Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of...
Transcript of Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of...
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STAT692, Wisconsin Rowe, MCW
The Fourier Transform in MRI/fMRI
Daniel B. Rowe, [email protected]
Department of BiophysicsDivision of Biostatistics
Graduate School of Biomedical Sciences
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STAT692, Wisconsin Rowe, MCW
Outline
➣ Introduction
➣ One Dimensional FT
• Time series constituents and Fourier spectrum.
➣ Two Dimensional FT
• An image constituents and Fourier spectrum.
➣ Two Dimensional MR Image Formation.
• k-space and MR Image Reconstruction.
➣ Two Dimensional MR Image Processing
• Smoothing/Edge detetection.
➣ One Dimensional fMRI Time Series
• fMRI time series Fourier spectrum and filtering.
➣ FMRI Time Series Statistics
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STAT692, Wisconsin Rowe, MCW
Outline
➣ Introduction
➣ One Dimensional FT
• Time series constituents and Fourier spectrum.
➣ Two Dimensional FT
• An image constituents and Fourier spectrum.
➣ Two Dimensional MR Image Formation.
• k-space and MR Image Reconstruction.
➣ Two Dimensional MR Image Processing
• Smoothing/Edge detetection.
➣ One Dimensional fMRI Time Series
• fMRI time series Fourier spectrum and filtering.
➣ FMRI Time Series Statistics
![Page 4: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/4.jpg)
STAT692, Wisconsin Rowe, MCW
Outline
➣ Introduction
➣ One Dimensional FT
• Time series constituents and Fourier spectrum.
➣ Two Dimensional FT
• An image constituents and Fourier spectrum.
➣ Two Dimensional MR Image Formation.
• k-space and MR Image Reconstruction.
➣ Two Dimensional MR Image Processing
• Smoothing/Edge detetection.
➣ One Dimensional fMRI Time Series
• fMRI time series Fourier spectrum and filtering.
➣ FMRI Time Series Statistics
![Page 5: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/5.jpg)
STAT692, Wisconsin Rowe, MCW
Outline
➣ Introduction
➣ One Dimensional FT
• Time series constituents and Fourier spectrum.
➣ Two Dimensional FT
• An image constituents and Fourier spectrum.
➣ Two Dimensional MR Image Formation
• k-space and MR Image Reconstruction.
➣ Two Dimensional MR Image Processing
• Smoothing/Edge detetection.
➣ One Dimensional fMRI Time Series
• fMRI time series Fourier spectrum and filtering.
➣ FMRI Time Series Statistics
![Page 6: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/6.jpg)
STAT692, Wisconsin Rowe, MCW
Outline
➣ Introduction
➣ One Dimensional FT
• Time series constituents and Fourier spectrum.
➣ Two Dimensional FT
• An image constituents and Fourier spectrum.
➣ Two Dimensional MR Image Formation
• k-space and MR Image Reconstruction.
➣ Two Dimensional MR Image Processing
• Smoothing/Edge detetection.
➣ One Dimensional fMRI Time Series
• fMRI time series Fourier spectrum and filtering.
➣ FMRI Time Series Statistics
![Page 7: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/7.jpg)
STAT692, Wisconsin Rowe, MCW
Outline
➣ Introduction
➣ One Dimensional FT
• Time series constituents and Fourier spectrum.
➣ Two Dimensional FT
• An image constituents and Fourier spectrum.
➣ Two Dimensional MR Image Formation
• k-space and MR Image Reconstruction.
➣ Two Dimensional MR Image Processing
• Smoothing/Edge detetection.
➣ One Dimensional fMRI Time Series
• fMRI time series Fourier spectrum and filtering.
➣ FMRI Time Series Statistics
![Page 8: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/8.jpg)
STAT692, Wisconsin Rowe, MCW
Outline
➣ Introduction
➣ One Dimensional FT
• Time series constituents and Fourier spectrum.
➣ Two Dimensional FT
• An image constituents and Fourier spectrum.
