Preservation of Piecewise Constancy under TV Regularization with Rectilinear Anisotropy

TL;DR

This paper extends a known result about the preservation of piecewise constancy under TV regularization from 2D to n-dimensional spaces, using a new proof involving averaging operators and subgradients.

Contribution

It generalizes the preservation of piecewise constancy under anisotropic TV regularization from 2D to higher dimensions with a novel proof technique.

Findings

Piecewise constancy is preserved in higher dimensions under TV regularization.

Averaging operators map subgradients of the anisotropic TV seminorm.

The proof technique can be applied to analyze TV regularization properties.

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.