Transcript Slide 1

2D FT Review
MP/BME 574
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
2D Sampling/Discrete-Space Signal
 N1
N1
x(n1, n 2) where
 n1 
2
2
 N2
N2
 n2 
2
2
n2
n1
2D Functions
•
•
•
•
Impulses
Step Sequences
Separable Sequences
Periodic Sequences
Line Impulse
n1  0
1
 T (n1 )  
0 otherwise
n2
n1
2D step
n1  0
1
uT (n1 )  
0 otherwise
n2
n1
2D step
n2  0
1
uT (n2 )  
0 otherwise
n2
n1
2D Step
n2
1 n1 , n2  0
u(n1 , n2 )  
0 otherwise
n1
Separable Sequences
1 n1 , n2  0
u (n1 , n2 )  
0 otherwise
n2
Separableif ,
x(n1 , n2 )  f (n1 ) g (n2 )
e.g. u (n1 , n2 )  u (n1 )u (n2 )
n1
Periodic Sequences
x(n1 , n2 )
 x(n1  N1 , n2 )
n2
 x(n1 , n2  N 2 ) for all (n1 , n2 )
periodic with period N1  N 2
n1
2D Convolution
y(n1 , n2 ) 

 x(k , k )h(n k , n
k  
1
2
1
1
2
 k2 )
h(k1,k2)
x(k1,k2)
k2
k2
(3)
(1)
k1
(4)
(2)
k1
2D Convolution
y(n1 , n2 ) 

 x(k , k )h(n k , n
k  
1
2
1
1
2
 k2 )
h(-k1,-k2)
x(k1,k2)
k2
k2
(2)
k1
(4)
(1)
(3)
k1
2D Convolution
y(n1 , n2 ) 

 x(k , k )h(n k , n
k  
1
2
1
1
2
 k2 )
h(4-k1,3-k2)
x(k1,k2)
k2
k2
3-k2
(2)
(4)
(1)
(3)
k1
4-k1
k1
2D Convolution
y(n1 , n2 ) 

 x(k , k )h(n k , n
k  
1
2
1
2
 k2 )
h(n1-k1,n2-k2)
x(k1,k2)
k2
1
k2
(2)
(2)
(1)
(4)
(4)
(1)
(3)
(3)
n2
k1
n1-1
k1
2D Convolution
y(n1 , n2 ) 

 x(k , k )h(n k , n
k  
1
2
1
(3)
2
 k2 )
h(n1-k1,n2-k2)
y(n1,n2)
n2
1
k2
(2)
(1)
(7) (7) (4)
(4)
(4)
(6)
(4)
(6)
(1)
(2) n
1
(3) (3)
(3)
n2
n1-1
k1
2D Convolution
y(n1 , n2 ) 

 x(k , k )h(n k , n
k  
1
2
1
(3)
2
 k2 )
h(n1-k1,n2-k2)
y(n1,n2)
n2
1
k2
(2)
(1)
(7) (7) (4)
(4)
(4)
(10)
(6)
(4)
(6)
(1)
(2) n
1
(3) (3)
(3)
n2
n1-1
k1
2DFT Imaging in MRI
MP/BME 574
Abbe’s Theory of Image Formation
From Meyer-Arendt
No Magnetic
Field
=
Random
Orientation
No Net
Magnetization
Dipole Moments from Entire Sample
Magnetic
Field (B0)
Magnetic
Field (B0)
m
m
Positive
Orientation
Negative
Orientation
Precession
  
TorqueN  m  Bo
 L   Bo
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
d
em f (t )  
 V (t )
dt
 
   B  dS
Coil Area
Example
z
B1(t)
y
x
B1 (t )   B1 sin t xˆ  B1 cost yˆ

Coil Area
V (t )  
L/2

 B1 sin t xˆ  B1 cost yˆ  dS  B sin t 
d
d sin t 2
B
L  BL2 cost
dt
dt
L/2
 dydz xˆ
L / 2 L / 2
Example
z
B1(t)
y
x
B1 (t )   B1 sin t xˆ  B1 cost yˆ

Coil Area
V (t )  
L/2

 B1 sin t xˆ  B1 cost yˆ  dS B cost 
d
d cost 2
 B
L  BL2 sin t
dt
dt
L/2
 dxdz yˆ
L / 2 L / 2
Complex Voltage/Signal:
General Case
Vo cos( t   )  Re(Vo e j ( t  ) )
V (t )coil1  Re(BL2 cos t  iBL2 sin  t )
V (t )coil2  Im(BL2 cos t  iBL2 sin  t )
 V (t )  2 BL2 ei t
rf-excitation
By reciprocity,
  
d
3
V (t )  
B
(
r
)