➣ Two Dimensional MR Image Formation
• k-space and MR Image Reconstruction.
➣ Two Dimensional MR Image Processing
• Smoothing/Edge detetection.
➣ One Dimensional fMRI Time Series
• fMRI time series Fourier spectrum and filtering.
➣ FMRI Time Series Statistics
![Page 9: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/9.jpg)
STAT692, Wisconsin Rowe, MCW
Introduction
Place volunteer/patient into MRI scanner.
MCW GE 3T Long Bore
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STAT692, Wisconsin Rowe, MCW
Introduction
In MRI we image a real-valued 3D object, R(x, y, z).Lattice of volume elements, voxels.
! >Volume Slice
Consider a single slice R(x, y).
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STAT692, Wisconsin Rowe, MCW
Introduction
In MRI we aim to image a real-valued object, R(x, y).Di!erent tissues have di!erent magnetic properties yielding contrast.
T !2 weighted image
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT
The Complex-Valued (Discrete) Fourier Transform (n=256, TR=2s)
32 64 96 128 160 192 224 2569
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6
10.8
11
32 64 96 128 160 192 224 256−3
−2
−1
0
1
2
3
10 " cos 0/512 Hz 3 " sin 8/512 Hz
32 64 96 128 160 192 224 256−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
32 64 96 128 160 192 224 256−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
sin 32/512 Hz cos 4/512 Hz
32 64 96 128 160 192 224 256
6
7
8
9
10
11
12
13
14
Sum
No noise added!
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT-Continuous
The FT of a continuous function f(x) is
F (k) =
! +"
#"f(x)e#i2!kx dx
also denoted as F{f(x)} and its inverse to be
f(x) =
! +"
#"F (k)e+i2!kx dk
also denoted as F#1{F (k)}.
Don’t forget thatei" = cos(!) + i sin(!) .
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT-Continuous
F (k) =
! "
#"f(x)[cos(2"kx) ! i sin(2"kx)] dx
=
! "
#"f(x) cos(2"kx) dx ! i
! "
#"f(x) sin(2"kx) dx
= FC(k) ! iFS(k)
FC(k) =
! "
#"
"
#$
j
Aj cos(2"#jx) +$
j
Bj sin(2"#jx)
%
& cos(2"kx) dx
FS(k) =
! "
#"
"
#$
j
Aj cos(2"#jx) +$
j
Bj sin(2"#jx)
%
& sin(2"kx) dx
The cos() sin() and sin() cos() cross terms are zero.Nonzero values at constituent frequencies where Aj and Bj nonzero.
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT-Continuous
Fourier Transform properties.Property Function Transform
Linearity af(x) + bg(x) aF (k) + bG(k)
Similarity f(ax) 1|a|F (ka)
Shifting f(x ! a) e#i2!kaF (k)
Derivative d!f(x)dx! (i2"k)#F (k)
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT-Continuous
Convolution of functions f(x) and g(x) is defined as
f(x) " g(x) =
! +"
#"f(!) g(x ! !) d! .
Further
F {f(x) · g(x)} = F (k) " G(k) ,
and
F {f(x) " g(x)} = F (k) · G(k) .
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT-Continuous
Convolution properties.f(x) ! g(x) = g(x) ! f(x) commutative
f(x) ! [g(x) ! h(x)] = [f(x) ! g(x)] ! h(x) associative
f(x) ! [g1(x) + g2(x)] = f(x) ! g1(x) + f(x) ! g2(x) distributive
d f(x)∗g(x)dx = d f(x)
dx ! g(x) = f(x) ! d g(x)dx derivative
h(x # x0) = f(x # x0) ! g(x) = f(x) ! g(x # x0) shift
if h(x) = f(x) ! g(x)
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT-Discrete
The (finite) Discrete FT can be derived from the continuous FT(with assumptions).
F (p!k)' () *Complex
=n#1$
q=#n
f(q!x)' () *Complex
e#i2"pq
2n
for p = !n, . . . , n ! 1
f(q!x)' () *Complex
=1
2n
n#1$
p=#n
F (p!k)' () *Complex
ei2"pq
2n
for q = !n, . . . , n ! 1 .
