2D FT Imaging MP/BME 574
Dec 23, 2015
2D FT Imaging
MP/BME 574
Frequency Encoding
T
yy
yy
dssGtk
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Time (t)
Temporal Frequency (f)
FT
Proportionality
Position (x, or y)
FT
Proportionality
Spatial Frequency (k)
2D Fast GRE Imaging
Gy
RF
Gx
TE
Dephasing/ Rewinder
Dephasing/ Rewinder
Shinnar-LaRoux RF
Phase Encode
Asymmetric Readout
Gz
TR = 6.6 msec
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
2D FT
y
xk
k
Start
Finish
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y
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kynTG
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3D FT
y
z
k
k
kx
Tscan =Ny Nz TR NEX
i.e. Time consuming!
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
Case II
FT
k-space: Image Space:
kz
ky
Case III
FT
k-space: Image Space:
Methods: Sampling
kz
ky
Case II Nyquist Case III Corner
Case II: Zero-filled
FT
k-space: Image Space:kz
ky
kz
ky
Case III: Zero-Filled
FT
k-space: Image Space:
Methods: Sampling
Case II: Nyquist Zero-filled Case III: Corner Zero-filled
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
)()(),(1
1)(
2121 kHkHkkHe
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sep
k
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)(),(
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21 22
21
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Point Spread Functions
Un-windowed: Radial Window:
),( 21 nnhsep ),( 21 nnhradial
Ref Corners Radial
),(),(),( 212121 nnfnnhnng sep ),(),(),( 212121 nnfnnhnng radial ),( 21 nnf
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
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
Experimental Results
= 45º= 45º
0 Degrees
45 Degrees
Methods: Point response function
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
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
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
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
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
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
Fain SB, et al., MRM 42 (1999)
The Hankel Transform
TR
kkM
tkkk
TRt
ekttb
zy
zy
kn
)(
)(
2
/
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
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
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
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
Fain SB, et al., MRM 42 (1999)
Experiment: FOVz Reduction
13 cm X 6.4 cm
13 cm X 4.0 cm
Z
Y
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
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
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
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
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
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
2D FT
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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')
2D FT
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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')
2D FT
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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')
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')