M
(
r
,
t
)
d
r
1

dt Sample

S (r , t )   o
t






3


M
(
r
,
t
)
sin(

B
t
)
dt
B
(
r
)
sin(

t

i

(
r
))

B
(
r
)
cos(

t

i

(
r
))
d
r


xy
1
1x
o
o
1y
o
o


Sample  0

Rotating Frame
Lab Frame
After Haacke, 1999
Quadrature Conversion in MRI
(and Ultrasound) Signal Processing
Received
Radio Frequency
Echo Signal x(t)
(fc = 10MHz; 40MS/s)
X
LPF
xc(t)
I—Channel
2 cos ct
-p/2 Phase Shift
-2 sin ct
X
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?
Spatial Encoding
  
Torque N  m  B1
 o   Bo

fL 
Bo
2
 
 
Consider m ore general B(r )  [ Bo  G  r ]k
 Bz Bz Bz
ˆj 
where G 
iˆ 
kˆ
x
y
z
Slice Selection
Ideal, non-selective rf:
S(t) =rect(t/Dt) B1ideal(t)
B1ideal  Bxye

 i ( ot o ( r ))
S non (t )  Bxy (r )e
    Bxy (r )e

i ( ot o ( r ))

i ( ot o ( r ))
rect (t / Dt )
rect (t / Dt )dt   Bxy (r ) Dt
Non-selective rf-pulse
Entire Volume
Excited
FTdemo: Rect modulated Cosine
FTdemo: Rect modulated Cosine
Spatial Encoding Gradients
z
B(r)
r
y
x
 

Consider more general B(r )  [ Bo  G  r ]kˆ
 Bz Bz Bz
ˆj 
where G 
iˆ 
kˆ
x
y
z
Slice Selection
Selective rf:
Ssel(t) = sinc(t/t) rect(t/Dt) B1ideal(t)
Apply spatial gradient simultaneous to rf-pulse.


 Df  f ( zmax )  f ( zmin )  ( Bo  Gz zmax )  ( Bo  Gz zmin )
2
2

 Gz Dz
2
1 f  Df / 2
rect ( f / Df )  
0 otherwise
S sel (r , t )  Bxy (r )e i (ot  ( r )) rect (t / Dt ) Df sinc (Dft )
1
t
Slice Selective rf-pulse
Slice of width Dz
Excited
FTdemo: Cosine modulated Sinc
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
Spatial Encoding Gradients
z
B(r)
r
y
x
 

Consider more general B(r )  [ Bo  G  r ]kˆ
 Bz Bz Bz
ˆj 
where G 
iˆ 
kˆ
x
y
z
Frequency Encoding

f ( x) 
( Bo  Gx x)
2

Df 
Gx FOVx
2
f, B
Df
B=Bo
FOVx
xmin
xmax
For com m on
bandwidthsGx 1Gauss/ cm
Frequency Encoding
Recall Lab 2, Problem 4: Piano Keyboard
Time varying signal
2
1
0
-1
-2
0
0.002
0.004
0.006
Time (s)
0.008
0.01
0.012
80
60
40
20
0
-1000
-800
-600
-400
…
A, 220 Hz Middle C
-200
0
200
Temporal frequency (Hz)
400
600
800
1000
…
E, 660 Hz
Frequency Encoding
Time (t)

f ( x) 
( Bo  Gx x)
2

Df 
Gx FOVx
2
FT
Proportionality
Temporal
Frequency (f)
Position (x)
Frequency Encoding
sdet (t )  e  iBot  M xy ( x) e  iGx xt dx
s (t )   M xy ( x) e
 iG x xt
dx
Gx
iG xt
M xy ( x) 
s (t ) e
dt

