performed using mammograms from the Digital Database for Screening Mammography (DDSM). The fractal dimension (FD) of mammographic regions of interest (ROIs) was calculated

Georgia D Tourassi; David M Delong; Carey E Floyd Jr

2006-01-01

192

Fundamental thermodynamic relations are formulated based on the equation of physical adsorption on microporous fractal solids, proposed previously and derived from the Polanyi-Dubinin theory of volume filling of micropores. A new adsorption isotherm and corresponding adsorption heat equations are verified using the experimental data published by Dubinin and Polstyanov of benzene and cyclohexane adsorption and adsorption heat on three microporous

Artur P. Terzyk; Piotr A. Gauden; Gerhard Rychlicki; Roman Wojsz

1998-01-01

193

NASA Astrophysics Data System (ADS)

Lipase B from Candida Antarctica (also known as Candida antarctica lipase B or CALB) was immobilized onto titanium dioxide (TiO 2) in a buffer-free, bidistilled aqueous medium. The adsorption isotherm was determined by UV-vis analysis of supernatant solution at 280 nm, revealing that in 7 h 98% of the theoretical lipase monolayer on the TiO 2 (with 45.7 m 2/g BET area) was achieved. Samples withdrawn from the supernatant immobilization medium were analyzed by Fourier-transform infrared spectroscopy in the 1700-1600 cm -1 range (where the Amide I Proteins band appears) in order to obtain quantitative information on the evolution of the secondary-structure elements of the protein. The analysis performed revealed that lipase conformation suffers only minor changes during its adsorption onto TiO 2. However, water associated to the lipase may interact of several ways with the surface of the hydrated oxide. Characterization of the immobilized biocatalyst (CALB/TiO 2) implied SEM, fractal dimension analysis and FTIR techniques. A proposal of lipase-hydrated oxide interaction is presented.

Foresti, M. L.; Valle, G.; Bonetto, R.; Ferreira, M. L.; Briand, L. E.

2010-01-01

194

Fractal Pore Structure Model and Multilayer Fractal Adsorption in Shale

NASA Astrophysics Data System (ADS)

The complex structure and surface property of porous media have significant impact on its accumulation and adsorption capacity. Based on the fractal theory, this paper presents a fractal pore structure model for shales. The effect of different pore structures on fractal dimension is discussed, and the influence of fractal dimension and pore size distribution on porosity is also analyzed. It is shown that the fractal dimension D decreases with the increase of structure parameter q/m for a certain pore diameter ratio, and porosity has positive relationship with fractal dimension. This paper also presents a multilayer fractal adsorption model which takes into account the roughness of adsorption surface by using fractal theory. With the introduction of pseudo-saturated vapor pressure in the supercritical temperature condition, the proposed adsorption model can be applied into a wider range of temperature. Based on the low-pressure nitrogen adsorption and methane isothermal adsorption experiments, the effect of fractal dimension on the adsorption behavior of shales is discussed. Fractal dimension has significant impact on the surface adsorption property and adsorption layer number n. The monolayer saturated adsorption volume Vm increases with the increase of D, while parameter C has the opposite variation trend. Finally, the optimal combination of fractal parameters for describing pore structure of shale samples is selected.

Zhang, Liehui; Li, Jianchao; Tang, Hongming; Guo, Jingjing

2014-09-01

195

Retarded hydrodynamic properties of fractal clusters.

Fractal clusters are commonly encountered when working with the stability and the aggregation of colloidal suspensions. In spite of the number of studies that have focused on their stationary hydrodynamic properties, no information is currently known on their retarded hydrodynamic properties. The objective of this work is to close this gap. Clusters with a broad range of fractal dimension values, generated via Monte-Carlo simulations have been analyzed. A rigorous model based on multipole expansion of time-dependent Stokes equations has been developed, and then the full cluster resistance matrix as a function of the frequency has been computed. An attempt has been made to extend Basset, Boussinesque and Oseen equations to fractal clusters, but it was found that the corresponding hydrodynamic radius needs to be a function of frequency. In the case of translational motion, the cluster hydrodynamic radius loses any structural information at high frequencies, becoming independent of the fractal dimension, but depending only on its mass. A simplified model, based on an extension of Kirkwood-Rieseman approach has also been developed. This allows one to perform calculations for clusters with arbitrary masses and fractal dimensions, with good accuracy and very low computational time. It is the first time that the frequency dependence of hydrodynamic properties of complex non-spherical objects has been investigated. PMID:24935184

Lattuada, Marco

2014-09-01

196

Fractal protein structure revisited: Topological, kinetic and thermodynamic relationships

NASA Astrophysics Data System (ADS)

The present work explored the definitions and calculations of fractal dimensions in protein structures and the corresponding relationships with the protein class, secondary structure contents, fold type as well as kinetic and thermodynamic parameters like the folding and unfolding rate, the folding-unfolding free energy and others. The results showed a positive correlation of some fractal exponents with the kinetic and thermodynamic variables even considering the effect of the protein length. On the other hand the influences of secondary structures types, especially the turn conformation are significant as well as the fractal exponent profiles according to class and fold types.

Tejera, E.; Machado, A.; Rebelo, I.; Nieto-Villar, J.

2009-11-01

197

NASA Technical Reports Server (NTRS)

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.

Wiscombe, W.

1999-01-01

198

Electromagnetism on Anisotropic Fractals

We derive basic equations of electromagnetic fields in fractal media which are specified by three indepedent fractal dimensions {\\alpha}_{i} in the respective directions x_{i} (i=1,2,3) of the Cartesian space in which the fractal is embedded. To grasp the generally anisotropic structure of a fractal, we employ the product measure, so that the global forms of governing equations may be cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving the {\\alpha}_{i}'s. First, a formulation based on product measures is shown to satisfy the four basic identities of vector calculus. This allows a generalization of the Green-Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Amp\\`ere laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell's electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers.

Martin Ostoja-Starzewski

2011-06-08

199

Map of fluid flow in fractal porous medium into fractal continuum flow.

This paper is devoted to fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. Hydrodynamics of fractal continuum flow is developed on the basis of a self-consistent model of fractal continuum employing vector local fractional differential operators allied with the Hausdorff derivative. The generalized forms of Green-Gauss and Kelvin-Stokes theorems for fractional calculus are proved. The Hausdorff material derivative is defined and the form of Reynolds transport theorem for fractal continuum flow is obtained. The fundamental conservation laws for a fractal continuum flow are established. The Stokes law and the analog of Darcy's law for fractal continuum flow are suggested. The pressure-transient equation accounting the fractal metric of fractal continuum flow is derived. The generalization of the pressure-transient equation accounting the fractal topology of fractal continuum flow is proposed. The mapping of fluid flow in a fractally permeable medium into a fractal continuum flow is discussed. It is stated that the spectral dimension of the fractal continuum flow d(s) is equal to its mass fractal dimension D, even when the spectral dimension of the fractally porous or fissured medium is less than D. A comparison of the fractal continuum flow approach with other models of fluid flow in fractally permeable media and the experimental field data for reservoir tests are provided. PMID:23004869

Balankin, Alexander S; Elizarraraz, Benjamin Espinoza

2012-05-01

200

Fractal analysis of time varying data

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.

Vo-Dinh, Tuan (Knoxville, TN); Sadana, Ajit (Oxford, MS)

2002-01-01

201

Self-avoiding walk on fractal complex networks: Exactly solvable cases

NASA Astrophysics Data System (ADS)

We study the self-avoiding walk on complex fractal networks called the (u ,v ) -flower by mapping it to the N -vector model in a generating function formalism. First, we analytically calculate the critical exponent ? and the connective constant by a renormalization-group analysis in arbitrary fractal dimensions. We find that the exponent ? is equal to the displacement exponent, which describes the speed of diffusion in terms of the shortest distance. Second, by obtaining an exact solution for the (u ,u ) -flower, we provide an example which supports the conjecture that the universality class of the self-avoiding walk on graphs is not determined only by the fractal dimension.

Hotta, Yoshihito

2014-11-01

202

Fractal and multifractal properties of a family of fractal networks

NASA Astrophysics Data System (ADS)

In this work, we study the fractal and multifractal properties of a family of fractal networks introduced by Gallos et al (2007 Proc. Nat. Acad. Sci. USA 104 7746). In this fractal network model, there is a parameter e which is between 0 and 1, and allows for tuning the level of fractality in the network. Here we examine the multifractal behavior of these networks, the dependence relationship of the fractal dimension and the multifractal parameters on parameter e. First, we find that the empirical fractal dimensions of these networks obtained by our program coincide with the theoretical formula given by Song et al (2006 Nature Phys. 2 275). Then from the shape of the ?(q) and D(q) curves, we find the existence of multifractality in these networks. Last, we find that there exists a linear relationship between the average information dimension

Li, Bao-Gen; Yu, Zu-Guo; Zhou, Yu

2014-02-01

203

The structure and dynamics of an aggregation have been studied when the aggregate grows from a lattice gas with a nonzero gas density ng. At low ng and for a short length scale up to xi, the structure of the aggregation is fractal and similar to the diffusion-limited aggregation (DLA). For a large length scale it is compact and has

Makio Uwaha; Yukio Saito

1989-01-01

204

Fractal antenna engineering: the theory and design of fractal antenna arrays

A fractal is a recursively generated object having a fractional dimension. Many objects, including antennas, can be designed using the recursive nature of a fractal. In this article, we provide a comprehensive overview of recent developments in the field of fractal antenna engineering, with particular emphasis placed on the theory and design of fractal arrays. We introduce some important properties

Douglas H. Werner; R. L. Haupt; P. L. Werner

1999-01-01

205

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.

Mossotti, Victor G.; Eldeeb, A. Raouf; Oscarson, Robert

1998-01-01

206

The nature of fractals and the use of fractals instead of classical scaling concepts to describe the irregular surfaces, structures, and processes exhibited by physiological systems are described. The mathematical development of fractals is reviewed, and examples of natural fractals are cited. Relationships among power laws, noise, and fractal time signals are examined

William Deering; Bruce J. West

1992-01-01

207

Fractal Geometry of Architecture

NASA Astrophysics Data System (ADS)

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.

Lorenz, Wolfgang E.

208

NASA Astrophysics Data System (ADS)

Atechnique is proposed which uses fractal analysis for the non- traumatic and non-invasive quantification of trabecular bone density in the mandible using standard dental radiographs. Binary images of trabecular bone patterns are derived from digitized radiographic images. Fractal analysis is then used to calculate the Hausdorif dimension (D) of the binary image patterns. Variations in D calculated with this method can be correlated with known cases of systemic osteoporosis to establish normal and abnormal ranges for the value of D.

Doyle, Michael D.; Rabin, Harold; Suri, Jasjit S.

1991-04-01

209

Testing Fractal Methods on Observed and Simulated Solar Magnetograms

NASA Technical Reports Server (NTRS)

The term "magnetic complexity" has not been sufficiently quantified. To accomplish this, we must understand the relationship between the observed magnetic field of solar active regions and fractal dimension measurements. Using data from the Marshall Space Flight Center's vector magnetograph ranging from December 1991 to July 2001, we compare the results of several methods of calculating a fractal dimension, e.g., Hurst coefficient, the Higuchi method, power spectrum, and 2-D Wavelet Packet Analysis. In addition, we apply these methods to synthetic data, beginning with representations of very simple dipole regions, ending with regions that are magnetically complex.

Adams, M.; Falconer, D. A.; Lee, J. K.; Jones, C.

2003-01-01

210

Routes to fractality and entropy in Liesegang systems

NASA Astrophysics Data System (ADS)

Liesegang bands are formed when solutions of co-precipitate ions interdiffuse in a 1D gel matrix. In a recent study [R. F. Sultan, Acta. Mech. Sin. 27, 119 (2011)], Liesegang patterns have been characterized as fractal structures. In addition to experimentally obtained patterns, geometric Liesegang patterns were constructed in conformity with the well-known empirical laws. Both mathematical fractal dimensions and box count dimensions for images of PbF2 and PbI2 Liesegang patterns have been calculated. Liesegang patterns can also be described by the entropy state function, and categorized as more or less ordered structures. We revisit the relation between entropy and fractal dimension, and apply it to simulated geometrical Liesegang patterns. We have resort to three different routes for the estimation of the entropy of a Liesegang pattern. The HarFA software enabled the calculation of the Hausdorff dimension and the topological entropy, then the information dimension and the Shannon entropy. In a third pathway, analytical calculations were carried out by estimating the probability of occurrence of a fractal element or coverage. The product of Shannon entropy and Boltzmann constant yields the thermodynamic entropy. The values for PbF2 and PbI2 Liesegang patterns attained the order of magnitude of the reported Third Law entropies, but yet remained lower, in conformity with the more ordered Liesegang structures.

Kalash, Leen; Sultan, Rabih

2014-06-01

211

NSDL National Science Digital Library

The "Mountains of Fractals" article in the Math DL develops algorithms to produce coastlines and mountains in two dimensions by adapting mathematical ideas related to the construction of such fractals as Koch's curve. EJS is used to create a hands-on activity that allows a reader to create a coastline with a rubberband, six-sided die, and thumb tacks. Java applications allow for exploration of these algorithms and the influence of their associated parameters. After discussing 2D fractal mountains, this article extends the 2D algorithm to produce 3D mountains. Finally, mathematical issues in random number generation are discussed. More specifically, linear congruential generators are considered and shown to be suitable as a random number generator for the 3D fractal landscape algorithm. The use of fractal landscapes in movies is also discussed.

Chartier, Tim

2009-09-11

212

Fractal structures and processes

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.}

Bassingthwaighte, J.B.; Beard, D.A.; Percival, D.B.; Raymond, G.M. [National Simulation Resource, Department of Bioengineering, University of Washington, Seattle, Washington 98195 (United States)

1996-06-01

213

Roughness Perception of Haptically Displayed Fractal Surfaces

NASA Technical Reports Server (NTRS)

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.

Costa, Michael A.; Cutkosky, Mark R.; Lau, Sonie (Technical Monitor)

2000-01-01

214

Fractal analysis of remotely sensed images: A review of methods and applications

Mandelbrot's fractal geometry has sparked considerable interest in the remote sensing community since the publication of his highly influential book in 1977. Fractal models have been used in several image processing and pattern recognition applications such as texture analysis and classification. Applications of fractal geometry in remote sensing rely heavily on estimation of the fractal dimension. The fractal dimension (D)

W. Sun; G. Xu; P. Gong; S. Liang

2006-01-01

215

Fractal analysis of circulating platelets in type 2 diabetic patients.

This paper investigates the use of computerized fractal analysis for objective characterization by means of transmission electron microscopy of the complexity of circulating platelets collected from healthy individuals and from type 2 diabetic patients, a pathologic condition in which platelet hyperreactivity has been described. Platelet boundaries were extracted by means of automatically image analysis. Local fractal dimension by box counting (measure of geometric complexity) was automatically calculated. The results showed that the platelet boundary observed by electron microscopy is fractal and that the shape of the circulating platelets is significantly more complex in the diabetic patients in comparison to healthy subjects (p < 0.01), with 100% correct classification. In vitro activated platelets from healthy subjects show an analogous increase of geometric complexity. Computerized fractal analysis of platelet shape by transmission electron microscopy can provide accurate, quantitative, data to study platelet activation in diabetes mellitus. PMID:25335814

Bianciardi, G; Tanganelli, I

2014-10-20

216

Objective Mammography is a widely used screening tool for the early detection of breast cancer. One of the commonly missed signs of\\u000a breast cancer is architectural distortion. The purpose of this study is to explore the application of fractal analysis and\\u000a texture measures for the detection of architectural distortion in screening mammograms taken prior to the detection of breast\\u000a cancer.\\u000a \\u000a \\u000a \\u000a Materials

Rangaraj M. Rangayyan; Shormistha Prajna; Fábio J. Ayres; J. E. Leo Desautels

2008-01-01

217

Metamaterial model of fractal time

While numerous examples of fractal spaces may be found in various fields of science, the flow of time is typically assumed to be one-dimensional and smooth. Here we present a metamaterial-based physical system, which can be described by effective three-dimensional (2+1) Minkowski spacetime. The peculiar feature of this system is that its time-like variable has fractal character. The fractal dimension of the time-like variable appears to be D=2.

Igor I. Smolyaninov

2012-03-02

218

Higuchi Dimension of Digital Images

There exist several methods for calculating the fractal dimension of objects represented as 2D digital images. For example, Box counting, Minkowski dilation or Fourier analysis can be employed. However, there appear to be some limitations. It is not possible to calculate only the fractal dimension of an irregular region of interest in an image or to perform the calculations in a particular direction along a line on an arbitrary angle through the image. The calculations must be made for the whole image. In this paper, a new method to overcome these limitations is proposed. 2D images are appropriately prepared in order to apply 1D signal analyses, originally developed to investigate nonlinear time series. The Higuchi dimension of these 1D signals is calculated using Higuchi's algorithm, and it is shown that both regions of interests and directional dependencies can be evaluated independently of the whole picture. A thorough validation of the proposed technique and a comparison of the new method to the Fourier dimension, a common two dimensional method for digital images, are given. The main result is that Higuchi's algorithm allows a direction dependent as well as direction independent analysis. Actual values for the fractal dimensions are reliable and an effective treatment of regions of interests is possible. Moreover, the proposed method is not restricted to Higuchi's algorithm, as any 1D method of analysis, can be applied. PMID:21931854

Ahammer, Helmut

2011-01-01

219

Anomalous thermal conduction in one dimension: a quantum calculation.

In this paper, we study the thermal conductivity of an anharmonically coupled chain of atoms. Numerical studies using classical dynamics have shown that the conductivity of a chain with nearest neighbor couplings diverges with chain length L as L(alpha); earlier studies found alpha approximately = 0.4 under a range of conditions, but a recent study on longer chains claims alpha = 1/3. Analytically, this problem has been studied by calculating the relaxation rate gamma(q) of the normal modes of vibration as a function of its wave vector q. Two theoretical studies of classical chains, one using the mode-coupling formulation and the other the Boltzmann equation method, led to gamma(q) proportional to q(5/3), which is consistent with alpha = 0.4. Here we study the problem for a quantum anharmonic chain with quartic anisotropy. We develop a low-temperature expansion for gamma(q) and find that, in the regime Dirac's constant omega(q) < k(B)T, gamma(q) is proportional to q(5/3)T2, where omega(q) is the frequency of the mode. In our analysis, the relaxation arises due to umklapp scattering processes. We further evaluate the thermal conductivity of the chain using the Kubo formula, which enables us to take into account the transport relaxation time through vertex corrections for the current-current correlator. This calculation also yields alpha = 0.4. PMID:17930004

Santhosh, G; Kumar, Deepak

2007-08-01

220

Fractal sets of dual topological quantum numbers

The universality classes of the quantum Hall transitions are considered in terms of fractal sets of dual topological quantum numbers filling factors, labelled by a fractal or Hausdorff dimension defined into the interval $1 < h < 2$ and associated with fractal curves. We show that our approach to the fractional quantum Hall effect-FQHE is free of any empirical formula

Wellington da Cruz

2003-01-01

221

Collision Frequencies between Fractal Aggregates and Small

Engineering, University of Arizona, Tucson, Arizona 85721 Three groups of aggregates with fractal dimensionsCollision Frequencies between Fractal Aggregates and Small Particles in a Turbulently Sheared Fluid.85 Âµm) in a Jar- test (paddle-mixing) device. The collision rates between these fractal aggregates (200

222

FRACTAL STRUCTURES IN DYADIC DIOPHANTINE APPROXIMATION

FRACTAL STRUCTURES IN DYADIC DIOPHANTINE APPROXIMATION JOHAN NILSSON Faculty of Engineering Centre prove that the set {x S : 2nx c, n > 0} is a fractal set whose Hausdorff dimension depends continuously on c and is constant on intervals which form a set of Lebesgue measure 1. Hence it has a fractal

Nilsson, Johan

223

Target Detection Using Fractal Geometry

NASA Technical Reports Server (NTRS)

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.

Fuller, J. Joseph

1991-01-01

224

Fractal geometry can be used to analyze shape and patterns in brain images. With this study we use fractals to analyze T1 data of patients affected by schizophrenia or bipolar disorder, with the aim of distinguishing between healthy and pathological brains using the complexity of brain structure, in particular of grey matter, as a marker of disease. 39 healthy volunteers, 25 subjects affected by schizophrenia and 11 patients affected by bipolar disorder underwent an MRI session. We evaluated fractal dimension of the brain cortex and its substructures, calculated with an algorithm based on the box-count algorithm. We modified this algorithm, with the aim of avoiding the segmentation processing step and using all the information stored in the image grey levels. Moreover, to increase sensitivity to local structural changes, we computed a value of fractal dimension for each slice of the brain or of the particular structure. To have reference values in comparing healthy subjects with patients, we built a template by averaging fractal dimension values of the healthy volunteers data. Standard deviation was evaluated and used to create a confidence interval. We also performed a slice by slice t-test to assess the difference at slice level between the three groups. Consistent average fractal dimension values were found across all the structures in healthy controls, while in the pathological groups we found consistent differences, indicating a change in brain and structures complexity induced by these disorders. PMID:25633275

Squarcina, Letizia; De Luca, Alberto; Bellani, Marcella; Brambilla, Paolo; Turkheimer, Federico E; Bertoldo, Alessandra

2015-02-21

225

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 ?(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

Guthold, M.; Liu, W.; Stephens, B.; Lord, S. T.; Hantgan, R. R.; Erie, D. A.; Taylor, R. M.; Superfine, R.

2004-01-01

226

NSDL National Science Digital Library

This page, created by the Goudreau Museum of the Mathematics in Art and Science, provides a tool to draw and customize fractals. The fractals available are the fern, the gingerbread man, the Sirpinski Set, and the Mandelbrot Set. This is a great resource for instructors looking for an interactive activity that introduces fractals to their students.

