Browsing by Author "Bunge, Astrid"

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    The Diamond Laplace for Polygonal and Polyhedral Meshes
    (The Eurographics Association and John Wiley & Sons Ltd., 2021) Bunge, Astrid; Botsch, Mario; Alexa, Marc; Digne, Julie and Crane, Keenan
    We introduce a construction for discrete gradient operators that can be directly applied to arbitrary polygonal surface as well as polyhedral volume meshes. The main idea is to associate the gradient of functions defined at vertices of the mesh with diamonds: the region spanned by a dual edge together with its corresponding primal element - an edge for surface meshes and a face for volumetric meshes. We call the operator resulting from taking the divergence of the gradient Diamond Laplacian. Additional vertices used for the construction are represented as affine combinations of the original vertices, so that the Laplacian operator maps from values at vertices to values at vertices, as is common in geometry processing applications. The construction is local, exactly the same for all types of meshes, and results in a symmetric negative definite operator with linear precision. We show that the accuracy of the Diamond Laplacian is similar or better compared to other discretizations. The greater versatility and generally good behavior come at the expense of an increase in the number of non-zero coefficients that depends on the degree of the mesh elements.
  • No Thumbnail Available
    Item
    A Survey on Discrete Laplacians for General Polygonal Meshes
    (The Eurographics Association and John Wiley & Sons Ltd., 2023) Bunge, Astrid; Botsch, Mario; Bousseau, Adrien; Theobalt, Christian
    The Laplace Beltrami operator is one of the essential tools in geometric processing. It allows us to solve numerous partial differential equations on discrete surface meshes, which is a fundamental building block in many computer graphics applications. Discrete Laplacians are typically limited to standard elements like triangles or quadrilaterals, which severely constrains the tessellation of the mesh. But in recent years, several approaches were able to generalize the Laplace Beltrami and its closely related gradient and divergence operators to more general meshes. This allows artists and engineers to work with a wider range of elements which are sometimes required and beneficial in their field. This paper discusses the different constructions of these three ubiquitous differential operators on arbitrary polygons and analyzes their individual advantages and properties in common computer graphics applications.