Cubical cohomology ring of 3D photographs

Abstract

Cohomology groups and the 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. This could facilitate efficient algorithms for the computation of topological invariants in the image context. In this article, we present a constructive process, made up by several algorithms, to compute the cohomology ring of 3D binary-valued digital photographs represented by cubical complexes. Starting from a cubical complex Q that represents such a 3D picture whose foreground has one connected component, we first compute the homological information on the boundary of the object, ∂Q, by an incremental technique; using a face reduction algorithm, we then compute it on the whole object; finally, applying explicit formulas for cubical complexes (without making use of any additional triangulation), the cohomology ring is computed from such information