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

本文研究了高阶张量场几何分析的理论和算法，以及它们在高效曲面网格划分中的应用。重建是从输入网格生成新网格以提高其质量的过程，它有许多应用，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。基于下表面固有对称性的重构可以提供更逼真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，其分析可以在几何网格重新划分之外的广泛应用，如固体和流体动力学、电磁、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。首先，研究人员探索了表面上的高阶张量场与正则或近正则旋转对称之间的联系。其次，研究了高阶张量场的基本几何和拓扑性质。最后，将这些知识应用于获得高效且高度可控的几何网格重新划分算法。为了探索张量和对称性之间的联系，研究者借用了张量分解的现有结果，并将它们推广到近正则对称和混合对称。从矢量场和张量场分析的工作扩展到高阶张量场，并将微分几何和动力学系统等概念应用于高阶张量分析。作为研究的结果，张量分析和网格重划分算法被集成到一个系统中。此外，这些算法的实现，特别是那些支持高阶张量分析的算法，被编译到库中，以便于集成到需要高阶张量分析的主机应用程序中。该系统及其辅助图书馆将会向公众发放。就其对教育的影响而言，重新网格化系统和张量分析被纳入理工科本科生的课程。

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.