Download Curve Preservation

Delaunay Meshing for Piecewise
Smooth Complexes
Tamal K. Dey
The Ohio State U.
Joint work: Siu-Wing Cheng, Joshua Levine, Edgar A. Ramos
Piecewise Smooth Complexes
Sharp Edges
Non-manifold
Department of Computer Science and Engineering
2/22
Piecewise Smooth Complexes
• D is a piecewise smooth complex (PSC) if
• Each k-dimensional element is a manifold and
compact subset of a smooth (C2) k-manifold, 0≤k≤2.
• The k-th stratum, Dk : set of k-dim elements of D.
• D0 – vertices, D1 – 1-faces, D2 – 2-faces.
• D≤k = D0  …  Dk.
• D satisfies usual reqs for being a complex.
• Interiors of elements are disjoint and for σ  D, bd σ  D.
• For any σ,  D, either σ   =  or σ    D .
Department of Computer Science and Engineering
3/22
Delaunay refinement : History
• Chew89, Ruppert92, Shewchuk98 (Linear domains with
no small angle)
• Cohen-Steiner-Verdiere-Yvinec02, Cheng-Dey-RamosRay04 (polyhedral domains with small angle)
• Chew93 (surface without guarantees)
• Cheng-Dey-Edelsbrunner-Sullivan01 (skin surfaces)
• Boissonnat-Oudot03 and Cheng-Dey-Ramos-Ray04
(smooth surface)
• Boissonnat-Oudot06 (Lipschitz surfaces)
• Oudot-Rineau-Yvinec06 (Volumes)
Department of Computer Science and Engineering
4/22
Basics of Delaunay Refinement
• Chew 89, Ruppert 92, Shewchuk 98
• 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).
Department of Computer Science and Engineering
5/22
Challenges for PSC
• Topology
• Polyhedral case (input conformity,topology trivial).
• Curved elements (topology is an issue).
• Topological Ball Property (TBP) was used for smooth
manifolds [BO03,CDRR04].
• We need extended TBP for nonmanifolds.
• Nonsmoothness
• Lipschitz surfaces [BO06], Remeshing [DLR05].
• Small angles
• Delaunay refinement is hard [CP03, CDRR05, PW04].
Department of Computer Science and Engineering
6/22
Topological Ball Property
• For a weighted point set
S, let Vor S and Del S
denote the weighted
Voronoi and Delaunay
diagrams.
• S has the TBP for σDi if
σ intersects any k-face in
Vor S either in emptyset
or in a closed topological
(i+k-3)-ball.
Department of Computer Science and Engineering
7/22
CW-Complexes
• A CW-complex R is a collection of closed
(topological) balls whose interiors are
pairwise disjoint and whose boundaries
are the union of other closed balls in R.
• Our algorithm builds a CW-complex,
Vor S||D|, to satisfy an extended
TBP[ES97].
Department of Computer Science and Engineering
8/22
Extended TBP
• S  |D| has the extended TBP
(eTBP) for D if there is a CWcomplex R with |R| = |D| s.t.
• (C1) The restricted Voronoi face
F  |D| is the underlying space
of a CW-complex R’  R.
• (C2) The closed balls in R’ are
incident to a unique closed ball
bF  R.
• (C3) If bF is a j-ball then bF  bd
F is a (j-1)-sphere.
• (C4) Each k-ball in R’, except bF,
intersects bd F in a (k-1)-ball.
Department of Computer Science and Engineering
9/22
Extended TBP
• For a 1- or 2-face σ, let Del S|σ denote
the Delaunay subcomplex restricted to σ.
• Del S||Di| = σDi Del S|σ.
• Del S||D| = σD Del S|σ.
• Theorem. If S has the eTBP for D then
the underlying space of Del S||D| is
homeomorphic to |D| [ES97].
Department of Computer Science and Engineering
10/22
Feature Size
• For analysis, we require a
feature size which is 1Lipschitz and non-zero.
• For any x  |D|, let f(x) =
min{m(x), g(x)}.
• For any σ  D, f() is 1-Lipschitz
over int σ.
• For δ  (0,1] and x  |D|,
• if x  D0, lfsδ(x) = δf(x).
