An Introduction to Image Reconstruction, Processing, and their Effects in FMRI
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1 An Introduction to Image Reconstruction, Processing, and their Effects in FMRI Daniel B. Rowe Program in Computational Sciences Department of Mathematics, Statistics, and Computer Science Marquette University August 19, 2015 NIHR21 NS
2 SAMSI 2015 CCNS Working Group on Image Reconstruction and Processing Thursdays 2 pm ET 2
3 Outline Introduction FMRI and fcmri have been utilized with amazing precision. Image Reconstruction Voxels are not directly measured (k-space). Reconstructed! Image Processing Images are processed for enhancement & artifact reduction. Implications Effects of image reconstruction & processing? Mean, Var, Corr? Discussion We need to be careful and know what is done to our data 3
4 Introduction Words of Wisdom: Ideally we should model and analyze the original data that we measure, not a processed version of our data. Don t change the data to fit the model, change the model to fit the data. We should statistically analyze all of our data and not delete half of it for convenience. Favorite Phrase: Analyzed raw preprocessed data. 4
5 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,
6 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 cm FOV 2mm 2mm In-Plane R Are we ahead of the data with our analyses and interpretations? I 6
7 Image 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. 7
8 Image Reconstruction (n=256, Δt=2 s) 10cos(2π0/512t) 3sin(2π8/512t) 1cos(2π32/512t) 1sin(2π4/512t) Acos(2πνt) Asin(2πνt) 8
9 Image Reconstruction (n=256, Δt=2 s) 10cos(2π0/512t) 3sin(2π8/512t) sum 1cos(2π32/512t) 1sin(2π4/512t) Acos(2πνt) Asin(2πνt) 9
10 Image Reconstruction represent = sum 10
11 Image Reconstruction represent = +i sum 11
12 Image Reconstruction represent = +i a +i sum 12
13 Image Reconstruction represent 1cos(2π32/512t) 10cos(2π0/512t) = 1cos(2π32/512t) +i a +i = +i sum 3sin(2π8/512t) 1sin(2π4/512t) 1sin(2π4/512t) 3sin(2π8/512t) Freq Lines R = cos freqs I = sine freq Intensity = amps 13
14 Image Reconstruction (FOV=192 mm) (n x =n y =96, Δx=Δ y=2 mm) 10cos(2π0/96x) 1.5cos(2π8/96x) sin(2π24/96y) cos(2π4/96x+2π4/96y) 14
15 Image Reconstruction (FOV=192 mm) (n x =n y =96, Δx=Δ y=2 mm) 10cos(2π0/96x) 1.5cos(2π8/96x) sum sin(2π24/96y) cos(2π4/96x+2π4/96y) 15
16 Image Reconstruction (FOV=192 mm) (n x =n y =96, Δx=Δ y=2 mm) The machine Fourier encodes the image. Measure spatial freq. +i Image 16
17 Image Reconstruction (FOV=192 mm) (n x =n y =96, Δx=Δ y=2 mm) The machine Fourier encodes the image. Measure spatial freq. + i +i +i FT matrix Image FT matrix 17
18 Image Reconstruction (FOV=192 mm) (n x =n y =96, Δx=Δ y=2 mm) The machine Fourier encodes the image. Measure spatial freq. 10cos(2π0/96x) cos(2π4/96x +cos(2π4/96y) 1.5cos(2π8/96x) + i +i +i = +i sin(2π24/96y) FT matrix image FT matrix spatial frequencies 18
19 Image Reconstruction (FOV=192 mm) (n x =n y =96, Δx=Δ y=2 mm) We inverse Fourier transform spatial freqs to generate image. +i spatial frequencies 19
20 Image Reconstruction (FOV=192 mm) (n x =n y =96, Δx=Δ y=2 mm) We inverse Fourier transform spatial freqs to generate image. + i +i +i FT matrix spatial frequencies FT matrix 20
21 Image Reconstruction (FOV=192 mm) (n x =n y =96, Δx=Δ y=2 mm) We inverse Fourier transform spatial freqs to generate image. + i +i +i = +i FT matrix spatial frequencies FT matrix image 21
22 Image Reconstruction (FOV=192 mm) (n x =n y =96, Δx=Δ y=2 mm) We inverse Fourier transform spatial freqs to generate image. + i +i +i exp(i ) FT matrix spatial frequencies FT matrix Phase discarded. Ask for the other ½ of YOUR data. image 22
