2D Fourier Transform
Fourier Transform
∞ ∞
G(kx ,k y ) = F [ g( x, y )] =
Bioengineering 280A
Principles of Biomedical Imaging
Fall Quarter 2014
MRI Lecture 3
∫ ∫ g(x, y)e
− j 2 π ( kx x +ky y )
dxdy
−∞ −∞
Inverse Fourier Transform
∞ ∞
g(x, y) =
∫ ∫ G(k ,k )e
x
y
j 2 π ( kx x +ky y )
dkx dk y
−∞ −∞
€
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
Phasor Diagram
Imaginary
∞
G(kx ) =
∫ g(x)exp(− j2πk x )dx
Vector
sum
x
−∞
x
θ = −2 πkx x
-13
Real
€
θ = −2 πkx x
€
kx = 1; x = 0
2πkx x = 0
€
€
x = 1/2
2πkx x = π /2
2πkx x = π
€
x = 3/ 4
2π k x x = 3π / 2
x
x
Vector
sum
13
€
θ = −π /2
θ =0
TT. Liu, BE280A, UCSD Fall 2014
€
x = 1/4
θ = −π
θ = −3π / 2
TT. Liu, BE280A, UCSD Fall 2014
€
1
Phase
MY
!
!
!
M (r,t) = M (r,0)eϕ ( r ,t )
MX
Phase = angle of the magnetization phasor
Frequency = rate of change of angle (e.g. radians/sec)
Phase = time integral of frequency
!
θ = Δϕ (r,t)
t


ϕ ( r ,t ) = − ∫ 0 ω ( r , τ )dτ

= −ω 0 t + Δϕ ( r ,t )
Where the incremental phase due to the gradients is
t


Δϕ (r ,t) = − ∫ Δω( r ,τ )dτ
0
t  

= − ∫ γG ( r ,τ ) ⋅ r dτ
€
0
Hanson 2009
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
€
Time-Varying Gradient Fields
Signal Equation
The transverse magnetization is then given by Signal from a volume




M( r ,t) = M( r ,0)e−t /T2 ( r )eϕ ( r ,t )

t


= M( r ,0)e−t /T2 ( r )e− jω 0 t exp − j ∫ o Δω ( r ,t ) dτ

= M( r ,0)e

−t /T2 ( r ) − jω 0 t
e
(
)


exp(− jγ ∫ G(τ ) ⋅ r dτ )
t
o

∫ M(r,t)dV
= ∫ ∫ ∫ M(x, y,z,0)e
sr (t) =
V
x
€
y
z

−t /T2 ( r ) − jω 0 t
e
t 

exp − jγ ∫ G(τ ) ⋅ r dτ dxdydz
(
)
o
For now, consider signal from a slice along z and drop
the T2 term. Define m(x, y) ≡ ∫ z +Δz / 2 M( r,t)dz
0
z 0 −Δz / 2
To obtain t 

sr (t) = ∫€x ∫ y m(x, y)e− jω 0 t exp − jγ ∫ o G(τ ) ⋅ r dτ dxdy
(
)
€
TT. Liu, BE280A, UCSD Fall 2014
€
TT. Liu, BE280A, UCSD Fall 2014
2
Signal Equation
MR signal is Fourier Transform
Demodulate the signal to obtain s(t) = e jω 0 t sr (t)
=
x
t 

