school of something faculty of other “complementary parameterization and forward solution...
TRANSCRIPT
School of somethingFACULTY OF OTHER
“Complementary parameterization and forward solution method”
Robert G AykroydUniversity of Leeds, [email protected]
Introduction
Image reconstruction and analysis
Image problems are everywhere, for example:
• Geophysics
• Industrial process monitoring
• Medicine
with an enormous range of modalities, for example:
• Electrical
• Magnetic
• Seismic
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Introduction
Ingredients of imaging problems
Data collection style:
• Direct or
projection
• Focused or
blurred
• Low noise or
high noise
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Introduction
Problem solution (by least squares)
Inverse problemsWell posed if:
1. A solution exists
2.The solution is unique
3.The solution depends continuously on the data
otherwise it is an inverse problem.
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Introduction
Problem solution (regularized least squares)
Standard image reconstruction aims to:• Find a single solution
• Use smallest amount of regularization
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Bayesian paradigm
Equivalent statistical model
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Bayesian paradigm
Links between approaches:
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So, what has been gained?• Some new notation, vocabulary…
• A statistical interpretation…
• Confidence/credible intervals etc.
• Option of using other modelling and estimation approaches
Bayesian paradigm
Ten good reasons:• Flexible approach
• Driven by practical issues
• Different model parameterization options
• Wide choice of prior descriptions
• Alternative numerical methods
• Stochastic optimization
• Sampling approaches, e.g. Markov chain Monte Carlo
• Varied solution summaries
• Credible intervals
• Hypothesis testing
• Fun! 8
Case study: liquid mixing
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Perspex cylinder:• 14cm diameter• 30cm high
Three rings of 16 electrodes:• 30mm high• 6mm wide
Here only bottom ring usedand only alternate electrodes
The reference electrode is earthed
Contact impedances created on electrodes
Case study: liquid mixing
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Aim: Given boundary voltages estimate interior conductivity pattern
These are related by:
•This, forward, problem is very difficult requiring substantial numerical calculations
•Traditionally use pixel-based solvers, e.g. Finite element method
• Large numbers of elements lead to large computational burden but proven solvers available – e.g. EIDORS
•Still scope for novel prior models and output summary
Case study: liquid mixing
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Other priors:
• contact impedances, flow movement etc.
Prior models:
Outputs:
• An image (plus contact impedances etc.)
Case study: liquid mixing
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Prior knowledge:
True conductivity distribution:
• Not smooth, piecewise constant
• Object and background
Model as a binary object:
• Two conductivities
• Object grown around a centre
Numerical methods: Still use mesh-based FEM (what about BEM?)
Output: Centre and size — plus an image
Case study: liquid mixing
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Posterior reconstructions though time
Case study: liquid mixing
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Posterior estimates though time
Conductivity contrast Size of object Centre
Case study: hydrocyclone
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• A hydrocyclone can be used to separate liquid-phase substances of differing densities, e.g. water and oil.
• Centrifuges the less dense material (water) to the outside, leaving the denser oil in the core
• Water and oil now separate entities and are removed from hydrocyclone
• If conditions on output purity are not met, the output is recycled to achieve optimum water/oil separation
• System may also intervene by changing input pressure to optimize separation effectiveness
Case study: hydrocyclone
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Feed
Overflow
Underflow
Model parameters• Core centre• Core size• Electrical conductivity
Ideal for boundary element method
Case study: hydrocyclone
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True conductivity distribution Model parameters• Core centre• Core size• Electrical conductivity
BEM has few elements compared to FEM —hence fast and simple!
Case study: hydrocyclone
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Centre: radius and angle
Conductivity and size
Image from posterior estimates
Case study: hydrocyclone
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Conductivity and size
Posterior credible regions
Centre: radius and angle
Case study: hydrocyclone
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Posterior credible regions for the boundary
Conclusions
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Intelligent and flexible parameterisation
• Pixelization not always appropriate
• Incorporating a priori knowledge avoids solving full problem
Dependence on regularization removed
• Regularization included in model, not inverse solution
• Further prior information can still be included
Well-matched forward solver
• Exploiting parameterization
• Leads to faster and simpler algorithms
Conclusions
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Final message:
• It is sometimes said that, “regularization introduces bias” — this is not a true statement!
• Remember, “all models are wrong” (GEP Box). Similarly, all regularization is wrong—then we might say that it is best to use the smallest amount of regularization possible…
• Alternatively, we can say that “all models are approximations” (T Tarpey), adding that all regularization introduces further approximation does not sound too bad?
• Using a good model and good regularization is better than using a bad model.
• Some models are useful... and some regularization is useful… but some combinations are more useful than others…
