Inverse problems with second-order Total Generalized Variation constraints
This work addresses image reconstruction challenges in computational imaging, but it appears incremental as it extends TGV from denoising to inverse problems.
The paper tackles the problem of solving ill-posed linear inverse problems, such as image recovery from blurred and noisy data, by using second-order Total Generalized Variation (TGV) constraints, showing existence and stability of solutions.
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.