Topological approach to mathematical morphology

For low—bit—rate coding of video sequences, specifically in the context of the MPEG—4 proposal, morphological approaches proved to be highly attractive (see e.g. [16]). During a series of coding experiments performed at Siemens Research Laboratory in München [9] the authors felt that there are some deficiencies of theory which need investigation. The aim of this paper is to sketch a theory which allows to understand the relationships between three classes of discrete concepts, namely discrete topology, discrete morphology and discrete metrics. Mathematical morphology is based on topologies for systems of subsets of a set [13]. The topology of the underlying set enters only indirectly. Therefore such concepts as connectedness of sets can cause difficulties if treated purely morphologically. These conceptual difficulties became especially apparent, when structures and algorithms were used practically which simultaneously involve both subset topologies and connectedness of subsets as is the case e.g. in watershed segmentation [16].