• if x  int |Di|, for i ≥ 1,
lfsδ(x) = max{δf(x), maxybd|Di|
{lfsδ(y)-||x-y||}}.
Department of Computer Science and Engineering
11/22
Protecting D1
1. Any 2 adjacent balls on a 1-face must
overlap significantly without
containing each others centers
2. No 3 balls have a common
intersection
3. For a point p  σ  D1, if we enlarge
any protecting ball Bp by a factor c ≤
8, forming B’:
1. B’ intersects σ in a single curve, and
intersects all   D2 adjacent to σ in
a topological disk.
2. For any q in B’  σ, the tangent
variation between p and q is
bounded.
3. For any q in B’   (  D2 adjacent
to σ), the normal variation between
p and q is bounded.
Department of Computer Science and Engineering
12/22
Admissible Point Sets
• Protecting balls are turned into weighted points
• We call a point set S admissible if
• S contains all weighted points placed on D1.
• Other points in S are unweighted and they lie outside
of the protecting balls (the weighted points).
• We maintain an admissible point set at each
step of the algorithm.
Department of Computer Science and Engineering
13/22
D1 conformation
• Lemma. Let S is an admissible point set. For a
1-face σ, if p and q are adjacent weighted
vertices spanning segment σpq on σ then Vpq is
the only Voronoi facet which intersects σpq and
it does so exactly once.
Department of Computer Science and Engineering
14/22
Meshing PSCs
• Meshing algorithm uses four tests to
detect eTBP violations.
• Upon violation, we insert points outside of
protected balls of weighted vertices.
Department of Computer Science and Engineering
15/22
Test 1: Multi-Intersection(q,σ)
• For a point q S on a 2face σ, find a triangle t
 Del S|σ incident to q
s.t. Vt intersects σ
multiple times.
• If no t exists, return
null, otherwise return
the furthest (weighted)
intersection point from
q.
q
Department of Computer Science and Engineering
16/22
Test 2: Normal-Deviation(q,σ,Θ)
• For a point q  S on a 2-face σ, check
nσ(p), nσ(q) < Θ for all points p  Vq|σ.
• 2ω ≤ Θ ≤ /6.
• If so return null.
• Otherwise return a point p where
nσ(p), nσ(q) = Θ .
Department of Computer Science and Engineering
17/22
Test 3: Infringement(q,σ)
• We say q is infringed w.r.t. σ if
• σ is a 2-face containing q s.t. pq 
Del S|σ for some p  σ.
• σ is a 2-face and there is a 1-face
in bd σ containing q and a nonadjacent vertex p s.t. pq  Del S|σ.
• For q  S  σ, return null if q is not infringed,
otherwise let pq be the infringing edge.
• If the boundary edges of Vpq intersect int σ, return any
intersection point.
• Else, Vpq  σ is a collection of closed curves, return a
critical point of Vpq  σ in a direction parallel to Vpq.
Department of Computer Science and Engineering
18/22
Test 4: No-Disk(q,σ)
• If the star of q in Del S|σ is
a topological disk, return
null.
• Otherwise, find the triangle
t  Del S|σ incident to q
which has the furthest
(weighted) intersection
point in Vt|σ from q and
return the intersection
point.
Department of Computer Science and Engineering
19/22
Meshing Algorithm
1. Protect elements in D≤1 with weighted points.
Insert a point in each element of D2 outside of
protected regions. Let S be this point set.
2. For any σ  D2 and point q  S  σ:
• If Infringed(q,σ), Multi-Intersection(q,σ),
Normal-Deviation(q,σ,Θ), or No-Disk(q,σ) (checked
in that order) return a point x, insert x into S.
3. Repeat 2. until no points are inserted.
4. Return Del S|D.
Department of Computer Science and Engineering
20/22
Admissibility is Invariant
Lemma. The algorithm never attempts to insert a
point in any protecting ball
• Since no 3 weighted
points intersect,
• all surface points
(intersections of dual
Voronoi edges and D) lie
outside of every protecting
ball
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21/22
Initialization
• The algorithm
must initialize
with a few points
from each patch
in D2
• Otherwise,
components can
be missed.
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22/22
Termination
• Each point x inserted is Ω(lfsδ(x)) away
from all other points.
• Standard packing argument follows.
