Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem

  • Jan Lellmann
  • Frank Lenzen
  • Christoph Schnörr
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6819)


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.


Discrete Solution Optimality Bound Continuous Domain Convex Relaxation Coarea Formula 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jan Lellmann
    • 1
  • Frank Lenzen
    • 1
  • Christoph Schnörr
    • 1
  1. 1.Image and Pattern Analysis Group & HCI Dept. of Mathematics and Computer ScienceUniversity of HeidelbergGermany

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