Shape Constrained Regularisation by Statistical Multiresolution for Inverse Problems: Asymptotic Analysis

TL;DR

This paper introduces a new regularisation method for ill-posed inverse problems that combines convex penalties with statistical multiresolution techniques, providing adaptive and consistent solutions in noisy data scenarios.

Contribution

It develops a novel regularisation approach using statistical multiresolution and convex penalties, with proven consistency and convergence rates for a broad class of problems.

Findings

Proves general consistency and convergence rates in Bregman-divergence framework

Demonstrates local adaptivity in signal and image reconstruction

Applicable to a wide range of penalty functionals

Abstract

This paper is concerned with a novel regularisation technique for solving linear ill-posed operator equations in Hilbert spaces from data that is corrupted by white noise. We combine convex penalty functionals with extreme-value statistics of projections of the residuals on a given set of sub-spaces in the image-space of the operator. We prove general consistency and convergence rate results in the framework of Bregman-divergences which allows for a vast range of penalty functionals. Various examples that indicate the applicability of our approach will be discussed. We will illustrate in the context of signal and image processing that the presented method constitutes a locally adaptive reconstruction method.