Accurate approximation of the distributions of the 3D Poisson-Voronoi typical cell geometrical features

TL;DR

This paper derives exact distributions for key geometrical features of 3D Poisson-Voronoi cells using scaling properties and Monte Carlo simulations, providing highly accurate parametric approximations.

Contribution

It introduces a method leveraging the Poisson process scaling property to derive distributions for all intensities and uses simulations to identify the best parametric fits.

Findings

Derived distribution formulas for geometrical features at any intensity

Monte Carlo simulations closely approximate the true distributions

Parametric fits are highly accurate for practical use

Abstract

Although Poisson-Voronoi diagrams have interesting mathematical properties, there is still much to discover about the geometrical properties of its grains. Through simulations, many authors were able to obtain numerical approximations of the moments of the distributions of more or less all geometrical characteristics of the grain. Furthermore, many proposals on how to get close parametric approximations to the real distributions were put forward by several authors. In this paper we show that exploiting the scaling property of the underlying Poisson process, we are able to derive the distribution of the main geometrical features of the grain for every value of the intensity parameter. Moreover, we use a sophisticated simulation program to construct a close Monte Carlo based approximation for the distributions of interest. Using this, we also determine the closest approximating distributions within the mentioned frequently used parametric classes of distributions and conclude that these approximations can be quite accurate.