How fast can the chord length distribution decay?

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

doi:10.1239/aap/1308662490

Cited by 6 publications

(2 citation statements)

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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, [54][55][56][57] or cellular systems, like Voronoi tessellations. 58,59 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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“…Much literature in EVT has been dedicated to the determination and estimation of the threshold above which the observations can be modeled by a GPD, via standard POT methods (mean excess plot method, Hill estimator, QQ-estimator; see e.g. [13,16,17,18,19,20]). The difficulty faced when applying these methods is that they are graphical ad hoc approaches.…”

Section: Introductionmentioning

confidence: 99%

A new unsupervised threshold determination for hybrid models

Debbabi¹,

Kratz

2014

2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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A Gauss-GPD hybrid model that links a Gaussian distribution to a Generalized Pareto Distribution (GPD) is considered for asymmetric heavy tailed data. The paper proposes a new unsupervised iterative algorithm to find successively the junction point between the two distributions and to estimate the hybrid model parameters. Simulation results show that this method provides a reliable position for the junction point, as well as an accurate estimation of the GPD parameters, which improves results when compared with other methods. Another advantage of this approach is that it can be adapted to any hybrid model.

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2013

Stochastic Geometry and Its Applications

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How fast can the chord length distribution decay?

Cited by 6 publications

References 19 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

A new unsupervised threshold determination for hybrid models

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