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    Discrete 2-Tensor Fields on Triangulations

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    Date
    2014
    Author
    Goes, Fernando de
    Liu, Beibei
    Budninskiy, Max
    Tong, Yiying
    Desbrun, Mathieu
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    Abstract
    Geometry 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 surfaces
    BibTeX
    @article {10.1111:cgf.12427,
    journal = {Computer Graphics Forum},
    title = {{Discrete 2-Tensor Fields on Triangulations}},
    author = {Goes, Fernando de and Liu, Beibei and Budninskiy, Max and Tong, Yiying and Desbrun, Mathieu},
    year = {2014},
    publisher = {The Eurographics Association and John Wiley and Sons Ltd.},
    ISSN = {1467-8659},
    DOI = {10.1111/cgf.12427}
    }
    URI
    http://dx.doi.org/10.1111/cgf.12427
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    • 33-Issue 5
    • SGP14: Eurographics Symposium on Geometry Processing (CGF 33-5)

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    Eurographics Association copyright © 2013 - 2023 
    Send Feedback | Contact - Imprint | Data Privacy Policy | Disable Google Analytics
    Theme by @mire NV
    System hosted at  Graz University of Technology.
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