Surface area and volume of excursion sets observed on point cloud based polytopic tessellations

TL;DR

This paper develops an unbiased estimator for the surface area and volume of excursion sets of smooth random fields observed on point cloud tessellations, accounting for bias invariance and providing statistical properties.

Contribution

It introduces a bias formula for surface area estimation that depends only on dimension and proposes an unbiased estimator applicable to general tessellations.

Findings

Bias in surface area estimates is invariant to observation point locations.

An explicit bias formula depending solely on dimension is derived.

A joint CLT for surface area and volume estimates is established.

Abstract

The excursion set of a $C^2$ smooth random field carries relevant information in its various geometric measures. From a computational viewpoint, one never has access to the continuous observation of the excursion set, but rather to observations at discrete points in space. It has been reported that for specific regular lattices of points in dimensions 2 and 3, the usual estimate of the surface area of the excursions remains biased even when the lattice becomes dense in the domain of observation. In the present work, under the key assumptions of stationarity and isotropy, we demonstrate that this limiting bias is invariant to the locations of the observation points. Indeed, we identify an explicit formula for the bias, showing that it only depends on the spatial dimension $d$. This enables us to define an unbiased estimator for the surface area of excursion sets that are approximated by general tessellations of polytopes in $\mathbb{R}^d$, including Poisson-Voronoi tessellations. We also establish a joint central limit theorem for the surface area and volume estimates of excursion sets observed over hypercubic lattices.