mri data processing and reconstruction via chirp z-transform
DESCRIPTION
This is the slide show for my defense of the graduation dissertation.TRANSCRIPT
MRI Data Processing
and Reconstruction via
Chirp z-Transform
Author: Huiming Dong
Supervisor: Shouliang Qi, Ph. D
Introduction
m-SPRITE Imaging Sequence [Balcom et al.]Derive from SPI sequences family [Emid et al.]
Pure phase encoding
Acquire multiple FID points after each RF excitation (See Fig. 1)
Particularly useful in fast-relaxation nuclei imaging
Fig. 1 m-SPRITE Imaging Sequence
Department of Biomedical Engineering, Northeastern University
Introduction
m-SPRITE Imaging Sequence [Balcom et al.]Derive from SPI sequences family [Emid et al.]
Pure phase encoding
Acquire multiple FID points after each RF excitation (See Fig. 1)
Particularly useful in fast-relaxation nuclei imaging
Fig. 1 m-SPRITE Imaging Sequence
Department of Biomedical Engineering, Northeastern University
Introduction
m-SPRITE Imaging Sequence [Balcom et al.]Derive from SPI sequences family [Emid et al.]
Pure phase encoding
Acquire multiple FID points after each RF excitation (See Fig. 1)
Particularly useful in fast-relaxation nuclei imaging
Fig. 1 m-SPRITE Imaging Sequence
Department of Biomedical Engineering, Northeastern University
Introduction
m-SPRITE Imaging Sequence [Balcom et al.]Derive from SPI sequences family [Emid et al.]
Pure phase encoding
Acquire multiple FID points after each RF excitation (See Fig. 1)
Particularly useful in fast-relaxation nuclei imaging
Fig. 1 m-SPRITE Imaging Sequence
Department of Biomedical Engineering, Northeastern University
Introduction
m-SPRITE Imaging Sequence [Balcom et al.]Derive from SPI sequences family [Emid et al.]
Pure phase encoding
Acquire multiple FID points after each RF excitation (See Fig. 1)
Particularly useful in fast-relaxation nuclei imaging
Fig. 1 m-SPRITE Imaging Sequence
Department of Biomedical Engineering, Northeastern University
K-Space Data Acquired Utilizing m-SRITE TechniqueSampled in a non-uniform pattern (See Fig. 2)
Challenge the conventional FFT reconstruction methods
The k-space can be separated into Nt different uniformly sampled k-spaces
Each k-space per se has a different FOV size
Reconstruct respectively gives a low SNR
Introduction
Fig. 2 Non-Uniformly Sampled Data
Department of Biomedical Engineering, Northeastern University
K-Space Data Acquired Utilizing m-SRITE TechniqueSampled in a non-uniform pattern (See Fig. 2)
Challenge the conventional FFT reconstruction methods
The k-space can be separated into Nt different uniformly sampled k-spaces
Each k-space per se has a different FOV size
Reconstruct respectively gives a low SNR
Introduction
Fig. 2 Non-Uniformly Sampled Data
Department of Biomedical Engineering, Northeastern University
K-Space Data Acquired Utilizing m-SRITE TechniqueSampled in a non-uniform pattern (See Fig. 2)
Challenge the conventional FFT reconstruction methods
The k-space can be separated into Nt different uniformly sampled k-spaces
Each k-space per se has a different FOV size
Reconstruct respectively gives a low SNR
Introduction
Fig. 2 Non-Uniformly Sampled Data
Department of Biomedical Engineering, Northeastern University
K-Space Data Acquired Utilizing m-SRITE TechniqueSampled in a non-uniform pattern (See Fig. 2)
Challenge the conventional FFT reconstruction methods
The k-space can be separated into Nt different uniformly sampled k-spaces
Each k-space per se has a different FOV size
Reconstruct respectively gives a low SNR
Introduction
Fig. 2 Non-Uniformly Sampled Data
Department of Biomedical Engineering, Northeastern University
Chirp z-Transform (CZT) [Rabiner et al.]A generalization of DFT
Evaluate signals on arbitrary contour on the z-plane (See Fig. 3)
The length of resultant signal can be set to any value for different practical applications
Computational complexity: Klog2K
1
0
)()(N
n
nkznxkX k
k AWz
0
0
jeAA
0
0
jeWW
Fig. 3 Unit Circle on the z-Plane