There are some assumptions here.The constituent frequencies do not change in time.The constituent frequencies are # 1/(2!x).
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT-Discrete
The DFT can be represented as
+
,f1...
fn
-
. = "̄
+
,y1...
yn
-
.
n $ 1 n $ n n $ 1
Complex Complex Complex (Real)
(fR + ifI) =/"̄R + i "̄I
0(yR + iyI)
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT-Discrete
The Complex-Valued (Discrete) Fourier Transform (n=256, TR=2s)
32 64 96 128 160 192 224 2569
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6
10.8
11
32 64 96 128 160 192 224 256−3
−2
−1
0
1
2
3
10 " cos 0/512 Hz 3 " sin 8/512 Hz
32 64 96 128 160 192 224 256−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
32 64 96 128 160 192 224 256−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
sin 32/512 Hz cos 4/512 Hz
32 64 96 128 160 192 224 256
6
7
8
9
10
11
12
13
14
Sum
No noise added!
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT-Discrete
The Complex-Valued (Discrete) Fourier Transform (n=256, TR=2s)
("̄R +i "̄I) *(yR +i yI) = (fR fI)
32
64
96
128
160
192
224
256 + i
32
64
96
128
160
192
224
256
Represent complex-valued time series as an image.
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT-Discrete
The Complex-Valued (Discrete) Fourier Transform (n=256, TR=2s)
("̄R +i "̄I) *(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
Pre-multiply by complex-valued forward Fourier Matrix as an image.
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT-Discrete
The Complex-Valued (Discrete) Fourier Transform (n=256, TR=2s)
("̄R +i "̄I) *(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
There are lines at the frequency locations.Real part (image) represents constituent cosine frequencies.Imaginary part (image) represents constituent sine frequencies.The intensity of the lines represents amplitude of that frequency.
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STAT692, Wisconsin Rowe, MCW
One Dimensional FT-Discrete
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 0.1875
0
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: 0Hz, .0078Hz Sines: .0156Hz, .0625Hz
32 64 96 128 160 192 224 2569
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6
10.8
11
32 64 96 128 160 192 224 256−3
−2
−1
0
1
2
3
32 64 96 128 160 192 224 256−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
32 64 96 128 160 192 224 256−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
10 ∗ cos 0/512 Hz 3 ∗ sin 8/512 Hz sin 32/512 Hz cos 4/512 Hz
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STAT692, Wisconsin Rowe, MCW
Two Dimensional FT-Discrete
24 48 72 96
24
48
72
9624 48 72 96
24
48
72
96
Cos Cos
24 48 72 96
24
48
72
9624 48 72 96
24
48
72
96
Sin Cos
24 48 72 96
24
48
72
96
1= cos + cos + sin + cos
FOV=192 mm, mat=96$96, vox=2 mm3
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STAT692, Wisconsin Rowe, MCW
Two Dimensional FT-Discrete
The Complex-Valued 2D (Discrete) Fourier Transform("̄yR + i"̄yI) * (YR + iYI) * ("̄xR + i"̄xI)
T = (FR + iFI)
24 48 72 96
24
48
72
96
+ i
32 64 96 128
32
64
96
128
FOV=192 mm, mat=96$96, vox=2 mm3
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STAT692, Wisconsin Rowe, MCW
Two Dimensional FT-Discrete
The Complex-Valued 2D (Discrete) Fourier Transform("̄yR + i"̄yI) * (YR + iYI) * ("̄xR + i"̄xI)
T = (FR + iFI)
32 64 96 128
32
64
96
12824 48 72 96
24
48
72
9632 64 96 128
32
64
96
128
+ i * + i * + i =
32 64 96 128
32
64
96
12832 64 96 128
32
64
96
12832 64 96 128
32
64
96
128
FOV=192 mm, mat=96$96, vox=2 mm3
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STAT692, Wisconsin Rowe, MCW
Two Dimensional FT-Discrete
The Complex-Valued 2D (Discrete) Fourier Transform("̄yR + i"̄yI) * (YR + iYI) * ("̄xR + i"̄xI)T = (FR + iFI)
32 64 96 128
32
64
96
12824 48 72 96
24
48
72
9632 64 96 128
32
64
96
128−.25 −.17 0 .17 .25
.25
.17
0
−.17
−.25
