Journal of Mathematical Imaging and Vision

, Volume 50, Issue 3, pp 261–285 | Cite as

Collapses and Watersheds in Pseudomanifolds of Arbitrary Dimension

  • Jean Cousty
  • Gilles Bertrand
  • Michel Couprie
  • Laurent Najman
Article

Abstract

This work is settled in the framework of abstract simplicial complexes. We propose a definition of a watershed and of a collapse (i.e., a homotopic retraction) for maps defined on pseudomanifolds of arbitrary dimension. Then, we establish two important results linking watersheds and homotopy. The first one generalizes a property known for distance transforms in a continuous setting to any map on pseudomanifolds: a watershed of any map is a subset of an ultimate collapse of the support of this map. The second result establishes, through an equivalence theorem, a deep link between watershed and collapse of maps: any watershed of any map can be straightforwardly obtained from an ultimate collapse of this map, and conversely any ultimate collapse of the initial map straightforwardly induces a watershed.

Keywords

Watershed Segmentation Collapse  Topology preservation Simplicial complex  Pseudomanifold  

Notes

Acknowledgments

This work received funding from the Agence Nationale de la Recherche, contract ANR-2010-BLAN-0205-03.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Jean Cousty
    • 1
  • Gilles Bertrand
    • 1
  • Michel Couprie
    • 1
  • Laurent Najman
    • 1
  1. 1.Laboratoire d’Informatique Gaspard-Monge, Université Paris-Est, A3SI ESIEENoisy-le-Grand CedexFrance

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