Introduction
Department of Biomedical Engineering, Northeastern University
Chirp z-Transform (CZT) [Rabiner et al.]A generalization of DFT
Evaluate signals on arbitrary contour on the z-plane (See Fig. 3)
The length of resultant signal can be set to any value for different practical applications
Computational complexity: Klog2K
1
0
)()(N
n
nkznxkX k
k AWz
0
0
jeAA
0
0
jeWW
Fig. 3 Unit Circle on the z-Plane
Introduction
Department of Biomedical Engineering, Northeastern University
Chirp z-Transform (CZT) [Rabiner et al.]A generalization of DFT
Evaluate signals on arbitrary contour on the z-plane (See Fig. 3)
The length of resultant signal can be set to any value for different practical applications
Computational complexity: Klog2K
1
0
)()(N
n
nkznxkX k
k AWz
0
0
jeAA
0
0
jeWW
Fig. 3 Unit Circle on the z-Plane
Introduction
Department of Biomedical Engineering, Northeastern University
Chirp z-Transform (CZT) [Rabiner et al.]A generalization of DFT
Evaluate signals on arbitrary contour on the z-plane (See Fig. 3)
The length of resultant signal can be set to any value for different practical applications
Computational complexity: Klog2K
1
0
)()(N
n
nkznxkX k
k AWz
0
0
jeAA
0
0
jeWW
Fig. 3 Unit Circle on the z-Plane
Introduction
Department of Biomedical Engineering, Northeastern University
Chirp z-Transform (CZT) [Rabiner et al.]A generalization of DFT
Evaluate signals on arbitrary contour on the z-plane (See Fig. 3)
The length of resultant signal can be set to any value for different practical applications
Computational complexity: Klog2K
1
0
)()(N
n
nkznxkX k
k AWz
0
0
jeAA
0
0
jeWW
Fig. 3 Unit Circle on the z-Plane
Introduction
Department of Biomedical Engineering, Northeastern University
FOV ScalingDFT of a signal evaluates a signal on the whole unit circle on the z-plane
CZT can evaluate the signal on a part of the unit circle (See Fig. 4)
Method
max
)1(
t
t
FOV
NFOV
FOV
FOVT act
act
T
act
des
10 A )1()(
2
120 T
FOV
FOVFOV
act
desact
10 WNc
TNc
FOV
FOV
act
des 2/)2(0
Fig. 4 Evaluating Contour
Department of Biomedical Engineering, Northeastern University
FOV ScalingDFT of a signal evaluates a signal on the whole unit circle on the z-plane
CZT can evaluate the signal on a part of the unit circle (See Fig. 4)
Method
max
)1(
t
t
FOV
NFOV
FOV
FOVT act
act
T
act
des
10 A )1()(
2
120 T
FOV
FOVFOV
act
desact
10 WNc
TNc
FOV
FOV
act
des 2/)2(0
Fig. 4 Evaluating Contour
Department of Biomedical Engineering, Northeastern University
FOV ScalingDFT of a signal evaluates a signal on the whole unit circle on the z-plane
CZT can evaluate the signal on a part of the unit circle (See Fig. 4)
Method
max
)1(
t
t
FOV
NFOV
FOV
FOVT act
act
T
act
des
10 A )1()(
2
120 T
FOV
FOVFOV
act
desact
10 WNc
TNc
FOV
FOV
act
des 2/)2(0
Fig. 4 Evaluating Contour
Department of Biomedical Engineering, Northeastern University
DFT Reconstruction for m-SPRITE MRI DataNumerous complex computation
Cannot be efficiently implemented by FFT algorithms
Require revised FFT or interpolation methods for reconstruction
Method
1
0
1
0
))()((T GN
u
N
v
j
pm eutvGsx ))(
)()(
)((maxmaxmax t
ut
G
vG
x
xN
pmG
2/)(/)()(/)(2 uTNjNumTjuvTjNuvmTj GTC
)()12
)(2
1()
)()(1
2)(
2
1(
max
uTN
v
N
mN
t
ut
N
v
N
mN
GC
G
p
GC
G
dxexks xkj 2)( mn
cxjk
N
n
nm eksx 2
1
0
)(
Department of Biomedical Engineering, Northeastern University
DFT Reconstruction for m-SPRITE MRI DataNumerous complex computation
Cannot be efficiently implemented by FFT algorithms
Require revised FFT or interpolation methods for reconstruction
Method
1
0
1
0
))()((T GN
u
N
v
j
pm eutvGsx ))(
)()(
)((maxmaxmax t
ut
G
vG
x
xN
pmG
2/)(/)()(/)(2 uTNjNumTjuvTjNuvmTj GTC
)()12
)(2
1()
)()(1
2)(
2
1(
max
uTN
v
N
mN
t
ut
N
v
N
mN
GC
G
p
GC
G
dxexks xkj 2)( mn
cxjk
N
n
nm eksx 2
1
0
)(
Department of Biomedical Engineering, Northeastern University
DFT Reconstruction for m-SPRITE MRI DataNumerous complex computation
Cannot be efficiently implemented by FFT algorithms