2009-05-07

227

Fractal behavior in magnetoconductance in coupled quantum dot systems

Fractal behavior in magnetoconductance fluctuations in coupled quantum dots has been studied by means of exact and statistical self-similarity. The fractal dimensions from the different features are not coincident exactly but show the similar gate voltage dependences, where the values increase with increasing negative gate voltage. Moreover, results of statistical fractal dimensions obtained from two types of dot-array samples show

Nobuyuki Aoki; Li-Hung Lin; Takahiro Morimoto; Takahiko Sasaki; Jun-Feng Song; Koji Ishibashi; Jonathan P. Bird; Agung Budiyono; Katsuhiro Nakamura; Takahisa Harayama; Yuichi Ochiai

2004-01-01

228

Fractal Musicand Fractal Music Lab

NSDL National Science Digital Library

This first website offers a collection of fractal music using images created by G.W.F. Albrecht. The technology and mathematics which this presentation draws on is described on the second website. The second website, developed by David Strohbeen, offers some basic information about fractals and fractal music. He has also posted some samples of his music and invites visitors to download software for creating fractal music and to submit their own compositions.

2007-12-12

229

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

Frankel, A.

1991-01-01

230

Electromagnetism on anisotropic fractal media

NASA Astrophysics Data System (ADS)

Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows a generalization of the Green-Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Ampère laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell's electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.

Ostoja-Starzewski, Martin

2013-04-01

231

Fractal Weyl law for Linux Kernel architecture

NASA Astrophysics Data System (ADS)

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 ? ? 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.

Ermann, L.; Chepelianskii, A. D.; Shepelyansky, D. L.

2011-01-01

232

Calculating excess chemical potentials using dynamic simulations in the fourth dimension

A general method for computing excess chemical potentials is presented. The excess chemical potential of a solute or ligand molecule is estimated from the potential of mean-force (PMF) calculated along a nonphysical fourth spatial dimension, {ital w}, into which the molecule is gradually inserted or from which it is gradually abstracted. According to this {open_quotes}4D-PMF{close_quotes} (four dimensional) scheme, the free energy difference between two limiting states defines the excess chemical potential: At w={plus_minus}{infinity}, the molecule is not interacting with the rest of the system, whereas at w=0, it is fully interacting. Use of a fourth dimension avoids the numerical instability in the equations of motion encountered upon growing or shrinking solute atoms in conventional free energy perturbation simulations performed in three dimensions, while benefiting from the efficient sampling of configurational space afforded by PMF calculations. The applicability and usefulness of the method are illustrated with calculations of the hydration free energy of simple Lennard-Jones (LJ) solutes, a water molecule, and camphor, using molecular dynamics simulations and umbrella sampling. Physical insight into the nature of the PMF profiles is gained from a continuum treatment of short- and long-range interactions. The short-range barrier for dissolution of a LJ solute in the added dimension provides an apparent surface tension of the solute. An approximation to the long-range behavior of the PMF profiles is made in terms of a continuum treatment of LJ dispersion and electrostatic interactions. Such an analysis saves the need for configurational sampling in the long-range limit of the fourth dimension. The 4D-PMF method of calculating excess chemical potentials should be useful for neutral solute and ligand molecules with a wide range of sizes, shapes, and polarities. {copyright} {ital 1999 American Institute of Physics.}

Pomes, R. [Theoretical Biology and Biophysics Group T-10, Mail Stop K710, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)] [Theoretical Biology and Biophysics Group T-10, Mail Stop K710, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States); Eisenmesser, E.; Post, C.B. [Department of Medical Chemistry, Purdue University, West Lafayette, Indiana 47901-1333 (United States)] [Department of Medical Chemistry, Purdue University, West Lafayette, Indiana 47901-1333 (United States); Roux, B. [Departements de Physique et de Chimie, Universite de Montreal, C.P. 6128, Succursalle Centre-Ville, Montreal (Quebec), H3C 3J7 (CANADA)] [Departements de Physique et de Chimie, Universite de Montreal, C.P. 6128, Succursalle Centre-Ville, Montreal (Quebec), H3C 3J7 (CANADA)

1999-08-01

233

NASA Astrophysics Data System (ADS)

We have studied index-coupled semiconductor distributed feedback (DFB) lasers and compared three types including continuously chirped gratings, fractally mixed gratings and fine-pitched gratings. Using transfer matrix model calculations self-consistently combined with the set of rate equations, important static and dynamic laser properties are compared and it is found that they can be designed to be nearly identical above laser threshold for the three types of lasers. Therefore, the grating configuration involving the lowest effort and production cost can be selected: continuously chirped DFB lasers implemented by bent waveguides. These gratings are very attractive since they show high performance and have the potential of production at lower cost.

Hillmer, H.; Prott, C.; Römer, F.; Hansmann, S.

2004-07-01

234

NSDL National Science Digital Library

This is a web site to support a first course in fractal geometry for students without a strong mathematical background. It covers a wide range of topics in fractals, modern dynamics, and chaos. Each of the topics contains examples of fractals in the arts, humanities, or social sciences. The site also contains lesson plans and software that can be used for a broad range of classes.

Frame, Michael; Mandelbrot, Benoit

2004-11-30

235

NSDL National Science Digital Library

Students investigate shapes that grow and change using an iterative process. Fractals are characterized by self-similarity, smaller sections that resemble the larger figure. From NCTM's Illuminations.

Mathematics, Illuminations N.

2009-11-23

236

Edge detection and image segmentation of space scenes using fractal analyses

NASA Technical Reports Server (NTRS)

A method was developed for segmenting images of space scenes into manmade and natural components, using fractal dimensions and lacunarities. Calculations of these parameters are presented. Results are presented for a variety of aerospace images, showing that it is possible to perform edge detections of manmade objects against natural background such as those seen in an aerospace environment.

Cleghorn, Timothy F.; Fuller, J. J.

1992-01-01

237

Fractal nature of hydrocarbon deposits. 2. Spatial distribution

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.

Barton, C.C.; Schutter, T.A; Herring, P.R.; Thomas, W.J. (Geological Survey, Denver, CO (United States)); Scholz, C.H. (Columbia Univ., Palisades, NY (United States))

1991-03-01

238

Fractal dynamics of earthquakes

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).

Bak, P.; Chen, K. [Brookhaven National Lab., Upton, NY (United States). Dept. of Physics

1995-05-01

239

Fractal dynamics of bioconvective patterns

NASA Technical Reports Server (NTRS)

Biologically generated cellular patterns, sometimes called bioconvective patterns, are found to cluster into aggregates which follow fractal growth dynamics akin to diffusion-limited aggregation (DLA) models. The pattern formed is self-similar with fractal dimension of 1.66 +/-0.038. Bioconvective DLA branching results from thermal roughening which shifts the balance between ordering viscous forces and disordering cell motility and random diffusion. The phase diagram for pattern morphology includes DLA, boundary spokes, random clusters, and reverse clusters.

Noever, David A.

1991-01-01

240

ERIC Educational Resources Information Center

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)

Dewdney, A. K.

1991-01-01

241

ERIC Educational Resources Information Center

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.…

Osler, Thomas J.

1999-01-01

242

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 ?, the fractional differencing parameter d of the autoregressive fractionally integrated moving average methodology, the power exponent ? 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

Stadnitski, Tatjana

2012-01-01

243

Fractal sets of dual topological quantum numbers

The universality classes of the quantum Hall transitions are considered in\\u000aterms of fractal sets of dual topological quantum numbers filling factors,\\u000alabelled by a fractal or Hausdorff dimension defined into the interval $1 < h <\\u000a2$ and associated with fractal curves. We show that our approach to the\\u000afractional quantum Hall effect-FQHE is free of any empirical formula

Wellington da Cruz

2003-01-01

244

Ocular fluid ferning test and fractals.

The ferning test can be regarded as a crystallization process obtained by simple water subtraction. Such a process is feasible for every kind of ocular fluid (i.e. tears and aqueous, vitreous and subretinal fluids). The ferning test can be described in terms of fractal geometry, as the image characteristics related to this test are consistent with three of the main properties of fractals: self-similarity, fractal dimension and lacunarity. PMID:8259264

Battaglia Parodi, M; Giusto, D D

1993-01-01

245

Fractal analysis of the galaxy distribution in the redshift range 0.45 ? z ? 5.0

NASA Astrophysics Data System (ADS)

This paper performs a fractal analysis of the galaxy distribution and presents evidence that it can be described as a fractal system within the redshift range of the FORS Deep Field (FDF) galaxy survey data. The fractal dimension D was derived by means of the galaxy number densities calculated by Iribarrem et al. (2012) using the FDF luminosity function parameters and absolute magnitudes obtained by Gabasch et al. (2004, 2006) in the spatially homogeneous standard cosmological model with ?m0 = 0.3, ??0 = 0.7 and H0 = 70 kms-1Mpc-1. Under the supposition that the galaxy distribution forms a fractal system, the ratio between the differential and integral number densities ? and ?? obtained from the red and blue FDF galaxies provides a direct method to estimate D and implies that ? and ?? vary as power-laws with the cosmological distances, feature which provides a second method for calculating D. The luminosity distance dL, galaxy area distance dG and redshift distance dz were plotted against their respective number densities to calculate D by linear fitting. It was found that the FDF galaxy distribution is better characterized by two single fractal dimensions at successive distance ranges, that is, two scaling ranges in the fractal dimension. Two straight lines were fitted to the data, whose slopes change at z ? 1.3 or z ? 1.9 depending on the chosen cosmological distance. The average fractal dimension calculated using ?? changes from < D > = 1 .4-0.6+0.7 to < D > = 0 .5-0.4+1.2 for all galaxies. Besides, D evolves with z, decreasing as the redshift increases. Small values of D at high z mean that in the past galaxies and galaxy clusters were distributed much more sparsely and the large-scale structure of the universe was then possibly dominated by voids.

Conde-Saavedra, G.; Iribarrem, A.; Ribeiro, Marcelo B.

2015-01-01

246

Talbot image of two-dimensional fractal grating

NASA Astrophysics Data System (ADS)

Talbot effect of two-dimensional fractal grating built by square aperture arrays is studied theoretically and experimentally in this paper. The amplitude fractal gratings are produced by use of the spatial light modulator, and the diffraction intensity distributions of fractal gratings with different fractal level in Fresnel diffraction field are measured with the help of the two-dimensional CCD. Talbot images of fractal gratings with 1-level and 2-level fractal are obtained in practical experiment. The analytic expression of Fresnel diffraction intensity of the fractal gratings is derived through decomposing fractal gratings into the sum of many periodic gratings. Theoretic results predict the self-image of fractal grating reappears at some certain distance. The numerical calculations also show the Talbot image and the fractional Talbot image of fractal grating. These results may extend the application of fractal grating in the optical processing of information and laser measurement.

Teng, Shuyun; Wang, Junhong; Li, Furui; Zhang, Wei

2014-03-01

247

Calculation of the dimensions of drug-polymer devices based on diffusion parameters.

The release kinetics of a polymeric-controlled release device are determined by its geometry and dimensions. A method to calculate the required size and shape of diffusion-controlled dosage forms to achieve a particular release profile is presented. The diffusion parameters are determined for various drugs (theophylline, diltiazem hydrochloride and caffeine) with thin ethyl cellulose (EC) films, containing different plasticizers [dibutyl sebacate (DBS) and acetyl tributyl citrate (ATBC)]. Computer simulations are then used to predict the drug release kinetics from various dosage forms (e.g. microparticles and cylinders). The practical benefit of these simulations is to optimize the geometry and dimensions of a controlled release device without the need of experimental studies. To verify the theoretical predictions, the release kinetics of theophylline from EC/ATBC microparticles of different size have also been determined experimentally. Good agreement is found between theory and experiment, proving the validity of the presented method. PMID:9649350

Siepmann, J; Ainaoui, A; Vergnaud, J M; Bodmeier, R

1998-07-01

248

Calculations were performed with the CTH and HULL finite difference wavecodes to evaluate computational capabilities for predicting depth and diameter of target cavities produced in high velocity penetration events. The calculations simulated selected tests in a set of armor penetration experiments conducted by the US Army Ballistic Research Laboratory and reported earlier in the literature. The tests and simulations involved penetration of semi-infinite targets by long rod projectiles over a range of impact velocities from 1.3 to 4.5 km/sec. Comparisons are made between the calculated and measured dimensions of the target cavities, and the sensitivity of the predicted results to target property variations is investigated. 9 refs., 18 figs., 3 tabs.

Kmetyk, L.N.; Yarrington, P.

1989-05-01

249

Large-dimension configuration-interaction calculations of positron binding to the group-II atoms

The configuration-interaction (CI) method is applied to the calculation of the structures of a number of positron binding systems, including e{sup +}Be, e{sup +}Mg, e{sup +}Ca, and e{sup +}Sr. These calculations were carried out in orbital spaces containing about 200 electron and 200 positron orbitals up to l=12. Despite the very large dimensions, the binding energy and annihilation rate converge slowly with l, and the final values do contain an appreciable correction obtained by extrapolating the calculation to the l{yields}{infinity} limit. The binding energies were 0.00317 hartree for e{sup +}Be, 0.0170 hartree for e{sup +}Mg, 0.0189 hartree for e{sup +}Ca, and 0.0131 hartree for e{sup +}Sr.

Bromley, M. W. J.; Mitroy, J. [Department of Physics, San Diego State University, San Diego, California 92182 (United States); Faculty of Technology, Charles Darwin University, Darwin NT 0909 (Australia)

2006-03-15

250

Analyzing and modeling fractal intensity point processes.

Fractal intensity point processes--doubly stochastic point processes with a fractal waveform intensity process--are required to describe the discharge patterns recorded from the auditory and visual systems. The Fano factor--the ratio of the variance of the number of events in an interval to the mean of this number--captures the self-similar characteristics of the intensity via two quantities: fractal dimension and fractal time. The fractal dimension is the exponent of the asymptotic power law behavior of the Fano factor with interval duration. The fractal time delineates long-term fractal behavior from short-term characteristics of the data. The average rate and self-similarity parameter of the intensity process, absolute and relative refractory effects, and serial dependence all modify the fractal time. To generate fractal intensity point processes, stochastic fractal processes are derived by applying memoryless, nonlinear transformations to fractional Gaussian noise. The intensity's amplitude distribution in combination with the Fano factor form criteria to choose the transformation that best describes data. PMID:8326063

Kumar, A R; Johnson, D H

1993-06-01

251

Using fractal analysis to assess how species perceive landscape structure

To develop a species-centered definition of ‘landscapes,’ I suggest using a fractal analysis of movement patterns to identify the scales at which organisms are interacting with the patch structure of the landscape. Significant differences in the fractal dimensions of movement patterns of two species indicate that the species may be interacting with the patch structure at different scales. Fractal analysis

1994-01-01

252

Generalized fractal analysis and its applications to engineering surfaces

Fractal analysis previously developed for surface characterization is generalized and an intrinsic length unit a in this analysis has been taken as lateral resolution of the measuring instrument ?. This generalized analysis allows the surface characterization in terms of two fractal parameters-fractal dimension D and amplitude coefficient C, which, in theory, are instrument independent and unique for each surface. A

Suryaprakash Ganti; Bharat Bhushan

1995-01-01

253

Self-similar Fractals: Projections, Sections and Percolation

Self-similar Fractals: Projections, Sections and Percolation Kenneth Falconer University of St Andrews, Scotland, UK Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation #12;Summary Â· Self-similar sets Â· Hausdorff dimension Â· Projections Â· Fractal percolation Â· Sections or slices

Falconer, Kenneth

254

Fractal analysis is a method of characterizing complex shapes such as the trabecular structure of bone. Numerous algorithms for estimating fractal dimension have been described, but the Fourier power spectrum method is particularly applicable to self-affine fractals, and facilitates corrections for the effects of noise and blurring in an image. We found that it provided accurate estimates of fractal dimension

Geoffrey Dougherty; Geoffrey M. Henebry

2001-01-01

255

NSDL National Science Digital Library

This is one of the best online resources about fractals, and is "meant to support a first course in fractal geometry for students without especially strong mathematical preparation." The site is incredibly deep, providing everything from the most basic definitions and non-technical discussions to involved mathematical formulations. Interactive Java applets, downloadable software for the PC and Macintosh, and laboratory activities are also presented. A particularly interesting section of the site explores about 100 places in nature and society where fractals are found.

Frame, Michael; Mandelbrot, Benoit; Neger, Nial

256

FRACTAL ANTENNA FOR PASSIVE UHF RFID APPLICATIONS

This paper addresses the design of fractal antennas placed onto dielectric object in the UHF RFID band and introduces a tag antenna conflguration of simple geometry having impedance tuning capability. Through the paper, the dimensions of the fractal antenna are optimized to improve the impedance matching with the chip impedance. The tag performance changes are studied when it is placed

Saber Helmy Zainud-Deen; Hend Abd El-Azem Malhat; Kamal Hassan Awadalla

2009-01-01

257

A route to fractal DNA-assembly

Crystallization is periodic self-assembly on the molecular scale. Individual DNA components have been used several times to achieve self-assembled crystalline arrangements in two dimensions. The design of a fractal system is a much more difficult goal to achieve with molecular components. We present DNA components whose cohesive portions are compatible with a fractal assembly. These components are DNA parallelograms that

ALESSANDRA CARBONE; NADRIAN C. SEEMAN

2002-01-01

258

The use of renormalization group techniques on fragmentation problems is examined. The equations which represent fractals and the size-frequency distributions of fragments are presented. Method for calculating the size distributions of asteriods and meteorites are described; the frequency-mass distribution for these interplanetary objects are due to fragmentation. The application of two renormalization group models to fragmentation is analyzed. It is

D. L. Turcotte

1986-01-01

259

Breaks in Fractal Scaling of Real and Synthetic Earthquake Catalogues

NASA Astrophysics Data System (ADS)

Earthquake generation within the crust is the result of a series of complicated spacio-temporal interactions between different tectonic blocks and units. The end product of the process is a function of both long term deterministic-chaotic processes in a regional scale and short-term Self-Organized Critical (SOC) processes of a local nature [e.g. McCloskey and Bean,1994; ]. In the past three to four decades many models of seismicity have been developed [e.g. Burridge and Knopoff, 1967; Huang and Turcotte, 1990; McCloskey, 1993; Ben-Zion, 1996] trying to model the observed patterns of earthquake generation and seismicity. Some of these studies have shown that it is possible to reproduce the main features of the real earthquake populations. In this study the fractal dimension of SCSN, JMA and ISC seismicity catalogues have been studied. the aim was to see whether all the different sizes of earthquakes within a catalogue (i.e. a single spacio-temporal window) belong to the same population and whether any breaks in fractal scaling exists within the catalogue concerned. Furthermore, a selection of synthetic earthquake models were analyzed with the same approach to determine whether they are able to reproduce the same results as empirical ones. Subsequent analysis of the data have revealed several distinct breaks in fractal scaling of earthquakes of different magnitudes. In other words, it emerged that small and large earthquakes in each catalogue are obeying different fractal dimensions hence belonging to different earthquake populations. It is possible to associate one of the breaks, observed in the SCSN catalogue to the average thickness of the seismogenic crust of California ( ˜ 15 km as calculated by Nazareth and Hauksson, 2004). With the same technique used for the empirical catalogues, three different synthetic catalogues [McCloskey, 1993; Ben-Zion, 1996; Khademi and McCloskey, this study ] were analyzed. Results have shown that all the models are able to predict the fractal dimension of the empirical catalogues to some degree and further, two latter models are able to simulate the breaks in the fractal scaling. The fractal dimensions from models F, U, M and A of Ben-Zion [1996] are generally in good agreement with observed dimensions of empirical catalogues, though the observed breaks in the empirical catalogues cannot be seen in these synthetic models. However, a gradual decrease in the fractal dimension with increasing treshold magnitude can be observed. The McCloskey [1993] Chaotic-SOC hierarchical model, with reasonable accuracy, predicts both the fractal dimensions and dimension breaks, observed in the empirical catalogues. The model's success in the prediction of behaviour of empirical data is particularly due to the combination of low-dimensional chaotic behaviour of bigger blocks (i.e. larger events) and high-dimensional SOC behaviour of smaller blocks (i.e. small events resulting from activity of smaller portions of the main fault or adjacent minor discontinuities). And finally the Khademi-McCloskey (KMC) model is able to reproduce both the dimension and one of the breaks of scaling. But, the model is unable to produce more than one break in scaling (i.e. to distinguish more than two earthquake populations within the same dataset). It is concluded that hierarchical earthquake models (though with some modifications) can be used to extend the temporally limited empirical catalogues to much longer time spans and to overcome the temporal limitations of the existing empirical catalogues.

Khademi, M. H.; McCloskey, J.

2004-12-01

260

It is presented how the new synergetic approach on the basis of fractional measuring, fractal dimension, scaling, fractals and deterministic chaos has been developed in IREE RAS as applied to problems of modern nonlinear radio physics and radio engineering since the eighties of XX century. Researches have been traditionally performed in the framework of modern fundamental interdisciplinary project “Fractal radiophysics

A. A. Potapov

2010-01-01

261

Self-avoiding walk on fractal complex networks: Exactly solvable cases.

We study the self-avoiding walk on complex fractal networks called the (u,v)-flower by mapping it to the N-vector model in a generating function formalism. First, we analytically calculate the critical exponent ? and the connective constant by a renormalization-group analysis in arbitrary fractal dimensions. We find that the exponent ? is equal to the displacement exponent, which describes the speed of diffusion in terms of the shortest distance. Second, by obtaining an exact solution for the (u,u)-flower, we provide an example which supports the conjecture that the universality class of the self-avoiding walk on graphs is not determined only by the fractal dimension. PMID:25493847

Hotta, Yoshihito

2014-11-01

262

Fractal geometry of aggregate snowflakes revealed by triple-wavelength radar measurements

NASA Astrophysics Data System (ADS)

Radar reflectivity measurements from three different wavelengths are used to retrieve information about the shape of aggregate snowflakes in deep stratiform ice clouds. Dual-wavelength ratios are calculated for different shape models and compared to observations at 3, 35, and 94 GHz. It is demonstrated that many scattering models, including spherical and spheroidal models, do not adequately describe the aggregate snowflakes that are observed. The observations are consistent with fractal aggregate geometries generated by a physically based aggregation model. It is demonstrated that the fractal dimension of large aggregates can be inferred directly from the radar data. Fractal dimensions close to 2 are retrieved, consistent with previous theoretical models and in situ observations.

