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

Abstract

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 non-convex 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 which are key for an efficient overall algorithm. Our method neither requires a priori knowledge of the gray or color levels nor of 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 a within reasonable time. In other words, it can be used whenever classical Tikhonov regularization is possible.

Discrete Mumford–Shah on Graph for Mixing Matrix Estimation

“…Efficient iterative strategies have been designed and a recent work by Cai and Steidl [14] establishes the link between a thresholded-ROF strategy and the MS model. The second class relies on the Blake-Zisserman model [15][16][17] which handles the piecewise smoothness by considering a different approximation of MS known as the discrete membrane energy. Convergence to local minimizers was proved for the 1D formulation in [16].…”

“…Convergence to local minimizers was proved for the 1D formulation in [16]. However, the major drawbacks of the extensions to image processing derived in [15] and [17] are the weak convergence guarantees, and the lack of flexibility in the data-term choice, whose proximity operator should have a closed form expression. Finally, for handling with the exact MS model, which estimates jointly the contours and the image, we can refer to Ambrosio-Tortorelli alternatives [18][19][20], at the price of a huge computational time, or the strategy we proposed in [21].…”

Fast Segmentation From Blurred Data in 3D Fluorescence Microscopy

“…Future directions of research include the adaption to further imaging setups such as magnetic particle imaging and the adaption to the (piecewise smooth) Mumford-Shah functional as in [68].…”

Smoothing for signals with discontinuities using higher order Mumford–Shah models

“…It is straightforward to devise an algorithm for (P k,β,γ ) of complexity O(N 3 ). For the first order problem (P 1,β,γ ), we proposed an O(N 2 ) algorithm [30] which utilizes a fast computation scheme for the approximation errors proposed by Blake [9]. Unfortunately, as that scheme is based on algebraic recurrences, a generalization to arbitrary orders of k seems difficult.…”