Team geometrica

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Section: New Results

Mesh Generation and Geometry Processing

2D Centroidal Voronoi Tessellations with Constraints

Participants : Pierre Alliez, Olivier Devillers.

In collaboration with Jane Tournois (previously PhD student in our group, and now post-doc at TU Vienna).

We tackle the problem of constructing 2D centroidal Voronoi tessellations with constraints through an efficient and robust construction of bounded Voronoi diagrams, the pseudo-dual of the constrained Delaunay triangulation [23] . We exploit the fact that the cells of the bounded Voronoi diagram can be obtained by clipping the ordinary ones against the constrained Delaunay edges. The clipping itself is efficiently computed by identifying for each constrained edge the (connected) set of triangles whose dual Voronoi vertex is hidden by the constraint. The resulting construction is amenable to Lloyd relaxation so as to obtain a centroidal tessellation with constraints.

Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

Participant : Pierre Alliez.

In collaboration with Daniel Sieger and Mario Botsch from Bielefeld University (Germany).

We present a 2D mesh improvement technique that optimizes Voronoi diagrams for their use in polygonal finite element computations [32] . Starting from a centroidal Voronoi tessellation of the simulation domain we optimize the mesh by minimizing a carefully designed energy functional that effectively removes the major reason for numerical instabilities—short edges in the Voronoi diagram. We evaluate our method on a 2D Poisson problem and demonstrate that our simple but effective optimization achieves a significant improvement of the stiffness matrix condition number. See Figure 1 .

Figure 1. CVT (top) and optimized mesh (bottom) for Lake Superior using a K -Lipschitz sizing function with K = 0.7 . The underlying Delaunay triangulation contains 4036 triangles. The condition number reduces from 371877 to 190.
IMG/superior-cvt
IMG/superior-dd

Robust Surface Reconstruction from Raw Point Sets

Participants : Pierre Alliez, David Cohen-Steiner.

In collaboration with Fernando de Goes, Patrick Mullen and Mathieu Desbrun from Caltech.

We propose a modular framework for robust 3D reconstruction from unorganized, unoriented, noisy, and outlier-ridden geometric data [20] . We gain robustness and scalability over previous methods through an unsigned distance approximation to the input data followed by a global stochastic signing of the function. An isosurface reconstruction is finally deduced via a sparse linear solve. We show with experiments on large, raw, geometric datasets that this approach is scalable while robust to noise, outliers, and holes. The modularity of our approach facilitates customization of the pipeline components to exploit specific idiosyncracies of datasets, while the simplicity of each component leads to a straightforward implementation. See Figure 2 .

Figure 2. Plaster Hand. Data scanned with a Kreon laser scanner mounted on an articulated arm; the 1.8M point sampling is very anisotropic as it was obtained by manual sweeping of a 1D contact sensor. Top: input point set (with a big hole at the bottom and others due to occlusions between the fingers), point set and 2D cut of unsigned function, same 2D cut with nearby edges of the coarse mesh Im1 $\#120132 $ , same 2D cut alone, and full $ \epsilon$ -band. Middle: 2D cuts of sign guess (red for inside, blue for outside and white uncertain), confidence (which decreases in the holes), signed function after smoothing, isosurface of the robust unsigned function obtained by marching tetrahedra in the lattice mesh, and same isosurface superimposed with input points. Bottom: views of the reconstructed surface obtained by Delaunay refinement without and with points added, and cut view of the $ \epsilon$ -band with the reconstructed isosurface of the signed function inside, with and without the input points.
IMG/hand

3D Periodic Meshes

Participants : Manuel Caroli, Mikhail Bogdanov, Monique Teillaud.

In collaboration with Vissarion Fisikopoulos, Department of Informatics and Telecommunications, University of Athens.

We show how the computation of 3D periodic triangulations can be used in combination of a surface mesh generation method to compute meshes of triply-periodic surfaces (see Figure 3 ). For smooth surfaces, a sufficiently refined output mesh is guaranteed to be both homeomorphic to the surface and geometrically close to it [50] .

Figure 3. Meshing of triply-periodic surfaces: The Schwarz-p surface (left) and a surface used in bone scaffolding (right, data by courtesy of Maarten Moesen, Department of Metallurgy and Materials Engineering, K.U. Leuven)
IMG/schwarz_p_8IMG/segments01_8

We are working on the extension to volume meshing (Figure 4 ).

Figure 4. Meshing of triply-periodic volumes.
IMG/schwarz_p_mesh_3_03IMG/segments_mesh_3_03

Feature Preserving Mesh Generation from 3D Point Clouds

Participants : Nader Salman, Mariette Yvinec.

In collaboration with Quentin Mérigot, Stanford University.

We address the problem of generating quality surface triangle meshes from 3D point clouds sampled on piecewise smooth surfaces. Using a feature detection process based on the covariance matrices of Voronoi cells, we first extract from the point cloud a set of sharp features. Our algorithm also runs on the input point cloud a reconstruction process, such as Poisson reconstruction, providing an implicit surface. A feature preserving variant of a Delaunay refinement process is then used to generate a mesh approximating the implicit surface and containing a faithful representation of the extracted sharp edges. See figure 5 . Such a mesh provides an enhanced trade-off between accuracy and mesh complexity. The whole process is robust to noise and made versatile through a small set of parameters which govern the mesh sizing, approximation error and shape of the elements. We demonstrate the effectiveness of our method on a variety of models including laser scanned datasets ranging from indoor to outdoor scenes [21] .

Figure 5. Top left:the point clouds (coutesy of INPG). Top right: the extracted features. Bottom left: a mesh of the implicit surface reconstructed by the Poisson reconstruction. Bottom right: the ouput of our feature preserving mesh generation.
IMG/church

Polygon Mesh Processing

Participant : Pierre Alliez.

In collaboration with Mario Botsch, Leif Kobbelt, Mark Pauly and Bruno Lévy.

Geometry processing, or mesh processing, is a fast-growing area of research that uses concepts from applied mathematics, computer science, and engineering to design efficient algorithms for the acquisition, reconstruction, analysis, manipulation, simulation, and transmission of complex 3D models. Applications of geometry processing algorithms already cover a wide range of areas from multimedia, entertainment, and classical computer-aided design, to biomedical computing, reverse engineering, and scientific computing. Over the last several years, triangle meshes have become increasingly popular, as irregular triangle meshes have developed into a valuable alternative to traditional spline surfaces. This book [39] discusses the whole geometry processing pipeline based on triangle meshes. The pipeline starts with data input, for example, a model acquired by 3D scanning techniques. This data can then go through processes of error removal, mesh creation, smoothing, conversion, morphing, and more. The authors detail techniques for those processes using triangle meshes.


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