23 Image Reconstruction (FOV=192 mm) (n x =n y =96, Δx=Δ y=2 mm) We inverse Fourier transform spatial freqs to generate image. + i +i +i = +i FT matrix spatial frequencies FT matrix image Illustrative Example 23
24 Image Reconstruction We get k-space measurements by changing gradients Adapted from Haacke et al., R-I Points taking complex-valued measurements over time. 24
25 Image Reconstruction We can stack freq. rows of reals over rows of imaginaries, 25
26 Image Reconstruction We can stack freq. rows of reals over rows of imaginaries, make one IFT reconstruction matrix from the two, 26
27 Image 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. 27
28 Image 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 SENSE/GRAPPA Simultaneous Multi-Slice 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. 28
29 Image Processing f k-space 29
30 Image Processing f 1 Z Zero-Filling k-space 30
31 Image Processing O k f 1 A Z Apodization Zero-Filling k-space 31
32 Image Processing O R O k f 1 Ω A Z IFT Reconstruction Apodization Zero-Filling k-space 32
33 Image Processing O I OR O k f 1 S Ω A Z Smoothing IFT Reconstruction Apodization Zero-Filling k-space 33
34 Image Processing y O I OR O k 1 S Ω A Z = Processed Image Smoothing IFT Reconstruction Apodization Zero-Filling k-space y=of 34
35 Image Processing y O I OR O k f 1 S Ω A Z 64 = 128 Processed Image Smoothing IFT Reconstruction Apodization Zero-Filling k-space y=of 35
36 Image Processing y O I OR O k f 1 S Ω A Z 64 = 128 Processed Image Smoothing IFT Reconstruction Apodization Zero-Filling k-space y=of 36
37 Image 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? 37
38 Implications In statistics, we know the rule that says: If a vector f has a mean δ, and a covariance Γ, Then ν=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. 38
39 Implications Operators, O. S Ω A Z Ω ΩZ SΩAZ 39
40 SΩAZ ΩZ SAMSI CCNS Opening Workshop True Values Implications Mean, µ=of. R I M P 40
41 Implications Correlation matrix and correlation image x5 image cor(ν) = 25x25 correlation matrix x5 correlation image 41
42 SΩAZ ΩZ SAMSI CCNS Opening Workshop True Values Implications Correlation, R=D -1/2 ΣD -1/2. R R R R I RI RR IR R R RI II M 2 +1 TH=
43 Reconstruction Parallel Imaging In-Plane Acceleration SENSE/GRAPPA ky kx AΔky Δkx NC 4 Accelerated Acquisition (A=2) 43
44 Reconstruction SENSE coil 1 sub-sampled k-space coil 1 aliased image combined image coil coil coil coil coil 4 coil N C 4 A
45 Reconstruction/Processing SENSE u OP I U P CS Ok 0 u p 0 image image Vector, complex permute to by folded voxel reconstruct N c =4 images k-space vector of N c images 45
46 Implications SENSE reconstruction induces long-range correlation. 4 fold real correlation mag squared correlation 24 real/imaginary correlation R-R I-I R-I M Center voxel fold Image 1 Theoretical A=3 with smoothing Image 1 Image 2 Image 1 Image 2 Image 1 Image 2 Basically multiplying voxel values v t by same 3 numbers over time to lead to correlated voxels a b c 46
47 Implications SENSE Reconstruction induces long-range correlation. Experimental Results SENSE A=3 smoothed R-R I-I R-I M 2 +1 R-R I-I R-I M
48 Reconstruction GRAPPA sub-sampled k-space Interpolate k-space coil 1 +i combined image coil i coil i coil i N C 4 A
49 Reconstruction/Processing GRAPPA G R -G I O I C P G I GR P Ok single image Vector, complex inverse FT reconstruction linear combination permute to by folded voxel GRAPPA interpolation k-space vector of N c images 49
50 Implications GRAPPA reconstruction induces long-range correlation. R-R I-I R-I M 2 R-R I-I R-I M 2 GRAPPA A=2 with smoothing 50
51 Implications GRAPPA reconstruction induces long-range correlation. R-R I-I R-I M 2 +1 R-R I-I R-I M 2 GRAPPA A=3 with smoothing 51-1
