Tamal K. Dey The Ohio State University Surface and Volume Meshing with Delaunay Refinement

2/52 Department of Computer and Information Science Polyhedral Volumes and Surface Input PLC Final Mesh QualMesh based on Cheng-Dey-Ramos-Ray 04 (solved small angle problem effectively)

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

4/52 Department of Computer and Information Science Polygonal surface

5/52 Department of Computer and Information Science Polyhedral surface Given a polyhedron P find the Delaunay mesh Extra vertices allowed. Conforming : Each input edge is the union of some mesh edges. Each input facet is the union of some mesh triangles. Quality guarantees.

6/52 Department of Computer and Information Science Radius-edge ratio: R - circumradius of a triangle. – shortest edge length. Quality Measures

7/52 Department of Computer and Information Science Quality Tetrahedra ThinFlat …… Bounded radius-edge-ratio: Sliver

8/52 Department of Computer and Information Science Related work Delaunay refinement Bounded radius-edge ratio [Chew 89, Ruppert 92, Shewchuk 98]. Disallow acute input angles. Allow small angles Effective implementation [Shewchuk 00, Murphy et al. 00, Cohen-Steiner et al. 02]. No quality guarantee. Allow small angles. [Cheng and Poon 03] [Cheng, Dey, Ramos and Ray 04]

9/52 Department of Computer and Information Science Related work [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.

10/52 Department of Computer and Information Science Main Result A simpler Delaunay meshing algorithm Local feature size needed only at the sharp vertices. No spherical surfaces to mesh. Quality Guarantees Most tetrahedra have bounded radius-edge ratio. Skinny tetrahedra will be provably close to the acute input angles. Quality Meshing for Polyhedra with Small Angles [Cheng, Dey, Ramos, Ray]

11/52 Department of Computer and Information Science Voronoi/Delaunay

12/52 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).

13/52 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.

14/52 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.

15/52 Department of Computer and Information Science Local Feature Size Local feature size: radius of smallest ball that intersects two disjoint input elements. Lipschitz property:

16/52 Department of Computer and Information Science Small angle problem

17/52 Department of Computer and Information Science SOS-split [Cohen-Steiner et al. 02] Sharp vertex protection

18/52 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.

19/52 Department of Computer and Information Science Algorithm Protect sharp vertices Construct a Delaunay mesh. Loop: Split encroached subsegments and non-Delaunay subfacets. 2-expansion of diametrical ball of sharp segments. (Radius = O( f(center) ) ) Refinement: Eliminate skinny triangle Keep their circumcenters outside We do not want to compute f (center)

20/52 Department of Computer and Information Science Refinement Split encroached subsegments and non- Delaunay subfacets. Let c be the circumcenter of a skinny triangle. If c lies inside the protecting ball of a sharp vertex or sharp subsegment then do nothing Else if c encroaches a subsegment or subfacet split it. Else insert c.

21/52 Department of Computer and Information Science Positions of skinny triangle

22/52 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 tetrahedra have bounded radius-edge ratio. Any skinny tetrahedron is at a distance from some sharp vertex or some point on a sharp edge.

23/52 Department of Computer and Information Science Results

24/52 Department of Computer and Information Science Results

25/52 Department of Computer and Information Science R/L Distribution ModelR/L >2.2 Anchor Rail Wiper Cutter Simple Box Ushape Mesh Test

26/52 Department of Computer and Information Science Dihedral Angle Distribution ModelDihedral Angle [0-5](5-10](10-15](15-30]>30 Anchor Rail Wiper Cutter Simple Box Ushape Mesh Test

Quality Meshing with Weighted Delaunay Refinement by Cheng-Dey 02 Meshing Polyhedra with Sliver Exudations

28/52 Department of Computer and Information Science History Bern, Eppstein, Gilbert 94 - Quadtree meshing (Non-Delaunay) Shewchunk 98 – Extended Delaunay refinement to 3D (Slivers remain) Cheng, Dey, Edelsbrunner, Facello, Teng Silver exudation (no boundary) Li, Teng Silver exudation with boundary (randomized extending Chew)

29/52 Department of Computer and Information Science Definitions Input is a PLC with no acute input angle. f(x): Local feature size x f(x) Weighted point: Weighted distance: If

30/52 Department of Computer and Information Science Weighted Delaunay Smallest orthospheres, orthocenters, orthoradius Weighted Delaunay tetrahedra

31/52 Department of Computer and Information Science Silver Exudation Delaunay refinement guarantees tetrahedra with bounded radius- edge-ratio Vertices are pumped with weights Sliver Theorem: Given a periodic point set V and a Delaunay triangulation of V with radius-edge ratio  , there exists  0 >0 and  0 >0 and a weight assignment in [0,  2 N(v) 2 ] for each vertex v in V such that  (  )   0 and  (  )>  0 for each tetrahedron  in the weighted Delaunay triangulation of V.

