On the ROF Model in Rectilinear Anisotropy: Piecewise Constant Approximation and Universal Minimality

TL;DR

This paper analyzes the approximation quality of the ROF model in rectilinear anisotropy, providing convergence rates for piecewise constant approximations and revealing a universal minimality property of the ROF minimizer.

Contribution

It establishes new convergence rates for the ROF minimizer's approximation by piecewise constant functions and introduces a universal minimality property extending previous results.

Findings

Convergence rate of $O(h^{1/2 - q'/2q})$ in general dimensions

Improved convergence rate of $O(h^{1/2 - 1/2q})$ in dimension 1

Universal minimality property of the ROF minimizer for a broad class of convex functionals

Abstract

We prove that the $L^2$ distance between the minimizer of the $\ell^1$-anisotropic Rudin-Osher-Fatemi (ROF) functional and its minimizer over the space of piecewise constant functions on a rectilinear grid is $\mathcal{O}(h^{\frac12 - \frac{q'}{2q}})$, where $h$ is the grid's mesh size and the datum belongs to $L^q$, $q \ge 2$. These convergence rates are valid in any dimension $d\ge 1$. However, in dimension $d = 1$ they can be further improved to $\mathcal{O}(h^{\frac12 - \frac{1}{2q}})$. To establish the error bounds, $L^q$ estimates of the ROF minimizer in terms of the datum are critical. Such estimates are particular cases of a universal minimality property of the ROF minimizer derived in the second part of the paper. There it is shown, in both the finite-dimensional and infinite-dimensional settings, that the minimizer simultaneously minimizes a broad class of convex functionals over a neighbourhood of the datum arising in the convex dual of the ROF problem. This extends previous results of similar type about taut strings and the ROF problem.