Delaunay Triangulation

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Delaunay Triangulation Application

• Course Presentation by
• Wei-Chao Chen
• Oct. 29, 1998

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Outline

• Applications of Delaunay Triangulation
• Data-Dependent Triangulation
• Constrained Delaunay Triangulation
• Mesh Generation using Delaunay Refinement
• 3-D Delaunay Triangulation

Rhythmes, Sonia Delaunay
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Applications

• Geoscientific Modelling
• Collect data about spatial objects and domains.
• Categorize features in soils or ocean.
• Unique triangulation, adaptive user control.

Lattuada et al. (http//www.iah.bbsrc.ac.uk/phd/gi
sruk95.html)
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Applications

• Volume visualization for 3D scalar function
• Given a set of 3D points with scalar value,
  visualize the dataset, one or more level with
  transparency.
• Requires intensive computation power.
• Delaunay Triangulation is used to interpolate the
  scalar function between given vertices (Delaunay
  Interpolation)
• Ray-Tracing image refinement
• Use Delaunay Triangulation, interpolate by using
  convolution and integral.

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Applications

• Medical Imaging, Feature Extraction
• Input A medical image such as CT or MRI.
• Process Find vertices on the edges.
• Output The medial axis and feature surface by
  combining Voronoi diagram and Delaunay
  triangulation.

Robinson. (http//noodle.med.yale.edu/robinson/)
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Data-Dependent Triangulation

• For applications like height fields and terrain,
  we want to have triangulation to approximate the
  true surface condition.
• Taking the height into account, we may get a
  better triangulation than merely doing the
  Delaunay Triangulation in 2-D.
• Dyn et al. Minimizing the size of normal
  derivative discontinuities across edges of the
  triangulation will provide dramatic improvement.

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Data-Dependent Triangulation

• Quak, Schumaker Given a set of vertices in 3D,
  find the triangulation that is closest to the
  actual surface (assuming cubic spline fitting).
• Specify an energy function and flip edge
  recursively to obtain the minimal energy.
• The actual surface is globally C1 across
  vertices.
• The vertices lies on the actual surface.
• Delaunay Triangulation is actually a
  triangulation for certain energy function.

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Data-Dependent Triangulation

• Algorithm 1
• Given any triangulation of vertices, calculate
  the Bernstein-Bézier representation, compute the
  energy of the associated surface over each
  triangles.
• Sort the energies computed on the previous step
  in decreasing order.
• From the triangle with biggest energy, swap the
  edges to reduce the energy.
• Algorithm 2
• Use conventional Delaunay Triangulation and
  replace angle-optimal swap test into
  energy-optimal swap test.

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Data-Dependent Triangulation

• Results
• Both algorithms yield similar results.
• Smooth surfaces Only limited reduction in energy
  function is observed.
• Non-smooth surfaces Substantial decrease in
  energy is found. However, this does not imply
  the resulting surface is visually more pleasing.

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Constrained Delaunay Triangulation

• Given a set of constraining edges, find an
  optimal triangulation including the constraining
  edges.
• User can specify the edges that must appear in
  the triangulation.
• Standard Delaunay Triangulation always
  triangulate the convex hull.
• Constraining edges are given in the form of
  planar straight line graph (PSLG).
• There are no intersection among these edges. We
  can avoid this limitation by converting
  intersections into vertices.

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Constrained Delaunay Triangulation

• Conforming Delaunay Triangulation
• Given a set of vertices vi, find intersections
  vj between PSLG and Delaunay Triangulation of
  vi.
• Do the Delaunay Triangulation on vertices
  (vi?vj)
• Drawback Too many vertices! O(m2n)

Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)
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Constrained Delaunay Triangulation

• Constrained Delaunay Triangulation
• Given a set of vertices vi, triangulate using
  Delaunay Triangulation.
• Delete all the triangles that overlap
  constraining edges. Retriangulate both sides of
  these edges.

Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)
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Constrained Delaunay Triangulation

• Triangulation by splitting constraining edges
• Do the Delaunay Triangulation for vi.
• Find constraining edges not in the triangulation.
  If the diametral circle of the edge contains
  some points (encroached), split this edge by
  adding a vertice in the center of the segment.

Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)
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Delaunay Refinement

• Delaunay Triangulation does not guarantee the
  actual minimum angle of triangles.
• We can refine the triangulation by adding more
  vertices.
• Specify the maximal acceptable aspect ratio for
  triangles. Add vertex for skinny triangles.
• Split edge when the triangle is too skinny.
• Guarantee the quality of generated triangle mesh.
  (Chew, Ruppert, Shewchuk)

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Delaunay Refinement

• Input PSLG ej and vertices vi

Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)

• Steps
• 1. Obtain Delaunay Triangulation of vertices
  vi.
• Divide conquer algorithm (Lee and Schachter) or
• Plane-sweep algorithm (Fortune) or
• Incremental insertion algorithm (Lawson).

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Delaunay Refinement
Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)

• 2. Do the constrained Delaunay Triangulation on
  triangles generated at previous step.

Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)
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Delaunay Refinement

• 3. Delete triangles outside the concavity of
  PSLG.
• User define the triangle to be deleted by
  assigning a point inside the Constrained Delaunay
  Triangulation, but outside the concavity.
• Run the point location algorithm, find the
  triangles, recursively delete until hits the
  boundary.

Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)
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Delaunay Refinement

• 4. Correct Bad Triangles in the triangulation
  generated from previous step.
• Find triangles with large aspect ratio (skinny
  triangles).
• For each triangle, find the circle it inscribes,
  and add the center of the circle as a vertex of
  the mesh.
• Recursively flip edge for inserted vertex.
• If the circle encroaches upon any segment
• Delete the vertex. Split the segments this circle
  encroaches upon.

Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)
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Delaunay Refinement
Running of Delaunay Refinement
algorithm. Highlighted segments are segments
before split, or triangles before vertex
insertion.
Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)
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3-D Delaunay Triangulation

• We can do triangulation for higher dimensions by
  using similar technique of 2-D Delaunay
  Triangulation.
• 3-D Replace circles by spheres, triangles by
  tetrahedrons.

Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)
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3-D Delaunay Triangulation

• Input Set of vertices, constraining edges
  facets.
• Steps
• For all constraining edges, make a diametral
  sphere.
• If some vertices encroach (inside) the sphere,
  split this edge by adding a vertex in the middle.

Step 1.
Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)
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3-D Delaunay Triangulation

• For all constraining facets, find its equatorial
  sphere.
• If the equatoral sphere are encroached by some
  vertices, add the center of the sphere as a new
  vertex.

Step 2.
Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)
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3-D Delaunay Triangulation

• Find all skinny tetrahedrons, find the spheres
  they inscribe.
• Add the center of the spheres as a vertex.
• Skinny criteria The ratio of the sphere radius
  to shortest tetrahedron edge.

Step 3.
Schewchuk. (http//www.cs.cmu.edu/jrs/jrspapers.h
tml)
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References

• D Dobkin et al., Special Issue on Computational
  Geometry. Proceedings of the IEEE, Vol. 80, No.9,
  Sep. 1992.
• Applications for Delaunay Triangulation can be
  found on http//www.ics.uci.edu/eppstein/gina/del
  aunay.html
• N. Max, P. Hanrahan et al., Area and Volumn
  Coherence for Efficient Visualization of 3D
  Scalar Functions, http//www.llnl.gov/graphics/doc
  s/VolViz90.pdf
• E. Quak, L. Schumaker, Cublic Spline Fitting
  using Data Dependent Triangulations. Computer
  Aided Geometric Design, Vol.7, 1990, pp293-301
• J. Ruppert. A Delaunay Refinement Algorithm for
  Quality 2-Dimensional Mesh Generation. J.
  Algorithms, pp.548-585, May 1995
  http//science.nas.nasa.gov/Pubs/TechReports/RNRre
  ports/jruppert/RNR-94-002/RNR-94-002.html
• J. Shewchuk, Selected Papers on Triangulation,
  http//www.cs.cmu.edu/jrs/jrspapers.html