Post on 31-Dec-2015
description
2/52Department 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)
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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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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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Delaunay Refinement for 2D point sets
R/l > 1.0
30 degree
R
l
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Local Feature Size
• Local feature size: radius of smallest ball that intersects two disjoint input elements.
• Lipschitz property:
( ) ( )f x f y x y
( )f x
x
min min 0xf f x
x
f(x)
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Delaunay Refinement with Boundary
Conforming but still not Gabriel
>f(x)
x
Circumcenter of skinny triangle encroaching edge./ 2L R / 2R l
/ 2L R l
L R
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Polyhedral Volumes and Surface
[Shewchuk 98]
Input PLC Final Mesh
• No input angle is less than 90 degree
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Quality of Tetrahedra
Thin Flat
……
radius-edge-ratio: 0 L
R0 03,
V
L
Sliver
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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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Refinement Steps
• Compute Delaunay of vertices
Do the splits in the following order:• Split encroached subsegments • Split encroached subfacets • Let c be the circumcenter of a skinny
tetrahedron• if c encroaches a subsegment or subfacet
split it. • Else insert c.
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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.
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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]
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Main Result
• Quality Meshing for Polyhedra with Small Angles [Cheng, Dey, Ramos, Ray 04]
• 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.
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SOS-split
[Cohen-Steiner et al. 02]
Sharp vertex protection
( ) / 4f u
u
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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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QualMesh 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/tetrahedra• Keep their circumcenters outside
We do not want to compute f (center)
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Refinement Cont..
• Split encroached subsegments and non-Delaunay subfacets.
• Let c be the circumcenter of a skinny triangle/tetrahedra.
• 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.
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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.
f xx x
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R/L Distribution
Model R/L
0.6-1.4 1.4-2.2 >2.2
Anchor 1779 1009 30
Rail 471 128 0
Wiper 1851 630 4
Cutter 1340 777 0
Simple Box 2580 896 43
Ushape 764 195 2
Mesh Test 806 267 0
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Dihedral Angle DistributionModel Dihedral Angle
[0-5] (5-10] (10-15]
(15-30]
>30
Anchor 6 62 115 1006 2173
Rail 1 4 10 152 435
Wiper 7 37 50 695 1773
Cutter 11 20 82 635 1368
Simple Box
13 45 113 1124 2248
Ushape 2 15 25 267 620
Mesh Test 1 19 46 302 716
Quality Meshing with Weighted Delaunay Refinement by Cheng-Dey 02
Meshing Polyhedra with Sliver Exudations
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History
• Bern, Eppstein, Gilbert 94 - Quadtree meshing (Non-
Delaunay)
• Cheng, Dey, Edelsbrunner, Facello, Teng 2000 - Silver
exudation (no boundary)
• Li, Teng 2001 - Silver exudation with boundary (randomized
extending Chew)
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Weighted points and distances
X
ˆ ˆ( , ) 0, orthogonalx y
• Weighted point: • Weighted distance: • If
orthogonalan further th,0)ˆ,ˆ( yxorthogonaln closer tha,0)ˆ,ˆ( yx
222)ˆ,ˆ( YXyxyx
ˆ ( , )x x X orthogonal,0)ˆ,ˆ( yx
x
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Weighted Delaunay• Smallest orthospheres, orthocenters, orthoradius
• Weighted Delaunay tetrahedra
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Silver Exudation• Delaunay refinement guarantees tetrahedra with bounded
radius-edge-ratio
• Vertices are pumped with weights
Sliver Theorem [Cheng-Dey-Edelsbrunner-Facello-Teng]:
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,N(v)] for each
vertex v in V such that () 0 and ()>0 for each tetrahedron in the weighted
Delaunay triangulation of V.
[0, ( )],N v ]21,0[
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QMESH algorithm1. 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
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Guarantees
• Theorem (Termination): QMESH terminates with a graded mesh.
