Section: New Results

Optimization

Participant : Marion Foare.

A new proximal method for joint image restoration and edge detection with the Mumford-Shah model

Joint work with Nelly Pustelnik (CNRS, Physics Lab of ENS de Lyon) and Laurent Condat (CNRS, GIPSA Lab) [24].

In this paper, we propose an adaptation of the PAM algorithm to the minimization of a nonconvex functional designed for joint image denoising and contour detection. This new functional is based on the Ambrosio–Tortorelli approximation of the well-known Mumford–Shah functional. We motivate the proposed approximation, offering flexibility in the choice of the possibly non-smooth penalization, and we derive closed form expression for the proximal steps involved in the algorithm. We focus our attention on two types of penalization: 1-norm and a proposed quadratic-1 function. Numerical experiments show that the proposed method is able to detect sharp contours and to reconstruct piecewise smooth approximations with low computational cost and convergence guarantees. We also compare the results with state-of-the-art re-laxations of the Mumford–Shah functional and a recent discrete formulation of the Ambrosio–Tortorelli functional.

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

Joint work with Nelly Pustelnik (CNRS, Physics Lab of ENS de Lyon) and Laurent Condat (CNRS, GIPSA Lab) [51].

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.

Discrete Mumford-Shah on graph for mixing matrix estimation

Joint work with Yacouba Kaloga (Physics Lab of ENS de Lyon), Nelly Pustelnik (CNRS, Physics Lab of ENS de Lyon) and Pablo Jensen (CNRS, Physics Lab of ENS de Lyon) [53].

The discrete Mumford-Shah formalism has been introduced for the image denoising problem, allowing to capture both smooth behavior inside an object and sharp transitions on the boundary. In the present work, we propose first to extend this formalism to graphs and to the problem of mixing matrix estimation. New algorithmic schemes with convergence guarantees relying on proximal alternating minimization strategies are derived and their efficiency (good estimation and robustness to initialization) are evaluated on simulated data, in the context of vote transfer matrix estimation.