Require revised FFT or interpolation methods for reconstruction
Method
1
0
1
0
))()((T GN
u
N
v
j
pm eutvGsx ))(
)()(
)((maxmaxmax t
ut
G
vG
x
xN
pmG
2/)(/)()(/)(2 uTNjNumTjuvTjNuvmTj GTC
)()12
)(2
1()
)()(1
2)(
2
1(
max
uTN
v
N
mN
t
ut
N
v
N
mN
GC
G
p
GC
G
dxexks xkj 2)( mn
cxjk
N
n
nm eksx 2
1
0
)(
Department of Biomedical Engineering, Northeastern University
DFT Reconstruction for m-SPRITE MRI DataNumerous complex computation
Cannot be efficiently implemented by FFT algorithms
Require revised FFT or interpolation methods for reconstruction
Method
1
0
1
0
))()((T GN
u
N
v
j
pm eutvGsx ))(
)()(
)((maxmaxmax t
ut
G
vG
x
xN
pmG
2/)(/)()(/)(2 uTNjNumTjuvTjNuvmTj GTC
)()12
)(2
1()
)()(1
2)(
2
1(
max
uTN
v
N
mN
t
ut
N
v
N
mN
GC
G
p
GC
G
dxexks xkj 2)( mn
cxjk
N
n
nm eksx 2
1
0
)(
Department of Biomedical Engineering, Northeastern University
CZT Reconstruction Method for m-SPRITE MRI DataSeparate one non-uniform k-space into Nt uniformly sampled k-space
Reconstruct each k-space obtained in last step and scale FOVs via CZT simultaneously
Sum all results together (i.e., signal averaging)
Spatial resolution improvement
SNR improvement
Method
c
GNuvmTjuvTj
N
vpp eeutvGsuTutvGsCZTu
/)(2)(1
0
))()(()](),()(([)(Image
1
0
/)(2)(1
0
1
0
))()(()(ImageeSingleImagT
c
GT N
u
NuvmTjuvTjN
v
p
N
u
eeutvGsu
High similarity can be found, except the phase angle
Department of Biomedical Engineering, Northeastern University
CZT Reconstruction Method for m-SPRITE MRI DataSeparate one non-uniform k-space into Nt uniformly sampled k-space
Reconstruct each k-space obtained in last step and scale FOVs via CZT simultaneously
Sum all results together (i.e., signal averaging)
Spatial resolution improvement
SNR improvement
Method
c
GNuvmTjuvTj
N
vpp eeutvGsuTutvGsCZTu
/)(2)(1
0
))()(()](),()(([)(Image
1
0
/)(2)(1
0
1
0
))()(()(ImageeSingleImagT
c
GT N
u
NuvmTjuvTjN
v
p
N
u
eeutvGsu
High similarity can be found, except the phase angle
Department of Biomedical Engineering, Northeastern University
CZT Reconstruction Method for m-SPRITE MRI DataSeparate one non-uniform k-space into Nt uniformly sampled k-space
Reconstruct each k-space obtained in last step and scale FOVs via CZT simultaneously
Sum all results together (i.e., signal averaging)
Spatial resolution improvement
SNR improvement
Method
c
GNuvmTjuvTj
N
vpp eeutvGsuTutvGsCZTu
/)(2)(1
0
))()(()](),()(([)(Image
1
0
/)(2)(1
0
1
0
))()(()(ImageeSingleImagT
c
GT N
u
NuvmTjuvTjN
v
p
N
u
eeutvGsu
High similarity can be found, except the phase angle
Department of Biomedical Engineering, Northeastern University
CZT Reconstruction Method for m-SPRITE MRI DataSeparate one non-uniform k-space into Nt uniformly sampled k-space
Reconstruct each k-space obtained in last step and scale FOVs via CZT simultaneously
Sum all results together (i.e., signal averaging)
Spatial resolution improvement
SNR improvement
Method
c
GNuvmTjuvTj
N
vpp eeutvGsuTutvGsCZTu
/)(2)(1
0
))()(()](),()(([)(Image
1
0
/)(2)(1
0
1
0
))()(()(ImageeSingleImagT
c
GT N
u
NuvmTjuvTjN
v
p
N
u
eeutvGsu
High similarity can be found, except the phase angle
Department of Biomedical Engineering, Northeastern University
CZT Reconstruction Method for m-SPRITE MRI DataSeparate one non-uniform k-space into Nt uniformly sampled k-space
Reconstruct each k-space obtained in last step and scale FOVs via CZT simultaneously
Phase Correction
Sum all results together (i.e., signal averaging)
Spatial resolution improvement
SNR improvement
Method
1
0
/)(2)(1
0
1
0
))()(()(ImageeSingleImagT
c
GT N
u
NuvmTjuvTjN
v
p
N
u
eeutvGsu
1
0
)()](),()(([TN
u
mjpm euTutvGsCZTx
)(1
0
/)(2)(1
0
))()((
mjN
u
NuvmTjuvTjN
vp eeeutvGs
T
c
G
TN
uT )(
2
)(uTNG
Department of Biomedical Engineering, Northeastern University
CZT Reconstruction Method for m-SPRITE MRI DataSeparate one non-uniform k-space into Nt uniformly sampled k-space
Reconstruct each k-space obtained in last step and scale FOVs via CZT simultaneously
Phase Correction
Sum all results together (i.e., signal averaging)
