1. Real-time Bayesian soft X-ray tomography on WEST
Presented by :
Tianbo Wang Phd student of Ghent University / SWIP
Promoted by :
Dr. Didier Mazon IRFM/CEA
Prof. Geert Verdoolaege Ghent University
Dr. Jakob Svensson IPP Greifswald
31 March 2017

2. Summary GEM Soft X-ray diagnostic on WEST Bayesian method
Gaussian process tomography with preliminary result
Conclusion perspectives

3. Soft X-ray diagnostic on WEST
75 Vertical line of sight
128 Horizontal line of sight
Minor radius (mm)
100 × 100 pixels
16mm × 16 mm pixel size
Major radius (mm)
Ref. GEM detectors for WEST and potential application for heavy impurity transport studie, D. Mazona, A. Jardina

4. Tomography and inverse problem
forward problem
Deductive, predictive : answer is finite and sure to know
inverse problem
Inductive, inferential : infinite amount of answer with a big uncertainty

5. Bayesian probability inference
A little bit probability :
𝑝 𝐴𝐵 =𝑝 𝐴 𝐵 𝑝 𝐵 =𝑝 𝐵 𝐴 𝑝(𝐴)
𝑝 𝐴 𝐵 = 𝑝 𝐵 𝐴 𝑝(𝐴) 𝑝(𝐵) ~𝑝 𝐵 𝐴 𝑝 𝐴 𝑛𝑜𝑛−𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑
A parameter (emissivity)
B test (measurement)
p A B posterior (emissivity knowing measurement)
p B A likelihood (measurement knowing emissivity)
p(A) prior (assumption on emissvity)
p(B) evidence = 𝑝 𝐴,𝐵 𝑑𝐴= 𝑝 𝐵 𝐴 𝑝 𝐴 𝑑𝐴
Evidence here can be considered as a normalisation factor which is independent of A, in another word, it’s usually ignored in our inference.
Bayes formula

6. Bayesian probability inference
The underlying logic:
Posterior information:
What’s the emissivity probability distribution that with the measurements we’ve got?
What’s the most possible emissivity depending on the measurements, and how sure we are?
Prior information:
How many emissivity distribution there could be?
How is it distributed?
𝑝 𝑒𝑚𝑖𝑠𝑠𝑖𝑣𝑖𝑡𝑦 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 ~ 𝑝 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 𝑒𝑚𝑖𝑠𝑠𝑖𝑣𝑖𝑡𝑦 𝑝(𝑒𝑚𝑖𝑠𝑠𝑖𝑣𝑖𝑡𝑦)
Likelihood Forward model:
If we have already known the emissivity, how would it be the measurement?
How is it distributed, if we consider the random uncertainty or noise?

7. Gaussian process tomography formulation
Byesian Formula:
𝑓 𝑁 emissivity
𝑑 𝑀 measurement
𝜃 hyperparameters (factors of smoothness)
Ref. J Svensson, Nonparametric tomography using Gaussian Processes, JET Internal report, EFDA-JET-PR(11)24, 2011

8. Gaussian process tomography
Modelling :
Why Gaussian process?
I will tell you later.
Ref. J Svensson, Nonparametric tomography using Gaussian Processes, JET Internal report, EFDA-JET-PR(11)24, 2011

9. Gaussian process tomography formulation
Prior (Gaussain process):
Mean of prior
Assumption of most probable emissivity (zeros)
Covariance of prior
𝑘 𝑆𝐸 = 𝜎 𝑓 2 exp⁡(− 𝑑 𝜎 𝑙 2 ), 𝑑= 𝑟 − 𝑟 ′
Squared exponential
covariance function
𝑟 location of pixel
𝑑 distance between pixels
𝝈 𝒇 basic variance value
𝝈 𝒍 length scale
Hyperparameters of smoothness, but how to choose them?
Ref. J Svensson, Nonparametric tomography using Gaussian Processes, JET Internal report, EFDA-JET-PR(11)24, 2011