Stein, T. H. M.; Westbrook, C. D.; Nicol, J. C.

2015-01-01

263

ERIC Educational Resources Information Center

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)

Jurgens, Hartmut; And Others

1990-01-01

264

A linear capacitor structure using fractal geometries is described. This capacitor exploits both lateral and vertical electric fields to increase the capacitance per unit area. Compared to standard parallel-plate capacitors, the parasitic bottom-plate capacitance is reduced. Unlike conventional metal-to-metal capacitors, the capacitance density increases with technology scaling. A classic fractal structure is implemented with 0.6-?m metal spacing, and a factor

H. Samavati; A. Hajimiri; A. R. Shahani; G. N. Nasserbakht; T. H. Lee

1998-01-01

265

Fractal signatures in the aperiodic Fibonacci grating.

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. PMID:24784044

Verma, Rupesh; Banerjee, Varsha; Senthilkumaran, Paramasivam

2014-05-01

266

Quantum statistical mechanics on infinitely ramified fractals

NASA Astrophysics Data System (ADS)

I present the thermodynamics of identical particles confined in infinitely ramified, exactly self-similar fractals, such as the Sierpinski carpet (in 2D) and the Menger sponge (in 3D). Recent results from analysis on fractals have established that the heat kernel associated with the Laplacian on such fractals satisfy, in the short-time regime, a scaling relation with exponent dS/2 (where dS is the spectral dimension) modulated by log-periodic oscillations. I explain how such a scaling affects the partition function, and the resultant thermodynamics associated with blackbody radiation [1], Casimir effect, and electrons in the fractal box.

Chen, Joe P.

2011-03-01

267

Effect of fractal silver electrodes on charge collection and light distribution in semiconducting

Effect of fractal silver electrodes on charge collection and light distribution in semiconducting of differing size scales. Here we test the effect of fractal silver electrodes on light distribution and charge deposited onto electrochemically grown fractal silver structures (5000 nm Ã? 500 nm; fractal dimension of 1

Osterloh, Frank

268

NSDL National Science Digital Library

This webpage is dedicated to all things fractals, and is organized and updated by Professor Paul Bourke. Visitors will find all kinds of useful tools for discovering and learning about fractals including: Fractal forms found by using Google Earth, Natural Fractals in Grand Canyon National Park, Introduction to fractals, a gallery of fractals, and much, much more.

Bourke, Paul

2007-08-15

269

Optics on a fractal surface and the photometry of the regoliths

NASA Astrophysics Data System (ADS)

The light scattered by a rough surface is calculated in a model where the surface is simulated by a mathematical fractal of dimension (D(H) between 2 and 3) and fractal density in the projected area towards the observer rho(H) (rho(H) between 0 and 1). The reflectance on such a surface is calculated in the special case of a 'hemispherical' fractal, in both the geometric optics approximation and a more general diffraction regime. By using a two-parameter phase function (single scattering albedo omega-sub-0 and asymmetry parameter g-sub-0), and including multiple scattering, this four-parameter model is found to reproduce within a good accuracy the phase function of several classes of atmosphereless bodies in the solar system, in good agreement with previous photometric models. The main effect of the diffraction is to reduce the width of the opposition surge by roughly a factor of 2. Another prediction of the model is that the single-scattering contribution due to the fractal part of the surface can be reduced, for nonzero phase angle, to an arbitrarily small amount, for high enough fractal dimension and density. This effect could give a new interpretation of the strong opposition effect observed on some objects, and also of the very low brightness of many solar system bodies.

Drossart, P.

1993-05-01

270

A new fractal feature, the Directional Fractal Curve (DFC), defined over an arc of 180° and composed of 90 fractal dimensions\\u000a determined at intervals of arc of 2°, is developed to account for the anisotropic property of a fractal texture. The DFC algorithm\\u000a is first applied to two images with different textural patterns one without directional preference and one with

C. F. Jiang; A. P. Avolio

1997-01-01

271

Applications of fractal analysis to physiology

This review describes approaches to the analysis of fractal properties of physiological observations. Fractals are useful to describe the natural irregularity of physiological systems because their irregularity is not truly random and can be demonstrated to have spatial or temporal correlation. The concepts of fractal analysis are introduced from intuitive, visual, and mathematical perspectives. The regional heterogeneities of pulmonary and myocardial flows are discussed as applications of spatial fractal analysis, and methods for estimating a fractal dimension from physiological data are presented. Although the methods used for fractal analyses of physiological data are still under development and will require additional validation, they appear to have great potential for the study of physiology at scales of resolution ranging from the microcirculation to the intact organism. PMID:1885430

Glenny, Robb W.; Robertson, H. Thomas; Yamashiro, Stanley; Bassingthwaighte, James B.

2010-01-01

272

Turbulence, Fractals, and the Solar Granulation

We give a brief, mostly qualitative introduction into the topics of convection and turbulence, and their description, mainly\\u000a referring to laboratory experiments. Following Mandelbrot (1967), the concept of the fractal dimension is introduced and some\\u000a earlier results of measurements of the fractal dimension in laboratory turbulence are discussed. Next, we address the question\\u000a whether hints of turbulence have been observed

P. Brandt; R. Greimel; E. Guenther; W. Mattig

1991-01-01

273

NASA Astrophysics Data System (ADS)

We investigate a process of random walks of a point particle on a two-dimensional square lattice of size n×n with periodic boundary conditions. A fraction p?20% of the lattice is occupied by holes (p represents macroporosity). A site not occupied by a hole is occupied by an obstacle. Upon a random step of the walker, a number of obstacles, M, can be pushed aside. The system approaches equilibrium in (nlnn)2 steps. We determine the distribution of M pushed in a single move at equilibrium. The distribution F(M) is given by M? where ?=-1.18 for p=0.1, decreasing to ?=-1.28 for p=0.01. Irrespective of the initial distribution of holes on the lattice, the final equilibrium distribution of holes forms a fractal with fractal dimension changing from a=1.56 for p=0.20 to a=1.42 for p=0.001 (for n=4,000). The trace of a random walker forms a distribution with expected fractal dimension 2.

Burdzy, Krzysztof; Ho?yst, Robert; Pruski, ?ukasz

2013-05-01

274

Radar Backscattering Computations for Fractal-Shaped Snowflakes

We propose a model for large snowflakes based on the fractal nature of their particle shapes. Monte Carlo simulations were conducted to make particles with a fractal dimension of 1.8 to 2.4. The roundness parameter for the projected images of the modeled particles was derived, and an average roundness of approximately 0.4 for particles with fractal dimension 2.1 was indicated;

Hiroshi ISHIMOTO

2008-01-01

275

Fun with Fractals Dr. Bori Mazzag

sequences are colored #12;Creating a fractal Â Recursion with pictures Generating the Koch snowflake #12;Creating a fractal Â Recursion with pictures Generating the Koch snowflake Generating Sierpinski's gasket #12;Calculating the perimenter of the the Koch snowflake #12;Calculating the perimenter

Mazzag, Borbala "Bori"

276

Fractals related to Pascal's triangle

A precise definition of a fractalFpr1 derived from Pascal's triangle modulopr (p prime) is given. The number of nonzero terms in the firstps lines of Pascal's triangle modulopr is computed. From this result the Hausdorff dimension and Hausdorff measure ofFpr1 are deduced. The nonself-similarty ofFpr1,r=2, is also discussed.

I. Jiménez Calvo; J. Muñoz Masqué

1996-01-01

277

Fractal Characterization of Hyperspectral Imagery

NASA Technical Reports Server (NTRS)

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.

Qiu, Hon-Iie; Lam, Nina Siu-Ngan; Quattrochi, Dale A.; Gamon, John A.

1999-01-01

278

A fractal analysis of CT liver images for the discrimination of hepatic lesions: a comparative study

A quantitative study for the discrimination of different hepatic lesions is presented in this paper. The study is based on the fractal analysis of CT liver images in order to estimate their fractal dimension and to differentiate normal liver parenchyma from hepatocellular carcinoma. Four fractal dimension estimators have been implemented throughout this work; three well-established methods and a novel implementation

C.-P. A. Sariyanni; P. Asvestas; G. K. Matsopoulos; K. S. Nikita; A. S. Nikita; D. Kelekis

2001-01-01

279

Estimating the cut-off in the fractal scaling of fractured concrete

An experimental study of the fractal scaling of concrete fracture is described. A concrete beam subjected to three-point bending in flexure was loaded to failure and the fracture surface viewed at various magnifications using a scanning electron microscope (SEM). The fractal dimension was then computed for 50 samples using the variable bandwidth technique. The results indicate that the fractal dimension

L. T. Dougan; P. S. Addison

2001-01-01

280

Fractal humic acids in aqueous suspensions at various concentrations, ionic strengths, and pH values

The fractal nature and fractal dimension of soil and peat humic acids were measured in dilute aqueous suspensions at various concentrations, ionic strengths, and pH values. The turbidimetric technique was used, in association with particle size analysis and scanning electron microscopy (SEM). The main objectives of the work were to relate the fractal dimension to the underlying morphological features and

N. Senesi; F. R. Rizzi; P. Dellino; P. Acquafredda

1997-01-01

281

Perturbative calculations in space-time having extra dimensions: The 6D single axial box anomaly

NASA Astrophysics Data System (ADS)

A detailed investigation about the 6D single axial box anomalous amplitude is presented. The superficial degree of divergence involved, in the one-loop perturbative calculations, is quadratic and the corresponding theory is nonrenormalizable. In spite of this, we show that the phenomenon of anomaly can be clearly characterized in a completely analogous way as that of 4D single axial triangle anomaly. The required calculations are made within the context of a novel calculational strategy where the amplitudes are not modified in intermediary steps. Divergent integrals are, in fact, not really solved. Adequate representations for the internal propagators are adopted according to the degree of divergence involved, so that when the last Feynman rule is taken (integration over the loop momentum) all the dependence on the internal (arbitrary) momenta are placed only in finite integrals. In the divergent structures emerging, no physical parameter is present and such objects are not really integrated. Only very general properties are assumed for such quantities which are universal (all space-time dimensions). The consistency of the perturbative calculations fixes some relations among the divergent integrals so that all the potentially ambiguous terms can be automatically removed. In spite of the absence of ambiguities, the emerging results allow us to give a clear and transparent description of the anomaly. The present investigation confirms the point of view stated by the same prescription for the well-known 2D axial-vector (AV) two-point and 4D single (AVV) and triple (AAA) axial-vector anomalies: the anomalous amplitudes need not be assumed as ambiguous quantities to allow an adequate description of the anomalies. We show also that a surprising, but natural, connection between the coupling of fermions with a pseudoscalar tensor field is found. In addition, we show that the crucial mathematical aspects of the problem are deeply related to a recently arisen controversy involving the evaluation of the Higgs Boson decay and the question of unicity in the dimensional regularization.

Fonseca, M. V. S.; Dallabona, G.; Battistel, O. A.

2014-11-01

282

Lacunarity of fractal superlattices: a remote estimation using wavelets

The lacunarity provides a useful parameter for describing the distribution of gap sizes in discrete self-similar (fractal) superlattices and is used in addition to the similarity dimension to describe fractals. We show here that lacunarity, as well as the similarity dimension, can be remotely estimated from the wavelet analysis of superlattices impulse response. As a matter of fact, the skeleton

Y. Laksari; H. Aubert; D. L. Jaggard; J. Y. Tourneret

2005-01-01

283

Statistical Error in a Chord Estimator of Correlation Dimension: the ``RULE of Five''

NASA Astrophysics Data System (ADS)

The statistical precision of a chord method for estimating fractal dimension from a correlation integral is derived. The optimal chord length is determined, and a comparison is made to other estimators. These calculations use the approximation that all pairwise distances between the points are statistically independent; the adequacy of this approximation is assessed numerically. The chord method provides a very quick and easy dimension estimate which is only slightly less precise than the optimal estimator. Keywords: correlation dimension, statistical error

Theiler, James; Lookman, Turab

1993-06-01

284

Multi-Scale Fractal Analysis of Image Texture and Pattern

NASA Technical Reports Server (NTRS)

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.

Emerson, Charles W.; Lam, Nina Siu-Ngan; Quattrochi, Dale A.

1999-01-01

285

NSDL National Science Digital Library

Fractal images made for the most part using a software application called Flarium24. Galleries contain about 15 images each and should be viewed in hi-color or truecolor settings. Tilable images that can be downloaded?for wallpaper are also available.

Forum, Math; Webb, Sharon

2000-01-01

286

The possibility of fractal benzenoids built upon a (deterministically) regular self-similar structure are proposed. Of various possibilities considered a simpler class of “trigonal” structures is identified. Possible directed synthetic schemes are described. Further, Kekulé structure counts are made as well as conjugated-circuits estimates of resonance energies. Results, for the general trigonal class as well as for particular cases are given.

D. J. Klein; M. J. Cravey; G. E. Hite

1991-01-01

287

93FRACTAL SOLIDS, PRODUCT MEASURES AND FRACTIONAL WAVE EQUATIONSLi - Ostoja-Starzewski ABSTRACT. This paper builds on the recently begun extension of continuum thermomechanics to fractal porous media that are specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d and a resolution length

Ostoja-Starzewski, Martin

288

Elasticity of fractal materials using the continuum model with non-integer dimensional space

NASA Astrophysics Data System (ADS)

Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of elasticity equations for non-integer dimensional space, and its solutions for the equilibrium case of fractal materials are suggested. Elasticity problems for fractal hollow ball and cylindrical fractal elastic pipe with inside and outside pressures, for rotating cylindrical fractal pipe, for gradient elasticity and thermoelasticity of fractal materials are solved.

Tarasov, Vasily E.

2015-01-01

289

NSDL National Science Digital Library

This site displays a new fractal image every day. It also includes images of many types of fractals, such as strange attractors and Julia sets, and it also has a section on the fractals found in nature.

Sprott, Julien

2011-08-02

290

NSDL National Science Digital Library

Spanky Fractal Database: fractal images, programs, documents, papers, code examples, and other fractal related material. Submitted by contributors or hunted down from various nooks and crannies on the net. Enjoy and discover.

291

Characterizing Hyperspectral Imagery (AVIRIS) Using Fractal Technique

NASA Technical Reports Server (NTRS)

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.

Qiu, Hong-Lie; Lam, Nina Siu-Ngan; Quattrochi, Dale

1997-01-01

292

NASA Astrophysics Data System (ADS)

This study is an attempt of a semi-automatic geomorphological GIS analysis based on morphometric indices. In the study, 10-m-resolution Digital Elevation Models (DEMs) are used to assess the neotectonic signals regarding the recent topographic developments and to attach additional significance to active tectonics in the Bingöl basin area. The methodology incorporates the determination of the structural similarities of the faults in the basins using fractal concepts and the application of four morphometric indices (Stream Power Index (SP), Multi-resolution Index of Valley Bottom Flatness (MRVBF), Terrain Ruggedness Index (TRI) and Valley Width-to-Height Ratio (Vf)). In order to detect the deviation from spatial randomness of the applied indices, the weighted correlation coefficient Moran’s I is used and the results are interpreted at a confidence interval of 99%. The spatial distribution of integrated index values is evaluated with the tectonically active fault zones in order to determine the probable activity and the structural deformation in the basin. The applied methodology reveals that the fractal analysis of the fault lines and the spatial analysis of the morphometric indices proved to be effective tools in analyzing the tectonic influence of the fault system on the basin area. Within the basin area, a relatively lesser degree of tectonic activity is observed, in contrast with the high tectonic activity outside the basin.

Sarp, Gulcan

2014-07-01

293

NSDL National Science Digital Library

This applet lets students explore connections among geometry, measurement and number patterns. Students select one of four preset fractal processes, including the Koch snowflake and the Sierpinski triangle, and observe the stages of complexity. The number of parts and an area or linear measure are shown in a table. The tool supports the lesson How Many Triangles Can You Construct? in the unit Building with Triangles (cataloged separately).

2011-01-01

294

Multi-Scale Fractal Analysis of Image Texture and Pattern

NASA Technical Reports Server (NTRS)

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.

Emerson, Charles W.; Lam, Nina Siu-Ngan; Quattrochi, Dale A.

1999-01-01

295

Computerized analysis of mammographic parenchymal patterns using fractal analysis

NASA Astrophysics Data System (ADS)

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.

Li, Hui; Giger, Maryellen L.; Huo, Zhimin; Olopade, Olufunmilayo I.; Chinander, Michael R.; Lan, Li; Bonta, Ioana R.

2003-05-01

296

Microtopographic Inspection and Fractal Analysis of Skin Neoplasia

NASA Astrophysics Data System (ADS)

Early detection of skin cancer is fundamental to a successful treatment. Changes in the shape, including the relief, of skin lesions are an indicator of a possible malignity. Optical microtopographic inspection of skin lesions can be used to identify diagnostic patterns of benign and malign skin' lesions. Statistical parameters like the mean roughness (Ra) may allow the discrimination between different types of lesions and degree of malignity. Fractal analysis of bi-dimensional and 3D images of skin lesions can validate or complement that assessment by calculation of its fractal dimensions (FD). On the study herein reported the microtopographic inspection of the skin lesions were performed using the optical triangulation based microtopographer developed at the Physics Department of the University of Minho, MICROTOP.03.MFC. The patients that participated in this research work were men and women older than 15 years with the clinical and histopathology diagnoses of: melanoma, basocellular carcinoma, epidermoide carcinoma, actinic keratosis, keratoacantosis and benign nevus. Latex impressions of the lesions were taken and microtopographically analyzed. Characteristic information for each type of studied lesion was obtained. For melanoma it was observed that on the average these tumors present an increased roughness of around 67 percent compared to the roughness of the healthy skin. This feature allows the distinction from other tumors as basocellular carcinoma (were the roughness increase was in the average of 49 percent) and benign lesions as the epidermoide cyst (37 percent) or the seborrhea keratosis (4 percent). Tumor size and roughness are directly proportional to the grade of malignality. The characterization of the fractal geometry of 2D (histological slides) and 3D images of skin lesions was performed by obtaining its FD evaluated by means of the Box counting method. Results obtained showed that the average fractal dimension of histological slide images (FDh) corresponding to some neoplasia is higher (1.334+/-0.072) than those for healthy skin (1.091+/-0.082). A significant difference between the fractal dimensions of neoplasia and healhty skin (>0.001) was registered. The FD of microtopography maps (FDm) can also distinguish between healthy and malignant tissue in general (2.277+/-0.070 to 2.309+/-0.040), but not discriminate the different types of skin neoplasias. The combination of the rugometric evaluation and fractal geometry characterization provides valuable information about the malignity of skin lesions and type of lesion.

Costa, Manuel F. M.; Hipolito, Alberto Valencia; Gutierrez, Gustavo Fidel; Chanona, Jorge; Gallegos, Eva Ramón

2008-04-01

297

In this work, the effect of the electromagnetic radiation generated by mobile phone, on the heart rate variability (HRV) has been investigated using correlation dimension calculation which is a nonlinear analysis method. The 17 volunteer subjects participated to our work and the experiment is designed as three periods and each period have 7 minutes. The electrocardiogram (ECG) signals were recorded

Derya Yilmaz; Metin Yildiz

2009-01-01

298

On the synthesis of fractal radiation patterns

NASA Astrophysics Data System (ADS)

The fundamental relationship between self-similar, that is, fractal, arrays and their ability to generate radiation patterns which possess fractal features is examined in this paper. The theoretical foundation and design procedures are developed for using fractal arrays to synthesize fractal radiation patterns having certain desired characteristics. A family of functions, known as generalized Weierstrass functions, are shown to play a pivotal role in the theory of fractal radiation pattern synthesis. These functions are everywhere continuous but nowhere differentiable and exhibit fractal behavior at all scales. It will be demonstrated that the array factor for a nonuniformly but symmetrically spaced linear array can be expressed in terms of a Weierstrass partial sum (band-limited Weierstrass function) for an appropriate choice of array element spacings and excitations. The resulting fractal radiation patterns from these arrays possess structure over a finite range of scales. This range of scales can be controlled by the number of elements in the array. For a fixed array geometry, the fractal dimension of the radiation pattern may be varied by changing the array current distribution. A general and highly flexible synthesis technique is introduced which is based on the theory of Fourier-Weierstrass expansions. One of the appealing attributes of this synthesis technique is that it provides the freedom to select an appropriate generating function, in addition to the dimension, for a desired fractal radiation pattern. It is shown that this synthesis procedure results in fractal arrays which are composed of a sequence of self-similar uniformly spaced linear subarrays. Finally, a synthesis technique for application to continuous line sources is presented which also makes use of Fourier-Weierstrass expansions.

Werner, D. H.; Werner, P. L.

1995-01-01

299

Fractal characterization of inhomogeneous geophysical measuring networks

The measuring stations of most in situ geophysical networks are spatially distributed in a highly inhomogeneous manner, being mainly concentrated on continents and population centers. When inhomogeneity occurs over a wide range of scales in a spatial dimension E, it can be characterized by a fractal dimension Dm. For measuring networks, Dm will usually be less than E. A world

S. Lovejoy; D. Schertzer; P. Ladoy

1986-01-01

300

Fractal aspects of elastin supramolecular organization.

