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)对生物学、医学、机器视觉和其他科学与工程领域中出现的数字图像进行精确的几何描述；(2)提供与模型无关的统计分析，以用于识别、区分和诊断。一个具体的应用是区分人体的正常器官和病变器官。在这些例子中，人们可以参考青光眼和某些类型的精神分裂症的诊断基于形状的变化。该项目将特别关注和深入分析的一个主题，涉及帕金森病、阿尔茨海默病、精神分裂症、自闭症等疾病引起的大脑皮层白质几何结构的变化及其进展。在图形学、机器人等领域的重要应用也将被探索。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水平上观察器官的内部功能。例如，在皮层的白质中，可以测量水分子的3x3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表现出明显的各向异性，而疾病引起的扰动导致每个此类位置的各向异性减少。这是扩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。还有其他的。到目前为止，还没有一种统计方法可以精确地将这种各向异性的减少与导致这种减少的特定疾病联系起来。本项目将用黎曼流形及其测地线的元素来表示白质中的主要神经结构。作为一项具体任务，该项目将在沿神经结构的正定矩阵的对齐空间上选择适当的度量张量。研究的主要目标是提供一种基于Fre'chet方法的非参数统计方法，用于鉴别和诊断，进一步扩展在早期NSF支持下进行的研究，并在新的方向上进行研究。在一个完全不同的方向上，该项目的一个理论目标是为在测地线距离下的frechet平均值的唯一性提供广泛的条件。这些条件在统计应用中是必需的，但在具有正曲率的黎曼流形中没有足够的普遍性。这种独特性对于图形和机器人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.