A theory of binary digital pictures

Abstract

We study 2- and 3-dimensional digital geometry in the context of almost arbitrary adjacency relations. (Previous authors have based their work on particular adjacency relations). We define a binary digital picture to be a pair whose components are a set of lattice-points and an adjacency relation on the whole lattice. We show how a wide class of digital pictures have natural “continuous analogs.” This enables us to use methods of continuous topology in studying digital pictures. We are able to prove general results on the connectivity of digital borders, which generalize results that have appeared in the literature. In the 3-dimensional case we consider the possibility of using a uniform relation on the whole lattice. (In the past authors have used different types of adjacency for “object” and “background.”)