A Statistical Multiscale Framework for Poisson Inverse Problems
Nowak, Robert David
Kolaczyk, Eric D.
Poisson processes; inverse problems; multiscale analysis; imaging
This paper describes a statistical modeling and analysis method for linear inverse problems involving Poisson data based on a novel multiscale framework. The framework itself is founded upon a multiscale analysis associated with recursive partitioning of the underlying intensity, a corresponding multiscale factorization of the likelihood (induced by this analysis), and a choice of prior probability distribution made to match this factorization by modeling the "splits" in the underlying partition. The class of priors used here has the interesting feature that the "non-informative" member yields the traditional maximum likelihood solution; other choices are made to re prior belief as to the smoothness of the unknown intensity. Adopting the expectation-maximization (EM) algorithm for use in computing the MAP estimate corresponding to our model, we find that our model permits remarkably simple, closed-form expressions for the EM update equations. The behavior of our EM algorithm is examined, and it is shown that convergence to the global MAP estimate can be guaranteed. Applications in emission computed tomography and astronomical energy spectral analysis demonstrate the potential of the new approach.