T. J. Peters, UConn Computer Science & Engineering Research & Education Topology & Animation :...

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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 selfintersections, 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 Selfintersection
Contemporary Computational Influences
• Edelsbrunner: geometry & topology
• Sethian: Marching methods, topology changes
• Blackmore: differential sweeps
• Carlsson, Zomordian : Algebraic
Mappings and Equivalences
Knots and selfintersections
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 & reuse 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!
Selfintersect?
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!!)
• Preprint 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?
• Selfintersection
2
TEA Authoring Tools for DVFX
• Timechecker like spellchecker – runs in background; not intrusive!
– very expensive if missed.
• Parametric redesign; similar to CAGD PTC
• Integrate with VFX.
Visualization for Simulations
• Animation `onthefly’.
• 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 computerbased 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/frenchtoast.jpg
• www.knotplot.com
• http://domino.research.ibm.com/comm/pr.nsf/pages/rscd.bluegenepicaa.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: dimensionindependent technology
• Provably correct temporal antialiasing
• Portability of animation to differing displays
• Efficient compression and decompression
Nbhd_1 about curve.