Showing papers on "Fractal dimension published in 2007"
TL;DR: In this article, a total of 21 planar fractal grids pertaining to three different fractal families have been used in two different wind tunnels to generate turbulence The resulting turbulent flows have been studied using hot wire anemometry.
Abstract: A total of 21 planar fractal grids pertaining to three different fractal families have been used in two different wind tunnels to generate turbulence The resulting turbulent flows have been studied using hot wire anemometry Irrespective of fractal family, the fractal-generated turbulent flows and their homogeneity, isotropy, and decay properties are strongly dependent on the fractal dimension Df≤2 of the grid, its effective mesh size Meff (which we introduce and define) and its ratio tr of largest to smallest bar thicknesses, tr=tmax∕tmin With relatively small blockage ratios, as low as σ=25%, the fractal grids generate turbulent flows with higher turbulence intensities and Reynolds numbers than can be achieved with higher blockage ratio classical grids in similar wind tunnels and wind speeds U The scalings and decay of the turbulence intensity u′∕U in the x direction along the tunnel’s center line are as follows (in terms of the normalized pressure drop CΔP and with similar results for v′∕U and w′∕U)
304 citations
TL;DR: It is argued 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.
Abstract: 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).
298 citations
TL;DR: The capacity dimension, information dimension and correlation dimension are applied to classify various fault types and evaluate various fault conditions of rolling element bearing, and the classification performance of each fractal dimension and their combinations are evaluated by using SVMs.
248 citations
TL;DR: In this paper, the authors compare the weathering advance rate of basalt (wDβ), where D and β indicate the scale at which the rate is determined and surface area measured, respectively, from the laboratory to the watershed scales.
229 citations
TL;DR: A study of four methods to compute the fractal dimension of the contours of breast masses, including the ruler method and the box counting method applied to 1D and 2D representations of thecontours, which observed to complement the other shape factors.
Abstract: Fractal analysis has been shown to be useful in image processing for characterizing shape and gray-scale complexity. Breast masses present shape and gray-scale characteristics that vary between benign masses and malignant tumors in mammograms. Limited studies have been conducted on the application of fractal analysis specifically for classifying breast masses based on shape. The fractal dimension of the contour of a mass may be computed either directly from the 2-dimensional (2D) contour or from a 1-dimensional (1D) signature derived from the contour. We present a study of four methods to compute the fractal dimension of the contours of breast masses, including the ruler method and the box counting method applied to 1D and 2D representations of the contours. The methods were applied to a data set of 111 contours of breast masses. Receiver operating characteristics (ROC) analysis was performed to assess and compare the performance of fractal dimension and four previously developed shape factors in the classification of breast masses as benign or malignant. Fractal dimension was observed to complement the other shape factors, in particular fractional concavity, in the representation of the complexity of the contours. The combination of fractal dimension with fractional concavity yielded the highest area (Az) under the ROC curve of 0.93; the two measures, on their own, resulted in Az values of 0.89 and 0.88, respectively.
177 citations
TL;DR: In this paper, a differential mobility analyzer (DMA) and a Couette CPMA were used to measure the effective density and fractal dimension of particles emitted from a light-duty diesel vehicle fitted with a diesel oxidation catalyst (DOC), which was found that at high engine loads, the DOC increased in temperature, sulphate levels in the particulate matter increased, and a transient nucleation mode was observed.
177 citations
TL;DR: In this article, a two-scale continuum model is used to simulate reactive dissolution of carbonate rocks in radial flow and the three main types of patterns observed in linear and radial flow experiments, namely, compact, wormhole and uniform patterns are numerically simulated.
174 citations
TL;DR: In this article, the authors report on the filtration behavior, scanning electron microscopy (SEM) and gas permeability of multi-wall carbon mats (buckypapers).
159 citations
26 Dec 2007
TL;DR: Local fractal features that are evaluated densely are developed and it is shown that the local fractal dimension is invariant to local bi-Lipschitz transformations whereas its extension is able to correctly distinguish between fundamental texture primitives.
