Discrete 2-Tensor Fields on Triangulations

dc.contributor.authorGoes, Fernando deen_US
dc.contributor.authorLiu, Beibeien_US
dc.contributor.authorBudninskiy, Maxen_US
dc.contributor.authorTong, Yiyingen_US
dc.contributor.authorDesbrun, Mathieuen_US
dc.contributor.editorThomas Funkhouser and Shi-Min Huen_US
dc.description.abstractGeometry processing has made ample use of discrete representations of tangent vector fields and antisymmetric tensors (i.e., forms) on triangulations. Symmetric 2-tensors, while crucial in the definition of inner products and elliptic operators, have received only limited attention. They are often discretized by first defining a coordinate system per vertex, edge or face, then storing their components in this frame field. In this paper, we introduce a representation of arbitrary 2-tensor fields on triangle meshes. We leverage a coordinate-free decomposition of continuous 2-tensors in the plane to construct a finite-dimensional encoding of tensor fields through scalar values on oriented simplices of a manifold triangulation. We also provide closed-form expressions of pairing, inner product, and trace for this discrete representation of tensor fields, and formulate a discrete covariant derivative and a discrete Lie bracket. Our approach extends discrete/finite-element exterior calculus, recovers familiar operators such as the weighted Laplacian operator, and defines discrete notions of divergence-free, curl-free, and traceless tensors-thus offering a numerical framework for discrete tensor calculus on triangulations. We finally demonstrate the robustness and accuracy of our operators on analytical examples, before applying them to the computation of anisotropic geodesic distances on discrete surfacesen_US
dc.description.seriesinformationComputer Graphics Forumen_US
dc.publisherThe Eurographics Association and John Wiley and Sons Ltd.en_US
dc.titleDiscrete 2-Tensor Fields on Triangulationsen_US