Decomposing cavities in digital volumes into products of cycles

Abstract

The homology of binary 3{dimensional digital images (digi- tal volumes) provides concise algebraic description of their topology in terms of connected components, tunnels and cavities. Homology gener- ators corresponding to these features are represented by nontrivial 0{ cycles, 1{cycles and 2{cycles, respectively. In the framework of cubical representation of digital volumes with the topology that corresponds to the 26{connectivity between voxels, we introduce a method for algorith- mic computation of a coproduct operation that can be used to decom- pose 2{cycles into products of 1{cycles (possibly trivial). This coproduct provides means of classifying di erent kinds of cavities; in particular, it allows to distinguish certain homotopically non-equivalent spaces that have isomorphic homology. We de ne this coproduct at the level of a cubical complex built directly upon voxels of the digital image, and we construct it by means of the classical Alexander-Whitney map on a sim- plicial subdivision of faces of the voxels.