Statistical Analysis of Image Reconstructed Fully-Sampled and Sub-Sampled fmri Data

Transcription

1 Statistical Analysis of Image Reconstructed Fully-Sampled and Sub-Sampled fmri Data Daniel B. Rowe Program in Computational Sciences Department of Mathematics, Statistics, and Computer Science Marquette University February 1, 2016 NIHR21 NS

2 3 Outline Introduction Varied methods to produce fmri images with varied properties. Reconstruction Voxels are not directly measured (k-space). Reconstructed! Processing Images are processed for enhancement & artifact reduction. Implications Effects of image reconstruction & processing? Mean, Var, Corr? Discussion How was our data was produced and what was done to it? 2

3 Introduction In fmri and fcmri, there has been an amazing amount of advanced analysis and interpretations presented, but little attention has been paid to what the data truly are. Fox et al, PNAS,

4 Introduction In general, reconstructed GRE EPI images look like below. How do we get from the below to the previous activation? And the below isn t even our original measurements mm FOV 2.5 mm 2 In-Plane R Are we ahead of the data with our analyses and interpretations? I 4

5 Reconstruction In fmri and MRI, the measurements taken by the machine are an array of complex-valued spatial frequencies. This array of complex-valued spatial frequencies need to be reconstructed into an image for us to see, analyze, and interpret. The array of complex-valued spatial frequencies are reconstructed into an image via the inverse Fourier transform. So lets briefly remind ourselves what the FT and IFT are. 5

6 Coil wraps around so uniform sensitivity. Reconstruction Single Coil Acquisition Coil measures k-space. Coil ky kx Δky Δkx Unaccelerated Acquisition (A=1). Kumar et al: JMR, 18(1):69-83,

7 Reconstruction (FOV=240 mm) (n x =n y =96, Δx=Δ y=2.5 mm) We inverse Fourier transform spatial freqs to generate image i spatial frequencies 7

8 Reconstruction (FOV=240 mm) (n x =n y =96, Δx=Δ y=2.5 mm) We inverse Fourier transform spatial freqs to generate image. + i +i +i IFT matrix spatial frequencies IFT matrix 8

9 Reconstruction (FOV=240 mm) (n x =n y =96, Δx=Δ y=2.5 mm) We inverse Fourier transform spatial freqs to generate image. + i +i +i = +i IFT matrix spatial frequencies IFT matrix image 9

10 Reconstruction (FOV=240 mm) (n x =n y =96, Δx=Δ y=2.5 mm) We inverse Fourier transform spatial freqs to generate image. + i +i +i exp(i ) IFT matrix spatial frequencies IFT matrix Phase discarded. Ask for the other ½ of YOUR data. image 10

11 Reconstruction The machine Fourier encodes the image. Measure spatial freq i +i +i = +i 0 FT matrix image FT matrix spatial frequencies 11

12 R Reconstruction We can stack freq. rows of reals over rows of imaginaries, I 12

13 R Reconstruction We can stack freq. rows of reals over rows of imaginaries, make one IFT reconstruction matrix from the two, I f R f I 13

14 R Reconstruction We can stack freq. rows of reals over rows of imaginaries, make one IFT reconstruction matrix from the two, to get the rows of reals over rows of imaginaries. I y R f R y I f I 14

15 Processing Many processing operations are performed by the scanner, by physicists, and by engineers before statistical analysis. + i k-space Processing Nyquist Ghost Correction Static B0 Field Correction Zero Fill Interpolation Non-Cartesian Interpolation Ramp Sampling Interpolation Homodyne Interpolation Apodization And many more Image Reconstruction 2D inverse Fourier transform In-Plane SENSE/GRAPPA Through-Plane SENSE Image Processing Image Smoothing Global Normalization Motion Correction And many more Time Series Processing Filtering Smoothing Dynamic B0 Correction Slice Timing And many more Show ones in blue. 15

16 Reconstruction We can stack freq. rows of reals over rows of imaginaries, make one IFT reconstruction matrix from the two, to get the rows of reals over rows of imaginaries. y R f R O I O k y I f I 16

17 Processing f k-space 17

18 Processing y O I OR O k f 1 1 S Ω A Z = Processed Image Smoothing IFT Reconstruction Apodization Zero-Filling k-space y=of 18

19 Processing We measure an array of complex-valued numbers, perform complex-valued image reconstruction to this array, to generate complex-valued images in real and imaginary, along the way, there is complex-valued image processing. What are the implications of what was done to the data? 19

