“Complementary parameterization and forward solution method”

1. “Complementary parameterization and forward solution method” Robert G Aykroyd University of Leeds, r.g.aykroyd@leeds.ac.uk

2. 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

3. Introduction • Ingredients of imaging problems • Data collection style: • Direct or • projection • Focused or • blurred • Low noise or • high noise

4. Introduction Problem solution (by least squares) • Inverse problems • Well posed if: • A solution exists • The solution is unique • The solution depends continuously on the data • otherwise it is an inverse problem.

5. Introduction Problem solution (regularized least squares) • Standard image reconstruction aims to: • Find a single solution • Use smallest amount of regularization

6. Bayesian paradigm • Equivalent statistical model

7. Bayesian paradigm • Links between approaches: • So, what has been gained? • Some new notation, vocabulary… • A statistical interpretation… • Confidence/credible intervals etc. • Option of using other modelling and estimation approaches

8. 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!

9. Case study: liquid mixing • Perspex cylinder: • 14cm diameter • 30cm high • Three rings of 16 electrodes: • 30mm high • 6mm wide • Here only bottom ring used • and only alternate electrodes • The reference electrode is earthed • Contact impedances created on electrodes

10. Case study: liquid mixing • 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

11. Case study: liquid mixing Prior models: • Other priors: • contact impedances, flow movement etc. • Outputs: • An image (plus contact impedances etc.)

12. Case study: liquid mixing • 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

13. Case study: liquid mixing Posterior reconstructions though time

14. Case study: liquid mixing Posterior estimates though time Conductivity contrast Size of object Centre

15. Case study: hydrocyclone • 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

16. Overflow Feed Underflow Case study: hydrocyclone • Model parameters • Core centre • Core size • Electrical conductivity Ideal for boundary element method

17. Case study: hydrocyclone True conductivity distribution • Model parameters • Core centre • Core size • Electrical conductivity BEM has few elements compared to FEM —hence fast and simple!

18. Case study: hydrocyclone Centre: radius and angle Image from posterior estimates Conductivity and size

19. Case study: hydrocyclone Posterior credible regions Conductivity and size Centre: radius and angle

20. Case study: hydrocyclone Posterior credible regions for the boundary

21. Conclusions • 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

22. Conclusions • 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…

23. The End… Robert G Aykroyd University of Leeds, r.g.aykroyd@leeds.ac.uk