T. J. Peters, UConnComputer Science & Engineering

Research & Education

Topology & Animation : Science & Technology

Topology

(from the Greek τόπος, “place”, and λόγος, “study”)

is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation.

http://en.wikipedia.org/wiki/Topology

Not to be confused with topography.

Topology --- Mobius Strip

Topography

(from Greek τόπος topo-, "place", and γράφω graphia, "writing")

is the study of Earth's surface shape and features or those of planets, moons, and asteroids. It is also the description of such surface shapes and features (especially their depiction in maps).

http://en.wikipedia.org/wiki/Topography

Topography – Contour lines hiking maps versus Google maps

Stowe

Google Maps

Different static viewing options  

Topology --- Change

Topography – Contour lines Static

KnotPlot: www.knotplot.com

Unknot or Trefoil?

Demo A: Unknown1 & Unknown2  

Projection of Knot  

For a closed curve, c, if there exists some projection such that there are no self-intersections, then c is the unknot.

Proof: ?

T. J. Peters, Kerner Graphics & UConn

Knots & Molecules in Animation, Simulation & Visualization

TEA & ToAST

T. J. PetersKerner Graphics

Topologically Encoded Animation (TEA)

Trefoil Knot

3D Rotation

Encode: Rot_0, Rot_1, …, Rot_n  

More Aggressive Moves

• Not just rigid body motion

• Deform shape

• Preserve crucial characteristics

1.682 Megs

1.682 Megs

Homeomorphism is not enough

F : X Y,

such that F is

1. continuous,2. 1 – 13. onto4. and has a continuous inverse.

Two Frames with Different Topology

Instantaneous Self-intersection

Contemporary Computational Influences

• Edelsbrunner: geometry & topology

• Sethian: Marching methods, topology changes

• Blackmore: differential sweeps

• Carlsson, Zomordian : Algebraic

Mappings and Equivalences

Knots and self-intersections

Piecewise Linear (PL) Approximation

My Scientific Emphasis

Isotopy & Animation

F : X x [0,1] Y,

such that for each

t in [0,1]

F : X x t is a homeomorphism.

We take Y to be 3D space.

Little reuse or modification

“Plus, we love to blow things up.”

Kerner Graphics: Digital Visual Effects (DVFX)

KERNER OPTICALKERNER OPTICAL

DVFX vs `Blowing things up’

• Modify & re-use vs destroy.

• But explosions are hard, for now.

• Provide path for integration.

EagleEye

Moore Dissertation 2006

Efficient algorithm for ambient isotopic PL approximation for

Bezier curves of degree 3.

Now scale & accelerate.

PL Approximation for Graphics –

Animation & Visualization(also for Engineeing Design)

Unknot

BadApproximation!

Self-intersect?

Good Approximation!

Respects Embedding:

Curvature (local) &Separation (global)

Error bounds!! =>Nbhd_2 about curve.

But recognizing unknot in NP (Hass, L, P, 1998)!!

Compression: TEA File (<1KB vs 1.7 Megs)

Bezier degree = 3, with Control points 0.0 0.0 0.0 4.293 4.441 0.0 8.777 5.123 1.234 12.5 0.0 0.0

Perturbation vectors; constraint on each vector 1 24.1 0.0 0.0 ; 26.4 1 -12.5 0.0 5.0 ; 18.1 2 -2.1 -2.4 -3.1 ; 9.0 1 -11.6 0.0 -1.9 ; 14.0

Compression: TEA File (<1KB vs 1.7 Megs)

HighResolution

LowResolution

Compression vs Decompression

• Compression, Phase I.

• Decompression, Phase II.

• Phase IB Project with Kerner Technologies??

Portability for Display

• Ipod to Big Screen by parameters.

• 3D TV. (Prototype in San Rafael.)

Dimension Independence

• Compute – Minimum separation distance.

– Minimum radius of curvature.

– Take minimum.

• Tubular neighborhood:– Constant radius = limit.

– Adaptive options?

Stadium CurveCurvature & MSD

Tubular Neighborhood for Stadium Curve

Computing

• Curvature – calculus problem

• Minimum Separation Distance:– Candidate line segments.

– Nearly normal at both ends.

– Newton’s Method to converge.

Infinitely many good seeds

Symmetry & Performance

• Important for animation.

• Not used in initial test cases.

• Role for PGPU’s (updates!!)

• Pre-print 09– www.cse.uconn.edu/~tpeters

Comparison

• XC, RFR, EC, JD 07

• Singularity

• Solver [GE+97]

• Multiple objects

• KG folk 09

• Critical points (C )

• Newton, PGPU?

• Self-intersection

2

TEA Authoring Tools for DVFX

• Time-checker like spell-checker – runs in background; not intrusive!

– very expensive if missed.

• Parametric re-design; similar to CAGD PTC

• Integrate with VFX.

Visualization for Simulations

• Animation `on-the-fly’.

• No human in the loop.

• Recall update issue (fast!!).

Time and Topology

Protein folding Data VolumeVisualize in real time !

Geometry

Slow with errors

Topology

Fast & correct – but scale?

Versus-------- ---------

K. E. Jordan (IBM), L. E. Miller (UConn), E.L.F. Moore (UConn), T. J. Peters (UConn), A. C. Russell (UConn)

Artifacts?

• The Need for Verifiable Visualization– Kirby and Silva, IEEE CG&A, 08– What confidence (or error measures) can be

assigned to a computer-based prediction of a complex event?

– CFD: colorful faulty dynamics

• “First, do no harm”

• “Primarily, don’t introduce artifacts.”

Graphics Supercomputers?

• PGPU:– Programmable Graphics Processing Unit– GPU

• System level

• Put pixels on screen

– Generic programming interface

• Nvidia– Quadro FX 5800 , 4 GB on board memory

– Use of CUDA and possible interns

Conclusions

• Time can be modeled continuously while frames remain discrete.

• Difference between

– Perturb then approximate versus

– Approximate then perturb.

Quotes & Interpretation

• “You can’t rush art.”, Woody, Toy Story 2

• “Time is money”.

• Correct math to make the most money.

Overview References• Modeling Time and Topology for Animation

and Visualization …., [JMMPR], TCS08

• Computation Topology Workshop, Summer Topology Conference, July 14, ‘05, Special Issue of Applied General Topology, 2007

• Open Problems in Topology II, 2007 [BP]

• NSF, Emerging Trends in Computational Topology, 1999, xxx.lanl.gov/abs/cs/9909001

Acknowledgements:$$$• SBIR: TEA, IIP -0810023 .

• SGER: Computational Topology for Surface Reconstruction, CCR - 0226504.

• Computational Topology for Surface Approximation, FMM - 0429477.

• IBM Faculty & Doctoral Awards

• Nvidia: boards, more pending

• UCRF: machines

• Investigator’s responsibility, not sponsors’.

Acknowledgements: Images

• http://se.inf.ethz.ch/people/leitner/erl_g

• http://www.koshermealstogo.com/images/french-toast.jpg

• www.knotplot.com

• http://domino.research.ibm.com/comm/pr.nsf/pages/rscd.bluegene-picaa.html

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

• blog.liverpoolmuseums.org.uk/graphics/lottie_sleigh.jpg

Challenges --- (Audacious?)

Another: Inner Life of a Cell – XVIVO for Harvard

TEA: dimension-independent technology

• Provably correct temporal antialiasing

• Portability of animation to differing displays

• Efficient compression and decompression

Nbhd_1 about curve.