Surface and Volume Meshing with Delaunay Refinement

Tamal K. Dey The Ohio State University

Surface and Volume Meshing with Delaunay Refinement

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)


Implicit surface

F: R3 => R, Σ = F-1(0)


Polygonal surfacePolygonal surface


Voronoi/Delaunay


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).


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


Delaunay Refinement for 2D point sets

R/l > 1.0

30 degree

R

l


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)


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


Polyhedral Volumes and Surface

[Shewchuk 98]

Input PLC Final Mesh

• No input angle is less than 90 degree


Quality of Tetrahedra

Thin Flat

……

radius-edge-ratio: 0 L

R0 03,

V

L

Sliver


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


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.


Child-Parent and insertion radii

> 2.0


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.


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]


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.


Small angle problem


SOS-split

[Cohen-Steiner et al. 02]

Sharp vertex protection

( ) / 4f u

u


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.


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)


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.


Positions of skinny triangle/tet


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


Results


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


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


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)


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


Weighted Delaunay• Smallest orthospheres, orthocenters, orthoradius

• Weighted Delaunay tetrahedra


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[


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


Guarantees

• Theorem (Termination): QMESH terminates with a graded mesh.

• Theorem (Conformity): No weighted-subsegment or weighted-subfacet is encroached upon the

completion of QMESH


• 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


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


Example - Arm

Input PLCSlivers

Sliver Removal Final Mesh


Example - Cap

Input PLC Slivers



Example - Propellant

Input PLC Slivers



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


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.

Cheng-Dey-Ramos-Ray 04

Delaunay Meshing for Implicit Surfaces


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


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]


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.


Restricted DelaunayRestricted Delaunay

• Del Q|G :- Collection of Delaunay simplices whose corresponding dual Voronoi face intersects G.


Delaunay Refinement (Chew)


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


Strategy

• Topology Sampling :

Grow a sample P by insertion until the Topological Ball Property is satisfied.

• Geometry Sampling:

Quality. Smoothness.


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.


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


Voronoi Edge Lemma Justification

Edge not parallel to normal

Almost normal edge


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


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


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


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


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


Voronoi Edge Test

VVEDGEEDGE ( ) ( ) If intersects Σ

in two or more points, return

the point furthest from .

qe Ve

q q

[Edge Lemma]


Topological Disk TestTopoDiskK ( )TopoDiskK ( ) If is not a

topological disk, return furthest point in edge-surface intersections.

qq


Voronoi Facet

• Facet with more than one topological interval.

u

v

F

[Facet Lemma II]


Voronoi CellVoronoi CellIf is not a 2-manifold with a single boundary

then TopoDISK () will take care of it.

pG V

[Cell Lemma]


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


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.


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


Algorithm

DELMESH (Σ) SampleTopology(P) Quality(P) Smooth(P) Continue till no point is

added.


Guarantees

• Output surface is homeomorphic to Σ.

• Each triangle has a guaranteed aspect ratio.

• Smooth triangulation.

• Size of P is asymptotically optimal.


Results


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


Non-SmoothnessNon-Smoothness

• Input G is piecewise-linear.• Non-smoothness is a challenge.• Delaunay refinement for polyhedron is not

a viable choice.


Delaunay refinementDelaunay refinement


AssumptionsAssumptions

• G approximates a smooth .

• G is -flat w.r.t .

• Many designed surfaces, reconstructed surfaces are -flat.

p

p( ){f p

pn

pn

( , )

( , )


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.


FFCYCLECYCLE( )( )

If has a cycle, return the point in furthest from .

Voronoi FacetqF V

F G F Gq


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


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)


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


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


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


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


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( ) ( )


TerminationTermination

• Theorem :Theorem : If satisfies a bounding condition with respect to then it will terminate.

and

k,

54 10 , 0.1 0.02k


Results


ResultsResults


Meshing a equipotential surface

data: courtesy to Alan Saalfeld

V=21014, F=42024 V=21507, F=42904V=2141, F=4278


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

Surface and Volume Meshing with Delaunay Refinement

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