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Delaunay Based Shape Reconstruction from Large Data. Tamal K. Dey, Joachim Giesen and James Hudson Ohio State University. Surface Reconstruction. Reconstruction. A sample. Local feature size and sampling Amenta-Bern-Eppstein. Medial axis Local feature size f(p). -sampling
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Delaunay Based Shape Reconstruction from Large Data Tamal K. Dey, Joachim Giesen and James Hudson Ohio State University
Surface Reconstruction Reconstruction A sample
Local feature size and samplingAmenta-Bern-Eppstein • Medial axis • Local feature size f(p) • -sampling • d(p)/f(p)
Reconstruction • Functional approach • Tangent plane [HDeDDMS92] • Natural Neighbors [BC00] • Voronoi/Delaunay filtering • Alpha shapes [EM94] • Crust [AB98] • Cocone [ACDL00]
Surface and Voronoi Diagram • Restricted Voronoi • Restricted Delaunay • Poles • Cocone Space spanned by vectors making angle q /8 with horizontal
Cocone Algorithm • Compute VP • Compute Boundary samples (Dey-Giesen 2001) • Filter triangles whose duals intersect cocones • Extract a manifold using prune and walk
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.
Large Data • Octree subdivision
Cracks • Cracks appear in surface computed from octree boxes
Padding • Include a fraction from the neighbors to form the extended box
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
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
Experiments • 733 MHz Pentium III, 512 MB RAM, 10GB disk • C++, CGAL code for Voronoi/Delaunay
Lucy25 3.5 million points, 198 mints
David’s Head 2 mil points, 93 minutes
St. Mathew’s Head 3.4 mil points, 150 minutes
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
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