Inverse Scale Space Iterations for Non-Convex Variational Problems: The Continuous and Discrete Case

TL;DR

This paper extends inverse scale space methods to non-convex variational problems using lifted Bregman iterations, providing theoretical conditions and experimental validation for both continuous and discrete cases.

Contribution

It introduces a novel lifted Bregman iteration framework that applies to non-convex variational problems, broadening the scope of inverse scale space methods.

Findings

Conditions identified for subgradients where lifted iteration reduces to standard Bregman iteration.

Experimental results demonstrate effectiveness in both convex and non-convex cases.

Framework applicable in continuous and discrete settings.

Abstract

Non-linear filtering approaches allow to obtain decompositions of images with respect to a non-classical notion of scale, induced by the choice of a convex, absolutely one-homogeneous regularizer. The associated inverse scale space flow can be obtained using the classical Bregman iteration with quadratic data term. We apply the Bregman iteration to lifted, i.e. higher-dimensional and convex, functionals in order to extend the scope of these approaches to functionals with arbitrary data term. We provide conditions for the subgradients of the regularizer -- in the continuous and discrete setting -- under which this lifted iteration reduces to the standard Bregman iteration. We show experimental results for the convex and non-convex case.