+ i * + i * + i = + i
32 64 96 128
32
64
96
12832 64 96 128
32
64
96
12832 64 96 128
32
64
96
128−.25 −.17 0 .17 .25
.25
.17
0
−.17
−.25
FOV=192 mm, mat=96$96, vox=2 mm3
![Page 29: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/29.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional FT-Discrete
−.25 −.17 0 .17 .25
.25
.17
0
−.17
−.25−.25 −.17 0 .17 .25
.25
.17
0
−.17
−.25
Real k-space (Cosines) Imaginary k-space (Sines)
24 48 72 96
24
48
72
9624 48 72 96
24
48
72
9624 48 72 96
24
48
72
9624 48 72 96
24
48
72
96
cos cos sin cos
![Page 30: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/30.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional FT-Discrete
−.25 −.17 0 .17 .25
.25
.17
0
−.17
−.25−.25 −.17 0 .17 .25
.25
.17
0
−.17
−.25
Real k-space (Cosines) Imaginary k-space (Sines)
Note: Rotate bottom half (complex conjugate) up to get top!Hermetian symmetry (property).
![Page 31: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/31.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Formation
So why do we need Fourier Transforms?
![Page 32: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/32.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Formation
So why do we need Fourier Transforms?
In MRI/fMRI our measurements are not voxel values!
![Page 33: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/33.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Formation
So why do we need Fourier Transforms?
In MRI/fMRI our measurements are not voxel values!
Our measurements are spatial frequencies!
−.25 −.17 0 .17 .25
.25
.17
0
−.17
−.25−.25 −.17 0 .17 .25
.25
.17
0
−.17
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Real k-space (Cosines) Imaginary k-space (Sines)
![Page 34: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/34.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Formation
How do we get spatial frequencies?
![Page 35: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/35.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Formation
How do we get spatial frequencies?
We apply Gx & Gy magnetic field gradients to encodethen we measure the complex-valued DFT of the object.
![Page 36: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/36.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Formation
How do we get spatial frequencies?
We apply Gx & Gy magnetic field gradients to encodethen we measure the complex-valued DFT of the object.
Images are formed (Reconstructed) by a 2D IFT
![Page 37: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/37.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Formation
(a) Gradient Echo-EPI Pulse Sequence
−4 −3 −2 −1 0 1 2 3
−4
−3
−2
−1
0
1
2
3
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.
![Page 38: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/38.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Formation
F (kx, ky) = FR(kx, ky) + iFI(kx, ky), the complex-valued DFT of object
(a) real: 96 × 96 (b) imaginary: 96 × 96
FOV=192 mm, mat=96$96, vox=2 mm3
![Page 39: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/39.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Formation
complex-valued 2D IFT("yR + i"yI) * (FR + iFI) * ("xR + i"xI)T = (YR + iYI)
+ i * + i * + i = + i
FOV=192 mm, mat=96$96, vox=2 mm3
![Page 40: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/40.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Formation
Due to the imperfect Fourier encoding, the IFT reconstructedobject is complex-valued, Y (x, y) = YR(x, y) + iYI(x, y).
(a) Real image, yR (b) Imaginary image, yI
FOV=192 mm, mat=96$96, vox=2 mm3
![Page 41: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/41.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Formation
Most fMRI studies transform from real-imaginary rectangular coordinatesto magnitude-phase polar coordinates, $(x, y) = m(x, y)ei$(x,y).
