Multidimensional Stochastic Sampling: Unified Theory and Point Process Applications

Multidimensional stochastic sampling is crucial for many applications that rely on the reconstruction of spatial fields from observations gathered by distributed sensing devices, including crowd-sensing and Internet-of-Things, as well as recent applications for metaverse and tactile communications. This paper establishes a unified foundation for multidimensional stochastic sampling by bridging sampling and point processes theories considering interactions within the samples distribution, uncertainties on measurements and sample positions, and interpolation filtering. The optimal linear shift-invariant interpolator and the reconstruction mean square error are determined for a vast class of point processes, including determinantal (repulsion) and Cox (attraction). We establish that sampling processes with the same intensity function can be ordered, in terms of increasing reconstruction error, as determinantal, Poisson, and Cox. We introduce the Beurling spectrum of a point process to facilitate the analysis. From the general framework, known sampling theory results are obtained as special cases and new results are derived involving Ginibre and Neyman-Scott point processes, as well as randomly-populated lattices.