A Proof of the Tree of Shapes in n-D

Thierry Géraud (1) , Nicolas Boutry (1) , Sébastien Crozet (1) , Edwin Carlinet (1) , Laurent Najman (2)

1 LRDE - Laboratoire de Recherche et de Développement de l'EPITA
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

In this paper, we prove that the self-dual morphological hierarchical structure computed on a n-D gray-level wellcomposed image u by the algorithm of Géraud et al. [1] is exactly the mathematical structure defined to be the tree of shape of u in Najman et al [2]. We recall that this algorithm is in quasi-linear time and thus considered to be optimal. The tree of shapes leads to many applications in mathematical morphology and in image processing like grain filtering, shapings, image segmentation, and so on.

Mots clés: Mathematical Morphology; Tree of Shapes; Self-Dual Operators; Well-Composedness; Algorithms

Domaines: Mathématique discrète [cs.DM]; Algorithme et structure de données [cs.DS]; Traitement des images [eess.IV]; Topologie géométrique [math.GT]

Thierry Géraud, Nicolas Boutry, Sébastien Crozet, Edwin Carlinet, Laurent Najman. A Proof of the Tree of Shapes in n-D. 2022. ⟨hal-03690353⟩