New Insight on Digital Topology

Abstract

The traditional approach of digital topology consists of using two different kinds of neighborhood for the black and white pixels of a binary image, and consequently two kinds of connectedness. In this paper, we are proposing to define connectedness in terms of a bounded subcollection of sets and to analyze the topological aspect of a binary image in an expanded domain in which it is sufficient to consider only one kind of connectedness. In the first part, we recall the definitions of neighborhood and connectedness of the traditional digital topology approach. In the second part, we define the notions of “bounded space”, “connected bounded space” and of “connected subset of a bounded space”. In the last part, we introduce two image operators (a dilation and an erosion) that produce expanded images whose connectedness is analyzed in relation to a bounded space obtained from the invariance domain of an opening. We show how the traditional two kinds of connectedness can be derived from this analysis.