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The excursion set of a $C^2$ smooth random field carries relevant informationin its various geometric measures. From a computational viewpoint, one neverhas access to the continuous observation of the excursion set, but rather toobservations at discrete points in space. It has been reported that forspecific regular lattices of points in dimensions 2 and 3, the usual estimateof the surface area of the excursions remains biased even when the latticebecomes dense in the domain of observation. In the present work, under the keyassumptions of stationarity and isotropy, we demonstrate that this limitingbias is invariant to the locations of the observation points. Indeed, weidentify an explicit formula for the bias, showing that it only depends on thespatial dimension $d$. This enables us to define an unbiased estimator for thesurface area of excursion sets that are approximated by general tessellationsof polytopes in $\mathbb{R}^d$, including Poisson-Voronoi tessellations. Wealso establish a joint central limit theorem for the surface area and volumeestimates of excursion sets observed over hypercubic lattices.

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