1. School of something FACULTY OF OTHER “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 2

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

4. Introduction Problem solution (by least squares) Inverse problems Well 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. 4

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

6. Bayesian paradigm Equivalent statistical model 6

7. Bayesian paradigm Links between approaches: 7 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! 8

9. Case study: liquid mixing 9 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 10 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 11 Other priors: contact impedances, flow movement etc. Prior models: Outputs: An image (plus contact impedances etc.)

12. Case study: liquid mixing 12 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 13 Posterior reconstructions though time

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

15. Case study: hydrocyclone 15 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. Case study: hydrocyclone 16 Feed Overflow Underflow Model parameters Core centre Core size Electrical conductivity Ideal for boundary element method

17. Case study: hydrocyclone 17 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 18 Centre: radius and angle Conductivity and size Image from posterior estimates

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

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

21. Conclusions 21 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 22 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. School of something FACULTY OF OTHER The End… Robert G Aykroyd University of Leeds, r.g.aykroyd@leeds.ac.uk