The research studies two problems, each exploiting redundancy of representations in mathematics and engineering, and develops new methods to recover a signal from a nonlinear processing scheme. The first problem is related to signal reconstruction from magnitudes of a redundant linear representation (the so-called phase retrieval problem). The second problem involves a geometric analysis of frames and connections to deep problems in mathematics (such as the Kadison-Singer problem). This analysis leads to faster methods that offer better quality and resolution of the reconstructed signals in applications from X-ray crystallography, data communication on fiber optics, and speech processing. Undergraduate and graduate students involved in this project are trained for a globally competitive STEM workforce by learning to develop new mathematical tools to solve real-world problems.
Phaseless Reconstruction and Geometric Analysis of Frames is a three-year, $106K grant.