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Transcript of 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality...
![Page 1: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/1.jpg)
2D FT Imaging
MP/BME 574
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Frequency Encoding
T
yy
yy
dssGtk
generalIn
TGk
0
)()(
,
Time (t)
Temporal Frequency (f)
FT
Proportionality
Position (x, or y)
FT
Proportionality
Spatial Frequency (k)
![Page 3: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/3.jpg)
2D Fast GRE Imaging
Gy
RF
Gx
TE
Dephasing/ Rewinder
Dephasing/ Rewinder
Shinnar-LaRoux RF
Phase Encode
Asymmetric Readout
Gz
TR = 6.6 msec
![Page 4: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/4.jpg)
Summary
• Frequency encoding– Bandwidth of precessing frequencies
• Phase– Incremental phase in image space
• Implies shift in k-space
• Entirely separable– 1D column-wise FFT– 1D row-wise FFT
![Page 5: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/5.jpg)
2D FT
y
xk
k
Start
Finish
22
,
)()(
)()(
0
0
yy
y
T
yy
t
xx
Nn
Nwhere
kynTG
dssGtk
dssGtk
![Page 6: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/6.jpg)
3D FT
y
z
k
k
kx
Tscan =Ny Nz TR NEX
i.e. Time consuming!
![Page 7: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/7.jpg)
Zero-padding/Sinc Interpolation
• Recall that the sampling theorem – Restoration of a compactly supported (band-
limited) function– Equivalent to convolution of the sampled
points with a sinc function
![Page 8: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/8.jpg)
Case II
FT
k-space: Image Space:
kz
ky
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Case III
FT
k-space: Image Space:
Methods: Sampling
kz
ky
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Case II Nyquist Case III Corner
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Case II: Zero-filled
FT
k-space: Image Space:kz
ky
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kz
ky
Case III: Zero-Filled
FT
k-space: Image Space:
Methods: Sampling
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Case II: Nyquist Zero-filled Case III: Corner Zero-filled
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Apodization
• Rect windowing implies covolution with a truncated sinc function leading to Gibbs’ Ringing
• Desire to smooth the windowing function so as to diminish ringing.– Gaussian is one option discussed by Prof.
Holden– MRI often uses “Fermi” Filter:
;a)./beta))-)exp((abs(x+1./(1 = f
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)()(),(1
1)(
2121 kHkHkkHe
kH
sep
k
),(),(),(
)(),(
212121
21 22
21
kkFkkHkkG
kHkkHkkkrradialradial
r
![Page 16: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/16.jpg)
Point Spread Functions
Un-windowed: Radial Window:
),( 21 nnhsep ),( 21 nnhradial
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Ref Corners Radial
),(),(),( 212121 nnfnnhnng sep ),(),(),( 212121 nnfnnhnng radial ),( 21 nnf
![Page 18: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/18.jpg)
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
-5 15 35
Angle (degrees)
Re
solu
tio
n (
mm
/lp
)
Not windowed
Windowed
Cosine reference
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
-5 15 35
Angle (degrees)
Re
solu
tio
n (
mm
/lp
)
Not Windowed
Windowed
Cosine reference
Angular Dependence w/o Zero-filling
)cos(
)cos(max
max
xres
x
res
k
k
r
x
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Angular Dependence with Zero-filling
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
-5 15 35
Angle (degrees)
Res
olut
ion
(mm
/lp)
Not windowed
Windowed
Cosine reference
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Experimental Results
= 45º= 45º
0 Degrees
45 Degrees
Methods: Point response function
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Summary
• Samples in 2D k-space represent 2D sinusoids at specific harmonics and at specific rotation angles
• Interpolation by zero-filling leads to:– Reduced partial volume artifact– Increased spatial resolution at specific angles
