Inhomogeneous Poisson Sampling of Finite-Energy Signals With Uncertainties in ${\mathbb{R}}^{d}$

Spatiotemporal signal reconstruction from samples randomly gathered in a multidimensional space with uncertainty is a crucial problem for a variety of applications. Such a problem generalizes the reconstruction of a deterministic signal and that of a stationary random process in one dimension, which was first addressed by Whittaker, Kotelnikov, and Shannon. In this work we analyze multidimensional random sampling with uncertainties jointly accounting for signal properties (signal spectrum and spatial correlation) and for sampling properties (inhomogeneous sample spatial distribution, sample availability, and non-ideal knowledge of sample positions). The reconstructed signal spectrum and the signal reconstruction accuracy are derived as a function of signal and sampling properties. It is shown that some of these properties expand the signal spectrum while others modify the spectrum without expansion. The signal reconstruction accuracy is first determined in a general case and then specialized for cases of practical interests. The optimal interpolator function is derived and asymptotic results are obtained to show the impact of sampling non-idealities. The analysis is corroborated by verifying that previously known results can be obtained as special cases of the general one and by means of a case study accounting for various settings of signal and sample properties.