Cups Products in Z2-Cohomology of 3D Polyhedral Complexes

Let $I=({\mathbb{Z}}^3,26,6,B)$ be a 3D digital image, let $Q(I)$ be the associated cubical complex and let $\mathrm{\partial }Q(I)$ be the subcomplex of $Q(I)$ whose maximal cells are the quadrangles of $Q(I)$ shared by a voxel of $B$ in the foreground – the object under study – and by a voxel of ${\mathbb{Z}}^3\setminus B$ in the background – the ambient space. We show how to simplify the combinatorial structure of $\mathrm{\partial }Q(I)$ and obtain a 3D polyhedral complex $P(I)$ homeomorphic to $\mathrm{\partial }Q(I)$ but with fewer cells. We introduce an algorithm that computes cup products on $H^{\ast }(P(I);{\mathbb{Z}}_2)$ directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in ${\mathbb{R}}^3$.