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
(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.
Not to be confused with topography.
Topology --- Mobius Strip
(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).
Topography – Contour lines hiking maps versus Google maps
Different static viewing options
Topology --- Change
Topography – Contour lines Static
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.
T. J. Peters, Kerner Graphics & UConn
Knots & Molecules in Animation, Simulation & Visualization
TEA & ToAST
T. J. PetersKerner Graphics
Topologically Encoded Animation (TEA)
Encode: Rot_0, Rot_1, …, Rot_n
More Aggressive Moves
• Not just rigid body motion
• Deform shape
• Preserve crucial characteristics
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
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.
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)
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)
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.)
• 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
• 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
• XC, RFR, EC, JD 07
• Solver [GE+97]
• Multiple objects
• KG folk 09
• Critical points (C )
• Newton, PGPU?
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 !
Slow with errors
Fast & correct – but scale?
K. E. Jordan (IBM), L. E. Miller (UConn), E.L.F. Moore (UConn), T. J. Peters (UConn), A. C. Russell (UConn)
• 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.”
• 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
• 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’.
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.