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.

物体的形状或图像可以通过有限的地标或位置上的位置数字记录，称为k-ad。旋转和K-ADS翻译的轨道空间是尺寸和形状的空间。如果包括旋转和翻译的缩放，则在K-Adsis在形状空间上所得的转换组下的轨道空间。这些是里曼尼亚歧管的例子，有些具有奇异性。这里提出的一种方法是针对基于其随机样本的一般可区分或riemannian歧管M上的分布进行推断，就是要在一个或多个切线空间上对M.的图像进行多变量分析。该项目从两种观点（即intrinsic and ofteric）探索了稳健测试和置信区域的一致性和渐近分布理论。 这项研究的一种动机来自于为医学诊断和生物形态学目的确定变形或形状变化。也立即应用于机器视觉和模式识别。这项研究对这些和许多其他领域有重大影响。该项目的另一个重要目标是培训学生的新开发方法，从而导致通过这项研究获得的知识传播，并创建了一群技术人员和专家来应用它。