This is the slide show for my defense of the graduation dissertation.

1. MRI Data Processing
and Reconstruction via
Chirp z-Transform
Author: Huiming Dong
Supervisor: Shouliang Qi, Ph. D

2. 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

3. 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

4. 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

5. 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

6. 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

7. Introduction
K-Space Data Acquired Utilizing m-SRITE Technique
Sampled 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
Fig. 2 Non-Uniformly Sampled Data
Department of Biomedical Engineering, Northeastern University

8. Introduction
K-Space Data Acquired Utilizing m-SRITE Technique
Sampled 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
Fig. 2 Non-Uniformly Sampled Data
Department of Biomedical Engineering, Northeastern University

9. Introduction
K-Space Data Acquired Utilizing m-SRITE Technique
Sampled 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
Fig. 2 Non-Uniformly Sampled Data
Department of Biomedical Engineering, Northeastern University

10. Introduction
K-Space Data Acquired Utilizing m-SRITE Technique
Sampled 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
Fig. 2 Non-Uniformly Sampled Data
Department of Biomedical Engineering, Northeastern University

11. Introduction
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
N 1
n
X (k )   x ( n) z k zk  AW  k
n 0
A  A0 e j0
W  W0e  j0
Fig. 3 Unit Circle on the z-Plane
Department of Biomedical Engineering, Northeastern University

12. Introduction
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
N 1
n
X (k )   x ( n) z k zk  AW  k
n 0
A  A0 e j0
W  W0e  j0
Fig. 3 Unit Circle on the z-Plane
Department of Biomedical Engineering, Northeastern University

13. Introduction
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
N 1
n
X (k )   x ( n) z k zk  AW  k
n 0
A  A0 e j0
W  W0e  j0
Fig. 3 Unit Circle on the z-Plane
Department of Biomedical Engineering, Northeastern University

14. Introduction
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
N 1
n
X (k )   x ( n) z k zk  AW  k
n 0
A  A0 e j0
W  W0e  j0
Fig. 3 Unit Circle on the z-Plane
Department of Biomedical Engineering, Northeastern University

15. Introduction
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
N 1
n
X (k )   x ( n) z k zk  AW  k
n 0
A  A0 e j0
W  W0e  j0
Fig. 3 Unit Circle on the z-Plane
Department of Biomedical Engineering, Northeastern University

16. Method
FOV Scaling
DFT 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)
FOVdes FOV ( NT  1) t
T   act
FOVact FOVact t max
1 ( FOVact  FOVdes )
A0  1  0  2    (1  T )
2 FOVact
FOVdes T
W0  1 0  (2  ) / Nc  2
FOVact Nc
Fig. 4 Evaluating Contour
Department of Biomedical Engineering, Northeastern University

17. Method
FOV Scaling
DFT 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)
FOVdes FOV ( NT  1) t
T   act
FOVact FOVact t max
1 ( FOVact  FOVdes )
A0  1  0  2    (1  T )
2 FOVact
FOVdes T
W0  1 0  (2  ) / Nc  2
FOVact Nc
Fig. 4 Evaluating Contour
Department of Biomedical Engineering, Northeastern University

18. Method
FOV Scaling
DFT 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)
FOVdes FOV ( NT  1) t
T   act
FOVact FOVact t max
1 ( FOVact  FOVdes )
A0  1  0  2    (1  T )
2 FOVact
FOVdes T
W0  1 0  (2  ) / Nc  2
FOVact Nc
Fig. 4 Evaluating Contour
Department of Biomedical Engineering, Northeastern University

19. Method
DFT Reconstruction for m-SPRITE MRI Data
Numerous complex computation
Cannot be efficiently implemented by FFT algorithms
Require revised FFT or interpolation methods for reconstruction
N c 1
s(k )     x e j 2xk
dx   x m    s (k
n 0
n )e 2jkn xm
NT 1N G 1 xm G (v) t p (u )
  xm     s(G(v)t (u ))e  j   NG ( )( )( )
u 0 v 0
p xmax Gmax tmax
m 1 2v t p (u ) m 1 2v
  N G (  )(  1)( )  N G (  )(  1)T (u )
NC 2 NG t max NC 2 NG
  j 2vmT(u) / NC  jvT (u)  jmT (u) / NT  jNGT (u) / 2
Department of Biomedical Engineering, Northeastern University