• Theorem (Conformity): No weighted-subsegment or weighted-subfacet is encroached upon the
completion of QMESH
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• Weight property[]: each weight u N(u)• Ratio property []: orthoradius-edge-ratio is at most .
• Lemma : 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 : 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: 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
V̂
V̂V̂
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Size Optimality
• Output vertices
• Output tetrahedra
• Any mesh of D with bounded aspect ratio must have
tetrahedra
• Theorem :
The output size of QMESH is within a constant factor of the size of any mesh
of bounded aspect ratio for the same domain.
D xf
dxn
3)(
D xf
dxkmnOm
3)(),(
3( )D
dxk
f x
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Example - Arm
Input PLCSlivers
Sliver Removal Final Mesh
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Example - Cap
Input PLC Slivers
Sliver Removal Final Mesh
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Example - Propellant
Input PLC Slivers
Sliver Removal Final Mesh
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TimeRatio=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
Anchor 0 0 2 / 4.95 0
Arm 10 / 0.75 1 / 2.9 26 / 0.75 2 / 4.66
Cap 6 / 0.00 0 9 / 0.0005 1 / 4.59
Cavity 3 / 2.31 0 6 / 2.31 2 / 4.52
Chair 1 / 1.36 0 6 / 1.36 2 / 4.18
House 4 / 2.02 1 / 2.02 5 / 2.02 1 / 2.02
L-shape 0 0 0 0
Nalcola 0 0 2 / 4.80 0
OurHouse 0 0 1 / 4.83 0
Propellant 9 / 1.10 1 / 1.97 33 / 1.10 13 / 0.87
Table 1 / 2.99 0 2 / 2.99 0
TeaTable 0 0 0 0
Tfire 1 / 2.11 0 2 / 2.11 0
Wrench 24 / 0.007 0 29 / 0.007 1 / 4.36
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Extending sliver exudations to polyhedra
with small angles Cheng-Dey-Ray 2005 (Meshing Roundtable 2005)
• 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.
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Implicit surfaces
• Surface Σ is given by an implicit equation E(x,y,z)=0
• Surface is smooth, compact, without any boundary
: ( ) ,E x n Ex 0
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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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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.
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Restricted DelaunayRestricted Delaunay
• Del Q|G :- Collection of Delaunay simplices whose corresponding dual Voronoi face intersects G.
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Topological Ball PropertyTopological 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.
k
( 1)k
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Strategy
• Topology Sampling :
Grow a sample P by insertion until the Topological Ball Property is satisfied.
• Geometry Sampling:
Quality. Smoothness.
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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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Voronoi Edge
• Edge Lemma : If intersects
Σ twice or more or tangentially, the farthest is
> k f(p) away from all points.
e V p
e
E dgeSurface e E x a x a x( , ): ( ) , . , . 0 1 11 2
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Voronoi Edge Lemma Justification
Edge not parallel to normal
Almost normal edge
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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.
F V p
F
L CC
F
e V p
e
C ritC urve F E x a x
n ax
( , ): ( ) , . ,
( ) . ( , , )
0 1
0 0 1 0
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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.
V p
e V p
e
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Voronoi Cells (0-,1-boundary)
Single boundary but not simplyconnected (Silhouette takes care)
Component inside ( taken care by critical pts.)
C ritSurf d n d E xx( , ): , ( ) 0 0
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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.
J x n dd x . 0
V p J d
d n p
q J d
V p
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Silhouette Computation
C ritS ilh d d E x
G x n d n G x dx x
( , , ): ( ) ,
( ) . , ( ( )).
0
0 0S ilhF acet F d E x
a x n dx
( , , ): ( ) ,
. , .
0
1 0
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Voronoi Edge Test
VVEDGEEDGE ( ) ( ) If intersects Σ
in two or more points, return
the point furthest from .
qe Ve
q q
[Edge Lemma]
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Topological Disk TestTopoDiskK ( )TopoDiskK ( ) If is not a
topological disk, return furthest point in edge-surface intersections.
