Hilbert Space Frames and their applications

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This award supports the research program of the Principal Investigator in the area of "frame theory," a subject concerned with the mathematical representation of signals, images, and information in general. This project concerns applications of Hilbert space frame theory---one of the most active and applied subjects in mathematics today, with applications to pure mathematics, applied mathematics, engineering, medicine, computer science, and more. Frame theory is fundamental to the digitalizing of information for processing and storing. One part of this project will develop new methods for "phase retrieval," which has applications to X-ray crystallography, electron microscopy, astronomical imaging, optics, quantum physics, and more. Phase retrieval will even be needed to align the mirrors in the new James Webb Space Telescope scheduled for launch in 2018. Another part of this research project involves developing "fusion frames" for modern applications. Recent advances in hardware technology have enabled the economic production and deployment of a large number of low-cost components, which through collaboration enable reliable and efficient operation. Wireless sensor networks have emerged as a new technology with the potential to enable cost-effective and reliable surveillance. Radar imaging is moving away from single platform to multiple platforms that cooperate to achieve better performance. Recent advances in sensor hardware provide an unprecedented capability to detect the production, deployment, and use of chemical and biological weapons. All of these applications involve a large number of data streams, which need to be integrated at a central processor---often at multiple levels of information fusion. Fusion frames are being designed to deal with these problems.Phase retrieval is the century-old problem of reconstructing a function (such as a signal or an image) from the magnitude of its Fourier transform. Its origin comes from the fact that in many scenarios, detectors can only record intensity measurements and lose the phases. This project is pushing phase retrieval in several relatively new directions using frame theory. Determining the minimal number of measurements needed to ensure that a signal or image can be uniquely reconstructed from its intensity measurements (without any prior knowledge) will provide crucial insight into designing detectors. This project will also use an innovative combination of tools from time-frequency analysis, quantum physics, and discrete geometry to provide novel measurement designs that are backed up by a rigorous mathematical theory and efficient numerical algorithms. Another direction of the research will be to extend the existing approach used for phase retrieval, which uses vector-based measurements, to higher-rank projections. This will have important applications to crystal twinning. Fusion frames offer great promise to a host of problems surrounding information fusion but need much better designs, algorithms, and much more generality to fit current needs. This project will address all of these problems to bring fusion frames into the mainstream.