• Role of Apodization window– Increases SNR – Decreases ringing artifact– Choice effects the angular symmetry of the PSF
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Point response function due to time-dependent contrast
• Example showing mapping on contrast-enhanced signal to model the point response function– Predict attainable resolution – Application to carotid artery MR angiography
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Fain SB, Bernstein MA, Huston J III, Riederer SJ
Point Spread Function (PSF) Analysis
• Step 1: Measure enhancement curves in patients
• Step 2: Map enhancement curves to k-space
• Step 3: Transform result to image space to obtain the point
spread function
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Fain SB, et al., MRM 42 (1999)
Step 1: Enhancement Model
Fitted Two Phase Gamma VariateC
on
tras
t E
nh
ance
men
t
Time (sec)0 10 20 30 40 50 60 70 80
0
20
40
60
80
100
120
140
160
Composite FitFirst Pass Fit
Residual FitMeasured Data
/)( knekttb
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Fain SB, et al., MRM 42 (1999)
Start
y
z
k
Finish
Overall ImageContrast
High DetailInformation
SampledPoints
k
Step 2: Mapping to k-Space
)(2 tkkk
TRt
zy
![Page 26: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/26.jpg)
Fain SB, et al., MRM 42 (1999)
Step 2: Mapping to k-Space
Spatial Frequency (cycles/mm)
Co
ntr
ast
En
han
ce
me
nt
(se
c)-1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.01
0.02
0.03
0.04
0.05
0.06
k-Space Weighting
Composite k-SpaceFirst Pass Only
Residual
Measured Data
![Page 27: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/27.jpg)
Fain SB, et al., MRM 42 (1999)
The Hankel Transform
TR
kkM
tkkk
TRt
ekttb
zy
zy
kn
)(
)(
2
/
![Page 28: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/28.jpg)
Fain SB, et al., MRM 42 (1999)
Step 3: Transform to Image Space
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.01
0
0.01
0.02
0.03
0.04
0.05
Analytical PSF for Fitted Curve
PS
F A
mp
litu
de
(mm
-sec
2 )-1
Radius (mm)
Composite PSF
First Pass PSF
Residual PSF
Image Contrast
Spatial Resolution
![Page 29: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/29.jpg)
Fain SB, et al., MRM 42 (1999)
Analysis: Spatial Resolution
FWHM 2FOV y FOV z TR
1
Full Width at Half Maximum (FWHM) of the Point Spread Function is given by:
where,FOVy and FOVz are the phase encoding Fields of ViewTR is the repetition time1 is the time to peak enhancement of the bolus curve
![Page 30: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/30.jpg)
Fain SB, et al., MRM 42 (1999)
PSF Dependence on Acquisition Time
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Radius (mm)
PS
F a
mp
litu
de
(mm
2 -sec
)-1
Dependence of PSF on Acquisition Time
Infinite ScanTacq = 230 secTacq = 90 secTacq = 50 secTacq = 10 sec
Tacq = Acquisition Time in seconds
10
50
90
230 sec, Maximum Spatial Resolution
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Fain SB, et al., MRM 42 (1999)
213 sec1.2 1.6
2.0
2.6 3.2
4.2
5.2
Z
Y
10 sec50 sec
90 secLine Pairs/mm
Acquisition Time (sec)
PSF Dependence on Acquisition Time
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Fain SB, et al., MRM 42 (1999)
Experiment: FOVz Reduction
13 cm X 6.4 cm
13 cm X 4.0 cm
Z
Y
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Fain SB, et al., MRM 42 (1999)
Carotid and Vertebral Arteries: Acquisition Parameters
– FOV: 22 cm (S/I) X 15 cm (R/L) X 6 cm (A/P)– Matrix: 256 X 168 X 40-44– Acquired Voxel: 0.9 mm X 0.9 mm X 1.4 mm
– 2X Zerofilling in all three directions– TR/TE 6.6 msec/1.4 msec– Acquisition Time: 44-51 seconds– 20 cc Gd
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Fain SB, et al., MRM 42 (1999)
Left Carotid Artery Stenosis: Reconstruction at Multiple Time Points
33 sec22 sec11 sec 44 secAcquisition Time:
X
Z
X
Z
Coronal MIP, Full Data Set:
MIP Reprojec-
tions
![Page 35: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/35.jpg)
Fain SB, et al., MRM 42 (1999)
Right Carotid Artery Stenosis: Reconstruction at Multiple Time Points
11 sec 22 sec 33 sec 44 secAcquisition Time:
