Lecture 4 MR: 2D Projection Reconstruction, 2D FT

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

2. MR Review (2) Signal equation related to k-space Signal Equation 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. 90° t y x t Timing Diagrams RF time over which data acquisition occurs Gx: Readout gradient Gy: Phase-encoding gradient Gz: Slice-encoding (or slice-selective) gradient constant gradient Gx t t0 t1 t2 t3 Object is a square box of water S(t) Receiver signal

4. 90° t t0 t1 t2 t3 y t1 t0 t3 x t2 kx Timing Diagrams: Time related to position in k-space RF Gx t Object is a square box of water Signal from t1 to t3 is the F.T. of the projection at angle 0, formed by the line integrals along y Signal

5. 90° t t0 t1 t2 t3 y t1 t0 t3 x t2 kx Timing Diagrams: Time related to position in k-space RF Gx t Object is a square box of water Signal from t1 to t3 is the F.T. of the projection at angle 0, formed by the line integrals along y Signal (Slide repeated without animation)

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

7. Timing Diagrams: Time related to position in k-space (2) 90° RF At t1, , we are at this point in k-space. Gy t t1 Gx ky kx

8. 90° 2D Projection Reconstruction (2D PR): Single-sided t RF where Gcos() Gx t Repeat at various  Gsin() Gy t DAQ t Data Acquisition  is considered in radians/G here. 4257 Hz/G often used also ky  Called single side measurement. kx

9.  2D Projection Reconstruction (2D PR):Double-sided Called double sided measurement. ky RF t Reconstruction: convolution back projection or filtered back projection kx Gx t Gy t Called double-sided measurement. DAQ t

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

11. 2D Fourier Transform (4) – 2 sided Readout or frequency-encode gradient (stays the same) Gx t By far, the 2 sided 2D FT is the most popular. Phase-encode gradient (varies) Gy t # 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) RF t ky Gx t kx Gy t DAQ t

13. 2D Fourier Transform (2) 2D Fourier Transform: (2D FT or Spin Warp) 1) RF t Gx t ky kx Gy t Reconstruction: 2D FFT

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

15. Review: Phase Encoding 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. ky kx

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