On Regularization via Frame Decompositions with Applications in Tomography

TL;DR

This paper investigates regularization techniques for linear ill-posed problems using frame decompositions, providing convergence proofs and applying these methods to tomography with Radon transforms.

Contribution

It introduces a general framework for regularization via frame decompositions, extending classical methods and establishing convergence and rate results.

Findings

Proves convergence of regularization methods using frame decompositions.

Derives convergence rates under different parameter choice rules.

Applies the theoretical results to tomography with Radon transform.

Abstract

In this paper, we consider linear ill-posed problems in Hilbert spaces and their regularization via frame decompositions, which are generalizations of the singular-value decomposition. In particular, we prove convergence for a general class of continuous regularization methods and derive convergence rates under both a-priori and a-posteriori parameter choice rules. Furthermore, we apply our derived results to a standard tomography problem based on the Radon transform.