Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem

TL;DR

This paper establishes optimality bounds for a convex relaxation approach to image partitioning, providing theoretical guarantees on the quality of solutions obtained from the relaxed problem.

Contribution

It introduces the first known optimality bounds in the continuous setting for a variational relaxation of multiclass image partitioning problems.

Findings

Probabilistic rounding yields near-optimal integral solutions.

Provides an a priori upper bound on the objective function.

Connects the approach to the coarea formula as an interpretation.

Abstract

We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation methods for finite-dimensional problems. While for the latter case several optimality bounds are known, to our knowledge no such bounds exist in the continuous setting. We provide such a bound by analyzing a probabilistic rounding method, showing that it is possible to obtain an integral solution of the original partitioning problem from a solution of the relaxed problem with an a priori upper bound on the objective, ensuring the quality of the result from the viewpoint of optimization. The approach has a natural interpretation as an approximate, multiclass variant of the celebrated coarea formula.