Generalized Roof Duality for Multi-Label Optimization: Optimal Lower Bounds and Persistency

  • Thomas Windheuser
  • Hiroshi Ishikawa
  • Daniel Cremers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7577)


We extend the concept of generalized roof duality from pseudo-boolean functions to real-valued functions over multi-label variables. In particular, we prove that an analogue of the persistency property holds for energies of any order with any number of linearly ordered labels. Moreover, we show how the optimal submodular relaxation can be constructed in the first-order case.


multi-label higher-order roof duality MRF computer vision 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Thomas Windheuser
    • 1
    • 2
  • Hiroshi Ishikawa
    • 2
  • Daniel Cremers
    • 1
  1. 1.Department of Computer ScienceTechnische Universität MünchenMunichGermany
  2. 2.Department of Computer Science and EngineeringWaseda UniversityTokyoJapan

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