Collaborative Research: Compression of Geometry Datasets

这是一个由DMS-0138458、DMS-0221642和DMS-0221669下的货运计划资助的合作项目。这是一个信息时代。无论是科学计算还是逆向工程，无论是遥感还是医学，每天都会产生令人难以置信的分辨率和精致细节的数据集。这些数据集通常具有几何结构，其中包含关于数据及其应用的重要信息。这种几何数据集的有用性取决于我们有效处理它们的能力，无论是用于存储、传输、视觉显示、关联还是与来自其他模式的数据进行注册。压缩是所有这些应用中共同的关键问题，目前的数据处理技术不能提供应用所要求的高效和几何逼真的表示。事实上，一个令人满意的数据处理平台不会通过当前技术的渐进进步来创建，而是通过对如何用固有几何表示大数据集的基础研究来创建的。为了追求必要的进步，已经组建了一支由研究人员和应用程序开发人员组成的团队，他们代表了包括数学、计算机科学和工程在内的几个学科的专业知识。作为这项研究的驱动力，已经选择了特别有针对性的应用程序(DTED、逆向工程、物理模拟)来指导最关键和最有意义的问题的制定，以及解决这些问题的测试方法。该团队正在对数据集的几何表示和压缩进行基础研究，方法是建立新的数学理论，包括确定性模型、适当的应用特定误差度量(例如Haussdorf距离而不是Lp范数)、基于Kolmogorov熵确定最佳压缩的信息理论，以及开发用于表示在熵意义上接近最优的数据集的非线性方法。

DMS-0138458Peter SchroderDMS-0221642Ronald A. DevoreDMS-0221669Mathieu DesbrunThis is a collaborative project funded by the CARGO program underDMS-0138458, DMS-0221642, and DMS-0221669. This is the Age of Information. Whether it be in scientific computation or reverse engineering, in remote sensing or medicine, data sets of incredible resolution and exquisite detail are createddaily. These data sets often have geometric structure which contains important information about the data and its application. The usefulness of such geometric datasets rests on our ability to process them efficiently, whether it be for storage, transmission, visual display, correlation, or registration against data from other modalities. Compression is the common critical issue in allthese applications.Current data processing technology does not provide the efficient and geometrically faithful representations demanded by applications. In fact, a satisfactory data processing platform will not be created by incremental advances of current technology but rather through a fundamental investigation of how to represent large data sets with inherent geometry. To pursue the necessary advances a team of researchers and application developers representing expertise fromseveral disciplines including mathematics, computer science, and engineering has been assembled. As drivers of the research, specially targeted applications (DTED, reverse engineering, physical simulation) have been selected to guide the formulation of the most critical and meaningful problems, as well as testing approaches to solving them. The team is carrying out fundamental investigations on the representation and compression of data sets with geometry by building new mathematical theoryincluding deterministic models, appropriate application specific error metrics (e.g., Haussdorf distance ratherthan Lp norms), an information theory based on Kolmogorov entropy to determine the optimal compression, and the development of nonlinear methods for representing the data sets which are near optimal in the entropy sense.