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

物体的形状或图像可以通过物体上有限数量的k个地标或位置（称为k-ad）进行数字记录。k-ads旋转平移下的轨道空间为大小形状空间。如果一个包含旋转和平移的缩放，那么在k上产生的一组变换下的轨道空间就是形状空间。这些是黎曼流形的例子，有些有奇点。本文提出的一种方法是基于一般可微流形或黎曼流形M上的随机样本来推断其分布，该方法是在所谓的到M的一个或多个切空间上的Log映射下对图像进行多元分析。除了这种内在方法外，还研究了基于欧几里德空间嵌入的计算强度较小的外在过程。该项目从内在和外在两个角度探讨了流形上鲁棒检验和置信区域的一致性和渐近分布理论。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，需要识别变形或形状变化。机器视觉和模式识别也有直接的应用。本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.