20 Implications In statistics, we know the rule that says: If a vector f has a mean δ, and a covariance Γ, Then y=of has a mean μ=oδ, and a covariance Σ=OΓO T. Then Σ can converted into a correlation matrix R=D -1/2 ΣD -1/2. 1/2 Where D 1/ diag( ). Assume k-space measurements independent so Γ=I. 20

21 Implications Operators, O. S Ω A Z

22 P M BIRS Neuroimaging Data Analysis Implications Mean, µ=of. O=Ω O=ΩZ O=ΩA O=ΩAZ O=SΩAZ 22

23 Implications Correlation matrix and correlation image x5 image cor(y) = 25x25 correlation matrix x5 correlation image 23

24 Implications Correlation, R=D -1/2 ΣD -1/2. R R R RR IR R R RI II +1-1 TH=

25 Coil 4 BIRS Neuroimaging Data Analysis Coil local so non uniform sensitivity. Reconstruction Multi-Coil Acquisition Coil 1 k y k x Coil 2 Δky Δkx Each coil measures k-space. Coil 3 N C =4, A=1 Hyde et al.: JMR, 70: , Pruessmann et al.: Proc. ISRM, 579,

26 Coil 4 BIRS Neuroimaging Data Analysis Coil local so non uniform sensitivity. Reconstruction Multi-Coil Acquisition S 1 S 4 Coil 1 v S 2 Coil 2 Coil 3 Each coil measures k-space. S 3 N C =4, A=1 26

27 Coil 4 BIRS Neuroimaging Data Analysis Coil local so non uniform sensitivity. Reconstruction Multi-Coil Acquisition a 4 =S 4 v S 1 a 1 =S 1 v Coil 1 S 4 v S 2 Coil 2 Coil 3 Each coil measures k-space. a 3 =S 3 v S 3 a 2 =S 2 v N C =4, A=1 27

28 Reconstruction SENSE Measured Coil Images Estimated Sensitivities Combined Image ^ v & N C =4, A=1 28

29 Coil 4 BIRS Neuroimaging Data Analysis Coil local so non uniform sensitivity. Reconstruction Multi-Coil Acquisition S 11 S 12 S 13 Coil 1 v 2 S 41 v 1 S 21 S 42 v 2 Coil 2 S 22 S 43 Coil 3 v 3 S 23 Each coil measures k-space. S 31 S 32 N C =4, A=3 S 33 29

30 Coil 4 BIRS Neuroimaging Data Analysis Coil local so non uniform sensitivity. Reconstruction Multi-Coil Acquisition S 11 S 12 a 4 =S 41 v 1 +S 42 v 2 +S 43 v 3 a 1 =S 11 v 1 +S 12 v 2 +S 13 v 3 S 13 Coil 1 v 2 S 41 v 1 S 21 S 42 v 2 Coil 2 S 22 S 43 Coil 3 v 3 S 23 Each coil measures k-space. S 31 S 32 N C =4, A=3 a 3 =S 31 v 1 +S 43 v 2 +S 33 v 3 a 2 =S 21 v 1 +S 22 v 2 +S 23 v 3 S 33 30

31 Reconstruction SENSE Measured Coil Images Estimated Sensitivities Separated Combined Image ^ v 1 & ^v 2 ^ v 3 N C =4, A=3 31

32 Reconstruction/Processing SENSE Image vector Coil 4 Coil 1 Coil 3 Coil 2 u OP I U P CS Ok 0 u p 0 permute to by folded voxel reconstruct N c =4 images k-space vector of N c images 32

33 Implications In statistics, we know the rule that says: If a vector f has a mean δ, and a covariance Γ, Then y=of has a mean μ=oδ, and a covariance Σ=OΓO T. Then Σ can converted into a correlation matrix R=D -1/2 ΣD -1/2. 1/2 Where D 1/ diag( ). Assume k-space measurements independent so Γ=I. 33

34 Implications SENSE induces long-range in-plane correlation. fold Theoretical Results SENSE A=3 smoothed R-R I-I R-I M 2 +1 Center voxel fold -1 Basically multiplying voxel values a t by same 3 numbers over time t to lead to correlated voxels. a b c 34

35 Implications SENSE Reconstruction induces long-range correlation. Experimental Results SENSE A=3 smoothed +1 R-R I-I R-I M 2 R-R I-I R-I M 2 Karaman,Nencka,Bruce,Rowe: Brain Connect, 4: ,