m(x, y)exp − jγ ∫ o G(τ ) ⋅ r dτ dxdy
(
)
∫ m(x, y)exp(− jγ ∫ [G (τ )x + G (τ )y ]dτ )dxdy
∫ ∫
y
∫ ∫ m(x, y)exp(− j2π (k (t)x + k (t)y ))dxdy
= M ( k (t),k (t))
s(t) =
t
=
∫
=
∫ ∫
x
y
x
Where
y
y
(
)
γ
2π
γ
ky (t) =
2π
x
y
x
m(x, y)exp − j2π ( kx (t)x + k y (t)y ) dxdy
kx (t) =
€
x
o
x
y
y
= F [ m(x, y)] k
x (t ),ky (t )
t
∫ G (τ )dτ
0
x
€
t
∫ G (τ )dτ
0
y
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
€
K-space
At each point in time, the received signal is the Fourier
transform of the object
s(t) = M ( k x (t),k y (t)) = F [ m(x, y)] k (t ),k (t )
x
t
t1
y
γ
2π
γ
ky (t) =
2π
kx (t) =
t
∫ G (τ )dτ
0
x
kx
t
∫ G (τ )dτ
0
kx (t1 )
y
Thus, the gradients control our position in k-space. The
design
€ of an MRI pulse sequence requires us to
efficiently cover enough of k-space to form our image.
TT. Liu, BE280A, UCSD Fall 2014
t2
ky
evaluated at the spatial frequencies:
€
K-space trajectory
Gx(t)
γ
2π
€
kx (t) =
€
€
kx (t 2 )
t
∫ G (τ )dτ
0
x
TT. Liu, BE280A, UCSD Fall 2014
3
Units
Gx(t) = 1 Gauss/cm
Spatial frequencies (kx, ky) have units of 1/distance.
Most commonly, 1/cm
Gradient strengths have units of (magnetic field)/
distance. Most commonly G/cm or mT/m
γ/(2π) has units of Hz/G or Hz/Tesla. γ t
kx (t) =
Gx (τ )dτ
2π 0
Example
t
t2 = 0.235ms
ky
kx
kx (t1 )
∫
kx (t 2 )
γ t
∫ Gx (τ )dτ
2π 0
= 4257Hz /G ⋅1G /cm ⋅ 0.235 ×10−3 s
=1 cm −1
kx (t2 ) =
= [Hz /Gauss][Gauss /cm][sec]
= [1/cm]
€
TT. Liu, BE280A, UCSD Fall 2014
€
1 cm
TT. Liu, BE280A, UCSD Fall 2014
€
€
K-space trajectory
Gx(t)
k-space
ky
t
Gy(t)
t1
ky (t 4 )
t2
ky (t 3 )
€
t3
ky
kx
t4
kx (t€1 )
€
kx (t 2 )
€
kx
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
4
K-space trajectory
Gx(t)
ky
t
Gy(t)
t1
t2
kx
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
Spin-Warp
Gx(t)
RF
Spin-Warp Pulse Sequence
ky
Gx(t)
t1
Gy(t)
Gy(t)
kx
Gy(t)
TT. Liu, BE280A, UCSD Fall 2014
ky
TT. Liu, BE280A, UCSD Fall 2014
5
Sampling in k-space
Thomas Liu, BE280A,
UCSD,UCSD
Fall 2008
TT. Liu, BE280A,
Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
Aliasing
Aliasing
Slower
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
∆Bz(x)=Gxx
Faster
6
Fourier Sampling
Intuitive view of Aliasing
FOV
F
kx = 2 /FOV
Sample
Instead of sampling the
signal, we sample its Fourier
Transform
€
???
kx = 1/FOV
F-1
€
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
Fourier Sampling -- Inverse
Transform
Fourier Sampling
(1 /Δkx) comb(kx /Δkx)
=
Χ
kx
Δkx
Δkx
GS (kx ) = G(k x )
# k &
1
comb% x (
Δkx
$ Δkx '
∞
*
= G(k x ) ∑δ (k x − nΔkx )
n=−∞
∞
=
=
1/Δkx
∑ G(nΔk )δ (k
x
x
− nΔk x )
n=−∞
