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



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 


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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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