“Vector-Field Processing on Triangle Meshes” by Desbrun and Tong

  • ©Mathieu Desbrun and Yiying Tong



Entry Number: 14


    Vector-Field Processing on Triangle Meshes

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    Basic knowledge of linear algebra and vector calculus. A background in differential geometry is recommended, but not mandatory.


    Who Should Attend
    Graduate students, researchers, and developers interested in geometry processing.

    While scalar fields on surfaces remain staples of geometry processing, the use of tangent vector fields has steadily grown over the last two decades. They are crucial to encoding directions and sizing on surfaces in tasks such as texture synthesis, non-photorealistic rendering, digital grooming, and meshing. There are, however, a variety of discrete representations of tangent vector fields on triangle meshes, and each approach offers different trade-offs among simplicity, efficiency, and accuracy depending on the targeted application.

    This course reviews the three main families of discretizations used to design computational tools for vector-field processing on triangle meshes: face-based, edge-based, and vertex-based representations. In the process of reviewing the computational tools offered by these representations, the course reviews a large body of recent developments in vector-field processing in discrete differential geometry. It also discusses the theoretical and practical limitations of each type of discretization and covers increasingly common extensions such as $n$-direction and $n$-vector fields.

    While the course focuses on explaining the key approaches to practical encoding (including data structures) and manipulation (including discrete operators) of finite-dimensional vector fields, important differential geometric notions are also covered. As is often the case in discrete differential geometry, the discrete picture will be used to illustrate deep continuous concepts such as covariant derivatives, metric connections, or Bochner Laplacians.

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