Topological Watershed Transform
Laurent Najman


The watershed transform was introduced by S. Beucher and C. Lantuejoul for image segmentation, and is now used as a fundamental step in many powerful segmentation procedures.
Gilles Bertrand, Michel Couprie and Laurent Najman have developed a formal framework, allowing to study properties of discrete watershed operators, and in particular of the topologial watershed. This original approach to the watershed consists in modifying the original image by lowering (until idempotence) some points while preserving the connectivity of each lower cross-section. We will show that several important conservation properties (and especially a property concerning a notion of "contrast" proved to be important in practice) are not satisfied by most watershed algorithms. On the contrary, the topological watershed is the only one to satisfy those properties, and is also the subject of several non-trivial theorems.
We will present some equivalence theorems for topological watershed, some of them link the topological watershed and the component tree (also known as the min-tree, for which we have proposed a quasi-linear algorithm) and allow to design quasi-linear algorithms for computing the topological watershed. Remarkably, the component tree is an essential tool to filter the image: it is used for instance for an efficient implementation of the connected filters, h-minima, dynamics filtering, ..., tools that are often associated to the watershed for morphological image segmentation.

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