Regularization of inverse problems by filtered diagonal frame decomposition

Abstract

Inverse problems are at the heart of many practical problems such as medical image reconstruction or non-destructive evaluation. A characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this paper, we introduce and analyze the concept of filtered diagonal frame decomposition, which extends the classical filtered singular value decomposition (or spectral filtering) to the case of frames. The use of frames as generalized singular systems allows for a better adaption to a given class of potential solutions of the inverse problem. This is also beneficial for problems where the SVD is not available analytically. We show that filtered diagonal frame decompositions provide convergent regularization methods. Moreover, we derive convergence rates under source conditions and prove order optimality when the frame under consideration is a Riesz basis. Our analysis applies to unbounded and bounded forward operators. As a practical application of our tools we study filtered diagonal frame decompositions for inverting the Radon transform as an unbounded operator on L 2 ( R 2 ) .