# Showing papers on "Fractal dimension published in 2001"

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27 Sep 2001TL;DR: In this article, the authors present a mathematical model for chaotic multidimensional flows and fractal dimension calculation based on the Lyapunov exponents and the Hamiltonian chaos.

Abstract: Preface 1. Introduction 2. One-dimensional maps 3. Nonchaotic multidimensional flows 4. Dynamical systems theory 5. Lyapunov exponents 6. Strange attractors 7. Bifurcations 8. Hamiltonian chaos 9. Time-series properties 10. Nonlinear prediction and noise reduction 11. Fractals 12. Calculation of fractal dimension 13. Fractal measure and multifractals 14. Nonchaotic fractal sets 15. Spatiotemporal chaos and complexity A. Common chaotic systems B. Useful mathematical formulas C. Journals with chaos and related papers Bibliography Index

1,651 citations

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TL;DR: In this paper, the authors provide guidelines for the accurate and practical estimation of exponents and fractal dimensions of natural fracture systems, including length, displacement and aperture power law exponents.

Abstract: Scaling in fracture systems has become an active field of research in the last 25 years motivated by practical applications in hazardous waste disposal, hy- drocarbon reservoir management, and earthquake haz- ard assessment. Relevant publications are therefore spread widely through the literature. Although it is rec- ognized that some fracture systems are best described by scale-limited laws (lognormal, exponential), it is now recognized that power laws and fractal geometry provide widely applicable descriptive tools for fracture system characterization. A key argument for power law and fractal scaling is the absence of characteristic length scales in the fracture growth process. All power law and fractal characteristics in nature must have upper and lower bounds. This topic has been largely neglected, but recent studies emphasize the importance of layering on all scales in limiting the scaling characteristics of natural fracture systems. The determination of power law expo- nents and fractal dimensions from observations, al- though outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these tech- niques and suggest guidelines for the accurate and ob- jective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after crit- ical appraisal of published studies, to show a wide vari- ation, frequently spanning the theoretically possible range. Extrapolations from one dimension to two and from two dimensions to three are found to be nontrivial, and simple laws must be used with caution. Directions for future research include improved techniques for gathering data sets over great scale ranges and more rigorous application of existing analysis methods. More data are needed on joints and veins to illuminate the differences between different fracture modes. The phys- ical causes of power law scaling and variation in expo- nents and fractal dimensions are still poorly understood.

982 citations

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TL;DR: In this article, the authors present a review of scattering and absorption of light by fractal aggregates, which are typically diffusion limited cluster aggregates with fractal dimensions of D.

Abstract: This paper presents a review of scattering and absorption of light by fractal aggregates. The aggregates are typically diffusion limited cluster aggregates (DLCA) with fractal dimensions of D

921 citations

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TL;DR: This study demonstrates that a careful selection of fractal dimension algorithm is required for specific applications, and the most common methods of estimating the fractaldimension of biomedical signals directly in the time domain are analyzed and compared.

Abstract: The fractal dimension of a waveform represents a powerful tool for transient detection. In particular, in analysis of electroencephalograms and electrocardiograms, this feature has been used to identify and distinguish specific states of physiologic function. A variety of algorithms are available for the computation of fractal dimension. In this study, the most common methods of estimating the fractal dimension of biomedical signals directly in the time domain (considering the time series as a geometric object) are analyzed and compared. The analysis is performed over both synthetic data and intracranial electroencephalogram data recorded during presurgical evaluation of individuals with epileptic seizures. The advantages and drawbacks of each technique are highlighted. The effects of window size, number of overlapping points, and signal-to-noise ratio are evaluated for each method. This study demonstrates that a careful selection of fractal dimension algorithm is required for specific applications.

413 citations

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TL;DR: In this paper, the authors introduce simple stochastic models which allow for any combination of fractal dimension and Hurst exponent, and synthesize images from these models, with arbitrary fractal properties and power-law correlations.

Abstract: Fractal behavior and long-range dependence have been observed in an astonishing number of physical systems. Either phenomenon has been modeled by self-similar random functions, thereby implying a linear relationship between fractal dimension, a measure of roughness, and Hurst coefficient, a measure of long-memory dependence. This letter introduces simple stochastic models which allow for any combination of fractal dimension and Hurst exponent. We synthesize images from these models, with arbitrary fractal properties and power-law correlations, and propose a test for self-similarity.

