intrinsic voxel correlation in fmri daniel b. rowe, ph.d ...daniel/pubs/rowetalkjsm06.pdf · jsm...
TRANSCRIPT
JSM 06’ Rowe, MCW
Intrinsic voxel correlation in fMRI
Daniel B. Rowe, [email protected]
Joint with R.G. Hoffmann
Department of BiophysicsDivision of Biostatistics
Graduate School of Biomedical Sciences
8701 Watertown Plank Rd.Milwaukee, WI 53226
JSM 06’ Rowe, MCW
Outline
Complex Image Reconstruction Part I
• Complex voxels from complex k-space measurements
Complex Statistical Activation Method Part I
• Real-Valued: Magnitude-Only, Phase-Only
• Complex-Valued: Magnitude & Phase
Complex Image Reconstruction Part II
• Relating voxels to k-space measurements
• Statistical Implications
Complex Statistical Activation Methods Part II
• fMRI activation directly from k-space measurements
Discussion
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
The Complex-Valued (Discrete) Fourier Transform (n=256, TR=2s)
32 64 96 128 160 192 224 2569
9.2
9.4
9.6
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32 64 96 128 160 192 224 256−3
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−1
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10, cos 0/512 Hz 3 ∗ sin 8/512 Hz
32 64 96 128 160 192 224 256−1
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32 64 96 128 160 192 224 256−1
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sin 32/512 Hz cos 4/512 Hz
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Sum
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
The Complex-Valued (Discrete) Fourier Transform (n=256, TR=2s)The DFT process is to make the data a vector y = (y1, ..., yn)′
Form the forward Fourier matrix Ω = ΩR + iΩI
f1...
fn
=
(ΩR + i ΩI
)
y1...
yn
n× 1 n× n n× 1
Complex Complex Real
(fR + ifI) =(ΩR + i ΩI
)(yR + iyI)
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
The Complex-Valued (Discrete) Fourier Transform (TR=2s)
(ΩR +i ΩI) *(yR +i yI) = (fR +i fI)
64 128 192 256
64
128
192
256 + i 64 128 192 256
64
128
192
256 *
32
64
96
128
160
192
224
256 + i
32
64
96
128
160
192
224
256 =
−.25
−.125
0
.125
.25 + i
−.25
−.125
0
.125
.25
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
The Complex-Valued (Discrete) Fourier Transform (TR=2s)
−.25
−.125
0
.125
.25
−.25
−.125
0
.125
.25
Cosines Sines
−0.25 −0.1875 −0.125 −0.0625 0 0.0625 0.125 0.18750
128
256
384
512
−0.25 −0.1875 −0.125 −0.0625 0 0.0625 0.125 0.1875
−128
−64
0
64
128
Cosines Sines
32 64 96 128 160 192 224 2569
9.2
9.4
9.6
9.8
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10.2
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10.6
10.8
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32 64 96 128 160 192 224 256−3
−2
−1
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1
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3
32 64 96 128 160 192 224 256−1
−0.8
−0.6
−0.4
−0.2
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0.2
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32 64 96 128 160 192 224 256−1
−0.8
−0.6
−0.4
−0.2
0
0.2
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0.6
0.8
1
10, cos 0/512 Hz 3 ∗ sin 8/512 Hz sin 32/512 Hz cos 4/512 Hz
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
The Complex-Valued 2D (Discrete) Fourier Transform
24 48 72 96
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9624 48 72 96
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9624 48 72 96
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96
∑= cos + cos + sin + cos
FOV=192 mm, mat=96×96, vox=2 mm3
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
The Complex-Valued 2D (Discrete) Fourier Transform(ΩyR + iΩyI) * (RR + iRI) * (ΩxR + iΩxI)T = (SR + iSI)
32 64 96 128
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12824 48 72 96
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9632 64 96 128
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64
96
128−.25 −.17 0 .17 .25
.25
.17
0
−.17
−.25
+ i * + i * + i = + i
32 64 96 128
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12832 64 96 128
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12832 64 96 128
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128−.25 −.17 0 .17 .25
.25
.17
0
−.17
−.25
FOV=192 mm, mat=96×96, vox=2 mm3
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
The Complex-Valued 2D (Discrete) Fourier Transform
−.25 −.17 0 .17 .25
.25
.17
0
−.17
−.25−.25 −.17 0 .17 .25
.25
.17
0
−.17
−.25
Real k-space Imaginary k-space
Note: Rotate bottom half up to get top!