32/52 Department of Computer and Information Science Encroachments again Subsegment encroachments Subfacet encroachments p Weight assignments Vertex Gap Property : For each vertex u in V, the weight of u used for encroachment checking or pumping is at most  0 2 f(u) 2 and the Euclidean nearest neighbor distance of u in V is at least 2  0 f(u). p

33/52 Department of Computer and Information Science Locations of Centers Lemma 3.1: Suppose that the vertex gap property holds. If no weighted- subsegment or weighted-subfacet is encroached, no weighted vertex intersects a segment or a facet that does not contain p. Lemma 3.2: Suppose that the vertex gap property holds. (i)A weighted-subsegment contains its orthocenter. (ii)If no weighted-subsegment is encroached, a facet contains the orthocenter of any weighted-subfacet on it. (iii)If no weighted-subsegment or weighted-subfacet is encroached, the input domain contains the orthocenter of any weighted Delaunay tetrahedron inside the input domain.

34/52 Department of Computer and Information Science Encroachments and Projection Lemma 3.3: If the vertex gap property holds, then for any weighted-subsegment ab on an edge e of P, ab cannot be encroached by any vertex that lies on an edge adjacent to e or a facet adjacent but non-incident to e. Lemma 3.4: Let abc be a weighted-subfacet on a facet F of P. If there is no encroached weighted-subsegment, then abc cannot be encroached by any vertex that lies on a facet adjacent to F or an edge adjacent but non-incident to F. Lemma 3.5: If no weighted-subsegment is encroached and encroaches upon some weighted-subfacet on a facet F, then there exists a weighted-subfacet h on F which is encroached upon by and h contains the orthogonal projection of p on F. p

35/52 Department of Computer and Information Science Q UAL M ESH algorithm 1.Compute the Delaunay triangulation of input vertices 2.Refine Rule 1: subsegment refinement Rule 2: subfacet refinement Rule 3: Tetrahedron refinement Rule 4: Weighted encroachment Check if weighted vertices encroach, if so refine. 3.Pump a vertex incident to silvers

36/52 Department of Computer and Information Science Insertion radii Parent-child: Type 3: a vertex of split tetrahedron Type 1, 2: Encroaching vertex r x is the distance from the nearest vertex in the current V Lemma 4.3: Suppose that the vertex gap property holds. Let x be an input vertex or a vertex inserted or rejected. Let p be the parent of x, if it exists. (i)If x is an input vertex or p is an input vertex, then (ii)Otherwise,

37/52 Department of Computer and Information Science Inter-vertex distances Lemma 5.2: Let x be a vertex of P or a vertex inserted or rejected by Q UAL M ESH. We have the following invariants for  0 > 4. (i)If x is a vertex of P or the parent of x is a vertex of P, then Otherwise, if x has type i, for 1  i  3, then r x  f(x)/C i. (ii)For any other vertex y that appears in V currently, (iii)If x is inserted by Q UAL M ESH, the vertex gap property holds afterwards.

38/52 Department of Computer and Information Science Guarantees Theorem 7.1 (Termination): Q UAL M ESH terminates with a graded mesh. Theorem 7.2 (Conformity): No weighted-subsegment or weighted-subfacet is encroached upon the completion of Q UAL M ESH

39/52 Department of Computer and Information Science Weight property[  ]: each weight u   N(u) Ratio property [  ]: orthoradius-edge-ratio is at most . Lemma 6.2: Let V be a finite point set. Assume that Del V has ratio property [  ], has weight property [  ], and the orthocenter of each tetrahedron in Del lies inside Conv V. Then Del has ratio property [  ’] for some constant  ’ depending on  and  Lemma 7.1: Assume that Del V has ratio property [  ]. The lengths of any two adjacent edges in K(V) is within a constant factor v depending on  and . Lemma 7.2: Assume that Del V has ratio property [  ]. The degree of every vertex in K(V) is bounded by some constant  depending on  and . No Sliver

40/52 Department of Computer and Information Science No Silver (continued) A silver remains incident to p only within a subinterval of width O(  0 ) during pumping. Lemma 7.2 says that the intervals of slivers can be made arbitrary small by choosing  0. if the weight interval contains subinterval [0,  2 N(  ) 2 ] for some  , exudation works.