243 citations

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TL;DR: In this article, the influence of surface roughness on the adhesion of elastic solids was studied and a partial detachment transition preceding a full detachment transition was found in the case of a self-affine fractal.

Abstract: We study the influence of surface roughness on the adhesion of elastic solids. Most real surfaces have roughness on many different length scales, and this fact is taken into account in our analysis. We consider in detail the case when the surface roughness can be described as a self-affine fractal, and show that when the fractal dimension Df>2.5, the adhesion force may vanish, or be at least strongly reduced. We consider the block-substrate pull-off force as a function of roughness, and find a partial detachment transition preceding a full detachment one. The theory is in good qualitative agreement with experimental data.

225 citations

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TL;DR: In this article, the effect of scale on the surface roughness of rock joints was investigated using a 3-D laser scanner with high accuracy and resolution on a silicon rubber replica of a natural rock joint surface.

Abstract: Accurate determination of surface roughness of rock joints at the large-scale is essential for proper rock mass characterization. Surface roughness of rock joints is commonly characterized using small samples. However, since roughness parameters of rock joints are scale-dependent and their descriptors change with scale, a systematic investigation has been carried out to understand the effect of scale on the surface roughness of rock joints. A silicon rubber replica, 1000 mm×1000 mm in size, was moulded in-situ from a natural rock joint surface and its surface was digitized in the laboratory using a 3-D laser scanner having high accuracy and resolution. The fractal parameters, i.e. the fractal dimension D and amplitude parameter A describing surface roughness of the replica, were calculated on the basis of the Roughness–Length Method. To investigate the scale-dependency of surface roughness of rock joints, ten sampling windows ranging in size from 100 mm×100 mm to 1000 mm×1000 mm were selected from the central part of the replica and their fractal parameters were calculated. The results show that both D and A are scale-dependent and their values decrease with increasing size of the sampling windows. This scale-dependency is limited to a certain size, defined as the stationarity threshold, and for sampling windows larger than the stationarity threshold, the estimated parameters remain almost constant. It is concluded that, for surface roughness to be accurately characterized on a laboratory scale or in the field, samples need to be equal to or larger than the stationarity limit. In this paper we have indicated the methodology for establishing the value for the stationarity limit for rock joints.

182 citations

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TL;DR: Practical methods that are now available to allow the determination of the complexity and scaling relationships of anatomical and physiological patterns, including fractal dimensionality are reviewed.

Abstract: Fractal analysis has already found widespread application in the field of neuroscience and is being used in many other areas. Applications are many and include ion channel kinetics of biological membranes and classification of neurons according to their branching characteristics. In this article we review some practical methods that are now available to allow the determination of the complexity and scaling relationships of anatomical and physiological patterns. The problems of describing fractal dimensions are discussed and the concept of fractal dimensionality is introduced. Several related methodological considerations, such as preparation of the image and estimation of the fractal dimensions from the data points, as well as the advantages and problems of fractal geometric analysis, are discussed.

174 citations

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TL;DR: In this paper, the porosity and permeability by scale dissolution and precipitation in porous media is described based on fractal attributes of the pores, realization of flow channels as a bundle of uniformly distributed mean-size cylindrical and tortuous hydraulic flow tubes, a permeability-porosity relationship conforming to Civan's power law flow units equation, and the pore surface scale precipitation and dissolution kinetics.

Abstract: Variation of porosity and permeability by scale dissolution and precipitation in porous media is described based on fractal attributes of the pores, realization of flow channels as a bundle of uniformly distributed mean-size cylindrical and tortuous hydraulic flow tubes, a permeability-porosity relationship conforming to Civan's power law flow units equation, and the pore surface scale precipitation and dissolution kinetics. Practical analytical solutions, considering the conditions of typical laboratory core tests and relating the lumped and phenomenological parameters, were derived and verified by experimental data. Deviations of the empirically determined exponents of the pore-to-matrix volume ratio compared to the Kozeny-Carman equation were due to the relative fractal dimensions of pore attributes of random porous media. The formulations provide useful insights into the mechanism of porosity and permeability variation by surface processes and accurate representation of the effect of scale on porosity and permeability by simpler lumped-parameter models.

