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.
Remeshing, the process of generating a higher quality mesh from a given input mesh, is becoming increasingly important in geometry processing and its applications in engineering and medicine. This project investigates remeshing of surfaces using higher-order tensor field processing. Several research areas are explored, from which both theoretical and empirical discoveries have resulted. The principal investigator (PI) and his collaborators investigate the problem of triangular remeshing using higher-order tensor fields. Unlike traditional triangulation approaches that are based on Delauney triangulation or centroidal Voronoi diagrams, the funded research is based on the notion of global parameterizations guided by a sixth-order tensor field that is carefully (and automatically) generated to be aligned with the features in the surfaces such as ridges and valleys. Moreover, the irregular vertices can be controlled by control in the singularities in the higher-order tensor fields. This leads to higher quality meshes than traditional approaches. While the main application is triangular remeshing, the theoretical investigation also applies to quadrangular remeshing, another popular topic that has drawn much interest from the geometry processing community. Moreover, the parameterizations generated this way lead to seamless tilings of surfaces using hexagons and quads, enabling seamless texture and geometry synthesis on surfaces using tillable patterns. The PI and his collaborators have also investigated the problem of mesh optimization, with a focus on the control of irregular vertices in the mesh. Irregular vertices can lead to undesirable effects in subsequent uses of meshes, especially when they are misplaced, i.e., not aligned with the curvature distribution in the surface. The researchers develop both theory and practical algorithms on the control of irregular vertices, in both quadrangular meshes and triangular meshes. In addition, the theory and editing operations for both triangular and quadrangular meshes can be placed under a unified framework, which shows the promises that the two types of meshes, while often preferred by different communities, share many fundamental properties. Holonomy is a mathematical concept characterizing the angular deficiency in a global parameterization that needs to be corrected in order to generate valid global parameterization (and remeshing) results. The PI and his collaborators develop parameterization approaches that take into account of holonomy correction, thus leading to seamless remeshing results. The funded research also provides high-quality, interactive visualization techniques for higher-order tensor fields that represent rotational symmetries on surfaces. There are three directions that are beyond the original scope of the research. First, the PI and his team of researchers provide efficient visualization techniques for asymmetric tensor fields, i.e., tensors that are not symmetric with respect to the transpose operator. Such analysis has applications in seismology, fluid and solid mechanics, and chemistry. The analysis and visualization of the researchers provide new insights to the understanding of vector fields by studying their spatial derivatives, i.e., tensors. This provides an entirely new direction for both vector and tensor field visualization, as well as new insights for domain applications such as seismology and mechanics. Second, the PI and his collaborators develop efficient algorithms to visualize vector field topology. As demonstrated by the PI’s previous work, vector fields, tensor fields, and higher-order tensor fields share some property in that they can all represent orientations. As such, the study of one object can be and has been adapted to others. The funded research leads to efficient algorithms to compute Morse decompositions of vector fields on surfaces. Morse decomposition, an important yet challenging concept from dynamical systems, can now be computed quickly enough to become practical to the domain scientists. Third, the research has targeted hexahedral remeshing. The PI and his collaborators create high-quality hexahedral meshes from an input unstructured mesh through the notion of polycubes. Meshes generated in this manner have few elements that are not hexahedra. Moreover, it is relatively easy to control the size variation and angular distortion in the mesh, both of which are important measures for the quality of a hexahedral mesh. The PI has collaborated with mathematicians, architects, civil engineers, and fellow computer scientists. The research results have been applied to oceanography and atmospheric flows as well as earthquake data. The publications resulted from the research are mostly with the prestigious ACM and IEEE societies. The software packages along with source codes developed for the research are freely available to the public on the PI's web page. One Ph.D. student has graduated and is now a faculty member at University of Houston. Another Ph.D. student is near graduation. Three MS students have graduated while working on the funded research. In addition, the award has allowed the PI to sponsor four undergraduate students for research experience and six high school summer interns. Most of these students are from underrepresented groups. The PI has also incorporated research results into classroom teaching of computer graphics, scientific visualization, and geometry processing.
