Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem
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.
Cite
@article{arxiv.1112.0974,
title = {Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem},
author = {Jan Lellmann and Frank Lenzen and Christoph Schnörr},
journal= {arXiv preprint arXiv:1112.0974},
year = {2011}
}