Applications of Moving Frames

移动框架的应用

• 批准号: 1108894
• 负责人: Peter Olver
• 金额: $ 34.16万
• 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: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1108894&HistoricalAwards=false
• 关键词: Applications; Moving; Frames

OlverDMS-1108894 This project is centered on the further development of the theoretical tools and extension of the range of applications of the equivariant method of moving frames, that was first introduced in the late 1990's by the principal investigator in collaboration with a visiting postdoc, and since developed in many directions by a number of research groups worldwide. Of particular note are: First, new directions in image processing, object recognition and shape matching, involving combinations and comparisons of moving frame-based signatures and joint invariant histograms. Second, analysis and classification of invariant variational problems and invariant curve and surface flows using moving frame techniques applied to the invariant variational bicomplex, motivated by applications in geometric (bio-)mechanics, image denoising and smoothing, interface dynamics, and integrable soliton differential equations. Third, applications of the newly completed moving frame-based structure theory for infinite-dimensional Lie pseudo-groups that arise as symmetry groups of a wide variety of physical systems, including fluid mechanics, gauge theories, and solitons. Fourth, a detailed analysis of the ramifications of a new and surprising phenomenon of "dispersive quantization" that has been recently shown to arise in very basic linear models of dispersive wave equations on periodic domains, as well as, previously, in quantum mechanical systems and optics, where it is known as the Talbot effect. The project is based on the exploitation of symmetry in a wide range of mathematics and its applications, through new, powerful methods that were inspired by classical tools coming from pure geometry. Besides further development of the underlying mathematical theory and tools, the primary focus is on three interconnected areas of application: First, object recognition in digital images from various sources, under various notions of when two objects can be considered the same: e.g., under rigid motions; under change in camera viewing angle; under deformations of a prescribed type, etc. Second, analysis of dynamical equations incorporating intrinsic physical and mathematical symmetries that arise in a wide range of applications in biomechanics and materials, fluid mechanics, image processing, and interface motions. Third, understanding and exploring a new, surprising and potentially important phenomenon, in which, under the assumption of periodicity, solutions are "fractalized" at irrational times and "quantized/localized" at rational times, that was recently shown to arise in many basic linearly dispersive wave models governing a very broad range of wave motions, including quantum mechanical systems.

OlverDMS-1108894 该项目的重点是进一步发展的理论工具和扩大的应用范围的等变方法的移动框架，这是第一次介绍了在20世纪90年代后期的首席研究员与来访的博士后合作，并在许多方向发展的一些研究小组在世界各地。 特别值得注意的是：首先，图像处理、对象识别和形状匹配的新方向，涉及基于移动帧的签名和联合不变直方图的组合和比较。 第二，分析和分类不变变分问题和不变曲线和曲面流使用移动框架技术应用于不变变分双复体，在几何（生物）力学，图像去噪和平滑，界面动力学，可积孤子微分方程的应用程序的动机。 第三，应用新完成的移动框架为基础的结构理论的无限维李伪群出现的各种物理系统，包括流体力学，规范理论和孤子的对称群。 第四，详细分析了一种新的令人惊讶的“色散量子化”现象的后果，这种现象最近被证明出现在周期域上色散波动方程的非常基本的线性模型中，以及以前在量子力学系统和光学中，在那里它被称为塔尔博特效应。 该项目基于在广泛的数学及其应用中利用对称性，通过新的，强大的方法，这些方法受到来自纯几何的经典工具的启发。 除了基础数学理论和工具的进一步发展外，主要重点是三个相互关联的应用领域：首先，在两个对象可以被认为是相同的各种概念下，来自各种来源的数字图像中的对象识别：例如，在刚性运动下;根据相机视角的变化;根据规定类型的变形，等等。第二，分析动力学方程，将内在的物理和数学对称性，出现在生物力学和材料，流体力学，图像处理和界面运动的广泛应用。 第三，理解和探索一个新的，令人惊讶的和潜在的重要现象，其中，在周期性的假设下，解决方案是“分形”在无理时间和“量化/本地化”在合理的时间，这是最近出现在许多基本的线性色散波模型，包括量子力学系统，管理非常广泛的波动。