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
€
7
Nyquist Condition
Aliasing
FOV (Field of View)
FOV (Field of View)
1/Δkx
1/Δkx
To avoid overlap, 1/Δkx> FOV, or equivalently, Δkx<1/FOV
TT. Liu, BE280A, UCSD Fall 2014
Aliasing occurs when 1/Δkx< FOV
TT. Liu, BE280A, UCSD Fall 2014
Aliasing Example
2D Comb Function
∞
comb(x, y) =
∞
∑ ∑δ (x − m, y − n)
m=−∞ n=−∞
∞
∞
=
∑ ∑δ (x − m)δ (y − n)
m=−∞ n=−∞
= comb(x)comb(y)
Δkx=1
€
1/Δkx=1
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
8
Scaled 2D Comb Function
2D k-space sampling
comb(x /Δx, y /Δy) = comb(x /Δx)comb(y /Δy)
∞
= ΔxΔy
∞
GS (kx ,k y ) = G(k x ,k y )
∑ ∑δ (x − mΔx)δ(y − nΔy)
m=−∞ n=−∞
# k
k &
1
comb%% x , y ((
Δk x Δk y
$ Δk x Δky '
∞
= G(k x ,k y ) ∑
∞
∑δ(k
x
− mΔk x ,k y − nΔk y )
m=−∞ n=−∞
€
∞
Δy
=
∞
∑ ∑ G(mΔk ,nΔk )δ (k
x
y
x
− mΔk x ,k y − nΔky )
m=−∞ n=−∞
€
Δx
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
Nyquist Conditions
1/∆kY> FOVY
=
*
1/∆kX> FOVX
1/∆kY
1/∆k
X
=
FOVY
FOVX
TT. Liu, BE280A, UCSD Fall 2014
1/∆kX
TT. Liu, BE280A, UCSD Fall 2014
9
Resolution
Windowing
Windowing the data in Fourier space GW ( kx ,k y ) = G( k x ,k y )W ( k x ,k y )
Results in convolution of the object with the inverse
€ transform of the window
gW ( x, y ) = g( x, y ) ∗ w(x, y)
€
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
Windowing Example
Effective Width
1
wE =
w(0)
" k %
" k %
W ( kx ,k y ) = rect$$ x ''rect$$ y ''
# W kx &
# W ky &
€
(
€
−∞
w(x)dx
Example
)
# k &,
# k &
w ( x, y ) = F −1+rect%% x ((rect%% y ((.
+*
$ W kx '
$ W ky '.= W kx W ky sinc (W kx x ) sinc W ky y
∫
∞
€
wE =
)
1 ∞
∫ sinc(W kx x)dx
1 −∞
= F [sinc(W kx x)]
(
gW ( x, y ) = g(x, y) ∗∗W kx W ky sinc (W kx x ) sinc W ky y
wE
kx = 0
% k (
1
=
rect'' x **
W kx
& W kx ) kx = 0
)
=
−
1
W kx
€
€
TT. Liu, BE280A, UCSD Fall 2014
1
W kx
1
W kx
€
TT. Liu, BE280A, UCSD Fall 2014
€
10
Resolution and spatial frequency
Windowing Example
With a window of width W kx the highest spatial frequency is W kx /2.
g(x, y) = [δ (x) + δ (x −1)]δ (y)
This corresponds to a spatial period of 2/W kx .
1
= Effective Width
W kx
(
gW ( x, y ) = [δ (x) + δ (x −1)]δ (y) ∗∗W kx W ky sinc (W kx x ) sinc W ky y
€
(
(
)
) ( )
) ( )
= W kx W ky [δ (x) + δ (x −1)] ∗ sinc (W kx x ) sinc W ky y
€
= W kx W ky sinc (W kx x ) + sinc (W kx (x −1)) sinc W ky y
€
€
W kx = 1
2
W kx
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
€
W kx = 1.5
€
W kx = 2
€
€
Sampling and Windowing
Sampling and Windowing
Sampling and windowing the data in Fourier space *
*
=
GSW ( kx ,k y ) = G( k x ,k y )
=
€
X
X
# k
# k
k &
k &
1
comb%% x , y ((rect%% x , y ((
