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1/43Department of Computer Science and Engineering
Delaunay Mesh Generation
Tamal K. Dey
The Ohio State University
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Delaunay Mesh Generation
• Automatic mesh generation with good quality.
• Delaunay refinements:• The Delaunay triangulation
lends to a proof structure.• And it naturally optimizes
certain geometric properties such as min angle.
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Input/Output
• Points P sampled from a surface S in 3D (don’t know S) Reconstruct : S A simplicial complex K, • (i) K has a geometric realization in 3D• (ii) |K| homeomorphic to S, • (iii) Hausdorff distance between |K| and S is small
• A smooth surface S(or a compact set):• Generate a point sample P from S• Generate a simplicial complex K with vert K=P and
satisfying (i), (ii), (iii).
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Surface Reconstruction
`
Point Cloud
Surface Reconstruction
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Medial Axis
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Local Feature Size (Smooth)
• Local feature size is calculated using the medial axis of a smooth shape.
• f(x) is the distance from a point to the medial axis
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Each x has a sample
within f(x) distance
-Sample[ABE98]
x
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Voronoi/Delaunay
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Normal and Voronoi Cells(3D) [Amenta-Bern SoCG98]
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Poles
P+
P-
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Normal Lemma
The angle between the pole vector
vp and the normal np is O().
P+
P-
np
vp
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Restricted Delaunay
• If the point set is sampled from a domain D.
• We can define the restricted Delaunay triangulation, denoted Del P|D.• Each simplex Del P|D is the dual
of a Voronoi face V that has a nonempty intersection with the domain D.
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Topological Ball Property (TBP)
• P has the TBP for a manifold S if each k-face in Vor P either does not intersect S or intersects in a topological (k-1)-ball.
• Thm (Edelsbrunner-Shah97 ) If P has the TBP then Del P|S is homeomorphic to S.
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Cocone (Amenta-Choi-D.-Leekha)
vp= p+ - p is the pole vector
Space spanned by vectors
within the Voronoi cell
making angle > 3/8 with
vp or -vp
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Cocone Algorithm
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Cocone Guarantees
Theorem:
Any point x S is within O( )e f(x) distance from a point in the output. Conversely, any point of output surface has a point x S within O(e)f(x) distance. Triangle normals make O(e) angle with true normals at vertices.
Theorem:
The output surface computed by Cocone from an e-sample is homeomorphic to the sampled surface for sufficiently small (<0.06)e .
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Meshing
• Input• Polyhedra• Smooth Surfaces• Piecewise-smooth Surfaces• Non-manifolds
&
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Basics of Delaunay Refinement
• Pioneered by Chew89, Ruppert92, Shewchuck98
• To mesh some domain D,1. Initialize a set of points P D, compute Del P.
2. If some condition is not satisfied, insert a point c from D into P and repeat step 2.
3.Return Del P|D.
• Burden is to show that the algorithm terminates (shown by a packing argument).
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Polyhedral Meshing
• Output mesh conforms to input:• All input edges meshed as a
collection of Delaunay edges.• All input facets are meshed with a
collection of Delaunay triangles.• Algorithms with angle
restrictions:• Chew89, Ruppert92, Miller-Talmor-
Teng-Walkington95, Shewchuk98.• Small angles allowed:
• Shewchuk00, Cohen-Steiner-Verdiere-Yvinec02, Cheng-Poon03, Cheng-Dey-Ramos-Ray04, Pav-Walkington04.
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Smooth Surface Meshing
• Input mesh is either an implicit surface or a polygonal mesh approximating a smooth surface
• Output mesh approximates input geometry, conforms to input topology:• No guarantees:
• Chew93.• Skin surfaces:
• Cheng-Dey-Edelsbrunner-Sullivan01.
• Provable surface algorithms:• Boissonnat-Oudot03 and Cheng-
Dey-Ramos-Ray04.• Interior Volumes:
• Oudot-Rineau-Yvinec06.