The supramolecular organisation of elastin and its soluble derivative alpha-elastin were studied by scanning and transmission electron microscopy. It was found a variety of different structures including filaments, fibrils, fibres, networks and dendritic, leaf-like forms. Self-similar patterns, extending for at least three orders of magnitude, were revealed, strongly suggesting the presence of fractal objects. The fractal dimension D was determined by using the box counting method. PMID:7669265

Tamburro, A M; De Stradis, A; D'Alessio, L

1995-06-01

301

Fractal nature of humic materials

Fractals are geometric representatives of strongly disordered systems whose structure is described by nonintegral dimensions. A fundamental tenet of fractal geometry is that disorder persists at any characterization scale-length used to describe the system. The nonintegral nature of these fractal dimensions is the result of the realization that a disordered system must possess more structural detail than an ordered system with classical dimensions of 1, 2, or 3 in order to accommodate this disorder within disorder.'' Thus from a fractal perspective, disorder is seen as an inherent characteristic of the system rather than as a perturbative phenomena forced upon it. Humic materials are organic substances that are formed by the profound alteration of organic matter in a natural environment. They can be operationally divided into 3 fractions; humic acid (soluble in base), fulvic acid (soluble in acid or base), and humin (insoluble in acid or base). Each of these fraction has been shown to be an extremely heterogeneous mixture. These mixtures have proven so intractable that they may represent the ultimate in molecular disorder. In fact, based on the characteristics that humic materials must possess in order to perform their functions in natural systems, it has been proposed that the fundamental chemical characteristic of a humic material is not a discrete chemical structure but a pronounced lack of order on a molecular level. If the fundamental chemical characteristic of a humic material is a strongly disordered nature, as has been proposed, then humic materials should be amenable to characterization by fractal geometry. The purpose of this paper is to test this hypothesis.

Rice, J.A. (South Dakota State Univ., Brookings, SD (United States). Dept. of Chemistry); Lin, J.S. (Oak Ridge National Lab., TN (United States))

1992-01-01

302

Fractal nature of humic materials

Fractals are geometric representatives of strongly disordered systems whose structure is described by nonintegral dimensions. A fundamental tenet of fractal geometry is that disorder persists at any characterization scale-length used to describe the system. The nonintegral nature of these fractal dimensions is the result of the realization that a disordered system must possess more structural detail than an ordered system with classical dimensions of 1, 2, or 3 in order to accommodate this ``disorder within disorder.`` Thus from a fractal perspective, disorder is seen as an inherent characteristic of the system rather than as a perturbative phenomena forced upon it. Humic materials are organic substances that are formed by the profound alteration of organic matter in a natural environment. They can be operationally divided into 3 fractions; humic acid (soluble in base), fulvic acid (soluble in acid or base), and humin (insoluble in acid or base). Each of these fraction has been shown to be an extremely heterogeneous mixture. These mixtures have proven so intractable that they may represent the ultimate in molecular disorder. In fact, based on the characteristics that humic materials must possess in order to perform their functions in natural systems, it has been proposed that the fundamental chemical characteristic of a humic material is not a discrete chemical structure but a pronounced lack of order on a molecular level. If the fundamental chemical characteristic of a humic material is a strongly disordered nature, as has been proposed, then humic materials should be amenable to characterization by fractal geometry. The purpose of this paper is to test this hypothesis.

Rice, J.A. [South Dakota State Univ., Brookings, SD (United States). Dept. of Chemistry; Lin, J.S. [Oak Ridge National Lab., TN (United States)

1992-03-01

303

Characterization of branch complexity by fractal analyses

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.

Alados, C.L.; Escos, J.; Emlen, J.M.; Freeman, D.C.

1999-01-01

304

We calculate the inclusive cross section of double Z-boson production within large extra dimensions at the Large Hadron Collider (LHC). Using perturbatively quantized gravity in the ADD (Arkani-Hamed, Dvali, Dimopoulos) model we perform a first order calculation of the graviton mediated contribution to the pp{yields}ZZ+x cross section. At low energies (e.g. Tevatron) this additional contribution is very small, making it virtually unobservable, for a fundamental mass scale above 2500 GeV. At LHC energies, however, the calculation indicates that the ZZ-production rate within the ADD model should differ significantly from the standard model if the new fundamental mass scale would be below 15000 GeV. A comparison with the observed production rate at the LHC might therefore provide direct hints on the number and structure of the extra dimensions.

Kober, Martin; Bleicher, Marcus [Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Max-von-Laue-Str. 1, 60438 Frankfurt (Germany); Koch, Benjamin [Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Max-von-Laue-Str. 1, 60438 Frankfurt (Germany); Frankfurt Institute for Advanced Studies (FIAS), Max-von-Laue-Str. 1, 60438 Frankfurt (Germany)

2007-12-15

305

Beauty of fractals design of fractal antennas

In this paper, beauty of fractals, some bases of its, processes, parameters and pictures of Mandelbrot and Julia sets are represented. Algorithms of computer-aided design of the dipole and planar fractal antennas of different orders are described. Current distribution, characteristics of impedance, distribution of near electromagnetic field and radiation of the fractal antennas are analyzed. Represented antennas structures may be

G. G. Chavka

2007-01-01

306

Fractal properties of surfaces have been explored by many investigators. Most have concluded that fractal characterisation is useful. This note questions the philosophy of using fractals to describe and control engineering surfaces. It concludes that the benefits are more virtual than real. The functional significance of fractal parameters is also examined and the overall question arises as to whether scale

D. J. Whitehouse

2001-01-01

307

Fuzzy fractals, chaos, and noise

To distinguish between chaotic and noisy processes, the authors analyze one- and two-dimensional chaotic mappings, supplemented by the additive noise terms. The predictive power of a fuzzy rule-based system allows one to distinguish ergodic and chaotic time series: in an ergodic series the likelihood of finding large numbers is small compared to the likelihood of finding them in a chaotic series. In the case of two dimensions, they consider the fractal fuzzy sets whose {alpha}-cuts are fractals, arising in the context of a quadratic mapping in the extended complex plane. In an example provided by the Julia set, the concept of Hausdorff dimension enables one to decide in favor of chaotic or noisy evolution.

Zardecki, A.

1997-05-01

308

Studying fractal geometry on submicron length scales by small-angle scattering

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.

Wong, P.; Lin, J.

1988-08-01

309

Fractal Property in the Light Curve of BL Lac Object S5 0716 + 714

NASA Astrophysics Data System (ADS)

In this paper, we compile the historical R-band data of S5 0716 + 714 from literature and obtain its fractal dimension by using a fractal method and then simulate the data with the Weierstrass-Mandelbrot (W-M) function. It is considered that the light curve has a fractal property.

Ou, J. W.; Zheng, Y. G.

2014-11-01

310

Collision Frequencies of Fractal Bacterial Aggregates with Small

Collision Frequencies of Fractal Bacterial Aggregates with Small Particles in a Sheared Fluid T E R, Girona, Spain, and Department of Civil and Environmental Engineering, The Pennsylvania State University aggregates of 3-300 Âµm diameter with an average fractal dimension of D ) 2.52. To determine the rate

311

CONTROL OF FLUID DYNAMICS WITH ENGINEERED FRACTALS - ADSORPTION PROCESS APPLICATIONS

Engineered fluid transporting fractals can be utilized for a broad range of fluid control applications. These applications include use as alternatives to turbulence, controlled formation of fluid geometry and rapid transition of effective fluid dimension. As applied to adsorption processes, fractals can be used to provide rapid and homogeneous distribution of fluids to form surfaces or rapid distribution and collection

MIKE KEARNEY

1999-01-01

312

Robust Methodology for Fractal Analysis of the Retinal Vasculature

We have developed a robust method to perform retinal vascular fractal analysis from digital retina images. The technique preprocesses the green channel retina images with Gabor wavelet transforms to enhance the retinal images. Fourier Fractal dimension is computed on these preprocessed images and does not require any segmentation of the vessels. This novel technique requires human input only at a

Mohd Zulfaezal Che Azemin; Dinesh Kant Kumar; Tien Y. Wong; Ryo Kawasaki; Paul Mitchell; Jie Jin Wang

2011-01-01

313

Fractal image analysis - Application to the topography of Oregon and synthetic images.

NASA Technical Reports Server (NTRS)

Digitized topography for the state of Oregon has been used to obtain maps of fractal dimension and roughness amplitude. The roughness amplitude correlates well with variations in relief and is a promising parameter for the quantitative classification of landforms. The spatial variations in fractal dimension are low and show no clear correlation with different tectonic settings. For Oregon the mean fractal dimension from a two-dimensional spectral analysis is D = 2.586, and for a one-dimensional spectral analysis the mean fractal dimension is D = 1.487, which is close to the Brown noise value D = 1.5. Synthetic two-dimensional images have also been generated for a range of D values. For D = 2.6, the synthetic image has a mean one-dimensional spectral fractal dimension D = 1.58, which is consistent with the results for Oregon. This approach can be easily applied to any digitzed image that obeys fractal statistics.

Huang, Jie; Turcotte, Donald L.

1990-01-01

314

Fractal simulation of the resistivity and capacitance of arsenic selenide

The temperature dependences of the ac resistivity R and ac capacitance C of arsenic selenide were measured more than four decades ago [V. I. Kruglov and L. P. Strakhov, in Problems of Solid State Electronics, Vol. 2 (Leningrad Univ., Leningrad, 1968)]. According to these measurements, the frequency dependences are R {proportional_to} {omega}{sup -0.80{+-}0.01} and {Delta}C {proportional_to} {omega}{sup -0.120{+-}0.006} ({omega} is the circular frequency and {Delta}C is measured from the temperature-independent value C{sub 0}). According to fractal-geometry methods, R {proportional_to} {omega}{sup 1-3/h} and {Delta}C {proportional_to} {omega}{sup -2+3/h}, where h is the walk dimension of the electric current in arsenic selenide. Comparison of the experimental and theoretical results indicates that the walk dimensions calculated from the frequency dependences of resistivity and capacitance are h{sub R} = 1.67 {+-} 0.02 and h{sub C} = 1.60 {+-} 0.08, which are in agreement with each other within the measurement errors. The fractal dimension of the distribution of conducting sections is D = 1/h = 0.6. Since D < 1, the conducting sections are spatially separated and form a Cantor set.

Balkhanov, V. K., E-mail: ballar@yandex.ru; Bashkuev, Yu. B. [Russian Academy of Sciences, Division of Physical Problems, Buryat Scientific Center, Siberian Branch (Russian Federation)

2010-03-15

315

Fractal characterization of neural correlates of consciousness

NASA Astrophysics Data System (ADS)

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.

Ibañez-Molina, A. J.; Iglesias-Parro, S.

2013-01-01

316

The structure of hematite aggregates in the presence of fairly monodisperse polyacrylic acid (PAA) with two different molecular weights (M{sub w} = 1.36 {times} 10{sup 6}, M{sub w}/M{sub n} = 1.53; M{sub w} = 3.69 {times} 10{sup 4}, M{sub w}/M{sub n} = 1.60) was studied using static light scattering (SLS). The fractal dimensions were calculated from the scattering exponents, after taking into account the finite size of aggregates, using exponential and Gaussian cutoff functions. Three flocculation regimes, namely, pre-DLA, DLA (diffusion-limited aggregation), and post-DLA, were defined based on the polymer concentration. In the DLA regime, fractal dimension values, D{sub f} = 1.84 {+-} 0.02 and 1.73 {+-} 0.02, were obtained using exponential and Gaussian cutoff functions, respectively. A fractal dimension of approximately 2.0 was found, as expected, in the pre-DLA regime (at PAA concentrations lower than the optimal dosage for a DLA regime) where the flocculation rate was reaction limited. In contrast, in the post-DLA regime, the flocculation was slow but the structure of aggregates was as tenuous as in the DLA regime with a fractal dimension D{sub f} {approx} 1.8. Moreover, for all three regimes, the D{sub f} values were independent of the molecular weights of PAA. The lower fractal dimension in post-DLA was probably due to the increased concentration of polymer chains between adjacent particles in aggregates. The steric hindrance favored tip-to-tip aggregation, leading to a more tenuous structure.

Ferretti, R.; Zhang, J.; Buffle, J. [Univ. of Geneva (Switzerland)] [Univ. of Geneva (Switzerland)

1998-12-15

317

Hexagonal and Pentagonal Fractal Multiband Antennas

NASA Technical Reports Server (NTRS)

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.

Tang, Philip W.; Wahid, Parveen

2005-01-01

318

Fractal parameters and vascular networks: facts & artifacts

Background Several fractal and non-fractal parameters have been considered for the quantitative assessment of the vascular architecture, using a variety of test specimens and of computational tools. The fractal parameters have the advantage of being scale invariant, i.e. to be independent of the magnification and resolution of the images to be investigated, making easier the comparison among different setups and experiments. Results The success of several commercial and/or free codes in computing the fractal parameters has been tested on well known exact models. Based on such a preliminary study, we selected the code Frac-lac in order to analyze images obtained by visualizing the angiogenetic process occurring in chick Chorio Allontoic Membranes (CAM), assumed to be paradigmatic of a realistic 2D vascular network. Among the parameters investigated, the fractal dimension Df proved to be the most robust estimator for CAM vascular networks. Moreover, only Df was able to discriminate between effective and elusive increases in vascularization after drug-induced angiogenic stimulations on CAMs. Conclusion The fractal dimension Df is likely to be the most promising tool for monitoring the effectiveness of anti-angiogenic therapies in various clinical contexts. PMID:18637183

Mancardi, Daniele; Varetto, Gianfranco; Bucci, Enrico; Maniero, Fabrizio; Guiot, Caterina

2008-01-01

319

Quantitative Assessment of Early Diabetic Retinopathy Using Fractal Analysis

OBJECTIVE—Fractal analysis can quantify the geometric complexity of the retinal vascular branching pattern and may therefore offer a new method to quantify early diabetic microvascular damage. In this study, we examined the relationship between retinal fractal dimension and retinopathy in young individuals with type 1 diabetes. RESEARCH DESIGN AND METHODS—We conducted a cross-sectional study of 729 patients with type 1 diabetes (aged 12–20 years) who had seven-field stereoscopic retinal photographs taken of both eyes. From these photographs, retinopathy was graded according to the modified Airlie House classification, and fractal dimension was quantified using a computer-based program following a standardized protocol. RESULTS—In this study, 137 patients (18.8%) had diabetic retinopathy signs; of these, 105 had mild retinopathy. Median (interquartile range) retinal fractal dimension was 1.46214 (1.45023–1.47217). After adjustment for age, sex, diabetes duration, A1C, blood pressure, and total cholesterol, increasing retinal vascular fractal dimension was significantly associated with increasing odds of retinopathy (odds ratio 3.92 [95% CI 2.02–7.61] for fourth versus first quartile of fractal dimension). In multivariate analysis, each 0.01 increase in retinal vascular fractal dimension was associated with a nearly 40% increased odds of retinopathy (1.37 [1.21–1.56]). This association remained after additional adjustment for retinal vascular caliber. CONCLUSIONS—Greater retinal fractal dimension, representing increased geometric complexity of the retinal vasculature, is independently associated with early diabetic retinopathy signs in type 1 diabetes. Fractal analysis of fundus photographs may allow quantitative measurement of early diabetic microvascular damage. PMID:18835945

Cheung, Ning; Donaghue, Kim C.; Liew, Gerald; Rogers, Sophie L.; Wang, Jie Jin; Lim, Shueh-Wen; Jenkins, Alicia J.; Hsu, Wynne; Li Lee, Mong; Wong, Tien Y.

2009-01-01

320

Fractal analysis of bone structure with applications to osteoporosis and microgravity effects

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.

Acharya, R.S.; Swarnarkar, V.; Krishnamurthy, R.; Hausman, E. [State Univ. of New York, Buffalo, NY (United States). Biomedical Imaging Group; LeBlanc, A.; Lin, C. [Baylor Coll. of Medicine, Houston, TX (United States); Shackelford, L. [National Aeronautics and Space Administration, Houston, TX (United States). Johnson Space Center

1995-12-31

321

Fractal Weyl law for three-dimensional chaotic hard-sphere scattering systems.

The fractal Weyl law connects the asymptotic level number with the fractal dimension of the chaotic repeller. We provide the first test for the fractal Weyl law for a three-dimensional open scattering system. For the four-sphere billiard, we investigate the chaotic repeller and discuss the semiclassical quantization of the system by the method of cycle expansion with symmetry decomposition. We test the fractal Weyl law for various symmetry subspaces and sphere-to-sphere separations. PMID:21230359

Eberspächer, Alexander; Main, Jörg; Wunner, Günter

2010-10-01

322

Deterministic fractals: Extracting additional information from small-angle scattering data

The small-angle scattering curves of deterministic mass fractals are studied and analyzed in momentum space. In the fractal region, the curve I(q)qD 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

A. Yu. Cherny; E. M. Anitas; V. A. Osipov; A. I. Kuklin

2011-01-01

323

Growth of the fractal patterns in Ni-Zr thin films during ion-solid interaction

An experimental study of the growth of fractal patterns during ion beam-solid interaction in the Ni-Zr alloy system is reported. The observed fractal patterns in the alloy films feature both isotropic and anisotropic characters depending on their growth surroundings. The fractal dimension of the isotropic fractal patterns is determined to be 1.4 +- 0.1. The possible growth mechanism of the observed patterns is discussed.

Huang, L.J.; Ding, J.R.; Li, H.; Liu, B.X.

1988-04-15

324

Fractal Analysis of Cervical Intraepithelial Neoplasia

Introduction Cervical intraepithelial neoplasias (CIN) represent precursor lesions of cervical cancer. These neoplastic lesions are traditionally subdivided into three categories CIN 1, CIN 2, and CIN 3, using microscopical criteria. The relation between grades of cervical intraepithelial neoplasia (CIN) and its fractal dimension was investigated to establish a basis for an objective diagnosis using the method proposed. Methods Classical evaluation of the tissue samples was performed by an experienced gynecologic pathologist. Tissue samples were scanned and saved as digital images using Aperio scanner and software. After image segmentation the box counting method as well as multifractal methods were applied to determine the relation between fractal dimension and grades of CIN. A total of 46 images were used to compare the pathologist's neoplasia grades with the predicted groups obtained by fractal methods. Results Significant or highly significant differences between all grades of CIN could be found. The confusion matrix, comparing between pathologist's grading and predicted group by fractal methods showed a match of 87.1%. Multifractal spectra were able to differentiate between normal epithelium and low grade as well as high grade neoplasia. Conclusion Fractal dimension can be considered to be an objective parameter to grade cervical intraepithelial neoplasia. PMID:25302712

Fabrizii, Markus; Moinfar, Farid; Jelinek, Herbert F.; Karperien, Audrey; Ahammer, Helmut

2014-01-01

325

Fractal symmetry of protein interior: what have we learned?

The application of fractal dimension-based constructs to probe the protein interior dates back to the development of the concept\\u000a of fractal dimension itself. Numerous approaches have been tried and tested over a course of (almost) 30 years with the aim\\u000a of elucidating the various facets of symmetry of self-similarity prevalent in the protein interior. In the last 5 years especially,\\u000a there

Anirban Banerji; Indira Ghosh

2011-01-01

326

Chaos, Fractals, and Polynomials.

ERIC Educational Resources Information Center

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)

Tylee, J. Louis; Tylee, Thomas B.

1996-01-01

327

ERIC Educational Resources Information Center

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)

Barton, Ray

1990-01-01

328

Estimating the dimension of a fracta.! point process Steven B. Lowen and Malvin C. Teich

. University, Department of Electrical Engineering New Yoik, NY 10027 ABSTRACT \\,\\Te discuss issues involved in estimating the dimension of a fractal point process. We first define the term fractal point process and natural examples of fractal point processes. 1. DEFINITION OF A FRACTAL POINT PROCESS Soie plienoniena

Teich, Malvin C.

329

NSDL National Science Digital Library

This first website offers a collection of fractal music using images created by G.W.F. Albrecht. The technology and mathematics which this presentation draws on is described on the second website. The second website, developed by David Strohbeen, offers some basic information about fractals and fractal music. He has also posted some samples of his music and invites visitors to download software for creating fractal music and to submit their own compositions.

330

Multi-Scale Fractal Analysis of Image Texture and Pattern

NASA Technical Reports Server (NTRS)

Fractals embody important ideas of self-similarity, in which the spatial behavior or appearance of a system is largely independent of scale. Self-similarity is defined as a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. An ideal fractal (or monofractal) curve or surface has a constant dimension over all scales, although it may not be an integer value. This is in contrast to Euclidean or topological dimensions, where discrete one, two, and three dimensions describe curves, planes, and volumes. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution. However, most geographical phenomena are not strictly self-similar at all scales, but they can often be modeled by a stochastic fractal in which the scaling and self-similarity properties of the fractal have inexact patterns that can be described by statistics. Stochastic fractal sets relax the monofractal self-similarity assumption and measure many scales and resolutions in order to represent the varying form of a phenomenon as a function of local variables across space. In image interpretation, pattern is defined as the overall spatial form of related features, and the repetition of certain forms is a characteristic pattern found in many cultural objects and some natural features. Texture is the visual impression of coarseness or smoothness caused by the variability or uniformity of image tone or color. A potential use of fractals concerns the analysis of image texture. In these situations it is commonly observed that the degree of roughness or inexactness in an image or surface is a function of scale and not of experimental technique. The fractal dimension of remote sensing data could yield quantitative insight on the spatial complexity and information content contained within these data. A software package known as the Image Characterization and Modeling System (ICAMS) was used to explore how fractal dimension is related to surface texture and pattern. The ICAMS software was verified using simulated images of ideal fractal surfaces with specified dimensions. The fractal dimension for areas of homogeneous land cover in the vicinity of Huntsville, Alabama was measured to investigate the relationship between texture and resolution for different land covers.

Emerson, Charles W.

1998-01-01

331

ERIC Educational Resources Information Center

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.…

Fraboni, Michael; Moller, Trisha

2008-01-01

332

Living cities have intrinsically fractal properties, in common with all living systems. The pressure to accommodate both the automobile and increased population growth led twentieth-century urbanists to impose anti-fractal geometrical typologies. The fractal properties of the traditional city were erased, with disastrous consequences for the urban fabric. To undo this damage, it is necessary to understand several things in some

Nikos A. Salingaros

333

Fractal interpretation of intermittency

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.

Hwa, R.C.

