On the Equivalence Between Hierarchical Segmentations and Ultrametric Watersheds

Authors:
Laurent Najman
Affiliations:
Laboratoire d'Informatique Gaspard-Monge, Equipe A3SI, ESIEE, Université Paris-Est, Paris, France
Venue:
Journal of Mathematical Imaging and Vision
Year:
2011

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Abstract

We study hierarchical segmentation in the framework of edge-weighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical segmentations. We end this paper by showing how to use the proposed framework in practice on the example of constrained connectivity; in particular it allows to compute such a hierarchy following a classical watershed-based morphological scheme, which provides an efficient algorithm to compute the whole hierarchy.