Collaborative Research: Compression of Geometry Datasets

dms -0138458、DMS-0221642和DMS-0221669下的CARGO项目资助的合作项目。这是信息时代。无论是科学计算还是逆向工程，无论是遥感还是医学，每天都有令人难以置信的分辨率和精美细节的数据集产生。这些数据集通常具有几何结构，其中包含有关数据及其应用的重要信息。这些几何数据集的有用性取决于我们有效处理它们的能力，无论是存储、传输、视觉显示、关联还是与其他模式的数据进行注册。压缩是所有这些应用程序中常见的关键问题。当前的数据处理技术不能提供应用程序所要求的高效和几何上忠实的表示。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式进步来创建，而是通过对如何用固有几何表示大型数据集的基础研究来创建。为了追求必要的进步，来自数学、计算机科学和工程等多个学科的研究人员和应用程序开发人员组成了一个团队。作为研究的驱动因素，特别有针对性的应用（DTED，逆向工程，物理模拟）已经被选择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。该团队正在通过建立新的数学理论，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规范），基于Kolmogorov熵的信息理论来确定最佳压缩，以及用于表示在熵意义上接近最优的数据集的非线性方法的发展，对几何数据集的表示和压缩进行基础研究。

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.