Computational topology for image understanding (topologie computationnelle pour la compréhension d'image)

点击翻译按钮获取中文摘要

My general goal is to develop an emerging research field called computational topology in the context of analysis and processing of multidimensional data. Rooted in both Computer Science and Mathematics, computational topology overlaps with computational geometry and combinatorial topology. It has applications in several domains of science and engineering, in particular, in imaging and multimedia. The specific goals of my proposal are to: - Develop the multi-parameter persistent homology introduced by Carlsson at al., also advanced by Frosini at al. In particular, investigate connections between reductions of the persistence diagram of a vector-valued measuring function and those of the cellular complex of a discrete Morse function. This study will offer new mathematical tools for realizing a system of observer-based 3D media contents retrieval. This is a joint project with collaborators from the universities of Bologna, Modena - Reggio Emilia, and Bishop's. - Apply the above concepts to complement my earlier work on detection and classification of singular zones in multidimensional data. The multi-parameter persistence will permit to extract image characteristics which persist in small changes in both spatial and dynamical resolution. - Extend homology reduction and coreduction algorithms to the cohomology, including the computation of its ring structure. This is a joint project with collaborators from the Jagiellonian University of Krakòw. The next step is to promote the use of these algorithms in computational electromagnetism. - Study modern extensions of the Morse theory to functions with non-isolated singularities in the context of imaging science.