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.