X
Z
X
Z
![Page 36: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/36.jpg)
Fain SB, et al., MRM 42 (1999)
Decreased FOV
1.0 mm 1.2 mm 1.6 mm 2.0 mm 2.6 mm
15 cm X 6.0 cm
20 cm X 6.0 cm
FOVy
13 cm X 6.4 cm
13 cm X 4.0 cm
FOVz
Z
Y
Z
Y
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Fain SB, et al., MRM 42 (1999)
Increased Scan Time
Z
Y
10 sec50 sec
90 sec213 sec1.2
1.6
2.0
2.6
3.2
4.2
5.2
![Page 38: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/38.jpg)
Partial k-Space Acquisition
• Means of accelerating image acquisition at the expense of minor artifacts– ¾ k-space– ½ k-space -> Hermetian symmetry
• Phase in the image space complicates matters– In practice, MR images have non-zero phase due to
magnetic field variations• Susceptibility• General field inhomogeneity
– “Homodyne” reconstruction required • Low spatial frequency estimation of the phase
![Page 39: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/39.jpg)
2D FT
y
xk
k
Start
Finish
22
,
)()(
)()(
0
0
yy
y
T
yy
t
xx
Nn
Nwhere
kynTG
dssGtk
dssGtk
![Page 40: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/40.jpg)
2D FT
y
xk
k
Start
Finish
22
,
)()(
)()(
0
0
yy
y
T
yy
t
xx
Nn
Nwhere
kynTG
dssGtk
dssGtk
2
N
4
N- where yy n
![Page 41: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/41.jpg)
FI = fftshift(fft(fftshift(I)));for i = 1:192,FI_34(i,:) =FI(i,:);endI_34 = fftshift(ifft(fftshift(FI_34)));figure;subplot(2,2,1),imagesc(abs(I_34));axis('image');colorbar;colormap('gray');title('Magnitude')subplot(2,2,2),imagesc(angle(I_34));axis('image');colorbar;colormap('gray');title('Phase')subplot(2,2,3),imagesc(abs(I-I_34));axis('image');colorbar;colormap('gray');title('Error')gtext('Three-quarter k-space')
![Page 42: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/42.jpg)
2D FT
y
xk
k
Start
Finish
22
,
)()(
)()(
0
0
yy
y
T
yy
t
xx
Nn
Nwhere
kynTG
dssGtk
dssGtk
2
N0 where yn
![Page 43: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/43.jpg)
for i = 1:129,FI_Herm(i,:) =FI(i,:);endI_Herm = fftshift(ifft(fftshift(FI_Herm)));figure;subplot(2,2,1),imagesc(abs(I_Herm2));axis('image');colorbar;colormap('gray');title('Magnitude')figure;subplot(2,2,1),imagesc(abs(I_Herm));axis('image');colorbar;colormap('gray');title('Magnitude')subplot(2,2,2),imagesc(angle(I_Herm));axis('image');colorbar;colormap('gray');title('Phase')subplot(2,2,3),imagesc(abs(I-I_Herm));axis('image');colorbar;colormap('gray');title('Error')gtext('One-half k-space')
![Page 44: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/44.jpg)
2D FT
y
xk
k
Start
Finish
22
,
)()(
)()(
0
0
yy
y
T
yy
t
xx
Nn
Nwhere
kynTG
dssGtk
dssGtk
2
N0
;12
N- where
y
''
y
n
n
![Page 45: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/45.jpg)
count2 = 128;for i = 130:256,FI_Herm(i,:) =conj(FI(count2,:));count2 = count2-1;endI_Herm2 = fftshift(ifft(fftshift(FI_Herm)));figure;subplot(2,2,1),imagesc(abs(I_Herm2));axis('image');colorbar;colormap('gray');title('Magnitude')save phase_phantomsubplot(2,2,2),imagesc(angle(I_Herm2));axis('image');colorbar;colormap('gray');title('Phase')subplot(2,2,3),imagesc(abs(I-I_Herm2));axis('image');colorbar;colormap('gray');title('Error')gtext('Hermetian k-space')
![Page 46: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/46.jpg)
![Page 47: 2D FT Imaging MP/BME 574. Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency.](https://reader034.fdocuments.us/reader034/viewer/2022051017/56649d9f5503460f94a8a96f/html5/thumbnails/47.jpg)
FIp = fftshift(fft(fftshift(IIII)));FIp_Herm = zeros(256);for i = 1:129,FIp_Herm(i,:) =FIp(i,:);endcount2 = 128;for i = 130:256,FIp_Herm(i,:) =conj(FIp(count2,:));count2 = count2-1;endIp_Herm = fftshift(ifft(fftshift(FIp_Herm)));figure;subplot(2,2,1),imagesc(abs(Ip_Herm));axis('image');colorbar;colormap('gray');title('Magnitude')subplot(2,2,2),imagesc(angle(Ip_Herm));axis('image');colorbar;colormap('gray');title('Phase')subplot(2,2,3),imagesc(abs(I-Ip_Herm));axis('image');colorbar;colormap('gray');title('Error')gtext('Attempt at Hermetian k-space for Image with Phase')