162 citations

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TL;DR: In this article, the authors applied multifractal techniques to characterize contrasting PSDs and to identify multifracfal parameters potentially useful for classification and prediction, which indicated that multifractals are promising descriptors of PSDs.

Abstract: A particle-size distribution (PSD) constitutes a fundamental soil property correlated to many other soil properties. Accurate representations of PSDs are, therefore, needed for soil characterization and prediction purposes. A power-law dependence of particle mass on particle diameter has been used to model soil PSDs, and such power-law dependence has been interpreted as being the result of a fractal distribution of particle sizes characterized with a single fractal dimension. However, recent studies have shown that a single fractal dimension is not sufficient to characterize a distribution for the entire range of particle sizes. The objective of this study was to apply multifractal techniques to characterize contrasting PSDs and to identify multifracfal parameters potentially useful for classification and prediction. The multifractal spectra of 30 PSDs covering a wide range of soil textural classes were analyzed. Parameters calculated from each multifractal spectrum were: (i) the Hausdorff dimension, f(α); (ii) the singularities of strength, α; (iii) the generalized fractal dimension, D q ; and (iv) their conjugate parameter the mass exponent, τ (q), calculated in the range of moment orders (q) of between -10 and -10 taken at 0.5 lag increments. Multifractal scaling was evident by an increase in the difference between the capacity D 0 and the entropy D 1 dimensions for soils with more than 10% clay content, Soils with <10% clay content exhibited single scaling. Our results indicate that multifractal parameters are promising descriptors of PSDs. Differences in scaling properties of PSDs should be considered in future studies.

157 citations

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TL;DR: In this paper, the surface fractal dimension of the pore-solid interface was measured by fitting two straight lines to the log-log plot and finding a crossover point at a scale of about 14 μm, forming the border between textural and structural fractality.

Abstract: There is general interest in quantifying soil structure in order to obtain physically based parameters relevant to transport processes. To measure the surface fractal dimension of the pore–solid interface we use approaches known from fractal geometry. The characteristics of this interface, expressed by its fractal dimension, are descriptors of the heterogeneity and complexity of soil structure. Samples of the Bt horizon of a Luvisol in loess were taken near Gottingen, Germany. To prepare thin sections, the material was dehydrated and embedded in resin. We obtained digital images at different magnifications from a field emission scanning electron microscope. Automatic image analysis was used to determine the corresponding surface fractal dimension by using the box counting and dilation methods, respectively. As the fractal dimension of a line ( D L ) within a plain has been measured, the surface fractal dimension D S is obtained by D S = D L +1 assuming isotropy. We strongly focussed the calculation of the fractal dimension from the measured data files. The decision as to which data should be included between the lower and upper cutoffs is of fundamental significance to the final result. For the upper cutoff, we followed the convention that the scale range should not exceed 30% of the characteristic length (object or image size). Data derived from outside both cutoffs reflect structural properties, either of pixels (lower cutoff) or of structuring elements (upper cutoff). Different methods were used to derive a mean surface fractal dimension for one magnification for (i) single images and (ii) each measurement step. Within the same range of scale, differences between the two methods (box counting and dilation) were smaller than the standard deviation of D S . In contrast to our expectations for a mathematical fractal, we found decreasing values for D S with increasing magnification. The values drift from D S =2.91 for a resolution of 2.44 μm/pixel to D S =2.58 for a resolution of 0.05 μm/pixel. By fitting two straight lines to the log–log plot, we found a crossover-point at a scale of about 14 μm, forming the border between textural and structural fractality. In addition, we will discuss further results obtained as well as possible sources of error.

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TL;DR: The fractal structure of star formation on large scales in disk galaxies is studied using the size distribution function of stellar aggregates in kiloparsec-scale star fields as mentioned in this paper.

