1. 2D FT Review
MP/BME 574

2. 1D to 2D Sampling Signal under analysis is periodic
Signal is ‘essentially bandlimited’
Sampling rate is high enough to satisfy Nyquist criterion
Other assumptions (for convenience)
Signal is sampled with uniformly spaced intervals

3. 2D Sampling/Discrete-Space Signal1 n n2

4. 2D Functions Impulses Step Sequences Separable Sequences
Periodic Sequences

5. Line Impulse 1 n n2

6. 2D step n2 1 n

7. 2D step 1 n n2

8. 2D Step n2 1 n

9. Separable Sequences n2 1 n

10. Periodic Sequences 1 n n2

11. 2D Convolution
h(k1,k2)
x(k1,k2) k1 k2
(3)
(4)
(1)
(2) k2 k1

12. 2D Convolution
h(-k1,-k2) k1 k2
x(k1,k2) k2
(2)
(1)
(4)
(3) k1

13. 2D Convolution h(4-k1,3-k2) x(k1,k2) k1 k2 k2 (2) (1) (4) (3) 3-k2 k1

14. 2D Convolution h(n1-k1,n2-k2) x(k1,k2) k1 k2 k2 (2) (1) (4) (3) (2)

15. 2D Convolution h(n1-k1,n2-k2) y(n1,n2) k2 n2 (2) (1) (4) (3) (7) (7)
(6)
(4)
(6) n1
(2) k1
(1)
(3)
(3)
n1-1

16. 2D Convolution h(n1-k1,n2-k2) y(n1,n2) k2 n2 (2) (1) (4) (3) (7) (7)
(6)
(10)
(4)
(6) n1
(2) k1
(1)
(3)
(3)
n1-1

17. 2DFT Imaging in MRI
MP/BME 574

18. Abbe’s Theory of Image Formation
From Meyer-Arendt

19. No Magnetic
Field =
No Net
Magnetization
Random
Orientation

20. Dipole Moments from Entire Sample
Magnetic
Field (B0)
Magnetic
Field (B0) m m
Positive
Orientation
Negative
Orientation

21. Precession

22. Precession and Electromotive Force (emf) or Voltage
emf derives from Faraday’s law
Time-dependent magnetic flux through a coil of wire
Induces current flow
Proportional to the magnetic field strength and the frequency of the field oscillation

23. Example x y z
B1(t)

24. Example x y z
B1(t)

25. Complex Voltage/Signal: General Case

26. rf-excitation By reciprocity, Lab Frame Rotating Frame
After Haacke, 1999

27. Quadrature Conversion in MRI (and Ultrasound) Signal Processing
Received
Radio Frequency
Echo Signal x(t)
(fc = 10MHz; 40MS/s) X
LPF
xc(t)
I—Channel
2 cos wct
-p/2 Phase Shift
-2 sin wct
Q—Channel
xs(t) X
LPF
In a high-end ultrasound/MR imaging system this conversion is done in the digital domain.
In a lower-end system the conversion is done in the analog domain.
Why?

28. Spatial Encoding

29. Slice Selection
Ideal, non-selective rf:
S(t) =rect(t/Dt) B1ideal(t)

30. Non-selective rf-pulse
Entire Volume Excited

31. FTdemo: Rect modulated Cosine

32. FTdemo: Rect modulated Cosine

33. Spatial Encoding Gradientsz
B(r) r y x

34. Slice Selection Selective rf:
Ssel(t) = sinc(t/t) rect(t/Dt) B1ideal(t)
Apply spatial gradient simultaneous to rf-pulse.

