The investigator and his colleagues study new methods to recover a signal from a nonlinear processing scheme. Recently two far-reaching discoveries have been made that connected the nonlinear information (magnitudes of frame coefficients) to certain scalar products in larger embedding spaces. Thus the initial problem, which is fundamentally nonlinear, is recast into a linear reconstruction problem coupled with a rank-one approximation problem. When the linear redundant representation is associated with a group representation (such as Weyl-Heisenberg, or windowed Fourier transform), then the relevant tensor operators inherit this invariance property. Thus a fast (nonlinear) reconstruction algorithm is possible. This approach suggests a new signal representation model, where signals are not represented simply by vectors in a Hilbert space, but rather by operators in a larger dimensional Hilbert-Schmidt like-space, similar to the quantum state theory. These methods use results from a wide range of mathematical areas such as harmonic analysis, operator theory, and polynomial algebras.
Results of this project have a practical application to areas such as signal processing, optical communication, quantum computing, and X-ray crystallography. Besides advancing the scientific understanding in applied harmonic analysis, this project broadens the two-way communication between mathematics and electrical engineering while promoting teaching, training and learning. The investigator is training graduate students for a globally competitive STEM force through his contacts with industry and international research labs. The project is supported by the Division of Mathematical Sciences and the Division of Computing and Communication Foundations.