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Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
DOE/NV/25946–1463
Computational Methods for Analyzing Fluid FlowDynamics from Digital Imagery
Aaron LuttmanNational Security Technologies, LLC
Joint work with Erik Bollt, Ranil Basnayake,Sean Kramer, and Nick Tufillaro.
This work was done by National Security Technologies, LLC, under Contract No. DE-AC52-06NA25946 with the U.S.
Department of Energy.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
You are watching dynamics when you watch a movie ofspatiotemporal dynamical systems.
Main Goal (long term): Performing computational dynamics analysis andquantifying uncertainty from vector fields computed directly from measured data.
Three Stories Here:
Optical flow based scientific priors – Variational approach to computing fluid flowfrom image data
Uncertainty Quantification – Estimating output errors both from estimates on theinput error but also from convergence estimates on regularization parameter.
Dynamics Analysis – Transfer operator methods and differentiable dynamicalsystems techniques for analyzing invariance and transport from optical flow fields.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
You are watching dynamics when you watch a movie ofspatiotemporal dynamical systems.
Main Goal (long term): Performing computational dynamics analysis andquantifying uncertainty from vector fields computed directly from measured data.
Three Stories Here:
Optical flow based scientific priors – Variational approach to computing fluid flowfrom image data
Uncertainty Quantification – Estimating output errors both from estimates on theinput error but also from convergence estimates on regularization parameter.
Dynamics Analysis – Transfer operator methods and differentiable dynamicalsystems techniques for analyzing invariance and transport from optical flow fields.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Gulf of Mexico, May 24, 2010, from HYCOM model.
Many kinds of dynamicsanalysis are performed fromtime-varying vector fields.t t
The kind of questions we discuss within dynamical systems language:– Where is transport easy or hard?– What are the coherent, invariant, and almost-invariant sets?
Answers to these questions can be addressed with computational tools:– Transfer Operator Methods– Finite Time Lyapunov Exponents (FTLEs) and Lagrangian Coherent Structures (LCS)
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Gulf of Mexico, May 24, 2010, from HYCOM model.
Can we get this “model-free”?
t t
These vector fields are generally computed from:– large-scale ocean models,– sometimes informed by floating sensors, LIDAR, etc.
We compute these vector fields from:– image observations via variational reconstruction,– Physics governing the system are estimated and incorporated into the cost functional
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
What kind of Global Analysis from Dynamical Systems Theory?FTLE Gulf of Mexico
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Goal is to perform similar global analysis based on observedspatiotemporal evolution – Dynamics from Movies
How it works:
I Objective function based on expected physics and informed scientific priorsI Variational optimization to compute vector fields from measured dataI Transport analysis proceeds with observations and priors
Assumptions and Limitations:
I Physics governing system processes are assumed and built into optimization, modelingincorporated into assumed form of the cost function.
I Best possible dynamical explanation of evolution rules results when assumed physicsaccurately describe actual physical processes.
I Remotely sensed data requires preprocessing for use in vision-based analysis.I In order to estimate uncertainty, it is necessary to have estimates of error in captured
data, which is very difficult for remotely sensed data.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Goal is to perform similar global analysis based on observedspatiotemporal evolution – Dynamics from Movies
How it works:
I Objective function based on expected physics and informed scientific priorsI Variational optimization to compute vector fields from measured dataI Transport analysis proceeds with observations and priors
Assumptions and Limitations:
I Physics governing system processes are assumed and built into optimization, modelingincorporated into assumed form of the cost function.
I Best possible dynamical explanation of evolution rules results when assumed physicsaccurately describe actual physical processes.
I Remotely sensed data requires preprocessing for use in vision-based analysis.I In order to estimate uncertainty, it is necessary to have estimates of error in captured
data, which is very difficult for remotely sensed data.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Mathematical Formulation for Computing Flow Fields
Compute Minimizer of
Eα(ψ) =
∫Ω
(It + div (I∇ψ))2 dΩ + αR(ψ),
whereI I(x , y , t) is the time-varying image data,I Ω ⊂ R2 is the image domain,I ψ is the potential of the flow field, i.e. ∇ψ = 〈u, v〉,I R(ψ) is an “appropriate” regularization, andI α is the regularization parameter.