Spatial resolution improvement
SNR improvement
Method
1
0
/)(2)(1
0
1
0
))()(()(ImageeSingleImagT
c
GT N
u
NuvmTjuvTjN
v
p
N
u
eeutvGsu
1
0
)()](),()(([TN
u
mjpm euTutvGsCZTx
)(1
0
/)(2)(1
0
))()((
mjN
u
NuvmTjuvTjN
vp eeeutvGs
T
c
G
TN
uT )(
2
)(uTNG
Department of Biomedical Engineering, Northeastern University
Image Scaling through CZTThe length of resultant signal can be set to any value for different practical applications
Can be implemented by simply set the parameter K in accordance with the scaling factor
Method
1
0
)()(N
n
nkznxkX k
k AWz
0
0
jeAA
0
0
jeWW
Fig. 3 Unit Circle on the z-Plane
Department of Biomedical Engineering, Northeastern University
Image Scaling through CZTThe length of resultant signal can be set to any value for different practical applications
Can be implemented by simply set the parameter K in accordance with the scaling factor
Method
1
0
)()(N
n
nkznxkX k
k AWz
0
0
jeAA
0
0
jeWW
Fig. 3 Unit Circle on the z-Plane
Department of Biomedical Engineering, Northeastern University
Image Scaling through CZTThe length of resultant signal can be set to any value for different practical applications
Can be implemented by simply set the parameter K in accordance with the scaling factor
Method
1
0
)()(N
n
nkznxkX k
k AWz
0
0
jeAA
0
0
jeWW
Fig. 3 Unit Circle on the z-Plane
Department of Biomedical Engineering, Northeastern University
Experiments and ParametersOriginal MRI data courtesy of James Rioux, University of New Brunswick, Canada
A fiber-reinforced polyester resin
Nt=25, Ng=64
FOV scaling
m-SPRITE data reconstruction
Image scaling
Result
Department of Biomedical Engineering, Northeastern University
Experiments and ParametersOriginal MRI data courtesy of James Rioux, University of New Brunswick, Canada
A fiber-reinforced polyester resin
Nt=25, Ng=64
FOV scaling
m-SPRITE data reconstruction
Image scaling
Result
Department of Biomedical Engineering, Northeastern University
Experiments and ParametersOriginal MRI data courtesy of James Rioux, University of New Brunswick, Canada
A fiber-reinforced polyester resin
Nt=25, Ng=64
FOV scaling
m-SPRITE data reconstruction
Image scaling
Result
Department of Biomedical Engineering, Northeastern University
FOV Scaling (CZT Versus Bilinear Interpolation)NcNclog2Nc
Scaling factor 0.89
Better spatial resolution and accuracy (See Fig. 5)
Result
Department of Biomedical Engineering, Northeastern University
Fig. 5 FOV Scaling by bilinear interpolation and CZT
FOV Scaling (CZT Versus Bilinear Interpolation)NcNclog2Nc
Scaling factor 0.89
Better spatial resolution and accuracy (See Fig. 5)
Result
Department of Biomedical Engineering, Northeastern University
Fig. 5 FOV Scaling by bilinear interpolation and CZT
FOV Scaling (CZT Versus Bilinear Interpolation)NcNclog2Nc
Scaling factor 0.89
Better spatial resolution and accuracy (See Fig. 5)
Result
Fig. 5 FOV Scaling by bilinear interpolation and CZT
Department of Biomedical Engineering, Northeastern University
FOV Scaling (CZT Versus Bilinear Interpolation)NcNclog2Nc
Scaling factor 0.89
Better spatial resolution and accuracy (See Fig. 5)
Result
Fig. 5 FOV Scaling by bilinear interpolation and CZT
Department of Biomedical Engineering, Northeastern University
m-SPRITE MRI Data ReconstructionHigher SNR (See Fig. 6)
Better apparent (spatial) resolution
Higher accuracy and less computational complexity
Result
Department of Biomedical Engineering, Northeastern University
Fig. 6 Reconstruction Results 1
m-SPRITE MRI Data ReconstructionHigher SNR (See Fig. 6)
Better apparent (spatial) resolution
Higher accuracy and less computational complexity
Result
Department of Biomedical Engineering, Northeastern University
Fig. 6 Reconstruction Results 1
m-SPRITE MRI Data ReconstructionHigher SNR
Better apparent (spatial) resolution (See Fig. 7)
Higher accuracy and less computational complexity
Result
Department of Biomedical Engineering, Northeastern University
Fig. 7 Reconstruction Results 2 [Rioux et al.]