Abstract: We address the problem of developing discriminative, yet invariant, features for texture classification. Texture variations due to changes in scale are amongst the hardest to handle. One of the most successful methods of dealing with such variations is based on choosing interest points and selecting their characteristic scales [Lazebnik et al. PAMI 2005]. However, selecting a characteristic scale can be unstable for many textures. Furthermore, the reliance on an interest point detector and the inability to evaluate features densely can be serious limitations. Fractals present a mathematically well founded alternative to dealing with the problem of scale. However, they have not become popular as texture features due to their lack of discriminative power. This is primarily because: (a) fractal based classification methods have avoided statistical characterisations of textures (which is essential for accurate analysis) by using global features; and (b) fractal dimension features are unable to distinguish between key texture primitives such as edges, corners and uniform regions. In this paper, we overcome these drawbacks and develop local fractal features that are evaluated densely. The features are robust as they do not depend on choosing interest points or characteristic scales. Furthermore, it is shown that the local fractal dimension is invariant to local bi-Lipschitz transformations whereas its extension is able to correctly distinguish between fundamental texture primitives. Textures are characterised statistically by modelling the full joint PDF of these features. This allows us to develop a texture classification framework which is discriminative, robust and achieves state-of-the-art performance as compared to affine invariant and fractal based methods.
156 citations
TL;DR: In this article, Lutz et al. showed that packing dimension can also be characterized in terms of gales, which are betting strategies that generalize martingales, and showed that the effective strong dimension of a set or sequence is at least as great as its effective dimension, with equality for sets or sequences that are sufficiently regular.
Abstract: The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff [Math. Ann., 79 (1919), pp. 157-179], and packing dimension, developed independently by Tricot [Math. Proc. Cambridge Philos. Soc., 91 (1982), pp. 57-74] and Sullivan [Acta Math., 153 (1984), pp. 259-277]. Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz [Proceedings of the 15th IEEE Conference on Computational Complexity, Florence, Italy, 2000, IEEE Computer Society Press, Piscataway, NJ, 2000, pp. 158-169] has recently proven a simple characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension, including constructive, computable, polynomial-space, polynomial-time, and finite-state dimensions. Work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual of—and every bit as simple as—the gale characterization of Hausdorff dimension. Effectivizing our gale characterization of packing dimension produces a variety of effective strong dimensions, which are exact duals of the effective dimensions mentioned above. In general (and in analogy with the classical fractal dimensions), the effective strong dimension of a set or sequence is at least as great as its effective dimension, with equality for sets or sequences that are sufficiently regular. We develop the basic properties of effective strong dimensions and prove a number of results relating them to fundamental aspects of randomness, Kolmogorov complexity, prediction, Boolean circuit-size complexity, polynomial-time degrees, and data compression. Aside from the above characterization of packing dimension, our two main theorems are the following. 1. If $\vec{\beta} = (\beta_0,\beta_1,\ldots)$ is a computable sequence of biases that are bounded away from 0 and $R$ is random with respect to $\vec{\beta}$, then the dimension and strong dimension of $R$ are the lower and upper average entropies, respectively, of $\vec{\beta}$. 2. For each pair of $\Delta^0_2$-computable real numbers $0 < \alpha \le \beta \le 1$, there exists $A \in {\rm E}$ such that the polynomial-time many-one degree of $A$ has dimension $\alpha$ in E and strong dimension $\beta$ in E. Our proofs of these theorems use a new large deviation theorem for self-information with respect to a bias sequence $\vec{\beta}$ that need not be convergent.
141 citations
TL;DR: In this article, the authors used electron tomography with a transmission electron microscope (TEM) to determine three-dimensional (3D) properties such as fractal dimension (D f ), radius of gyration (R g ), volume (V), surface area (As), and structural coefficient (k a ) for individual soot particles from the ambient air of an Asian dust (AD) episode and from a U.S. traffic source.
Abstract: [1] The morphologies of soot particles are both complex and important. They influence soot atmospheric lifetimes, global distributions, and climate impacts. Particles can have complex geometries with overlapping projecting parts and pores that are difficult to infer from the conventional techniques used to study them. We used electron tomography with a transmission electron microscope (TEM) to determine three-dimensional (3D) properties such as fractal dimension (D f ), radius of gyration (R g ), volume (V), surface area (As), and structural coefficient (k a ) for individual soot particles from the ambient air of an Asian dust (AD) episode and from a U.S. traffic source. The respective median values of D f are 2.4 and 2.2, of R g are 274 and 251 nm, ofA s /Vare 9.2 and 13.7 x 10 7 m -1 , and of k a are 0.67 and 0.71. The corresponding parameters, when calculated from 2D projections such as TEM images, are considerably less precise and commonly erroneous. Unlike other methods that have been used to derive fractal parameters, our method is applicable to particles of any D f . Using the 3D data, we estimate that mass-normalized scattering cross sections of our AD and traffic soot particles are respectively about 15 and 30 times greater than those of unaggregated spheres, which is the shape assumed in global models to estimate radiative forcing. Accurate 3D information can be used to compute more precise optical properties, which are important for estimating direct radiative forcing and improving our understanding of the climate impact of soot.
