Optimality bounds for a variational relaxation of the image partitioning problem

Authors:
Jan Lellmann;Frank Lenzen;Christoph Schnörr
Affiliations:
Dept. of Mathematics and Computer Science, University of Heidelberg;Dept. of Mathematics and Computer Science, University of Heidelberg;Dept. of Mathematics and Computer Science, University of Heidelberg
Venue:
EMMCVPR'11 Proceedings of the 8th international conference on Energy minimization methods in computer vision and pattern recognition
Year:
2011

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Abstract

Variational relaxations can be used to compute approximate minimizers of optimal partitioning and multiclass labeling problems on continuous domains. While the resulting relaxed convex problem can be solved globally optimal, in order to obtain a discrete solution a rounding step is required, which may increase the objective and lead to suboptimal solutions. We analyze a probabilistic rounding method and prove that it allows to obtain discrete solutions with an a priori upper bound on the objective, ensuring the quality of the result from the viewpoint of optimization. We show that the approach can be interpreted as an approximate, multiclass variant of the coarea formula.