New Insight on Digital Topology

Abstract

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. . IntroductionDigital topology provides the theoretical foundations for image analysis and more specifically advanced image segmentation.Traditionally, digital topology is based on the neighborhood concept. From it, other concepts, like connectedness and connected components, are derived.When the pixels of a binary image are arranged along lines and columns, the need for using two kinds of neighborhood, one for the black pixels and one for the white, appears in the early studies on digital topology.In this paper, we propose the notion of bounded space as first concept, instead of neighborhood. Then we can use the traditional topological approach (not digital) to define connectedness.Our contribution consists of showing that it is possible to derive the traditional 4-connectedness and 8-connectedness from a unique connectedness defined on an expanded image domain. This is obtained by using two expansion operators (one dilation and one erosion) and by studying connectedness in the expanded domain.In the first section, we recall the definitions of neighborhood and connectedness of the traditional digital topology approach, and we show the need for two definitions.