Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem

Authors:
Jan Lellmann;Frank Lenzen;Christoph Schnörr
Affiliations:
Image and Pattern Analysis Group & HCI, Dept. of Mathematics and Computer Science, University of Heidelberg, Heidelberg, Germany and Dept. of Applied Mathematics and Theoretical Physics, Universit ...;Image and Pattern Analysis Group & HCI, Dept. of Mathematics and Computer Science, University of Heidelberg, Heidelberg, Germany;Image and Pattern Analysis Group & HCI, Dept. of Mathematics and Computer Science, University of Heidelberg, Heidelberg, Germany
Venue:
Journal of Mathematical Imaging and Vision
Year:
2013

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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 several optimality bounds are known, to our knowledge no such bounds exist in the infinite-dimensional 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. The approach has a natural interpretation as an approximate, multiclass variant of the celebrated coarea formula.