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

Marion Foare (1, 2) , Nelly Pustelnik (1) , Laurent Condat (3)

1 Phys-ENS - Laboratoire de Physique de l'ENS Lyon
2 DANTE - Dynamic Networks : Temporal and Structural Capture Approach
3 GIPSA-AGPIG - GIPSA - Architecture, Géométrie, Perception, Images, Gestes

The Mumford–Shah model is a standard model in image segmentation and many approximations have been proposed in order to approximate it. The major interest of this functional is to be able to perform jointly 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.

Mots clés: nonsmooth optimization; Segmentation; inverse problems; restoration; proximal algorithms; noncon- vex optimization; Mumford–Shah; PALM

Domaines: Traitement du signal et de l'image [eess.SP]; Optimisation et contrôle [math.OC]

Marion Foare, Nelly Pustelnik, Laurent Condat. Semi-Linearized Proximal Alternating Minimization for a Discrete Mumford–Shah Model. 2018. ⟨hal-01782346v2⟩