Preservation of Piecewise Constancy under TV Regularization with Rectilinear Anisotropy

Abstract

A recent result by Lasica, Moll and Mucha about the ${\ell }^1$-anisotropic Rudin-Osher-Fatemi model in ${\mathbb{R}}^2$ asserts that the solution is piecewise constant on a rectilinear grid, if the datum is. By means of a new proof we extend this result to ${\mathbb{R}}^n$. The core of our proof consists in showing that averaging operators associated to certain rectilinear grids map subgradients of the ${\ell }^1$-anisotropic total variation seminorm to subgradients.