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Voronoi Facet
• Facet with more than one topological interval.
u
v
F
[Facet Lemma II]
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Voronoi CellVoronoi CellIf is not a 2-manifold with a single boundary
then TopoDISK () will take care of it.
pG V
[Cell Lemma]
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Four Tests Contd..• FacetCycle( ): X:= CritCurve(Σ,F), then
check if L intersects twice or more, return a point.
[Facet Lemma I].• Silhouette(Vp ):
X:=CritSilh(Σ,np,d). If , return a point
from X otherwise see facet intersection.
[Silhouette Lemma]
F V p
V Xp
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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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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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Algorithm
DELMESH (Σ) SampleTopology(P) Quality(P) Smooth(P) Continue till no point is
added.
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Guarantees
• Output surface is homeomorphic to Σ.
• Each triangle has a guaranteed aspect ratio.
• Smooth triangulation.
• Size of P is asymptotically optimal.
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Polygonal surfacesPolygonal surfaces[Dey-Ray 05][Dey-Ray 05]
Input:Input: Polygonized surface G approximating .
Output:Output: A vertex set Q where each vertex lies on G and triangulation T
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Non-SmoothnessNon-Smoothness
• Input G is piecewise-linear.• Non-smoothness is a challenge.• Delaunay refinement for polyhedron is not
a viable choice.
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AssumptionsAssumptions
• G approximates a smooth .
• G is -flat w.r.t .
• Many designed surfaces, reconstructed surfaces are -flat.
p
p( ){f p
pn
pn
( , )
( , )
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SurfRemesh
1. Initialize Q.2. Compute Vor Q.3. While (! Topology
Recovered)4. VEDGE().5. DISK().6. FCYCLE().7. VCELL().8. End while9. Output Del Q|G.
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FFCYCLECYCLE( )( )
If has a cycle, return the point in furthest from .
Voronoi FacetqF V
F G F Gq
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Voronoi CellVoronoi Cell
VVCELLCELL( )( ) If Euler number return the point in furthest
from .
p
# # # 1v e g
pG V p
Single boundary but not 2-disk
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Extending results from smooth case
Big empty balls acting as medial balls
If t=pqr has O(k)f(p) circumradius,
‹nt , np›=O(k)
provided lengths > √(6δ)
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Bounding Conditions
Condition 1: and .
Condition 2: [Amenta,Choi,Dey,Leekha ‘02]
61 8
k
k
1
48k
12 6 4 3 2k k k
1 4k k k
1 1 12sin sin sin 2sin
3k k k
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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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Geometric Approximation
• : is the circumradius of the triangle t.
• : is the ratio of the circumradius to shortest edge length of t . p
p
p
( ) min ,h p p p p p
( )r t
( )t
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RefinementRefinementGGEOMEOMRRECOVECOV( )( )1. For , if with
insert the intersection point .
TTRIANGLERIANGLE_Q_QUALUAL( )( )1. For , if with
insert the intersection point
Gp Q
pt
( )c dual t G
( ) 12pr t h k
( ) (1 8 )t k p Q pt
( )c dual t G
G
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Remeshing reconstructed Remeshing reconstructed surfacessurfaces
• 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].
( , ) 2( ) ( )
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TerminationTermination
• Theorem :Theorem : If satisfies a bounding condition with respect to then it will terminate.
and
k,
54 10 , 0.1 0.02k
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Meshing a equipotential surface
data: courtesy to Alan Saalfeld
V=21014, F=42024 V=21507, F=42904V=2141, F=4278
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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 Polygonal: surfremesh.html• Meshing a nonsmooth curved surface, remeshing
polygonal surface approximating a non-smooth surface is a challenge.
• Anisotropic meshing [CDRW05]• CGAL acknowledgement: www.cgal.org