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

Abstract

The excursion set of a ${\mathit{C}}^2$ 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 approximation of the surface area of the excursions does not converge when the lattice becomes dense in the domain of observation to the desired limit. In the present work, under the key assumptions of stationarity and isotropy, we demonstrate that this limiting factor is invariant to the locations of the observation points. Indeed, we identify an explicit formula for the correction factor, showing that it only depends on the spatial dimension d. This enables us to define an approximation for the surface area of excursion sets for general tessellations of polytopes in ${\mathbb{R}}^{\mathit{d}}$, including Poisson–Voronoi tessellations. We also establish a joint central limit theorem for the surface area and volume of excursion sets observed over hypercubic lattices.