The goal of this projects is to generate an underlying mathematical and computer model framework which help scientific and engineer communities to exploit homology tools (those related to the notion of n-dimensional hole) in a doxel-based 4D discrete ambiance. On the mathematical side, we will develop a homological algebra framework dealing with a new efficient representation schema for digital images created by the researchers Helena Molina-Abril and Pedro Real:Homological Spanning Forest.
We will demonstrate its computational nature and close connection to applications and will develop useful,sequential and also parallel software for computing topological information of 4D objects and images. By topological information, we mean Euler characteristic, Betti numbers, topological skeletons and Reeb graphs, representative cycles of homology generators, contractibility, transformability and classification of cycles, shortest paths and advanced homology and cohomology operations and invariants, (relative homology groups, digital fundamental groups, homotopy groups, cohomology algebra, homology A(infty) algebra, ...).
With this philosophy of geotopologically representing the object by means of trees, and linking in a strong way topology, parallelism, analysis and recognition, we want to progress in the issues of lossless compression, topology-based digital image proecessing, topological pattern recognition, simulation on images using a topology-based physical model and efficient transmission through internet channel.