TL;DR: In this article, a fractal model for resistance of flow through porous media is developed based on the fractal characters of porous media and on the pore-throat model for capillary.
TL;DR: In this article, a new numerical modeling technique called renormalized numerical simulation (RNS) is introduced to address this fundamental difficulty and avoid the need to resolve all the small-scale branches of the fractal.
TL;DR: In this paper, the capacity dimension of fractal flocs has been investigated in terms of a population balance equation that implements a novel description of floc geometry, and a comparison of experimental floc size distributions from settling column test with computed distributions for two hypotheses of capacity dimension (i.e., constant and variable) and two hypotheses for flocculation reactions (e.g., semi-stochastic and deterministic) are shown.
TL;DR: In this article, a continuous optical monitoring and a microscope with CCD camera was used to monitor aggregation under six different shear intensities and the results showed that the flocculation index (FI) decreased from 1.16 at 20-rpm to 0.25 at 250-rpm and the floc size decreased from 550μm to 150μm, meanwhile, the FI value showed a good correlation with the size.
TL;DR: In this article, an effective method for detecting cracks in plate structures based on fractal dimension analysis is presented, applied to the simulated fundamental vibration mode of a simply supported rectangular plate containing a crack parallel to one of its edges of arbitrary length, depth and location.
TL;DR: In this article, the authors analyzed the statistical properties of pre-failure damage based on acoustic emission events (AE) recorded during the catastrophic fracture of typical rock samples under differential compression.
TL;DR: In this article, the authors review methods for determining the fractal dimensions of earthquake epicentres and hypocentres, paying special attention to the problem of errors, biases and systematic effects.
Abstract: SUMMARY We review methods for determining the fractal dimensions of earthquake epicentres and hypocentres, paying special attention to the problem of errors, biases and systematic effects. Among effects considered are earthquake location errors, boundary effects, inhomogeneity of depth distribution and temporal dependence. In particular, the correlation dimension of earthquake spatial distribution is discussed, techniques for its evaluation presented, and results for several earthquake catalogues are analysed. We show that practically any value for the correlation dimension can be obtained if many errors and inhomogeneities in observational data as well as deficiencies in data processing are not properly considered. It is likely that such technical difficulties are intensified when one attempts to evaluate multifractal measures of dimension. Taking into account possible errors and biases, we conclude that the fractal dimension for shallow seismicity asymptotically approaches 2.20 ± 0.05 for a catalogue time span of decades and perhaps centuries. The value of the correlation dimension declines to 1.8‐1.9 for intermediate events (depth interval 71‐280 km) and to 1.5‐1.6 for deeper ones.
TL;DR: It is shown that the contact angle can vary with the drop volume according to a power law, and this was validated against values published in the literature.
TL;DR: In this article, a set of fractal and non-fractal soot particles (mobility diameters d m = 135 to 310 nm) were produced at three controlled fuel equivalence ratios (φ = 2.1, 3.5, and 4.5) by an ethylene/oxygen flame.
Abstract: Mobility-selected fractal and non-fractal soot particles (mobility diameters d m = 135 to 310 nm) were produced at three controlled fuel equivalence ratios (φ = 2.1, 3.5, and 4.5) by an ethylene/oxygen flame. Oleic acid (liquid) and anthracene (solid) coatings were alternately applied to the particles and removed. Simultaneous measurements with an Aerodyne aerosol mass spectrometer and a scanning mobility particle sizer yielded the particle mass, volume, density, composition, dynamic shape factor, fractal dimension, surface area, and the size and number of the primary spherules forming the fractal aggregate. For a given φ, the diameters of the primary spherules are approximately the same, independent of d m (15 nm, 35 nm, and 55 nm for φ = 2.1, 3.5, and 4.5, respectively). As the coating thickness on a particle increases, the dynamic shape factor decreases but d m remains constant until the particle reaches a spherical (for oleic acid) or non-fractal but irregular (for anthracene) shape. Under some condit...
TL;DR: In this paper, the authors reviewed and investigated the fractal scaling and self-similar connectivity behavior of scale-free networks in diverse aspects, and showed that the skeleton is a non-causal tree, either critical or supercritical.
