Invariant $\varphi$-minimal sets and total variation denoising on graphs

TL;DR

This paper extends the understanding of total variation denoising from one-dimensional signals to data on finite oriented graphs, revealing conditions under which minimizers preserve certain minimality properties and relating them to gradient flows.

Contribution

It generalizes invariant $ ext{phi}$-minimal set results to graph-based total variation and analyzes the relationship between ROF minimizers and gradient flows on graphs.

Findings

ROF minimizer on graphs shares minimality properties with 1D case

Replacing J with isotropic total variation loses the minimality property

Conditions for equivalence between ROF minimizer and gradient flow are identified

Abstract

Total variation flow, total variation regularization and the taut string algorithm are known to be equivalent filters for one-dimensional discrete signals. In addition, the filtered signal simultaneously minimizes a large number of convex functionals in a certain neighbourhood of the data. In this article we study the question to what extent this situation remains true in a more general setting, namely for data given on the vertices of a finite oriented graph and the total variation being $J(f) = \sum_{i,j} |f(v_i) - f(v_j)|$. Relying on recent results on invariant $\varphi$-minimal sets we prove that the minimizer to the corresponding Rudin-Osher-Fatemi (ROF) model on the graph has the same universal minimality property as in the one-dimensional setting. Interestingly, this property is lost, if $J$ is replaced by the discrete isotropic total variation. Next, we relate the ROF minimizer to the solution of the gradient flow for $J$. It turns out that, in contrast to the one-dimensional setting, these two problems are not equivalent in general, but conditions for equivalence are available.