This is a least-squares optimization, where the data fidelity is derived from thecontinuity equation. Assumption is that flow is the gradient of a potential.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Mathematical Formulation for Computing Flow Fields
Compute Minimizer of
Eα(ψ) =
∫Ω
(It + div (I∇ψ))2 dΩ + αR(ψ),
whereI I(x , y , t) is the time-varying image data,I Ω ⊂ R2 is the image domain,I ψ is the potential of the flow field, i.e. ∇ψ = 〈u, v〉,I R(ψ) is an “appropriate” regularization, andI α is the regularization parameter.
This is a least-squares optimization, where the data fidelity is derived from thecontinuity equation. Assumption is that flow is the gradient of a potential.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Regularization Principles
The goal is to compute Minimizer of
Eα(ψ) =
∫Ω
(It + div (I∇ψ))2 dΩ + αR(ψ),
where R(ψ) is an “appropriate” minimizer. What does this mean?
With R(ψ) as a component of the energy functional,
I for each fixed α > 0 there should exist a unique minimizer ψ∗α,
I the computation of ψ∗α should be stable with respect to errors in the
measurements,I given some sequence αn with αn → 0, the sequence ψ∗
αn shouldapproximate a minimizer of the unregularized functional E0(ψ), and
I the chosen regularizer should impose some kind of “regularity,” which shouldbe determined by prior scientific principles.
Problem: Regularizers that impose appropriate scientific priors rarely ensureuniqueness of minimizers.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Regularization Principles
The goal is to compute Minimizer of
Eα(ψ) =
∫Ω
(It + div (I∇ψ))2 dΩ + αR(ψ),
where R(ψ) is an “appropriate” minimizer. What does this mean?
With R(ψ) as a component of the energy functional,
I for each fixed α > 0 there should exist a unique minimizer ψ∗α,
I the computation of ψ∗α should be stable with respect to errors in the
measurements,I given some sequence αn with αn → 0, the sequence ψ∗
αn shouldapproximate a minimizer of the unregularized functional E0(ψ), and
I the chosen regularizer should impose some kind of “regularity,” which shouldbe determined by prior scientific principles.
Problem: Regularizers that impose appropriate scientific priors rarely ensureuniqueness of minimizers.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Regularization Principles
The goal is to compute Minimizer of
Eα(ψ) =
∫Ω
(It + div (I∇ψ))2 dΩ + αR(ψ),
where R(ψ) is an “appropriate” minimizer. What does this mean?
With R(ψ) as a component of the energy functional,
I for each fixed α > 0 there should exist a unique minimizer ψ∗α,
I the computation of ψ∗α should be stable with respect to errors in the
measurements,
I given some sequence αn with αn → 0, the sequence ψ∗αn should
approximate a minimizer of the unregularized functional E0(ψ), andI the chosen regularizer should impose some kind of “regularity,” which should
be determined by prior scientific principles.
Problem: Regularizers that impose appropriate scientific priors rarely ensureuniqueness of minimizers.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Regularization Principles
The goal is to compute Minimizer of
Eα(ψ) =
∫Ω
(It + div (I∇ψ))2 dΩ + αR(ψ),
where R(ψ) is an “appropriate” minimizer. What does this mean?
With R(ψ) as a component of the energy functional,
I for each fixed α > 0 there should exist a unique minimizer ψ∗α,
I the computation of ψ∗α should be stable with respect to errors in the
measurements,I given some sequence αn with αn → 0, the sequence ψ∗
αn shouldapproximate a minimizer of the unregularized functional E0(ψ), and
I the chosen regularizer should impose some kind of “regularity,” which shouldbe determined by prior scientific principles.
Problem: Regularizers that impose appropriate scientific priors rarely ensureuniqueness of minimizers.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Regularization Principles
The goal is to compute Minimizer of
Eα(ψ) =
∫Ω
(It + div (I∇ψ))2 dΩ + αR(ψ),
where R(ψ) is an “appropriate” minimizer. What does this mean?
