Topological Recognition And Computation Of 4d Digital Images Via Hsf Model
4D-HSF will deliver a framework to address the complex problems involving in generating, adapting and applying algebraic topology-based methods for the analysis and recognition of digital images up to dimension 4. In fact, the project will design specific tools for exploiting fractality (related to degrees of self-similarity) and homotopy (related to “holes” as n-dimensional loops”) within this discrete setting. 4D-HSF will provide a mathematical-computational framework of analysis/recognition, which helps the scientific and engineering communities to exploit homotopy tools in 4D digital image setting, to demonstrate its computational nature and close connection to applications, and to develop useful and efficient software to compute homotopy information using a topological representation model for 4D-objects, called “Homotopy Spanning Forest” (HSF, for short). The HSF representation can be modulated and manipulated at three different levels: (a) [pACC model] at the own (co)homotopy level, handling generalizations of the notion of abstract cell complex and elementary operations of cell collapse; (b) [chain-integral model] at (co)homology level, operating with suitable generalizations of the notion of chain-complex and maps between chain-complexes well-defined at homology level; (c) [fractal-topological model] at fractal level which is transversal to the both previous level and efficiently manages the local-global topological information. From an analytical perspective, the capacity of this flexible topological model will allow a fast sequential and parallel computation of topological invariants and features as simple as the Euler-Poincaré characteristic and as complex as the homotopy groups, going through by the determination of classical (co)homology operations, the cup product in cohomology or new notions of fractal-topological dimensions. At level of modelization and recognition, the 4D-HSF framework will allow to generate in an automatic way topological squeletons and textures, region-adjacency-graphs of segmentations at different levels-of-detail, exclusively using topological techniques and without requiring geometric or integral-differential calculus methods. In this way, the 4D-HSF framework can be perfectly added as an independent conceptual layer to existing models of computer aided design. From a practical perspective, several implementations for generating the HSF and computing the topological parameters must be tested. Taking into account that HSF generation can be implemented only with integer arithmetic, and due that it is predictable that the rest of concepts in the HSF framework comprises a huge degree of parallelism, it seems that the most appropriates devices for an efficient implementation are FPGAs. In this respect, the benefits of FPGAs in computation divided by power are obvious. However, due to the uncertainty of selecting the most convenient topological parameters for the recognition of 4D images, standard CPUs and GPGPUs may be most suitable for the recognition stages.
PI: Pedro Real Jurado
Type: Plan Estatal 2013-2016 Excelencia – Proyectos I+D
Funding by: Ministerio de Economía y Competitividad
Start date: 30-12-2016
End date: 29-12-2019
Fernando Díaz del Río
Helena Molina Abril.