tamal k. dey the ohio state university delaunay meshing of surfaces
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Tamal K. Dey The Ohio State University
Delaunay Meshing of Surfaces
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2/52Department of Computer and Information Science
Point Cloud Data Surface Reconstruction
`
Point Cloud
Surface Reconstruction
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3/52Department of Computer and Information Science
Voronoi Based Algorithms1. 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)
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4/52Department of Computer and Information Science
Medial axis
f(x) is the
distance
to medial axis
f(x)
Each x has a sample
within f(x) distance
Local Feature Size and ε-sample [ABE98]
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5/52Department 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?)
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7/52Department of Computer and Information Science
Polyhedral Surface (conforming)
Input PLC Output Mesh
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8/52Department 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).
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10/52Department of Computer and Information Science
Delaunay Refinement for Quality
• R/l = 1/(2sinθ)≥1/√3
• Choose a constant ≥ 1if R/l is greater than this constant, insert the circumcenter.
R
l
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11/52Department of Computer and Information Science
Delaunay Refinement for 2D Point Sets
R/l ≥ 1.0
30 degree
R
l
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14/52Department of Computer and Information Science
Polyhedral Volumes and Surface
[Shewchuk 98]
Input PLC Final Mesh
• No input angle is less than 90 degree
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15/52Department 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.
p
p
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18/52Department of Computer and Information Science
Small Angle Problem
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19/52Department of Computer and Information Science
SOS-split
[Cohen-Steiner et al. 02]
Sharp Vertex Protection
( ) / 4f u
u
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20/52Department 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.
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21/52Department 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.
f xx x
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22/52Department of Computer and Information Science
Results
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Delaunay Meshing for Smooth Surfaces
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26/52Department of Computer and Information Science
Implicit Surface
F: R3 => R, Σ = F-1(0)
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29/52Department 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.
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30/52Department of Computer and Information Science
Restricted DelaunayRestricted Delaunay
• Del Q|Σ :- Collection of Delaunay simplices whose corresponding dual Voronoi face intersects Σ.
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31/52Department of Computer and Information Science
Topological Ball PropertyTopological 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.
k
( 1)k
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32/52Department 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 = 0.06.
-- Termination is guaranteed by 2. -- Topology is guaranteed by 1 and
the termination.
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33/52Department of Computer and Information Science
Topological Disk TestTopoDiskK ( )TopoDiskK ( ) If is not a
topological disk, return furthest point in edge-surface intersections.
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35/52Department of Computer and Information Science
Topological Disk Test
TopoDiskK ( )TopoDiskK ( ) If is not a
topological disk, return furthest point in .
q
q
qG V
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36/52Department 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.
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37/52Department 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.
e
e
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38/52Department of Computer and Information Science
Results
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40/52Department of Computer and Information Science
Polyhedral Surfaces (non-Polyhedral Surfaces (non-conforming)conforming)[Dey-Li-Ray 05][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
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41/52Department of Computer and Information Science
AssumptionsAssumptions
• G approximates a smooth .
• G is -flat w.r.t .• Many designed
surfaces, reconstructed surfaces are -flat.
• Relation to Lipschitz surface (Boissonnat-Oudot 06)
p
p( ){f p
pn
pn
( , )
( , )
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43/52Department 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
k
p ( )kf p
54 10 , 0.1 0.02k
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45/52Department of Computer and Information Science
Results
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48/52Department 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 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