52 Reconstruction Simultaneous Multi-Slice (Multiband) The exact same math for SENSE and GRAPPA has been used for separating overlapping slices. 52
53 Reconstruction Simultaneous Multi-Slice (Multiband) Efforts have been made to decrease voxel signal overlap with some success. CAIPI FOV shifts
54 Reconstruction Simultaneous Multi-Slice (Multiband) Have developed the SPECS technique with CAIPI, extended to Hadamard encoding
55 3. Materials and Methods In order to demonstrate the SPECS Hadamard model with encoding with the aliasing, a T 2 * weighted digital phantom is generated with 720 TRs for 4 slices. For the optimal separation, a unique magnitude and phase is added to each slice, with an average SNR = 50. One voxel region in each slice, with the ocations rotating clockwise, has a block design task simulated of sixteen 22- second periods, added to its magnitude with a CNR = ½. In both models the initial off-task portion of the time-series is used for the calibration images in the slice separation. 55
56 Variance Phase Magnitude SAMSI CCNS Opening Workshop 4. Results Hadamard Correlation 56
57 Results SPECS-Hadamard 2 complex acquisitions A=1 A=2 A=4 TH=2 9 4 separated complex slices Complex-valued Activation -9 57
58 Results Expand processing to include Time Series image 1 vector OI1 1 O k 1 0 image 1 k-space vector image n vector 0 OIn n Okn image n k-space vector 58
59 Results ts 1 O T1 0 iv 1 0/1 permutation matrix reorder from by image to by voxel. 0 O Tn iv n ts n ordered by voxel Induced temporal autocorrelation. 59
60 Discussion This opens up the opportunity for complex-valued analysis! Complex-valued activation and/or Complex-valued correlation? Magnitude and Phase or equivalently Real and Imaginary Lengthening & Rotation with task! 60
61 Discussion Cartesian coordinates Real-Imaginary Polar Coordinates Magnitude-Phase n n Phase discarded! t=3 t=3 t=2 t=2 Real Imaginary t=1 Magnitude Phase t=1 61
62 Discussion There is biological information in the phase! Magnitude Image Phase discarded! Phase Image 62
63 Discussion There is biological information in the phase! Magnitude Image Phase discarded! Phase Image 63
64 Discussion Complex-valued fmri models can be developed. Imaginary Imaginary Imaginary Real Real Real Complex Magnitude w/ Constant Phase (CP) Activation Complex Magnitude and/or Phase (MP) Activation Real Magnitude-Only (MO/UP) Activation (Discard Phase) Real Phase-Only (PO) Activation (Discard Magnitude) 64
65 Discussion 3.0T GE LX 20s off+16 (8 s on 8 s off), 276 TRs 12 axial slices, 96 96, FOV = 24 cm TH = 2.5 mm, TR = 1 s, TE = 34.6 ms FA = 45, BW = 125 khz, ES =.708 ms Bi finger Breathing.167 Hz Information in the phase. 20s off+16 (8 s on 8 s off), 276 TRs 10 axial slices, 96 96, FOV = 24 cm TH = 2.5 mm, TR = 1 s, TE = 42.8 ms FA = 45, BW = 125 khz, ES =.768 ms 20s off+16 (8 s on 8 s off), 276 TRs 10 axial slices, 96 96, FOV = 24 cm, TH = 2.5 mm, TR = 1 s, TE = 42.8 ms FA = 45, BW = 125 khz, ES =. 768 ms Bi finger Breathing.167 Hz Bi finger Mouth.167 Hz 20s off+10 (8 s on 8 s off), 180 TRs 9 axial slices, 64 64, FOV = 24 cm TH = 3.8 mm, TR = 1 s, TE = 26.0 ms FA = 45, BW = 125 khz, ES =.680 ms Bi finger none MO PO MP 65
66 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. Our models should incorporate processing. 66
67 Thank You! This work is joint with former & current 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 Ms. Emily M. Paulson, Marquette University Mr. Kevin K. Liu, Marquette University 67
68 SAMSI 2015 CCNS Working Group on Image Reconstruction and Processing Thursdays 2 pm ET 68
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