41/52 Department of Computer and Information Science No Sliver (continued) Lemma 7.3: Let M be the mesh obtained at the end of step 2 of Q UAL M ESH. For any vertex v in M, its nearest neighbor distance is at most. Theorem 7.3: There is a constant  0 > 0 such that  (  ) >  0 for every tetrahedron  in the output mesh of Q UAL M ESH.

42/52 Department of Computer and Information Science Size Optimality Output vertices Output tetrahedra Any mesh of D with bounded aspect ratio must have tetrahedra Theorem 7.4: The output size of Q UAL M ESH is within a constant factor of the size of any mesh of bounded aspect ratio for the same domain.

43/52 Department of Computer and Information Science Example - Arm Input PLC Slivers Sliver RemovalFinal Mesh

44/52 Department of Computer and Information Science Example - Cap Input PLC Slivers Sliver RemovalFinal Mesh

45/52 Department of Computer and Information Science Example - Wrench Input PLC Slivers Sliver Removal Final Mesh

46/52 Department of Computer and Information Science Example - Propellant Input PLC Slivers Sliver Removal Final Mesh

47/52 Department of Computer and Information Science Time Ratio=2.2, Dihedral=3 , Factor=0.5Ratio=2.2, Dihedral=5 , Factor=0.5 # of slivers/min. dihedral angle after Skinny removal # of slivers/min. dihedral angle after Pumping # of slivers/min. dihedral angle after Skinny removal # of slivers/min. dihedral angle after Pumping Anchor002 / Arm10 / / / / 4.66 Cap 6 / / / 4.59 Cavity 3 / / / 4.52 Chair1 / / / 4.18 House4 / / / / 2.02 L-shape0000 Nalcola002 / OurHouse001 / Propellant9 / / / / 0.87 Table1 / / TeaTable0000 Tfire1 / / Wrench24 / / / 4.36

48/52 Department of Computer and Information Science Extending sliver exudations to polyhedra with small angles Carry on all steps for meshing polyhedra with small angles Add the sliver exudation step All tetrahedra except the ones near small angles have bounded aspect ratio. Cheng-Dey-Ray 2005 (Meshing Roundtable 2005)

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

50/52 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

51/52 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]

52/52 Department of Computer and Information Science Previous Work 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.

53/52 Department of Computer and Information Science Restricted Delaunay Del Q| G :- Collection of Delaunay simplices whose corresponding dual Voronoi face intersects G.

54/52 Department of Computer and Information Science Delaunay Refinement (Chew)

55/52 Department of Computer and Information Science Topological Ball Property A -dimensional Voronoi face intersects G in a -dimensional ball. Theorem : [ES’97] The underlying space of the complex Del Q| G is homeomorphic to G if Vor Q has the topological ball property.

56/52 Department of Computer and Information Science Strategy Topology Sampling : Grow a sample P by insertion until the Topological Ball Property is satisfied. Geometry Sampling: Quality. Smoothness.

57/52 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.

58/52 Department of Computer and Information Science Voronoi Edge Edge Lemma : If intersects Σ twice or more or tangentially, the farthest is > k f(p) away from all points.

59/52 Department of Computer and Information Science Voronoi Edge Lemma Justification Edge not parallel to normal Almost normal edge

60/52 Department of Computer and Information Science Voronoi Facet Facet Lemma I: If has a cycle of, then has a point > k f(p) away from all points. Facet Lemma II: If has two or more intervals, then s.t is > k f(p) away from all points.

61/52 Department of Computer and Information Science Voronoi Cells (>1 boundary) Cell Lemma(>1 boundary): If is a manifold with two or more boundary cycles, then with > k f(p) away from all points.

62/52 Department of Computer and Information Science Voronoi Cells (0-,1-boundary) Single boundary but not simply connected (Silhouette takes care) Component inside ( taken care by critical pts.)

63/52 Department of Computer and Information Science Silhouette Definition : Silhouette Lemma I: If has a single boundary and no pt with, then is a disk. Silhouette Lemma II: Any is > k f(p) away from all points.

64/52 Department of Computer and Information Science Silhouette Computation

65/52 Department of Computer and Information Science Voronoi Edge Test VorEdge( ): if e intersects Σ in two or more points (EdgeSurface), return the farthest point. [ Edge Lemma ]

66/52 Department of Computer and Information Science Voronoi Edge Test V EDGE ( ) If intersects Σ in two or more points, return the point furthest from. [ Edge Lemma ]

67/52 Department of Computer and Information Science Topological Disk Test TopoDiskK ( ) If is not a topological disk, return furthest point in edge-surface intersections.

