Delaunay Meshing for Piecewise Smooth Complexes

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Delaunay Meshing for Piecewise Smooth Complexes

• Tamal K. Dey
• The Ohio State U.
• Joint work Siu-Wing Cheng, Joshua Levine, Edgar
  A. Ramos

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Piecewise Smooth Complexes

• Sharp Edges

• Non-manifold

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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,
  0k2.
• The k-th stratum, Dk set of k-dim elements of
  D.
• D0 vertices, D1 1-faces, D2 2-faces.
• Dk D0 ? ? Dk.
• D satisfies usual reqs for being a complex.
• Interiors of elements are disjoint and for s ? D,
  bd s ? D.
• For any s,? ? D, either s ? ? ? or s ? ? ? D .

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Delaunay refinement History

• Chew89, Ruppert92, Shewchuk98 (Linear domains
  with no small angle)
• Cohen-Steiner-Verdiere-Yvinec02,
  Cheng-Dey-Ramos-Ray04 (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)

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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).

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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.

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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 s?Di if s intersects any k-face
  in Vor S either in emptyset or in a closed
  topological (ik-3)-ball.

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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 SD,
  to satisfy an extended TBPES97.

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Extended TBP

• S ? D has the extended TBP (eTBP) for D if
  there is a CW-complex 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.

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Extended TBP

• For a 1- or 2-face s, let Del Ss denote the
  Delaunay subcomplex restricted to s.
• Del SDi ?s?Di Del Ss.
• Del SD ?s?D Del Ss.
• Theorem. If S has the eTBP for D then the
  underlying space of Del SD is homeomorphic to
  D ES97.

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Feature Size

• For analysis, we require a feature size which is
  1-Lipschitz and non-zero.
• For any x ? D, let f(x) minm(x), g(x).
• For any s ? D, f() is 1-Lipschitz over int s.
• For d ? (0,1 and x ? D,
• if x ? D0, lfsd(x) df(x).
• if x ? int Di, for i 1,
• lfsd(x) maxdf(x), maxy?bdDi
  lfsd(y)-x-y.

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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 ? s ? D1, if we enlarge any
   protecting ball Bp by a factor c 8, forming B
4. B intersects s in a single curve, and intersects
   all ? ? D2 adjacent to s in a topological disk.
5. For any q in B ? s, the tangent variation
   between p and q is bounded.
6. For any q in B ? ? (? ? D2 adjacent to s), the
   normal variation between p and q is bounded.

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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.

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D1 conformation

• Lemma. Let S is an admissible point set. For a
  1-face s, if p and q are adjacent weighted
  vertices spanning segment spq on s then Vpq is
  the only Voronoi facet which intersects spq and
  it does so exactly once.

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Meshing PSCs

• Meshing algorithm uses four tests to detect eTBP
  violations.
• Upon violation, we insert points outside of
  protected balls of weighted vertices.

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Test 1 Multi-Intersection(q,s)

• For a point q? S on a 2-face s, find a triangle t
  ? Del Ss incident to q s.t. Vt intersects s
  multiple times.
• If no t exists, return null, otherwise return the
  furthest (weighted) intersection point from q.

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Test 2 Normal-Deviation(q,s,T)

• For a point q ? S on a 2-face s, check ?ns(p),
  ns(q) lt T for all points p ? Vqs.
• 2? T ?/6.
• If so return null.
• Otherwise return a point p where
• ?ns(p), ns(q) T .

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Test 3 Infringement(q,s)

• We say q is infringed w.r.t. s if
• s is a 2-face containing q s.t. pq ? Del Ss for
  some p ? s.
• s is a 2-face and there is a 1-face in bd s
  containing q and a non-adjacent vertex p s.t. pq
  ? Del Ss.

• For q ? S ? s, return null if q is not infringed,
  otherwise let pq be the infringing edge.
• If the boundary edges of Vpq intersect int s,
  return any intersection point.
• Else, Vpq ? s is a collection of closed curves,
  return a critical point of Vpq ? s in a direction
  parallel to Vpq.

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Test 4 No-Disk(q,s)

• If the star of q in Del Ss is a topological
  disk, return null.
• Otherwise, find the triangle t ? Del Ss incident
  to q which has the furthest (weighted)
  intersection point in Vts from q and return the
  intersection point.

