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

Abstract

The 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.

Discrete Mumford–Shah on Graph for Mixing Matrix Estimation

“…This has been relaxed by Bolte et al [26], who derived a proximal alternating linearized scheme (PALM). A hybrid version, named SL-PAM for semi-linearized proximal alternating direction method, has been proposed in [21], especially adapted to the resolution of the D-MS problem. The iterations of SL-PAM to solve (3) would read…”

“…where ∇ 1 denotes the gradient with respect to the first variable. In [21], a closed form expression has been derived for the update of e while, for the basic image denoising MS, the update of M only involves the proximity operator of L. When dealing with mixing matrix estimation, it is thus of almost importance of either having a closed form expression of the proximity operator involved in the update of M [k+1] or providing a new iterative scheme without such an updating step. However, the proximity operator of a sum of functions is known to have closed form expression for a very limited number of functions (see e.g.…”

“…However, the major drawbacks of the extensions to image processing derived in [15] and [17] are the weak convergence guarantees, and the lack of flexibility in the data-term choice, whose proximity operator should have a closed form expression. Finally, for handling with the exact MS model, which estimates jointly the contours and the image, we can refer to Ambrosio-Tortorelli alternatives [18][19][20], at the price of a huge computational time, or the strategy we proposed in [21]. Note that in the literature, the extension of MS to graphs relies on a Chan-Vese formulation [22].…”

“…In this work, we extend the Discrete MS-type (D-MS) model proposed in [21] for image denoising to mixing matrix estimation on graph. We propose to formulate the data-term as…”

“…is given in [21], so that Assumption 3.1 iii) is fulfilled by the proposed functional. This specific choice implies R to be a bounded below, proper, lower semi-continuous, and semi-algebraic function, and so does Ψ.…”