Regularized Non-local Total Variation and Application in Image Restoration

Abstract

In the usual non-local variational models, such as the non-local total variations (NLTV), the image is regularized by minimizing an energy term that penalizes gray-levels discrepancy between some specified pairs of pixels, a weight value is computed between these two pixels to penalize their dissimilarity. In this paper, we impose some regularity to those weight values. More precisely, we minimize a function involving a regularization term, analogous to an H 1 term, on weights. Doing so, the finite differences defining the image regularity depend on their environment. When the weights are difficult to define, they can be restored by the proposed stable regularization scheme.We provide all the details necessary for the implementation of a PALM algorithm with proved convergence. We illustrate the ability of the model to restore relevant unknown edges from the neighboring edges on an image inpainting problem. We also argue on inpainting, zooming and denoising problems that the model better recovers thin structures.

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

“…In this case, we consider u = (u R , u G , u B ), M = 3 and e ∈ R |E| common to the three components of u. We compare the proposed method with state-of-the-art approaches, including "ROF" minimization [23,24], the "MS relaxation" proposed in [27], the "Discrete AT" formulation [29] and the "NL-ROF" [28], although it is not designed for RGB-color images. Since the ROF minimization and the "NL-ROF" do not allow us to directly extract the contours, we compute them by thresholding the gradient of the estimate following [23,24].…”

“…Experiments show that this ROF-like coupling term is more robust to image gradients, but eliminates high-frequency content. More recently, Li et al [28] suggested to set e p = {e We can remark that, when dealing with multivariate images, contours can be defined either as similar edges through all the components, or as distinct edges, leading to a path K that may be different for all the components. In order to facilitate the understanding and the reading, we formulate Problem 1 in the context of similar edges.…”

“…They derive a heuristic algorithm, based on a convex relaxation of the functional they propose, and extract the contours by thresholding. Recently, Li et al [28] proposed a nonlocal ROF (NL-ROF) model, similar to the AT functional, where the gradient is computed in a weighted neighborhood. Convergence is proved, but the contours are obtained by post-processing the estimated image.…”

A New Proximal Method for Joint Image Restoration and Edge Detection with the Mumford-Shah Model

“…In [8] a finite element implementation of the AT functional was proposed, refining and realigning the meshing locally around the discontinuities but extremely costly for large images. Li et al proposed in [9] a functional similar to AT functional, where (4) is approximated with a quadratic penalization. The convergence is proved but it does not perform contour detection.…”

Symmetric Upwind Scheme for Discrete Weighted Total Variation

“…Context In this paper, we are interested in the discrete formulation of Total Variation (TV) in image processing and computer vision as a prior for piecewise constant images. This topic has been widely studied since it has been proposed by Rudin-Osher-Fatemi (ROF) [3] and remains an active research field; as an example, non-local formulations have been proposed in [4], [5] and a combination of TV with Non-Local means has been proposed in [6], all in order to preserve textures and thin structures. The Total-Generalized-Variation (TGV) studied in [7] and the method in [8] are variants that reduce the staircasing effect observed with TV.…”