Bayesian denoising of generalized poisson processes with finite rate of innovation

We investigate the problem of the optimal reconstruction of a generalized Poisson process from its noisy samples. The process is known to have a finite rate of innovation since it is generated by a random stream of Diracs with a finite average number of impulses per unit interval. We formulate the recovery problem in a Bayesian framework and explicitly derive the joint probability density function (pdf) of the sampled signal. We compare the performance of the optimal Minimum Mean Square Error (MMSE) estimator with common regularization techniques such as ℓ₁ and Log penalty functions. The simulation results indicate that, under certain conditions, the regularization techniques can achieve a performance close to the MMSE method.