signal equation k-space
TRANSCRIPT
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TT. Liu, BE280A, UCSD Fall 2015
Bioengineering 280A �Principles of Biomedical Imaging�
�Fall Quarter 2015�
MRI Lecture 3�
TT. Liu, BE280A, UCSD Fall 2015
TT. Liu, BE280A, UCSD Fall 2015
Signal EquationDemodulate the signal to obtain
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s(t) = e jω 0t sr(t)
= m(x,y)exp − jγ G (τ ) ⋅
r dτ
o
t∫( )y∫x∫ dxdy
= m(x,y)exp − jγ Gx (τ)x + Gy (τ)y[ ]dτo
t∫( )y∫x∫ dxdy
= m(x,y)exp − j2π kx (t)x + ky (t)y( )( )y∫x∫ dxdy
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kx (t) =γ2π
Gx (τ )dτ0
t∫
ky (t) =γ2π
Gy (τ )dτ0
t∫
Where
TT. Liu, BE280A, UCSD Fall 2015
K-space
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s(t) = M kx (t),ky (t)( ) = F m(x,y)[ ] kx ( t ),ky ( t )
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kx (t) =γ2π
Gx (τ )dτ0
t∫
ky (t) =γ2π
Gy (τ )dτ0
t∫
At each point in time, the received signal is the Fourier transform of the object
evaluated at the spatial frequencies:
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.
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TT. Liu, BE280A, UCSD Fall 2015
K-space trajectoryGx(t)
t
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kx (t) =γ2π
Gx (τ )dτ0
t∫
t1 t2
kx
ky
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kx (t1)
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kx (t2)
TT. Liu, BE280A, UCSD Fall 2015
UnitsSpatial frequencies (kx, ky) have units of 1/distance. Most commonly, 1/cmGradient strengths have units of (magnetic field)/distance. Most commonly G/cm or mT/mγ/(2π) has units of Hz/G or Hz/Tesla.
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kx (t) =γ2π
Gx (τ )dτ0
t
∫
= [Hz /Gauss][Gauss /cm][sec]= [1/cm]
TT. Liu, BE280A, UCSD Fall 2015
ExampleGx(t) = 1 Gauss/cm
t
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kx (t2) =γ
2πGx (τ )dτ
0
t
∫= 4257Hz /G ⋅1G /cm ⋅0.235×10−3 s=1 cm−1
kx
ky
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kx (t1)
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kx (t2)
t2 = 0.235ms
1 cm
TT. Liu, BE280A, UCSD Fall 2015
K-space trajectoryGx(t)
tt1 t2
ky
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kx (t1)
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kx (t2)
Gy(t)
t3 t4kx
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ky (t4 )
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ky (t3)
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TT. Liu, BE280A, UCSD Fall 2015
k-space
kx
ky
TT. Liu, BE280A, UCSD Fall 2015
TT. Liu, BE280A, UCSD Fall 2015
K-space trajectoryGx(t)
tt1 t2
ky
Gy(t)
kx
TT. Liu, BE280A, UCSD Fall 2015
Spin-WarpGx(t)
t1
ky
Gy(t)
kx
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TT. Liu, BE280A, UCSD Fall 2015
Spin-Warp Pulse Sequence
Gx(t)
Gy(t)
RF
kyGy(t)
TT. Liu, BE280A, UCSD Fall 2015
TT. Liu, BE280A, UCSD Fall 2015
K-space trajectories
kx
ky ky
kx
EPI Spiral
Credit: Larry FrankTT. Liu, BE280A, UCSD Fall 2015
kx
ky ky
kx
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TT. Liu, BE280A, UCSD Fall 2015
Thomas Liu, BE280A, UCSD, Fall 2008
Sampling in k-space
TT. Liu, BE280A, UCSD Fall 2015
Aliasing
TT. Liu, BE280A, UCSD Fall 2015
Aliasing
∆Bz(x)=Gxx FasterSlowerTT. Liu, BE280A, UCSD Fall 2015
Intuitive view of Aliasing
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kx =1/FOV
FOV
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kx = 2 /FOV
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TT. Liu, BE280A, UCSD Fall 2015
Fourier Sampling
Instead of sampling the signal, we sample its Fourier Transform
???
