The mathematical description of redundant linear representations of analog signals is called frame theory, and it is central to open problems in many fields of applied mathematics such as approximation theory and numerical as well as statistical analysis. In short, a frame is an (over)complete set of vectors that gives stable expansions. Previous works have measured the error correction capabilities of frames with means derived from geometric or functional analytic formulations. However, for purposes of signal communication and statistical analysis, information theory should provide the guiding principles for optimal design. The goal of the project is to close this gap by combining frame and information theory. Applying information-theoretic optimality criteria to frame design is expected to have a significant impact on all tasks that involve analog signals being transmitted or interpreted in the presence of noise. The investigator seeks to characterize and construct linear signal encoding and decoding maps for two specific areas of applications: Find frames with associated encoding and decoding maps that achieve the information-theoretic optimum for (1) error correction of digital streaming media and (2) classification of signals in noisy microscopy images.
Digital signal transmissions have revolutionized our daily lives, from cellular phones and Voice-over-Internet-Protocol telephony to high-definition television and other streaming media. The use of digitization helps suppress distortions typical for analog devices and allows seemingly faultless communication by incorporating redundancy, that is, repetetive information. However, at times the digital nature of error suppression leads to artifacts that do not resemble the graceful degradation we recall from analog transmissions. The reason for this deficiency is that the general-purpose digital communication protocols are not well-adapted to the properties of the encoded analog signal. The investigator aims to restore the possibility of graceful degradation to streaming media sent across the internet, or to wireless communications in unreliable environments. The optimal design of analog-digital encoding has similar implications for the decoding of analog signals, such as the extraction of information from microscopy images in biology. The reliable monitoring and classification of cell constituents in microscopy images is essential for measuring physiologic responses in cells (such as the stages in the cell cycle) and for detecting anomalous behavior that occurs as a result of disease. Classifying cell behavior is relevant for the evaluation of drugs with high-throughput assays, including the development of cancer-cell growth inhibitors.
