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Semi-Linearized Proximal Alternating Minimization for a Discrete Mumford-Shah Model

By Marion Foare, Nelly Pustelnik and Laurent Condat

Abstract

International audienceThe Mumford-Shah model is a standard model in image segmentation, and due to its difficulty , many approximations have been proposed. The major interest of this functional is to enable joint image restoration and contour detection. In this work, we propose a general formulation of the discrete counterpart of the Mumford-Shah functional, adapted to nonsmooth penalizations, fitting the assumptions required by the Proximal Alternating Linearized Minimization (PALM), with convergence guarantees. A second contribution aims to relax some assumptions on the involved functionals and derive a novel Semi-Linearized Proximal Alternated Minimization (SL-PAM) algorithm, with proved convergence. We compare the performances of the algorithm with several nonsmooth penalizations, for Gaussian and Poisson denoising, image restoration and RGB-color denoising. We compare the results with state-of-the-art convex relaxations of the Mumford-Shah functional, and a discrete version of the Ambrosio-Tortorelli functional. We show that the SL-PAM algorithm is faster than the original PALM algorithm, and leads to competitive denoising, restoration and segmentation results

Topics: Segmentation, nonconvex optimization, Mumford–Shah, Mumford-Shah, nonsmooth optimization, PALM, inverse problems, proximal algorithms, restoration, noncon- vex optimization, [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Publisher: 'Institute of Electrical and Electronics Engineers (IEEE)'
Year: 2019
DOI identifier: 10.1109/TIP.2019.2944561
OAI identifier: oai:HAL:hal-01782346v4
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