Delaunay Based Shape Reconstruction from Large Data

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Delaunay Based Shape Reconstruction from Large
Data

• Tamal K. Dey, Joachim Giesen and James Hudson
• Ohio State University

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Surface Reconstruction
Reconstruction
A sample
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Local feature size and samplingAmenta-Bern-Eppste
in

• Medial axis
• Local feature size f(p)

• ?-sampling
• ? ? d(p)/f(p)

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Reconstruction

• Functional approach
• Tangent plane HDeDDMS92
• Natural Neighbors BC00

• Voronoi/Delaunay filtering
• Alpha shapes EM94
• Crust AB98
• Cocone ACDL00

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Surface and Voronoi Diagram

• Restricted Voronoi
• Restricted Delaunay
• Poles
• Cocone

Space spanned by vectors making angle q? ?/8 with
horizontal
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Cocone Algorithm

• Compute VP
• Compute Boundary samples (Dey-Giesen 2001)
• Filter triangles whose duals intersect cocones
• Extract a manifold using prune and walk

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Why manifold extraction works?

• Candidate triangles are dual to Voronoi edges
  intersected by cocones.
• Each candidate triangle is small w.r.t. feature
  size.
• All restricted Delaunay triangles is in the set
  of candidate triangles.

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Large Data

• Octree subdivision

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Cracks

• Cracks appear in surface computed from octree
  boxes

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Padding

• Include a fraction from the neighbors to form
  the extended box

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Boundary sample detection
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Supercocone(P,D,l)

• Compute octree for P with D and extended box
  with l th subdivision
• For each box perform all steps of cocone on the
  extended set
• Extract a manifold

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Theoretical Justification

• Because of padding restricted Voronoi neighbors
  are included
• Normals are approximated
• Cocone computes the candidate triangles with two
  necessary properties
• Manifold extraction takes care of matching
• Even a local manifold extraction works in
  practice due to padding

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Surface matching
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Experiments

• 733 MHz Pentium III, 512 MB RAM, 10GB disk
• C, CGAL code for Voronoi/Delaunay

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Dragon data (time)
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Dragon data (memory)
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Dragon
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Blade data (time)
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Blade data(memory)
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Blade
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Lucy25 data(time)
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Lucy Data (memory)
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Lucy25
3.5 million points, 198 mints
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Experimental data
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Davids Head

2 mil points, 93 minutes
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St. Mathews Head

3.4 mil points, 150 minutes
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Parallel

• 10x2 450MHz Pentium II Xeon
• 512MB, 1GB swap
• one 733 MHz Pentium III, 512MB, 1.5 GB
• MPI, 10Mbit ethernet

14 mil points, 67 minutes
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Conclusions

• Introduced Reconstruction by local Voronoi
  computations.
• Large sample in the range of million points is
  doable.
• Parallel implementation.
• Softwares
• www.cis.ohio-state.edu/tamaldey/cocone.html