Boundaries in digital spaces: Basic theory
Gábor T. Herman · Machine intelligence and pattern recognition · 1996
We define a digital space to be a pair consisting of an arbitrary nonempty set V and a symmetric binary relation π on V with respect to which V is connected. The boundary between subsets O and Q of V is defined to be the set of those (c, d) in π for which c is in O and d is in Q. We demonstrate that in spite of the extremely general nature of this definition, a nontrivial theory of boundaries can be developed and, in particular, that results regarding the connectedness and separability properties of the interiors and exteriors of boundaries in classical digital topology (concerned with particular tessellations of some finite-dimensional Euclidean space) become corollaries of corresponding theorems in our general environment.