Theory and Application of Hilbert Space Frames

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

This project is concerned with a number of important problems related to phase retrieval and frame theory. In several engineering applications the phase of a signal is lost during processing. In the first part of the project the principal investigator (PI) will resolve several problems concerning phase retrieval. Phase retrieval has applications to x-ray crystallography, electron microscopy, astronomical imaging, optics, x-ray tomography, and more, such as in aligning the mirrors of the new James Webb Space Telescope. Combined with Fusion Frame Theory, phase retrieval helps to design sensor networks for detecting chemical, biological, radiological and nuclear weapons. The PI will also study fundamental problems in biangular tight frames. Biangular tight frames (BTF) have a myriad of applications including quantum state tomography, quantum cryptography, communication theory, spherical designs, and strongly regular graphs. This research will address some of the oldest problems in this area, such as the 150 year old Hadamard Conjecture, which are holding up the applications. This part of the project has applications to digital radio protocols designed to work in difficult (low signal-to-noise ratio plus multipath propagation) conditions on shortwave bands, balanced repeated replication, coded aperture spectrometry, feedback delay networks, Plackett-Burman design of experiments, among others. This project has a significant student training component, as several of the PI's Ph.D. students and one undergraduate student are all directly involved in the research. In the first part of this project the PI will study phase retrieval. A family of unit vectors does phase retrieval if every vector in the space is uniquely determined (up to a universal phase) by the modulus of its inner product with each of the unit vectors. The main goal here is to construct families of vectors doing phase retrieval and to find out the least number of vectors needed to do phase retrieval in each dimension. The answer is known for two- and three-dimensional real and complex spaces and the approach is to solve the problem in all dimensions by induction with respect to the dimension. The second part of this project deals with biangular tight frames, which are families of unit vectors with the property that the modulus of the inner product of any two distinct vectors takes one of two values. The goal is to construct infinite families of such sets. The PI has developed a new tool for this construction. There are many known equiangular tight frames (ETF) and the new tool will turn these ETFs into BTFs. The last part of the project aims to produce a solution to the celebrated Hadamard Conjecture using a new approach for this problem developed by the PI. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.