Cell Shape Analysis of Random Tessellations Based on Minkowski Tensors

Part of the Lecture Notes in Mathematics book series (LNM, volume 2177)


To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic processes and models. In the context of applied image analysis of structured synthetic and biological materials, this question is central to the problem of inferring information about the formation process from spatial measurements of the resulting random structure. This chapter addresses this question by a theory-based simulation study of cell shape indices derived from tensor-valued intrinsic volumes, or Minkowski tensors, for a variety of common tessellation models. We focus on the relationship between two indices: (1) the dimensionless ratio 〈V2∕〈A3 of empirical average cell volumes to areas, and (2) the degree of cell elongation quantified by the eigenvalue ratio 〈β10,2〉 of the interface Minkowski tensors W10,2. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and cell volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, Gibbs hard-core processes of spheres, and random sequential absorption processes as well as for Laguerre tessellations of configurations of polydisperse spheres, STIT-tessellations, and Poisson hyperplane tessellations. These data are complemented by experimental 3D image data of mechanically stable ellipsoid configurations, area-minimising liquid foam models, and mechanically stable crystalline sphere configurations. We find that, not surprisingly, the indices 〈V2∕〈A3 and 〈β10,2〉 are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of these shape indices between many of the tessellation models listed above. Therefore, given a realization of a tessellation (e.g., an experimental image), these shape indices are able to narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by considering density-resolved volume-shape correlations.


Point Process Voronoi Diagram Shape Index Voronoi Cell Packing Fraction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank Andy Kraynik, Markus Spanner, and Richard Schielein for their data of the monodisperse foam, the equilibrium hard-sphere liquids, and the crystalline sphere packings, respectively. We also thank Felix Ballani for his software simulating the typical cell of a Poisson hyperplane tessellation. We thank Markus Kiderlen for valuable discussions and suggestions. We also thank the German science foundation (DFG) for the grants SCHR1148/3, HU1874/3-2, LA965/6-2, and ME1361/11 awarded as part of the DFG-Forschergruppe “Geometry and Physics of Spatial Random Systems”.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Institut für Theoretische PhysikUniversität Erlangen-NürnbergErlangenGermany
  3. 3.Fachbereich MathematikTechnische Universität KaiserslauternKaiserslauternGermany
  4. 4.School of Engineering and Information TechnologyMurdoch UniversityMurdochAustralia

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