Computational Topology for Reconstruction of

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Computational Topology for Reconstruction of
Manifolds With Boundary
(Potential Applications to Prosthetic Design)
T. J. Peters, University of Connecticut Computer
Science Mathematics www.cse.uconn.edu/tpeters wit
h K. Abe, J. Bisceglio, A. C. Russell, T.
Sakkalis, D. R. Ferguson
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Problem in Approximation

• Input Set of unorganized sample points
• Approximation of underlying manifold
• Want
• Error bounds
• Topological fidelity

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Typical Point Cloud Data
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Subproblem in Sampling

• Sampling density is important
• For error bounds and topology

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Recent Overviews on Point Clouds

• Notices AMS,11/04, Discretizing Manifolds via
  Minimum Energy Points, bagels with red seeds
• Energy as a global criterion for shape (minimum
  separation of points, see examples later)
• Leading to efficient numerical algorithms
• SIAM News Point Clouds in Imaging, 9/04, report
  of symposium at Salt Lake City summarizing recent
  work of 4 primary speakers of .

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Recent Overviews on Point Clouds

• F. Menoti (UMn), compare with Gromov-Hausdorff
  metric, probabalistic
• D. Ringach (UCLA), neuroscience applications
• G. Carlsson (Stanford), algebraic topology for
  analysis in high dimensions for tractable
  algorithms
• D. Niyogi (UChi), pattern recognition

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Seminal Paper

• Surface reconstruction from unorganized points,
• H. Hoppe, T. DeRose, et al., 26 (2), Siggraph,
  92
• Modified least squares method.
• Initial claim of topological correctness.

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Modified Claim
The output of our reconstruction method produced
the correct topology in all the examples. We
are trying to develop formal guarantees on the
correctness of the reconstruction, given
constraints on the sample and the original surface
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Sampling Via Medial Axis

• Delauney Triangulation
• Use of Medial Axis to control sampling
• for every point x on F the distance from x to the
  nearest sampling point is at most 0.08 times the
  distance from x to MA(F)

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Medial Axis

• Defined by H. Blum
• Biological Classification, skeleton of object
• Grassfire method

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X
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Formal Definition Medial Axis
The medial axis of F, MA(F), is the closure of
the set of all points that have at least two
distinct nearest points on S.
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Sampling Via Medial Axis

• Nice Adaptive
• for every point x on F the distance from x to the
  nearest sampling point is at most 0.08 times the
  distance from x to MA(F)
• Bad
• Small change to surface can give large change to
  MA
• Distance from surface to MA can be zero

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Need for Positive Separation

• Differentiable surfaces,continuous 2nd
  derivatives
• Shift from MA to
• Curvature (local)
• Separation (global)

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Topological Equivalence Criterion?

• Alternative from knot theory
• KnotPlot
• Homeomorphism not strong enough

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Unknot
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Bad Approximation Why?
Separation?
Curvature?
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Good Approximation All Vertices on
Curve Respects Embedding
Via Curvature (local) Separation (global)
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Boundary or Not

• Surface theory no boundary
• Curve theory OK for both boundary no
  boundary

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Related Work

• D. Manocha (UNC), MA algorithms, exact arithmetic
• T. Dey, (OhSU), reconstruction with MA
• J. Damon (UNC, Math), skeletal alternatives
• K. Abe, J. Bisceglio, D. R. Ferguson, T. J.
  Peters, A. C. Russell, T. Sakkalis, for no
  boundary .

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Computational Topology Generalization

• D. Blackmore, sweeps, next week
• Different from H. Edelsbrunner emphasis on
  PL-approximations, some Morse theory.
• A. Zamorodian, Topology for Computing
• Computation Topology Workshop, Summer Topology
  Conference, July 14, 05, Denison.
• Digital topology, domain theory
• Generalizations, unifications?

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Acknowledgements, NSF

• I-TANGO Intersections --- Topology, Accuracy and
  Numerics for Geometric Objects (in Computer Aided
  Design), May 1, 2002, DMS-0138098.
• SGER Computational Topology for Surface
  Reconstruction, NSF, October 1, 2002, CCR -
  0226504.
• Computational Topology for Surface Approximation,
  September 15, 2004,
• FMM -0429477.

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