Minimal non-simple and minimal non-cosimple sets in binary images on cell complexes

Abstract

The concepts of weak component and simple 1 are generalizations, to binary images on the n-cells of n-dimensional cell complexes, of the standard concepts of "26-component" and "26-simple" 1 in binary images on the 3-cells of a 3D cubical complex; the concepts of strong component and cosimple 1 are generalizations of the concepts of "6-component" and "6-simple" 1. Over the past 20 years, the problems of determining just which sets of 1's can be minimal non-simple, just which sets can be minimal non-cosimple, and just which sets can be minimal non-simple (minimal non-cosimple) without being a weak (strong) fore-ground component have been solved for the 2D cubical and hexagonal, 3D cubical and face-centered-cubical, and 4D cubical complexes. This paper solves these problems in much greater generality, for a very large class of cell complexes of dimension ≤ 4. © Springer-Verlag Berlin Heidelberg 2006.