Sublabel-Accurate Discretization of Nonconvex Free-Discontinuity Problems

TL;DR

This paper introduces a systematic method for discretizing nonconvex free-discontinuity problems using sublabel-accurate convex relaxations, improving solution precision with fewer labels.

Contribution

It extends sublabel-accurate multilabeling to general convex and nonconvex regularizations and provides a systematic discretization approach for continuous convex relaxations.

Findings

More precise solutions with fewer labels.

Extension to general convex and nonconvex regularizations.

Application to vectorial Mumford-Shah functional.

Abstract

In this work we show how sublabel-accurate multilabeling approaches can be derived by approximating a classical label-continuous convex relaxation of nonconvex free-discontinuity problems. This insight allows to extend these sublabel-accurate approaches from total variation to general convex and nonconvex regularizations. Furthermore, it leads to a systematic approach to the discretization of continuous convex relaxations. We study the relationship to existing discretizations and to discrete-continuous MRFs. Finally, we apply the proposed approach to obtain a sublabel-accurate and convex solution to the vectorial Mumford-Shah functional and show in several experiments that it leads to more precise solutions using fewer labels.