Δk x Δk y
$ Δk x Δky '
$ W kx W ky '
Results in replication and convolution in object space. gSW ( x, y ) = W kx W ky g( x, y ) ∗∗comb(Δk x x,Δk y y) ∗∗sinc(W kx x)sinc(W ky y)
=
€
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
11
Sampling in ky
RF
Sampling in kx
ky
Δky
x
Gx(t)
RF Signal
Gy(t)
kx
Δk y =
Gyi
€
TT. Liu, BE280A, UCSD Fall 2014
γ
Gyiτ y
2π
€
ADC
I
Low pass
Filter
ADC
Q
cosω 0 t
x
τy
Low pass
Filter
sin ω 0 t
1
FOVy =
Δk y
One I,Q sample every Δt
Note: In practice, there are number of ways of
implementing this processing.
M= I+jQ
TT. Liu, BE280A, UCSD Fall 2014
€
€
Sampling in kx
Gx(t)
Resolution
1
1
1
δx =
=
=
γ
W kx 2k x,max
Gxrτ x
2π
Gx(t)
τ x
ky
Gxr
kx
t1
€
ADC
W ky
Gxr
€
W
kx
€
Δk x =
γ
GxrΔt
2π
δy =
1
FOVx =
Δk x
Δt
€
TT. Liu, BE280A, UCSD Fall 2014
1
1
1
=
=
γ
W ky 2k y,max
2Gyp τ y
2π
€
Gy(t)
Gyp
€
TT. Liu, BE280A, UCSD Fall 2014
€
τy
€
12
Example
Example
Readout Gradient :
1
δx =
γ
Gxrτ x
2π
Goal :
FOVx = FOVy = 25.6 cm
Gxr
δx = δy = 0.1 cm
Readout Gradient :
1
FOVx =
γ
GxrΔt
2π
Pick Δt = 32 µsec
1
1
Gxr =
=
6 −1 −1
−6
γ
FOVx
Δt (25.6cm)( 42.57 ×10 T s )( 32 ×10 s)
2π
= 2.8675 ×10−5 T/cm
= .28675 G/cm
€
1
1
=
−1 −1
γ
δx
Gxr (0.1cm)( 4257 G s )(0.28675 G/cm)
2π
= 8.192 ms
= N read Δt
where
FOVx
N read =
= 256
δx
τx =
t1
ADC
Gx(t)
τx
Gxr
€
Δt
1 Gauss = 1×10−4 Tesla
€
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
€
Example
Example
Phase - Encode Gradient :
1
FOVy =
γ
Gyiτ y
2π
Pick τ y = 4.096 msec
Gyi =
1
FOVy
γ
τy
2π
=
Phase - Encode Gradient :
1
γ
2Gyp τ y
2π
δy =
1
(25.6cm)( 42.57 ×10 6 T −1s−1 )( 4.096 ×10−3 s)
Gyp =
= 2.2402 ×10 -7 T/cm
= .00224 G/cm
δy 2
τy
where
FOVy
Np =
= 256
δy
Gyi
€
TT. Liu, BE280A, UCSD Fall 2014
1
1
=
−1 −1
-3
γ
τ y (0.1cm)( 4257 G s )( 4.096 ×10 s)
2π
= 0.2868 G/cm
N
= p Gyi
2
€
Gy(t)
Gyp
€
τy
TT. Liu, BE280A, UCSD Fall 2014
13
Sampling
Example
In practice, an even number
(typically power of 2) sample is
usually taken in each direction to
take advantage of the Fast Fourier
Transform (FFT) for reconstruction. Consider the k-space trajectory shown below. ADC samples are acquired at the points shown
with Δt = 10 µsec . The desired FOV (both x and y) is 10 cm and the desired resolution (both x
and y) is 2.5 cm. Draw the gradient waveforms required to achieve the k-space trajectory. Label
the waveform with the gradient amplitudes required to achieve the desired FOV and resolution.
Also, make sure to label the time axis correctly.
W ky
€
4/FOV
€
W
kx
ky
1/FOV
€
FOV
y
FOV/4
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
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