20. Method
DFT Reconstruction for m-SPRITE MRI Data
Numerous complex computation
Cannot be efficiently implemented by FFT algorithms
Require revised FFT or interpolation methods for reconstruction
N c 1
s(k )     x e j 2xk
dx   x m    s (k
n 0
n )e 2jkn xm
NT 1N G 1 xm G (v) t p (u )
  xm     s(G(v)t (u ))e  j   NG ( )( )( )
u 0 v 0
p xmax Gmax tmax
m 1 2v t p (u ) m 1 2v
  N G (  )(  1)( )  N G (  )(  1)T (u )
NC 2 NG t max NC 2 NG
  j 2vmT(u) / NC  jvT (u)  jmT (u) / NT  jNGT (u) / 2
Department of Biomedical Engineering, Northeastern University

21. Method
DFT Reconstruction for m-SPRITE MRI Data
Numerous complex computation
Cannot be efficiently implemented by FFT algorithms
Require revised FFT or interpolation methods for reconstruction
N c 1
s(k )     x e j 2xk
dx   x m    s (k
n 0
n )e 2jkn xm
NT 1N G 1 xm G (v) t p (u )
  xm     s(G(v)t (u ))e  j   NG ( )( )( )
u 0 v 0
p xmax Gmax tmax
m 1 2v t p (u ) m 1 2v
  N G (  )(  1)( )  N G (  )(  1)T (u )
NC 2 NG t max NC 2 NG
  j 2vmT(u) / NC  jvT (u)  jmT (u) / NT  jNGT (u) / 2
Department of Biomedical Engineering, Northeastern University

22. Method
DFT Reconstruction for m-SPRITE MRI Data
Numerous complex computation
Cannot be efficiently implemented by FFT algorithms
Require revised FFT or interpolation methods for reconstruction
N c 1
s(k )     x e j 2xk
dx   x m    s (k
n 0
n )e 2jkn xm
NT 1N G 1 xm G (v) t p (u )
  xm     s(G(v)t (u ))e  j   NG ( )( )( )
u 0 v 0
p xmax Gmax tmax
m 1 2v t p (u ) m 1 2v
  N G (  )(  1)( )  N G (  )(  1)T (u )
NC 2 NG t max NC 2 NG
  j 2vmT(u) / NC  jvT (u)  jmT (u) / NT  jNGT (u) / 2
Department of Biomedical Engineering, Northeastern University

23. Method
CZT Reconstruction Method for m-SPRITE MRI Data
Separate 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
N G 1
Image (u )  CZT [ s(G(v)t p (u ), T (u )]   s(G(v)t p (u))e jvT (u )e j 2vmT (u ) / N c
v 0
NT 1 NT 1N G 1
SingleImag e   Image (u)    s(G(v)t
u 0 u 0 v 0
p (u ))e jvT (u ) e  j 2vmT (u ) / Nc
High similarity can be found, except the phase angle
Department of Biomedical Engineering, Northeastern University

24. Method
CZT Reconstruction Method for m-SPRITE MRI Data
Separate 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
N G 1
Image (u )  CZT [ s(G(v)t p (u ), T (u )]   s(G(v)t p (u))e jvT (u )e j 2vmT (u ) / N c
v 0
NT 1 NT 1N G 1
SingleImag e   Image (u)    s(G(v)t
u 0 u 0 v 0
p (u ))e jvT (u ) e  j 2vmT (u ) / Nc
High similarity can be found, except the phase angle
Department of Biomedical Engineering, Northeastern University

25. Method
CZT Reconstruction Method for m-SPRITE MRI Data
Separate 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
N G 1
Image (u )  CZT [ s(G(v)t p (u ), T (u )]   s(G(v)t p (u))e jvT (u )e j 2vmT (u ) / N c
v 0
NT 1 NT 1N G 1
SingleImag e   Image (u)    s(G(v)t
u 0 u 0 v 0
p (u ))e jvT (u ) e  j 2vmT (u ) / Nc
High similarity can be found, except the phase angle
Department of Biomedical Engineering, Northeastern University

