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Topological Watershed Transform
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Laurent Najman
Abstract:

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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