35. Slice Selective rf-pulse
Slice of width Dz Excited

36. FTdemo: Cosine modulated Sinc

37. Summary Spin ½ nuclei will precess in a magnetic field Bo
Excite and receive signal with coils (antennae) by Faraday’s Law
Complex representation of real signals
Quadrature detection
Reciprocity
Spatial magnetic field gradients
Bandwidth of precessing “spins”
Non-Selective rf pulses using Fourier transform principles
Shift theorem etc… applies

38. Spatial Encoding Gradientsz
B(r) r y x

39. Frequency Encoding
f, B Df
B=Bo
xmin
xmax
FOVx

40. Frequency Encoding … … Recall Lab 2, Problem 4: Piano Keyboard
E, 660 Hz
A, 220 Hz
Middle C

41. Frequency Encoding Time (t) FT Temporal Frequency (f) Position (x)
Proportionality
Temporal Frequency (f)
Position (x)

42. Frequency Encoding

43. Frequency Encoding Spatial Frequency (k) Time (t) FT
Temporal Frequency (f) FT
Proportionality
Position (x)
Spatial Frequency (k)

44. Phase Encoding y
f, B Df
B=Bo
xmin
xmax
FOVx

45. Phase Encoding y
f, B Df
B=Bo
xmin
xmax
FOVx

46. Phase Encoding y
f, B Df
B=Bo
xmin
xmax
FOVx

47. Zero gradient for time, T
Phase Encoding y B
Zero gradient for time, T y

48. Positive gradient for time, T
Phase Encoding y B
Positive gradient for time, T y

49. Positive gradient for time, T
Phase Encoding y B
Positive gradient for time, T y

50. Frequency Encoding Spatial Frequency (k-ko) Time (t) FT FT
Proportionality FT FT
Proportionality
Temporal Frequency (f)
Position (x) e-igGyT

51. 2D Fast GRE Imaging Gx Gy Gz RF TR = 6.6 msec TE Phase Encode
Dephasing/
Rewinder
Asymmetric Readout Gy
Dephasing/ Rewinder Gz
Shinnar-
LaRoux RF RF
TR = 6.6 msec

52. 2D FT y x k n
Start
Finish
The elliptical centric view order is centric in both phase encode directions. Views are sampled based on their distance from the center of k-space. This leads to a radially symmetric distribution of contrast enhancement about the center of k-space.

53. 3D Fast GRE Imaging Gx Gy Gz RF TR = 6.6 msec TE Phase Encode
Dephasing/
Rewinder
Asymmetric Readout Gy
Dephasing/ Rewinder
Phase Encode Gz
Shinnar-
LaRoux RF RF
TR = 6.6 msec

54. 3D FT k z y x Tscan =Ny Nz TR NEX i.e. Time consuming! n
The elliptical centric view order is centric in both phase encode directions. Views are sampled based on their distance from the center of k-space. This leads to a radially symmetric distribution of contrast enhancement about the center of k-space.

55. Summary Frequency encoding Phase Entirely separable
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

56. Navigating in 2D k-Space
Goals
Improve your intuition
Specific examples
Effects of:
Apodization windowing
“Zero-Padding” or Sinc interpolation
Vendors refer to this as “ZIP”
Sampling the corners of k-Space

57. Elliptical Centric View Orderk z
High Detail
Information k y
Overall Image
Contrast
The elliptical centric view order is centric in both phase encode directions. Views are sampled based on their distance from the center of k-space. This leads to a radially symmetric distribution of contrast enhancement about the center of k-space.
Sampled
Points

58. MRI: Image AcquisitionFT FT
K-space
Image
space

59. Case I
Case II
Case III ky kz

60. Case I k-space: Image Space: kz ky DFT
Bernstein MA, Fain SB, and Riederer SJ, JMRI 14: (2001)

61. Case II
k-space: Image Space: kz ky FT

62. Case III
k-space: Image Space: kz ky FT

63. a b

64. 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

65. Recovering or “Restoring” f(x) from f(n):

66. Recovering or “restoring” f(x) from f(n):Dx

67. Recovering or “Restoring” f(x) from f(n):

68. Recovering or “restoring” f(x) from f(n):Dx

69. Recovering or “restoring” f(x) from f(n):
f(n’) where Dx

70. Case I k-space: Image Space: kz ky DFT
Bernstein MA, Fain SB, and Riederer SJ, JMRI 14: (2001)

71. Methods: Sampling
Case I: Zero-filled
k-space: Image Space: kz ky FT

72. Case II
k-space: Image Space: kz ky FT

73. Case II
k-space: Image Space: kz ky FT

74. Methods: Sampling
Case III
k-space: Image Space: kz ky FT

75. Methods: Sampling
Case III: Zero-Filled
k-space: Image Space: kz ky FT