Inverse Scale Space Iterations for Non-Convex Variational Problems Using Functional Lifting

TL;DR

This paper introduces an inverse scale space iteration method for non-convex variational problems by employing functional lifting and sublabel-accurate discretization, extending classical Bregman iteration techniques.

Contribution

It extends inverse scale space methods to non-convex energies using functional lifting and provides conditions under which lifted iteration aligns with standard Bregman iteration.

Findings

Experimental results demonstrate effectiveness on convex and non-convex problems.

The proposed method generalizes classical Bregman iteration to non-convex energies.

Conditions identified for subgradients ensure the lifted iteration reduces to standard Bregman iteration.

Abstract

Non-linear filtering approaches allow to obtain decompositions of images with respect to a non-classical notion of scale. The associated inverse scale space flow can be obtained using the classical Bregman iteration applied to a convex, absolutely one-homogeneous regularizer. In order to extend these approaches to general energies with non-convex data term, we apply the Bregman iteration to a lifted version of the functional with sublabel-accurate discretization. We provide a condition for the subgradients of the regularizer under which this lifted iteration reduces to the standard Bregman iteration. We show experimental results for the convex and non-convex case.