Geometric and Topological Analysis of Higher-Order Tensor Fields on Surfaces

表面高阶张量场的几何和拓扑分析

• 批准号: 0830808
• 负责人: Eugene Zhang
• 金额: $ 37.5万
• 依托单位: Oregon State University
• 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: 2008-09-15 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0830808&HistoricalAwards=false
• 关键词: Geometric; Topological; Analysis; Higher; Order

This research investigates theories and algorithms for the geometric analysis of higher-order tensor fields and their applications to efficient surface remeshing. Remeshing, the process of producing a new mesh from an input mesh to improve its quality, has many applications that include shape modeling and manufacturing, medical imaging, solid and fluid simulation, and architectural design. Remeshing based on the intrinsic symmetries in the underlying surface can afford more faithful shape representation and greater user control. Such surface symmetries can be represented by higher-order tensors, whose analysis can benefit a wide range of applications beyond geometry remeshing, such as solid and fluid dynamics, electromagnetism, weather prediction, tsunami and hurricane modeling, aircraft design and testing, biometrics, arts and entertainment, motion analysis and synthesis, and education.The project contains three research thrusts. First, the investigator explores the connections between higher-order tensor fields on a surface and regular or near-regular rotational symmetries. Second, the fundamental geometric and topological properties of higher-order tensor fields are studied. Finally, such knowledge is applied to obtaining efficient and highly controllable geometry remeshing algorithms. To explore the connection between tensors and symmetries, the investigator borrows results form existing work in tensor decomposition and extends them to near-regular and mixed symmetries. Work from vector and tensor field analysis is extended to higher-order tensor fields, and concepts such as differential geometry and dynamics systems are applied to higher-order tensor analysis. As a result of the research, the tensor analysis and remeshing algorithms are integrated into a system. In addition, the implementations of these algorithms, especially those supporting higher-order tensor analysis, are compiled into libraries to facilitate integration into host applications that requires higher-order tensor analysis. Both the system and its supporting libraries will be disseminated to the public. With respect to its impacts on education, the remeshing system and tensor analysis are integrated into the curriculum for undergraduate students majoring in science and engineering.

本研究探讨高阶张量场几何分析的理论与演算法，并将其应用于高效率的曲面网格重划。重新网格化是从输入网格生成新网格以提高其质量的过程，具有许多应用，包括形状建模和制造，医学成像，固体和流体模拟以及建筑设计。基于底层曲面中固有对称性的网格重划可以提供更忠实的形状表示和更好的用户控制。这种表面对称性可以用高阶张量来表示，其分析可以使几何重新网格化以外的广泛应用受益，例如固体和流体动力学、电磁学、天气预报、海啸和飓风建模、飞机设计和测试、生物识别、艺术和娱乐、运动分析和合成以及教育。该项目包括三个研究方向。首先，研究者探索表面上的高阶张量场与规则或近规则旋转对称性之间的联系。其次，研究了高阶张量场的基本几何和拓扑性质。最后，这些知识被应用于获得高效和高度可控的几何网格重划分算法。为了探索张量和对称性之间的联系，研究人员借用了张量分解中现有工作的结果，并将其扩展到近正则和混合对称性。工作从向量和张量场分析扩展到高阶张量场，和概念，如微分几何和动力学系统适用于高阶张量分析。作为研究的结果，张量分析和网格重构算法集成到一个系统中。此外，这些算法的实现，特别是那些支持高阶张量分析，被编译成库，以方便集成到主机应用程序，需要高阶张量分析。该系统及其支持图书馆将向公众推广。针对其对教育的影响，将网格重划系统和张量分析纳入理工科本科生课程。