Weighted Poisson Cells as Models for Random Convex Polytopes

  • Felix Ballani
  • Karl Gerald van den Boogaart


We introduce a parametric family for random convex polytopes in ℝ d which allows for an easy generation of samples for further use, e.g., as random particles in materials modelling and simulation. The basic idea consists in weighting the Poisson cell, which is the typical cell of the stationary and isotropic Poisson hyperplane tessellation, by suitable geometric characteristics. Since this approach results in an exponential family, parameters can be efficiently estimated by maximum likelihood. This work has been motivated by the desire for a flexible model for random convex particles as can be found in many composite materials such as concrete or refractory castables.


Random polygon Random polyhedron Poisson cell Crofton cell Exponential family  Gibbs distribution 

AMS 2000 Subject Classifications

60D05 62B05 60G55 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Baumstark V, Last G (2009) Gamma distributions for stationary Poisson flat processes. Adv Appl Prob (SGSA) 41:911–939CrossRefzbMATHMathSciNetGoogle Scholar
  2. Bickel PJ, Doksum KA (2007) Mathematical statistics: basic ideas and selected topics, 2nd edn, vol I. Prentice HallGoogle Scholar
  3. Calka P (2001) Mosaïques poissoniennes de l’espace euclidien. Une extension d’un résultat de R. E. Miles. CR Acad Sci I-Math 332:557–562zbMATHMathSciNetGoogle Scholar
  4. Calka P (2010) Tessellations. In: Kendall W, Molchanov I (eds) New perspectives in stochastic geometry. Oxford University PressGoogle Scholar
  5. Coster M, Chermant JL (2002) On a way to material models for ceramics. J Europ Cer Soc 35:1191–1203CrossRefGoogle Scholar
  6. De Pellegrin DV, Stachowiak GW (2005) Simulation of three-dimensional abrasive particles. Wear 258:208–216CrossRefGoogle Scholar
  7. Fleischer F, Gloaguen C, Schmidt V, Voss F (2009) Simulation of the typical Poisson–Voronoi–Cox–Voronoi cell. J Stat Comput Simul 79(7):939–957CrossRefzbMATHMathSciNetGoogle Scholar
  8. Gilks WR, Richardson S, Spiegelhalter DJ (1996) Introducing Markov chain Monte Carlo. In: Gilks WR, Richardson S, Spiegelhalter DJ (eds) Markov chain Monte Carlo in practice. Chapman & Hall, LondonCrossRefGoogle Scholar
  9. Hawkins AE (1993) The shape of powder-particle outlines. Research Studies Press, Taunton, Somerset, England and J. Wiley and Sons, New York, ChichesterGoogle Scholar
  10. He H, Guo Z, Stroeven P, Stroeven M, Sluys LJ (2010) Strategy on simulation of arbitrary-shaped cement grains in concrete. Image Anal Stereol 29:79–84CrossRefGoogle Scholar
  11. Hilhorst HJ (2009) Heuristic theory for many-faced d-dimensional Poisson–Voronoi cells. J Stat Mech P08003:1–14Google Scholar
  12. Hilhorst HJ, Calka P (2008) Random line tessellations of the plane: statistical properties of many-sided cells. J Stat Phys 132:627–647CrossRefzbMATHMathSciNetGoogle Scholar
  13. Hug D, Schneider R (2007) Asymptotic shapes of large cells in random tessellations. Geom Funct Anal 17:156–191CrossRefzbMATHMathSciNetGoogle Scholar
  14. Lantuéjoul C (2002) Geostatistical simulation. Springer, BerlinCrossRefzbMATHGoogle Scholar
  15. Lautensack C, Zuyev S (2008) Random Laguerre tessellations. Adv Appl Prob (SGSA) 40:630–650CrossRefzbMATHMathSciNetGoogle Scholar
  16. Lee Y, Fang C, Tsou YR, Lu LS, Yang CT (2009) A packing algorithm for three-dimensional convex particles. Granular Matter 11:307–315CrossRefzbMATHGoogle Scholar
  17. May JH, Smith RL (1982) Random polytopes: their definition, generation and aggregate properties. Math Program 24:39–54CrossRefzbMATHMathSciNetGoogle Scholar
  18. Miles RE (1971) Poisson flats in Euclidean spaces. Part II: homogeneous poisson flats and the complementary theorem. Adv Appl Prob 3:1–93CrossRefzbMATHMathSciNetGoogle Scholar
  19. Molchanov IS, Stoyan D (1996) Statistical models of random polyhedra. Commun Statist—Stoch Models 12(2):199–214CrossRefzbMATHMathSciNetGoogle Scholar
  20. Møller J, Zuyev S (1996) Gamma-type results and other related properties of poisson processes. Adv Appl Prob 28:662–673CrossRefGoogle Scholar
  21. Ohser J, Schladitz K (2009) 3D images of material structures. Wiley-VCH, WeinheimCrossRefGoogle Scholar
  22. Quenec’h JL, Chermant JL, Coster M, Jeulin D (1994) Liquid phase sintered materials modelling by random closed sets. In: Serra J, Soille P (eds) Mathematical morphology and its applications to image processing. Kluwer Academic Pub., Dordrecht, pp 225–232CrossRefGoogle Scholar
  23. Rao CR (1973) Linear statistical inference and its applications, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  24. Schneider R (1993) Convex bodies: the Brunn–Minkowski theory. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  25. Schneider R, Weil W (2008) Stochastic and integral geometry. Springer, BerlinCrossRefzbMATHGoogle Scholar
  26. Stoyan D, Stoyan H (1994) Fractals, random shapes and point fields. Wiley, ChichesterzbMATHGoogle Scholar
  27. Stoyan D, Davtyan A, Turetayev D (2002) Shape statistics for random domains and particles. In: Mecke K, Stoyan D (eds) Morphology of condensed matter. Physics and geometry of spatially complex systems. Lecture notes in physics. Springer, BerlinGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Felix Ballani
    • 1
  • Karl Gerald van den Boogaart
    • 1
    • 2
  1. 1.Institute for StochasticsTU Bergakademie FreibergFreibergGermany
  2. 2.Helmholtz Institute Freiberg for Resource TechnologyFreibergGermany

Personalised recommendations