A New Voronoi-based Reconstruction Algorithm

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A New Voronoi-based Reconstruction Algorithm

• CS 598 MJG
• Presented by Ivan Lee

N. Amenta, M. Bern, and M. Kamvysselis. In
Proceedings of SIGGRAPH 98, pp. 415-422, July
1998.
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What is Surface Reconstruction?

• Set of points in 3-d space
• Generate a mesh from the points

http//web.mit.edu/manoli/www/crust/crust.html
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What to talk about

• Previous Work
• Definitions
• The Crust Algorithm
• Comparison to Previous Work
• Further Research

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Previous work

• Alpha shapes
• Zero-set
• Delaunay Sculpting

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Alpha Shapes
Dey et al. 5

• Given a parameter, a, connect vertices within a
  units
• Subset of Delaunay triangulation
• Generalized convex hull

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Zero sets

• Using input points, define implicit signed
  distance function
• Distance function is interpolated and polygonized
  using marching cubes
• Approximation rather than interpolation
• e.g. Curless and Levoy paper

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Delaunay Sculpting

• Remove tetrahedra from Delaunay triangulation
• Associate values to tetrahedra and eliminate
  largest valued ones

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First, some definitions

• Voronoi cell
• A cell where all points in the cell are closer to
  a given sample point than any other point
• Voronoi diagram
• A space partitioned into Voronoi cells
• Voronoi vertex
• A point equidistant to d1 sample points in Rd

Amenta et al. 1
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Some more definitions

• Delaunay triangulation
• Dual of Voronoi diagram
• Each triangles circumcircle contains no other
  vertices

Amenta et al. 1

• Medial axis
• Set of points with more than one closest point

Amenta et al. 1
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And finally

• Poles
• Farthest Voronoi vertices for a sample point that
  are on opposite sides
• Crust
• Shell created to represent the surface

Amenta et al. 1
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On to the algorithm

• Compute the Voronoi diagram of S, where S is the
  set of sample points
• For each sample point, find the poles on opposite
  sides of the sample point
• Compute Delaunay triangulation of S U P, where P
  is the set of all poles
• Keep all triangles in which all three vertices
  are sample points

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On to the algorithm

• Delete triangles whose normals differ too much
  from the direction vectors from the triangle
  vertices to their poles
• Orient triangles consistently with its neighbors
  and remove sharp dihedral edges to create a
  manifold

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Advantages

• No need for experimental parameters in basic
  algorithm
• Not sensitive to distribution of points

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Disadvantages

• Sampling of points needs to be dense
• Undersampling causes holes
• Does not handle sharp edges
• Can be fixed by picking two farthest vertices as
  poles, regardless of being on opposite sides
• Boundaries cause problems
• But not always

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Comparison to Previous Work

• Alpha Shapes
• No need for experimental values
• Zero set
• Essentially low-pass filtering, lose information
• Delaunay sculpting
• Very similar to this algorithm

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Hull

• Command line implementation of Voronoi regions in
  C
• Downloadable at
• http//cm.bell-labs.com/netlib/voronoi/hull.html

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Proposed Future Research in 1998

• Fixing problems with boundaries and sharp edges
• Using sample points with normals
• Allows for sparser samplings
• Lossless mesh compression

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Whats happened since then?

• Co-cones (Amenta et al. 2)
• Cone with apex at sample point and aligned with
  poles
• Algorithm only requires one Voronoi diagram
  computation
• Eliminates normal trimming step
• Still does not support sharp edges

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Whats happened since then?

• The power crust (Amenta et al. 3)
• Use polar balls and power diagrams to separate
  the inside and outside of the surface
• Approximates medial axis

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Whats happened since then?

• Detecting Undersampling (Dey and Giesen 4)
• Fat Voronoi cells or dissimilarly oriented
  neighboring Voronoi cells imply undersampling.
  Add sample points to accommodate
• This accounts for sharp edges and boundaries
• Tight Co-cone
• After detecting undersampling, stitch up holes

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Summary

• New Crust Algorithm
• Advantages over previous algorithms
• Advancements to fix original crust algorithms
  flaws

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Thank you
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References

• 0 N. Amenta and M. Bern. Surface
  Reconstruction by Voronoi Filtering. Annual
  Symposium on Computational Geometry, pp. 39-48,
  1998.
• 1 N. Amenta, M. Bern, and M. Kamvysselis. A
  New Voronoi-Based Surface Reconstruction
  Algorithm. In Proceedings of SIGGRAPH 98, pp.
  415-422, July 1998.
• 2 N. Amenta, S. Choi, T. Dey, and N. Leekha. A
  Simple Algorithm for Homeomorphic Surface
  Reconstruction. Internation Journal of
  Computational Geometry and its Applications, vol.
  12 (1-2), pp. 125-141, 2002.
• 3 N. Amenta, S. Choi, and R. Kolluri. The
  Power Crust. ACM Symposium on Solid Modeling and
  Applications, pp 249-266, 2001.

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References

• 4 T. Dey and J. Giesen. Detecting
  Undersampling in Surface Reconstruction. In
  proceedings for 17th ACM Annual Symposium for
  Computational Geometry, pp. 257-263, 2001.
• 5 T. Dey and S. Goswami. Tight Cocone A
  Water-Tight Surface Reconstructor. In
  Proceedings for 8th ACM Symposium for Solid
  Modeling Applications, pp. 127-134, 2003.
• 6 T. Dey, J. Giesen, and M. John. Alpha-Shapes
  and Flow Shapes are Homotopy Equivalent. STOC
  03, 2003.
• 7 H. Edelsbrunner and E. Mücke.
  Three-dimensional Alpha Shapes. ACM Transactions
  on Graphics, 13(1)43-72, 1994.

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Questions and Discussion