Digital Particle Image Velocimetry using Splines in Tension
U. von Toussaint and S. Gori
Udo v. Toussaint, MaxEnt 2010,
[email protected]
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Overview
The problem
The approach I Results
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The approach II Results Conclusions Udo v. Toussaint, MaxEnt 2010,
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The problem Fusion: one of the possible solutions of the energy problem Several experiments are being built: ITER (France, B$), Japan, China, India South Korea Crucial: Plasma-Wall-Interaction: Protect plasma(!) & protect first wall Image of AUG (by V. Rohde)
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•Prevention or mitigation of unwanted events necessary Udo v. Toussaint, MaxEnt 2010,
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The problem Otherwise:
•Plasma disruptions
AUG tile (N. Endstrasser)
•First wall may deteriorate by arcs, droplets, self-sputtering
Event detection and event analysis required Challenges: short time scales, trajectories (3-D) required, multiple events Answer: fast imaging techniques – high speed video cameras 3-D reconstruction from multiple images required Udo v. Toussaint, MaxEnt 2010,
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The Approach •3-D Reconstruction from multiple views: - standard problem of computer vision - still challenging - fast algorithms (OpenCV) available - often easy identification of relevant pixels - e.g pinhole model for projection 3-D->2-D: Udo v. Toussaint, MaxEnt 2010,
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The Approach •3-D Reconstruction from multiple views: - every (u(ti),v(ti))-set provides information about positions r(ti) - Approximation: gaussian distribution: r(ti) with covariance C(ti) - Interpolate using e.g. cubic spline to obtain r(t)
Udo v. Toussaint, MaxEnt 2010,
[email protected]
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The Approach •3-D Reconstruction from multiple views: - every (u(ti),v(ti))-set provides information about positions r(ti) - Approximation: gaussian distribution: r(ti) with covariance C(ti) - Interpolate using e.g. cubic spline to obtain r(t):
Interest in v(t) and a(t)!
Udo v. Toussaint, MaxEnt 2010,
[email protected]
7
The Approach Check with literature: - Particle Image Velocimetry: ...differential quantities are difficult...noise amplification... ...the following finite differencing schemes.... (non-smoothing but unstable) - Other areas: ...differencing is more stable after smoothing regression using splines or low-order polynomials... [NR] (stable but systematic bias)
Needed: A model capable of both: Spline in tension (exponential spline)
Udo v. Toussaint, MaxEnt 2010,
[email protected]
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The Approach Exponential Splines: - Minimize bending energy of rod under tension (Rentrop 1980)
R. Fischer 2006
- Cubic spline and linear interpolation as limiting cases - Basis functions: {1,x,exp(λx),exp(-λx)} - Parameters: {x,f(x),λ,N} Do some Bayes (priors, marginals)... Udo v. Toussaint, MaxEnt 2010,
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Results I Exponential Splines: Test example using charged particle in complex E-B-field geometry and low noise:
Udo v. Toussaint, MaxEnt 2010,
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Results I Exponential Splines: Test example using charged particle in complex E-B-field geometry and low noise: acceleration too spiky! What is wrong?
Udo v. Toussaint, MaxEnt 2010,
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The Approach II Back to physics: • Spline (trajectory) is continous...but not necessarily differentiable velocity discontinous
• Finite force
infinite force
continous velocity
differentiable trajectory
Exponential splines are fine in velocity space
Udo v. Toussaint, MaxEnt 2010,
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12
The Approach II Exponential splines reloaded : t1
r x t 1= x 0∫t dt S vx t vx , t , x 0
•analytic integration is feasible
d a x t 1 = S vx t vx , t , x dt For all required times τ a numerically efficient formulation is possible:
T(τ)=W(ξ,τ,λ) v
Udo v. Toussaint, MaxEnt 2010,
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The Approach II Parameter: λx,λy,λz, tx,ty,tz, vx,vy,vz, Nx,Ny,Nz, r0 Prior-distributions: p(λi|I) = c/λi (within bounds) p(N|I) = uniform (< frames) p(r0|I) = uniform (inside observed volume) p(t|I) = for Δt>tMin uniform p(v|I) = uniform (within bounds) Likelihood:
Udo v. Toussaint, MaxEnt 2010,
[email protected]
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The Approach II •Numerics: a) MAP estimate of full parameter vector h using GC b) Marginalize v, followed by GC c) MCMC Bayes' theorem: p(h|d,Σ,I)=p(d|h,Σ,I) p(h|I)/p(d|I) Evidence p(d|I) computed from hessian at h* (without multiplicity) Subsequent model averaging Use of Hessian to sample multivariate hk (-> diagnostic purposes) Udo v. Toussaint, MaxEnt 2010,
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Results II Exponential Splines in velocity space: Test example using charged particle in complex E-B-field geometry and low noise:
Udo v. Toussaint, MaxEnt 2010,
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Results II Exponential Splines in velocity space: Test example using charged particle in complex E-B-field geometry and low noise: 3-D velocity 3-D acceleration
Udo v. Toussaint, MaxEnt 2010,
[email protected]
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Results II Exponential Splines in velocity space: Test example using charged particle in complex E-B-field geometry and low noise: 3-D velocity 3-D acceleration
Udo v. Toussaint, MaxEnt 2010,
[email protected]
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Conclusion and Outlook Project in progress... To-Do: - Prediction : Likely trajectories -> damage prevention/assessment - particle mass determination: defined external forces physics model of particle evaporation+plasma interaction
- particle splitting: detection and handling - real-time capability
Udo v. Toussaint, MaxEnt 2010,
[email protected]
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Conclusion and Outlook Summary: - Exponential splines: interesting generalisation of cubic splines - Proposed modelling in velocity space promising for problem of velocity and acceleration estimation - Possible application: Ubiquitous problems require estimation of derived quantities (Langmuir probes, TDS,...)
Think twice about your model: it matters
Udo v. Toussaint, MaxEnt 2010,
[email protected]
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Udo v. Toussaint, MaxEnt 2010,
[email protected]
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Experimental Design IBA: Experimental Design can address questions like:
- Accelerator time for 4 measurements: Which energies/set-up to choose?
- Expected information gain of additional measurements
- Benefit of additional sensors?
- Where to measure next (microbeam,x-y grid)?
Udo v. Toussaint, MaxEnt 2010,
[email protected]
©CEA, France
