2D FT Review

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. n2 n 1 2D Sampling/Discrete-Space Signal

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

5. n2 n 1 Line Impulse

6. n 1 2D step n2

7. n2 n 1 2D step

8. n 1 2D Step n2

9. n 1 Separable Sequences n2

10. n2 n 1 Periodic Sequences

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

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

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

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

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

16. (2) (1) (4) (3) 2D Convolution h(n1-k1,n2-k2) y(n1,n2) k2 n2 (7) (7) (4) (3) (4) n2 (6) (10) (6) (4) 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. z B1(t) y x Example

24. z B1(t) y x Example

25. Complex Voltage/Signal: General Case

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

27. X X Quadrature Conversion in MRI (and Ultrasound) Signal Processing Received Radio Frequency Echo Signal x(t) (fc = 10MHz; 40MS/s) LPF xc(t) I—Channel 2 cos wct -p/2 Phase Shift -2 sin wct Q—Channel xs(t) 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. r Spatial Encoding Gradients z B(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. r Spatial Encoding Gradients z B(r) y x

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

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

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

42. Frequency Encoding

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

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

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

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

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

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

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

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