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Computational Topology for Scientific Visualization

Computational Topology for Scientific Visualization. and Integration with Blue Gene L. T. J. Peters 2005 IBM Faculty Award www.cse.uconn.edu/~tpeters with E. L. F. Moore & J. Bisceglio. Rotate Molecule?. UMass, RasMol. Molecular Modeling?. Using Surfaces!. Joining Geometry.

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Computational Topology for Scientific Visualization

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  1. Computational Topology for Scientific Visualization and Integration with Blue Gene L T. J. Peters 2005 IBM Faculty Award www.cse.uconn.edu/~tpeters with E. L. F. Moore & J. Bisceglio

  2. Rotate Molecule?

  3. UMass, RasMol

  4. Molecular Modeling? Using Surfaces!

  5. Joining Geometry

  6. Dynamic Scientific Visualization (A Conservative Lower Bound) Approximately 11M translations per hour: 100 translations per frame, at 30 frames per second

  7. Documented Animation Issues • Geri’s game: along boundary joins. • Resolution was data-specific. • Short time span was favorable DeRose, Kass and Truong, Subdivision surfaces in character animation, SIGGRAPH '98

  8. Practical Animation Response • Accumulated error versus Maya alternative. • Used at BlueSky Studios (Ice Age II)

  9. Pragmatic Research Response • Mathematics for perturbing curves. • Generalize to surfaces.

  10. Approximation & Knots • Approximate & compare knot types: But recognizing unknot in NP (Hass, L, P, 1998)!! • Approximation as operation in geometric design • Preserve original knot type (even if unknown).

  11. Unknot

  12. Bad Approximation! Self-intersect?

  13. Good Approximation! Respects Embedding Via Curvature (local) Separation (global) (recognizing unknot in NP; Hass, L, P, 1998)

  14. Interpolation points* Nr(B) B • Construct the boundary of an open neighborhood Nr(B)of curve B • The boundary (a pipe surface) will have a radius r, with the following conditions* • no local self-intersections • no global self-intersections *

  15. Applications !

  16. Subdivision for graphics • Integration with sub-systems. • Generation of vertices. • Performance benefits. • Motion driven by chemistry and physics.

  17. The Class of Unknotted Spline Curves with Knotted Control Polygons • Planar Degree 10 Bézier Curve P0 P9 P8 P1 P10 P5 P3 P2 P7 P6 P4 • Note: the control polygon is self-intersecting

  18. Knot Projection Folk Lemma If a projection of a curve is non-self-intersecting, then the curve is unknotted.

  19. Spline Projection Done by projection of control points.

  20. The Class of Unknotted Spline Curves with Knotted Control Polygons • 3D Degree 10 Bézier Curve P9 P0 P8 P1 P10 P5 P3 P2 P7 P6 • Note: the control polygon is knotted

  21. Algorithm for Isotopic Subdivision (cubic) Subdividing B until its control polygon is contained in Nr(B). a. Compute number of subdivisions required* b. Test to ensure there are no self-intersections Nr(B) Pk+2 B Pk+1 Pk+2 lk+2 lk+1 Pk qk,f lk+3 lk qk,i Cubic: no local knotting *

  22. Algorithm for Isotopic Subdivision 1. Computing r for B Find minimum of a. separation distance [c(s) – c(t)] • c'(s) = 0 [c(s) – c(t)] • c'(t) = 0 b. radius of curvature 2r Cubic b-spline curve

  23. Min distance with Newton's method

  24. KnotPlot !

  25. Crucial Difference Known Dynamics Versus Real-time Response (molecular simulation) (surgery)

  26. Additional High Performance Issues Example: Blue Gene L, Macro-Molecule • Over 100,000 processors, with local geometry. • Join across all nodes (surfaces & curves). • Output to light-weight graphics clients raises bandwidth & architectural concerns. Andersson-Peters-Stewart, IJCGA 00 & CAGD 98

  27. Example:Seismic Data, P. Bording, MUN, IBM Faculty Award • Terabytes of point data. • Triangulation too data intensive. • Reduce by orders of magnitudes. • Spline approximation, with acceptable loss.

  28. Status • Only synthetic data. • Order of magnitude reduction. • Small loss. • Awaiting test data.

  29. Options • Local constraints. • Mathematically & algorithmically possible. • Need domain-specific information.

  30. Goals • Integrate Surface Approximation Provable Topological Dynamic Constraints • Apply to real-time, computer-assisted cardiac surgery.

  31. Credits • ROTATING IMMORTALITY • www.bangor.ac.uk/cpm/sculmath/movimm.htm • KnotPlot • www.cs.ubc.ca/nest/imager/ contributions/scharein/KnotPlot.html

  32. Acknowledgements, NSF • I-TANGO,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. • IBM Faculty Award, 2005

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