T. J. Peters2005 IBM Faculty Award

www.cse.uconn.edu/~tpeters

withE. L. F. Moore &

J. Bisceglio

Computational Topology for Scientific Visualization

and Integration with Blue Gene L

Rotate Molecule?

UMass, RasMol

Molecular Modeling?

Using Surfaces!

Joining Geometry

Dynamic Scientific Visualization

Approximately 11M translations per hour:

100 translations per frame,

at 30 frames per second

(A Conservative Lower Bound)

• 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

Documented Animation Issues

• Accumulated error versus Maya alternative.

• Used at BlueSky Studios (Ice Age II)

Practical Animation Response

• Mathematics for perturbing curves.

• Generalize to surfaces.

Pragmatic Research Response

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).

Unknot

BadApproximation!

Self-intersect?

Good Approximation!

Respects Embedding

Via

Curvature (local)

Separation (global)

(recognizing unknot in NP; Hass, L, P, 1998)

*

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

Applications !

Subdivision for graphics

• Integration with sub-systems.

• Generation of vertices.

• Performance benefits.

• Motion driven by chemistry and physics.

P8

P7

P6

P5

P4

P3

P2

P1

P10

P0P

9

➢ Planar Degree 10 Bézier Curve

➢ Note: the control polygon is self-intersecting

The Class of Unknotted Spline Curves with Knotted Control Polygons

Knot Projection Folk Lemma

If a projection of a curve is non-self-intersecting,

then

the curve is unknotted.

Spline Projection

Done by projection of control points.

➢ 3D Degree 10 Bézier Curve

➢ Note: the control polygon is knotted

The Class of Unknotted Spline Curves with Knotted Control Polygons

P0

P10

P9P

8

P7 P

6

P5

P3

P2

P1

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)

B

Pk

Pk+1

Pk+2

qk,i

lk

lk+1

lk+3

Pk+2

lk+2

qk,f

*

Cubic: no local knotting

2r

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 Cubic b-spline curve

Min distance with Newton's method

KnotPlot !

Crucial Difference

Known Dynamics

Versus

Real-time Response

(molecular simulation)

(surgery)

Additional High Performance Issues

• Over 100,000 processors, with local geometry.

• Join across all nodes (surfaces & curves).

• Output to light-weight graphics clients raises bandwidth & architectural concerns.

Example: Blue Gene L, Macro-Molecule

Andersson-Peters-Stewart, IJCGA 00 & CAGD 98

• Terabytes of point data.

• Triangulation too data intensive.

• Reduce by orders of magnitudes.

• Spline approximation, with acceptable loss.

Example:Seismic Data,

P. Bording, MUN, IBM Faculty Award

• Only synthetic data.

• Order of magnitude reduction.

• Small loss.

• Awaiting test data.

Status

• Local constraints.

• Mathematically & algorithmically possible.

• Need domain-specific information.

Options

• Integrate

Surface Approximation

Provable Topological Dynamic Constraints

• Apply to real-time, computer-assisted cardiac surgery.

Goals

Credits• ROTATING IMMORTALITY

– www.bangor.ac.uk/cpm/sculmath/movimm.htm

• KnotPlot– www.cs.ubc.ca/nest/imager/

contributions/scharein/KnotPlot.html

  

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