1991-12-01

334

The fractal measurement of experimental images of supersonic turbulent mixing layer

Flow visualization of supersonic mixing layer has been studied based on the high spatiotemporal resolution Nano-based Planar\\u000a Laser Scattering (NPLS) method in SML-1 wind tunnel. The corresponding images distinctly reproduced the flow structure of\\u000a laminar, transitional and turbulent region, with which the fractal measurement can be implemented. Two methods of measuring\\u000a fractal dimension were introduced and compared. The fractal dimension

Yuxin Zhao; Shihe Yi; Lifeng Tian; Lin He; Zhongyu Cheng

2008-01-01

335

Fractal Weyl laws for chaotic open systems

We present a result relating the density of quantum resonances for an open chaotic system to the fractal dimension of the associated classical repeller. The result is supported by numerical computation of the resonances of the system of n disks on a plane. The result generalizes the Weyl law for the density of states of a closed system to chaotic open systems.

W. T. Lu; S. Sridhar; Maciej Zworski

2003-05-30

336

Load-flow fractals draw clues to erratic behavior

Fractal images have been discovered in many areas of science and engineering in the past two decades, so it is not surprising that they have also appeared in power system literature. This article provides a brief introduction to fractals and presents some fractal patterns produced by Newton-Raphson load-flow calculations for a small power system. If one imagines performing load-flow calculations from a dense grid of initial conditions, the region of initial conditions that converge to a particular equilibrium point using the Newton-Raphson method is seen to have a fractal boundary.

Thorp, J.S.; Naqavi, S.A. [Cornell Univ., Ithaca, NY (United States)] [Cornell Univ., Ithaca, NY (United States)

1997-01-01

337

NASA Astrophysics Data System (ADS)

The bending and torsional degrees of freedom in S1 acetylene, C2H2, are subject to strong vibrational resonances and rovibrational interactions, which create complex vibrational polyad structures even at low energy. As the internal energy approaches that of the barrier to cis-trans isomerization, these energy level patterns undergo further large-scale reorganization that cannot be satisfactorily treated by traditional models tied to local minima of the potential energy surface for nuclear motion. Experimental spectra in the region near the cis-trans transition state have revealed these complicated new patterns. In order to understand near-barrier spectroscopic observations and to predict the detailed effects of cis-trans isomerization on the rovibrational energy level structure, we have performed reduced dimension rovibrational variational calculations of the S1 state. In this paper, we present the methodological details, several of which require special care. Our calculation uses a high accuracy ab initio potential surface and a fully symmetrized extended complete nuclear permutation inversion group theoretical treatment of a multivalued internal coordinate system that is appropriate for large amplitude bending and torsional motions. We also discuss the details of the rovibrational basis functions and their symmetrization, as well as the use of a constrained reduced dimension rovibrational kinetic energy operator.

Changala, P. Bryan

2014-01-01

338

We have created a second-order finite-difference solution to the anisotropic elastic wave equation in three dimensions and implemented the solution as an efficient Matlab script. This program allows the user to generate synthetic seismograms for three-dimensional anisotropic earth structure. The code was written for teleseismic wave propagation in the 1-0.1 Hz frequency range but is of general utility and can be used at all scales of space and time. This program was created to help distinguish among various types of lithospheric structure given the uneven distribution of sources and receivers commonly utilized in passive source seismology. Several successful implementations have resulted in a better appreciation for subduction zone structure, the fate of a transform fault with depth, lithospheric delamination, and the effects of wavefield focusing and defocusing on attenuation. Companion scripts are provided which help the user prepare input to the finite-difference solution. Boundary conditions including specification of the initial wavefield, absorption and two types of reflection are available. ?? 2005 Elsevier Ltd. All rights reserved.

Boyd, O.S.

2006-01-01

339

Rapid calculation method for Frenkel-type two-exciton states in one to three dimensions

NASA Astrophysics Data System (ADS)

Biexciton and two-exciton dissociated states of Frenkel-type excitons are well described by a tight-binding model with a nearest-neighbor approximation. Such two-exciton states in a finite-size lattice are usually calculated by numerical diagonalization of the Hamiltonian, which requires an increasing amount of computational time and memory as the lattice size increases. I develop here a rapid, memory-saving method to calculate the energies and wave functions of two-exciton states by employing a bisection method. In addition, an attractive interaction between two excitons in the tight-binding model can be obtained directly so that the biexciton energy agrees with the observed energy, without the need for the trial-and-error procedure implemented in the numerical diagonalization method.

Ajiki, Hiroshi

2014-07-01

340

Rapid calculation method for Frenkel-type two-exciton states in one to three dimensions.

Biexciton and two-exciton dissociated states of Frenkel-type excitons are well described by a tight-binding model with a nearest-neighbor approximation. Such two-exciton states in a finite-size lattice are usually calculated by numerical diagonalization of the Hamiltonian, which requires an increasing amount of computational time and memory as the lattice size increases. I develop here a rapid, memory-saving method to calculate the energies and wave functions of two-exciton states by employing a bisection method. In addition, an attractive interaction between two excitons in the tight-binding model can be obtained directly so that the biexciton energy agrees with the observed energy, without the need for the trial-and-error procedure implemented in the numerical diagonalization method. PMID:25053304

Ajiki, Hiroshi

2014-07-21

341

Fractal network model for contact conductance

The topography of rough surfaces strongly influences the conduction of heat and electricity between two surfaces in contact. Roughness measurements on a variety of surfaces have shown that their structure follows a fractal geometry whereby similar images of the surface appear under repeated magnification. Such a structure is characterized by the fractal dimension D, which lies between 2 and 3 for a surface and between 1 and 2 for a surface profile. This paper uses the fractal characterization of surface roughness to develop a new network model for analyzing heat conduction between two contacting rough surfaces. The analysis yields the simple result that the contact conductance h and the real area of contact A{sub t} are related as h {approximately} A{sub t}{sup D/2} where D is the fractal dimension of the surface profile. Contact mechanics of fractal surfaces has shown that A{sub t} varies with the load F as A{sub t} {approximately} F{sup {eta}} where {eta} ranges from 1 to 1.33 depending on the value of D. This proves that the ocnductance and load are related as h {approximately} F{sup {eta}D/2} and resolves the anomaly in previous investigations, which theoretically and experimentally obtained different values for the load exponent. The analytical results agreed well with previous experiments although there is a tendency for overprediction.

Majumdar, A. (Arizona State Univ., Tempe (United States)); Tien, C.L. (Univ. of California, Berkeley (United States))

1991-08-01

342

Recent investigations suggest that the mechanisms of heat transfer in bubbling gas fluidized beds exhibit characteristics consistent with chaos. Also, methods for simulating fractal time-series data have undergone significant development. In the present work, experimental time-series data were acquired using a constant temperature, platinum film heat flux probe flush mounted on a horizontal cylinder submerged in a bubbling gas fluidized bed. Analysis of the power spectra of the experimental signals suggests the existence of multi-fractal characteristics. Based on the fractal nature of chaotic systems, the multi-fractal Weierstrass-Mandelbrot (WM) function was used to simulate the scale-independent contribution to the local instantaneous heat transfer signals. The number of fractal components, the dimension of each fractal component and the range of frequencies associated with each fractal were determined from the power spectrum of an experimental signal. This information was subsequently used to simulate the multi-fractal contribution of an experimental signal.

Pence, D.V.; Beasley, D.E. [Clemson Univ., SC (United States)

1995-12-31

343

Subwavelength waveguiding and imaging with a one-dimensional array of metallic H-fractals

NASA Astrophysics Data System (ADS)

We demonstrate, both experimentally and theoretically, subwavelength waveguiding and imaging through a one-dimensional (1D) array of 3D metallic H-fractals. The waveguide formed by the fractal array is subwavelength in all cross-sectional dimensions, thereby allowing compact designs for guided propagation of long-wavelength EM waves. The underlying physics is governed by the fractal metallic wire structure that allows subwavelength resonances. The measured results indicate that such waveguides can provide low-loss propagation of EM waves with a decay length in the range 10-40 m, achieved through coupling of resonant dipoles/multipoles between neighboring H-fractals along the wave propagation direction. By treating the waveguide as a 1D photonic crystal, the calculated bandstructure shows that in the neighborhood of peak transmission frequencies the dispersion relations are relatively flat as a function of the wave vector, implying low group velocities. This is experimentally verified. Owing to the increased spatial information carried at such transmission frequencies, imaging becomes possible and this was observed by placing an obstacle at the inlet and scanning the local field intensity at the other end of the waveguide.

Xiao, Xiao; Yi, Xin; Hou, Bo; Wen, Weijia; Liu, Zhengyou; Shi, Jing; Sheng, Ping

2010-07-01

344

Analytical estimation of the correlation dimension of integer lattices

Recently [L. Lacasa and J. G\\'omez-Garde\\~nes, Phys. Rev. Lett. {\\bf 110}, 168703 (2013)], a fractal dimension has been proposed to characterize the geometric structure of networks. This measure is an extension to graphs of the so called {\\em correlation dimension}, originally proposed by Grassberger and Procaccia to describe the geometry of strange attractors in dissipative chaotic systems. The calculation of the correlation dimension of a graph is based on the local information retrieved from a random walker navigating the network. In this contribution we study such quantity for some limiting synthetic spatial networks and obtain analytical results on agreement with the previously reported numerics. In particular, we show that up to first order the correlation dimension $\\beta$ of integer lattices $\\mathbb{Z}^d$ coincides with the Haussdorf dimension of their coarsely-equivalent Euclidean spaces, $\\beta=d$.

Lacasa, Lucas

2014-01-01

345

Analytical estimation of the correlation dimension of integer lattices.

Recently [L. Lacasa and J. Gómez-Gardeñes, Phys. Rev. Lett. 110, 168703 (2013)], a fractal dimension has been proposed to characterize the geometric structure of networks. This measure is an extension to graphs of the so called correlation dimension, originally proposed by Grassberger and Procaccia to describe the geometry of strange attractors in dissipative chaotic systems. The calculation of the correlation dimension of a graph is based on the local information retrieved from a random walker navigating the network. In this contribution, we study such quantity for some limiting synthetic spatial networks and obtain analytical results on agreement with the previously reported numerics. In particular, we show that up to first order, the correlation dimension ? of integer lattices ?(d) coincides with the Haussdorf dimension of their coarsely equivalent Euclidean spaces, ??=?d. PMID:25554021

Lacasa, Lucas; Gómez-Gardeñes, Jesús

2014-12-01

346

Analytical estimation of the correlation dimension of integer lattices

Recently [L. Lacasa and J. G\\'omez-Garde\\~nes, Phys. Rev. Lett. {\\bf 110}, 168703 (2013)], a fractal dimension has been proposed to characterize the geometric structure of networks. This measure is an extension to graphs of the so called {\\em correlation dimension}, originally proposed by Grassberger and Procaccia to describe the geometry of strange attractors in dissipative chaotic systems. The calculation of the correlation dimension of a graph is based on the local information retrieved from a random walker navigating the network. In this contribution we study such quantity for some limiting synthetic spatial networks and obtain analytical results on agreement with the previously reported numerics. In particular, we show that up to first order the correlation dimension $\\beta$ of integer lattices $\\mathbb{Z}^d$ coincides with the Haussdorf dimension of their coarsely-equivalent Euclidean spaces, $\\beta=d$.

Lucas Lacasa; Jesús Gómez-Gardeñes

2014-07-07

347

Triangular constellations in fractal measures

NASA Astrophysics Data System (ADS)

The local structure of a fractal set is described by its dimension D, which is the exponent of a power-law relating the mass {\\cal N} in a ball to its radius \\varepsilon{:}\\ {\\cal N}\\sim \\varepsilon^D . It is desirable to characterise the shapes of constellations of points sampling a fractal measure, as well as their masses. The simplest example is the distribution of shapes of triangles formed by triplets of points, which we investigate for fractals generated by chaotic dynamical systems. The most significant parameter describing the triangle shape is the ratio z of its area to the radius of gyration squared. We show that the probability density of z has a phase transition: P(z) is independent of ? and approximately uniform below a critical flow compressibility \\beta_{\\text{c}} , which we estimate. For \\beta>\\beta_{\\text{c}} the distribution appears to be described by two power laws: P(z)\\sim z^{\\alpha_1} when 1\\gg z\\gg z_{\\text{c}}(\\varepsilon) , and P(z)\\sim z^{\\alpha_2} when z\\ll z_{\\text{c}}(\\varepsilon) .

Wilkinson, Michael; Grant, John

2014-09-01

348

Fractals: To Know, to Do, to Simulate.

ERIC Educational Resources Information Center

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)

Talanquer, Vicente; Irazoque, Glinda

1993-01-01

349

Black carbon fractal morphology and short-wave radiative impact: a modelling study

NASA Astrophysics Data System (ADS)

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 are other potentially important morphological features that have not been addressed in the present study, such as sintering and coating of freshly emitted black carbon by films of organic material. Finally, we found that the spectral variation of the absorption cross section of black carbon significantly deviates from a simple 1/? scaling law. We therefore discourage the use of single-wavelength absorption measurements in conjunction with a 1/? scaling relation in broadband radiative forcing simulations of black carbon.

Kahnert, M.; Devasthale, A.

2011-11-01

350

Black carbon fractal morphology and short-wave radiative impact: a modelling study

NASA Astrophysics Data System (ADS)

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 radiative impact studies. We emphasise that there are other potentially important morphological features that have not been addressed in the present study, such as sintering and coating of freshly emitted black carbon by films of organic material.

Kahnert, M.; Devasthale, A.

2011-08-01

351

Exploring Fractals in the Classroom.

ERIC Educational Resources Information Center

Describes an activity involving six investigations. Introduces students to fractals, allows them to study the properties of some famous fractals, and encourages them to create their own fractal artwork. Contains 14 references. (ASK)

Naylor, Michael

1999-01-01

352

NASA Technical Reports Server (NTRS)

Fractals are geometric or data structures which do not simplify under magnification. Fractal Image Compression is a technique which associates a fractal to an image. On the one hand, the fractal can be described in terms of a few succinct rules, while on the other, the fractal contains much or all of the image information. Since the rules are described with less bits of data than the image, compression results. Data compression with fractals is an approach to reach high compression ratios for large data streams related to images. The high compression ratios are attained at a cost of large amounts of computation. Both lossless and lossy modes are supported by the technique. The technique is stable in that small errors in codes lead to small errors in image data. Applications to the NASA mission are discussed.

Barnsley, Michael F.; Sloan, Alan D.

1989-01-01

353

Determining Effective Thermal Conductivity of Fabrics by Using Fractal Method

NASA Astrophysics Data System (ADS)

In this article, a fractal effective thermal conductivity model for woven fabrics with multiple layers is developed. Structural models of yarn and plain woven fabric are derived based on the fractal characteristics of macro-pores (gap or channel) between the yarns and micro-pores inside the yarns. The fractal effective thermal conductivity model can be expressed as a function of the pore structure (fractal dimension) and architectural parameters of the woven fabric. Good agreement is found between the fractal model and the thermal conductivity measurements in the general porosity ranges. It is expected that the model will be helpful in the evaluation of thermal comfort for woven fabric in the whole range of porosity.

Zhu, Fanglong; Li, Kejing

2010-03-01

354

Analysis of permeability for transient two-phase flow in fractal porous media

NASA Astrophysics Data System (ADS)

A relative permeability model for transient two-phase flow in fractal porous media is derived based on the fractal characteristics of pore size distribution and the assumption that porous media consists of capillary bundles. The functions in this model are tortuosity fractal dimension, pore fractal dimension, and maximum and minimum pore diameters. Every parameter has clear physical meaning without the use of empirical constants. Good agreement between model predictions and experimental data is obtained, the sensitive parameters that influence the relative permeability are specified and their effects on relative permeability are discussed.

Tan, Xiao-Hua; Li, Xiao-Ping; Liu, Jian-Yi; Zhang, Guang-Dong; Zhang, Lie-Hui

2014-03-01

355

Application of fractal geometry to damage development and brittle fracture in materials

The concepts of fractal geometry are applicable to a wide range of problems in materials science. One obvious application is the characterization of irregular surfaces (e.g., fracture surfaces) by means of a fractal dimension. Several papers on this subject have recently appeared. A somewhat less obvious use of the fractal dimension involves characterization of the fragmented (nonuniform) nature of microstructural features such as second phase particles and microcracks. This article utilizes fractal geometry to develop simple models for microcrack growth. Both stable and unstable growth are considered. These results are potentially applicable to a wide range of materials including composites, ceramics and structural steels.

Anderson, T.L.

1989-01-01

356

Fractal and Multifractal Analysis of Human Gait

NASA Astrophysics Data System (ADS)

We carried out a fractal and multifractal analysis of human gait time series of young and old individuals, and adults with three illnesses that affect the march: The Parkinson's and Huntington's diseases and the amyotrophic lateral sclerosis (ALS). We obtained cumulative plots of events, the correlation function, the Hurst exponent and the Higuchi's fractal dimension of these time series and found that these fractal markers could be a factor to characterize the march, since we obtained different values of these quantities for youths and adults and they are different also for healthy and ill persons and the most anomalous values belong to ill persons. In other physiological signals there is complexity lost related with the age and the illness, in the case of the march the opposite occurs. The multifractal analysis could be also a useful tool to understand the dynamics of these and other complex systems.

Muñoz-Diosdado, A.; del Río Correa, J. L.; Angulo-Brown, F.

2003-09-01

357

Dimensions of structurally stable and critical strange non-chaotic attractors

NASA Astrophysics Data System (ADS)

We provide evidence that the box-counting dimension of a structurally stable strange non-chaotic attractor (SNA) of pinched skew product type is equal to 2 by showing that it has non-negligible area. The argument presented is made more accurate in the study of a piecewise linear SNA. Furthermore we provide evidence that the fractal dimension of a critical SNA is not equal to 2, but in fact lies between 1 and 2. We numerically calculate the box-counting dimension for several critical SNAs, providing further evidence to support this conjecture.

Adamson, L. N. C.; Osbaldestin, A. H.

2015-01-01

358

NSDL National Science Digital Library

While some may know fractals primarily from their use in abstract painting and African art, fractals are important elements within the world of mathematics. For those who seek to learn more about the construction of fractals and their uses, this very nice Flash-enabled feature from Daniel Gries at Merrimack College will definitely come in handy. This particular Flash applet draws fractals by means of a recursive algorithm, using a simple "generator" that users draw in the space that it is provided. Before using the application, users may wish read the overview offered online, and also take the time to read the instructions thoroughly.

Gries, Daniel

359

Fourier transforms and fractals in the food and agricultural industry

NASA Astrophysics Data System (ADS)

Links between the fractal Hausdorff-dimension, the Fourier transform of 2D scenes and image segmentation by texture are discussed. It is shown that the fractal Hausdorff-dimension can be derived by integration of the intensity of the spatial frequency domain (i.e. the Fourier plane) over a set of different band-limited spatial filters. The difference between a computational and optical approach to determine the Hausdorff-dimension are shown, with advantages of both methods discussed. Possible future directions of research/improvements are mentioned. Natural and simulated scenes are considered which apply to a wide range of situations in the agricultural and food industry.

Zwiggelaar, Reyer; Bull, Christine R.

1994-11-01

360

Evaluation of Two Fractal Methods for Magnetogram Image Analysis

NASA Technical Reports Server (NTRS)

Fractal and multifractal techniques have been applied to various types of solar data to study the fractal properties of sunspots as well as the distribution of photospheric magnetic fields and the role of random motions on the solar surface in this distribution. Other research includes the investigation of changes in the fractal dimension as an indicator for solar flares. Here we evaluate the efficacy of two methods for determining the fractal dimension of an image data set: the Differential Box Counting scheme and a new method, the Jaenisch scheme. To determine the sensitivity of the techniques to changes in image complexity, various types of constructed images are analyzed. In addition, we apply this method to solar magnetogram data from Marshall Space Flight Centers vector magnetograph.

Stark, B.; Adams, M.; Hathaway, D. H.; Hagyard, M. J.

1997-01-01

361

Cut-out sets, fractal voids and cosmic structure

"Cut-out sets" are fractals that can be obtained by removing a sequence of disjoint regions from an initial region of d-dimensional euclidean space. Conversely, a description of some fractals in terms of their void complementary set is possible. The essential property of a sequence of fractal voids is that their sizes decrease as a power law, that is, they follow Zipf's law. We prove the relation between the box dimension of the fractal set (in d voids; namely, if the Zipf law exponent e is such that 1 void shapes, we prove that the corresponding cut-out set has box dimension d/e (d-1 < d/e < d). We explore the application of this result to the large scale distribution of matter in cosmology, in connection with ``cosmic foam'' models.

Jose Gaite

2006-03-21

362

Systematization is performed for the results of theoretical and experimental investigations that were obtained in radio physics and radio engineering with the help of fractal theory and the mathematical theory of fractional dimension and fractional operators with consideration for the scaling effects of real radio signals and electromagnetic fields. The methods are based on texture and fractal attributes with allowance

Alexander A. Potapov

2008-01-01

363

Fractal geometry of some Martian lava flow margins: Alba Patera

NASA Technical Reports Server (NTRS)

Fractal dimension for a few lava flow margins on the gently sloping flanks of Alba Patera were measured using the structured walk method. Fractal behavior was observed at scales ranging from 20 to 100 pixels. The upper limit of the linear part of log(margin length) vs. log(scale) profile correlated well to the margin length. The lower limit depended on resolution and flow properties.

Kauhanen, K.