10. Gaussian process tomography formulation
Hyperparameter inference or optimization is to find the most probable hyperparameters depending on the measurement we have already got.
𝑝 ℎ𝑦𝑝𝑒𝑟𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡
~ 𝑝 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 ℎ𝑦𝑝𝑒𝑟𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑝(ℎ𝑦𝑝𝑒𝑟𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟)
Uniform distribution, constant everywhere
𝑝 ℎ𝑦𝑝𝑒𝑟𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 ~ 𝑝 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 ℎ𝑦𝑝𝑒𝑟𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟
Evidence term
So,
Optimization of hyperparameter = maximizing evidence
Ref. J Svensson, Nonparametric tomography using Gaussian Processes, JET Internal report, EFDA-JET-PR(11)24, 2011

11. Gaussian process tomography formulation
Evidence 𝑝 𝐵 = 𝑝 𝐴,𝐵 𝑑𝐴= 𝑝 𝐵 𝐴 𝑝 𝐴 𝑑𝐴
𝑝 𝜃 𝑑 𝑀 ~ 𝑝 𝑑 𝑀 𝜃 = 𝑝 𝑑 𝑀 𝑓 𝑁 , 𝜃 𝑝 𝑓 𝑁 𝜃 𝑑 𝑓 𝑁
log 𝑝 𝜃 𝑑 𝑀 =− 1 2 {𝑀 log 𝜋 log Σ 𝑑 + 𝑅 𝑇 Σ 𝑑 𝑅 𝑑 𝑀 𝑇 (Σ 𝑑 + 𝑅 𝑇 Σ 𝑑 𝑅 ) −1 𝑑 𝑀 }
Covariance of measurement
~ Signal noise 5%
Mean of measurement
Matrix of pixel chord length 𝑅
Number of line of sight 𝑀
Ref. J Svensson, Nonparametric tomography using Gaussian Processes, JET Internal report, EFDA-JET-PR(11)24, 2011

12. Gaussian process tomography formulation（example）
An example:
sigma f is
length scale is mm
Asymetric hollow shape phantom test optimization

13. Gaussian process tomography formulation
Likelihood (Gaussain process):
Covariance of likelihood
~ Signal noise 5%
Mean of likelihood
~ Signal measurement
Matrix of pixel chord length 𝑅
Ref. J Svensson, Nonparametric tomography using Gaussian Processes, JET Internal report, EFDA-JET-PR(11)24, 2011

14. Gaussian process tomography formulization
Posterior (Gaussain process):
The product of two Gaussian Probability Density functions is also a Gaussian.
So we can directly write down the mean and covariance of posterior. The mathematical frame is simple enough to carry out the calculation in real-time.
Covariance of posterior
(description of uncertainty)
Mean of posterior
Inference result
Distribution of emissivity
Ref. J Svensson, Nonparametric tomography using Gaussian Processes, JET Internal report, EFDA-JET-PR(11)24, 2011

15. Bayesian SXR tomography on WEST Phantom test (Gaussian shape)
Vertical camera mearsurement
Horizontal camera mearsurement

16. Bayesian SXR tomography on WEST Phantom test (Gaussian shape)
Bayesian tomography recosntruction result

17. Bayesian SXR tomography on WEST Phantom test (Gaussian shape)
Bayesian tomography recosntruction result
Gaussian shape Phantom

18. Bayesian SXR tomography on WEST Phantom test (Gaussian shape)
𝑒𝑟𝑟𝑜𝑟= 𝑟𝑒𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑜𝑛−𝑝ℎ𝑎𝑛𝑡𝑜𝑚 max⁡(𝑝ℎ𝑎𝑛𝑡𝑜𝑚)
Maximum error ~ 6,9%
Reconstruction error map

19. Bayesian SXR tomography on WEST Phantom test (Gaussian shape)
Error map is not available in real shot
But uncertainty map is predictable
Uncertainty (color surface) and
error map (black contour)

20. Bayesian SXR tomography on WEST Phantom test (asymetric kidney shape)
Vertical camera measurement
Horizontal camera measurement

21. Bayesian SXR tomography on WEST Phantom test (asymetric kidney shape)
Bayesian tomography recosntruction result

22. Bayesian SXR tomography on WEST Phantom test (asymetric kidney shape)
Bayesian tomography reconstruction result
Asymetric kidney shape Phantom