36 Implications GRAPPA reconstruction induces long-range correlation. Experimental Results GRAPPA A=3 smoothed +1 R-R I-I R-I M 2 R-R I-I R-I M 2 Bruce: PhD Dissertation,

37 Coil 4 BIRS Neuroimaging Data Analysis Coil local so non uniform sensitivity. Reconstruction Multi-Coil Acquisition Simultaneous Multi-Slice (SMS) Coil 1 v 2 v 3 v 1 Coil 2 Coil 3 Each coil measures k-space. v j N C =4, A=3 j=1:a Larkman et al: JMRI, 13: ,

38 Coil 4 BIRS Neuroimaging Data Analysis Coil local so non uniform sensitivity. Reconstruction Multi-Coil Acquisition Simultaneous Multi-Slice (SMS) S 11 S 12 S 13 Coil 1 S 43 v 3 S 23 S 42 S 41 v 2 v 1 Coil 2 S 21 S 22 Coil 3 Each coil measures k-space. S 31 S 32 S 33 N C =4, A=3 v j,s kj j=1:a, k=1:n c 38

39 Coil 4 BIRS Neuroimaging Data Analysis Coil local so non uniform sensitivity. Reconstruction Multi-Coil Acquisition Simultaneous Multi-Slice (SMS) S 11 S 12 S 13 a 4 =S 41 v 1 +S 42 v 2 +S 43 v 3 a 1 =S 11 v 1 +S 12 v 2 +S 13 v 3 Coil 1 S 41 S 42 S 43 v 1 v 2 v 3 Coil 2 S 21 S 22 S 23 Coil 3 Each coil measures k-space. a 3 =S 31 v 1 +S 43 v 2 +S 33 v 3 a 2 =S 21 v 1 +S 22 v 2 +S 23 v 3 S 31 S 32 S 33 N C =4, A=3 v j,s kj,a k j=1:a, k=1:n c 39

40 Reconstruction SENSE SMS Measured Coil Images Estimated Sensitivities Separated Combined Images ^ v 1 & ^ v 2 ^ v 3 N C =4, A=3 40

41 Reconstruction/Processing SENSE SMS Image vector 4 u OP I U P CS Ok 0 u p 0 permute to by folded voxel reconstruct N c =4 images k-space vector of N c images 41

42 Implications In statistics, we know the rule that says: If a vector f has a mean δ, and a covariance Γ, Then y=of has a mean μ=oδ, and a covariance Σ=OΓO T. Then Σ can converted into a correlation matrix R=D -1/2 ΣD -1/2. 1/2 Where D 1/ diag( ). Assume k-space measurements independent so Γ=I. 42

43 Coil 4 Slice 3 Coil 3 Slice 2 Coil 2 Slice 1 Coil 1 BIRS Neuroimaging Data Analysis Implications SENSE induces long-range through-plane correlation. Coil Images Separated Images R1 R1 I1 R2 I2 R3 I3 I1 R2 smooth I2 R3 I3 43

44 Coil 4 Slice 3 Coil 3 Slice 2 Coil 2 Slice 1 Coil 1 BIRS Neuroimaging Data Analysis Implications SENSE induces long-range in-plane correlation. Coil Images Separated Images R1 R1 I1 R2 I2 R3 I3 I1 R2 smooth I2 R3 I3 44

45 Coil 4 Slice 3 Coil 3 Slice 2 Coil 2 Slice 1 Coil 1 BIRS Neuroimaging Data Analysis Implications SENSE induces long-range through-plane correlation. Coil Images Separated Images 2 M1 M1 M2 M smooth 2 M2 2 M3 45

46 Coil 4 Slice 3 Coil 3 Slice 2 Coil 2 Slice 1 Coil 1 BIRS Neuroimaging Data Analysis Implications SENSE induces long-range in-plane correlation. Coil Images Separated Images 2 M1 M1 M2 M smooth 2 M2 2 M3 46

47 Discussion Care needs to be taken when we obtain data. We should get data in its originally measured form. We should do any required processing ourselves. We should develop models that incorporate processing. We should use all of the data (magnitude and phase). 47

48 Thank You! This work is joint with former & current PhD students: Dr. Andrew S. Nencka, Medical College of Wisconsin Dr. Andrew D. Hahn, University of Wisconsin-Madison Dr. Iain P. Bruce, Duke University Dr. M. Muge Karaman, University if Illinois-Chicago Ms. Mary C. Kociuba, Marquette University Mr. Kevin K. Liu, Marquette University 48