Abstract: Fractal scaling and self-similar connectivity behaviour of scale-free (SF) networks are reviewed and investigated in diverse aspects. We first recall an algorithm of box-covering that is useful and easy to implement in SF networks, the so-called random sequential box-covering. Next, to understand the origin of the fractal scaling, fractal networks are viewed as comprising of a skeleton and shortcuts. The skeleton, embedded underneath the original network, is a spanning tree specifically based on the edge-betweenness centrality or load. We show that the skeleton is a non-causal tree, either critical or supercritical. We also study the fractal scaling property of the k-core of a fractal network and find that as k increases, not only does the fractal dimension of the k-core change but also eventually the fractality no longer holds for large enough k. Finally, we study the self-similarity, manifested as the scale-invariance of the degree distribution under coarse-graining of vertices by the box-covering method. We obtain the condition for self-similarity, which turns out to be independent of the fractality, and find that some non-fractal networks are self-similar. Therefore, fractality and self-similarity are disparate notions in SF networks.
01 Jun 2007
TL;DR: In this article, the usefulness of the fractal dimension for characterizing the spatial structure of the built-up surfaces within the periurban fringe of Brussels has been discussed empirically and theoretically.
Abstract: This paper aims at showing the usefulness of the fractal dimension for characterizing the spatial structure of the built-up surfaces within the periurban fringe. We first discuss our methodology and expectations in terms of operationality of the fractal dimension theoretically and geometrically. An empirical analysis is then performed on the southern periphery of Brussels (Brabant Wallon). The empirical analysis is divided into two parts: first, the effect of the size and shape of the windows on the fractal measures is empirically evaluated; this leads to a methodological discussion about the importance of the scale of analysis as well as the real sense of fractality. Second, we show empirically how far fractal dimension and density can look alike, but are also totally different. The relationship between density and fractality of built-up areas is discussed empirically and theoretically. Results are interpreted in an urban sprawl context as well as in a polycentric development of the peripheries. These analyses confirm the usefulness but also the limits of the fractal approach in order to describe the built-up morphology. Fractal analysis is a promising tool for describing the morphology of the city and forsimulating its genesis and planning.
TL;DR: A new description of three essential classes of fractal dimensions based on: (1) morphology, (2) entropy, and (3) transforms, all unified through the generalized-entropy-based RA©nyi fractal dimension spectrum is presented.
Abstract: Many scientific papers treat the diversity of fractal dimensions as mere variations on either the same theme or a single definition. There is a need for a unified approach to fractal dimensions for there are fundamental differences between their definitions. This paper presents a new description of three essential classes of fractal dimensions based on: (1) morphology, (2) entropy, and (3) transforms, all unified through the generalized-entropy-based RA©nyi fractal dimension spectrum. It discusses practical algorithms for computing 15 different fractal dimensions representing the classes. Although the individual dimensions have already been described in the literature, the unified approach presented in this paper is unique in terms of (1) its progressive development of the fractal dimension concept, (2) similarity in the definitions and expressions, (3) analysis of the relation between the dimensions, and (4) their taxonomy. As a result, a number of new observations have been made, and new applications discovered. Of particular interest are behavioral processes (such as dishabituation), irreversible and birth-death growth phenomena (e.g., diffusion-limited aggregates (DLAs), dielectric discharges, and cellular automata), as well as dynamical non-stationary transient processes (such as speech and transients in radio transmitters), multi-fractal optimization of image compression using learned vector quantization with Kohonen’s self-organizing feature maps (SOFMs), and multi-fractal-based signal denoising.
TL;DR: Fractal-based integrals were formulated to calculate such morphological parameters of vascular and bronchial trees as fluid conduit systems, including the individual branch flow rate, mean flow velocity, wall shear rate and stress, internal pressure, and circumferential tension.
Abstract: The branching systems in our body (vascular and bronchial trees) and those in the environment (plant trees and river systems) are characterized by a fractal nature: the self-similarity in the bifur...
TL;DR: In this paper, a power law function is proposed to describe changes in suspended floc fractal dimension during floc growth and is used to analyze its impact on floc structural parameters, settling velocity, and kinematics of aggregation and breakup.
