Contact and chord length distribution of a stationary Voronoi tessellation

  title={Contact and chord length distribution of a stationary Voronoi tessellation},
  author={Lothar Heinrich},
  journal={Advances in Applied Probability},
  pages={603 - 618}
  • L. Heinrich
  • Published 1 September 1998
  • Mathematics
  • Advances in Applied Probability
We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝ d in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point… 
In this paper, the parallel set ΞR of the facets ((d−1)-faces) of a stationary Poisson-Voronoi tessellation in ℝ2 and ℝ3 is investigated. An analytical formula for the spherical contact distribution
Numerical and Analytical Computation of Some Second-Order Characteristics of Spatial Poisson-Voronoi Tessellations
We describe and discuss the explicit calculation of the pair correlation function of the point process of nodes associated with a three-dimensional stationary Poisson – Voronoi tessellation.
Random Laguerre tessellations
A systematic study of random Laguerre tessellations, weighted generalisations of the well-known Voronoi tessellations, is presented. We prove that every normal tessellation with convex cells in
Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces
  • Y. Isokawa
  • Mathematics
    Advances in Applied Probability
  • 2000
We study Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces, and give explicit expressions for mean surface area, mean perimeter length, and mean number of vertices of their cells.
A Review on Analytic Formulae for Poisson Laguerre Tessellations
Random Voronoi tessellations are used as models for a wide range of cellular structures such as foams, polycrystalline materials, plant cells but also telecommunication networks or animal
We study Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces, and give explicit expressions for mean surface area, mean perimeter length, and mean number of vertices of their cells.
On the distribution of the spherical contact vector of stationary germ-grain models
We consider a stationary germ-grain model Ξ with convex and compact grains and the distance r(x) from x ε ℝ d to Ξ. For almost all points x ε ℝ d there exists a unique point p(x) in the boundary of Ξ
Asymptotics of the visibility function in the Boolean model
The aim of this paper is to give a precise estimate on the tail probability of the visibility function in a germ-grain model: this function is defined as the length of the longest ray starting at the
Random Sets: Models and Statistics
This paper surveys aspects of the theory of random closed sets, focussing on issues of practical and current interest. First, some historical remarks on this part of probability theory are made,
A stochastic geometric model for continuous local trends in soil variation


Contact and chord length distributions of the Poisson Voronoi tessellation
This paper presents the form of some characteristics of the Voronoi tessellation which is generated by a stationary Poisson process in . Expressions are given for the spherical and linear contact
On the Pair Correlation Function of the Point Process of Nodes in a Voronoi Tessellation
We give a representation of the second-order factorial moment measure of the point process of nodes (vertices of cells) associated with a stationary Voronoi tessellation in Rd. If the Voronoi
Generation of the typical cell of a non-poissonian Johnson-Mehl tessellation
We consider Johnson-Mehl tessellations generated by stationary independently marked (not necessarily Poissonian) point processes in d-dimensional Euclidean space. We first analyze the Palm
The basic structures of Voronoi and generalized Voronoi polygons
For each particle in an aggregate of point particles in the plane, the set of points having it as closest particle is a convex polygon, and the aggregate V of such Voronoi polygons tessellates the
Lectures on Random Voronoi Tessellations
This volume provides an introduction to random Voronoi tessellations by presenting a survey of the main known results and the directions in which research is proceeding, making this essentially a self-contained account in which no background knowledge of the subject is assumed.
Stereological analysis of the spatial Poisson–Voronoi tessellation
Stereological model tests and parameter estimators for the spatial Poisson–Voronoi tessellation are discussed. The tests aim to discriminate the Poisson–Voronoi tessellation from more regular or more
On the Spherical Contact Distribution of Stationary Random Sets
Using a ow conservation law we study the spherical contact distribution of stationary random sets in I R d .
Asymptotic Behaviour of an Empirical Nearest‐Neighbour Distance Function for Stationary Poisson Cluster Processes
Summary. For stationary POISSON cluster processes (PCP's) O on R the limit behaviour, as v(D) → ∞, of the quantity , where χ(x, r) = 1, if O(b(x, r)) = 1, and χ(x, r) = 0 otherwise, is studied. A
Asymptotic properties of minimum contrast estimators for parameters of boolean models
A method for the estimation of parameters of random closed sets (racs’s) in ℝd based on a single realization within a (large) convex sampling window is presented, shown to be strongly consistent and asymptotically normal when the sampling window expands unboundedly.