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Random Tessellations -- An Overview of Models

Claudia Redenbach, Christian Jung

TL;DR

This chapter surveys the broad landscape of random tessellations within stochastic geometry, detailing construction mechanisms from hyperplane cuts to Voronoi-based, Laguerre-weighted, STIT, Gibbs, and T-tessellation models. It emphasizes both analytic results and Monte Carlo approaches, and covers topics from model fitting, reconstruction, and simulation to specific model variants like dead leaves and iterated/tessellations. The work highlights interconnections among different tessellation families, provides practical guidance for simulation and estimation, and discusses applications to material science, biology, and spatial pattern modeling. Overall, it offers a cohesive framework for selecting, simulating, and fitting tessellation models to complex spatial structures.

Abstract

Random tessellations are a prominent class of models in stochastic geometry. In this chapter, we give an overview of mechanisms that have been used to formulate random tessellation models. First, the notion of a random tessellation and basic geometric characteristics of random tessellations are introduced. Then, several model classes are presented. This includes, but is not limited to, Voronoi tessellations and their weighted generalizations, hyperplane tessellations, and STIT tessellations. Simulation of the tessellation models and approaches for model fitting are also discussed.

Random Tessellations -- An Overview of Models

TL;DR

This chapter surveys the broad landscape of random tessellations within stochastic geometry, detailing construction mechanisms from hyperplane cuts to Voronoi-based, Laguerre-weighted, STIT, Gibbs, and T-tessellation models. It emphasizes both analytic results and Monte Carlo approaches, and covers topics from model fitting, reconstruction, and simulation to specific model variants like dead leaves and iterated/tessellations. The work highlights interconnections among different tessellation families, provides practical guidance for simulation and estimation, and discusses applications to material science, biology, and spatial pattern modeling. Overall, it offers a cohesive framework for selecting, simulating, and fitting tessellation models to complex spatial structures.

Abstract

Random tessellations are a prominent class of models in stochastic geometry. In this chapter, we give an overview of mechanisms that have been used to formulate random tessellation models. First, the notion of a random tessellation and basic geometric characteristics of random tessellations are introduced. Then, several model classes are presented. This includes, but is not limited to, Voronoi tessellations and their weighted generalizations, hyperplane tessellations, and STIT tessellations. Simulation of the tessellation models and approaches for model fitting are also discussed.
Paper Structure (31 sections, 7 theorems, 61 equations, 21 figures, 2 tables)

This paper contains 31 sections, 7 theorems, 61 equations, 21 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Stochastic geometry background
  3. Point and hyperplane processes
  4. Tessellations and their properties
  5. Random tessellations
  6. Hyperplane tessellations
  7. Voronoi tessellations and their generalizations
  8. Voronoi tessellations
  9. Weighted Voronoi tessellations
  10. Laguerre tessellations
  11. Johnson-Mehl tessellations
  12. Generalized balanced power diagrams
  13. Delaunay tessellations
  14. Gibbs tessellations
  15. Models for T-tessellations
  16. ...and 16 more sections

Key Result

theorem 1

Let $T$ be a stationary random tessellation in $\mathbb{R}^2$. If $T$ is face-to-face, the following mean value relations hold: If $T$ is normal, then $N_{02}=3$ and $N_{20}=6$. Then eq:TessChar2 and eq:TessChar1 yield

Figures (21)

  • Figure 1: Left: A normal tessellation in $\mathbb{R}^2$. Every vertex is contained in exactly three cells. Vertices are Y-shaped. Middle: A face-to-face, but not normal tessellation. Vertices belong to four cells and are X-shaped. Right: A tessellation that is not face-to-face. Intersections of cells are not necessarily edges of cells. Vertices are T-shaped.
  • Figure 2: Left: Point process $\Phi_0$ of vertices. Middle: Point process $\Phi_1$ of edge centers. The intensities $\gamma_0$ and $\gamma_1$ are the expected numbers of points of $\Phi_0$ and $\Phi_1$, respectively, in the unit square (grey). Right: The measure $M_1$ measures the total edge length in a given set $B$. The intensity $\mu_1$ is the expected total edge length (blue) in the unit square (grey).
  • Figure 3: Left: An I-segment in a tessellation with T-vertices. Middle: J-segments defined by the cells below the I-segment. Right: K-segments.
  • Figure 4: Isotropic Poisson line tessellation in $\mathbb{R}^2$ (left) and Poisson hyperplane tessellation in $\mathbb{R}^3$ (right). Poisson line tessellation with direction distribution concentrated on the coordinate directions (middle).
  • Figure 5: Illustration of the Voronoi growth process at three equidistant time points.
  • ...and 16 more figures

Theorems & Definitions (29)

  • definition 1: Point process
  • definition 2: Stationarity and isotropy
  • definition 3: Intensity measure
  • definition 4: Poisson process
  • definition 5: Hyperplane process
  • definition 6: Tessellation
  • definition 7: k-faces
  • definition 8: Face-to-face and normal tessellation
  • definition 9: X-, Y-, and T-vertices
  • definition 10: Random tessellation
  • ...and 19 more