AF: Small: Analyzing Spaces and Scalar Fields via Point Clouds

AF：小：通过点云分析空间和标量场

• 批准号: 1116258
• 负责人: Tamal Dey
• 金额: $ 49.98万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1116258&HistoricalAwards=false
• 关键词: AF; Small; Analyzing; Spaces; Scalar

Recent advances in scanning technology and general progress in science and engineering have made it easy to sample a geometry presented explicitly or implicitly. In some cases, sampling allows the data to be equipped with values of a scalar function defined on the sampled domain. The goal is to process the resulting point cloud data (PCD) for approximating different geometric and topological structures of the original space and/or functions defined on them. How can this be achieved is the main agenda of this award.The extraction of geometric and topological structures from a PCD begins with the understanding of these structures on the original sampled space. In some cases one needs to adapt existing mathematical concepts to fit the discretization inherent in algorithmic development. In others new analysis techniques and new algorithmic tools need to be developed. PIs expect the more of the latter to tame the curse of dimensionality and scale. The two PIs bring their expertise in computational geometry and computational topology to the board to conduct the project. A central problem in data analysis is to design computational methods for representing and retrieving information from massive and high-dimensional data. Various scientific, engineering, and social studies such as the ones in product development, medicine, economics, climate, disease control produce data that presumably sample a hidden parameter space sitting in a high dimension. They will benefit from the techniques developed in this project. By the very nature of the problems, this project will enhance a synergy between fields such as mathematics, theoretical computer science, and data analysis.Other than standard research forums, results from the project will be disseminated through course notes, tutorials, and web-pages to reach wider audience. Graduate students supported by the project will develop skills in mathematics and theoretical computer science, most notably in geometry and topology and also in writing robust, efficient and user-friendly software.

扫描技术的最新进展以及科学和工程的普遍进步使得对显式或隐式呈现的几何形状进行采样变得容易。在某些情况下，采样允许数据配备在采样域上定义的标量函数的值。其目标是处理所得到的点云数据（PCD），以近似原始空间的不同几何和拓扑结构和/或在其上定义的函数。如何实现这一点是该奖项的主要议程。从PCD中提取几何和拓扑结构始于对原始采样空间上这些结构的理解。在某些情况下，需要调整现有的数学概念，以适应算法开发中固有的离散化。在其他方面，需要开发新的分析技术和新的算法工具。PI希望后者更多地驯服维度和规模的灾难。这两名PI将他们在计算几何和计算拓扑方面的专业知识带到董事会来执行该项目。数据分析中的一个核心问题是设计计算方法来表示和检索海量高维数据中的信息。各种科学、工程和社会研究，如产品开发、医学、经济学、气候、疾病控制等，产生的数据可能会对高维中的隐藏参数空间进行采样。他们将受益于该项目中开发的技术。根据问题的本质，该项目将加强数学、理论计算机科学和数据分析等领域之间的协同作用。除了标准的研究论坛外，该项目的结果将通过课程笔记、教程和网页传播，以达到更广泛的受众。由该项目支持的研究生将培养数学和理论计算机科学方面的技能，特别是几何和拓扑学，以及编写强大，高效和用户友好的软件。