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Meshing Algorithm

• Protect elements in D1 with weighted points.
  Insert a point in each element of D2 outside of
  protected regions. Let S be this point set.
• For any s ? D2 and point q ? S ? s
• If Infringed(q,s), Multi-Intersection(q,s),
  Normal-Deviation(q,s,T), or No-Disk(q,s)
  (checked in that order) return a point x, insert
  x into S.
• Repeat 2. until no points are inserted.
• Return Del SD.

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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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Initialization

• The algorithm must initialize with a few points
  from each patch in D2
• Otherwise, components can be missed.

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Termination

• Each point x inserted is O(lfsd(x)) away from all
  other points.
• Standard packing argument follows.

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Topology Preservation

• To satisfy C1-C4 of eTBP, we show each Voronoi
  k-face F Vp1 ? ? Vp(4-k) has
• (P1) If F ? s ? ?, for s ? Dj, the intersection
  is a (kj-3)-ball
• (P2) There is a unique lowest dimensional sF s.t.
  p1, , p(4-k)?sF.
• (P3) F intersects sF and only incident elements
  of sF.
• Theorem. If S satisfies P1-P3 then S satisfies
  C1-C4 of eTBP.

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Feature Preservation

• hD ? Del SD can be constructed which
  respects each Di ES97.
• Thus hiDi ? Del SDi also a homeomorphism
  with vertex restrictions, ensuring that the
  nonsmooth features are preserved.

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Delaunay Refinement made practical for PSCs
S.-W. Cheng, Tamal K. Dey, Joshua Levine
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Definitions

• For a patch s ? Di,
• When sampled with S
• Del Ss is the Delaunay subcomplex restricted to
  s
• Skli Ss is the i-dimensional subcomplex of Del
  Ss,
• Skli Ss closure t
• t ? Del Ss is an i-simplex
• Skli SDi ?s ? Di Skli Ss

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Disk Condition

• For a point p on a 2-face s,
• UmbD(p) is the set of triangles in Skl2 SD2
  incident to p.
• Umbs(p) is the set of triangles in Skl2 Ss
  incident to p.
• Disk_Condition(p) requires
• UmbD(p) ?s, p ? s Umbs(p)
• For each s containing p, Umbs(p) is a 2-disk
  where p is in the interior iff p ? int s

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Meshing Algorithm

• DelPSC(D, r)
• Protect elements of D1.
• Mesh2Complex Repeatedly insert surface points
  for triangles in Skl2 Ss for some s if either
• Disk_Condition(p) violated for p ? s, or
• A triangle has orthoradius gt r.
• Mesh3Complex Repeatedly insert orthocenters of
  tetrahedra in Skl3 Ss for some s if
• A tetrahedra has orthoradius gt r and its
  orthocenter does not encroach any surface
  triangle in Skl2 SD2.
• Return ?i Skli SDi.

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Termination Properties

• Curve Preservation
• For each s ? D1, Skl1 Ss ? s. Two vertices are
  joined by an edge in Skl1 Ss iff they were
  adjacent in s.
• Manifold
• For 0 i 2, and s ? Di, Skli Ss is a manifold
  with vertices only in s. Further, bd Skli Ss
  Skli-1 Sbd s.
• For i3, the above holds when Skli Ss is
  nonempty after Mesh2Complex.
• Strata Preservation
• There exists some r gt 0 so that the output of
  DelPSC(D, r) is homeomorphic to D.
• This homeomorphism respects stratification.

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Voronoi Cells Intersect Discly

• Given a vertex p on a 2-face s, if
• Triangles incident to p in Skl2 Ss are small
  enough.
• Then,
• Vps is a topological disk,
• Any edge of Vps intersects s at most once, and
• Any facet of Vps which intersects s does so in
  an open curve.

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TBP holds globally

• if
• All triangles incident in Skl2 Ss are smaller
  than a bound for all 2-faces,
• Then
• TBP holds globally
• This leads to the proof of ETBP and moretopic of
  a new unpublished paper.

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Adjusting MaxRad Example
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Adjusting MaxRad Example
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Examples
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Examples
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Examples
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Examples
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Examples
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Examples
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Examples
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Examples
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Examples
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Examples
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Sharp Example
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Conclusions

• Delaunay meshing for PSC with guarantees.
• Feature preservation is an extra feature.
• Making computations easier, faster?
• Analyzing size complexity?