A Bayesian multiscale framework for Poisson inverse problems

This paper describes a maximum a posteriori (MAP) estimation method for linear inverse problems involving Poisson data based on a novel multiscale framework. The framework itself is founded on a carefully designed multiscale prior probability distribution placed on the “splits” in the multiscale partition of the underlying intensity, and it admits a remarkably simple MAP estimation procedure using an expectation-maximization (EM) algorithm. Unlike many other approaches to this problem, the EM update equations for our algorithm have simple, closed-form expressions. Additionally, our class of priors has the interesting feature that the “non-informative” member yields the traditional maximum likelihood solution; other choices are made to reflect prior belief as to the smoothness of the unknown intensity