Journal of Mathematical Imaging and Vision

, Volume 47, Issue 3, pp 239–257 | Cite as

Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem

  • Jan Lellmann
  • Frank Lenzen
  • Christoph Schnörr


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.


Convex relaxation Multiclass labeling Approximation bound Combinatorial optimization Total variation Linear programming relaxation 



This publication is partly based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Jan Lellmann
    • 1
    • 2
  • Frank Lenzen
    • 1
  • Christoph Schnörr
    • 1
  1. 1.Image and Pattern Analysis Group & HCI, Dept. of Mathematics and Computer ScienceUniversity of HeidelbergHeidelbergGermany
  2. 2.Dept. of Applied Mathematics and Theoretical PhysicsUniversity of Cambridge, Centre for Mathematical SciencesCambridgeUK

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