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

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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.