2
iff Gx t is equalto the conjugateof positionx.
x
Frequency Encoding
Spatial
Frequency (k)
Time (t)
In general,
Proportionality
FT
FT
t
k x (t )    Gx ( s )ds
Proportionality
Temporal
Frequency (f)
k x  Gx t
0
Position (x)
Phase Encoding
y
s (t )   M xy ( x) e
f, B
Df
B=Bo
FOVx
xmin
xmax
 iG x xt
dx dy
Phase Encoding
y
s (t )   M xy ( x) e
f, B
Df
B=Bo
FOVx
xmin
xmax
 iG x xt
dx dy
Phase Encoding
y
f, B
Df
s(t )   M xy ( x) e
 iGx xt  i ( y )
s(t )   M xy ( x) e
iGx xt iG y yT
B=Bo
FOVx
xmin
xmax
e
e
dx dy
dx dy
Phase Encoding
B
y
Zero gradient for
time, T
y
Phase Encoding
B
y
Positive gradient
for time, T
y
Phase Encoding
B
y
Positive gradient
for time, T
y
Frequency Encoding
Spatial
Frequency (k-ko)
Time (t)
In general,
Proportionality
FT
FT
T
k y (t )    G y ( s )ds
Proportionality
Temporal
Frequency (f)
k y  G yT
Position (x)
e-iGyT
0
2D Fast GRE Imaging
TE
Gx
Phase
Encode
Asymmetric
Readout
Gy
Dephasing/
Rewinder
Gz
ShinnarLaRoux RF
RF
TR = 6.6 msec
Dephasing/
Rewinder
2D FT
t
ky
Finish
k x (t )    Gx ( s )ds
0
T
n
kx
Start
k y (t )    G y ( s )ds
0
 G yT  nDky,
where
 Ny
2
n
Ny
2
3D Fast GRE Imaging
TE
Gx
Phase
Encode
Asymmetric
Readout
Gy
Phase
Encode
Gz
Dephasin
g/
Rewinder
ShinnarLaRoux RF
RF
TR = 6.6 msec
Dephasing/
Rewinder
3D FT
kz
Tscan =Ny Nz TR NEX
i.e. Time consuming!
ky
n
kx
Summary
• Frequency encoding
– Bandwidth of precessing frequencies
• Phase
– Incremental phase in image space
• Implies shift in k-space
• Entirely separable
– 1D column-wise FFT
– 1D row-wise FFT
Navigating in 2D k-Space
• Goals
– Improve your intuition
• Specific examples
– Effects of:
• Apodization windowing
• “Zero-Padding” or Sinc interpolation
– Vendors refer to this as “ZIP”
• Sampling the corners of k-Space
Elliptical Centric View Order
High Detail
Information
kz
ky
Sampled
Points
Overall Image
Contrast
MRI: Image Acquisition
K-space
FT
Image
space
FT
FT
Case I
kz
ky
Case II
Case III
Case I
k-space:
Image Space:
kz
DFT
ky
Bernstein MA, Fain SB, and Riederer SJ, JMRI 14: 270-280 (2001)
Case II
k-space:
Image Space:
kz
FT
ky
Case III
k-space:
Image Space:
kz
FT
ky
a
b
Zero-padding/Sinc Interpolation
• Recall that the sampling theorem
– Restoration of a compactly supported (bandlimited) function
– Equivalent to convolution of the sampled
points with a sinc function
Recovering or “Restoring” f(x)
from f(n):
2s’
1
Dx
F ( s)  F ( s) per  rect( s 2 s ')
sin(2s' x)
f ( x)   F ( s)  f (n)  2s'
2s' x
1
Recovering or “restoring” f(x)
from f(n): where N  n'  N
sin( 2s ' x )
2s'
2s ' x
f(n)
f(x)
2
Dx
F ( s)  F ( s) per  rect( s 2 s ')
sin(2s' x)
f ( x)   F ( s)  f (n)  2s'
2s' x
1
2
Recovering or “Restoring” f(x)
from f(n):
2s’’
1
Dx
F ( s)  F ( s) per  rect( s 2 s ')
sin(2s' x)
f ( x)   F ( s)  f (n)  2s'
2s' x
1
Recovering or “restoring” f(x)
from f(n):
sin( 2s ' ' x)
2s' '
2s ' ' x
f(n)
f(x)
Dx
F ( s)  F ( s) per  rect( s 2 s '')
sin(2s' ' x)
f ( x)   F ( s)  f (n)  2s' '
2s' ' x
1
Recovering or “restoring” f(x)
from f(n):
f(n’) where N  n'  N
Dx
F ( s)  F ( s) per  rect( s 2 s '')
sin(2s' ' x)
f ( x)   F ( s)  f (n)  2s' '
2s' ' x
1
Case I
k-space:
Image Space:
kz
DFT
ky
Bernstein MA, Fain SB, and Riederer SJ, JMRI 14: 270-280 (2001)
Methods: Sampling
Case I: Zero-filled
k-space:
Image Space:
kz
FT
ky
Case II
k-space:
Image Space:
kz
FT
ky
Case II
k-space:
Image Space:
kz
FT
ky
Methods: Sampling
Case III
k-space:
Image Space:
kz
FT
ky
Methods: Sampling
Case III: Zero-Filled
k-space:
Image Space:
kz
FT
ky