Photon counting in the presence of thermal noise: Estimation of a filtered point process in additive white Gaussian noise

A stochastic processes is considered which is consisted of a filtered Poisson process (a Poisson process passed through a linear filter) and an additive white Gaussian noise. Motivated by applications in single photon counting, the problem of estimating the Poisson process from the observations of this stochastic process is investigated under a low Poisson intensity regime. As the solution to this problem, a class of nonlinear causal filters is developed to compute the minimum mean squared error estimation of the Poisson process. Performance of this estimator is verified by numerical simulations.