1993-01-01

364

Lacunarity of multi-gap fractal superlattices using wavelet analysis

Multi-gap superlattices having the same fractal (similarity) dimension may differ in their distribution of gap sizes. This distribution is characterized by a fractal descriptor denoted the lacunarity. From the wavelet analysis of the superlattice's impulse response, we extract the lacunarity. We show that the skeleton of wavelet-transform modulus maxima overlaps two hierarchical structures in the time-scale domain: one allows the

A.-S. Saleh; H. Aubert; D. L. Jaggard

2001-01-01

365

Fractal geometry and fractal material behaviour in solids and structures

The present paper discusses certain methods which permit us to consider the influence of the fractal geometry and the fractal material behaviour in solid and structural mechanics. The method of fractal interpolation function is introduced and the fractal quantities (boundary geometry, interface geometry and stress-strain laws) are considered as the fixed points of a given set-valued transformation. Our first aim

P. D. Panagiotopoulos; O. K. Panagouli; E. S. Mistakidis

1993-01-01

366

Statistics of semiflexible self-avoiding trails on a family of two-dimensional compact fractals

NASA Astrophysics Data System (ADS)

We have applied the exact and Monte Carlo renormalization group (MCRG) method to study the statistics of semiflexible self-avoiding trails (SATs) on the family of plane-filling (PF) fractals. Each fractal of the family is compact, that is, the fractal dimension df is equal to 2 for all members of the PF family, which are enumerated by an odd integer b, 3\\le b\\lt \\infty . Varying values of the stiffness parameter s of trails from 1 to 0 (so that when s decreases the trail stiffness increases) we calculate exactly (for 3 <= b <= 7) and through the MCRG approach (for b <= 201) the sets of the critical exponents ? (associated with the mean squared end-to-end distances of SATs) and ? (associated with the total number of different SATs). Our results show that critical exponents are stiffness dependent functions, so that ?(s) is a monotonically decreasing function of s, for each studied b, whereas ?(s) displays a non-monotonic behavior for some values of b. On the other hand, by fixing the stiffness parameter s, our results show clearly that for highly flexible trails (with s = 1 and 0.9) ? is a non-monotonic function of b, while for stiffer SATs (with s <= 0.7) ? monotonically decreases with b. We also show that ?(b) increases with increasing b, independently of s. Finally, we compare the obtained SAT data with those obtained for the semiflexible self-avoiding walk (SAW) model on the same fractal family, and for both models we discuss behavior of the studied exponents in the fractal-to-Euclidean crossover region b\\to \\infty .

Živi?, I.; Elezovi?-Hadži?, S.; Miloševi?, S.

2011-10-01

367

Observation of two different fractal structures in nanoparticle, protein and surfactant complexes

NASA Astrophysics Data System (ADS)

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.

Mehan, Sumit; Kumar, Sugam; Aswal, V. K.

2014-04-01

368

Observation of two different fractal structures in nanoparticle, protein and surfactant complexes

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.

Mehan, Sumit, E-mail: sumit.mehan@gmail.com; Kumar, Sugam, E-mail: sumit.mehan@gmail.com; Aswal, V. K., E-mail: sumit.mehan@gmail.com [Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai-400085 (India)

2014-04-24

369

This paper examines whether a fractal cloud geometry can reproduce the emission-line spectra of active galactic nuclei (AGNs). The nature of the emitting clouds is unknown, but many current models invoke various types of magnetohydrodynamic confinement. Recent studies have argued that a fractal distribution of clouds, in which subsets of clouds occur in self-similar hierarchies, is a consequence of such

Mark Bottorff; Gary Ferland

2001-01-01

370

NASA Technical Reports Server (NTRS)

Fractals are objects that are generally self similar at all scales. Coastlines, mountains, river systems, planetary orbits and some mathematical objects are all examples of fractals. Bruno et al. used the structured walk model of Richardson to establish that lava flows are fractals and that lava flow morphology could be determined by looking at the fractal dimension of flow margins. They determined that Hawaiian a.a flows have fractal dimensions that range from 1.05 to 1.09 and that the pahoehoe lava flows have a fractal dimension from 1.13 to 1.23. We have analyzed a number of natural and simulated lava flow margins and find that the fractal dimension varies according to the number and length of rod lengths used in the structured walk method. The potential variation we find in our analyses is sufficiently large so that unambiguous determination of lava flow morphology is problematic for some flows. We suggest that the structured walk method can provide meaningful fractal dimensions if rod lengths employed in the analysis provide a best-fit residual of greater than 0.98, as opposed to the 0.95 cutoff used in previous studies. We also find that the use of more than 4 rod lengths per analysis also reduces ambiguity in the results.

Hudson, Richard K.; Anderson, Steven W.; McColley, Shawn; Fink, Jonathan H.

2004-01-01

371

Fractal Analysis of Prime Indian STOCK Market Indices

NASA Astrophysics Data System (ADS)

The purpose of the present work is to study the fractal behaviour of prime Indian stock exchanges, namely Bombay Stock Exchange Sensitivity Index (BSE Sensex) and National Stock Exchange (NSE). To analyze the monofractality of these indices we have used Higuchi method and Katz method separately. By applying Mutifractal Detrended Fluctuation Analysis (MFDFA) technique we have calculated the generalized Hurst exponents, multifractal scaling exponents and generalized multifractal dimensions for the present indices. We have deduced Hölder exponents as well as singularity spectra for BSE and NSE. It has been observed that both the stock exchanges are possessing self-similarity at different small ranges separately and inhomogeneously. By comparing the multifractal behaviour of the BSE and NSE indices, we have found that the second one exhibits a richer multifractal feature than the first one.

Samadder, Swetadri; Ghosh, Koushik; Basu, Tapasendra

2013-03-01

372

Landscape roughness analysis of Mt. Etna volcanic complex detected via fractal geometry

NASA Astrophysics Data System (ADS)

During the last years several aspects relevant to volcanic activity have been analyzed in fractal context. These studies have been aimed at identifying the power laws that govern the magma fragmentation processes and/or the classification of different geological processes. In this work we exploit the algorithm proposed by Di Martino et al. (2012) that allows retrieving the fractal dimension of a natural surface starting from its corresponding Synthetic Aperture Radar (SAR) image. Such an algorithm is based on an analytical model that links the stochastic characterization of a single SAR amplitude image to the fractal dimension of the observed surface, modeled via a fractional Brownian motion (fBm) process. The considered SAR image processing provides - as an output product - the pixel by pixel map of the fractal dimension of the scene observed by the sensor. Previous works demonstrated that the fractal dimension of lava flows is strictly connected to the natural surface roughness. Moreover, Pepe et al. (2012) showed the possibility of characterizing the single volcanic structures by means of the fractal dimension values retrieved from the corresponding SAR images. In the present work we consider a data-set of Cosmo-SkyMed high resolution images acquired over the Mt. Etna volcanic complex (South Italy), spanning the 2009 - 2011 time period. Starting from the SAR amplitude images of the considered data-set, we generated the corresponding fractal dimension maps that were subsequently co-registered each other, thus retrieving the fractal dimension time-series of the Mt. Etna volcano. Then, by averaging the so-computed fractal dimension maps with respect to time we generated a map of the mean fractal dimension of the investigated area. This procedure allows significantly improving the quality of the final fractal dimension map, as the average operation reduces the noise (due to the speckle effect on SAR images) present on each fractal map. Besides, the so-obtained mean 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.

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

373

Subwavelength waveguiding and imaging with a one-dimensional array of metallic H-fractals

We demonstrate, both experimentally and theoretically, subwavelength waveguiding and imaging through a one-dimensional (1D) array of 3D metallic H-fractals. The waveguide formed by the fractal array is subwavelength in all cross-sectional dimensions, thereby allowing compact designs for guided propagation of long-wavelength EM waves. The underlying physics is governed by the fractal metallic wire structure that allows subwavelength resonances. The measured

Xiao Xiao; Xin Yi; Bo Hou; Weijia Wen; Zhengyou Liu; Jing Shi; Ping Sheng

2010-01-01

374

Simulation of turbulent combustion flame feature based on fractal theory for SI engines

The flame structure of gasoline engine is complicated and has the characteristic of fractal geometry. A fractal combustion\\u000a model was used to simulate the engine working cycle. Based on this model, the fractal dimension and laminar flame surface\\u000a area of turbulent premixed flames were studied under different working conditions. The experimental system mainly includes\\u000a an optical engine and a set

Jun Zhang; Qing Du; Dongxian Song; Yanxiang Yang

2010-01-01

375

Analysis of root fractal characteristics in remote areas of the Taklimakan desert, China

Fractal geometry is a potential new approach to analyze the root architecture, which may offer improved ways to quantify and\\u000a summarize root system complexity as well as yield ecological and physiological insights into the functional relevance of specific\\u000a architectural patterns. Fractal analysis is a sensitive measure of root branching intensity and fractal dimension expresses\\u000a the “space filling” properties of a

Xiao-lin Yang; Xi-ming Zhang; Yi-ling Li; Shao-cai Li; Hai-long Sun

2008-01-01

376

Fractal nature of multiple shear bands in severely deformed metallic glass

We present an analysis of fractal geometry of extensive and complex shear band patterns in a severely deformed metallic glass. We show that the shear band patterns have fractal characteristics, and the fractal dimensions are determined by the stress noise induced by the interaction between shear bands. A theoretical model of the spatial evolution of multiple shear bands is proposed in which the collective shear bands slide is considered as a stochastic process far from thermodynamic equilibrium.

Sun, B. A.; Wang, W. H. [Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China)

2011-05-16

377

Soliton fractals in the Korteweg-de Vries equation.

We have studied the process of creation of solitons and generation of fractal structures in the Korteweg-de Vries (KdV) equation when the relation between the nonlinearity and dispersion is abruptly changed. We observed that when this relation is changed nonadiabatically the solitary waves present in the system lose their stability and split up into ones that are stable for the set of parameters. When this process is successively repeated the trajectories of the solitary waves create a fractal treelike structure where each branch bifurcates into others. This structure is formed until the iteration where two solitary waves overlap just before the breakup. By means of a method based on the inverse scattering transformation, we have obtained analytical results that predict and control the number, amplitude, and velocity of the solitary waves that arise in the system after every change in the relation between the dispersion and the nonlinearity. This complete analytical information allows us to define a recursive L system which coincides with the treelike structure, governed by KdV, until the stage when the solitons start to overlap and is used to calculate the Hausdorff dimension and the multifractal properties of the set formed by the segments defined by each of the two "brothers" solitons before every breakup. PMID:17995132

Zamora-Sillero, Elias; Shapovalov, A V

2007-10-01

378

Fractal analysis of the spatial distribution of earthquakes along the Hellenic Subduction Zone

NASA Astrophysics Data System (ADS)

The Hellenic Subduction Zone (HSZ) is the most seismically active region in Europe. Many destructive earthquakes have taken place along the HSZ in the past. The evolution of such active regions is expressed through seismicity and is characterized by complex phenomenology. The understanding of the tectonic evolution process and the physical state of subducting regimes is crucial in earthquake prediction. In recent years, there is a growing interest concerning an approach to seismicity based on the science of complex systems (Papadakis et al., 2013; Vallianatos et al., 2012). In this study we calculate the fractal dimension of the spatial distribution of earthquakes along the HSZ and we aim to understand the significance of the obtained values to the tectonic and geodynamic evolution of this area. We use the external seismic sources provided by Papaioannou and Papazachos (2000) to create a dataset regarding the subduction zone. According to the aforementioned authors, we define five seismic zones. Then, we structure an earthquake dataset which is based on the updated and extended earthquake catalogue for Greece and the adjacent areas by Makropoulos et al. (2012), covering the period 1976-2009. The fractal dimension of the spatial distribution of earthquakes is calculated for each seismic zone and for the HSZ as a unified system using the box-counting method (Turcotte, 1997; Robertson et al., 1995; Caneva and Smirnov, 2004). Moreover, the variation of the fractal dimension is demonstrated in different time windows. These spatiotemporal variations could be used as an additional index to inform us about the physical state of each seismic zone. As a precursor in earthquake forecasting, the use of the fractal dimension appears to be a very interesting future work. Acknowledgements Giorgos Papadakis wish to acknowledge the Greek State Scholarships Foundation (IKY). References Caneva, A., Smirnov, V., 2004. Using the fractal dimension of earthquake distributions and the slope of the recurrence curve to forecast earthquakes in Colombia. Earth Sci. Res. J., 8, 3-9. Makropoulos, K., Kaviris, G., Kouskouna, V., 2012. An updated and extended earthquake catalogue for Greece and adjacent areas since 1900. Nat. Hazards Earth Syst. Sci., 12, 1425-1430. Papadakis, G., Vallianatos, F., Sammonds, P., 2013. Evidence of non extensive statistical physics behavior of the Hellenic Subduction Zone seismicity. Tectonophysics, 608, 1037-1048. Papaioannou, C.A., Papazachos, B.C., 2000. Time-independent and time-dependent seismic hazard in Greece based on seismogenic sources. Bull. Seismol. Soc. Am., 90, 22-33. Robertson, M.C., Sammis, C.G., Sahimi, M., Martin, A.J., 1995. Fractal analysis of three-dimensional spatial distributions of earthquakes with a percolation interpretation. J. Geophys. Res., 100, 609-620. Turcotte, D.L., 1997. Fractals and chaos in geology and geophysics. Second Edition, Cambridge University Press. Vallianatos, F., Michas, G., Papadakis, G., Sammonds, P., 2012. A non-extensive statistical physics view to the spatiotemporal properties of the June 1995, Aigion earthquake (M6.2) aftershock sequence (West Corinth rift, Greece). Acta Geophys., 60, 758-768.

Papadakis, Giorgos; Vallianatos, Filippos; Sammonds, Peter

2014-05-01

379

Viscous fingering fractals in porous media

Gas displacing a high-viscosity fluid in a two-dimensional porous disk intrudes in the form of ramified fingers similar to the structures obtained in diffusion-limited aggregation. We find that the resulting finger structures are described by a fractal dimension D=1.62+\\/-0.04 consistent with D for diffusion-limited aggregation clusters. This result confirms the analogy between diffusion-limited aggregation and two-fluid displacement in porous media

Knut Jørgen Mløy; Jens Feder; Torstein Jøssang

1985-01-01

380

Ulam method and fractal Weyl law for Perron-Frobenius operators

NASA Astrophysics Data System (ADS)

We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show numerically that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent ? = d-1, where d is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we numerically find the Weyl exponent ? = d/2 where d is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation ? = d0/2 where d0 is the fractal dimension of the invariant sets. We also discuss the properties of eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law.

Ermann, L.; Shepelyansky, D. L.

2010-06-01

381

Fractal analysis of tidal channels in the Bah?´a Blanca Estuary (Argentina)

NASA Astrophysics Data System (ADS)

The fractal dimension ( D) was estimated for nine tidal channels depicted in thematic mapper (TM) Landsat-5 imagery to derive information about the degree of geomorphological control on a tidal channel network characteristic of the Bah?´a Blanca Estuary (Argentina). Two methods, box counting and contiguity, were used to estimate fractal dimensions for each tidal channel. All channels produced D values close to 1, meaning that they are self-affine fractal features. However, these fractal dimensions do not represent the meandering pattern complexity characteristic of the tidal channels analysed. Although both methods allowed for estimation of D, the contiguity method showed that three of the channels actually are not fractal but have sinusoidal characteristics, a condition that was not detected by the former method.

Angeles, Guillermo R.; Perillo, Gerardo M. E.; Piccolo, M. Cintia; Pierini, Jorge O.

2004-02-01

382

Single- and dual-fractal analysis of hydridization binding kinetics: Biosensor applications

The diffusion-limited hybridization kinetics of analyte in solution to a receptor immobilized on a biosensor or immunosensor surface is analyzed within a fractal framework. The data may be analyzed by a single- or a dual-fractal analysis. This was indicated by the regression analysis provided by sigmaplot. It is of interest to note that the binding rate coefficient and the fractal dimension both exhibit changes in the same direction for both the single-fractal and the dual-fractal analysis examples presented. For example, for a single-fractal analysis and for the hybridization of 10 nM 16*CFl (oligonucleotide) to 16*B immobilized via sulfosuccinimidyl-6-(biotinamido)-hexanoate and streptavidin using chemical and thermal regeneration, an increase in the fractal dimension, D{sub f} from 1.211 to 1.394, leads to an increase in the binding rate coefficient, k, from 86.53 to 100.0. An increase in the degree of heterogeneity on the biosensor surface leads to an increase in the binding rate coefficient. When a dual-fractal analysis was utilized, an increase in the fractal dimension value from D{sub f1} to D{sub f2} leads to an increase in the binding rate coefficient value from k{sub 1} to k{sub 2}.

Sadana, A. [Univ. of Mississippi, University, MS (United States). Chemical Engineering Dept.] [Univ. of Mississippi, University, MS (United States). Chemical Engineering Dept.; Vo-Dinh, T. [Oak Ridge National Lab., TN (United States). Advanced Monitoring Development Group] [Oak Ridge National Lab., TN (United States). Advanced Monitoring Development Group

1998-09-01

383

Self-organized stiffness in regular fractal polymer structures

NASA Astrophysics Data System (ADS)

We investigated membrane-like polymer structures of fractal connectivity such as Sierpinski gaskets and Sierpinski carpets applying the bond fluctuation model in three dimensions. Without excluded volume (phantom), both polymeric fractals obey Gaussian elasticity on larger scales determined by their spectral dimension. On the other hand, the swelling effect due to excluded volume is rather distinct between the two polymeric fractals: Self-avoiding Sierpinski gaskets can be described using a Flory-type mean-field argument. Sierpinski carpets having a spectral dimension closer to perfect membranes are significantly more strongly swollen than predicted. Based on our simulation results it cannot be excluded that Sierpinski carpets in athermal solvent show a flat phase on larger scales. We tested the self-consistency of Flory predictions using a virial expansion to higher orders. From this we conclude that the third virial coefficient contributes marginally to Sierpinski gaskets, but higher order virial coefficients are relevant for Sierpinski carpets.

Werner, Marco; Sommer, Jens-Uwe

2011-05-01

384

Self-organized stiffness in regular fractal polymer structures.

We investigated membrane-like polymer structures of fractal connectivity such as Sierpinski gaskets and Sierpinski carpets applying the bond fluctuation model in three dimensions. Without excluded volume (phantom), both polymeric fractals obey Gaussian elasticity on larger scales determined by their spectral dimension. On the other hand, the swelling effect due to excluded volume is rather distinct between the two polymeric fractals: Self-avoiding Sierpinski gaskets can be described using a Flory-type mean-field argument. Sierpinski carpets having a spectral dimension closer to perfect membranes are significantly more strongly swollen than predicted. Based on our simulation results it cannot be excluded that Sierpinski carpets in athermal solvent show a flat phase on larger scales. We tested the self-consistency of Flory predictions using a virial expansion to higher orders. From this we conclude that the third virial coefficient contributes marginally to Sierpinski gaskets, but higher order virial coefficients are relevant for Sierpinski carpets. PMID:21728562

Werner, Marco; Sommer, Jens-Uwe

2011-05-01

385

Thermodynamics of Fractal Universe

We investigate the thermodynamical properties of the apparent horizon in a fractal universe. We find that one can always rewrite the Friedmann equation of the fractal universe in the form of the entropy balance relation $ \\delta Q=T_h d{S_h}$, where $ \\delta Q $ and $ T_{h} $ are the energy flux and Unruh temperature seen by an accelerated observer just inside the apparent horizon. We find that the entropy $S_h$ consists two terms, the first one which obeys the usual area law and the second part which is the entropy production term due to nonequilibrium thermodynamics of fractal universe. This shows that in a fractal universe, a treatment with nonequilibrium thermodynamics of spacetime may be needed. We also study the generalized second law of thermodynamics in the framework of fractal universe. When the temperature of the apparent horizon and the matter fields inside the horizon are equal, i.e. $T=T_h$, the generalized second law of thermodynamics can be fulfilled provided the deceleration and the equation of state parameters ranges either as $-1 \\leq q thermodynamics can be secured in a fractal universe by suitably choosing the fractal parameter $\\beta$.

Ahmad Sheykhi; Zeinab Teimoori; Bin Wang

2012-10-29

386

Retinal Vascular Fractals and Cognitive Impairment

Background Retinal microvascular network changes have been found in patients with age-related brain diseases such as stroke and dementia including Alzheimer's disease. We examine whether retinal microvascular network changes are also present in preclinical stages of dementia. Methods This is a cross-sectional study of 300 Chinese participants (age: ?60 years) from the ongoing Epidemiology of Dementia in Singapore study who underwent detailed clinical examinations including retinal photography, brain imaging and neuropsychological testing. Retinal vascular parameters were assessed from optic disc-centered photographs using a semiautomated program. A comprehensive neuropsychological battery was administered, and cognitive function was summarized as composite and domain-specific Z-scores. Cognitive impairment no dementia (CIND) and dementia were diagnosed according to standard diagnostic criteria. Results Among 268 eligible nondemented participants, 78 subjects were categorized as CIND-mild and 69 as CIND-moderate. In multivariable adjusted models, reduced retinal arteriolar and venular fractal dimensions were associated with an increased risk of CIND-mild and CIND-moderate. Reduced fractal dimensions were associated with poorer cognitive performance globally and in the specific domains of verbal memory, visuoconstruction and visuomotor speed. Conclusion A sparser retinal microvascular network, represented by reduced arteriolar and venular fractal dimensions, was associated with cognitive impairment, suggesting that early microvascular damage may be present in preclinical stages of dementia. PMID:25298774

Ong, Yi-Ting; Hilal, Saima; Cheung, Carol Yim-lui; Xu, Xin; Chen, Christopher; Venketasubramanian, Narayanaswamy; Wong, Tien Yin; Ikram, Mohammad Kamran

2014-01-01

387

Reinforcement of rubber by fractal aggregates

NASA Astrophysics Data System (ADS)

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 ? when ?gg 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 ? quand ?gg 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.

Witten, T. A.; Rubinstein, M.; Colby, R. H.

1993-03-01

388

NSDL National Science Digital Library

Cynthia Lanius, a former mathematics teacher who currently serves as Technology Integration Specialist for Sinton Independent School District in Sinton, Texas, has posted numerous lessons online. This website features a Fractals Unit for elementary and middle school students (although adults are also welcome to enjoy the lesson). The lesson includes a discussion on why one might study fractals and then provides step-by-step explanations on how to make fractals using Java, along with some challenging mathematics questions to consider. Samples of student work are also posted. A section for teachers provides an overview of the unit objectives along with links to other resources and materials to use in the classroom.

