Nonparametric Statistics and Riemannian Geometry in Image Analysis: New Perspectives with Applications in Biology, Medicine, Neuroscience and Machine Vision

该项目旨在（1）生物学，医学，机器视觉和其他科学和工程领域中出现的数字图像的精确几何转换，以及（2）提供其模型无关的统计分析，用于识别，区分和诊断。一个具体的应用是区分人体中的正常器官和病变器官。在示例中，可以指基于形状变化的青光眼和某些类型的精神分裂症的诊断。该项目将特别深入研究和分析的一个主题，涉及帕金森病、老年痴呆症、精神分裂症、自闭症等引起的大脑皮层白色物质几何结构的变化，和他们的进步。 图形学、机器人等领域的重要应用，成像技术的进步使今天的科学家和医学专业人员能够在细胞水平和更高的水平上观察器官的内部功能。 例如，在皮质中的白色物质中，可以测量水分子的3 × 3扩散矩阵的系数。 在没有疾病或创伤的情况下，这些基质沿着组织良好的神经结构显示出明显的各向异性，而由于疾病引起的扰动导致每个这样的位置的各向异性降低。这是弥散张量成像扫描中可见的疾病引起的结构变化的一个方面。还有其他人 到目前为止，还没有统计方法可以精确地将这种各向异性的减少与引起它的特定疾病联系起来。本项目将用黎曼流形的元素及其测地线来表示白色物质中的主要神经结构。作为一个具体的任务，该项目将选择适当的度量张量的空间上的对齐的正定矩阵沿着神经结构。广泛的目标是提供一个非参数的统计方法的基础上弗雷切特手段的歧视和诊断，进一步扩展和新的方向的研究，进行了早期的NSF支持。在一个完全不同的方向，该项目的一个理论目标是提供广泛的条件下的唯一性的弗雷谢平均测地距离。这样的条件是统计应用所必需的，但对于具有正曲率的黎曼流形却没有足够的一般性。 这种独特性对图形学和机器人技术也有着令人惊讶的影响。

This project aims at (1) precise geometric depictions of digital images arising in biology, medicine, machine vision and other fields of science and engineering and (2) providing their model-independent statistical analysis for purposes of identification, discrimination and diagnostics. One specific application is to discriminate between a normal organ and a diseased one in the human body. Among examples, one may refer to the diagnosis of glaucoma and certain types of schizophrenia based on shape changes. A subject that the project will especially look at and analyze in depth, concerns changes in the geometric structure of the white matter in the brain's cortex brought about by Parkinson's disease, Alzheimers, schizophrenia, autism, etc., and their progression. Important applications in the fields of graphics, robotics, etc., will be explored as well.Advancements in imaging technology enable scientists and medical professionals today to view the inner functioning of organs at the cell level and beyond. For example, in the white matter in the cortex, the coefficients of the 3x3 diffusion matrix of water molecules can be measured. In the absence of a disease or trauma, these matrices show pronounced anisotropy along well organized neural structures, while perturbations due to a disease lead to a decrease in anisotropy in each such location. This is one aspect of the structural change due to a disease that is visible in the diffusion tensor imaging scans. There are others. So far there is no statistical methodology that can precisely associate such a decrease in anisotropy with the particular disease that causes it. The present project will represent the main neural structures in the white matter in terms of elements of a Riemannian manifold and their geodesics. As one specific task, the project will choose appropriate metric tensors on the space of alignments of positive definite matrices along neural structures. The broad goal is to provide a nonparametric statistical methodology based on Fre'chet means for discrimination and diagnostics, extending much further and in novel directions the research that was carried out under earlier NSF supports. In a completely different direction, one theoretical objective of the project is to provide broad conditions for uniqueness of the Fre'chet mean under a geodesic distance. Such conditions are required for statistical applications but are unavailable in adequate generality for Riemannian manifolds with positive curvature. This matter of uniqueness also has surprising implications, for graphics and robotics.