A completely random T-tessellation model and Gibbsian extensions

TL;DR

This paper introduces a new completely random model for T-tessellations based on Poisson lines, extends it with Gibbs variants for feature control, and provides a simulation algorithm for practical investigation.

Contribution

It proposes a novel random T-tessellation model with Gibbs extensions, translating key point process concepts to tessellations and enabling controlled feature modeling.

Findings

The model exhibits properties similar to Poisson point processes.

Gibbs variants allow feature control in T-tessellations.

A Metropolis-Hastings-Green algorithm for simulation is developed.

Abstract

In their 1993 paper, Arak, Clifford and Surgailis discussed a new model of random planar graph. As a particular case, that model yields tessellations with only T-vertices (T-tessellations). Using a similar approach involving Poisson lines, a new model of random T-tessellations is proposed. Campbell measures, Papangelou kernels and Georgii-Nguyen-Zessin formulae are translated from point process theory to random T-tessellations. It is shown that the new model shows properties similar to the Poisson point process and can therefore be considered as a completely random T-tessellation. Gibbs variants are introduced leading to models of random T-tessellations where selected features are controlled. Gibbs random T-tessellations are expected to better represent observed tessellations. As numerical experiments are a key tool for investigating Gibbs models, we derive a simulation algorithm of the Metropolis-Hastings-Green family.