2007-12-12

389

An Event-Driven Algorithm for Fractal Cluster S. Gonzalez, A. R. Thornton, S. Luding

. In contrast to the case of diffusion-limited aggregation (DLA), where df = 1.71 is found [7], we keep track dimensional gas: particles move freely until they collide and "stick" together irreversibly. These clusters aggregate into bigger structures in an isotropic way, forming fractal structures whose fractal dimension

Luding, Stefan

390

FRACTAL ANALYSIS TOOLS FOR CHARACTERIZING THE COLORIMETRIC ORGANIZATION OF DIGITAL IMAGES

FRACTAL ANALYSIS TOOLS FOR CHARACTERIZING THE COLORIMETRIC ORGANIZATION OF DIGITAL IMAGES Case. Abstract: The colorimetric organization of RGB color images is analyzed through the computation in the colorimetric domain for natural images with often non-integer fractal dimension over a certain range of scale

Chapeau-Blondeau, FranÃ§ois

391

L-band circular polarization microstrip antenna based on the narrow-slot fractal method

The resonant frequencies of traditional Koch island fractal patch antennas and their narrow-slot counterparts are investigated. The simulation shows that at the stage of higher than the first iteration, the narrow-slot fractal patch antenna has a better effect in decreasing the resonant frequency and reducing the antenna dimension, and this is partly due to its higher performance in increasing the

Nanbo Jin; Mingyan Fan; Xuexia Zhang

2003-01-01

392

DIFFERENCES BETWEEN INTESTINAL AND DIFFUSE TYPE OF GASTRIC CARCINOMA: A FRACTAL ANALYSIS

The histopathological classification of gastric cancer is a complex and difficult task. The main impediment is the heterogeneity of this tumors, frequently underestimated in clinicopathological research. Fractal geometry may help to explain this heterogeneity. In order to investigate the relationship between the complexity of the epithelial\\/connective interface (as determined by fractal dimension) and the malignancy of the gastric tumor we

Vlad Herlea; Bogdan Ivanov; Radu Dobrescu

393

Fractal properties of tremor and gas piston events observed at Kilauea Volcano, Hawaii

We study the fractal properties of shallow volcanic tremor and gas piston events associated with magma degassing at Kilauea Volcano, Hawaii, using data from two dense short-baseline arrays of seismographs deployed near the active crater of Puu Oo on the east rift of the volcano. We found an upper bound on the fractal dimension of a strange attractor common to

Bernard Chouet; Herbert R. Shaw

1991-01-01

394

Modeling of fractal patterns in matrix acidizing and their impact on well performance

This paper describes a model where wormholes, the primary feature of carbonate acidizing, are considered as fractals. The influences of acid volume, injection rate, fractal dimension, porosity, and the ratio of undamaged to damaged permeabilities on well performance are studied. Exact expressions of post-treatment skin effects are developed for vertical and horizontal wells.

Frick, T.P.; Kuermayr, M.; Economides, M.J.

1994-02-01

395

Distinct Kinetic Pathways Generate Organogel Networks with Contrasting Fractality and Thixotropic of Chemical and Biomolecular Engineering, UniVersity of Maryland, College Park, Maryland 20742-2111, and CEA. Values of Df, the mass fractal dimension of the microcrystalline self-assembled fibrillar networks

Raghavan, Srinivasa

396

A fractal transition in the two dimensional shear layer

NASA Technical Reports Server (NTRS)

The dependence of product generation with the Peclet and Reynolds number in a numerically simulated, reacting, two dimensional, temporally growing mixing layer is used to compute the fractal dimension of passive scalar interfaces. A transition from a low dimension of 4/3 to a higher one of 5/3 is identified and shown to be associated to the kinematic distortion on the flow field during the first pairing interaction. It is suggested that the structures responsible for this transition are non-deterministic, non-random, inhomogeneous fractals. Only the large scales are involved. No further transition is found for Reynolds numbers up to 20,000.

Jimenez, Javier; Martel, Carlos

1990-01-01

397

ERIC Educational Resources Information Center

Reports on a mathematical investigation of fractals and highlights the thinking involved, problem solving strategies used, generalizing skills required, the role of technology, and the role of mathematics. (ASK)

Clark, Garry

1999-01-01

398

NSDL National Science Digital Library

This online activity challenges students to explore the relationship between the number of triangles and the sum of the triangle perimeters in each of the first three iterations of the Sierpinski triangle fractal. The activity is one of 80 mathematical challenges featured on the Figure This! web site. In this activity, students are encouraged to use two problem-solving strategies: investigate a simpler problem and make a chart. For other sections of the activity, students find the general rule for determining the amount of paint needed to cover the increasing number of triangles in iterations of the Sierpinski triangle and investigate similar area and perimeter questions with square fractals. The activity includes information about self-similarity, a key characteristic of fractals, and about how fractals can model natural phenomena. Copyright 2005 Eisenhower National Clearinghouse

National Council of Teachers of Mathematics (NCTM)

2002-01-01

399

Fractal funcitons and multiwavelets

This paper reviews how elements from the theory of fractal functions are employed to construct scaling vectors and multiwavelets. Emphasis is placed on the one-dimensional case, however extensions to IR{sup m} are indicated.

Massopust, P.R.

1997-04-01

400

NSDL National Science Digital Library

This website introduces chaos and describes how it appears in animal populations and weather models. The site also describes fractals and explains the butterfly effect. Images provide representations of chaotic behavior.

2007-07-18

401

Tsallis information dimension of complex networks

NASA Astrophysics Data System (ADS)

The fractal and self-similarity properties are revealed in many complex networks. The information dimension is a useful method to describe the fractal and self-similarity properties of the complex networks. In order to show the influence of different parts in the complex networks to the information dimension, we have proposed a new information dimension based on the Tsallis entropy namely the Tsallis information dimension. The proposed information dimension is changed according to the scale which is described by the nonextensivity parameter q, and it is inverse with the nonextensivity parameter q. The existing information dimension is a special case of the Tsallis information dimension when q = 1. The Tsallis information dimension is a generalized information dimension of the complex networks.

Zhang, Qi; Luo, Chuanhai; Li, Meizhu; Deng, Yong; Mahadevan, Sankaran

2015-02-01

402

NSDL National Science Digital Library

This lesson is designed to introduce students to the idea of finding patterns in the generation of several different types of fractals. This lesson provides links to discussions and activities related to patterns and fractals as well as suggested ways to work them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

2011-01-20

403

Spatial log-periodic oscillations of first-passage observables in fractals.

For transport processes in geometrically restricted domains, the mean first-passage time (MFPT) admits a general scaling dependence on space parameters for diffusion, anomalous diffusion, and diffusion in disordered or fractal media. For transport in self-similar fractal structures, we obtain an expression for the source-target distance dependence of the MFPT that exhibits both the leading power-law behavior, depending on the Hausdorff and spectral dimension of the fractal, as well as small log-periodic oscillations that are a clear and definitive signal of the underlying fractal structure. We also present refined numerical results for the Sierpinski gasket that confirm this oscillatory behavior. PMID:23367911

Akkermans, Eric; Benichou, Olivier; Dunne, Gerald V; Teplyaev, Alexander; Voituriez, Raphael

2012-12-01

404

NASA Astrophysics Data System (ADS)

The space-time development of hadron-nucleus interactions is examined using bubble chamber and downstream particle identifier data from the hybrid spectrometer of Fermilab experiment E597. 5583 events representing 12 interactions are studied with conventional and fractal techniques. Comparisons are made to simulated events from the Lund Monte Carlo FRITIOF 1.6. Multiplicities are studied conventionally. Negative binomial descriptions of produced particle multiplicities are interpreted in terms of clusters and cascading and in terms of partial stimulated emission; forward-backward correlations, in terms of short- and long-range correlations and multiple scattering. Multiplicities are consistent with a multiple collision view of multiparticle production mechanisms and are investigated in terms of the number of collisions nu. Rapidity density fluctuations are studied fractally. The possibility of new dynamics is considered on the basis of event-by-event studies of spike phenomena, intermittency, and fractal dimensions. Results from these exploratory studies are consistent with predictions made for quark-gluon plasma transitions. 131 spike events are analyzed; intermittency is investigated with normalized factorial moments and cumulants; and fractal dimensions and correlations dimensions are calculated. Seagull effects and production region sizes from Bose-Einstein pion interferometry are also considered.

Mattingly, Margarita Claudia Krieghoff

405

Quantum critical behavior of the quantum Ising model on fractal lattices

NASA Astrophysics Data System (ADS)

I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpi?ski carpet, Sierpi?ski gasket, and Sierpi?ski tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpi?ski tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.

Yi, Hangmo

2015-01-01

406

Fractal physiology and the fractional calculus: a perspective.

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 statistical process. Control of physiologic complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. We emphasize the difference between homeostatic and allometric control mechanisms. Homeostatic control has a negative feedback character, which is both local and rapid. Allometric control, on the other hand, is a relatively new concept that takes into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the network variable. We hypothesize that allometric control maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks. Moreover, allometric control can often be described using the fractional calculus to capture the dynamics of complex physiologic networks. PMID:21423355

West, Bruce J

2010-01-01

407

Fractal Analysis of Aerosol Mass-Size Distribution

NASA Astrophysics Data System (ADS)

Fractal geometry has been widely used in description of complicated natural phenomena. The objectives of this study were (i) to apply fractal scaling for aerosols mass-size distribution, (ii) to study one and two-domain fractal analyses of aerosols mass-size distribution and compare these two approaches. A total of 20 of different elements were considered in which Mg, Al, Ca, Si, K, and Fe were the main compositions of atmospheric aerosols over the Mount Yulongxue Region, 15 km north of Lijiang in China, and account for more than 82% of a total of 20 elements. For one-domain fractal analysis, fractal dimension changed from 2.213 for Fe to 2.874 for Zn, and was significantly correlated with the total mass of aerosols of size less than or equal to 0.25 ?m (R2=0.98). The goodness of fit (R2) was in the range of 0.698 for Cu to 0.996 for Ca. Elements such as Mg and Ca showed one-fractal domain completely, whereas other elements indicated more than one fractal domain. For two-domain fractal analysis, D1 and D2 covered the small and large aerosol sizes, respectively. D1 changed from 1.093 for Mn to 2.748 for Zn, and D2 changed from 2.386 for Fe to 2.935 for Zn. The goodness of fit for two-domain fractal analysis was greater than 0.96. For all samples except Ca, D1 was less than D2, and for 16 elements, dc which was the cutoff of the whole domain was between 0 and 1 ?m. Acknowledgement The authors are thankful to Dr. Zhen-xing Shen, Department of Environmental Science and Engineering of Xi'an Jiaotong University, for providing the data set used in this study.

Ghanbarian-Alavijeh, Behzad; Liaghat, Abdolmajid

2010-05-01

408

Silica wet gels were prepared from hydrolysis of tetraethoxysilane (TEOS) with additions of sodium dodecyl sulfate (SDS). The surfactant was removed after gelation. Wet gels exhibited mass-fractal structure with mass-fractal dimension D (typically around 2.25) in a length scale extending from a characteristic size ? (typically about 10 nm) of the mass-fractal domains to a characteristic size a0 (typically between 0.3 and 0.4 nm) of the primary particles building up the fractal domains. ? increased while D and a0 diminished slightly as the SDS quantity increased. Aerogels with typical specific surface of 1000 m(2)/g and density of 0.20 g/cm(3) were obtained by supercritical drying of the wet gels after washing with ethanol and n-hexane. The pore volume and the mean pore size increased with the increase of the SDS quantity. The aerogels presented most of the mass-fractal characteristics of the original wet gels at large length scales and exhibited at a higher resolution level at about 0.7 nm a crossover to a mass-surface fractal structure, with apparent mass-fractal dimension Dm ? 2.4 and surface-fractal dimension Ds ? 2.6, as inferred from small-angle X-ray scattering (SAXS) and nitrogen adsorption data. PMID:25513729

Perissinotto, Amanda P; Awano, Carlos M; Donatti, Dario A; de Vicente, Fabio S; Vollet, Dimas R

2015-01-13

409

Mechanics on fractal bodies. Data compression using fractals

The present paper deals with two interrelated subjects: the definition of a ‘correct’ mechanics on fractal bodies and the data compression method using fractals and its application to the numerical analysis of problems in engineering. The first part uses theoretical results of the theory of Bessov spaces while the second part discusses certain elements of approximation of fractals by iterated

P. D. Panagiotopoulos; O. Panagouli

1997-01-01

410

Fractal Signatures in Analogs of Interplanetary Dust Particles

Interplanetary dust particles (IDPs) are an important constituent of the earth's 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. \\cite{volten2007}, to extract their morphological features. By evaluating the structure factor, we find that the aggregates are mass fractals with a mass fractal dimension $d_{m} \\simeq 1.75$. The same fractal dimension also characterizes clusters obtained from {\\it diffusion limited aggregation} (DLA). This suggests that the analogs are formed by an irreversible aggregation of stochastically-transported silicate particles

Katyal, Nisha; Puri, Sanjay

2014-01-01

411

Fractal structure of equipotential curves on a continuum percolation model

NASA Astrophysics Data System (ADS)

We numerically investigate the electric potential distribution over a two-dimensional continuum percolation model between the electrodes. The model consists of overlapped conductive particles on the background with an infinitesimal conductivity. Using the finite difference method, we solve the generalized Laplace equation and show that in the potential distribution, there appear quasi-equipotential clusters which approximately and locally have the same values as steps and stairs. Since the quasi-equipotential clusters have the fractal structure, we compute the fractal dimension of equipotential curves and its dependence on the volume fraction over [0,1]. The fractal dimension in [1.00, 1.246] has a peak at the percolation threshold pc.

Matsutani, Shigeki; Shimosako, Yoshiyuki; Wang, Yunhong

2012-12-01

412

The fractal structure of the ventral scales in legless reptiles

Surface constructs in snakes reflect desirable design traits for technical surface engineering. Their micro-textural patterns, however, do not lend themselves easily to unified analysis due to species-specific variations in surface geometry and topology. Fractal description is useful in this context since it accentuates the correspondence between patterns especially when responding to tribological phenomena. In this work we examine the surface construction of 14 snake species, representing five families, and evaluate the fractal dimension for each of the skins (both the dorsal and ventral sides) using three different computational algorithms. Our results indicate first that all of the examined species share a common fractal dimension (with a very small variation between species in the order 4-5%). This finding implies that despite the different micro-geometry of texture among species, the skin as a unit responds in a similar manner to many interfacial influences.

Abdel-Aal, Hisham A

2015-01-01

413

Changes in the concentration profiles of ?-carotene caused by diffusion through parenchymatic dried apple tissue were characterized by image and fractal analysis. Apple slices were dried by convection, and then impregnated with an aqueous ?-carotene solution. Scanning electron microscopy images of dried apple slices were captured and the fractal dimension (FD) values of the textures of the images were obtained (FDSEM). It was observed that the microstructure of the foodstuff being impregnated have an important effect on the impregnation phenomenon, generating irregular concentration profiles of ?-carotene, which are numerically described by the fractal dimension FDPROFILES and are related to the diffusion process during impregnation in dried edible tissue. PMID:25694678

Santacruz-Vázquez, Claudia; Santacruz-Vázquez, Verónica

2015-02-01

414

NASA Technical Reports Server (NTRS)

The rapid increase in digital data volumes from new and existing sensors necessitates the need for efficient analytical tools for extracting information. We developed an integrated software package called ICAMS (Image Characterization and Modeling System) to provide specialized spatial analytical functions for interpreting remote sensing data. This paper evaluates the three fractal dimension measurement methods: isarithm, variogram, and triangular prism, along with the spatial autocorrelation measurement methods Moran's I and Geary's C, that have been implemented in ICAMS. A modified triangular prism method was proposed and implemented. Results from analyzing 25 simulated surfaces having known fractal dimensions show that both the isarithm and triangular prism methods can accurately measure a range of fractal surfaces. The triangular prism method is most accurate at estimating the fractal dimension of higher spatial complexity, but it is sensitive to contrast stretching. The variogram method is a comparatively poor estimator for all of the surfaces, particularly those with higher fractal dimensions. Similar to the fractal techniques, the spatial autocorrelation techniques are found to be useful to measure complex images but not images with low dimensionality. These fractal measurement methods can be applied directly to unclassified images and could serve as a tool for change detection and data mining.

Lam, Nina Siu-Ngan; Qiu, Hong-Lie; Quattrochi, Dale A.; Emerson, Charles W.; Arnold, James E. (Technical Monitor)

2001-01-01

415

This paper extends the velocity dependent Krook (VDK) model, developed at NRL over the last 4 years, to two dimensions and presents a variety of calculations. One dimensional spherical calculations presented here investigate shock ignition. Comparing VDK calculations to a flux limit calculation shows that the laser profile has to be retuned and some gain is sacrificed due to preheat of the fuel. However, preheat is by no means a show stopper for laser fusion. The recent foil acceleration experiments at the University of Rochester Laboratory for Laser Energetics are modeled with two-dimensional simulations. The radial loss is very important to consider in modeling the foil acceleration. Once this is done, the VDK model gives the best agreement with the experiment.

Manheimer, Wallace; Colombant, Denis; Schmitt, Andrew J. [Laser Plasma Branch, Plasma Physics Division, Naval Research Laboratory, Washington, DC 20375 (United States)

2012-05-15

416

Fractal analysis of ultrasound images of carotid atherosclerotic plaque

In this paper, we investigate the possibility of using the fractal dimension to characterise carotid atheromatous plaques from B-mode ultrasound images. The images were obtained from ten symptomatic and nine asymptomatic subjects. Symptomatic subjects included patients with previous history of cerebral events, whereas asymptomatic ones had no evidence of any cerebral symptoms prior to the time of the investigation. For

P. Asvestas; S. Golemati; G. K. Matsopoulos; K. S. Nikita

2001-01-01

417

An upper bound on the area occupied by a fractal

Fractal images defined by an iterated function system (IFS) are specified by a finite number of contractive affine transformations. In order to plot the image specified by the transformations on the screen of a digital computer, it is necessary to determine a bounding area for the image. This paper derives a formula that expresses the dimensions of this bounding area

A. Edalat; David W. N. Sharp; R. L. While

1995-01-01

418

Modeling the scattering properties of mineral aerosols using concave fractal polyhedra.

The single-scattering properties of concave fractal polyhedra are investigated, with particle size parameters ranging from the Rayleigh to geometric-optics regimes. Two fractal shape parameters, irregularity and aspect ratio, are used to iteratively construct "generations" of irregular fractal particles. The pseudospectral time-domain (PSTD) method and the improved geometric-optics method (IGOM) are combined to compute the single-scattering properties of fractal particles over the range of size parameters. The effects of fractal generation, irregularity, and aspect ratio on the single-scattering properties of fractals are investigated. The extinction efficiency, absorption efficiency, and asymmetry factor, calculated by the PSTD method for fractal particles, with small-to-moderate size parameters, smoothly bridges the gap between those size parameters and size parameters for which solutions given by the IGOM may be used. Somewhat surprisingly, excellent agreement between values of the phase function of randomly oriented fractal particles calculated by the two numerical methods is found, not only for large particles, but in fact extends as far down in equivalent-projected-area size parameters as 25. The agreement in the case of other nonzero phase matrix elements is not as good at so small a size. Furthermore, the numerical results of ensemble-averaged phase matrix elements of a single fractal realization are compared with dust particle measurements, and good agreement is found by using the fractal particle model to represent data from a study of feldspar aerosols. PMID:23385901

Liu, Chao; Panetta, R Lee; Yang, Ping; Macke, Andreas; Baran, Anthony J

2013-02-01

419

Music critics have compared Bach's music to the precision of mathematics. What "mathematics" and what "precision" are the questions for a curious scientist. The purpose of this short note is to suggest that the mathematics is, at least in part, Mandelbrot's fractal geometry and the precision is the deviation from a log-log linear plot. PMID:11607061

Hsü, K J; Hsü, A J

1990-01-01

420

Fractal Analysis of AFM Images of the Surface of Bowman's Membrane of the Human Cornea.

The objective of this study is to further investigate the ultrastructural details of the surface of Bowman's membrane of the human cornea, using atomic force microscopy (AFM) images. One representative image acquired of Bowman's membrane of a human cornea was investigated. The three-dimensional (3-D) surface of the sample was imaged using AFM in contact mode, while the sample was completely submerged in optisol solution. Height and deflection images were acquired at multiple scan lengths using the MFP-3D AFM system software (Asylum Research, Santa Barbara, CA), based in IGOR Pro (WaveMetrics, Lake Oswego, OR). A novel approach, based on computational algorithms for fractal analysis of surfaces applied for AFM data, was utilized to analyze the surface structure. The surfaces revealed a fractal structure at the nanometer scale. The fractal dimension, D, provided quantitative values that characterize the scale properties of surface geometry. Detailed characterization of the surface topography was obtained using statistical parameters, in accordance with ISO 25178-2: 2012. Results obtained by fractal analysis confirm the relationship between the value of the fractal dimension and the statistical surface roughness parameters. The surface structure of Bowman's membrane of the human cornea is complex. The analyzed AFM images confirm a fractal nature of the surface, which is not taken into account by classical surface statistical parameters. Surface fractal dimension could be useful in ophthalmology to quantify corneal architectural changes associated with different disease states to further our understanding of disease evolution. PMID:25266935

T?lu, Stefan; Stach, Sebastian; Sueiras, Vivian; Ziebarth, Noël Marysa

2014-09-30

421

The influence of Hausdorff dimension on plasmonic antennas with Pascal's triangle geometry

We introduce fractal geometry to the common bowtie antenna and investigate the influence of a key fractal parameter, Hausdorff dimension, on the broadband spectral response of the antenna. Length scaling trends are presented for antennas having various Hausdorff dimensions. We show that antennas with Pascal's triangle geometry accommodate resonances that are red-shifted when compared to a standard bowtie antenna having

S. Sederberg; A. Y. Elezzabi

2011-01-01

422

Technology Transfer Automated Retrieval System (TEKTRAN)

In order to explore the effect of changes in plant communities and land use on soil properties, as a result of anthropogenic disturbances, we apply the theory of fractals and soil physics as a means to better quantify changes in particle-size distribution (PSD) and soil porosity. Fractal dimension a...