Abstract: The fractal structure of star formation on large scales in disk galaxies is studied using the size distribution function of stellar aggregates in kiloparsec-scale star fields. Archival Hubble Space Telescope images of 10 galaxies are Gaussian-smoothed to define the aggregates, and a count of these aggregates versus smoothing scale gives the fractal dimension. Fractal and Poisson models confirm the procedure. The fractal dimension of star formation in all of the galaxies is ~2.3. This is the same as the fractal dimension of interstellar gas in the Milky Way and nearby galaxies, suggesting that star formation is a passive tracer of gas structure defined by self-gravity and turbulence. Dense clusters such as the Pleiades form at the bottom of the hierarchy of structures, where the protostellar gas is densest. If most stars form in such clusters, then the fractal arises from the spatial distribution of their positions, giving dispersed star fields from continuous cluster disruption. Dense clusters should have an upper mass limit that increases with pressure, from ~103 M⊙ in regions like the solar neighborhood to ~106 M⊙ in starbursts.

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TL;DR: In this article, small-angle neutron scattering and dynamic and static light scattering measurements were used to probe the structures of aqueous and organic-solvent-based magnetic fluids comprising dispersed magnetite nanoparticles (∼10 nm in diameter) stabilized against flocculation by adsorbed alkanoic acid layers.

Abstract: Small-angle neutron scattering and dynamic and static light scattering measurements were used to probe the structures of aqueous and organic-solvent-based magnetic fluids comprising dispersed magnetite nanoparticles (∼10 nm in diameter) stabilized against flocculation by adsorbed alkanoic acid layers. A core−shell model fitted to a set of neutron scattering spectra obtained from contrast variation experiments allowed the determination of the iron oxide core size and size distribution, the thicknesses of the surfactant shells, and the spatial arrangement of the individual particles. The magnetic colloidal particles appear to form compact fractal clusters with a fractal dimension of 2.52 and a correlation length of ∼350 A in aqueous magnetic fluids, consistent with the structures of clusters observed directly using cryo-TEM (transmission electron microscopy), whereas chainlike clusters with a fractal dimension of 1.22 and a correlation length of ∼400 A were found for organic-solvent-based magnetic fluids. T...

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TL;DR: In this paper, a framework for the mechanics of solids, deformable over fractal subsets, is outlined, and an extension of the Gauss-Green theorem to fractional operators is proposed to demonstrate the duality principle for fractal media.

Abstract: The framework for the mechanics of solids, deformable over fractal subsets, is outlined. While displacements and total energy maintain their canonical physical dimensions, renormalization group theory permits to define anomalous mechanical quantities with fractal dimensions, i.e., the fractal stress [ σ * ] and the fractal strain [ e * ]. A fundamental relation among the dimensions of these quantities and the Hausdorff dimension of the deformable subset is obtained. New mathematical operators are introduced to handle these quantities. In particular, classical fractional calculus fails to this purpose, whereas the recently proposed local fractional operators appear particularly suitable. The static and kinematic equations for fractal bodies are obtained, and the duality principle is shown to hold. Finally, an extension of the Gauss–Green theorem to fractional operators is proposed, which permits to demonstrate the Principle of Virtual Work for fractal media.

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TL;DR: A lower bound of the box size is found and the reason for having it is provided and indicates the need for limiting the box sizes within certain bounds.

Abstract: Fractal geometry has gradually established its importance in the study of image characteristics. There are many techniques to estimate the dimensions of fractal surfaces. A famous technique to calculate fractal dimension is the grid dimension method popularly known as box-counting method. In this paper, we have found out a lower bound of the box size and provided the reason for having it. The study indicates the need for limiting the box sizes within certain bounds.

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TL;DR: In this paper, the authors used two-dimensional variogram surfaces to derive directionally dependent estimates of fractal dimension and found that fractal dimensions were greater in the downstream direction than in other directions suggesting that the effects of water working are to alter the level of surface organisation by increasing surface irregularity and hence roughness.

