1D: square wave

1
Fourier transforms
Phase map of pineapple slice under read-out
gradient, with phase-encode

• Fourier transforms
• 1D square wave
• 2D kx and ky
• Spatial encoding with gradients
• Common artifacts

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0
2
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
Fourier composition of square wave (v2)
4
Fourier (de)composition of a square wave
Fundamental frequency
Fundamental 1st harmonic
Fundamental 2 harmonics
Fundamental 3 harmonics
5
Fourier (de)composition of a square wave
16s
6
2D Fourier transform
7
original image
filtered with gaussian filter
filtered with hard filter
8
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
FLASH.m accidentally simulates an off-center
k-space
10
Good, clean k-space
11
K-space with small error near center
12
K-space with small error farther out
13
K-space with spike
14
Pulse sequence diagram 2D FLASH
Nrep resolution, phase-encode direction
Flip angle 7 deg.
K-space data
RF
GSS
PE table increments each repetition
GPE
TE 5ms
GRO
N res., read-out
DAC
15
Gradient echo
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)
Immediately after excitation, all the spins in a
sample are in phase
G 5.1kHz/cm
G 12mT/m
B
f
x
x
t 0 ?s
t 20 ?s
t 160 ?s
16
Gradient echo
B
G -12mT/m
Applying a gradient in the opposite direction
reverses this process
x
t 160 ?s
t 300 ?s
t 320 ?s
17
Applying a gradient produces a periodic spin
phase pattern
GRO
Movies arent 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/caolman/courses/Sprin
g2006/Lectures/Lecture7.zip
Magnitude of signal in RF coil
Real part of signal in RF coil
Imaginary component of signal in RF coil
18
The read-out signal is the 1D FFT of the sample
GRO
Magnitude of signal in RF coil
Real part of signal in RF coil
Imaginary component of signal in RF coil
19
Applying simultaneous gradients rotates the
coordinate system
GRO
GPE
20
Phase encoding allows independent spatial
frequency encoding on 2 axes
PE
Read gradient creates phase evolution while one
line of k-space is acquired
PE gradient imposes phase pattern on one axis
GPE
GRO
RO
Read "refocusing" gradient rewinds phase pattern
on another axis
21
Phase encoding allows independent spatial
frequency encoding on 2 axes
PE
Read gradient creates phase evolution while one
line of k-space is acquired
PE gradient imposes phase pattern on one axis
GPE
GRO
RO
Read "refocusing" gradient rewinds phase pattern
on another axis
22
Navigating k-space
Nrep resolution, phase-encode direction
Flip angle 7 deg.
K-space data
RF
GSS
PE table increments each repetition
GPE
TE 5ms
GRO
N res., read-out
DAC
23
Pulse sequence diagrams FLASH EPI
Nrep 32
Nrep 64
Flip angle 7 deg.
Flip angle 60 deg.
RF
GSS
PE table increments each repetition
GPE
TE 5ms
TE 30ms
GRO
N res., read-out
64 pts
64 pts
DAC
24
K-space trajectories FLASH EPI
FLASH (TE 5 ms)
EPI (TE 30 ms)
excitation
excitation
Read and phase pre-encode (refocusing)
Read and phase pre-encode (refocusing)
Phase blip
Read-out
Read-out
TE time between excitation and acquisition of DC
data point
25
K-space trajectories FLASH
FLASH (TE 5 ms)
Flip angle 7 deg.
RF
GSS
excitation
Read and phase pre-encode (refocusing)
PE table increments each repetition
GPE
TE 5ms
Read-out
GRO
N res., read-out
DAC