Preservation of Piecewise Constancy under TV Regularization with Rectilinear Anisotropy

A recent result by Łasica, 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.