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.

a) (b) (c) (d) Figure 1: Piecewise smooth reconstruction of a normal vector field on a digital shape, normal vectors are represented through the flat shading of faces according to illumination (top-left: perfect digitization / down-right: noisy digitization): (a) input digital shape V , (b) input normal vector field g obtained with digital integral invariant (II) method [CLL14] with r = 3, (c) output normal vector field u and (d) sharpfeatures v superposed in red. Perfect and noisy digitization results are obtained using the same parameters. Our approach smoothes the input normal vector field except across sharp features, which are precisely delineated, thin, and consistent with the smoothing. AbstractWe propose a novel method to regularize a normal vector field defined on a digital surface (boundary of a set of voxels). When the digital surface is a digitization of a piecewise smooth manifold, our method localizes sharp features (edges) while regularizing the input normal vector field at the same time. It relies on the optimisation of a variant of the Ambrosio-Tortorelli functional, originally defined for denoising and contour extraction in image processing [AT90]. We reformulate this functional to digital surface processing thanks to discrete calculus operators. Experiments show that the output normal field is very robust to digitization artifacts or noise, and also fairly independent of the sampling resolution. The method allows the user to choose independently the amount of smoothing and the length of the set of discontinuities. Sharp and vanishing features are correctly delineated even on extremely damaged data. Finally, our method can be used to enhance considerably the output of state-ofthe-art normal field estimators like Voronoi Covariance Measure [MOG11] or Randomized Hough Transform [BM12].

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