1. Lecture 4 MR: 2D Projection Reconstruction, 2D FT Longitudinal Magnetization returns to equilibrium as MR Review Transverse Magnetization Gradients’ effect on B-field B-field’s effect on frequency

2. MR Review (2) Signal Equation Signal equation related to k-space The MR signal is always telling us a point of the frequency domain expression for M, the Fourier Transform of m(x,y), the proton density of the image. The mapping between s(t) and the points in k-space is determined by the gradient waveforms.

3. Timing Diagrams G x : Readout gradient G y : Phase-encoding gradient G z : Slice-encoding (or slice-selective) gradient RF tGxGx time over which data acquisition occurs t0t0 t3t3 t1t1 t2t2 t 90° S(t) constant gradient t Receiver signal x y Object is a square box of water

4. Timing Diagrams: Time related to position in k-space GxGx t kxkx t3t3 t3t3 t1t1 t1t1 t0t0 t0t0 t2t2 t2t2 x y t 90° RF Signal Object is a square box of water Signal from t 1 to t 3 is the F.T. of the projection at angle 0, formed by the line integrals along y

5. Timing Diagrams: Time related to position in k-space GxGx t kxkx t3t3 t3t3 t1t1 t1t1 t0t0 t0t0 t2t2 t2t2 x y t 90° RF Signal Object is a square box of water Signal from t 1 to t 3 is the F.T. of the projection at angle 0, formed by the line integrals along y (Slide repeated without animation)

6. What gradient(s) are playing? Can you determine the G x (t) and G y (t) waveforms? 9PHFRENC.AVI

7. Timing Diagrams: Time related to position in k-space (2) At t 1,, we are at this point in k-space. tGyGy t1t1 kxkx kyky RF 90° GxGx

8. 2D Projection Reconstruction (2D PR): Single-sided t RF 90° tGxGx tGyGy tDAQ Data Acquisition kxkx kyky  G cos(  ) G sin(  ) Repeat at various  Called single side measurement.  is considered in radians/G here. 4257 Hz/G often used also where

9. 2D Projection Reconstruction (2D PR):Double-sided kxkx  Called double sided measurement. t tGxGx RF tGyGy tDAQ Called double-sided measurement. Reconstruction: convolution back projection or filtered back projection kyky

10. Object Domain In MR, S(t) gives a radial line in k-space. Central Section Theorem x’ y’ x y F.T. Interesting - Time signal gives spatial frequency information of m(x,y)

11. 2D Fourier Transform (4) – 2 sided By far, the 2 sided 2D FT is the most popular. tGxGx tGyGy Readout or frequency-encode gradient (stays the same) Phase-encode gradient (varies) # of steps: 128-512 In practice, I(x,y) is complex-valued. Displayed image is |I(x,y)|, not Re{I(x,y)} Theoretically: is a real image Practically: has a phase due to imperfect, inhomogenous B0 field

12. 2D Fourier Transform 2D Fourier Transform: (2D FT or Spin Warp) 1-Sided 1) t tGxGx RF tGyGy tDAQ kxkx kyky

13. 2D Fourier Transform (2) 2D Fourier Transform: (2D FT or Spin Warp) 1) t tGxGx RF tGyGy kxkx kyky Reconstruction: 2D FFT

14. 2D Fourier Transform (3) Let’s revisit the object domain - “Modified” Central Slice Theorem gives a projection of x y kxkx kyky

15. Review: Phase Encoding kyky kxkx Consider the 64 x 8 box to the right. A series of MR experiments as described above were performed. To simplify visualization, a 1D FFT was done on each experiment. The results are shown on the bottom where each row is a separate experiment with a different Y direction phase weighting.

16. 2D Fourier Transform (2) 2D Fourier Transform: (2D FT or Spin Warp) 1) t G x (t) RF 6GRADECH.AVI