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.