Team Geometrica

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Section: New Results

Keywords : Isotropic meshing, anisotropic meshing, triangle meshing, tetrahedral meshing, mesh optimization, level sets (level set, level set, level set).

Mesh Generation and Geometry Processing

Locally uniform anisotropic meshing

Participants : Jean-Daniel Boissonnat, Camille Wormser, Mariette Yvinec.

Anisotropic meshes are triangulations of a given domain in the plane or in higher dimensions, with elements elongated along prescribed directions. Anisotropic triangulations have shown particularly well suited to interpolation of functions or to numerical modeling. Following our previous investigations  [14] , we propose a new approach to anisotropic mesh generation relying on the notion of locally uniform anisotropic mesh  [23] . A locally uniform anisotropic mesh is a mesh designed such that the star around each vertex v coincides with the star that v would have if the metric on the domain was uniform and equal to the metric at v . This definition allows devising a simple refinement algorithm which relies on elementary predicates, and provides, after completion, an anisotropic mesh in dimensions 2 or 3. Prototypes have been implemented for both the 2D and the 3D cases (see Figures  1 and 2 ).

Figure 1.

Anisotropic mesh of a 2D domain with a close-up on the central part.

The red lines separate the zoom from the regular drawing and show the zoomed part.

IMG/aniso2D
Figure 2.

Anisotropic mesh of the surface of a torus.

This mesh has been generated by Yuanmi Chen during her internship.

IMG/anisotropic-torusIMG/anisotropic-torus-detail

Mesh Optimization

Participants : Pierre Alliez, Jane Tournois, Camille Wormser.

We are elaborating upon a mesh optimization technique designed to improve the quality of isotropic tetrahedron meshes. Our approach improves over the optimal Delaunay approach introduced by Chen in 2004, which consists of casting the mesh optimization problem as a function interpolation in 4D. In the original approach the optimization is performed by alternating vertex relocations and Delaunay connectivity updates. While the original approach keeps the boundary vertices fixed in order to avoid mesh shrinking, we relocate them in a consistent manner with the interior vertices by reproducing the so-called cospherical property. We also investigate the possibility to approximate the paraboloid instead of interpolating it using a regular triangulation, in order to reduce the number of slivers in the final mesh. At the intuitive level, this amounts to embed a sliver exudation process as part of the mesh optimization. Furthermore, we show how alternating batches of refinement with mesh optimization in a multilevel manner generates mesh with fewer Steiner vertices. Figure  3 shows an optimized mesh compared to a mesh generated by Delaunay refinement. We have not yet published this on-going work.

Figure 3. Top: Input polyhedral surface with sharp creases tagged. Middle: Isotropic tetrahedron mesh obtained by Delaunay refinement parameterized so as to control the shape of the elements and the boundary approximation error (no sliver exudation performed). The distribution of dihedral angles is shown to the left. Bottom: Optimized mesh with improved quality (dihedral angles in [11.13;150] degrees).
IMG/mesh_optimization

Conformal Parameterization

Participant : Pierre Alliez.

In collaboration with Patrick Mullen, Yiying Tong and Mathieu Desbrun from Caltech.

We propose a linear-algebra-based conformal parameterization technique to parameterize triangle mesh patches [21] . Unlike previous free-boundary linear methods we do not require point constraints to be added to the linear system, thus reducing distortion. While Laplacian eigenvectors have been proposed as a constraint-free approach to least-distorted maps in the context of manifold learning and graph drawing, we demonstrate that a better conformal parameterization can be found through a generalized eigenvalue problem as it minimizes a weighted conformal energy mostly insensitive to sampling irregularity of the original mesh (see Figure 4 ). We discuss the similarities and differences between our approach and previous work, and demonstrate numerical advantages of our spectral method on small and large meshes alike.

Figure 4. Sforza. On this mesh (50K vertices) with varying sampling rates (left), previous linear methods (top right, least squares conformal maps) fail to capture the symmetry of the mesh in the parameterization (solved in 4s). The two red dots depict the constrained vertices. In contrast, our spectral approach (bottom right) automatically computes a low-distortion conformal map (solved in 5.2s) without any constraints. Middle images depict the sforza mesh with a checkboard texture mapped using the parameterizations shown on the right.
IMG/param

Principal Component Analysis

Participants : Pierre Alliez, Sylvain Pion, Ankit Gupta.

Principal component analysis is a basic component of many geometric computing and processing algorithms. It is most commonly used on point sets, although applicable to other primitives as well through the computation of covariance matrices. In this work [49] we provide closed form formulas of covariance matrices for sets of 2D and 3D geometric primitives such as segments, circles, triangles, iso rectangles, spheres, tetrahedra and iso cuboids. We also derive covariance matrices for their dimensional variants such as disks, balls, etc. We discuss the flexibility and added value of the present approach by discussing its potential use in applications. Our implementation will be made available through the next release of the cgal library.


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