Cubical Cohomology Ring of 3D Pictures

Cubical cohomology ring of 3D photographs

Cohomology groups and the cohomology ring of threedimensional (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 computat...

Cohomology Ring of 3D Cubical Complexes

On the Cohomology of 3D Digital Images

We propose a method for computing the cohomology ring of three–dimensional (3D) digital binary–valued pictures. We obtain the cohomology ring of a 3D digital binary–valued picture I, via a simplicial complex K(I) topologically representing (up to isomorphisms of pictures) the picture I. The usefulness of a simplicial description of the “digital” cohomology ring of 3D digital binary– valued pict...

Computing The Cubical Cohomology Ring

The goal of this work is to establish a new algorithm for computing the cohomology ring of cubical complexes. The cubical structure enables an explicit recurrence formula for the cup product. We derive this formula and, next, show how to extend the Mrozek and Batko [7] homology coreduction algorithm to the cohomology ring structure. The implementation of the algorithm is a work in progress. Thi...

Cubical Cohomology Ring: Algorithmic Approach

A cohomology ring algorithm in a dimension-independent framework of combinatorial cubical complexes is developed with the aim of applying it to the topological analysis of high-dimensional data. This approach is convenient in the cup-product computation and motivated, among others, by interpreting pixels or voxels in digital images as cubes. The S-complex theory and so called co-reductions are ...