Symmetric Upwind Scheme for Discrete Weighted Total Variation

Abstract

This paper is devoted to the study of the discrete formulations of the weighted Total Variation (TV) based on upwind schemes that have been proposed for imaging problems in a local setting in [1] and in a non-local setting for graphs and point-clouds in [2]. We focus on two new symmetric formulations based on the 2 and ∞ norms respectively and propose a dedicated optimization algorithm to solve convex problems based on such TV penalties. We demonstrate the theoretical and practical interest of such formulations for image processing tasks.

Variational models for signal processing with Graph Neural Networks

“…Specifically, we are interested in this work in non-local Total-Variation on a graph, which has been thoroughly studied, e.g. for non-local signal processing [6,9,25], multi-scale decomposition of signals [12], point-cloud segmentation [18] and sparsification [26], using various optimization schemes (such as finite difference schemes [6], primal-dual [3], forward-backward [23], or cut pursuit [24]).…”

“…For q = 1, the regularization term is the Non-Local Total Variation (NL-TV), referred to as isotropic when p = 2 and anisotropic when p = 1. Other choice of norm might be useful: using p ≤ 1, as studied for instance for p = 0 in [26], results in sparsification of signals; using p = ∞ [25] is also useful when considering unbiased symmetric schemes.…”

Towards Off-the-grid Algorithms for Total Variation Regularized Inverse Problems

“…Many algorithms have been proposed to solve (1). With the notable exception of [17], most of them rely on the introduction of a fixed discrete grid, which yields reconstruction artifacts such as anisotropy or blur (see the experiments in [16]). On the other hand, it is known that some solutions of (1) are sums of a finite number of indicator functions of simply connected sets [4,5], yielding piecewise constant images.…”