Abstract: This paper is concerned with the application of fractal analysis to understand the structure of water-worked gravel-bed river surfaces. High resolution digital elevation models, acquired using digital photogrammetric methods, allowed the application of two-dimensional fractal methods. Previous gravel-bed river studies have been based upon sampled profiles and hence one-dimensional fractal characterisation. After basic testing that bed elevation increments are Gaussian, the paper uses two-dimensional variogram surfaces to derive directionally dependent estimates of fractal dimension. The results identify mixed fractal behavior with two characteristic fractal bands, one associated with the subgrain scale and one associated with the grain scale. The subgrain scale characteristics were isotropic and sensitive to decisions made during the data collection process. Thus, it was difficult to differentiate whether these characteristics were real facets of the surfaces studied. The second band was anisotropic and not sensitive to data collection issues. Fractal dimensions were greater in the downstream direction than in other directions suggesting that the effects of water working are to alter the level of surface organisation, by increasing surface irregularity and hence roughness. This is an important observation as it means that water-worked surfaces may have a distinct anisotropic signal, revealed when using a fractal type analysis.

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TL;DR: The presence of multifractility is confirmed, the legitimacy of the defined dimension is affirmed in the sense of the theoretical Hausdorff limit in as much as this limit can be reached with experimental data.

Abstract: A definition for the fractal dimension of a vascular tree is proposed based on the hemodynamic function of the tree and in terms of two key branching parameters: the asymmetry ratio of arterial bifurcations and the power law exponent governing the relation between vessel diameter and flow. Data from the cardiovascular system, which generally exhibit considerable scatter in the values of these two parameters, are found to produce the same degree of scatter in the value of the fractal dimension. When this scatter is explored for a multifractal pattern, however, it is found that the required collapse onto a single curve is achieved in terms of the coarse Holder exponent. Thus, the presence of multifractility is confirmed, and the legitimacy of the defined dimension is affirmed in the sense of the theoretical Hausdorff limit in as much as this limit can be reached with experimental data.

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15 Sep 2001TL;DR: In this article, the power law behavior of scattered light intensity as a function of scattering wave vector is observed in all cases and is suggestive of fractal structure, and the fractal dimensions obtained fall within the expected range of 1.8 to 2.3 observed for colloidal aggregates.

Abstract: Suspensions of a variety of different aluminum oxides have previously been shown to require very high concentrations of chloride and nitrate anions (>0.5 M) to induce rapid aggregation. This high stability has been accredited to the presence of surface forces considered to be due to the formation of highly charged Al13 polymeric species at slightly acidic pH's and aluminum oxyhydroxide gel formation under alkaline conditions. The effect of this stability on the structure of the resulting aggregates is investigated here using well-established static light-scattering techniques. Power law behavior of scattered light intensity as a function of scattering wave vector is observed in all cases and is suggestive of fractal structure. The fractal dimensions obtained fall within the expected range of 1.8 to 2.3 observed for colloidal aggregates but do not appear to follow the typical observations for colloids destabilized by indifferent electrolytes where lower fractal dimensions are associated with rapid (diffusion-limited) aggregation and higher fractal dimensions with slower (reaction-limited) aggregation. Indeed, relatively constant fractal dimensions (2.10 to 2.25) are observed over the range of salt concentrations at which the slow to rapid aggregation rate transformation occurs with, if anything, a slightly higher fractal dimension observed for higher aggregation rates. The presence of specifically binding sulfate anions appears to negate the strong near-distance repulsive forces leading to rapid aggregation at low (1 to 2 mM) sulfate concentrations. Significantly lower fractal dimensions (1.85 to 1.91) are observed for aggregates formed by destabilization using sulfate ions than obtained when chloride or nitrate are used with, again, an apparent slight increase in fractal dimension upon increasing aggregation rate.

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TL;DR: In this paper, the structures of cluster-cluster and particlecluster fractal-like aggregates were investigated and the morphological properties of aggregates undergoing partial sintering and restructuring were also investigated.