FOV=192 mm, mat=96×96, vox=2 mm3
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
Gx & Gy to encode, measure the complex-valued FT of the object.
S(kx, ky) =∫
R(x, y)e−i2π(xkx+yky)dx dy,
(a) EPI Pulse Sequence
−4 −3 −2 −1 0 1 2 3
−4
−3
−2
−1
0
1
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kx
k y
(b) k-Space Trajectory
Kumar, Welti and Ernst: NMR Fourier Zeugmatography, J. Magn. Reson. 1975
Haacke et al.: Magnetic Resonance Imaging: Physical Principles and Sequence Design, 1999.
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
S(kx, ky) = SR(kx, ky) + iSI(kx, ky), complex-valued FT of the object.
(a) real: 96 × 96 (b) imaginary: 96 × 96
Haacke et al.: Magnetic Resonance Imaging: Physical Principles and Sequence Design, 1999.
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
complex-valued 2D IFT(ΩyR + iΩyI) * (SR + iSI) * (ΩxR + iΩxI)T = (RR + iRI)
+ i * + i * + i = + i
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
Due to the imperfect Fourier encoding, the IFT reconstructedobject is complex-valued, R(x, y) = RR(x, y) + iRI(x, y).
(a) Real image, yRt (b) Imaginary image, yIt
Haacke et al.: Magnetic Resonance Imaging: Physical Principles and Sequence Design, 1999.
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
This occurs over time in fMRI and results in complex-valued imagesand voxel time course observations, yt = yRt + iyIt.
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
Most fMRI studies transform from real-imaginary rectangular coordinatesto magnitude-phase polar coordinates, ρ(x, y) = m(x, y)eiφ(x,y).
(a) Magnitude, mt =√
y2Rt + y2
It(b) Phase, φt = atan4(yIt/yRt)
Haacke et al.: Magnetic Resonance Imaging: Physical Principles and Sequence Design, 1999.
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part I
Collect a sequence of these reconstructed images over time.Form voxel time courses, yt = rte
iφt.
JSM 06’ Rowe, MCW
Complex Statistical Activation Method Part ITime series are complex-valued or bivariate with phase coupled means.
0
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Realtim
e
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t
yI
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03264961281601922242560
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0 32 64 96 128 160 192 224 2560
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time
Imag
inary
Imaginary
The yR and yI time courses have related vector length info!This is a time series from a actual human experimental data!
JSM 06’ Rowe, MCW
Complex Statistical Activation Method Part ITime series are complex-valued or bivariate with phase coupled means.
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yI
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r
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Magn
itude
Magnitude
32 64 96 128 160 192 224 256
−3
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Phas
e
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MO time courses only have vector length info!PO time courses only has vector angle info!Real-Imaginary or Magnitude-Phase time courses have all info!
JSM 06’ Rowe, MCW
Complex Statistical Activation Method Part I
Block-designed experiment: Off-On-Off-...-On-Off task
- Real
6
Imaginary
*
*
- Real
6
Imaginary
*
- Real
6
Imaginary
*
Real Magnitude-Only (MO/UP) Activation1,2,3
Real Phase-Only (PO) Activation4
Complex Magnitude w/ Constant Phase (CP) Activation5
Complex Magnitude &/or Phase (CM) Activation6
Complex Magnitude w/ Phase Regressor Activation7,8
1Bandettini et al.: MRM, 30:161-173, 1993. 2Friston et al.: HBM, 2:189-210, 1995.3Rowe and Logan: NeuroImage, 24:603-606, 2005. 4Rowe, Meller, and Hoffmann: Submitted, 20065Rowe and Logan: NeuroImage, 23:1078-1092, 2004. 6Rowe: NeuroImage, 25:1310-1324, 2005b.7Menon, MRM, 47:1-9, 2002. 8Nencka and Rowe: Submitted, 2006.
