Evaluation de la dimension fractale d'un graphe

Tricot, Claude; Quiniou, J.F.; Wehbi, D.; Roques‐Carmes, C.; Dubuc, B.

doi:10.1051/rphysap:01988002302011100

Cited by 61 publications

(26 citation statements)

References 1 publication

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“…where for the uncorrelated case, given by H = −1, it is known that d cp = 7/4 [66,67,71,72]. In addition to considering the scaling of M cp with L, we also determined the fractal dimension d cp using the yardstick method [82,83]. There, one measures the number of sticks S(m) of size m needed to follow the perimeter from one end to the other.…”

Section: Cluster Perimetersmentioning

confidence: 99%

Percolation with long-range correlated disorder

et al. 2013

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Long-range power-law correlated percolation is investigated using Monte Carlo simulations. We obtain several static and dynamic critical exponents as functions of the Hurst exponent H, which characterizes the degree of spatial correlation among the occupation of sites. In particular, we study the fractal dimension of the largest cluster and the scaling behavior of the second moment of the cluster size distribution, as well as the complete and accessible perimeters of the largest cluster. Concerning the inner structure and transport properties of the largest cluster, we analyze its shortest path, backbone, red sites, and conductivity. Finally, bridge site growth is also considered. We propose expressions for the functional dependence of the critical exponents on H.

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Section: Cluster Perimetersmentioning

confidence: 99%

Percolation with long-range correlated disorder

et al. 2013

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“…More specifically, when the behaviour of the variogram near the origin can be reduced to h j j a (linear in log-log scale), with h designating the distance vector, one finds in the ecological literature that the HB dimension is given by N À a 2 where N corresponds to the topological dimension of the graph of the function plus 1 (N ¼ D topo þ 1). However, following early works on so-called difference statistics (Tricot et al 1988;Dubuc et al 1989) and the geostatistical characterisation of fractal models of surfaces developed by Bruno and Raspa (1989), the correct estimator of D HB should in fact be based on the firstorder variogram also called the madogram, rather than on the traditional (second-order) variogram (see Appendix 1 for details). The madogram of a random function Z(x) is defined by…”

Section: Geostatistical Estimatorsmentioning

confidence: 99%

The duality of fractals: roughness and self-similarity

Bez

Bertrand

2010

Theor Ecol

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The fractal dimension (D HB ) is an interesting metrics because it is supposed to quantify by a single value, scale independence and roughness of ecological objects. However, we show here that those two properties may be quantified by a single dimension only in some specific cases. In general, a non-integer D HB quantifies only the roughness, and self-similarity needs to be evidenced or postulated by other means. Second, we revisit some aspects of the practical estimation of D HB . We recommend the use of madogram instead of variogram for estimations based on geostatistics. We propose a simplification of its estimation for 2D fields and discuss its possible relationship with self-similarity. We finally underline the problem of scale and resolution. Field data recorded during a scientific acoustic survey on the North Sea herring are used for illustrations. The paper concludes on a synthesis of practical recommendations to ecologists when using fractal dimension.

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“…The idea that the fractal dimension of trabecular bone might be related to bone architecture and bone strength is an appealing one, since the fractal dimension can be assessed from clinical images. To avoid the phenomenon of length described in (Tricot et al, 1988;Feder, 1988), the size of the boxes varies by a power of 2. To be consistent with the measure of S 2D , the slope was measured on the fourth first points of each Log-Log plot which corresponds to a spatial resolution between 12 μm and 96 μm.…”

Section: D Self-similarity: the Box Counting Methodsmentioning

confidence: 99%

Estimation of the 3D self-similarity parameter of trabecular bone from its 2D projection

Jennane

Harba

Lemineur

et al. 2007

Medical Image Analysis

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It has been shown that the analysis of two dimensional (2D) bone X-ray images based on the fractional Brownian motion (fBm) model is a good indicator for quantifying alterations in the three dimensional (3D) bone micro-architecture. However, this 2D measurement is not a direct assessment of the 3D bone properties. In this paper, we first show that S 3D , the self-similarity parameter of 3D fBm, is linked to S 2D , that of its 2D projection, by S 3D = S 2D -0.5. In the light of this theoretical result, we have experimentally examined whether this relation holds for trabecular bone. Twenty one specimens of trabecular bone were derived from frozen human femoral heads. They were digitized using a high resolution -CT. Their projections were simulated numerically by summing the data in the three orthogonal directions and both 3D and 2D self-similarity parameters were measured. Results show that the self-similarity of the 3D bone volumes and that of their projections are linked by the previous equation. This demonstrates that a simple projection provides 3D information about the bone structure. This information can be a valuable adjunct to the bone mineral density for the early diagnosis of osteoporosis.

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Evaluation de la dimension fractale d'un graphe

Cited by 61 publications

References 1 publication

Percolation with long-range correlated disorder

Percolation with long-range correlated disorder

The duality of fractals: roughness and self-similarity

Estimation of the 3D self-similarity parameter of trabecular bone from its 2D projection

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