A statistical multiscale framework for Poisson inverse problems

This paper describes a statistical multiscale modeling and analysis framewo rk for linear inverse problems involving Poisson data. The framework itself is founded upon a multiscale analysis associated with recursive partitioni ng of the underlying intensity, a corresponding multiscale factorization of the likelihood (induced by this analysis), and a choice of prior probabili ty distribution made to match this factorization by modeling the "splits" i n the underlying partition. The class of priors used here has the interesti ng feature that the "noninformative" member yields the traditional maximum- likelihood solution; other choices are made to reflect prior belief as to t he smoothness of the unknown intensity. Adopting the expectation-maximizati on (EM) algorithm for use id computing the maximum a posteriori (MAP) estim ate corresponding to our model, we find that our model permits remarkably s imple, closed-form expressions for the EM update equations. The behavior of our EM algorithm is examined, and it is shown that convergence to the glob al MAP estimate can be guaranteed. Applications in emission computed tomogr aphy and astronomical energy spectral analysis demonstrate the potential of the new approach.