Translation invariant diagonal frame decomposition of inverse problems and their regularization

TL;DR

This paper introduces the translation invariant diagonal frame decomposition (TI-DFD) as a novel regularization method for inverse problems, addressing artifacts caused by lack of translational invariance in wavelet-based approaches, and demonstrates its effectiveness through numerical examples.

Contribution

The paper develops the TI-DFD framework, generalizing SVD, and proves its effectiveness as a regularization method with optimal convergence rates for inverse problems.

Findings

TI-DFD eliminates wavelet artifacts in inverse problem solutions

Filtered TI-DFDs provide fast, accurate, and stable reconstructions

Numerical experiments confirm the theoretical advantages of TI-DFD

Abstract

Solving inverse problems is central to a variety of important applications, such as biomedical image reconstruction and non-destructive testing. These problems are characterized by the sensitivity of direct solution methods with respect to data perturbations. To stabilize the reconstruction process, regularization methods have to be employed. Well-known regularization methods are based on frame expansions, such as the wavelet-vaguelette (WVD) decomposition, which are well adapted to the underlying signal class and the forward model and furthermore allow efficient implementation. However, it is well known that the lack of translational invariance of wavelets and related systems leads to specific artifacts in the reconstruction. To overcome this problem, in this paper we introduce and analyze the translation invariant diagonal frame decomposition (TI-DFD) of linear operators as a novel concept generalizing the SVD. We characterize ill-posedness via the TI-DFD and prove that a TI-DFD combined with a regularizing filter leads to a convergent regularization method with optimal convergence rates. As illustrative example, we construct a wavelet-based TI-DFD for one-dimensional integration, where we also investigate our approach numerically. The results indicate that filtered TI-DFDs eliminate the typical wavelet artifacts when using standard wavelets and provide a fast, accurate, and stable solution scheme for inverse problems.