JSM 06’ Rowe, MCW
Complex Statistical Activation Method Part I
Block-designed experiment: Off-On-Off-...-On-Off task
- Real
6
Imaginary
*
**
- Real
6
Imaginary
*
- Real
6
Imaginary
*
Real Magnitude-Only (MO/UP) Activation1,2,3
Real Phase-Only (PO) Activation4
Complex Magnitude w/ Constant Phase (CP) Activation5
Complex Magnitude &/or Phase (CM) Activation6
Complex Magnitude w/ Phase Regressor Activation7,8
1Bandettini et al.: MRM, 30:161-173, 1993. 2Friston et al.: HBM, 2:189-210, 1995.3Rowe and Logan: NeuroImage, 24:603-606, 2005. 4Rowe, Meller, and Hoffmann: Submitted, 20065Rowe and Logan: NeuroImage, 23:1078-1092, 2004. 6Rowe: NeuroImage, 25:1310-1324, 2005b.7Menon, MRM, 47:1-9, 2002. 8Nencka and Rowe: Submitted, 2006.
JSM 06’ Rowe, MCW
Complex Statistical Activation Method Part I
Block-designed experiment: Off-On-Off-...-On-Off task
- Real
6
Imaginary
*
**
- Real
6
Imaginary
*
@@
@@@I
- Real
6
Imaginary
*
Real Magnitude-Only (MO/UP) Activation1,2,3
Real Phase-Only (PO) Activation4
Complex Magnitude w/ Constant Phase (CP) Activation5
Complex Magnitude &/or Phase (CM) Activation6
Complex Magnitude w/ Phase Regressor Activation7,8
1Bandettini et al.: MRM, 30:161-173, 1993. 2Friston et al.: HBM, 2:189-210, 1995.3Rowe and Logan: NeuroImage, 24:603-606, 2005. 4Rowe, Meller, and Hoffmann: Submitted, 20065Rowe and Logan: NeuroImage, 23:1078-1092, 2004. 6Rowe: NeuroImage, 25:1310-1324, 2005b.7Menon, MRM, 47:1-9, 2002. 8Nencka and Rowe: Submitted, 2006.
JSM 06’ Rowe, MCW
Complex Statistical Activation Method Part I
Block-designed experiment: Off-On-Off-...-On-Off task
- Real
6
Imaginary
*
**
- Real
6
Imaginary
*
@@
@@@I
- Real
6
Imaginary
*
CCCCCCCCCO
Real Magnitude-Only (MO/UP) Activation1,2,3
Real Phase-Only (PO) Activation4
Complex Magnitude w/ Constant Phase (CP) Activation5
Complex Magnitude &/or Phase (CM) Activation6
Complex Magnitude w/ Phase Regressor Activation7,8
1Bandettini et al.: MRM, 30:161-173, 1993. 2Friston et al.: HBM, 2:189-210, 1995.3Rowe and Logan: NeuroImage, 24:603-606, 2005. 4Rowe, Meller, and Hoffmann: Submitted, 20065Rowe and Logan: NeuroImage, 23:1078-1092, 2004. 6Rowe: NeuroImage, 25:1310-1324, 2005b.7Menon, MRM, 47:1-9, 2002. 8Nencka and Rowe: Submitted, 2006.
