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

对象的形状或图像可以通过被称为k-ad的对象上的有限数量的地标或位置来数字记录。K-ADS在旋转和平移下的轨道空间是大小和形状空间。如果包含旋转和平移的缩放，那么k-adds上的结果变换组下的轨道空间就是形状空间。这些是黎曼流形的例子，有些流形带有奇点。本文提出的一种推论一般可微流形或黎曼流形上分布的方法，基于流形M上的随机样本，是在一个或多个与M相切的空间上，在所谓的Log映射下对图像进行多元分析。该项目从内在和外在两个角度探索流形上的一致性和渐近分布理论，以进行稳健测试和置信度区域。这项研究的动机之一是为了医学诊断和生物形态学的目的识别畸形或形状变化。机器视觉和模式识别也出现了直接的应用。这项研究对这些领域和许多其他领域都有重大影响。该项目的另一个重要目标是对学生进行新开发方法方面的培训，从而传播通过这项研究获得的知识，并建立一批技术人员和专家来应用这些知识。

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.