Topological monsters in Z^3: A non-exhaustive bestiary

Nicolas Passat (1) , Michel Couprie (2) , Gilles Bertrand (2)

1 LSIIT - Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

Simple points in Z^n, and especially in Z^3, are the basis of several topology-preserving transformation methods proposed for image analysis (segmentation, skeletonisation, ...). Most of these methods rely on the assumption that the --iterative or parallel-- removal of simple points from a discrete object X necessarily leads to a globally minimal topologically equivalent sub-object of X (i.e. a subset Y which is topologically equivalent to X and which does not strictly include another set Z topologically equivalent to X). This is however false in Z^3, and more generally in Z^n. We illustrate this fact by presenting some topological monsters, i.e. some objects of Z^3 only composed of non-simple points, but which could however be reduced without altering their topology.

Domaines: Traitement des images [eess.IV]; Topologie géométrique [math.GT]

Nicolas Passat, Michel Couprie, Gilles Bertrand. Topological monsters in Z^3: A non-exhaustive bestiary. International Symposium on Mathematical Morphology (ISMM), 2007, Rio de Janeiro, France. pp.11-12. ⟨hal-00622180⟩