Abstract: While the fractal dimension of suspended flocs of cohesive sediment is known to vary with the shear rate, electrochemical properties of the sediment and environment, geometrical restructuring, and presence of organic matter, experimental data presented in this work suggest changes in fractal dimension also during floc genesis at constant sedimentological and hydraulic conditions. A power law function is proposed to describe these changes in floc fractal dimension during floc growth and is used to analyze its impact on floc structural parameters, settling velocity, and kinematics of aggregation and breakup. An analysis of this model for the fractal dimension highlights changes of approximately a factor of 2 or more in floc porosity and aggregation and breakup frequencies and of approximately 1 order of magnitude in floc excess density and settling velocity compared to values estimated with constant fractal dimension. The results from this model compare well with prior experimental data collected in situ (Khelifa and Hill, 2006; Manning and Dyer, 1999).
TL;DR: A synthetic fractal fracture network with a known D value was used to develop a new algorithm for the box-counting method that returns improved estimates of D that imply a greater degree of fracture connectivity and thus increased propensity for fracture flow and the transport of miscible or immiscible chemicals.
Abstract: given by N / rD , where N is the number of boxes containing one or more fractures and r is the box size, then the network is considered to be fractal. However, researchers are divided in their opinion about which is the best box-counting algorithm to use, or whether fracture networks are indeed fractals. A synthetic fractal fracture network with a known D value was used to develop a new algorithm for the box-counting method that returns improved estimates of D. The method is based on identifying the lower limit of fractal behavior (rcutoff) using the condition ds/dr ! 0, where s is the standard deviation from a linear regression equation fitted to log(N) versus log(r) with data for r < rcutoff sequentially excluded. A set of 7 nested fracture maps from the Hornelen Basin, Norway was used to test the improved method and demonstrate its accuracy for natural patterns. We also reanalyzed a suite of 17 fracture trace maps that had previously been evaluated for their fractal nature. The improved estimates of D for these maps ranged from 1.56 ± 0.02 to 1.79 ± 0.02, and were much greater than the original estimates. These higher D values imply a greater degree of fracture connectivity and thus increased propensity for fracture flow and the transport of miscible or immiscible chemicals.
TL;DR: It is found that such box-split allowance in the algorithm is a crucial ingredient necessary to obtain the fractal scaling for fractal networks; however, it is inessential for regular lattice and conventional fractal objects embedded in the Euclidean space.
Abstract: A random sequential box-covering algorithm recently introduced to measure the fractal dimension in scale-free (SF) networks is investigated. The algorithm contains Monte Carlo sequential steps of choosing the position of the center of each box; thereby, vertices in preassigned boxes can divide subsequent boxes into more than one piece, but divided boxes are counted once. We find that such box-split allowance in the algorithm is a crucial ingredient necessary to obtain the fractal scaling for fractal networks; however, it is inessential for regular lattice and conventional fractal objects embedded in the Euclidean space. Next, the algorithm is viewed from the cluster-growing perspective that boxes are allowed to overlap; thereby, vertices can belong to more than one box. The number of distinct boxes a vertex belongs to is, then, distributed in a heterogeneous manner for SF fractal networks, while it is of Poisson-type for the conventional fractal objects.
TL;DR: In this paper, the authors investigated the influence of the rough joint surface configuration on stress wave propagation in jointed rocks by means of SHPB technique and fractal geometry method and found that the rougher the joint surface was, the more permanent deformation occurred and the more attenuation stress wave took place.
22 Oct 2007
TL;DR: A computerized approach enhances and segments the retinal vasculature in digital fundus images with an accuracy of 94% in comparison to the gold standard of manual tracing through the measurement of fractal dimension.
Abstract: Complexity of the retinal vascular network is quantified through the measurement of fractal dimension. A computerized approach enhances and segments the retinal vasculature in digital fundus images with an accuracy of 94% in comparison to the gold standard of manual tracing. Fractal analysis was performed on skeletonized versions of the network in 40 images from a study of stroke. Mean fractal dimension was found to be 1.398 (with standard deviation 0.024) from 20 images of the hypertensives sub-group and 1.408 (with standard deviation 0.025) from 18 images of the non-hypertensives subgroup. No evidence of a significant difference in the results was found for this sample size. However, statistical analysis showed that to detect a significant difference at the level seen in the data would require a larger sample size of 88 per group.
TL;DR: In this paper, the self-preserving distribution for soot aggregate size evolves as the aerosol passes through the transition regime from free molecule to diffusion limited collision dynamics, experimentally studied over the range of fractal dimension from 1.9 to 2.5.