Collaborative Research: Nonparametric Theory on Manifolds of Shapes and Images, with Applications to Biology, Medical Imaging and Machine Vision

合作研究：形状和图像流形的非参数理论及其在生物学、医学成像和机器视觉中的应用

• 批准号: 0806011
• 负责人: Rabindra Bhattacharya
• 金额: $ 16万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0806011&HistoricalAwards=false
• 关键词: Collaborative; Research; Nonparametric; Theory; Manifolds

Much of the focus of this collaborative project is on the analysis of landmark based shapes in which a k-ad, i.e., a set of k points or landmarks on an object or a scene are observed in 2-D or 3-D, usually with expert help, for purposes of identification, discrimination, or diagnostics. Depending on the way the data are collected or recorded, the appropriate shape of an object is the maximal invariant specified by the space of orbits under a group G of transformations. In particular, Kendall's shape spaces of k-ads are invariant under scaling and Euclidean rigid motions. While this is a proper choice for many problems in biology and medical imaging, other notions of shape such as affine shape and projective shape are important in machine vision and bioinformatics. All these spaces are differentiable manifolds, often with natural Riemannian structures for measuring lengths and angles. The statistical analysis based on Riemannian structures is said to be intrinsic. In other cases, proper distances are sought via an equivariant embedding of the manifold M in a vector space E. Corresponding statistical analysis is called extrinsic. Finding proper Riemannian structures and equivariant embeddings is one of the objectives of this project, which is crucial for the statistical inference proposed. Establishing broad conditions for the existence of the Fre´chet mean, as the unique minimizer of the Fre´chet function the expected squared distance from a Q-distributed random shape is important for statistical inference; and it is a goal of the project to pursue, especially for intrinsic analysis where it has remained an outstanding open problem from the inception of shape theory. Reconstruction of a scene from two (ormore) aerial photographs taken from a plane is one of the research problems in affine shape analysis. Potential applications of projective shape analysis proposed here include face recognition and robotics-for robots to visually recognize a scene.Statistical analysis of data on geometric objects, or manifolds, is an exciting and challenging field of research, where statistical theory and differential geometry are inextricably intertwined, and implementation requires innovative algorithms and high speed computation. The project proposed here deals with the development of nonparametric methodology in this context, which must also resolve associated geometric issues and problems of implementation. Past progress in this field by the PIs has laid the foundation for the present project. The statistical analysis proposed has wide ranging applications, especially in biology and bioinformatics, health sciences, and machine vision. Under this project, the PIs plan to train both undergraduate and graduate students, as well as at least one postdoctoral fellow, in theory and in its practical implementation. This continues much further the present activities of the PIs in this regard. In addition, computational algorithms and codes are made available on websites to create and disseminate this research and its applications.

该协作项目的大部分重点是分析基于里程碑的形状，其中K-AD（即在2-D或3-D中观察到一个K点或地标在对象或场景上的K点或地标，通常在专家帮助下，出于识别，歧视或诊断的目的。根据收集或记录数据的方式，对象的适当形状是由转换组G组下轨道空间指定的最大不变性。特别是，在缩放和欧几里得刚性运动下，肯德尔的k-ads形状空间是不变的。尽管这是针对生物学和医学成像许多问题的适当选择，但其他形状（例如仿射形状和投射形状）的注意事项在机器视觉和生物信息学中很重要。所有这些空间都是可区分的歧管，通常具有自然的riemannian结构，用于测量长度和角度。基于黎曼结构的统计分析被认为是内在的。在其他情况下，通过在矢量空间中的歧管M的等效嵌入来感知适当的距离。相应的统计分析称为外部。找到适当的Riemannian结构和等效嵌入是该项目的目标之一，这对于提出的统计推断至关重要。建立频率平均值的广泛条件，因为freechet函数的独特最小化器与Q分布的随机形状的预期平方距离对于统计推断很重要。它是该项目购买的目标，特别是对于内在分析，它仍然是形状理论的启动。从飞机上拍摄的两个（或更多）航空照片的场景重建是仿射形状分析中的研究问题之一。此处提出的投射形状分析的潜在应用包括面部识别和机器人的机器人，以视觉识别场景。对几何对象或多种流形的数据进行统计分析是一个令人兴奋的研究领域，其中统计理论和差异几何形状是密不可分的，并且实施需要创新的Algorith和高速计算。此处提出的项目涉及在这种情况下非参数方法的发展，这还必须解决相关的几何问题和实施问题。 PIS在这一领域的过去进步为本项目奠定了基础。提出的统计分析具有广泛的应用，特别是在生物学和生物信息学，健康科学和机器视觉方面。在这个项目下，PIS计划在理论上及其实际实施培训本科生和研究生以及至少一个博士后研究员。在这方面，这进一步延续了PI的当前活动。此外，在网站上提供了计算算法和代码，以创建和传播这项研究及其应用程序。