The topological characterization of adjacency and neighborhood relations between pixels or voxels in digital imaging is discussed in this paper. Some new research is presented along with a review of the author’s philosophy, which he has explored in previous work.

The first new result is a characterization of a discrete topological space in terms of a system of point neighborhoods satisfying some axioms, in particular on the thinness and idempotence of the frontier of a set. The second is a study of the possible choices of foreground/background adjacency pairs in digital space; it is shown that they have a “topological equivalent” (in a precise mathematical sense) if and only if exactly one of the two adjacencies is the maximal one. Even then, some requirements, like the satisfaction of the discrete Jordan theorem, compel the usual choice of opposite adjacencies (4 and 8 in 2D, 6 and 26 in 3D, and so on).

The rest of the paper consists mainly of reminders of known facts about cellular complexes and of the necessity to conceptually embed the digital space into a cellular complex, thanks to the introduction of inter-voxels with half-integer coordinates.

The bibliography shows the narrow framework of this paper: it consists mostly of the author’s own works, some by Kong and Klette, and topology classics. French studies on discrete image topology and Swedish ones on noncubic voxels are completely ignored.