The field of image analysis studies the topology and geometry of digital images and data sets. The subfield of combinatorial image analysis studies their combinatorial properties. Algebraic topology is a field of mathematics concerned with the study of deformation-invariant properties of geometrical objects. Our recently created Andalusian research group FQM-369 “Combinatorial Image Analysis” develops methods and tools for combinatorial image analysis, which apply the theory of computational topology. In this talk, we will introduce the three main tasks in which our group is involved:

(1) Well-composed cell complexes. The 2D manifold that is the surface bounding a real 3D object, might appear to be non-manifold in the geometric realization of the cubical complex Q(I) associated to a discrete representation of the object after the acquisition process. This task deals with the construction of a cell complex K(I) that is homotopy equivalent to Q(I) and whose boundary surface is a union of 2D manifolds.

(2) Cup products on general cell complexes. The cup product on cohomology encodes more information than homology, but has traditionally been computed only for cubical and simplicial complexes. Recently our group developed techniques that allow to compute the cup product on cell complexes X, where X is a quotient, a Cartesian product, or the result of merging cells of highest dimension.

(3) Extending persistent homology. The incremental algorithm for persistent homology by Edelsbrunner et al., is currently the de facto standard for extracting topological information, especially Betti numbers, when an object is seen as a sequence of additions from a point cloud data. A first aim of this task is to extend persistent homology to cell complexes obtained from the given one by merging, removing cells or edge contractions.

Collaborators: Maria Jose Jimenez and Belen Medrano (Applied Math Dept., University of Seville, Spain), Walter G. Kropatsch (PRIP group, Vienna University of Technology, Austria), Adrian Ion (PRIP group and Institute of Science and Technology, Austria), Ron Umble (Math. Dept., Millersville University of Pennsylvania, USA).