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Quasilinear algorithms for the topological watershed
 JOURNAL OF MATHEMATICAL IMAGING AND VISION
, 2005
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Watershed Cuts: Minimum Spanning Forests and the Drop of Water Principle
 IEEE TRANS. PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2009
"... We study the watersheds in edgeweighted graphs. We define the watershed cuts following the intuitive idea of drops of water flowing on a topographic surface. We first establish the consistency of these watersheds: they can be equivalently defined by their “catchment basins” (through a steepest des ..."
Abstract

Cited by 43 (22 self)
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We study the watersheds in edgeweighted graphs. We define the watershed cuts following the intuitive idea of drops of water flowing on a topographic surface. We first establish the consistency of these watersheds: they can be equivalently defined by their “catchment basins” (through a steepest descent property) or by the “dividing lines ” separating these catchment basins (through the drop of water principle). Then we prove, through an equivalence theorem, their optimality in terms of minimum spanning forests. Afterward, we introduce a lineartime algorithm to compute them. To the best of our knowledge, similar properties are not verified in other frameworks and the proposed algorithm is the most efficient existing algorithm, both in theory and practice. Finally, the defined concepts are illustrated in image segmentation leading to the conclusion that the proposed approach improves, on the tested images, the quality of watershedbased segmentations.
Fusion graphs: merging properties and watersheds
"... This paper deals with mathematical properties of watersheds in weighted graphs linked to region merging methods, as used in image analysis. In a graph, a cleft (or a binary watershed) is a set of vertices that cannot be reduced, by point removal, without changing the number of regions (connected com ..."
Abstract

Cited by 10 (6 self)
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This paper deals with mathematical properties of watersheds in weighted graphs linked to region merging methods, as used in image analysis. In a graph, a cleft (or a binary watershed) is a set of vertices that cannot be reduced, by point removal, without changing the number of regions (connected components) of its complement. To obtain a watershed adapted to morphological region merging, it has been shown that one has to use the topological thinnings introduced by M. Couprie and G. Bertrand. Unfortunately, topological thinnings do not always produce thin clefts. Therefore, we introduce a new transformation on vertex weighted graphs, called Cwatershed, that always produces a cleft. We present the class of perfect fusion graphs, for which any two neighboring regions can be merged, while preserving all other regions, by removing from the cleft the points adjacent to both. An important theorem of this paper states that, on these graphs, the Cwatersheds are topological thinnings and the corresponding divides are thin clefts. We propose a lineartime immersionlike monotone algorithm to compute Cwatersheds on perfect fusion graphs, whereas, in general, a lineartime topological thinning algorithm does not exist. Finally, we derive some characterizations of perfect fusion graphs based on thinness properties of both Cwatersheds and topological watersheds.
Discrete region merging and watersheds
, 2007
"... This paper summarizes some results of the authors concerning watershed divides and their use in region merging schemes. The first aspect deals with properties of watershed divides that can be used in particular for hierarchical region merging schemes. We introduce the mosaic to retrieve the altitude ..."
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This paper summarizes some results of the authors concerning watershed divides and their use in region merging schemes. The first aspect deals with properties of watershed divides that can be used in particular for hierarchical region merging schemes. We introduce the mosaic to retrieve the altitude of points along the divide set. A desirable property is that, when two minima are separated by a crest in the original image, they are still separated by a crest of the same altitude in the mosaic. Our main result states that this is the case if and only if the mosaic is obtained through a topological thinning. The second aspect is closely related to the thinness of watershed divides. We present fusion graphs, a class of graphs in which any region can be always merged without any problem. This class is equivalent to the one in which watershed divides are thin. Topological thinnings do not always produce thin divides, even on fusion graphs. We also present the class of perfect fusion graphs, in which any pair of neighbouring regions can be merged through their common neighborhood. An important theorem states that the divides of any ultimate topological thinning are thin on any perfect fusion graph.