423

A Fractal Subgrid-scale Model of Turbulence

NASA Astrophysics Data System (ADS)

We discuss how empirical observations of self-affinity of turbulent fields can be ex- ploited to formulate a fractal closure for the subgrid-scale stresses (Scotti &Mene- veau, Physica D 1999). We discuss applications of this approach in the case of ran- domly forced Burgers equation in one dimension. Initial applications, so far restricted to isotropic turbulence, have highlighted important features that need to be included alongside fractal descriptions. Specifically, successful models need to include dynam- ical phase information and eigenvector alignment trends. Recent field observations in ABL using arrays of sonic anemometers (Higgins et a. 2002) provides new insights into these alignment trends.

Meneveau, C.

424

Growth of a self-assembled monolayer by fractal aggregation

Atomic force microscope images show that self-assembled monolayers of octadecyltrichlorosilane form on mica by nucleating isolated, self-similar domains. With increasing coverage, the fractal dimension of the growing domains evolves from 1.6 to 1.8. At higher coverage, continued growth is limited by adsorption from solution. Monte Carlo simulations that include, for the first time, adsorption as well as surface diffusion qualitatively reproduce both the growth kinetics and evolution of fractal structure, much better than a two-dimensional diffusion-limited-aggregation model.

Schwartz, D.K.; Steinberg, S.; Israelachvili, J.; Zasadzinski, J.A.N. (Department of Chemical and Nuclear Engineering, University of California Santa Barbara, Santa Barbara, California 93106 (United States))

1992-12-07

425

Fractal patterns of insect movement in microlandscape mosaics

How individuals move, whether in short-term searching behavior or long-term dispersal influences the probability that individuals will experience physiological stress or encounter appropriate habitat, potential mates, prey, or predators. Because of variety and complexity, it is often difficult to make sense of movements. Because the fractal dimension of a movement pathway is scale independent, however, it may provide a useful measure for comparing dissimilar taxa. The authors use fractal measures to compare the movement pathways of individual beetles occupying semiarid shortgrass steppe in north-central Colorado. 20 refs., 1 fig., 1 tab.

Wiens, J.A. [Colorado State Univ., Fort Collins, CO (United States); Crist, T.O. [Colorado State Univ., Fort Collins, CO (United States)]|[Miami Univ., Oxford, OH (United States); With, K.A. [Colorado State Univ., Fort Collins, CO (United States)]|[Oak Ridge National Lab., TN (United States); Milne, B.T. [Univ. of New Mexico, Albuquerque, NM (United States)

1995-03-01

426

Fractal analysis of Xylella fastidiosa biofilm formation

NASA Astrophysics Data System (ADS)

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.

Moreau, A. L. D.; Lorite, G. S.; Rodrigues, C. M.; Souza, A. A.; Cotta, M. A.

2009-07-01

427

Fractal geometry for atmospheric correction and canopy simulation

NASA Astrophysics Data System (ADS)

Global climate modeling needs a good parameterization of the vegetative surface. Two of the main important parameters are the leaf area index (LAI) and the fraction of absorbed photosynthetically active radiation (FPAR). In order to derive these values from space and airborne spectral radiance measurements one needs information on the actual atmospheric state as well as good canopy models. First we have developed a retrieval method for the optical depth to perform an atmospheric correction of remote sensing data. The atmospheric influence reduces the global image contrast and acts as a low pass filter. We found that the autocorrelation function [ACF(lambda )(h)] of the image depends on the global image contrast C and on the fractal dimension s. Using multiple regression the spectral optical depth in the visible range can be estimated from C and s with an absolute accuracy of 0.021. This method was applied and tested for a number of rural TM scenes. Atmospheric correction allows us to calculate the canopy reflectance from the image data. The relationships between the canopy reflectance and LAI or FPAR can be determined from canopy radiative transfer modeling. Row and shadowing effects influence the bi-directional reflectance distribution function (BRDF) since the leaves and stems are real 3D objects. In order to use a ray tracer for 3D radiative transfer simulation the canopy should be described by simple shapes (discs, cylinders) and polygones. Lindenmayer systems which are based on the ideas of fractal geometry allow the construction of plants and trees in this way. We have created simple artificial plants and arranged them into rows to study shadowing and row effects and compute the BRDF in various spectral channels.

Tornow, Carmen

1996-06-01

428

Error Assessment in Modeling with Fractal Brownian Motions

NASA Astrophysics Data System (ADS)

To model a given time series F(t) with fractal Brownian motions (fBms), it is necessary to have appropriate error assessment for related quantities. Usually the fractal dimension D is derived from the Hurst exponent H via the relation D = 2-H, and the Hurst exponent can be evaluated by analyzing the dependence of the rescaled range <|F(t + ?) - F(t)|> on the time span ?. For fBms, the error of the rescaled range not only depends on data sampling but also varies with H due to the presence of long term memory. This error for a given time series then can not be assessed without knowing the fractal dimension. We carry out extensive numerical simulations to explore the error of rescaled range of fBms and find that for 0 < H < 0.5, |F(t + ?) - F(t)| can be treated as independent for time spans without overlap; for 0.5 < H < 1, the long term memory makes |F(t + ?) - F(t)| correlated and an approximate method is given to evaluate the error of <|F(t + ?) - F(t)|>. The error and fractal dimension can then be determined self-consistently in the modeling of a time series with fBms.

Qiao, Bingqiang; Liu, Siming

2013-12-01

429

Fractal and geostatistical methods for modeling of a fracture network

The modeling of fracture networks is useful for fluid flow and rock mechanics studies. About 6600 fracture traces were recorded on drifts of a uranium mine in a granite massif. The traces have an extension of 0.20-20 m. The network was studied by fractal and by geostatistical methods but can be considered neither as a fractal with a constant dimension nor a set of purely randomly located fractures. Two kinds of generalization of conventional models can still provide more flexibility for the characterization of the network: (a) a nonscaling fractal model with variable similarity dimension (for a 2-D network of traces, the dimension varying from 2 for the 10-m scale to 1 for the centimeter scale, (b) a parent-daughter model with a regionalized density; the geostatistical study allows a 3-D model to be established where: fractures are assumed to be discs; fractures are grouped in clusters or swarms; and fracturation density is regionalized (with two ranges at about 30 and 300 m). The fractal model is easy to fit and to simulate along a line, but 2-D and 3-D simulations are more difficult. The geostatistical model is more complex, but easy to simulate, even in 3-D.

Chiles, J.P.

1988-08-01

430

In this investigation, five experimental data sets are used to evaluate the ability of the EGSnrc Monte Carlo code to calculate the change in chamber response associated with changes in wall material and cavity dimension at 60Co energies. Calculations of the ratios of response per unit mass of air as a function of cavity volume for walls ranging from polystyrene to lead are generally within 1%-3% of experiments. A few exceptions, which are discussed, include 20%-30% discrepancies with experiments involving lead-walled chambers used by Attix et al. [J. Res. Natl. Bur. Stand. 60, 235-243 (1958)] and Cormack and Johns [Radiat. Res. 1, 133-157 (1954)], and 5% discrepancies for the graphite chamber of Attix et al. (relative to data for other wall materials). Simulations of the experiment by Whyte [Radiat. Res. 6, 371-379 (1957)], which varied cavity air pressure in a large cylindrical chamber, are generally within 0.5% (wall/electrode materials ranging from beryllium to copper). In all cases, the agreement between measurements and EGSnrc calculations is much better when the response as a function of cavity height or air pressure is considered for each wall material individually. High-precision measurements [Burns et al., Phys. Med. Biol. 52, 7125-7135 (2007)] of the response per unit mass as a function of cavity height for a graphite chamber are also accurately reproduced, and validate previous tests of the transport mechanics of EGSnrc. Based on the general agreement found in this work between corresponding experimental results and EGSnrc calculations it can be concluded that EGSnrc can reliably be used to calculate changes in response with changes in various wall materials and cavity dimensions at 60Co energies within a accuracy of a few percent or less. PMID:19175120

La Russa, Daniel J; Rogers, D W O

2008-12-01

431

NASA Astrophysics Data System (ADS)

Microchannel heat sink with high heat transfer coefficients has been extensively investigated due to its wide application prospective in electronic cooling. However, this cooling system requires a separate pump to drive the fluid transfer, which is uneasy to minimize and reduces their reliability and applicability of the whole system. In order to avoid these problems, valveless piezoelectric pump with fractal-like Y-shape branching tubes is proposed. Fractal-like Y-shape branching tube used in microchannel heat sinks is exploited as no-moving-part valve of the valveless piezoelectric pump. In order to obtain flow characteristics of the pump, the relationship between tube structure and flow rate of the pump is studied. Specifically, the flow resistances of fractal-like Y-shape branching tubes and flow rate of the pump are analyzed by using fractal theory. Then, finite element software is employed to simulate the flow field of the tube, and the relationships between pressure drop and flow rate along merging and dividing flows are obtained. Finally, valveless piezoelectric pumps with fractal-like Y-shape branching tubes with different fractal dimensions of diameter distribution are fabricated, and flow rate experiment is conducted. The experimental results show that the flow rate of the pump increases with the rise of fractal dimension of the tube diameter. When fractal dimension is 3, the maximum flow rate of the valveless pump is 29.16 mL/min under 100 V peak to peak (13 Hz) power supply, which reveals the relationship between flow rate and fractal dimensions of tube diameter distribution. This paper investigates the flow characteristics of valveless piezoelectric pump with fractal-like Y-shape branching tubes, which provides certain references for valveless piezoelectric pump with fractal-like Y-shape branching tubes in application on electronic chip cooling.

Huang, Jun; Zhang, Jianhui; Wang, Shouyin; Liu, Weidong

2014-05-01

432

Drag and near wake characteristics of flat plates normal to the flow with fractal edge geometries

NASA Astrophysics Data System (ADS)

Past results have suggested that the drag coefficient and the shedding frequencies of regular polygon plates all fall within a very narrow band of values. In this study, we introduce a variety of length scales into the perimeter of a square plate and study the effects this has on the wake characteristics and overall drag. The perimeter of the plate can be made as long as allowed by practical constraints with as many length scales as desired under these constraints without changing the area of the plate. A total of eight fractal-perimeter plates were developed, split into two families of different fractal dimensions all of which had the same frontal area. It is found that by increasing the number of fractal iterations and thus the perimeter, the drag coefficient increases by up to 7%. For the family of fractal plates with the higher dimension, it is also found that when the perimeter increases above a certain threshold the drag coefficient drops back again. Furthermore, the shedding frequency remains the same but the intensity of the shedding decreases with increasing fractal dimension. The size of the wake also decreases with increasing fractal dimension and has some dependence on iteration without changing the area of the plate.

Nedi?, J.; Ganapathisubramani, B.; Vassilicos, J. C.

2013-12-01

433

This aim of this study was to assess the discriminatory value of fractal and grey level co-occurrence matrix (GLCM) analysis methods in standard microscopy analysis of two histologically similar brain white mass regions that have different nerve fiber orientation. A total of 160 digital micrographs of thionine-stained rat brain white mass were acquired using a Pro-MicroScan DEM-200 instrument. Eighty micrographs from the anterior corpus callosum and eighty from the anterior cingulum areas of the brain were analyzed. The micrographs were evaluated using the National Institutes of Health ImageJ software and its plugins. For each micrograph, seven parameters were calculated: angular second moment, inverse difference moment, GLCM contrast, GLCM correlation, GLCM variance, fractal dimension, and lacunarity. Using the Receiver operating characteristic analysis, the highest discriminatory value was determined for inverse difference moment (IDM) (area under the receiver operating characteristic (ROC) curve equaled 0.925, and for the criterion IDM?0.610 the sensitivity and specificity were 82.5 and 87.5%, respectively). Most of the other parameters also showed good sensitivity and specificity. The results indicate that GLCM and fractal analysis methods, when applied together in brain histology analysis, are highly capable of discriminating white mass structures that have different axonal orientation. PMID:24967845

Pantic, Igor; Dacic, Sanja; Brkic, Predrag; Lavrnja, Irena; Pantic, Senka; Jovanovic, Tomislav; Pekovic, Sanja

2014-10-01

434

Fractals in geology and geophysics

NASA Technical Reports Server (NTRS)

The definition of a fractal distribution is that the number of objects N with a characteristic size greater than r scales with the relation N of about r exp -D. The frequency-size distributions for islands, earthquakes, fragments, ore deposits, and oil fields often satisfy this relation. This application illustrates a fundamental aspect of fractal distributions, scale invariance. The requirement of an object to define a scale in photograhs of many geological features is one indication of the wide applicability of scale invariance to geological problems; scale invariance can lead to fractal clustering. Geophysical spectra can also be related to fractals; these are self-affine fractals rather than self-similar fractals. Examples include the earth's topography and geoid.

Turcotte, Donald L.

1989-01-01

435

Elasticity of Fractal Inspired Interconnects.

The use of fractal-inspired geometric designs in electrical interconnects represents an important approach to simultaneously achieve large stretchability and high aerial coverage of active devices for stretchable electronics. The elastic stiffness of fractal interconnects is determined analytically in this paper. Specifically, the elastic energy and the tensile stiffness for an order n fractal interconnect of arbitrary shape are obtained, and are verified by the finite element analysis and experiments. PMID:25183293

Su, Yewang; Wang, Shuodao; Huang, YongAn; Luan, Haiwen; Dong, Wentao; Fan, Jonathan A; Yang, Qinglin; Rogers, John A; Huang, Yonggang

2014-09-01

436

A fractal analysis of pathogen detection by biosensors

NASA Astrophysics Data System (ADS)

A fractal analysis is presented for the detection of pathogens such as Franscisela tularensis, and Yersinia pestis (the bacterium that causes plague) using a CANARY (cellular analysis and notification of antigens risks and yields) biosensor (Rider et al., 2003). In general, the binding and dissociation rate coefficients may be adequately described by either a single- or a dual-fractal analysis. An attempt is made to relate the binding rate coefficient to the degree of heterogeneity (fractal dimension value) present on the biosensor surface. Binding and dissociation rate coefficient values obtained are presented. The kinetics aspects along with the affinity values presented are of interest, and should along with the rate coefficients presented for the binding and the dissociation phase be of significant interest in help designing better biosensors for an application area that is bound to gain increasing importance in the future.

Doke, Atul M.; Sadana, Ajit

2006-05-01

437

Temperature induced smoothing of initially fractal grain boundaries

Recently the effect of serrated or rugged grain boundaries on the mechanical properties of alloys and the numerical characterization of such a geometrically irregular microstructure by means of the concept of fractal geometry has attracted great attention. It has been reported that the generation of serrated or rugged grain boundaries, e.g. by cold work or heat treatment, is one of the most effective methods to improve the high-temperature strength of alloys, especially the creep rupture properties. In the present paper, for the first time, measurements of the change in the roughness of initially fractal grain boundaries after annealing are presented. The experimental results are discussed on the basis of a coarsening model for self-similar interfaces, which predicts a dependency of the smoothing kinetics of the grain boundaries on their initially fractal dimension.

Streitenberger, P.; Foerster, D.; Kolbe, G.; Veit, P. [Otto-von-Guericke-Univ. Magdeburg (Germany). Inst. fuer Experimentelle Physik] [Otto-von-Guericke-Univ. Magdeburg (Germany). Inst. fuer Experimentelle Physik

1996-01-01

438

and natural fractures were investigated in this study using an X-Ray CT Scanner. Fractal dimension, D, and amplitude parameter, A, of fracture aperture approaches a constant value with increased sampling area, similar to the behavior of fracture roughness...

Kim, Tae Hyung

2009-05-15

439

The bending and torsional degrees of freedom in S{sub 1} acetylene, C{sub 2}H{sub 2}, are subject to strong vibrational resonances and rovibrational interactions, which create complex vibrational polyad structures even at low energy. As the internal energy approaches that of the barrier to cis-trans isomerization, these energy level patterns undergo further large-scale reorganization that cannot be satisfactorily treated by traditional models tied to local minima of the potential energy surface for nuclear motion. Experimental spectra in the region near the cis-trans transition state have revealed these complicated new patterns. In order to understand near-barrier spectroscopic observations and to predict the detailed effects of cis-trans isomerization on the rovibrational energy level structure, we have performed reduced dimension rovibrational variational calculations of the S{sub 1} state. In this paper, we present the methodological details, several of which require special care. Our calculation uses a high accuracy ab initio potential surface and a fully symmetrized extended complete nuclear permutation inversion group theoretical treatment of a multivalued internal coordinate system that is appropriate for large amplitude bending and torsional motions. We also discuss the details of the rovibrational basis functions and their symmetrization, as well as the use of a constrained reduced dimension rovibrational kinetic energy operator.

Changala, P. Bryan, E-mail: bryan.changala@colorado.edu [Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (United States)

2014-01-14

440

Fractals in physiology and medicine

NASA Technical Reports Server (NTRS)

The paper demonstrates how the nonlinear concepts of fractals, as applied in physiology and medicine, can provide an insight into the organization of such complex structures as the tracheobronchial tree and heart, as well as into the dynamics of healthy physiological variability. Particular attention is given to the characteristics of computer-generated fractal lungs and heart and to fractal pathologies in these organs. It is shown that alterations in fractal scaling may underlie a number of pathophysiological disturbances, including sudden cardiac death syndromes.

Goldberger, Ary L.; West, Bruce J.

1987-01-01

441

Investigating Fractal Geometry Using LOGO.

ERIC Educational Resources Information Center

Discusses dimensionality in Euclidean geometry. Presents methods to produce fractals using LOGO. Uses the idea of self-similarity. Included are program listings and suggested extension activities. (MVL)

Thomas, David A.

1989-01-01

442

Fractals in biology and medicine

NASA Technical Reports Server (NTRS)

Our purpose is to describe some recent progress in applying fractal concepts to systems of relevance to biology and medicine. We review several biological systems characterized by fractal geometry, with a particular focus on the long-range power-law correlations found recently in DNA sequences containing noncoding material. Furthermore, we discuss the finding that the exponent alpha quantifying these long-range correlations ("fractal complexity") is smaller for coding than for noncoding sequences. We also discuss the application of fractal scaling analysis to the dynamics of heartbeat regulation, and report the recent finding that the normal heart is characterized by long-range "anticorrelations" which are absent in the diseased heart.

Havlin, S.; Buldyrev, S. V.; Goldberger, A. L.; Mantegna, R. N.; Ossadnik, S. M.; Peng, C. K.; Simons, M.; Stanley, H. E.

1995-01-01

443

Fractal multifiber microchannel plates

NASA Technical Reports Server (NTRS)

The construction and performance of microchannel plates (MCPs) made using fractal tiling mehtods are reviewed. MCPs with 40 mm active areas having near-perfect channel ordering were produced. These plates demonstrated electrical performance characteristics equivalent to conventionally constructed MCPs. These apparently are the first MCPs which have a sufficiently high degree of order to permit single channel addressability. Potential applications for these devices and the prospects for further development are discussed.

Cook, Lee M.; Feller, W. B.; Kenter, Almus T.; Chappell, Jon H.

1992-01-01

444

Diophantine Approximations on Fractals

We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove\\u000a that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure)\\u000a contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0, 1]2, possessing some

Manfred Einsiedler; Lior Fishman; Uri Shapira

2011-01-01

445

Fractal packing and highly irregular shaped particles increase the mechanical properties of rocks and building materials. This suggests that fractal methods are good tools for modeling particle mixes with efficient properties like maximum strength and maximum surface area or minimum porosity and minimum permeability. However gradings and packings are calculated by ``Euclidean'' disk models and sphere models. Surprisingly even the

C. A. Hecht

2000-01-01

446

The Fractal Behavior of Crystal Distribution of la Gloria Pluton, Chile

NASA Astrophysics Data System (ADS)

We utilize fractal analysis to study the spatial distributions of crystals in a 10 Ma granitic intrusion (La Gloria pluton) located in the central Chilean Andes. Previous work determined the crystal size distribution (CSD) and anisotropy of magnetic susceptibility (AMS) tensors throughout this pluton. Using orthogonal thin sections oriented along the AMS tensor axes, we have applied fractal analysis in three magmatic crystal families: plagioclase, ferromagnesian minerals (biotite and amphibole), and Fe-Ti oxides (magnetite with minor ilmenite). We find that plagioclase and ferromagnesian minerals have a Semi-logarithmic CSD (S-CSD), given by: log(n/n0)= -L/C (1) where n [mm-4], n0 [mm-4], L [mm] and C [mm] are crystal density, intercept (nucleation density; L=0), size of crystals (three axes) and characteristic length, respectively. In contrast, Fe-Ti oxides have a Fractal CSD (F-CSD, power law size distribution), given by: log(n)= - Dn log(L) + n1 (2) where Dn and n1 [log(mm-4)] are a non-dimensional proportionality constant and the logarithm of the initial crystallization density (n1 = log(n(L=1 mm))), respectively. Finally, we calculate the fractal dimension (D0) by applying the box-counting method on each crystal thin section image, using: log(N) = -D0 log(?) (3) where N and ? are the number of boxes occupied by minerals and the length of the square box, respectively. Results indicate that D0 values (eq. 3) are well defined for all minerals, and are higher for plagioclase than for ferromagnesian minerals and lowest for Fe-Ti oxides. D0 values are correlated with n0 and -1/C for S-CSD (eq. 1), and with n1 values for F-CSD (eq. 2). These correlations between fractal dimensions with CSD parameters suggest crystal growth follows a fractal behaviour in magmatic systems. Fractal behaviour of CSD means that the spatial distribution of crysta