Invariant φ-Minimal Sets and Total Variation Denoising on Graphs.

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 neighborhood of the data. In this article we study the question of to what extent this situation remains true in a more general setting, namely for data given on the vertices of an oriented graph and the total variation being J(f) = Sigma(i,j) vertical bar f (v(i)) - f (v(i))vertical bar. Relying on recent results on invariant phi-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.