Signal Processing with Computational Topology

Abstract

The objective of this research is to analyze the mathematical methods of signal processing on abstract geometric spaces of the given sensor network, develop an algorithm on a tangible software development environment, and visualize the usefulness of those algorithms. Specifically, the scope of this research covers the notions of persistence in computational topology and how this notion can be observed in certain mathematical features of the given network. Moreover, we will be analyzing sheaves constructed over a network of sensors in the topological space, from the point of persistence modules and persistent sheaf cohomology, visualizing the persistent sheaf cohomology over barcode diagrams. The research objective is analyzed in detail in the introduction. In the first part of the main body, we have analyzed the mathematical background and previous research works in this domain to derive insight into developing an algorithm. In the second section of the main body, discussed and developed the algorithms, based on the computational topology theoretic approaches. In this paper, we will develop the algorithms to construct simplicial complexes, obtain filtration, construct sheaves, and compute persistent sheaf cohomology. The resulting barcode diagrams are analyzed in terms of certain notions, e.g. robustness, stability, and the Betti numbers. In the end, the primary findings will be succinctly expressed in several points.