Collaborative Research: Multiscale Proximity Algorithms for Optimization Problems Arising from Image/Signal Processing

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Restoring images or signals from limited available data is required in variety of applications, including parallel magnetic resonance imaging in medical applications and fingerprint and face recognition in security identification. Such image or signal reconstruction problems are often modeled as large-scale optimization problems. This research project aims to develop more efficient computational algorithms for solving these optimization problems. Results of this project are anticipated to have an impact in practical applications. In particular, the numerical schemes under development are expected to support medical imaging research and assist in improving the accuracy of clinical decisions. Senior undergraduate and graduate students are trained in the course of this project. This research project aims to develop multiscale proximity algorithms for optimization problems arising in image or signal processing. Signal processing problems of practical importance, such as incomplete data recovery, compressive sensing, and matrix completion, are modeled as optimization problems that have non-differentiable objective functions. A signal of interest naturally has a hierarchical structure or allows itself to be sparsely represented in a multiscale analysis. Multiscale analysis, as a convenient tool for computation, however, is mainly used to sparsify the underlying signal in formulating optimization problems; it is not fully exploited in development of efficient algorithms for optimization problems. In this project, to make systematic use of the hierarchical structure that exists in optimization problems of interest, the investigators will synthesize and combine multiscale analysis and proximity algorithms to solve the problems in an accurate and computationally efficient way.