Cups Products in Z2-Cohomology of 3D Polyhedral Approximations

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