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

Abstract

International audienceSimple 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