Abstract: The structures of both cluster–cluster and particle–cluster fractal-like aggregates were investigated in the present study. Statistically significant populations of numerically simulated aggregates having appropriate fractal properties and prescribed number of primary particles per aggregate were generated in order to characterize three-dimensional morphological properties of aggregates, such as fractal dimension, fractal pre-factor, coordination number distribution function, and distribution of angles between triplets. Effects of aggregation mechanisms (i.e., cluster–cluster or particle–cluster) and aggregate size were taken into consideration. In addition, the morphological properties of aggregates undergoing partial sintering and restructuring were also investigated. To fulfill these objectives, aggregates were initially built without considering sintering or restructuring effects. Partial sintering of primary particles was then considered by introducing a penetration coefficient that allows touching particles to approach each other. Restructuring of aggregates was modeled during the process of building the cluster–cluster aggregates. For each pair of clusters that were attached together due to the normal aggregation procedure, a further mechanism was included that allowed the cluster to collapse until a more compact and stable position was achieved. The population studied was composed of ca. 450 simulated aggregates having a number of primary particles per aggregate between 8 and 1024. Calculations were performed for aggregates having a penetration coefficient in the range of 0–0.25 with and without restructuring. The following properties were investigated: fractal dimension, fractal pre-factor, coordination number distribution function, angle between triplets, and aggregate radius of gyration.

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TL;DR: In this paper, the authors estimate Levy-stable (fractal) distributions that can accurately account for skewness, kurtosis, and fat tails of the returns.

Abstract: It is argued that the study of the correct specification of returns distributions has attractive implications in financial economics. This study estimates Levy-stable (fractal) distributions that can accurately account for skewness, kurtosis, and fat tails. The Levy-stable family distributions are parametrized by the Levy index (α), 0 < (α), ≤ 2, and include the normal distribution as a special case (α = 2). The Levy index, α, is the fractal dimension of the probability space. The unique feature of Levy-stable family distributions is the existence of a relationship between the fractal dimension of the probability space andthe fractal dimensionof the time series. This relationshipis simply expressed in terms of Hurst exponent (H), i.e. α = 1/ H. In addition, Hurst exponent is related to long-memory effects. Thus, estimating the Levy index allows us to determine long-memory effects. The immediate practical implication of the present work is that on the one hand we estimate the shape of returns distributions...

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TL;DR: The fractal signatures of small regions of interest (32x32 pixels), computed in the frequency domain after corrections for imaging system noise and MTF, were able to characterize the texture of vertebral trabecular bone in CT images.

Abstract: 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 for synthesized fractal images. For natural texture images fractality is limited to a range of scales, and the fractal dimension as a function of spatial frequency presents as a fractal signature. We found that the fractal signature was more successful at discriminating between these textures than either the global fractal dimension or other metrics such as the mean width and root-mean-square width of the spectral density plots. Different natural textures were also readily distinguishable using lacunarity plots, which explicitly characterize the average size and spatial organization of structural sub-units within an image. The fractal signatures of small regions of interest (32x32 pixels), computed in the frequency domain after corrections for imaging system noise and MTF, were able to characterize the texture of vertebral trabecular bone in CT images. Even small differences in texture due to acquisition slice thickness resulted in measurably different fractal signatures. These differences were also readily apparent in lacunarity plots, which indicated that a slice thickness of 1 mm or less is necessary if essential architectural information is not to be lost. Since lacunarity measures gap size and is not predicated on fractality, it may be particularly useful for characterizing the texture of trabecular bone.

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La Roche College

^{1}TL;DR: This study compared three algorithms on the same series of bone biopsies, to obtain the Kolmogorov, Minkowski–Bouligand, and mass‐radius fractal dimensions.

Abstract: Trabecular bone has been reported as having two-dimensional (2-D) fractal characteristics at the histological level, a finding correlated with biomechanical properties However, several fractal dimensions (D) are known and computational ways to obtain them vary considerably This study compared three algorithms on the same series of bone biopsies, to obtain the Kolmogorov, Minkowski-Bouligand, and mass-radius fractal dimensions The relationships with histomorphometric descriptors of the 2-D trabecular architecture were investigated Bone biopsies were obtained from 148 osteoporotic male patients Bone volume (BV/TV), trabecular characteristics (TbN, TbSp, TbTh), strut analysis, star volumes (marrow spaces and trabeculae), inter-connectivity index, and Euler-Poincare number were computed The box-counting method was used to obtain the Kolmogorov dimension (D(k)), the dilatation method for the Minkowski-Bouligand dimension (D(MB)), and the sandbox for the mass-radius dimension (D(MR)) and lacunarity (L) Logarithmic relationships were observed between BV/TV and the fractal dimensions The best correlation was obtained with D(MR) and the lowest with D(MB) Lacunarity was correlated with descriptors of the marrow cavities (ICI, star volume, TbSp) Linear relationships were observed among the three fractal techniques which appeared highly correlated A cluster analysis of all histomorphometric parameters provided a tree with three groups of descriptors: for trabeculae (TbTh, strut); for marrow cavities (Euler, ICI, TbSp, star volume, L); and for the complexity of the network (TbN and the three D's) A sole fractal dimension cannot be used instead of the classic 2-D descriptors of architecture; D rather reflects the complexity of branching trabeculae Computation time is also an important determinant when choosing one of these methods