Sample
F-1
F
TT. Liu, BE280A, UCSD Fall 2015
Fourier Sampling
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GS (kx ) =G(kx )1Δkx
comb kxΔkx
#
$ %
&
' (
=G(kx ) δ(n=−∞
∞
∑ kx − nΔkx )
= G(nΔkx )δ(n=−∞
∞
∑ kx − nΔkx )
kx
(1 /Δkx) comb(kx /Δkx)
Δkx
TT. Liu, BE280A, UCSD Fall 2015
Fourier Sampling -- Inverse Transform
Χ =
*
1/Δkx
Δkx
=
TT. Liu, BE280A, UCSD Fall 2015
Nyquist Condition
FOV (Field of View)
1/Δkx
To avoid overlap, 1/Δkx> FOV, or equivalently, Δkx<1/FOV
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TT. Liu, BE280A, UCSD Fall 2015
Aliasing
FOV (Field of View)
1/Δkx
Aliasing occurs when 1/Δkx< FOV
TT. Liu, BE280A, UCSD Fall 2015
Aliasing Example
Δkx=1
1/Δkx=1
TT. Liu, BE280A, UCSD Fall 2015
2D Comb Function
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comb(x,y) = δ(x −m,y − n)n=−∞
∞
∑m=−∞
∞
∑
= δ(x −m)δ(y − n)n=−∞
∞
∑m=−∞
∞
∑= comb(x)comb(y)
TT. Liu, BE280A, UCSD Fall 2015
Scaled 2D Comb Function
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comb(x /Δx,y /Δy) = comb(x /Δx)comb(y /Δy)
= ΔxΔy δ(x −mΔx)δ(y − nΔy)n=−∞
∞
∑m=−∞
∞
∑
Δx
Δy
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TT. Liu, BE280A, UCSD Fall 2015
2D k-space sampling
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GS (kx,ky ) =G(kx,ky )1
ΔkxΔkycomb kx
Δkx,kyΔky
#
$ % %
&
' ( (
=G(kx,ky ) δ(n=−∞
∞
∑ kx −mΔkx,ky − nΔky )m=−∞
∞
∑
= G(mΔkx,nΔky )δ(n=−∞
∞
∑ kx −mΔkx,ky − nΔky )m=−∞
∞
∑
TT. Liu, BE280A, UCSD Fall 2015
X =
* =
1/∆k
TT. Liu, BE280A, UCSD Fall 2015
Nyquist Conditions
FOVX
FOVY
1/∆kX
1/∆kY
1/∆kY> FOVY
1/∆kX> FOVX
TT. Liu, BE280A, UCSD Fall 2015
Windowing
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GW kx,ky( ) =G kx,ky( )W kx,ky( )
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gW x,y( ) = g x,y( )∗w(x,y)
Windowing the data in Fourier space
Results in convolution of the object with the inverse transform of the window
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TT. Liu, BE280A, UCSD Fall 2015
Resolution
TT. Liu, BE280A, UCSD Fall 2015
Windowing Example
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W kx,ky( ) = rect kxWkx
"
# $ $
%
& ' ' rect
kyWky
"
# $ $
%
& ' '
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w x,y( ) = F −1 rect kxWkx
#
$ % %
&
' ( ( rect
kyWky
#
$ % %
&
' ( (
)
* + +
,
- . .
=WkxWky
sinc Wkxx( )sinc Wky
y( )
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gW x,y( ) = g(x,y)∗∗WkxWky
sinc Wkxx( )sinc Wky
y( )
TT. Liu, BE280A, UCSD Fall 2015
Effective Width
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wE =1
w(0)w(x)dx
−∞
∞
∫
wE
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wE =11
sinc(Wkxx)dx
−∞
∞
∫
= F sinc(Wkxx)[ ]
kx = 0
=1Wkx
rect kxWkx
%
& ' '
(
) * * kx = 0
=1Wkx
Example
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1Wkx
€
−1Wkx
TT. Liu, BE280A, UCSD Fall 2015
Resolution and spatial frequency
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2Wkx
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With a window of width Wkx the highest spatial frequency is Wkx
/2.This corresponds to a spatial period of 2/Wkx
.
