An algorithmic framework for Mumford–Shah regularization of inverse problems in imaging

Kilian Hohm, Martin Storath, Andreas Weinmann
2015 Inverse Problems  
The Mumford-Shah model is a very powerful variational approach for edge preserving regularization of image reconstruction processes. However, it is algorithmically challenging because one has to deal with a non-smooth and nonconvex functional. In this paper, we propose a new efficient algorithmic framework for Mumford-Shah regularization of inverse problems in imaging. It is based on a splitting into specific subproblems that can be solved exactly. We derive fast solvers for the subproblems
more » ... the subproblems which are key for an efficient overall algorithm. Our method neither requires a priori knowledge on the gray or color levels nor on the shape of the discontinuity set. We demonstrate the wide applicability of the method for different modalities. In particular, we consider the reconstruction from Radon data, inpainting, and deconvolution. Our method can be easily adapted to many further imaging setups. The relevant condition is that the proximal mapping of the data fidelity can be evaluated within reasonable time. In other words, it can be used whenever classical Tikhonov regularization is possible.
doi:10.1088/0266-5611/31/11/115011 fatcat:gdces3rkkvdulkoqyo2ty52ndy