A Bayesian Multi-Scale Approach to Poisson Inverse Problems

Keywords: EM Algorithm, Recursive Partitioning

Abstract: We describe a method of estimation for Poisson intensity functions in linear inverse problems, based on a novel Bayesian multi-scale framework. Beginning with the standard likelihood-based model, the intensity to be recovered is decomposed with respect to a recursive dyadic partition, from which results a multi-scale analysis of the complete-data space. A prior distribution is specified by modeling the ``splits'' in the underlying partition in such a way as to encourage ``sparseness''. Due to a factorization permitted by the likelihood and the nature of our prior, a remarkably simple maximum {\it a posteriori} (MAP) estimation procedure results using the expectation-maximization (EM) algorithm. In particular, the E- and M-steps in our algorithm have expressions that are closed-form and easily interpretable. It can be shown that under certain interesting conditions on the hyper-parameters our EM algorithm will converge to a unique, global maximum. We illustrate the performance of our approach using datasets from the fields of medical imaging and astrophysics.