Inverting Poisson-Laguerre tessellations

2026-06-01

AI Summary

This study addresses the inverse problem of recovering the weighted generator points from an observed Poisson–Laguerre tessellation. The authors propose a novel inversion framework that achieves uniform convergence by progressively expanding the observation window to approximate the original generators, while fully characterizing all equivalent weighted configurations that yield identical Laguerre tessellations. The approach integrates tools from stochastic geometry, random tessellation theory, and nonparametric estimation to enable effective reconstruction of the underlying weighted point process. Numerical experiments confirm the accuracy of the inversion method and demonstrate its successful application in nonparametric estimation of the weight distribution function.

Abstract

While it is well-known how to compute the cells of a Laguerre tessellation for a given set of weighted generator points, it is not obvious how to invert a Laguerre tessellation. That is, given that one observes a Laguerre tessellation, how can one retrieve the weighted generators corresponding to the observed cells. In this paper, we consider inversion of a class of random Laguerre tessellations known as Poisson-Laguerre tessellations. The weighted generators of observed cells of a Poisson-Laguerre tessellation are of interest because knowledge of these weighted generators is useful for statistical inference of Poisson-Laguerre tessellations. For general Laguerre tessellations we provide a characterization of all configurations of weighted generator points which yield the same Laguerre tessellation. For Poisson-Laguerre tessellations we propose a method for consistent inversion, meaning that as one observes the tessellation through increasing observation windows, a closer approximation of the original weighted generators can be obtained. In a simulation study we examine both performance of the inversion procedure, as well as the use of the obtained approximated weighted generators for nonparametrically estimating the weight distribution function corresponding to a Poisson-Laguerre tessellation.

Problem

• Poisson-Laguerre tessellation
• inversion
• weighted generators
• Laguerre tessellation
• statistical inference

Innovation

• Laguerre tessellation
• Poisson-Laguerre tessellation
• inversion
• weighted generators
• nonparametric estimation