SONAR SIGNATURE ENHANCEMENT USING QUASI-PERIODIC FACTORIZATION TECHNIQUES

This thesis is a part of a larger endeavor to yield a topological classification technique for use in many signal-processing settings. The first problem we encountered was a lack of data with minimal artifacts that was easily adjustable during reproduction. To date we have developed and tested a simple ray-trace simulator that handles double bounce reflections and we have verified its output against real experiments and vice-versa. Following this success, we decided to improve this simulator with a more flexible, boundary element method solver. This boundary element method solver yields sonar data that is from replicated experiments of the larger project, Dr. Robinson’s ‘Homological Features for Sonar Target Classification,’ so as allow us to perform parallel analysis to the real data. There, our technical approach includes using intrinsic, low-dimensional persistent homological features pulled from “raw” (high-dimensional) sonar echoes on which one can employ machine-learning techniques. Further, the boundary element method solver allowed us to verify the application of the theorem presented in (Ghrist and Robinson) in the form of the Quasi-Periodic Low Pass Filter algorithm from (Robinson "A Topological Lowpass Filter for Quasiperiodic Signals"). The theorem states that a set of geometric target symmetries is present in the sonar data and the algorithm allows us to find this set of symmetries. However, as currently implemented, the algorithm is highly constrained so we verified its effectiveness by applying it to the trivial quasi-periodic factorization of a mapping from the group of SO(3) rotations to R^9. Then, we used output from the boundary element method solver, noisy sonar simulations using targets each by a random element of SO(3), to extend the application of the quasi-periodic factorization algorithm closer to real data implementation. Ultimately this demonstrated the readiness of the algorithm for detecting geometrical symmetries in targets in real sonar data, contributing to the progress of topological analysis in signal processing.