(a) Magnitude, m =!
y2R + y2
I(b) Phase, ! = atan4(yI/yR)
FOV=192 mm, mat=96$96, vox=2 mm3
![Page 42: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/42.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Formation
Most fMRI studies transform from real-imaginary rectangular coordinatesto magnitude-phase polar coordinates, $(x, y) = m(x, y)ei$(x,y).
(a) Magnitude, m =!
y2R + y2
I(b) Phase, ! = atan4(yI/yR)
![Page 43: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/43.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Processing
There are two basic image (filtering) processing categories.
1) Image smoothing
2) Image sharpening
Both can be performed with the DFT.
![Page 44: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/44.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Processing
For image processing, we first define a kernel (AKA mask).A 3 $ 3 kernel with weights denoted by w’s.
w1 w2 w3w4 w5 w6w7 w8 w9
We take this kernel and move it around the image.
A new image is made by summing the product of the kernel weightswith the pixel intensity values under the kernel.
The kernel weights typically sum to unity.
![Page 45: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/45.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Processing
u(x, y)1/16 1/8 1/161/8 1/4 1/81/16 1/8 1/16
Y (x, y)· · · · · · · · ·· · · · · · · · ·· · · · · · · · ·· · · 202 198 207 · · ·qy · · 195 186 201 · · ·· · · 211 189 208 · · ·· · · · · · · · ·· · · · · · · · ·· · · · qx · · · ·
Make new image Z with value at (qx, qy) that is
Z(qx, qy) = 1/16 " 202 + 1/8 " 198 + ... + 1/8 " 189 + 1/16 " 208
![Page 46: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/46.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Processing
The previous procedure of moving the kernel around andmaking new voxel values is the definition of convolution!
Z(qx, qy) =m$
s=#m
n$
r=#n
Y (qx ! r, qy ! s)u(r, s)
where n and m are the x and y dimensions of Y .u is zero padded to be of the same dimension as Y .
Image filtering (Smoothing) is computationally faster in freq space.
![Page 47: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/47.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Processing-Smoothing
Image Real Kernel Real Image FT Real Kernel FT Real
Image Imag. Kernel Imag. Image FT Imag. Kernel FT Imag.
kernel=
.0000 .0000 .0000 .0001 .0001 .0001 .0000 .0000 .0000
.0000 .0000 .0004 .0017 .0026 .0017 .0004 .0000 .0000
.0000 .0004 .0041 .0154 .0239 .0154 .0041 .0004 .0000
.0001 .0017 .0154 .0581 .0906 .0581 .0154 .0017 .0001
.0001 .0026 .0239 .0906 .1412 .0906 .0239 .0026 .0001
.0001 .0017 .0154 .0581 .0906 .0581 .0154 .0017 .0001
.0000 .0004 .0041 .0154 .0239 .0154 .0041 .0004 .0000
.0000 .0000 .0004 .0017 .0026 .0017 .0004 .0000 .0000
.0000 .0000 .0000 .0001 .0001 .0001 .0000 .0000 .0000
![Page 48: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/48.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Processing-Smoothing
Image FT Real Kernel FT Real Prod. FT Real IFT Prod. Real
Image FT Imag. Kernel FT Imag. Prod. FT Imag. IFT Prod. Imag.
![Page 49: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/49.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Processing-Smoothing
Original Image Smoothed Image
FOV=192 mm, mat=96$96, vox=2 mm3
![Page 50: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/50.jpg)
STAT692, Wisconsin Rowe, MCW
Two Dimensional MR Image Processing-Smoothing
Smothing Caveats:1) Increases/Induces local voxel correlation
512 1024 1536 2048 2560 3072 3584 4096
512
1024
1536
2048
2560
3072
3584
4096 −1
−0.8
−0.6
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0
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1
512 1024 1536 2048 2560 3072 3584 4096
512
1024
1536
2048
2560
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4096 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
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1
Original Image Corr Smoothed Image Corr2) t-statistics need to be renormalized
K =21
wj under independence
![Page 51: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/51.jpg)
STAT692, Wisconsin Rowe, MCW
One Dimensional fMRI Time Series
In fMRI we get complex-valued images over timeand voxel time course observations, yt = yRt + iyIt.
