Applications of frames to problems in mathematics and engineering

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

Casazza DMS-1008183 The investigator works on several fundamental questions concerning applications of Hilbert space frames. Much of the work concerns "fusion frames," which provide a natural framework for performing hierarchical data processing. Recent advances in hardware technology have enabled the economic production and deployment of a large number of low-cost components, which in combination enable reliable and efficient operation. Across many disciplines there is a fundamental shift from centralized information processing to distributed or network-wide information processing. Data communication is shifting from point-to-point communication to packet transport over wide area networks where network management is distributed and the reliability of individual links is less critical. Radar imaging is moving away from single platforms to multiple platforms that cooperate to achieve better performance. Wireless sensor networks are emerging as a new technology that potentially enables cost-effective and reliable surveillance. All these applications involve a large number of data streams, which need to be integrated at a central processor. Fusion frames are a recent development designed precisely for these applications. The investigator develops concrete constructions of fusion frames and the algorithms needed for their implementation. Working directly with engineers guarantees that advances here have Immediate implementation. The techniques are also applied to produce Grassmannian frames for quantum physics -- especially Quantum information theory, quantum state tomography, and quantum cryptography. Finally, frame theory has shown that the 1959 Kadison-Singer Problem is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics, and engineering. This project develops the needed frames for resolving what is now one of the most significant open problems in mathematics. We have now entered a new era where the Defense Department, Homeland Security, and industry must process amounts of data that overwhelm even current computer capacities. Especially for defense, it is critical to be able to process these data in real time, which is becoming more and more difficult. The problems caused by this "information overload" are escalating rapidly every year and need to be addressed immediately and quickly. Therefore, it is of vital national interest to automate as much of the information fusion process as possible. But at this time, in the area of information fusion the fundamental mathematics necessary for such an automated system is far behind the current needs. The investigator and his students develop the mathematics necessary for tackling these critical needs. This includes methods for storing and processing information in parallel -- i.e. dividing it into multiple subsystems for simpler processing and then "fusing" the results to get the desired outcomes. It also includes developing distributed sensing networks that can keep an area under surveillance while taking into account severe physical limitations of the systems such as low communication bandwidth, limited signal processing power, limited battery life, or the topography of the surveillance area. Such an automated system must be able to combine a large number of sources of quite disparate information while leaving almost no room for error. The investigator and his students address these needs as well as try to anticipate the next level of difficulties that will arise.