Boundaries in digital planes

The importance of topological connectedness properties in processing digital pictures is well known. A natural way to begin a theory for this is to give a definition of connectedness for subsets of a digital plane which allows one to prove a Jordan curve theorem. The generally accepted approach to this has been a non-topological Jordan curve theorem which requires two different definitions, 4-connectedness, and 8-connectedness, one for the curve and the other for its complement. Earlier, the authors introduced a purely topological context for a digital plane and proved a Jordan curve theorem. The present paper gives a topological proof of the non-topological Jordan curve theorem mentioned above and extends the previous work of the authors by considering some questions associated with image processing: How do more complicated curves separate the digital plane into connected sets? Conversely given a partition of the digital plane into connected sets, what are the boundaries like and how can we recover them? The authors’ construction gives a unified answer to these questions. The crucial step in making the present approach topological is to utilize a natural connected topology on a finite, totally ordered set; the topologies on the digital spaces are then just the associated product topologies. Furthermore, this permits the authors to define path, arc, and curve as certain continuous functions on such a parameter interval.