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TL;DR: In this paper, a simple constitutive law is derived for determining the fractal dimension of an aggregate, resulting from a coagulation event between aggregates with different fractal dimensions.

Abstract: Fractal aggregate coagulation is described within a general framework of multivariate population dynamics. The effect of aggregate morphology on the coagulation rate, is taken into account explicitly, introducing in addition to aggregate particle size, the aggregate fractal dimension, as a second independent variable. A simple constitutive law is derived for determining the fractal dimension of an aggregate, resulting from a coagulation event between aggregates with different fractal dimensions. An efficient Monte Carlo method was implemented to solve the resulting bivariate Brownian coagulation equation, in the limits of continuum and free molecular flow regimes. The results indicate that as the population mean fractal dimension goes from its initial value towards its asymptotic value, the distribution of fractal dimension remains narrow for both flow regimes. The evolution of the mean aggregate size in the continuum regime is found to be nearly independent of aggregate morphology. In the free molecular regime however, the effects of aggregate morphology, as embodied in its fractal dimension, become more important. In this case the evolution of the aggregate size distribution cannot be described by the traditional approach, that employs a constant fractal dimension.

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TL;DR: In this article, a new pattern recognition algorithm based on multiple intensity clips was developed, which assures an optimal adaptation to the solar structure under study, and is demonstrated by application to the intensity structure of solar granulation near the disk center, both speckle reconstructed and not.

Abstract: We have developed a new pattern-recognition algorithm based on multiple intensity clips which assures an optimal adaptation to the solar structure under study. The shapes found at higher clip levels are repeatedly extended to lower levels, thus filling more and more of the observed intensity contours. Additionally, at each intensity threshold new shapes, exceeding the level, are integrated. The number and height of the levels can be optimized making this `multiple level tracking' algorithm (MLT) superior to commonly used Fourier-based recognition techniques (FBR). The capability of MLT is demonstrated by application to the intensity structure of solar granulation near the disk center, both speckle reconstructed and not. Comparisons with Doppler maps prove its reliability. The granular pattern recognized by MLT differs essentially from that obtained with FBR. Elongated `snake-like' granules do not occur with MLT and, consequently, the perimeter-area distribution exhibits only a marginal `second branch' of higher fractal dimension, which dramatically diminishes the better the MLT pattern matches the granular structure. The final distribution obtained with optimized parameters has a single fractal dimension near 1.1, making the question of a `critical size', a `second branch', and the often discussed dimension of 4/3; highly questionable. This result is equally obtained from application of MLT to the corresponding Doppler velocity map of granular up-flows. In contrast, the pattern of down-flows contains some elongated `snake-like' structures with higher fractal dimension. We also use the new algorithm to recognize speckle-reconstructed limb faculae, which MLT separates from their granular surroundings, and compare both, granules and faculae, using large statistical samples. The facular grains near cosθ=57° exhibit a slightly different ellipticity than the (geometrically foreshortened) adjacent granules. However, small facular grains are more round than small granules and larger grains are more similar to granules.

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Rice University

^{1}TL;DR: In this paper, different collision frequency kernel calculation schemes were evaluated to simulate experimentally determined latex particle aggregation kinetics, and the intermediate collision scheme proposed by Veerepanneni and Wiesner (J. Colloid Interface Sci. 177, 45-57 (1996)) is in agreement with experimental aggregation kinetic if considered along with experimentally derived fractal dimension evolution.

