Holes and Genus of 2D and 3D Digital Images

Abstract

"Hole" has been a confusing idea in the 3D digital literature. We replace counting holes by the clear geometrical idea of counting non-separating cuts, and show that this gives the Betti number b1, while b0 counts components and b2 cavities. Connected sets with equal b1 and b2 must match topologically when b1 = 0 (implying simple connectedness). When b1 ≠ 0, contrary to digital folklore, they need not. This paper is a conceptually self-contained introduction for computer scientists to these numbers of 2D and 3D images, and to other topological features such as Euler and linking numbers.