An algorithm for minimizing the Mumford-Shah functional

TLDR

Existing Mumford‑Shah minimization methods are non‑convex, whereas this work presents the first convex‑relaxation‑based algorithm. The paper revisits the Mumford‑Shah functional and proposes an algorithm to minimize its convex relaxation via functional lifting. The authors develop an efficient primal‑dual projection algorithm that converges to the minimizer of the lifted convex relaxation. The algorithm produces initialization‑independent solutions that yield smooth approximations while preserving signal discontinuities, as confirmed by experiments.

Abstract

In this work we revisit the Mumford-Shah functional, one of the most studied variational approaches to image segmentation. The contribution of this paper is to propose an algorithm which allows to minimize a convex relaxation of the Mumford-Shah functional obtained by functional lifting. The algorithm is an efficient primal-dual projection algorithm for which we prove convergence. In contrast to existing algorithms for minimizing the full Mumford-Shah this is the first one which is based on a convex relaxation. As a consequence the computed solutions are independent of the initialization. Experimental results confirm that the proposed algorithm determines smooth approximations while preserving discontinuities of the underlying signal.