Collaborative Research: Statistical Analysis on Manifolds: A Nonparametric Approach for Shapes and Images

合作研究：流形统计分析：形状和图像的非参数方法

• 批准号: 0406143
• 负责人: Rabindra Bhattacharya
• 金额: $ 16.24万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406143&HistoricalAwards=false
• 关键词: Collaborative; Research; Statistical; Analysis; Manifolds

The shape or image of an object may be recorded digitally by a finite number k of landmarks or positions on the object, called a k-ad. The space of orbits under rotation and translation of k-ads is the size-and-shape space. If one includes scaling with rotation and translation, then the space of orbits under the resulting group of transformations on the k-adsis the shape space. These are examples of Riemannian manifolds, some with singularities. One approach proposed here for inference about a distribution on a general differentiable or Riemannian manifold M, based on a random sample from it, is to carry out a multivariate analysis of the image under the so-called Log map on one or more tangent spaces to M. Apart from this intrinsic approach, less computation intensive extrinsic procedures based on embeddings in Euclidean spaces are investigated. The project explores consistency and asymptotic distribution theory on manifolds for robust tests and confidence regions from both points of view--intrinsic and extrinsic. One motivation for this study comes from the need to identify deformations or shape changes for purposes of medical diagnosis and biomorphology. Immediate applications also arise to machine vision and pattern recognition. There are significant impacts of this research on these and many other fields. Another important goal of this project is to train students in the newly developed methodologies, leading to the dissemination of knowledge gained through this research and the creation of a body of technicians and experts to apply it.

对象的形状或图像可以通过对象上的有限数量k个界标或位置（称为k-ad）来数字地记录。k-ads在旋转和平移下的轨道空间是大小和形状空间。如果一个包括旋转和平移的缩放，那么在k-adis上的结果变换组下的轨道空间是形状空间。这些是黎曼流形的例子，有些带有奇点。这里提出的一种方法是基于一个随机样本来推断一般可微流形或黎曼流形M上的分布，即在一个或多个M的切空间上的所谓对数映射下对图像进行多变量分析。除了这种内在的方法，较少的计算密集型的外部程序的基础上嵌入在欧几里得空间进行了研究。该项目从内在和外在两个角度探讨了流形上的一致性和渐近分布理论，用于稳健测试和置信区域。 这项研究的一个动机来自于需要识别医疗诊断和生物形态学目的的变形或形状变化。机器视觉和模式识别也有直接的应用。这项研究对这些领域和许多其他领域产生了重大影响。该项目的另一个重要目标是培训学生掌握新开发的方法，从而传播通过这项研究获得的知识，并建立一个应用这些知识的技术人员和专家队伍。