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Computational Topology on Approximated Manifolds

Explore theory, algorithms, and applications in Computational Topology on Approximated Manifolds for surface reconstruction and approximation. Learn about sampling density, topological fidelity, and the role of medial axis in achieving accurate approximations. Discover the importance of curvature, separation, and knot equivalence in creating good approximations. Gain insights into ambient isotopy and tubular neighborhoods for stable approximations on large data sets.

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Computational Topology on Approximated Manifolds

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  1. Computational Topology on Approximated Manifolds (with Applications) T. J. Peters, University of Connecticut www.cse.uconn.edu/~tpeters K. Abe, J. Bisceglio, A. C. Russell

  2. Outline: Topology & Approximation • Theory • Algorithms • Applications

  3. Role for Animation Towards Mathematical Discovery • ROTATING IMMORTALITY • www.bangor.ac.uk/cpm/sculmath/movimm.htm • Möbius Band in the form of a Trefoil Knot • Animation makes 3D more obvious • Simple surface here • Spline surfaces joined along boundaries

  4. Problem in Approximation • Input: Set of unorganized sample points • Approximation of underlying manifold • Want • Error bounds • Topological fidelity

  5. Typical Point Cloud Data

  6. Subproblem in Sampling • Sampling density is important • For error bounds and topology

  7. 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 ….

  8. 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.

  9. 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

  10. 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) • Approximation is homeomorphic to original. (Amenta & Bern)

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

  12. KnotPlot!!

  13. Unknot

  14. Bad Approximation Why? Separation? Curvature?

  15. Why Bad? No Intersections! Changes Knot Type Now has 4 Crossings

  16. Good Approximation All Vertices on Curve Respects Embedding Via Curvature (local) Separation (global)

  17. Summary – Key Ideas • Curves • Don’t be deceived by images (3D !) • Crossings versus self-intersections • Local and global arguments • Knot equivalence via isotopy

  18. Initial Assumptionson a 2-manifold, M • Without boundary • 2nd derivatives are continuous (curvature) • Improved to ambient isotopy (Amenta, Peters, Russell)

  19. T

  20. Theorem: Any approximation of F in T such that each normal hits one point of W is ambient isotopic to F. Proof: Similar to flow on normal field. Comment: Points need not be on surface. (noise!)

  21. Tubular Neighborhoods and Ambient Isotopy • Its radius defined by ½ minimum • all radii of curvature on 2-manifold • global separation distance. • Estimates, but more stable than medial axis.

  22. Medial Axis • H. Blum, biology, classification by skeleton • Closure of the set of points that have at least 2 nearest neighbors on M

  23. X

  24. Large Data Set ! Partitioned Stanford Bunny

  25. 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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