With R(ψ) as a component of the energy functional,
I for each fixed α > 0 there should exist a unique minimizer ψ∗α,
I the computation of ψ∗α should be stable with respect to errors in the
measurements,I given some sequence αn with αn → 0, the sequence ψ∗
αn shouldapproximate a minimizer of the unregularized functional E0(ψ), and
I the chosen regularizer should impose some kind of “regularity,” which shouldbe determined by prior scientific principles.
Problem: Regularizers that impose appropriate scientific priors rarely ensureuniqueness of minimizers.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Regularization Principles
The goal is to compute Minimizer of
Eα(ψ) =
∫Ω
(It + div (I∇ψ))2 dΩ + αR(ψ),
where R(ψ) is an “appropriate” minimizer. What does this mean?
With R(ψ) as a component of the energy functional,
I for each fixed α > 0 there should exist a unique minimizer ψ∗α,
I the computation of ψ∗α should be stable with respect to errors in the
measurements,I given some sequence αn with αn → 0, the sequence ψ∗
αn shouldapproximate a minimizer of the unregularized functional E0(ψ), and
I the chosen regularizer should impose some kind of “regularity,” which shouldbe determined by prior scientific principles.
Problem: Regularizers that impose appropriate scientific priors rarely ensureuniqueness of minimizers.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Example Regularization Schemes
Scheme Description Strengths and Priors
R1(ψ) =∫
Ω |ψ|2 dΩ L2 norm of potential Coercivity, Smoothness
R2(ψ) =∫
Ω |∇ψ| dΩ Total Variation of potential Sparse flow
R3(ψ) =∫
Ω(ψxx − ψyy )2 Engineering strain tensor Does not penalize rigid+(ψxy + ψyx )2 motion+ψ2
yxx + ψ2xyy dΩ
R4(ψ) =∫
Ω(ψxx + ψyy )2 Rigid motion flow Smoothness, does not(ψxy − ψyx )2 dΩ penalize hyperbolic flow
R5(ψ) =∫
Ω(ψxx − ψyy )2 Rigid motion flow Smoothness, does not(ψyx − ψxy )2 dΩ penalize rotational flow
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Notes on Quantifying Uncertainty
There are several different components to uncertainty quantificationwithin this framework:
I estimating errors in image data (hard),
I quantifying how the dynamics in the images – which are 2Dprojections of 3D phenomena – relate to the actual dynamics ofthe system (very hard), and
I computing the propagation of errors through the reconstructionalgorithm (not as hard, but must be done for each regularizationscheme independently).
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Notes on Quantifying Uncertainty
There are several different components to uncertainty quantificationwithin this framework:
I estimating errors in image data (hard),
I quantifying how the dynamics in the images – which are 2Dprojections of 3D phenomena – relate to the actual dynamics ofthe system (very hard), and
I computing the propagation of errors through the reconstructionalgorithm (not as hard, but must be done for each regularizationscheme independently).
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Notes on Quantifying Uncertainty
There are several different components to uncertainty quantificationwithin this framework:
I estimating errors in image data (hard),
I quantifying how the dynamics in the images – which are 2Dprojections of 3D phenomena – relate to the actual dynamics ofthe system (very hard), and
I computing the propagation of errors through the reconstructionalgorithm (not as hard, but must be done for each regularizationscheme independently).
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Applications to Oceanic Flow:Sea Surface Temperature
A PDE ocean model is seeded withinfrared satellite measurements andevolved forward in time, periodicallyassimilating new measurements. SSTand surface salinity are two of the mostimportant quantities for ocean flow pre-diction and the associated atmosphericeffects.
The data is given at 1-hour intervals for
seven days in August 2010.
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
SST Off Oregon Coast – Global Analysis from Observed Movies
Vision – Service – Partnership Managed and Operated by National Security Technologies, LLC Nevada National Security Site
Computational Methods for Analyzing Fluid Flow Dynamics from Digital Imagery
Thank you!
Feel free to contact me if you have any questions or comments, or forreferences, offprints, and preprints.
