The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges GEOMETRICA Period 2002-2006 Jean-Daniel Boissonnat Evaluation seminar November 14, 2006 Evaluation seminar GEOMETRICA Period 2002-2006
The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges The team Permanent researchers Temporary employees J-D. Boissonnat (head) F . Chazal (U. Bourgogne) F . Cazals 11 Ph.D. Students, 3 Post Doc., O. Devillers 2 Engineers M. Yvinec M. Teillaud ( GALAAD 2003-2005) P . Alliez (2001) S. Pion (2003) D. Cohen-Steiner (2004) Evaluation seminar GEOMETRICA Period 2002-2006
The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Context and overall objectives 1/2 Context ◮ Processing (massive) geometric data is mandatory ◮ A theory for approximating geometric objects is lacking ◮ Complexity and robustness issues are critical Focus: Effective Non-Linear Computational Geometry ◮ Geometric Algorithms ◮ Geometric Calculus ◮ Geometric Approximation Evaluation seminar GEOMETRICA Period 2002-2006
The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Context and overall objectives 2/2 Our approach ◮ Provide mathematical and algorithmic foundations ◮ Validate our theoretical advances through extensive experimental research ◮ Develop the CGAL library and use it to disseminate our results Evaluation seminar GEOMETRICA Period 2002-2006
The team Sampling, meshing and reconstruction Context and overall objectives Ridges A selection of scientific achievements Topological persistence Software development and technology transfer Delaunay triangulations and Voronoi diagrams New challenges Compact data structures Some selected results Sampling, meshing and reconstruction ◮ Sampling theorems and algorithms zoom ◮ Mesh generation zoom Feature extraction ◮ Ridge extraction zoom ◮ Stability of geometric features zoom Data Structures ◮ Delaunay triangulation and Voronoi diagrams zoom ◮ Compact data structures zoom next Evaluation seminar GEOMETRICA Period 2002-2006
The team Sampling, meshing and reconstruction Context and overall objectives Ridges A selection of scientific achievements Topological persistence Software development and technology transfer Delaunay triangulations and Voronoi diagrams New challenges Compact data structures Sampling theory for compact sets in R n ◮ Well-chosen offsets of sampled compact subsets of R n have the correct homotopy type if the sampling is dense enough. ◮ Automatic scale selection is possible using the critical function of the point cloud. ◮ The sampling theory is based on non-smooth analysis of distance functions. SoCG 06 Evaluation seminar GEOMETRICA Period 2002-2006
The team Sampling, meshing and reconstruction Context and overall objectives Ridges A selection of scientific achievements Topological persistence Software development and technology transfer Delaunay triangulations and Voronoi diagrams New challenges Compact data structures Restricted Delaunay triangulation back ◮ If S is C 1 , 1 or Lipschitz surface and E a sufficiently dense sample of S , RDT( E ) is a good approximation of S ◮ Can be turned into an approximation algorithm, assuming knowledge about the reach of S (or k -Lipschitz radius) Graph. Models 05, SOCG 06, S. Oudot’s PhD thesis Evaluation seminar GEOMETRICA Period 2002-2006
The team Sampling, meshing and reconstruction Context and overall objectives Ridges A selection of scientific achievements Topological persistence Software development and technology transfer Delaunay triangulations and Voronoi diagrams New challenges Compact data structures Surface meshing by Delaunay refinement ◮ A simple and incremental algorithm for sampling and meshing surfaces ◮ Produces a good approximation of the surface ◮ Produces sparse samples of optimal size ◮ Produces facets with good aspect ratio ◮ General (black box) model of surfaces basic primitive: intersection line segment-surface ◮ Usable in various situations implicit surfaces, remeshing, reconstruction simplification of large data sets SGP 03, Graph. Models 05, SOCG 05-06, Oudot’s Ph.D. thesis Evaluation seminar GEOMETRICA Period 2002-2006