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23/22
Topology Preservation
• To satisfy C1-C4 of eTBP, we
show each Voronoi k-face F =
Vp1  …  Vp(4-k) has:
• (P1) If F  σ ≠ , for σ  Dj, the
intersection is a (k+j-3)-ball
• (P2) There is a unique lowest
dimensional σF s.t. p1, …, p(4-k)σF.
• (P3) F intersects σF and only
incident elements of σF.
• Theorem. If S satisfies P1-P3
then S satisfies C1-C4 of eTBP.
Department of Computer Science and Engineering
24/22
Feature Preservation
• h:|D|  |Del S|D| can be constructed which
respects each Di [ES97].
• Thus hi:|Di|  |Del S|Di| also a homeomorphism
with vertex restrictions, ensuring that the
nonsmooth features are preserved.
Department of Computer Science and Engineering
25/22
Delaunay Refinement made
practical for PSCs
S.-W. Cheng, Tamal K. Dey, Joshua Levine
Definitions
• For a patch σ  Di,
• When sampled with S
• Del S|σ is the Delaunay
subcomplex restricted to σ
• Skli S|σ is the i-dimensional
subcomplex of Del S|σ,
• Skli S|σ = closure { t |
t  Del S|σ is an i-simplex}
• Skli S|Di = σ  Di Skli S|σ
Department of Computer Science and Engineering
27/22
Disk Condition
•
For a point p on a 2-face σ,
•
•
•
UmbD(p) is the set of triangles
in Skl2 S|D2 incident to p.
Umbσ(p) is the set of triangles
in Skl2 S|σ incident to p.
Disk_Condition(p) requires:
i. UmbD(p) = σ, p  σ Umbσ(p)
ii. For each σ containing p,
Umbσ(p) is a 2-disk where p is
in the interior iff p  int σ
Department of Computer Science and Engineering
28/22
Meshing Algorithm
DelPSC(D, r)
1. Protect elements of D≤1.
2. Mesh2Complex – Repeatedly insert surface points for
triangles in Skl2 S|σ for some σ if either
1. Disk_Condition(p) violated for p  σ, or
2. A triangle has orthoradius > r.
3. Mesh3Complex – Repeatedly insert orthocenters of
tetrahedra in Skl3 S|σ for some σ if
1. A tetrahedra has orthoradius > r and its orthocenter does not
encroach any surface triangle in Skl2 S|D2.
4. Return i Skli S|Di.
Department of Computer Science and Engineering
29/22
Termination Properties
1. Curve Preservation
•
For each σ  D1, Skl1 S|σ  σ. Two vertices are joined by an
edge in Skl1 S|σ iff they were adjacent in σ.
2. Manifold
•
•
For 0 ≤ i ≤ 2, and σ  Di, Skli S|σ is a manifold with vertices
only in σ. Further, bd Skli S|σ = Skli-1 S|bd σ.
For i=3, the above holds when Skli S|σ is nonempty after
Mesh2Complex.
3. Strata Preservation
•
•
There exists some r > 0 so that the output of DelPSC(D, r)
is homeomorphic to D.
This homeomorphism respects stratification.
Department of Computer Science and Engineering
30/22
Voronoi Cells Intersect “Discly”
• Given a vertex p on a 2-face σ, if
• Triangles incident to p in Skl2 S|σ are small
enough.
• Then,
• Vp|σ is a topological disk,
• Any edge of Vp|σ intersects σ at most once,
and
• Any facet of Vp|σ which intersects σ does so
in an open curve.
Department of Computer Science and Engineering
31/22
TBP holds globally
• if
• All triangles incident in Skl2 S|σ are smaller
than a bound for all 2-faces,
• Then
• TBP holds globally
• This leads to the proof of ETBP and
more…topic of a new unpublished paper.
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32/22
Adjusting MaxRad Example
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33/22
Adjusting MaxRad Example
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34/22
Examples
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Examples
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Examples
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37/22
Examples
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38/22
Examples
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39/22
Examples
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Examples
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Examples
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Examples
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43/22
Examples
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44/22
Sharp Example
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45/22
Conclusions
• Delaunay meshing for PSC with
guarantees.
• Feature preservation is an extra `feature’.
• Making computations easier, faster?
• Analyzing size complexity?
Department of Computer Science and Engineering
46/22