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

图像分析中的非参数统计和黎曼几何：在生物学、医学、神经科学和机器视觉中应用的新视角

• 批准号: 1406872
• 负责人: Rabindra Bhattacharya
• 金额: $ 12万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406872&HistoricalAwards=false
• 关键词: Nonparametric; Statistics; Riemannian; Geometry; Image

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.

该项目旨在（1）在生物学，医学，机器视觉和其他科学和工程领域产生的数字图像的精确几何描绘，以及（2）提供其独立于模型的统计分析，以识别识别，歧视和诊断。一种特定的应用是区分正常器官和人体中的患病器官。在示例中，可能是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这个主题，这是帕金森氏病，阿尔茨海默氏病，精神分裂症，自闭症等带来的大脑皮层中白质的几何结构的变化及其进展。 还将探索图形，机器人技术等领域的重要应用。成像技术的预测使科学家和医学专业人员当今能够在细胞层及以后查看器官的内部功能。 例如，在皮质中的白质中，可以测量水分子的3x3扩散基质的系数。 在没有疾病或创伤的情况下，这些矩阵在组织良好的神经结构上显示出明显的各向异性，而疾病引起的扰动导致每个此类位置的各向异性下降。这是由于疾病在扩散张量成像扫描中可见的疾病而引起的结构变化的一个方面。还有其他。 到目前为止，还没有统计学方法可以将这种各向异性的减少与引起它的特定疾病相关。本项目将根据riemannian歧管及其大地测量的要素来代表白质中的主要神经结构。作为一项特定的任务，该项目将在沿神经结构的正确定矩阵对齐空间上选择适当的度量张量。广泛的目标是提供一种基于歧视和诊断方法的非参数统计方法，以进一步扩展，并在新颖的方向扩展了NSF支持下进行的研究。在完全不同的方向上，该项目的一个理论目标是为在大地距离下的频率平均值提供广泛的条件。这种条件是统计应用所必需的，但对于带正弯曲的Riemannian歧管，无法获得足够的一般性。 这个独特性问题也对图形和机器人技术具有令人惊讶的含义。