Geometric properties of solutions to the total variation denoising problem

TL;DR

This paper investigates the geometric properties of solutions to total variation denoising, defining the extended support region and proving that denoised images are constant outside a shrinking tube around this support, with implications for edge preservation.

Contribution

It introduces a precise mathematical definition of extended support in TV denoising and proves the image is constant outside a small tube around this support, with explicit results for certain shapes.

Findings

Extended support can be explicitly determined in practical cases.

Denoised images are constant outside a shrinking tube around the extended support.

Edges are tightly clustered around the extended support for calibrable sets.

Abstract

This article studies the denoising performance of total variation (TV) image regularization. More precisely, we study geometrical properties of the solution to the so-called Rudin-Osher-Fatemi total variation denoising method. The first contribution of this paper is a precise mathematical definition of the "extended support" (associated to the noise-free image) of TV denoising. It is intuitively the region which is unstable and will suffer from the staircasing effect. We highlight in several practical cases, such as the indicator of convex sets, that this region can be determined explicitly. Our second and main contribution is a proof that the TV denoising method indeed restores an image which is exactly constant outside a small tube surrounding the extended support. The radius of this tube shrinks toward zero as the noise level vanishes, and are able to determine, in some cases, an upper bound on the convergence rate. For indicators of so-called "calibrable" sets (such as disks or properly eroded squares), this extended support matches the edges, so that discontinuities produced by TV denoising cluster tightly around the edges. In contrast, for indicators of more general shapes or for complicated images, this extended support can be larger. Beside these main results, our paper also proves several intermediate results about fine properties of TV regularization, in particular for indicators of calibrable and convex sets, which are of independent interest.