Cubical Cohomology Ring of 3D Photographs

TL;DR

This paper introduces formulas and algorithms to directly compute the cohomology ring of 3D digital images represented as cubical complexes, avoiding triangulation and enabling efficient topological analysis of voxel-based objects.

Contribution

It provides a novel method for directly computing the cohomology ring of 3D cubical complexes from digital images without triangulation, improving efficiency.

Findings

Efficient formulas for cohomology ring computation on cubical complexes.

Incremental technique for boundary cohomology computation.

Face reduction algorithm for whole object cohomology.

Abstract

Cohomology and cohomology ring of three-dimensional (3D) objects are topological invariants that characterize holes and their relations. Cohomology ring has been traditionally computed on simplicial complexes. Nevertheless, cubical complexes deal directly with the voxels in 3D images, no additional triangulation is necessary, facilitating efficient algorithms for the computation of topological invariants in the image context. In this paper, we present formulas to directly compute the cohomology ring of 3D cubical complexes without making use of any additional triangulation. Starting from a cubical complex $Q$ that represents a 3D binary-valued digital picture whose foreground has one connected component, we compute first the cohomological information on the boundary of the object, $\partial Q$ by an incremental technique; then, using a face reduction algorithm, we compute it on the whole object; finally, applying the mentioned formulas, the cohomology ring is computed from such information.