Holes and Genus of 2D and 3D Digital Images

"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 b 1, while b 0 counts components and b 2 cavities. Connected sets with equal b 1 and b 2 must match topologically when b 1 = 0 (implying simple connectedness). When b 1 ≠ 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.