39-Issue 5

Permanent URI for this collection

https://diglib.eg.org/handle/10.2312/2632926

Browse

Browsing 39-Issue 5 by Subject "Computational geometry"

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Interpolated Corrected Curvature Measures for Polygonal Surfaces
    (The Eurographics Association and John Wiley & Sons Ltd., 2020) Lachaud, Jacques-Olivier; Romon, Pascal; Thibert, Boris; Coeurjolly, David; Jacobson, Alec and Huang, Qixing
    A consistent and yet practically accurate definition of curvature onto polyhedral meshes remains an open problem. We propose a new framework to define curvature measures, based on the Corrected Normal Current, which generalizes the normal cycle: it uncouples the positional information of the polyhedral mesh from its geometric normal vector field, and the user can freely choose the corrected normal vector field at vertices for curvature computations. A smooth surface is then built in the Grassmannian R3xS2 by simply interpolating the given normal vector field. Curvature measures are then computed using the usual Lipschitz-Killing forms, and we provide closed-form formulas per triangle. We prove a stability result with respect to perturbations of positions and normals. Our approach provides a natural scale-space for all curvature estimations, where the scale is given by the radius of the measuring ball. We show on experiments how this method outperforms state-of-the-art methods on clean and noisy data, and even achieves pointwise convergence on difficult polyhedral meshes like digital surfaces. The framework is also well suited to curvature computations using normal map information.
  • Thumbnail Image
    Item
    Properties of Laplace Operators for Tetrahedral Meshes
    (The Eurographics Association and John Wiley & Sons Ltd., 2020) Alexa, Marc; Herholz, Philipp; Kohlbrenner, Max; Sorkine-Hornung, Olga; Jacobson, Alec and Huang, Qixing
    Discrete Laplacians for triangle meshes are a fundamental tool in geometry processing. The so-called cotan Laplacian is widely used since it preserves several important properties of its smooth counterpart. It can be derived from different principles: either considering the piecewise linear nature of the primal elements or associating values to the dual vertices. Both approaches lead to the same operator in the two-dimensional setting. In contrast, for tetrahedral meshes, only the primal construction is reminiscent of the cotan weights, involving dihedral angles.We provide explicit formulas for the lesser-known dual construction. In both cases, the weights can be computed by adding the contributions of individual tetrahedra to an edge. The resulting two different discrete Laplacians for tetrahedral meshes only retain some of the properties of their two-dimensional counterpart. In particular, while both constructions have linear precision, only the primal construction is positive semi-definite and only the dual construction generates positive weights and provides a maximum principle for Delaunay meshes. We perform a range of numerical experiments that highlight the benefits and limitations of the two constructions for different problems and meshes.
  • Thumbnail Image
    Item
    Topology-Aware Surface Reconstruction for Point Clouds
    (The Eurographics Association and John Wiley & Sons Ltd., 2020) Brüel-Gabrielsson, Rickard; Ganapathi-Subramanian, Vignesh; Skraba, Primoz; Guibas, Leonidas J.; Jacobson, Alec and Huang, Qixing
    We present an approach to incorporate topological priors in the reconstruction of a surface from a point scan. We base the reconstruction on basis functions which are optimized to provide a good fit to the point scan while satisfying predefined topological constraints. We optimize the parameters of a model to obtain a likelihood function over the reconstruction domain. The topological constraints are captured by persistence diagrams which are incorporated within the optimization algorithm to promote the correct topology. The result is a novel topology-aware technique which can (i) weed out topological noise from point scans, and (ii) capture certain nuanced properties of the underlying shape which could otherwise be lost while performing surface reconstruction. We show results reconstructing shapes with multiple potential topologies, compare to other classical surface construction techniques, and show the completion of real scan data.