A Tutorial on Well-Composedness

Nicolas Boutry (1) , Thierry Géraud (2) , Laurent Najman (1)

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

Due to digitization, usual discrete signals generally present topological paradoxes, such as the connectivity paradoxes of Rosenfeld. To get rid of those paradoxes, and to restore some topological properties to the objects contained in the image, like manifoldness, Latecki proposed a new class of images, called well-composed images, with no topological issues. Furthermore , well-composed images have some other interesting properties: for example, the Euler number is locally computable, boundaries of objects separate background from foreground, the tree of shapes is well-defined, and so on. Last, but not the least, some recent works in mathematical morphology have shown that very nice practical results can be obtained thanks to well-composed images. Believing in its prime importance in digital topology, we then propose this state-of-the-art of well-composedness, summarizing its different flavours, the different methods existing to produce well-composed signals, and the various topics that are related to well-composedness.

Mots clés: Discrete surfaces; Mathematical morphology; Well-composedness; Digital topology; Critical configurations; Manifolds

Domaines: Topologie algébrique [math.AT]; Mathématique discrète [cs.DM]

Nicolas Boutry, Thierry Géraud, Laurent Najman. A Tutorial on Well-Composedness. Journal of Mathematical Imaging and Vision, 2018, 60 (3), pp.443-478. ⟨10.1007/s10851-017-0769-6⟩. ⟨hal-01609892v2⟩