23. Bayesian SXR tomography on WEST Phantom test (asymetric kidney shape)
𝑒𝑟𝑟𝑜𝑟= 𝑟𝑒𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑜𝑛−𝑝ℎ𝑎𝑛𝑡𝑜𝑚 max⁡(𝑝ℎ𝑎𝑛𝑡𝑜𝑚)
Maximum error ~ 12%
Big errors usually at uncovered zone or strong emissivity zone
Reconstruction error map

24. Bayesian SXR tomography on WEST Phantom test (asymetric kidney shape)
Error map is not available in real shot
But uncertainty map is predictable
Uncertainty (color surface) and
error map (black contour)

25. Bayesian SXR tomography on WEST Phantom test (asymetric hollow shape)
Vertical camera mearsurement
Horizontal camera mearsurement

26. Bayesian SXR tomography on WEST Phantom test (asymetric hollow shape)
Bayesian tomography recosntruction result

27. Bayesian SXR tomography on WEST Phantom test (asymetric hollow shape)
Bayesian tomography recosntruction result
Asymetric hollow shape Phantom

28. Bayesian SXR tomography on WEST Phantom test (asymetric hollow shape)
𝑒𝑟𝑟𝑜𝑟= 𝑟𝑒𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑜𝑛−𝑝ℎ𝑎𝑛𝑡𝑜𝑚 max⁡(𝑝ℎ𝑎𝑛𝑡𝑜𝑚)
Maximum error ~ 15%
Reconstruction error map

29. Bayesian SXR tomography on WEST Phantom test (asymetric hollow shape)
Error map is not available in real shot
But uncertainty map is predictable
Uncertainty (color surface) and
error map (black contour)

30. GPT Phantom Test

31. MFI Phantom Test

32. Bayesian probability inference
# Courtesy of Jakob Svensson : Culham summer school lecture
Measurement
Measurements
results
result
Inference using multiple diagnostics (Minerva framework)
Developped by Seed eScience Research

33. conclusion
A new bayesian soft X-ray tomography algorithm is being developped for WEST.
Gaussian process tomography is an optimum candidate for real-time tomography, sevral different shape phantom tests have been carried out with a good accuracy.
The algorithm is fully independent, the inference result can be compared to the other related diagnostics.
Tomography result can be improved by using extra-information, typically equilibrum magnetic flux surface.
Combine multiple diagnostics tomography is possible, for example: bolometer, SXR, HXR joint inference.

34. Back up slides

35. Measurement and result
# Courtesy of Jakob Svensson : Culham summer school lecture
Data: A person gets a positive result on a HIV test. The test is 99% accurate. What is the probability that he has HIV? Answer: the test does not tell us whether he has HIV or not unless we make assumptions on the probability that he has HIV before he does the test. In numbers: about 0.1% of the UK population have HIV. Of 1000 tested, about 10 will score positive due to the 1% error, and about 1 will actually have HIV. So answer under those assumptions is: about 9%. If we assume that the (prior) probability for the person to have HIV is 0.1% (the proportion HIV infected in the UK) – then we can answer the question. Without the probability assumption we can say...nothing. With the probability assumption we can reason consistently.

36. Measurement and result
# Courtesy of Jakob Svensson : Culham summer school lecture

37. Gaussian process tomography formulation
Optimization:
To avoid over-fitting which gives a too complex reconstruction structure; or under- fitting which the reconstuction is not suitable enough to explain the measurement, the optimization process is necessary.
over-simple optimized over-fitting
In the frame of Gaussian Process, the regularization is carried out by covariance function of pixel variables. Two hyperparameters: 𝝈 𝒇 basic variance factor and 𝝈 𝒍 length scale factor should be defined.
𝑘 𝑆𝐸 = 𝜎 𝑓 2 exp⁡(− 𝑑 𝜎 𝑙 2 ), 𝑑= 𝑟 − 𝑟 ′
Ref. J Svensson, Nonparametric tomography using Gaussian Processes, JET Internal report, EFDA-JET-PR(11)24, 2011