Inverse problems with second-order Total Generalized Variation constraints

TL;DR

This paper explores the use of second-order Total Generalized Variation (TGV) in solving ill-posed linear inverse problems, demonstrating existence, stability, and application to image recovery from blurred, noisy data.

Contribution

It extends TGV to second order in inverse problems, providing theoretical guarantees and practical application insights.

Findings

Proves existence and stability of TGV-based solutions.

Applies TGV to image deblurring with noise.

Shows improved image reconstruction quality.

Abstract

Total Generalized Variation (TGV) has recently been introduced as penalty functional for modelling images with edges as well as smooth variations. It can be interpreted as a "sparse" penalization of optimal balancing from the first up to the $k$-th distributional derivative and leads to desirable results when applied to image denoising, i.e., $L^2$-fitting with TGV penalty. The present paper studies TGV of second order in the context of solving ill-posed linear inverse problems. Existence and stability for solutions of Tikhonov-functional minimization with respect to the data is shown and applied to the problem of recovering an image from blurred and noisy data.