An Algorithm for Second Order Mumford--Shah Models Based on a Taylor Jet Formulation

Abstract

Mumford--Shah models are well-established and powerful variational tools for the regularization of noisy data. In the case of images this includes regularizing both the edge set as well as the image values itself. Thus, these models may be used as a basis for a segmentation pipeline or for smoothing the data. In this paper we consider higher order Mumford--Shah functionals which penalize the deviation from piecewise polynomials instead of piecewise constant functions as first order Mumford--Shah functionals do. Minimizing Mumford--Shah functionals, which are nonsmooth and nonconvex functionals, are NP hard problems. Compared with first order Mumford--Shah functionals, numerically solving higher order models is even more challenging, and in contrast to work on more theoretical aspects there are only very few works dealing with the algorithmic side. In this paper, we propose a new algorithmic framework for second order Mumford--Shah regularization. It is based on a proposed reformulation of higher order Mumford--Shah problems in terms of Taylor jets and a corresponding discretization. Using an ADMM approach, we split the discrete jet-based problem into subproblems which we can solve efficiently, noniteratively, and exactly. We derive numerically stable and fast solvers for these subproblems. In summary, we obtain an efficient overall algorithm. Our method requires a priori knowledge on neither the gray or color levels nor the shape of the discontinuity set of a solution. We demonstrate the applicability of the proposed methods in various numerical experiments. In particular, we quantitatively and qualitatively compare the proposed scheme with the algorithms proposed in the literature.