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

TL;DR

This paper investigates Bayesian hierarchical models for linear inverse problems, demonstrating convergence of estimates to Mumford--Shah functional minimizers and introducing a new existence result for these minimizers.

Contribution

It introduces a hierarchical solution method for signal restoration that converges to Mumford--Shah minimizers and proves a new existence theorem for these minimizers.

Findings

Maximum a posteriori estimates converge to Mumford--Shah minimizers.

Conditional mean estimates also converge under certain conditions.

A new existence theorem for Mumford--Shah minimizers is established.

Abstract

The Bayesian methods for linear inverse problems is studied using hierarchical Gaussian models. The problems are considered with different discretizations, and we analyze 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 behavior of the noise as discretization becomes finer. We show that the maximum a posteriori and conditional mean estimates converge under different conditions.