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

2/52 Department of Computer and Information Science Point Cloud Data Surface Reconstruction ` Point Cloud Surface Reconstruction

3/52 Department of Computer and Information Science Voronoi Based Algorithms 1.Alpha-shapes (Edelsbrunner, Mück 94) 2.Crust (Amenta, Bern 98) 3.Natural Neighbors (Boissonnat, Cazals 00) 4.Cocone (Amenta, Choi, Dey, Leekha, 00) 5.Tight Cocone (Dey, Goswami, 02) 6.Power Crust (Amenta, Choi, Kolluri 01) 7.Distance function (Edelsbrunner 95, Giesen 02, Chazal, Lieutier,Cohen-Steiner 06)

4/52 Department of Computer and Information Science Medial axis f(x) is the distance to medial axis f(x) f(x) Each x has a sample within  f(x) distance Local Feature Size and ε-sample [ABE98]

5/52 Department of Computer and Information Science 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?)

6/52 Department of Computer and Information Science Polyhedral Surface (conforming) Input PLC Output Mesh QualMesh based on Cheng-Dey-Ramos-Ray 04 (solved small angle problem effectively)

7/52 Department of Computer and Information Science Polyhedral Surface (conforming) Input PLC Output Mesh

8/52 Department of Computer and Information Science 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).

9/52 Department of Computer and Information Science Implicit Surface F: R 3 => R, Σ = F -1 (0)

10/52 Department of Computer and Information Science 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.

11/52 Department of Computer and Information Science Delaunay Refinement for 2D Point Sets R/l ≥ degree R l

12/52 Department of Computer and Information Science Local Feature Size Local feature size: radius of smallest ball that intersects two disjoint input elements. Lipschitz property: x f(x)

13/52 Department of Computer and Information Science Delaunay Refinement with Boundary Conforming but still not Gabriel >f(x) x Circumcenter of skinny triangle encroaching edge. L R

14/52 Department of Computer and Information Science Polyhedral Volumes and Surface [Shewchuk 98] Input PLC Final Mesh No input angle is less than 90 degree

15/52 Department of Computer and Information Science 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.

16/52 Department of Computer and Information Science Polyhedral Surface with Any Angle Small angles allowed Conforming : Each input edge is the union of some mesh edges. Each input facet is the union of some mesh triangles. Quality guarantees.

17/52 Department of Computer and Information Science History No quality guarantee Effective implementation [Shewchuk 00, Murphy et al. 00, Cohen-Steiner et al. 02]. Quality guarantee [Cheng and Poon 03] Complex. Protect input segments with orthogonal balls. Need to mesh spherical surfaces. Expensive. Compute local feature/gap sizes at many points. [Cheng, Dey, Ramos and Ray 04]

18/52 Department of Computer and Information Science Small Angle Problem

19/52 Department of Computer and Information Science SOS-split [Cohen-Steiner et al. 02] Sharp Vertex Protection

20/52 Department of Computer and Information Science 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.

21/52 Department of Computer and Information Science 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.

22/52 Department of Computer and Information Science Results

Delaunay Meshing for Smooth Surfaces

Cheng-Dey-Ramos-Ray 04 Delaunay Meshing for Smooth Surfaces

25/52 Department of Computer and Information Science Implicit Surfaces Surface Σ is given by an implicit equation E(x,y,z)=0 Surface is smooth, compact, without any boundary

26/52 Department of Computer and Information Science Implicit Surface F: R 3 => R, Σ = F -1 (0)

27/52 Department of Computer and Information Science Medial axis f(x) is the distance to medial axis f(x) f(x) Each x has a sample within  f(x) distance Local Feature Size and ε-sample [ABE98]

28/52 Department of Computer and Information Science Previous Work Non Delaunay : Plantinga-Vegter 04 Chew 93: first Delaunay refinement for surfaces Cheng-Dey-Edelsbrunner-Sullivan 01: Skin surface meshing, Ensure topological ball property by feature size Boissonnat-Oudot 03: General implicit surfaces, Ensure TBP with local feature size Cheng-Dey-Ramos-Ray 04: General implicit surface, no feature size computation.

29/52 Department of Computer and Information Science 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.

30/52 Department of Computer and Information Science Restricted Delaunay Del Q| Σ :- Collection of Delaunay simplices whose corresponding dual Voronoi face intersects Σ.

31/52 Department of Computer and Information Science 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.

32/52 Department of Computer and Information Science Building Sample P 1.If topological ball property is not satisfied insert a point p in P. 2.Argue each point p is inserted > k f(p) away from all other points where k = Termination is guaranteed by Topology is guaranteed by 1 and the termination.

33/52 Department of Computer and Information Science Topological Disk Test TopoDiskK ( ) If is not a topological disk, return furthest point in edge-surface intersections.

34/52 Department of Computer and Information Science Topological Disk Test TopoDiskK ( ) If is not a topological disk, return furthest point in. [Facet Lemma II]

35/52 Department of Computer and Information Science Topological Disk Test TopoDiskK ( ) If is not a topological disk, return furthest point in.

36/52 Department of Computer and Information Science Topology Sampling 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.

37/52 Department of Computer and Information Science 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.

38/52 Department of Computer and Information Science Results

39/52 Department of Computer and Information Science Polyhedral surface (non-conforming)

40/52 Department of Computer and Information Science Polyhedral Surfaces (non-conforming) [Dey-Li-Ray 05] Input: Input: Polyhedral surface G approximating. Output: Output: A vertex set Q where each vertex lies on G and triangulation T

41/52 Department of Computer and Information Science 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)

42/52 Department of Computer and Information Science SurfRemesh 1.Initialize Q. 2.Compute Vor Q. 3.While (! Topology Recovered) 4. V EDGE (). 5. D ISK (). 6. F CYCLE (). 7. V CELL (). 8.End while 9.Output Del Q| G.

43/52 Department of Computer and Information Science Sparse Sampling and Termination Theorem:Theorem: If and are sufficiently small, such that each intersection point is away from all other points. and

44/52 Department of Computer and Information Science Remeshing reconstructed surfaces If P is an -sample, then the reconstructed surface with Delaunay methods (Cocone) are -flat for and. A simple algorithm for homeomorphic surface reconstruction [Amenta, Choi, Dey and Leekha ’ 02].

45/52 Department of Computer and Information Science Results

46/52 Department of Computer and Information Science Results

47/52 Department of Computer and Information Science Results

48/52 Department of Computer and Information Science 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 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: