Applications of Moving Frames

该项目主要集中在理论工具的进一步发展和移动框架的等变方法的应用范围的扩展，该方法是在20世纪90年代末由首席研究员与访问博士后合作首次引入的，此后由世界各地的许多研究小组在许多方向上发展。特别值得注意的是：首先，图像处理，物体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。其次，通过几何（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的应用，将移动框架技术应用于不变变分双复合体，分析和分类不变变分问题以及不变曲线和曲面流动。第三，新完成的基于运动框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“色散量子化”这一令人惊讶的新现象的后果，该现象最近被证明出现在周期域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被称为塔尔博特效应的现象。该项目是基于对称在广泛的数学及其应用的开发，通过新的，强大的方法，灵感来自纯几何的经典工具。除了进一步发展基础数学理论和工具外，主要重点是三个相互关联的应用领域：首先，来自各种来源的数字图像中的物体识别，在不同的概念下，当两个物体可以被认为是相同的：例如，在刚性运动下；在摄像机视角变化下；在规定类型等的变形下。其次，分析包含内在物理和数学对称性的动力学方程，这些方程在生物力学和材料、流体力学、图像处理和界面运动中广泛应用。第三，理解和探索一个新的、令人惊讶的、潜在重要的现象，在周期性的假设下，解在非理性时间被“分形”，在理性时间被“量子化/局域化”，最近在许多基本的线性色散波模型中出现，这些模型控制着非常广泛的波动，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.