Applications of Moving Frames

OlverDMS-1108894这个项目的中心是运动标架等变法理论工具的进一步发展和应用范围的扩展，该方法于20世纪90年代末由首席研究员S与一名访问博士后合作首次引入，此后由世界各地的许多研究小组在多个方向上发展起来。特别值得注意的是：第一，图像处理、目标识别和形状匹配的新方向，涉及基于运动帧的签名和联合不变直方图的组合和比较。第二，在几何(生物)力学、图像去噪和平滑、界面动力学和可积孤子微分方程等方面的应用，利用移动标架技术对不变变分问题和不变曲线曲面流动进行分析和分类。第三，新完成的基于运动标架的结构理论在无限维李伪群中的应用，这些伪群是作为各种物理系统的对称群出现的，包括流体力学、规范理论和孤子。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的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，它被称为Talbot效应。该项目的基础是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，主要的重点是三个相互关联的应用领域：第一，在不同来源的数字图像中的对象识别，在不同的概念下，当两个对象可以被认为是相同的：例如，在刚性运动下；在相机视角改变的情况下；在规定类型的变形下，等等。第二，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中出现的包含内在物理和数学对称性的动力学方程的分析。第三，理解和探索一种新的、令人惊讶的潜在重要现象，即在周期性的假设下，解在非理性时间被“分裂”，在合理时间被“量化/局部化”，这一现象最近被证明出现在许多基本的线性色散波模型中，这些模型管理着非常广泛的波动，包括量子力学系统。

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.