2013
DOI: 10.1007/978-3-642-38294-9_12
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Playing with Kruskal: Algorithms for Morphological Trees in Edge-Weighted Graphs

Abstract: Abstract. The goal of this paper is to provide linear or quasi-linear algorithms for producing some of the various trees used in mathemetical morphology, in particular the trees corresponding to hierarchies of watershed cuts and hierarchies of constrained connectivity. A specific binary tree, corresponding to an ordered version of the edges of the minimum spanning tree, is the key structure in this study, and is computed thanks to variations around Kruskal algorithm for minimum spanning tree.

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Cited by 69 publications
(97 citation statements)
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References 17 publications
(31 reference statements)
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“…These links open the way towards a family of efficient algorithms, based on Kruskal minimum spanning tree algorithms, for computing morphological hierarchies. These algorithms are presented in the companion paper [18]. Furthermore, the links established in this paper invites us to bridge hierarchical processing coming from different family of hierarchies.…”
Section: Resultsmentioning
confidence: 94%
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“…These links open the way towards a family of efficient algorithms, based on Kruskal minimum spanning tree algorithms, for computing morphological hierarchies. These algorithms are presented in the companion paper [18]. Furthermore, the links established in this paper invites us to bridge hierarchical processing coming from different family of hierarchies.…”
Section: Resultsmentioning
confidence: 94%
“…An important consequence of our results is the design of efficient algorithms based on Kruskal minimum spanning tree algorithm to compute these morphological hierarchies in quasi linear-time. These algorithms are presented in [18].…”
Section: Ph(g) Mh(g) Ph(t ) Mh(t )mentioning
confidence: 99%
“…In particular, in [20], an extensive assessment based on the framework of [4] shows that the hierarchical method performs at least as well as its non-hierarchical counterpart while providing at once all the possible scales. The results of this article constitute the theoretical basis of the methods presented in the aforementioned references [12,14,35,19,21,22]. It also opens the door towards new hierarchical image analysis.…”
Section: Introductionmentioning
confidence: 72%
“…The preprocessing step runs in linear time with respect to the number of nodes of the considered tree. The tree-based representation of a hierarchy on V is made of at most 2|V | − 1 nodes since a hierarchy on V contains at most 2|V | − 1 distinct regions: |V | singletons and |V | − 1 regions built from merging two regions of lower levels (see, e.g., [35]). Thus, the preprocessing step runs in O(|V |) time complexity.…”
Section: Saliency Map Algorithmsmentioning
confidence: 99%
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