Other secondary sources Jonathan Shewchuk lecture notes on mesh generation. Surface reconstruction survey by Cazals and Giesen. Chapter on meshing surfaces by Boissonnat, Cohen-Steiner, Mourrain, Rote, and Vegter.
2D Curve Reconstruction Blue Delaunay edges reconstruct the curve, pink triangulate interior/exterior. Many algorithms, with proofs.
Sliver tetrahedra In 3D, some Voronoi vertices are not near medial axis …
Sliver tetrahedra …. even when samples are arbitrarily dense. Interior Voronoi balls
Poles Interior polar balls Subset of Voronoi vertices, the poles, approximate medial axis. Amenta & Bern, 98 “Crust” papers
Sampling Requirement -sample: distance from any surface point to nearest sample is at most small constant times distance to medial axis. Zero at sharp corners – uh-oh.
Sampling Requirement Intuition: dense sampling where curvature is high or near features.
Kinds of Results Assuming input sampling is dense enough, then output triangulation will be homeomorphic to, and close to, the original surface. Usually also demonstrate robustness by implementation.
Algorithms and Software Examine Delaunay triangles Amenta and Bern, Crust Amenta, Choi, Dey and Leekha, Cocone Dey & Goswami, (water)-Tight Cocone Dey & Giesen, undersampling errors Inside/Outside Boissonnat, sculpting Boissonnat and Cazals, Natural neighbor Amenta, Choi and Kolluri, Power crust Kolluri, Shewchuk, O’Brien, Spectral
Distance function Distance from nearest sample. Giesen and John, 01,02
Distance function flow Consdier uphill flow …. Idea: interior is part that flows to interior maxima.
Distance function Max and (some) saddle points. Compute flow combinatorially using Delaunay/Voronoi
Distance Functions are Pretty Stable Distance functions of similar (Hausdorff) sets are similar –Maxima lie near near-maxima (points with small generalized gradient)
Gradient Flow Algorithms Giesen and John Edelsbrunner Wrap….
Geomagic Founded by Herbert Edelsbrunner. Leading system on the market.
Other Companies Dessault – Catia – Andrei Liutier as “resident genius”, includes Nearest- neighbor reconstruction? Imageware, RapidForm, ScanTo3D. Bottom line - They all know we’re out here, but we are not integral to their business.
What’s really used in graphics… Poisson algorithm - Kazhdan, Bolitho, Hoppe ‘06. Define gradient at boundaries, solve PDE on octree to fill space, take level-set of implicit function.
Why? Noise Noisy data sources are increasingly important. Computing DT of whole point cloud is overkill. Persistence is really not the answer. Averaging in 3D is faster and better. Distance-like functions (Chazal talk)?
Why? Delaunay bottleneck 3D Delaunay triangulation O(n 2 ), O(n) in practice, but still slow. Attali, Boissonnat, Lieutier ‘03 O(n lg n) DT complxity Funke & Ramos, ‘02, Funke & Milosavljevic ‘07, O(n lg n) thinning and then reconstructing. Cheng, Jin, Lau, this conference. More practical O(n lg n).
For comparison… Delaunay of 1 million 3D points ~ 1 minute. GPU octree: 18 milliseconds GPU k-NN: answer 1 million 50-NN queries/second (based on Bern, Chan reduction to sorting) A., Li, Simons, Parkaravor, Abbasinejad, Owens
What to work on? Fast octree-based algorithms with proofs -> surface meshing algorithms. Prove results about what people already do in practice. Work on other problems related to building objects from data! Eg, alignment (= matching)
Surface meshing Adapt planar techniques to surfaces. Chew
Restricted Delaunay Triangulation Edelsbrunner and Shah, ‘96, showed closed-ball property: if every rVor cell is a disk, rVoD is homeomorphic to surface. 3D Voronoi diagram restricted to 2D surface. Delaunay is dual.
Kind of results Surface can be covered with well- shaped triangles, and the number of triangles is O(minimal). Requires the input surface boundary to have no sharp angle; otherwise algorithm may not terminate!
Delaunay refinement Smooth - Chew –Boissonnat and Oudot –Cheng, Dey, Ramos and Ray Piecewise-smooth –Rineau and Yvinec –Cheng, Dey and Ramos –Cheng, Dey and Levine (software!)
Edge Protection Place strings of barely- intersecting balls along edges; mesh faces by Delaunay refinement. Dey&Levine
Comment Local feature size is overkill for just surface meshing.
Sliver tetrahedra Are NOT eliminated by optimizing circumradius/shortest edge. This is OK for finite volume methods (Miller, Talmor, Teng and Walkington, STOC ’95, mesh a Poisson-disk point set). But not OK for finite element methods!
Sliver removal Sliver exudation, ‘00, Cheng, Dey, Edelsbrunner, Facello and Teng. Adjust weights of mesh vertices to squeeze out slivers. Dihedral guaranteed to be bounded away from zero. Randomized perturbation, Chew ‘97 and Li and Teng ‘01.
Isosurface Stuffing Octree-based method, Labelle and Shewchuk ‘07. Dihedral angles bounded between 10.7 o and 164.8 o Requires: smooth manifold boundary, uniform sizing on boundary. NOT DELAUNAY.
Free Tet Meshing Software Several algorithms implemented in CGAL - Stéphane Tayeb, Yvinec, L. Rineau, Alliez and Tournois. TetGen, Hang Si, Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Some day…Pyramid, Shewchuk.
Industry/Government Ansys – Sells simulation capability, not meshes. Many CAD systems, eg. SolidWorks. Sandia organizes International Meshing Roundtable. This is very incomplete.
What to work on? …you’re asking me?... Stuff I didn’t talk about –Anisotropic meshing (Canas & Gortler, this conference) –Quad/hex meshing Digital differential geometry? Get out and meet people.
Conclusions Real problems, real science/industry, real impact. Theoretical structures and results, and software. Bridging the gap to practice is an ongoing challenge, not necessarily our top priority.