How fast can the chord length distribution decay?

Demichel, Yann; Estrade, Anne; Kratz, Marie; Samorodnitsky, Gennady

doi:10.1017/s0001867800004961

Cited by 4 publications

(3 citation statements)

References 12 publications

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“…Further analytic calculations of intercept lengths, possibly aided by results from queuing theory, could include additional explicit formulas or improved approximations for other models of porous media, like Gaussian random fields, or cellular systems, like Voronoi tessellations . Going beyond the Poisson process and overlapping grains could help to deepen our understanding about the information content of the MIL analysis.…”

Section: Discussionmentioning

confidence: 99%

Mean-intercept anisotropy analysis of porous media. I. Analytic formulae for anisotropic Boolean models

2017

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Purpose: Structure-property relations, which relate the shape of the microstructure to physical properties such as transport or mechanical properties, need sensitive measures of structure. What are suitable fabric tensors to quantify the shape of anisotropic heterogeneous materials? The mean intercept length is among the most commonly used characteristics of anisotropy in porous media, for example, of trabecular bone in medical physics. Methods:We analyze the orientation-biased Boolean model, a versatile stochastic model that represents microstructures as overlapping grains with an orientation bias towards a preferred direction. This model is an extension of the isotropic Boolean model, which has been shown to truthfully reproduce multi-functional properties of isotropic porous media. We explain the close relationship between the concept of intersections with test lines to the elaborate mathematical theory of queues, and how explicit results from the latter can be directly applied to characterize microstructures.Results: In this series of two papers, we provide analytic formulas for the anisotropic Boolean model and demonstrate often overlooked conceptual shortcomings of this approach. Queuing theory is used to derive simple and illustrative formulas for the mean intercept length. It separates into an intensity-dependent and an orientation-dependent factor. The global average of the mean intercept length can be expressed by local characteristics of a single grain alone. Conclusions:We thus identify which shape information about the random process the mean intercept length contains. The connection between global and local quantities helps to interpret observations and provides insights to the possibilities and limitations of the analysis. In the second paper of this series, we discuss, based on the findings in this paper, severe short-comings of the mean intercept analysis for (bone-)microstructure characterization. We will suggest alternative and better defined sensitive anisotropy measures from integral geometry.Keywords: shape analysis, fabric tensors, mean intercept length, mean chord length, anisotropic porous media This article is protected by copyright. All rights reserved. Accepted ArticleFabric tensors characterize the complex microstructure in both natural and man-made materials 1 , for example, in geology 2 , granular matter 3 , foams 4 , rough surfaces 5 , solids with cracks 6 , and trabecular bone 7-9 . The aim is to gain physical insight via a better understanding of the geometrical properties, for example, relating the mechanical properties of the material to its microstructure 2,10,11 . A common example from medical physics is the prediction of the mechanical stability or transport and migration properties of trabecular (or cancellous) bone by analyzing its complex structure 12-14 ; see Fig. 1. Scalar measures, which are rotation-and translation-invariant quantities, like the volume fraction of the material or the area of the interface as the simplest example, may accurately describe isotr...

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Section: Discussionmentioning

confidence: 99%

Mean-intercept anisotropy analysis of porous media. I. Analytic formulae for anisotropic Boolean models

2017

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“…During the final budget year, G. Samorodnitsky worked on the analysis of stochastic systems, concentrating on the effects of long range dependence and heavy tails. A surprising onclusion from the work "How Fast Can the Chord-Length Distribution Decay" with Demichel, Estrade and Kratz [4] is that in the threshold model for the bi-phasic media the effect of memory on the size of the phases is minimal. On the other hand, long memory has a major effect on the length of long strange segments and on the ruin probabilities for moving average processes, as was shown in a joint work with Ghosh, "Long strange segments, ruin probabilities and the effect of memory on moving average processes" [5].…”

Section: Names Of Personnel Receiving Masters Degreesmentioning

confidence: 99%

Probabilistic and Statistical Modeling of Complex Systems Exhibiting Long Range Dependence and Heavy Tails

Resnick¹,

Samorodnitsky²

2010

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“…Vamvakos and Anantharam [VA98] showed that the long-range dependence of a point process is preserved by a leaky bucket flow control mechanism for data traffic. A study focused on the long-range dependence of multidimensional random sets is the recent work of Demichel, Estrade, Kratz, and Samorodnitsky [DEKS11], who studied whether random sets having power-law decaying chord length distributions, closely related to the covariance function of the random set, can be generated as a level set of a Gaussian random field-they found that in wide generality (merely assuming that the underlying Gaussian field is mixing), this is not possible.…”

Section: Is the Power-law Covariance Decay Of The Proposed Boolean Momentioning

confidence: 99%

Hard-Core Thinnings of Germ‒Grain Models with Power-Law Grain Sizes

Kuronen

Leskelä

2013

Adv. Appl. Probab.

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Random sets with long-range dependence can be generated using a Boolean model with power-law grain sizes. We study thinnings of such Boolean models which have the hard-core property that no grains overlap in the resulting germ-grain model. A fundamental question is whether long-range dependence is preserved under such thinnings. To answer this question we study four natural thinnings of a Poisson germ-grain model where the grains are spheres with a regularly varying size distribution. We show that a thinning which favors large grains preserves the slow correlation decay of the original model, whereas a thinning which favors small grains does not. Our most interesting finding concerns the case where only disjoint grains are retained, which corresponds to the well-known Matérn type I thinning. In the resulting germ-grain model, typical grains have exponentially small sizes, but rather surprisingly, the long-range dependence property is still present. As a byproduct, we obtain new mechanisms for generating homogeneous and isotropic random point configurations having a power-law correlation decay.

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How fast can the chord length distribution decay?

Cited by 4 publications

References 12 publications

Mean-intercept anisotropy analysis of porous media. I. Analytic formulae for anisotropic Boolean models

Mean-intercept anisotropy analysis of porous media. I. Analytic formulae for anisotropic Boolean models

Probabilistic and Statistical Modeling of Complex Systems Exhibiting Long Range Dependence and Heavy Tails

Hard-Core Thinnings of Germ‒Grain Models with Power-Law Grain Sizes

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