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.

Constructive Links between Some Morphological Hierarchies on Edge-Weighted Graphs

“…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.…”

“…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].…”

Hierarchical Segmentations with Graphs: Quasi-flat Zones, Minimum Spanning Trees, and Saliency Maps

“…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.…”

“…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.…”

“…Conversely, thanks to Theorem 1, one can work on a saliency map or, thanks to Theorem 4, on the weights of a minimum spanning tree and explicitly computes the hierarchy in the end. In [12,35,27], a resulting saliency map is computed before a possible extraction of the associated hierarchy of watersheds. In [19], a basic transformation that consists of modifying one weight on a minimum spanning tree according to some criterion is considered.…”