m-SPRITE MRI Data Reconstruction (CZT Versus DRS Method)Higher SNR
Better apparent (spatial) resolution
Higher accuracy and less computational complexity (See Fig. 8)
Dutt, Rokhlin and Sarty method [Dutt et al. and Sarty et al.]
Result
Department of Biomedical Engineering, Northeastern University
Nt SNR
1 10.9049
4 16.2190
9 21.7201
12 25.3492
16 28.2785
25 33.3223
Table I Table of SNR
Fig. 8 Running Time and Accuracy
[Rioux et al.]
Image Scaling (Bilinear Interpolation vs FFT Zero Filling vs CZT)Non-integer scaling factor
No significant advantages (See Fig. 9)
Result
Department of Biomedical Engineering, Northeastern University
Standard Interpolation CZTFFT Zero
Filling
d 0.2451 0.4361 0.2519
r 0.0606 0.1674 0.1128
Table II Table of Rescaled Image Quality
Fig. 9 Rescaled Images
Image Scaling (Bilinear Interpolation vs FFT Zero Filling vs CZT)Non-integer scaling factor
No significant advantages (See Fig. 9)
Result
Department of Biomedical Engineering, Northeastern University
Standard Interpolation CZTFFT Zero
Filling
d 0.2451 0.4361 0.2519
r 0.0606 0.1674 0.1128
Table II Table of Rescaled Image Quality
Fig. 9 Rescaled Images
Image Scaling (Bilinear Interpolation vs FFT Zero Filling vs CZT)Non-integer scaling factor
No significant advantages (See Fig. 9)
Result
Department of Biomedical Engineering, Northeastern University
Standard Interpolation CZTFFT Zero
Filling
d 0.2451 0.4361 0.2519
r 0.0606 0.1674 0.1128
Table II Table of Rescaled Image Quality
Fig. 9 Rescaled Images
Accuracy
Spatial resolution
SNR
Computational level
To be discovered
Conclusion
Department of Biomedical Engineering, Northeastern University
Accuracy
Spatial resolution
SNR
Computational level
To be discovered
Conclusion
Department of Biomedical Engineering, Northeastern University
1
0
/)(2)(1
0
1
0
))()(()(ImageeSingleImagT
c
GT N
u
NuvmTjuvTjN
v
p
N
u
eeutvGsu
1
0
)()](),()(([TN
u
mjpm euTutvGsCZTx
)(1
0
/)(2)(1
0
))()((
mjN
u
NuvmTjuvTjN
vp eeeutvGs
T
c
G
TN
uT )(
2
)(uTNG
Accuracy
Spatial resolution
SNR
Computational level
To be discovered
Conclusion
Department of Biomedical Engineering, Northeastern University
Fig. 7 Reconstruction Results 2 [Rioux et al.]
Accuracy
Spatial resolution
SNR
Computational level
To be discovered
Conclusion
Department of Biomedical Engineering, Northeastern University
Fig. 6 Reconstruction Results 1
Accuracy
Spatial resolution
SNR
Computational level
To be discovered
Conclusion
Department of Biomedical Engineering, Northeastern University
Fig. 8 Running Time and Accuracy [Rioux et al.]
Accuracy
Spatial resolution
SNR
Computational level
To be discovered
Conclusion
Department of Biomedical Engineering, Northeastern University
This study only shines very limited lights on the scenery of CZT applications on MRI research and a
majestic panorama of its applications is expected to be discovered unremittingly.