26. Method
CZT Reconstruction Method for m-SPRITE MRI Data
Separate 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
N G 1
Image (u )  CZT [ s(G(v)t p (u ), T (u )]   s(G(v)t p (u))e jvT (u )e j 2vmT (u ) / N c
v 0
NT 1 NT 1N G 1
SingleImag e   Image (u)    s(G(v)t
u 0 u 0 v 0
p (u ))e jvT (u ) e  j 2vmT (u ) / Nc
High similarity can be found, except the phase angle
Department of Biomedical Engineering, Northeastern University

27. Method
CZT Reconstruction Method for m-SPRITE MRI Data
Separate 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
NT 1 NT 1N G 1
SingleImag e   Image (u)    s(G(v)t
u 0 u 0 v 0
p (u ))e jvT (u ) e  j 2vmT (u ) / Nc
NT 1
  xm    CZT [ s(G(v)t p (u ), T (u )]e j (m  )
u 0
NT 1 N G 1
   s(G(v)t p (u))e jvT (u ) e  j 2vmT (u ) / N e j (m  )
c
u 0 v 0
T (u )  N G T (u )
 
NT 2
Department of Biomedical Engineering, Northeastern University

28. Method
CZT Reconstruction Method for m-SPRITE MRI Data
Separate 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
NT 1 NT 1N G 1
SingleImag e   Image (u)    s(G(v)t
u 0 u 0 v 0
p (u ))e jvT (u ) e  j 2vmT (u ) / Nc
NT 1
  xm    CZT [ s(G(v)t p (u ), T (u )]e j (m  )
u 0
NT 1 N G 1
   s(G(v)t p (u))e jvT (u ) e  j 2vmT (u ) / N e j (m  )
c
u 0 v 0
T (u )  N G T (u )
 
NT 2
Department of Biomedical Engineering, Northeastern University

29. Method
Image Scaling through CZT
The 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
N 1
X (k )   x ( n) z k  n zk  AW  k
n 0
A  A0 e j0
W  W0e  j0
Fig. 3 Unit Circle on the z-Plane
Department of Biomedical Engineering, Northeastern University

30. Method
Image Scaling through CZT
The 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
N 1
X (k )   x ( n) z k  n zk  AW  k
n 0
A  A0 e j0
W  W0e  j0
Fig. 3 Unit Circle on the z-Plane
Department of Biomedical Engineering, Northeastern University

31. Method
Image Scaling through CZT
The 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
N 1
X (k )   x ( n) z k  n zk  AW  k
n 0
A  A0 e j0
W  W0e  j0
Fig. 3 Unit Circle on the z-Plane
Department of Biomedical Engineering, Northeastern University

32. Result
Experiments and Parameters
Original 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
Department of Biomedical Engineering, Northeastern University

33. Result
Experiments and Parameters
Original 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
Department of Biomedical Engineering, Northeastern University

34. Result
Experiments and Parameters
Original 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
Department of Biomedical Engineering, Northeastern University

35. Result
FOV Scaling (CZT Versus Bilinear Interpolation)
NcNclog2Nc
Scaling factor 0.89
Better spatial resolution and accuracy (See Fig. 5)
Fig. 5 FOV Scaling by bilinear interpolation and CZT
Department of Biomedical Engineering, Northeastern University

36. Result
FOV Scaling (CZT Versus Bilinear Interpolation)
NcNclog2Nc
Scaling factor 0.89
Better spatial resolution and accuracy (See Fig. 5)
Fig. 5 FOV Scaling by bilinear interpolation and CZT
Department of Biomedical Engineering, Northeastern University

37. Result
FOV Scaling (CZT Versus Bilinear Interpolation)
NcNclog2Nc
Scaling factor 0.89
Better spatial resolution and accuracy (See Fig. 5)
Fig. 5 FOV Scaling by bilinear interpolation and CZT
Department of Biomedical Engineering, Northeastern University

38. Result
FOV Scaling (CZT Versus Bilinear Interpolation)
NcNclog2Nc
Scaling factor 0.89
Better spatial resolution and accuracy (See Fig. 5)
Fig. 5 FOV Scaling by bilinear interpolation and CZT
Department of Biomedical Engineering, Northeastern University

