In this paper, we present a new edge detection model based on proximal unfolded neural networks. The architecture relies on unfolding proximal Blake–Zisserman iterations, leading to a composition of two blocks: a smoothing block and an edge detection block. We show through simulations that the proposed approach efficiently eliminates irrelevant details while retaining key edges and significantly improves performance with respect to state-of-the-art strategies. Additionally, our architecture is significantly lighter than recent learning models designed for edge detection in terms of number of learnable parameters and inference time.

This work is dedicated to joint image restoration and contour detection considering the Ambrosio-Tortorelli functional. Two proximal alternating minimization schemes with convergence guarantees are provided, PALM-AT and SL-PAM-AT, as well as closed-form expressions of the involved proximity operators. A thorough numerical study is conducted in order to evaluate the performance of both numerical schemes as well as comparisons to state-of-the-art Mumford-Shah strategies.

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.

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 this letter, 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) is evaluated on simulated data, in the context of vote transfer matrix estimation.

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: ℓl-norm and a proposed quadratic-f. 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 relaxations of the Mumford-Shah functional and a recent discrete formulation of the Ambrosio-Tortorelli functional.

Essential image processing and analysis tasks, such as image segmentation, simplification and denoising, can be conducted in a unified way by minimizing the Mumford-Shah (MS) functional. Although seductive, this minimization is in practice difficult because it requires to jointly define a sharp set of contours and a smooth version of the initial image. For this reason, various relaxations of the original formulations have been proposed, together with optimisation methods. Among these, the Ambrosio-Tortorelli (AT) parametric functional is of particular interest, because minimizers of AT can be shown to converge to a minimizer of MS. However this convergence is difficult to achieve numerically using standard finite difference schemes. Indeed, with AT, discontinuities need to be represented explicitly rather than implicitly. In this work, we propose to formulate AT using the full framework of Discrete Calculus (DC), which is able to sharply represent discontinuities thanks to a more sophisticated topological framework. We present our proposed formulation, its resolution, and results on synthetic and real images. We show that we are indeed able to represent sharp discontinuities and as a result significantly better stability to noise, compared with finite difference schemes.

Abstract We 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‐of‐the‐art normal field estimators like Voronoi Covariance Measure [ MOG11 ] or Randomized Hough Transform [ BM12 ].