Team Geometrica

Overall Objectives
Scientific Foundations
Application Domains
New Results
Contracts and Grants with Industry
Other Grants and Activities

Section: New Results

Mesh Generation and Geometry Processing

Mesh Optimization

Participants : Pierre Alliez, Jane Tournois.

In collaboration with Camille Wormser from ETH Zürich and Mathieu Desbrun from Caltech.

We propose a practical approach to isotropic tetrahedron meshing of 3D domains bounded by piecewise smooth surfaces. Building upon recent theoretical and practical advances, our algorithm interleaves Delaunay refinement and mesh optimization to generate quality meshes that satisfy a set of user-defined criteria. This interleaving is shown to be more conservative in number of Steiner point insertions than refinement alone, and to produce higher quality meshes than optimization alone (see Figure 1 ). A careful treatment of boundaries and their features is presented, offering a versatile framework for designing quality isotropic tetrahedron meshes [24] .

Figure 1. Isotropic tetrahedron mesh generated by interleaving Delaunay refinement and optimization. The distribution of tetrahedron dihedral angles is shown (here no angle is lower than 16 degrees).

Sliver Removal

Participants : Pierre Alliez, Jane Tournois.

In collaboration with Rahul Srinivasan, IIT Bombay.

Isotropic tetrahedron meshes generated by Delaunay refinement algorithms are known to contain a majority of well-shaped tetrahedra, as well as spurious sliver tetrahedra. As the slivers hamper stability of numerical simulations we aim at removing them while keeping the Delaunay property of the triangulation for simplicity. The solution which explicitly perturbs the slivers through random vertex relocation and Delaunay connectivity update is very effective but slow. In this paper we present a perturbation algorithm which favors deterministic over random perturbations. The added value is an improved efficiency and effectiveness. Our experimental study applies the proposed algorithm to meshes obtained by Delaunay refinement as well as to carefully optimized meshes [35] .

Feature Preserving Delaunay Mesh Generation from 3D Multi-Material Images

Participants : Jean-Daniel Boissonnat, Dobrina Boltcheva, Mariette Yvinec.

Generating realistic geometric models from 3D segmented images is an important task in many biomedical applications. Segmented 3D images impose particular challenges for meshing algorithms because they contain multi-material junctions forming features such as surface patches, edges and corners. The resulting meshes should preserve these features to ensure the visual quality and the mechanical validity of the models. We present a feature preserving Delaunay refinement algorithm which can be used to generate high-quality tetrahedral meshes from segmented images. See Figure 2 . The idea is to explicitly sample corners and edges from the input image and to constrain the Delaunay refinement algorithm to preserve these features in addition to the surface patches. Our experimental results on segmented medical images have shown that, within a few seconds, the algorithm outputs a tetrahedral mesh in which each material is represented as a consistent submesh without gaps and overlaps. The optimization property of the Delaunay triangulation makes these meshes suitable for the purpose of realistic visualization or finite element simulations [27] , [15] .

Figure 2. Meshes generated from a segmented liver image representing 4 anatomical liver regions. The first raw shows meshes obtained with the usual Delaunay refinement algorithm. The second raw shows meshes generated with our feature preserving extension. The 3rd column shows some internal interfaces between the anatomical regions.

Discrete Critical Values: a General Framework for Silhouettes Computations

Participants : Frédéric Chazal, Nicolas Montana.

This work has been done in collaboration with A. Lieutier (Dassault Systèmes).

Many shapes resulting from important geometric operations in industrial applications such as Minkowski sums or volume swept by a moving object can be seen as the projection of higher dimensional objects. When such a higher dimensional object is a smooth manifold, the boundary of the projected shape can be computed from the critical points of the projection. In this work, using the notion of polyhedral chains introduced by Whitney, we introduce a new general framework to define an analogous of the set of critical points of piecewise linear maps defined over discrete objects that can be easily computed. We illustrate our results by showing how they can be used to compute Minkowski sums of polyhedra and volumes swept by moving polyhedra [19] , [61] .


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