On the Cohomology of 3D Digital Images

TL;DR

This paper introduces a method to compute the cohomology ring of 3D digital images using simplicial complexes, enabling topological analysis and visualization of digital structures.

Contribution

It presents a novel approach to determine the cohomology ring of 3D digital images via simplicial complexes, with implementation and visualization tools.

Findings

Successfully computes cohomology rings of 3D images

Visualizes (co)cycles and cup products

Demonstrates method on example images

Abstract

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 pictures is tested by means of a small program visualizing the different steps of the method. Some examples concerning topological thinning, the visualization of representative (co)cycles of (co)homology generators and the computation of the cup product on the cohomology of simple pictures are showed.