Abstract: Aggregate structure, i.e., fractal dimension, has a coupled geometric and hydrodynamic impact on the aggregate collision frequencies. In this work, we evaluated different collision frequency kernel calculation schemes to simulate experimentally determined latex particle aggregation kinetics. Light scattering was used to simultaneously measure aggregate size and fractal dimension throughout the latex aggregation experiments. The intermediate collision scheme proposed by Veerepanneni and Wiesner ( J. Colloid Interface Sci. 177, 45–57 (1996)) is in agreement with experimental aggregation kinetics if considered along with experimentally determined fractal dimension evolution. Aggregate restructuring was found to be an important mechanism in explaining aggregation kinetics.

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TL;DR: In this article, a stereoscopic scanning electron microscopic (SEM) method based on surface areas tallied over a much wider range of measurement scales was used to evaluate the fractal characteristics of fracture surfaces of cement pastes and mortars.

Abstract: A stereoscopic scanning electron microscopic (SEM) method, based on surface areas tallied over a much wider range of measurement scales than has been used in the past, was used to evaluate the fractal characteristics of fracture surfaces of cement pastes and mortars. Fracture surfaces of cement pastes exhibit two distinct fractal regimes: a regime of low fractal dimension (ca. 2.02) at low magnification scales and a significantly higher fractal dimension (ca. 2.12) at higher magnifications. The crossover occurs at a scale dimension slightly greater than 1 μm 2 . Neither water/binder (w/b) ratio nor the presence of silica fume appears to influence these fractal characteristics. Cement paste areas exposed on mortar fracture surfaces are identical with those of cement paste specimens. In contrast to the paste areas, areas of crushed sand exposed on mortar fracture surfaces display a single fractal domain of significantly higher fractal dimension, around 2.20.

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TL;DR: In this paper, different fractal models of porous solid are described and compared, and fractal dimension values calculated from the different models are consistent between them, which can be used as stone pore system descriptor and, in addition, allow the differentiation between weathered and unweathered stones.

Abstract: Different fractal models of porous solid are described and compared. Fractal dimension, following the different methodologies proposed to evaluate fractal dimensions, of different unweathered stones and building stones were measured using mercury porosimetry in a pore diameter range between 100 and 0.0065 μm. In all cases, fractal behaviour expands more than 1.5 orders of magnitude. In general, fractal dimension values calculated from the different models are consistent between them. Fractal dimension can be used as stone pore system descriptor and, in addition, allows the differentiation between weathered and unweathered stones.

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01 Dec 2001TL;DR: A modified approach to estimating fractal dimensions that is less sensitive to lighting conditions and provides information about the orientation of an image under consideration is proposed.

Abstract: Facial feature extraction is an important step in many applications such as human face recognition, video conferencing, surveillance systems, human computer interfacing etc. The eye is the most important facial feature. A reliable and fast method for locating the eye pairs in an image is vital to many practical applications. A new method for locating eye pairs based on valley field detection and measurement of fractal dimensions is proposed. Possible eye candidates in an image with a complex background are identified by valley field detection. The eye candidates are then grouped to form eye pairs if their local properties for eyes are satisfied. Two eyes are matched if they have similar roughness and orientation as represented by fractal dimensions. A modified approach to estimating fractal dimensions that is less sensitive to lighting conditions and provides information about the orientation of an image under consideration is proposed. Possible eye pairs are further verified by comparing the fractal dimensions of the eye-pair window and the corresponding face region with the respective means of the fractal dimensions of the eye-pair windows and the face regions. The means of the fractal dimensions are obtained based on a number of facial images in a database. Experiments have shown that this approach is fast and reliable.

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TL;DR: Water-sorption measurements confirm that interface smoothing is due predominantly to the water condensing in the most strongly curved asperities rather than covering the surface with a wetting film of uniform thickness, where a Porod behavior typical of smooth interfaces is observed instead.

Abstract: Small-angle x-ray and neutron scattering are used to characterize the surface roughness and porosity of a natural rock which are described over three decades in length scales and over nine decades in scattered intensities by a surface fractal dimension D=2.68{+-}0.03 . When this porous medium is exposed to a vapor of a contrast-matched water, neutron scattering reveals that surface roughness disappears at small scales, where a Porod behavior typical of smooth interfaces is observed instead. Water-sorption measurements confirm that such interface smoothing is due predominantly to the water condensing in the most strongly curved asperities rather than covering the surface with a wetting film of uniform thickness.