1. Psy 8960, Fall ‘06 Fourier transforms1 –1D: square wave –2D: k x and k y Spatial encoding with gradients Common artifacts Phase map of pineapple slice under read-out gradient, with phase-encode --  0

2. Psy 8960, Fall ‘06 Fourier transforms2 Key terms K-space (with units of inverse length) Phase encode vs. read-out direction (k-space axes) –Read-out = frequency encode FLASH vs. EPI (types of pulse sequences, loosely speaking)

3. Psy 8960, Fall ‘06 Fourier transforms3 Fourier composition of square wave (v2)

4. Psy 8960, Fall ‘06 Fourier transforms4 Fourier (de)composition of a square wave Fundamental frequency: Fundamental + 1 st harmonic: Fundamental + 2 harmonics: Fundamental + 3 harmonics:

5. Psy 8960, Fall ‘06 Fourier transforms5 Fourier (de)composition of a square wave 16s

6. Psy 8960, Fall ‘06 Fourier transforms6 2D Fourier transform

7. Psy 8960, Fall ‘06 Fourier transforms7 original imagefiltered with gaussian filterfiltered with hard filter

8. Psy 8960, Fall ‘06 Fourier transforms8 Fourier relationships Big step size in one domain = small FOV in the other Large extent (FOV) in one domain = small step size in the other Multiplication in one domain = convolution in the other Symmetry in one domain = no imaginary part in the other

9. Psy 8960, Fall ‘06 Fourier transforms9 FLASH.m accidentally simulates an off-center k-space

10. Psy 8960, Fall ‘06 Fourier transforms10 Good, clean k-space

11. Psy 8960, Fall ‘06 Fourier transforms11 K-space with small error near center

12. Psy 8960, Fall ‘06 Fourier transforms12 K-space with small error farther out

13. Psy 8960, Fall ‘06 Fourier transforms13 K-space with spike

14. Psy 8960, Fall ‘06 Fourier transforms14 Pulse sequence diagram: 2D FLASH N rep = resolution, phase- encode direction N = res., read-out RF G SS G PE G RO DAC PE table increments each repetition Flip angle ~ 7 deg. TE ~ 5ms K-space data

15. Psy 8960, Fall ‘06 Fourier transforms15 Gradient echo Immediately after excitation, all the spins in a sample are in phase When a gradient is applied, the spins begin to pick up a phase difference The phase depends on both space and time (and gradient strength) t = 0  st = 20  st = 160  s G = 12mT/m B x G = 5.1kHz/cm f x --  0

16. Psy 8960, Fall ‘06 Fourier transforms16 Gradient echo Applying a gradient in the opposite direction reverses this process t = 160  st = 300  st = 320  s G = -12mT/m B x --  0

17. Psy 8960, Fall ‘06 Fourier transforms17 Applying a gradient produces a periodic spin phase pattern G RO --  0 Real part of signal in RF coil Magnitude of signal in RF coil Imaginary component of signal in RF coil Movies aren’t linked. Similar movies can be generated by uncommenting the “imagesc …” line in FLASH.m, or the originals can be found at http://vision.psych.umn.edu/~caol man/courses/Spring2006/Lecture s/Lecture7.zip http://vision.psych.umn.edu/~caol man/courses/Spring2006/Lecture s/Lecture7

18. Psy 8960, Fall ‘06 Fourier transforms18 The read-out signal is the 1D FFT of the sample G RO --  0 Real part of signal in RF coil Magnitude of signal in RF coil Imaginary component of signal in RF coil

19. Psy 8960, Fall ‘06 Fourier transforms19 Applying simultaneous gradients rotates the coordinate system G RO G PE --  0

20. Psy 8960, Fall ‘06 Fourier transforms20 Phase encoding allows independent spatial frequency encoding on 2 axes G PE G RO PE gradient imposes phase pattern on one axis Read "refocusing" gradient rewinds phase pattern on another axis Read gradient creates phase evolution while one line of k-space is acquired PE RO --  0

21. Psy 8960, Fall ‘06 Fourier transforms21 Phase encoding allows independent spatial frequency encoding on 2 axes G PE G RO PE gradient imposes phase pattern on one axis Read "refocusing" gradient rewinds phase pattern on another axis Read gradient creates phase evolution while one line of k-space is acquired PE RO --  0

22. Psy 8960, Fall ‘06 Fourier transforms22 Navigating k-space N rep = resolution, phase- encode direction N = res., read-out RF G SS G PE G RO DAC PE table increments each repetition Flip angle ~ 7 deg. TE ~ 5ms K-space data

23. Psy 8960, Fall ‘06 Fourier transforms23 Pulse sequence diagrams: FLASH & EPI N = res., read-out RF G SS G PE G RO DAC PE table increments each repetition Flip angle ~ 7 deg. TE ~ 5ms N rep = 32 Flip angle ~ 60 deg. TE ~ 30ms N rep = 64 64 pts

24. Psy 8960, Fall ‘06 Fourier transforms24 K-space trajectories: FLASH & EPI FLASH (TE ~ 5 ms)EPI (TE ~ 30 ms) excitation Read and phase pre-encode (refocusing) Read-out Read and phase pre-encode (refocusing) excitation Phase blip TE: time between excitation and acquisition of DC data point

25. Psy 8960, Fall ‘06 Fourier transforms25 K-space trajectories: FLASH FLASH (TE ~ 5 ms) excitation Read and phase pre-encode (refocusing) Read-out N = res., read-out RF G SS G PE G RO DAC PE table increments each repetition Flip angle ~ 7 deg. TE ~ 5ms