39. Result
m-SPRITE MRI Data Reconstruction
Higher SNR (See Fig. 6)
Better apparent (spatial) resolution
Higher accuracy and less computational complexity
Fig. 6 Reconstruction Results 1
Department of Biomedical Engineering, Northeastern University

40. Result
m-SPRITE MRI Data Reconstruction
Higher SNR (See Fig. 6)
Better apparent (spatial) resolution
Higher accuracy and less computational complexity
Fig. 6 Reconstruction Results 1
Department of Biomedical Engineering, Northeastern University

41. Result
m-SPRITE MRI Data Reconstruction
Higher SNR
Better apparent (spatial) resolution (See Fig. 7)
Higher accuracy and less computational complexity
Fig. 7 Reconstruction Results 2 [Rioux et al.]
Department of Biomedical Engineering, Northeastern University

42. Result
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.]
Table I Table of SNR
Nt SNR
1 10.9049
4 16.2190
9 21.7201
12 25.3492
16 28.2785
25 33.3223
Fig. 8 Running Time and Accuracy
[Rioux et al.]
Department of Biomedical Engineering, Northeastern University

43. Result
Image Scaling (Bilinear Interpolation vs FFT Zero Filling vs CZT)
Non-integer scaling factor
No significant advantages (See Fig. 9)
Table II Table of Rescaled Image Quality
FFT Zero
Standard Interpolation CZT
Filling
d 0.2451 0.4361 0.2519
r 0.0606 0.1674 0.1128
Fig. 9 Rescaled Images
Department of Biomedical Engineering, Northeastern University

44. Result
Image Scaling (Bilinear Interpolation vs FFT Zero Filling vs CZT)
Non-integer scaling factor
No significant advantages (See Fig. 9)
Table II Table of Rescaled Image Quality
FFT Zero
Standard Interpolation CZT
Filling
d 0.2451 0.4361 0.2519
r 0.0606 0.1674 0.1128
Fig. 9 Rescaled Images
Department of Biomedical Engineering, Northeastern University

45. Result
Image Scaling (Bilinear Interpolation vs FFT Zero Filling vs CZT)
Non-integer scaling factor
No significant advantages (See Fig. 9)
Table II Table of Rescaled Image Quality
FFT Zero
Standard Interpolation CZT
Filling
d 0.2451 0.4361 0.2519
r 0.0606 0.1674 0.1128
Fig. 9 Rescaled Images
Department of Biomedical Engineering, Northeastern University

46. Conclusion
Accuracy
Spatial resolution
SNR
Computational level
To be discovered
Department of Biomedical Engineering, Northeastern University

47. Conclusion
Accuracy
Spatial resolution
SNR
Computational level
To be discovered
NT 1 NT 1N G 1
SingleImag e   Image (u)    s(G(v)t
u 0 u 0 v 0
p (u ))e jvT (u ) e  j 2vmT (u ) / Nc
NT 1
  xm    CZT [ s(G(v)t p (u ), T (u )]e j (m  )
u 0
NT 1 N G 1
   s(G(v)t p (u))e jvT (u ) e  j 2vmT (u ) / N e j (m  )
c
u 0 v 0
T (u )  N G T (u )
 
NT 2
Department of Biomedical Engineering, Northeastern University

48. Conclusion
Accuracy
Spatial resolution
SNR
Computational level
To be discovered
Fig. 7 Reconstruction Results 2 [Rioux et al.]
Department of Biomedical Engineering, Northeastern University

49. Conclusion
Accuracy
Spatial resolution
SNR
Computational level
To be discovered
Fig. 6 Reconstruction Results 1
Department of Biomedical Engineering, Northeastern University

50. Conclusion
Accuracy
Spatial resolution
SNR
Computational level
To be discovered
Fig. 8 Running Time and Accuracy [Rioux et al.]
Department of Biomedical Engineering, Northeastern University

51. Conclusion
Accuracy
Spatial resolution
SNR
Computational level
To be discovered
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.
Department of Biomedical Engineering, Northeastern University