Computer algebra algorithms and image set compression and encryption

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We propose to study problems from two distinct areas of computer science: computer algebra and image processing.Computer algebra is the study of algorithms which perform mathematical computations symbolically rather than numerically.  Because of its ability to handle symbolic quantities and its lack of numerical errors, computer algebra has many applications in mathematical research, engineering, and mathematical education.  Despite advances in this area, many problems remain impractical to solve.  A common issue encountered in computer algebra is the so-called "intermediate expression swell."  In many problems, the size of the input is small but the size of the intermediate results during the calculations can grow significantly.  If this growth is not controlled, an algorithm cannot finish its computation due to time and memory constraints.  The goal in the proposed research in computer algebra is to design and analyze algorithms to control this growth for a variety of problems in computer algebra.  This study will improve our understanding of such problems and provide practical improvements to the current algorithms.In a separate direction, we propose to study issues related to the storage of large image sets.  We plan to investigate compression methods to reduce storage requirements, as well as encryption algorithms for security.  We are also interested in the interaction among compression, encryption, and efficient image retrieval.  The results from this study  have many applications as the use of large image databases have become more common (e.g. medical and satellite images).