Cups Products in Z2-Cohomology of 3D Polyhedral Complexes
This work addresses a computational bottleneck in topological data analysis for 3D digital images, offering an incremental improvement in efficiency for researchers in computational topology and image processing.
The paper tackles the problem of computing cup products in the Z2-cohomology of 3D polyhedral complexes by simplifying the combinatorial structure of digital images to reduce cell count, resulting in an algorithm that directly computes these products from the combinatorics and is applicable to any polyhedral complex in R3.
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 $\mathbb{R}^3$.