Decomposing Cavities in Digital Volumes into Products of Cycles

The homology of binary 3–dimensional digital images (digital volumes) provides concise algebraic description of their topology in terms of connected components, tunnels and cavities. Homology generators 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 algorithmic computation of a coproduct operation that can be used to decompose 2–cycles into products of 1–cycles (possibly trivial). This coproduct provides means of classifying different kinds of cavities; in particular, it allows to distinguish certain homotopically non-equivalent spaces that have isomorphic homology. We define 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 simplicial subdivision of faces of the voxels.