Filters for Spatial Point Processes

Abstract

We study the general problem of estimating a “hidden” point process $\mathbf{X}$, given the realization of an “observed” point process $\mathbf{Y}$ (possibly defined in different spaces) with known joint distribution. We characterize the posterior distribution of $\mathbf{X}$ under marginal Poisson and Gauss–Poisson priors and when the transformation from $\mathbf{X}$ to $\mathbf{Y}$ includes thinning, displacement, and augmentation with extra points. These results are then applied in a filtering context when the hidden process evolves in discrete time in a Markovian fashion. The dynamics of $\mathbf{X}$ considered are general enough for many target tracking applications.