The team Sampling, meshing and reconstruction Context and overall objectives Ridges A selection of scientific achievements Topological persistence Software development and technology transfer Delaunay triangulations and Voronoi diagrams New challenges Compact data structures Evaluation seminar GEOMETRICA Period 2002-2006
The team Sampling, meshing and reconstruction Context and overall objectives Ridges A selection of scientific achievements Topological persistence Software development and technology transfer Delaunay triangulations and Voronoi diagrams New challenges Compact data structures Surface reconstruction Evaluation seminar GEOMETRICA Period 2002-2006
The team Sampling, meshing and reconstruction Context and overall objectives Ridges A selection of scientific achievements Topological persistence Software development and technology transfer Delaunay triangulations and Voronoi diagrams New challenges Compact data structures Delaunay refinement for 3D meshes ◮ Coupling the surface mesher with a 3D Delaunay refinement process IMRT 05 ◮ interleaved refinement of surfaces and volumes ◮ multi-volumes domain are meshed at once ◮ with optionally region-dependent granularity Rineau’s Ph.D thesis Evaluation seminar GEOMETRICA Period 2002-2006
The team Sampling, meshing and reconstruction Context and overall objectives Ridges A selection of scientific achievements Topological persistence Software development and technology transfer Delaunay triangulations and Voronoi diagrams New challenges Compact data structures Variational meshing Minimize (locally) � � ρ ( x ) || x − x j || 2 dx E = x ∈ R j j = 1 .. k Lloyd iteration alternates: - partition w.r.t. sites (Voronoi tiling) - move sites to centroids Evaluation seminar GEOMETRICA Period 2002-2006
The team Sampling, meshing and reconstruction Context and overall objectives Ridges A selection of scientific achievements Topological persistence Software development and technology transfer Delaunay triangulations and Voronoi diagrams New challenges Compact data structures Variational meshing back Evaluation seminar GEOMETRICA Period 2002-2006
The team Sampling, meshing and reconstruction Context and overall objectives Ridges A selection of scientific achievements Topological persistence Software development and technology transfer Delaunay triangulations and Voronoi diagrams New challenges Compact data structures Ridges [SGP 03, CAGD 05-06, IJCGA 05, Pouget’s Ph.D. Thesis] Difficulties : ◮ Differential quantities up to 4th order Def.: extrema of ◮ Singularities: umbilics + self-intersections principal curvatures ◮ Non global orientability of principal along curvature lines directions Contributions : ◮ On meshes: guaranteed approximations ◮ On parametric surfaces: novel implicit characterization as P ( u , v ) = 0 ◮ On polynomial surfaces: certified algorithms Evaluation seminar GEOMETRICA Period 2002-2006
The team Sampling, meshing and reconstruction Context and overall objectives Ridges A selection of scientific achievements Topological persistence Software development and technology transfer Delaunay triangulations and Voronoi diagrams New challenges Compact data structures Extraction on parametric surfaces : illustration back h ( u , v ) = 116 u 4 v 4 − 200 u 4 v 3 + 108 u 4 v 2 − 24 u 4 v − 312 u 3 v 4 + 592 u 3 v 3 − 360 u 3 v 2 + 80 u 3 v + 252 u 2 v 4 − 504 u 2 v 3 + 324 u 2 v 2 − 72 u 2 v − 56 uv 4 + 112 uv 3 − 72 uv 2 + 16 uv . • P ( u , v ) : bivariate polynomial of total degree 84, of degree 43 in u , degree 43 in v with 1907 terms and coefficients with up to 53 digits. • Certified extraction < 10 minutes Evaluation seminar GEOMETRICA Period 2002-2006
The team Sampling, meshing and reconstruction Context and overall objectives Ridges A selection of scientific achievements Topological persistence Software development and technology transfer Delaunay triangulations and Voronoi diagrams New challenges Compact data structures Topological persistence How to distinguish “features” from “noise”? ◮ track the creation/destruction of components/loops... in sub-level sets as “sea level” increases ◮ set of “persistence intervals” ≃ life-spans of features ◮ “features” yield long intervals, “noise” yields short ones Evaluation seminar GEOMETRICA Period 2002-2006
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