Set Reconstruction by Voronoi cells

TL;DR

This paper analyzes the accuracy of Poisson--Voronoi approximations of sets in Euclidean space, providing exact asymptotics for the expected symmetric difference volume as the intensity grows large.

Contribution

It derives exact asymptotics for the expected symmetric difference volume between a set and its Poisson--Voronoi approximation, including moment estimates.

Findings

Asymptotic formula for $ ext{E}[ ext{Vol}(A riangle A_ ext{eta})]$ as $ ext{lambda} o abla$

Moment estimates for $ ext{Vol}(A_ ext{eta})$ and $ ext{Vol}(A riangle A_ ext{eta})$

Results applicable to sets with finite volume and perimeter

Abstract

For a Borel set $A$ and a homogeneous Poisson point process $\eta$ in $\R^d$ of intensity $\lambda >0$, define the Poisson--Voronoi approximation $ A_\eta$ of $A$ as a union of all Voronoi cells with nuclei from $\eta$ lying in $A$. If $A$ has a finite volume and perimeter we find an exact asymptotic of $\E\Vol(A\Delta A_\eta)$ as $\lambda\to\infty$ where $\Vol$ is the Lebesgue measure. Estimates for all moments of $\Vol(A_\eta)$ and $\Vol(A\Delta A_\eta)$ together with their asymptotics for large $\lambda$ are obtained as well.