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part II
complex-valued 2D F FT(ΩyR + iΩyI) * (RR + iRI) * (ΩxR + iΩxI)
T = (SR + iSI)
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+ i * + i * + i = + i
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JSM 06’ Rowe, MCW
Complex Image Reconstruction Part II
complex-valued 2D I FT(ΩyR + iΩyI) * (SR + iSI) * (ΩxR + iΩxI)
T = (RR + iRI)
2 4 6 8
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+ i * + i * + i = + i
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JSM 06’ Rowe, MCW
Complex Image Reconstruction Part II
s =vec(STR, ST
I ) Ω =
[ΩR −ΩIΩI ΩR
]
s =
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Ω =16 32 48 64
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64 , − 16 32 48 64
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16 32 48 64
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64 , 16 32 48 64
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s =(
sRsI
)ΩR = [(ΩyR ⊗ ΩxR)− (ΩyI ⊗ ΩxI)]
ΩI = [(ΩyR ⊗ ΩxI) + (ΩyI ⊗ ΩxR)]
⊗ is Kronecker product that multiplies each element of 1st matrix argument with entire 2nd matrix argument
JSM 06’ Rowe, MCW
Complex Image Reconstruction Part II
r = Ω ∗ s
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128 ∗
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JSM 06’ Rowe, MCW
Complex Image Reconstruction Part II
r =
(rRrI
)→ RT = vec(r) = (RT
R, RTI )
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128 →
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↓← ←↓ RC = RR + i RI
→ 2 4 6 8
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+ i 2 4 6 8
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JSM 06’ Rowe, MCW
Statistical Implications
Showed that r = Ωs.
If E(s) = s0 and var(s) = Γ
then E(r) = Ωs0 and var(r) = ΩΓΩT
And vise versa.
We know we have spatially correlated voxels (the r’s).
This can only result from correlated k-space measurements (the s’s)
JSM 06’ Rowe, MCW
Statistical Implications
Could part of the correlation between voxels be fromtemporally correlated k-space measurements along the EPI trajectory?
−4 −3 −2 −1 0 1 2 3
−4
−3
−2
−1
0
1
2
3
kx
k y
k-Space Trajectory
Is (−3,−4) correlated with (−4,−4)?Is (−2,−4) correlated with (−3,−4)?
JSM 06’ Rowe, MCW
Statistical Implications
Could part of the correlation between voxels be fromk-space adjustments?
−4 −3 −2 −1 0 1 2 3
−4
−3
−2
−1
0
1
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kx
k y
k-Space Trajectory
Shifting of k-space odd/even lines?
Evens left and odds right through timeinduces k-space correlation!
Effect on voxel spatial correlation?
JSM 06’ Rowe, MCW
Statistical Implications
Could part of the correlation between voxels be frompartial k-space acquisition?
−4 −3 −2 −1 0 1 2 3
−4
−3
−2
−1
0
1
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kx
k y
Partial k-Space Acquisition
Acquistion of k-space botom 1/2 lines?
Make up top 1/2 from bottom 1/2!
Maybe k-space correlation correlatedsince same numbers.Effect on voxel spatial correlation?
JSM 06’ Rowe, MCW
Statistical Implications
What about Apodization, AKA k-space smoothing?
k-Space Apodization
Effect on voxel spatialcorrelation?
Increase correlation ofhigher spatial frequencies?
Bernstein, King, Zhou: Handbook of MRI Pulse Sequences, Academic Press, 2004.
”an outstanding reference source that covers all the important aspects of pulse sequence design and implementation.”–G.H. Glover, in NMR IN BIOMEDICINE (2005)
JSM 06’ Rowe, MCW
Statistical Implications
Sample k-space Correlation (along EPI trajectory) 8× 8 imagevar(s) var(r) var(m)
16 32 48 64 80 96 112 128
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12816 32 48 64 80 96 112 128
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12816 32 48 64
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64−1
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Real-Imaginary k-space Real-Imaginary Voxels Magnitude Voxels
Some good correlations ∼ .3 in magnitude data.Maybe mising out on correlations seen in complex?Should decompose covariance into constituent parts.var(s) = Γ = ΓP + ΓN + ΓI
var(r) = ΩΓΩT = ΩΓPΩT + ΩΓNΩT + ΩΓIΩT
JSM 06’ Rowe, MCW
Discussion
Described Complex Image Reconstruction.
Related voxels to k-space measurements.
Not shown fMRI activation directly from k-space measurements!
Further research is needed.
To be continued ...
JSM 06’ Rowe, MCW
Thank You.
Could complex-valued activation be the future of fMRI analysis?Compute activation directly from original k-space measurements!
Collaborators:Mr. Andy NenckaDr. Brent LoganDr. Ray Hoffmann
Colleagues:Dr. Jim HydeDr. Shi-Jiang LiDr. Andrzej Jesmanowicz