![Page 52: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/52.jpg)
STAT692, Wisconsin Rowe, MCW
One Dimensional fMRI Time Series
Collect a sequence of these reconstructed images over time.Form voxel time courses, yt = rte
i$t.
![Page 53: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/53.jpg)
STAT692, Wisconsin Rowe, MCW
One Dimensional fMRI Time Series
Collect a sequence of these reconstructed images over time.Form voxel time courses, yt = rte
i$t.
![Page 54: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/54.jpg)
STAT692, Wisconsin Rowe, MCW
One Dimensional fMRI Time SeriesTime series are complex-valued or bivariate with phase coupled means.
0
0.2
0.4
0.6
0.8
1
1.2
00.5
1
0326496128160192224256
Imaginary
Realtime
yR
t
yI
Magnitude
r
Phase φ
03264961281601922242560
0.2
0.4
0.6
0.8
1
1.2
1.4
time
Real
Real: Task related changes!
0 32 64 96 128 160 192 224 2560
0.2
0.4
0.6
0.8
1
time
Imag
inary
Imaginary: Task related changes!
The yR and yI time courses have related vector length info!This is a time series from a actual human experimental data!
![Page 55: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/55.jpg)
STAT692, Wisconsin Rowe, MCW
One Dimensional fMRI Time SeriesTime series are complex-valued or bivariate with phase coupled means.
0
0.2
0.4
0.6
0.8
1
1.2
00.5
1
0326496128160192224256
Imaginary
Realtime
yR
t
yI
Magnitude
r
Phase φ
0 32 64 96 128 160 192 224 2561.5
1.55
1.6
1.65
1.7
1.75
time
Magn
itude
Magnitude: Task related magnitude changes!
32 64 96 128 160 192 224 256−3
−2
−1
0
1
2
3
time
Phase
Phase: Often relatively constant temporally.
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!
![Page 56: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/56.jpg)
STAT692, Wisconsin Rowe, MCW
One Dimensional fMRI Time SeriesTime series are complex-valued or bivariate with phase coupled means.
0
0.2
0.4
0.6
0.8
1
1.2
00.5
1
0326496128160192224256
Imaginary
Realtime
yR
t
yI
Magnitude
r
Phase φ
0 32 64 96 128 160 192 224 2561.5
1.55
1.6
1.65
1.7
1.75
time
Magn
itude
Magnitude: Task related magnitude changes!
32 64 96 128 160 192 224 256
0.6
0.62
0.64
0.66
0.68
0.7
0.72
0.74
0.76
0.78
time
Phase
Phase: Task related phase changes!
Real-Imaginary or Magnitude-Phase time courses have all info!Recent work indicates that phase time courses may exhibit TRPCsMenon, 2002; Hoogenrad et al., 1998; Borduka et al., 1999; Chow et al., 2006;
![Page 57: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/57.jpg)
STAT692, Wisconsin Rowe, MCW
One Dimensional fMRI Time Series
Block-designed experiment: O!-On-O!-...-On-O! task
! Real
"
Imaginary
#############$
#########$#####$
! Real
"
Imaginary
#########$
%%%%%%%%%%%%%&
'''''''''(
! Real
"
Imaginary
#########$
%%%%%%%%%&
))
)))*
➣ Complex Magnitude w/ Constant Phase (CP) Activation1,2
➣ Complex Magnitude &/or Phase (CM) Activation3
➣ Real Magnitude-Only (MO/UP) Activation4,5
➣ Real Phase-Only (PO) Activation6
1Rowe and Logan: NeuroImage, 23:1078-1092, 2004. 2Rowe: NeuroImage 25:1124-1132, 2005a.3Rowe: NeuroImage, 25:1310-1324, 2005b. 4Bandettini et al.: Magn Reson Med, 30:161-173, 1993.5Friston et al.: Hum Brain Mapp, 2:189-210, 1995. 6Rowe, Meller, & Ho!mann: J Neuro Meth, in press, 2006.
