Applications of Moving Frames

移动框架的应用

• 批准号: 0103944
• 负责人: Peter Olver
• 金额: $ 16.04万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103944&HistoricalAwards=false
• 关键词: Applications; Moving; Frames

0103944OlverThe project will focus on the applications of the proposer's new equivariant theory of moving frames for general Lie group actions. Particular emphasis will be in three key applied directions: analysis and applications of invariant variational problems and invariant partial differential equations in geometry and physics, the design of symmetry-preserving numerical algorithms for approximating differential invariants and integrating invariant differential equations, and object recognition and symmetry detection in computer vision based on differential invariant and noise-resistant joint invariant signatures. The project will include further development of the underlying moving frame theory, particularly in the case of infinite-dimensional Lie pseudo-groups, and the formulation of a proper geometrical foundation for multivariate numerical approximation and interpolation.The recognition and exploitation symmetry is an essential tool in modern mathematics and its applications. This project will continue to develop a new, powerful geometric approach, known as moving frames, to systems that have continuous symmetry. The modern moving frame theory developed by the PI has already witnessed a remarkable range of new applications, including computer vision, for object recognition and symmetry detection, a geometric approach to classical algebra and invariant theory, as well as the design of numerical integration schemes that preserve the underlying symmetry of the problem to be solved. The combination of analytical, geometrical, and numerical advances, coupled with practical applications has proved to be a particularly potent blend of theory and practical tools. This research project will continue the rapid development and application of the moving frame method, concentrating on theoretical developments tied to applications in computer vision, in geometry and physics, and in symmetry-based numerical integration methods

0103944 olver该项目将重点关注提议者新的模棱两可理论的应用程序，以进行一般谎言小组行动。特定的重点将在三个关键的应用方向上：分析和不变的变分问题和不变的部分微分方程在几何和物理学中的分析，对对称性的数值算法的设计，用于近似差异不变性的数值算法，以不变的差异方程式和对象识别型和对象识别型和对象识别型和对象识别型和对象识别型和对象识别型和对象识别型的差异和差异化。该项目将包括进一步发展基础移动框架理论，特别是在无限维二谎言伪组的情况下，以及为多变量数值近似和插值的适当几何基础制定。识别和利用对称性是现代数学及其应用中的必不可少的工具。 该项目将继续将一种新的，强大的几何方法（称为移动帧）开发为具有对称性的系统。 PI开发的现代移动框架理论已经见证了一系列新应用，包括计算机视觉，用于对象识别和对称性检测，一种对经典代数和不变理论的几何方法，以及保留要解决问题的潜在对称性的数值集成计划的设计。 被证明是理论和实用工具的特别有效的融合，结合了分析，几何和数值进步的结合。该研究项目将继续进行移动框架方法的快速发展和应用，重点关注与计算机视觉，几何和物理学以及基于对称的数值集成方法相关的理论发展。