On the cohomology of 3D digital images

Abstract

We propose a method for computing the cohomology ring of three-dimensional (3D) digital binaryvalued 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.

Region-based segmentation of 2D and 3D images with tissue-like P systems

“…Segmentation can be seen as a topologybased process in Digital Imagery, attending to the topological characteristics (connected components and holes, mainly) of each object defined by one unique label. To have at hand representation models for digital images trying to capture full topological and geometrical information of them (González-Díaz and Real, 2005;González-Díaz et al, 2009a,b) could be of help for getting this last goal.…”

Simplicial Relative Cohomology Rings of Digital Images

“…Gonzalez-Diaz and Real [15] have their 14-adjacency algorithm to compute cup products on the simplicial complex. The advantage of this method is tried via a small program visualizing the several steps.…”

Homological optimality in Discrete Morse Theory through chain homotopies

“…The underlying idea is to classify chain homotopies as gradient vector fields on a finite cell complex and vice versa. In order to do this, we use a chain homotopy operator (see González-Dí az and Real (2005), González-Díaz et al (2009)) at algebraic level, and its associated graph-based representation (homological spanning forest or, HSF for short Molina-Abril and Real (2009)) at combinatorial level.…”