1. What to measure when measuring noise in MRI Santiago Aja-Fernández sanaja@tel.uva.es Antwerpen 2013

2. Noise in MRI 2 Motivation: Many papers and methods to estimate noise out of MRI data. Noise estimation vs. SNR estimation In single coil systems, variance of noise is a “good” measure. Complex systems: what are we really measuring? Is the variance of noise still valid?

3. Noise in MRI 3 Outline 1.Noise in MR acquistions 2.Basic Models –Rician –Non-central chi 3.More Complex Models –Correlated multiple coils –Parallel MRI: SENSE and GRAPPA 4.The nc-chi example –Non-stationary noise –Effective values 5.Other models 6.Conclusions

4. Noise in MRI 4

5. 5

6. 6 1- Noise in MRI

7. Noise in MRI 7 ScannerK-space X-space (complex) magnitude F -1 |. | Signal acquisition (Single coil)

8. Noise in MRI 8 K-space X-space (complex) magnitude F -1 |. | Acquisition Noise (single coil) Complex Gaussian    Complex Gaussian   n =    /|  | Rician   n

9. Noise in MRI 9 9 F -1 Signal acquisition (Multiple coil)

10. Noise in MRI 10 F -1 Acquisition noise (Multiple coil) Complex Gaussian    i Complex Gaussian   i =    i /|  |  12 22 33 44 11 22 33 44  23  34  14  12  14  34  23 11

11. Noise in MRI 11 Acquisition noise: Noise in receiving coils is complex Gaussian (assuming no post-processing) Noise in x-space related to noise in k-space Final distribution will depend on the reconstruction

12. Noise in MRI 12 2- Basic Noise Models

13. Noise in MRI 13 K-space X-space (complex) magnitude F -1 |. | Rician Model Complex Gaussian    Complex Gaussian   n =    /|  | Rician   n

14. Noise in MRI 14 Complex Magnitude Rician Rayleigh Rician Model

15. Noise in MRI 15 RicianGaussian nn Meaning of  n ? SNR=A /  n

16. Noise in MRI 16 Non-central chi model Complex Gaussian   i =    i /|  | 11 22 33 44  12  14  34  23 Particular case:   i=   j =   n  ij =0

17. Noise in MRI 17 Non-central chi model Complex Gaussian   n | nn nn nn nn Sum of Squares Non central chi L,   n |

18. Noise in MRI 18 Non-central chi model Non Central Chi Central Chi

19. Noise in MRI 19 Basic models: Rician: relation between  2 n and variance of noise in Gaussian complex data. Nc-chi: also relation between  2 n and Gaussian Variance. Nc-chi: many times, interesting parameter  2 n ·L E{M(x) 2 }=A(x) 2 +2L·  2 n SNR=A(x)/L 1/2 ·  n Usually: equivalence between L·  2 n (nc-chi) and  2 n (Rician).

20. Noise in MRI 20 Example: Conventional approach Rician Noncentral-chi

21. Noise in MRI 21 Example: LMMSE filter Rician Noncentral-chi

22. Noise in MRI 22 3- More Complex Models

23. Noise in MRI 23 Limitation of the nc-model: Only valid if: Same variance of noise in each coil No correlation between coils No acceleration Reconstruction done with sum of squares Real acquisitions. Correlated, different variances Accelerated Reconstructed with different methods

24. Noise in MRI 24 A- Effect of Correlations 11 22 33 44  12  14  34  23 Sum of Squares Nc-chi no longer valid!!!

25. Noise in MRI 25 Nc-chi approximation if effective parameters are used Relative errors in the PDF for central-chi approximation as a function of the correlation coefficient. A) Using effective parameters. B) Using the original parameters.

26. Noise in MRI 26 B- Sampling of the k-space:

27. Noise in MRI 27 In real acquisitions: Different variances and correlations → effective values and approximated PDF. Subsampling → modification of   i in x-space Reconstruction method → output may be Rician or an approximation of nc-chi The variance of noise (   i ) becomes x- dependant →   i (x)

28. Noise in MRI 28 Survey of statistical models for MRI

29. Noise in MRI 29 4- Example: the non stationary nc-chi approximation

30. Noise in MRI 30 Nc-chi approximation 11 22 33 44  12  14  34  23 Sum of Squares Nc-chi approximation if effective parameters are used

31. Noise in MRI 31 Effective number of coils as a function of the coefficient of correlation. A) Absolute value. B) Relative Value.

32. Noise in MRI 32  eff (x) L eff (x)

33. Noise in MRI 33 Params.  eff (x) and L eff (x) both depend on x. Good news: L eff (x)·   eff (x)= tr(  )=   1 +…+   L =L· The product is a constant: L eff (x)·   eff (x)=L·   n

34. Noise in MRI 34 Implications: Equivalence between effective and real parameters Product: easy to estimate L eff (x)·   eff (x)= L·   n =mode { E{M L 2 (x)} x } / 2 In some problems: only product needed: I 2 (x)= E{M L 2 (x)} x - 2 L·   n NOTE: If effective values are used for   eff (x), also for L eff (x)·   eff (x)·L → Wrong!!!

35. Noise in MRI 35 Implications: My point of view:   eff (x) and L eff (x) should be seen as a single parameter:   L = L eff (x) ·   eff (x) and therefore   eff (x) =   L / L eff (x) Some applications do need   eff (x)

36. Noise in MRI 36 Param.   eff (x) now depends on position. The dependence can be bounded. SNR → 0   B =   n (1+ (L-1)) SNR → ∞   S =   n (1+ (L-1)) Total:   eff (x)=(1-  (x))   S +  (x)   B with  (x)=(SNR 2 (x)+1) -1 X-dependant noise:

37. Noise in MRI 37 BB SS Where is noise estimated? What to estimate:   eff (x),   n,   B,   S …

38. Noise in MRI 38 Implications: Estimation over the background → underestimation of noise. Use of a single value of  n → error is most areas. Noise is higher in the high SNR. Main source of non-stationarity: correlation between coils. High correlation: lower effective coils and higher noise. Possible source of error:   eff (x)·L

39. Noise in MRI 39 5- Other models

40. Noise in MRI 40 SENSE: (non stationary Rician)   R (x) x-dependant noise   n =   K (r / |  |)

41. Noise in MRI 41  eff (x) GRAPPA: nc-chi approx. L eff (x)    eff (x) L eff (x) ) 1/2   n =   K / (r |  |) Product   eff (x)· L eff (x) is not a constant

42. Noise in MRI 42 6- Conclusions

43. Noise in MRI 43 Be sure what you need in your application. Follow the whole reconstruction pipeline to be sure which is your “original” noise. Useful: simplified models to make process easier. Be sure how noise affects the different slides. Conclusions:

44. Noise in MRI 44 Questions?

45. What to measure when measuring noise in MRI Santiago Aja-Fernández sanaja@tel.uva.es Antwerpen 2013