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STAT692, Wisconsin Rowe, MCW
One Dimensional fMRI Time Series
Let’s consider the magnitude of the time series and its FT.
32 64 96 128 160 192 224 256
1.5
1.55
1.6
1.65
1.7
1.75
−0.4963 −0.3773 −0.2584 −0.1394 −0.0204 0.0985 0.2175 0.3364 0.4554−10
−5
0
5
10
15
20
−0.4963 −0.3773 −0.2584 −0.1394 −0.0204 0.0985 0.2175 0.3364 0.45540
2
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14
16
18
20
TS Mag. TS FT real TS FT Mag.
32 64 96 128 160 192 224 256−1
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−0.4963 −0.3773 −0.2584 −0.1394 −0.0204 0.0985 0.2175 0.3364 0.4554−10
−5
0
5
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15
20
−0.4963 −0.3773 −0.2584 −0.1394 −0.0204 0.0985 0.2175 0.3364 0.4554−3
−2
−1
0
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TS Phase TS FT Imag TS FT Phase
![Page 59: Daniel B. Ro w e, Ph.D. db ro w e@mc w.edu Depa rtment of ...pages.stat.wisc.edu/~mchung/teaching/MIA/lectures/... · Place volunt eer/pat ient int o MRI scanner. MC W GE 3T Long](https://reader033.fdocuments.net/reader033/viewer/2022050323/5f7d755364fdab153e09c5c7/html5/thumbnails/59.jpg)
STAT692, Wisconsin Rowe, MCW
One Dimensional fMRI Time Series
Let’s filter some FT frequencies of the time series.
−0.4963 −0.3773 −0.2584 −0.1394 −0.0204 0.0985 0.2175 0.3364 0.45540
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TS FT Mag. TS FT Mag. filt
−0.4963 −0.3773 −0.2584 −0.1394 −0.0204 0.0985 0.2175 0.3364 0.4554−3
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TS FT Phase TS FT Phase101,102,110,118,152,160,168,169
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STAT692, Wisconsin Rowe, MCW
One Dimensional fMRI Time Series
Let’s filter some FT frequencies of the time series.
−0.4963 −0.3773 −0.2584 −0.1394 −0.0204 0.0985 0.2175 0.3364 0.45540
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−0.4963 −0.3773 −0.2584 −0.1394 −0.0204 0.0985 0.2175 0.3364 0.4554−10
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TS FT Mag. TS FT real TS Mag.
−0.4963 −0.3773 −0.2584 −0.1394 −0.0204 0.0985 0.2175 0.3364 0.4554−3
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TS FT Phase TS FT Imag TS Phase101,102,110,118,152,160,168,169
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STAT692, Wisconsin Rowe, MCW
One Dimensional fMRI Time Series
32 64 96 128 160 192 224 256
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Original Time Series Filtered Time Series
This filtering will reduce your residual variance!A smaller variance means larger activation statistics!But you have changed the temporal autocorrelation!
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STAT692, Wisconsin Rowe, MCW
FMRI time series statistics
Activation statistic (measure of association) is computed in every voxel.
Many ways to compute activation statistics. Magnitude vs. Complex
Activation is another topic all to itself! Happy to return.
Need to separate activation signal from noise!
Thresholding and the multiple comparisons problem.
Thresholding is another topic all to itself!Unthresh Activation
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PCE Activation
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FDR Activation
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FWE Activation
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STAT692, Wisconsin Rowe, MCW
Summary
➣ Introduction
➣ One Dimensional FT• Time series constituents and Fourier spectrum.
➣ Two Dimensional FT• An image constituents and Fourier spectrum.
➣ Two Dimensional MR Image Formation• k-space and MR Image Reconstruction.
➣ One Dimensional fMRI time series• fMRI time series Fourier spectrum and filtering.
➣ FMRI time series statistics
Thank You.