Delaunay Meshing of Surfaces

1. Delaunay Meshing of Surfaces Tamal K. Dey The Ohio State University

2. ` Point Cloud Data Surface Reconstruction Point Cloud Surface Reconstruction

3. Voronoi Based Algorithms • Alpha-shapes (Edelsbrunner, Mück 94) • Crust (Amenta, Bern 98) • Natural Neighbors (Boissonnat, Cazals 00) • Cocone (Amenta, Choi, Dey, Leekha, 00) • Tight Cocone (Dey, Goswami, 02) • Power Crust (Amenta, Choi, Kolluri 01) • Distance function (Edelsbrunner 95, Giesen 02, Chazal, Lieutier,Cohen-Steiner 06)

4. f(x) • Medial axis • f(x) is the distance to medial axis Local Feature Size and ε-sample [ABE98] • Each x has a sample within f(x) distance

5. Reconstruction Guarantees • Given an ε-sample from a smooth, compact surface without boundary, the output piecewise linear surface has the exact topology (homeomorphic/isotopic) and approximate geometry (Hausdorff distance O(ε)f(x)) if ε<0.06. • Curve and Surface Reconstruction : Algorithms with Mathematical Analysis, Cambridge University Press (2006?)

6. Polyhedral Surface (conforming) Input PLC Output Mesh

7. Basics of Delaunay Refinement Chew 89, Ruppert 95 • Maintain a Delaunay triangulation of the current set of vertices. • If some property is not satisfied by the current triangulation, insert a new point which is locally farthest. • Burden is on showing that the algorithm terminates (shown by packing argument).

8. Delaunay Refinement for Quality • R/l = 1/(2sinθ)≥1/√3 • Choose a constant ≥ 1 if R/l is greater than this constant, insert the circumcenter.

9. Delaunay Refinement for 2D Point Sets R/l ≥ 1.0 30 degree l R

10. No input angle is less than 90 degree Polyhedral Volumes and Surface[Shewchuk 98] Input PLC Final Mesh

11. Delaunay Refinement for Input Conformity • Diametric ball of a subsegment empty. • If encroached by a point p, insert the midpoint. • Subfacets: 2D Delaunay triangles of vertices on a facet. • If diametric ball of a subfacet encroached by a point p, insert the center.

12. Small Angle Problem

13. Sharp Vertex Protection SOS-split [Cohen-Steiner et al. 02]

14. Subfacet Splitting • Trick to stop indefinite splitting of subfacets in the presence of small angles is to split only the non-Delaunay subfacets. • It can be shown that the circumradius of such a subfacet is large when it is split.

15. Summary of Results • A simpler algorithm and an implementation. • Local feature size needed at only the sharp vertices. • No spherical surfaces to mesh. • Quality guarantees • Most triangles have bounded radius-edge ratio. • Any skinny triangle is at a distance from some sharp vertex or some point on a sharp edge.

16. Results

17. Delaunay Meshing for Smooth Surfaces

18. Implicit Surface F: R3 => R, Σ = F-1(0)

19. Two Work • Boissonnat-Oudot 03: General implicit surfaces, Ensure TBP with local feature size • Cheng-Dey-Ramos-Ray 04: General implicit surface, no feature size computation.

20. Restricted Delaunay • Del Q|Σ:- Collection of Delaunay simplices whose corresponding dual Voronoi face intersects Σ.

21. Topological Ball Property • A -dimensional Voronoi face intersects in Σ a -dimensional ball. • Theorem : [ES’97] The underlying space of the complex Del Q|Σ is homeomorphic to Σ if Vor Q has the topological ball property.

22. Building Sample P • If topological ball property is not satisfied insert a point p in P. • Argue each point p is inserted > k f(p) away from all other points where k = 0.06. -- Termination is guaranteed by 2. -- Topology is guaranteed by 1 and the termination.

23. TopologicalDisk Test TopoDiskK ( ) If is not a topological disk, return furthest point in edge-surface intersections.

24. TopologicalDisk Test TopoDiskK ( ) If is not a topological disk, return furthest point in .

25. Topology(P): If VorEdge, TopoDisk, FacetCycle or Silhouette in order inserts a new point in P. Continue till no new point is inserted. Return P. Topology Lemma: If P includes critical points of Σ and Topology(P) terminates then topological ball property is satisfied. Distance Lemma I: Each inserted point p is > k f(p) away from all other points. Topology Sampling

26. Geometry Sampling • Quality(P): If a triangle t has ρ(t) > (1+k)2 , insert where e = dual t. • Smoothing(P): If two adjacent triangles make sharp edge, insert where e = dual t. • Distance Lemma II: Each point is > k f(p) away from all other points.

27. Results

28. Output: A vertex set Q where each vertex lies on G and triangulation T Polyhedral Surfaces (non-conforming)[Dey-Li-Ray 05] Input: Polyhedral surface G approximating .

29. Assumptions • G approximates a smooth . • G is -flat w.r.t . • Many designed surfaces, reconstructed surfaces are -flat. • Relation to Lipschitz surface (Boissonnat-Oudot 06)

30. Sparse Sampling and Termination • Theorem:If and are sufficiently small, such that each intersection point is away from all other points. and

31. Results

32. Conclusions • Different algorithms for Delaunay meshing of surfaces/volumes in different input forms • All of them have theoretical guarantees • The implementations can be downloaded from http://www.cse.ohio-state.edu/~tamaldey/ Cocone: cocone.html Polyhedra: qualmesh.html Polyhedra (nonconforming): surfremesh.html • Meshing a nonsmooth curved surface [BO06], remeshing polygonal surface with small angles. • Anisotropic meshing [CDRW06] • CGAL acknowledgement: www.cgal.org