68/52 Department of Computer and Information Science Voronoi Facet Facet with more than one topological interval. u v F [Facet Lemma II]

69/52 Department of Computer and Information Science Topological Disk Test TopoDiskK ( ) If is not a topological disk, return furthest point in. [Facet Lemma II]

70/52 Department of Computer and Information Science Voronoi Cell If is not a 2-manifold with a single boundary then TopoD ISK () will take care of it. [Cell Lemma]

71/52 Department of Computer and Information Science Topological Disk Test TopoDiskK ( ) If is not a topological disk, return furthest point in. [Facet Lemma II] [Cell Lemma]

72/52 Department of Computer and Information Science Four Tests Contd.. FacetCycle( ): X:= CritCurve(Σ,F), then check if L intersects twice or more, return a point. [Facet Lemma I]. Silhouette(V p ): X:=CritSilh(Σ,n p,d). If, return a point from X otherwise see facet intersection. [Silhouette Lemma]

73/52 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.

74/52 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.

75/52 Department of Computer and Information Science Algorithm D EL M ESH (Σ) SampleTopology(P) Quality(P) Smooth(P) Continue till no point is added.

76/52 Department of Computer and Information Science Guarantees Output surface is homeomorphic to Σ. Each triangle has a guaranteed aspect ratio. Smooth triangulation. Size of P is asymptotically optimal.

77/52 Department of Computer and Information Science Results

78/52 Department of Computer and Information Science Polygonal surfaces [Dey-Ray 05] Input: Input: Polygonized surface G approximating. Output: Output: A vertex set Q where each vertex lies on G and triangulation T

79/52 Department of Computer and Information Science Prior Work Delaunay Refinement [Chew ’89, Ruppert ’95 Shewchuk ’98, many others]. Chew [1993]. Cheng, Dey, Edelsbrunner and Sullivan[2001]. Boissonnat and Oudot [2003]. Sampling and meshing a surface with guaranteed topology and geometry (Cheng, Dey, Ramos, Ray)[2004].

80/52 Department of Computer and Information Science Non-Smoothness Input G is piecewise-linear. Non-smoothness is a challenge. Delaunay refinement for polyhedron is not a viable choice.

81/52 Department of Computer and Information Science Delaunay refinement

82/52 Department of Computer and Information Science Assumptions G approximates a smooth. G is -flat w.r.t. Many designed surfaces, reconstructed surfaces are -flat.

83/52 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.

84/52 Department of Computer and Information Science F CYCLE ( ) If has a cycle, return the point in furthest from. Voronoi Facet

85/52 Department of Computer and Information Science Voronoi Cell If is not a 2-manifold with a single boundary then D ISK () will take care of it.

86/52 Department of Computer and Information Science Voronoi Cell V CELL ( ) If Euler number return the point in furthest from. Single boundary but not 2- disk

87/52 Department of Computer and Information Science Extending results from smooth case Big empty balls acting as medial balls If t=pqr has O(k)f(p) circumradius, ‹n t, n p ›=O(k) provided lengths > √(6 δ)

88/52 Department of Computer and Information Science Bounding Conditions Condition 1: and. Condition 2: [Amenta,Choi,Dey,Leekha ‘02]

89/52 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

90/52 Department of Computer and Information Science Geometric Approximation : is the circumradius of the triangle t. : is the ratio of the circumradius to shortest edge length of t.

91/52 Department of Computer and Information Science Refinement G EOM R ECOV ( ) 1.For, if with insert the intersection point. T RIANGLE _Q UAL ( ) 1.For, if with insert the intersection point

92/52 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].

93/52 Department of Computer and Information Science Termination Theorem :Theorem : If satisfies a bounding condition with respect to then it will terminate. and

94/52 Department of Computer and Information Science Results

95/52 Department of Computer and Information Science Results

96/52 Department of Computer and Information Science Results

97/52 Department of Computer and Information Science Meshing a equipotential surface data: courtesy to Alan Saalfeld V=21014, F=42024 V=21507, F=42904 V=2141, F=4278

98/52 Department of Computer and Information Science Conclusions Different algorithms for Delaunay meshing of surfaces in different input forms All of them have theoretical guarantees The implementations can be downloaded from Cocone: cocone.html Polyhedra: qualmesh.html Polygonal: surfremesh.html Meshing a nonsmooth curved surface, remeshing polygonal surface approximating a non-smooth surface is a challenge. Anisotropic meshing [CDRW05] CGAL acknowledgement: