Characterization and Estimation of the Variations of a Random Convex Set by Its Mean n-Variogram: Application to the Boolean Model

  • Saïd RahmaniEmail author
  • Jean-Charles Pinoli
  • Johan Debayle
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9389)


In this paper we propose a method to characterize and estimate the variations of a random convex set \(\varXi _0\) in terms of shape, size and direction. The mean n-variogram \(\gamma ^{(n)}_{\varXi _0}{:}(u_1\cdots u_n)\mapsto \mathbb {E}[\nu _d(\varXi _0\cap (\varXi _0-u_1)\cdots \cap (\varXi _0-u_n))]\) of a random convex set \(\varXi _0\) on \( \mathbb {R}^d\) reveals information on the \(n^{th}\) order structure of \(\varXi _0\). Especially we will show that considering the mean n-variograms of the dilated random sets \(\varXi _0\oplus rK \) by an homothetic convex family \(rK_{r>0}\), it’s possible to estimate some characteristic of the \(n^{th}\) order structure of \(\varXi _0\). If we make a judicious choice of K, it provides relevant measures of \(\varXi _0\). Fortunately the germ-grain model is stable by convex dilatations, furthermore the mean n-variogram of the primary grain is estimable in several type of stationary germ-grain models by the so called n-points probability function. Here we will only focus on the Boolean model, in the planar case we will show how to estimate the \( n^{th}\) order structure of the random vector composed by the mixed volumes \(^t(A(\varXi _0),W(\varXi _0,K))\) of the primary grain, and we will describe a procedure to do it from a realization of the Boolean model in a bounded window. We will prove that this knowledge for all convex body K is sufficient to fully characterize the so called difference body of the grain \(\varXi _0\oplus \breve{\varXi _0}\). we will be discussing the choice of the element K, by choosing a ball, the mixed volumes coincide with the Minkowski’s functional of \(\varXi _0\) therefore we obtain the moments of the random vector composed of the area and perimeter \(^t(A(\varXi _0),U(\varXi ))\). By choosing a segment oriented by \(\theta \) we obtain estimates for the moments of the random vector composed by the area and the Ferret’s diameter in the direction \(\theta \), \(^t((A(\varXi _0), H_{\varXi _0}(\theta ))\). Finally, we will evaluate the performance of the method on a Boolean model with rectangular grain for the estimation of the second order moments of the random vectors \(^t(A(\varXi _0),U(\varXi _0))\) and \(^t((A(\varXi _0), H_{\varXi _0}(\theta ))\).


Boolean model Geometric covariogram Mixed volumes Particle size distribution n points set probability Random set Shape variations 


  1. 1.
    Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties, vol. 16. Springer, New York (2002)zbMATHGoogle Scholar
  2. 2.
    Jeulin, D.: Random texture models for material structures. Stat. Comput. 10(2), 121–132 (2000)CrossRefGoogle Scholar
  3. 3.
    Galerne, B.: Modèles d’image aléatoires et synthèse de texture. Ph.D. thesis, Ecole normale supérieure de Cachan-ENS Cachan (2010)Google Scholar
  4. 4.
    Peyrega, C.: Prediction des proprietes acoustiques de materiaux fibreux heterogenes a partir de leur microstructure 3D. Ph.D. thesis, École Nationale Supérieure des Mines de Paris (2010)Google Scholar
  5. 5.
    Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and Its Applications. Wiley, Hoboken (2013)CrossRefzbMATHGoogle Scholar
  6. 6.
    Molchanov, I.S.: Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester (1997)zbMATHGoogle Scholar
  7. 7.
    Miles, R.E.: Estimating aggregate and overall characteristics from thick sections by transmission microscopy. J. Microsc. 107(3), 227–233 (1976)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Heinrich, L.: Asymptotic properties of minimum contrast estimators for parameters of Boolean models. Metrika 40(1), 67–94 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Galerne, B.: Computation of the perimeter of measurable sets via their covariogram. Applications to random sets. Image Anal. Stereol. 30(1), 39–51 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Heinrich, L.: Asymptotic properties of minimum contrast estimators for parameters of boolean models. Metrika 40(1), 67–94 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Michielsen, K., De Raedt, H.: Integral-geometry morphological image analysis. Phys. Rep. 347(6), 461–538 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory, vol. 151. Cambridge University Press, Cambridge (2013)CrossRefzbMATHGoogle Scholar
  13. 13.
    Stoyanov, J.: Krein condition. In: Hazewinkel, M. (ed.) Encyclopedia of Mathematics. Springer, Dordrecht (2001). ISBN 978-1-55608-010-4Google Scholar
  14. 14.
    Akhiezer, N., Kemmer, N.: The Classical Moment Problem and Some Related Questions in Analysis, vol. 5. Oliver & Boyd, Edinburgh (1965)Google Scholar
  15. 15.
    Yau, S.-T.: Institute for Advanced Study (Seminar on differential geometry), no. 102. Princeton University Press, Princeton (1982)Google Scholar
  16. 16.
    Rivollier, S., Debayle, J., Pinoli, J-C.: Analyse morphométrique d’images à tons de gris par diagrammes de forme. In: RFIA 2012 (Reconnaissance des Formes et Intelligence Artificielle) (2012). ISBN 978-2-9539515-2-3Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Saïd Rahmani
    • 1
    Email author
  • Jean-Charles Pinoli
    • 1
  • Johan Debayle
    • 1
  1. 1.École Nationale Supérieure des Mines de Saint Etienne, SPIN/LGF UMR CNRS 5307Saint-EtienneFrance

Personalised recommendations