Hierarchical models in statistical inverse problems and the Mumford-Shah functional

The Bayesian methods for linear inverse problems are studied using hierarchical Gaussian models. The problems are considered with different discretizations, and we analyse the phenomena which appear when the discretization becomes finer. A hierarchical solution method for signal restoration problems is introduced and studied with arbitrarily fine discretization. We show that the maximum a posteriori estimate converges to a minimizer of the Mumford-Shah functional, up to a subsequence. A new result regarding the existence of a minimizer of the Mumford-Shah functional is proved. Moreover, we study the inverse problem under different assumptions on the asymptotic behaviour of the noise as discretization becomes finer. We show that the maximum a posteriori and conditional mean estimates converge under different conditions. This paper concentrates on the results regarding the maximum a posteriori estimate. The convergence results of the conditional mean estimate are proven in Helin (2009 Inverse Problems Imaging 3 4).