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Sampling Theorem
Theorem (Boissonat-Oudot 2005):
If P S is a discrete sample of a smooth surface S so that each x where a Voronoi edge intersects S lies within ef(x) distance from a sample, then for e<0.09, the restricted Delaunay triangulation Del P|S has the following properties:
(i) It is homeomorphic to S (even isotopic embeddings).
(ii) Each triangle has normal aligning within O(e) angle to the surface normals
(iii) Hausdorff distance between S and Del P|S is O(e2) of the local feature size.
Theorem:(Amenta-Bern 98, Cheng-Dey-Edelsbrunner-Sullivan 01)
If P S is a discrete e-sample of a smooth surface , Sthen for e< 0.09 the restricted Delaunay triangulation Del P|S has the following properties:
Sampling Theorem Modified
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Basic Delaunay Refinement
1. Initialize a set of points P S, compute Del P.
2. If some condition is not satisfied, insert a point c from S into P and repeat step 2.
3. Return Del P|S.
Surface Delaunay Refinement
2. If some Voronoi edge intersects S at x with d(x,P)> ef(x) insert x in P.
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Difficulty
• How to compute f(x)?• Special surfaces such as
skin surfaces allow easy computation of f(x) [CDES01]
• Can be approximated by computing approximate medial axis, needs a dense sample.
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A Solution
• Replace d(x,P)< ef(x) with d(x,P)<l, an user parameter
• But, this does not guarantee any topology
• Require that triangles around vertices form topological disks
• Guarantees that output is a manifold
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A Solution
1. Initialize a set of points P S, compute Del P.
2. If some Voronoi edge intersects M at x with d(x,P)>ef(x) insert x in P, and repeat step 2.
2. (b)If restricted triangles around a vertex p do not form a topological disk, insert furthest x where a dual Voronoi edge of a triangle around p intersects S.
3. Return Del P|S
2. (a) If some Voronoi edge intersects S at x with d(x,P)> l insert x in P, and repeat step 2(a).
Algorithm DelSurf(S,l)
X=center of largest Surface Delaunay ballx
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A MeshingTheorem
Theorem:
The algorithm DelSurf produces output mesh with the following guarantees:
(i) The output mesh is always a 2-manifold
(ii) If l is sufficiently small, the output mesh satisfies topological and geometric guarantees:
1. It is related to S with an isotopy. 2. Each triangle has normal aligning within O(l) angle to the
surface normals
3. Hausdorff distance between S and Del P|S is O(l2) of the local feature size.
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Implicit surface
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Remeshing
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PSCs – A Large Input Class[Cheng-D.-Ramos 07]
• Piecewise smooth complexes (PSCs) include:• Polyhedra• Smooth Surfaces• Piecewise-smooth Surfaces• Non-manifolds
&
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Protecting Ridges
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DelPSC Algorithm[Cheng-D.-Ramos-Levine 07,08]
DelPSC(D, λ)1. Protect ridges of D using protection balls. 2. Refine in the weighted Delaunay by turning the balls
into weighted points.
1.Refine a triangle if it has orthoradius > l.2.Refine a triangle or a ball if disk condition is violated3.Refine a ball if it is too big.
3. Return i Deli S|Di
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Guarantees for DelPSC
1. Manifold• For each σ D2, triangles in Del
S|σ are a manifold with vertices only in σ. Further, their boundary is homeomorphic to bd σ with vertices only in σ.
2. Granularity• There exists some λ > 0 so that
the output of DelPSC(D, λ) is homeomorphic to D.
• This homeomorphism respects stratification, For 0 ≤ i ≤ 2, and σ Di, Del S|σ is homemorphic to σ too.
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Reducing λ
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Examples
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Examples
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Some Resources• Software available from
http://www.cse.ohio-state.edu/~tamaldey/cocone.html
http://www.cse.ohio-state.edu/~tamaldey/delpsc.html
http://www.cse.ohio-state.edu/~tamaldey/locdel.html
Open : Reconstruct piecewise smooth surfaces, non-manifolds
Open: Guarantee quality of all tetrahedra in volume meshing
A book Delaunay Mesh Generation: w/ S.-W. Cheng, J. Shewchuk (2012)
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Thank You!