In obtaining a tractable solution to the problem of extracting a minimal partition from hierarchy or tree by dynamic programming, we introduce the braids of partition and h-increasing energies, the former extending the solution space from a hierarchy to a larger set, the latter describing the family of energies, for which one can obtain the solution by a dynamic programming. We also provide the singularity condition for the existence of unique solution, leading to the definition of the energetic lattice. The paper also identifies various possible braids in literature and how this structure relaxes the segmentation problem.


Hierarchies Dynamic programming Optimization Lattice 


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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Centre de robotiqueMINES ParisTech, PSL-Research UniversityParisFrance
  2. 2.A3SI-ESIEE LIGMUniversité Paris-EstParisFrance

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