International Journal of Computer Vision

, Volume 104, Issue 3, pp 241–269 | Cite as

Discrete and Continuous Models for Partitioning Problems

  • Jan Lellmann
  • Björn Lellmann
  • Florian Widmann
  • Christoph Schnörr


Recently, variational relaxation techniques for approximating solutions of partitioning problems on continuous image domains have received considerable attention, since they introduce significantly less artifacts than established graph cut-based techniques. This work is concerned with the sources of such artifacts. We discuss the importance of differentiating between artifacts caused by discretization and those caused by relaxation and provide supporting numerical examples. Moreover, we consider in depth the consequences of a recent theoretical result concerning the optimality of solutions obtained using a particular relaxation method. Since the employed regularizer is quite tight, the considered relaxation generally involves a large computational cost. We propose a method to significantly reduce these costs in a fully automatic way for a large class of metrics including tree metrics, thus generalizing a method recently proposed by Strekalovskiy and Cremers (IEEE conference on computer vision and pattern recognition, pp. 1905–1911, 2011).


Multi-class labeling Segmentation  Partitioning problem Graph cut Convex relaxation  Variational methods 



The second and third author were supported by Engineering and Physical Sciences Research Council (EPSRC)-Project EP/H016317/1. 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 2013

Authors and Affiliations

  • Jan Lellmann
    • 1
  • Björn Lellmann
    • 2
  • Florian Widmann
    • 2
  • Christoph Schnörr
    • 3
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK
  2. 2.Department of ComputingImperial College LondonLondonUK
  3. 3.Image and Pattern Analysis Group & HCI, Department of Mathematics and Computer ScienceUniversity of HeidelbergHeidelbergGermany

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