Computer algebra algorithms and image set compression and encryption

我们建议从计算机科学的两个截然不同的领域来研究问题：计算机代数和图像处理。计算机代数是研究以符号而不是数值来执行数学计算的算法。由于其处理符号量的能力和它没有数值错误的能力，计算机代数在数学研究、工程和数学教育中有许多应用。尽管这一领域取得了进展，但许多问题仍然难以解决。在计算机代数中遇到的一个常见问题是所谓的中间表达式膨胀。在许多问题中，输入的大小很小，但计算过程中中间结果的大小可能会显著增加。如果这种增长得不到控制，算法就会因为时间和内存的限制而无法完成计算。计算机代数研究的目标是为计算机代数中的各种问题设计和分析算法来控制这种增长。这项研究将提高我们对这些问题的理解，并对现有算法进行实际改进。在另一个方向上，我们建议研究与大图像集存储相关的问题。*我们计划研究压缩方法以减少存储需求，此外，我们还对压缩、加密和高效图像检索之间的相互作用感兴趣。随着大型图像数据库的使用变得越来越普遍(例如医疗和卫星图像)，这项研究的结果可能会有许多应用。

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).