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1Wkx
= Effective Width
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TT. Liu, BE280A, UCSD Fall 2015
Windowing Example
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g(x,y) = δ(x) + δ(x −1)[ ]δ(y)
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gW x,y( ) = δ(x) + δ(x −1)[ ]δ(y)∗∗WkxWky
sinc Wkxx( )sinc Wky
y( )=Wkx
Wkyδ(x) + δ(x −1)[ ]∗ sinc Wkx
x( )( )sinc Wkyy( )
=WkxWky
sinc Wkxx( ) + sinc Wkx
(x −1)( )( )sinc Wkyy( )
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Wkx=1
€
Wkx= 2
€
Wkx=1.5
TT. Liu, BE280A, UCSD Fall 2015
Sampling and Windowing
X =
* =
X
* =
TT. Liu, BE280A, UCSD Fall 2015
Sampling and Windowing
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GSW kx,ky( ) =G kx,ky( ) 1ΔkxΔky
comb kxΔkx
,kyΔky
#
$ % %
&
' ( ( rect
kxWkx
,kyWky
#
$ % %
&
' ( (
€
gSW x,y( ) =WkxWkyg x,y( )∗∗comb(Δkxx,Δkyy)∗∗sinc(Wkx
x)sinc(Wkyy)
Sampling and windowing the data in Fourier space
Results in replication and convolution in object space.
TT. Liu, BE280A, UCSD Fall 2015
Sampling and Windowing∆kY< 1/(FOVY)
∆kX< 1/(FOVX)
Δkx
Δky kx
ky
Wky>1/δy
Wkx
Wkx>1/δx
Wky
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TT. Liu, BE280A, UCSD Fall 2015 GE Medical Systems 2003 TT. Liu, BE280A, UCSD Fall 2015
Sampling in ky
kx
ky
Gx(t)
Gy(t)
RF
Δky
τy
Gyi
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Δky =γ2π
Gyiτ y
€
FOVy =1Δky
TT. Liu, BE280A, UCSD Fall 2015
Sampling in kx
x Low pass Filter ADC
x Low pass Filter ADC€
cosω0t
€
sinω0t
RF Signal
One I,Q sample every Δt
M= I+jQ
I
Q
Note: In practice, there are number of ways of implementing this processing.
TT. Liu, BE280A, UCSD Fall 2015
Sampling in kx
kx
ky
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Δkx =γ2π
GxrΔt
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FOVx =1Δkx
Gx(t)
t1
ADC
Gxr
Δt
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TT. Liu, BE280A, UCSD Fall 2015
Resolution
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δx =1Wkx
= 12kx,max
= 1γ
2πGxrτ x
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Wkx€
Wky
Gx(t)Gxr
€
τ x
€
δy =1Wky
= 12ky,max
= 1γ
2π2Gypτ y
Gy(t)
τy€
Gyp
TT. Liu, BE280A, UCSD Fall 2015
Example
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Goal :FOVx = FOVy = 25.6 cmδx = δy = 0.1 cm
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Readout Gradient :
FOVx =1
γ2π
GxrΔt
Pick Δt = 32 µsec
Gxr =1
FOVxγ
2πΔt
=1
25.6cm( ) 42.57 ×106T−1s−1( ) 32 ×10−6 s( )
= 2.8675 ×10−5 T/cm = .28675 G/cm
1 Gauss = 1×10−4 Tesla
t1
ADC
Gxr
Δt
TT. Liu, BE280A, UCSD Fall 2015
Example
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Readout Gradient :
δx =1
γ2π
Gxrτ x
τ x =1
δxγ
2πGxr
=1
0.1cm( ) 4257 G−1s−1( ) 0.28675 G/cm( )
= 8.192 ms = NreadΔtwhere
Nread =FOVxδx
= 256
Gx(t)
Gxr
€
τ x
TT. Liu, BE280A, UCSD Fall 2015
Example
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Phase - Encode Gradient :
FOVy =1
γ2π
Gyiτ y
Pick τ y = 4.096 msec
Gyi =1
FOVyγ
2πτ y
=1
25.6cm( ) 42.57 ×106T−1s−1( ) 4.096 ×10−3 s( )
= 2.2402 ×10-7 T/cm = .00224 G/cm
τy
Gyi
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TT. Liu, BE280A, UCSD Fall 2015
Example
€
Phase - Encode Gradient :δy =
1γ
2π2Gypτ y
Gyp =1
δy 2 γ2π
τ y
=1
0.1cm( ) 4257 G−1s−1( ) 4.096 ×10-3 s( ) = 0.2868 G/cm
=Np
2Gyi
where
Np =FOVyδy
= 256
Gy(t)
τy€
Gyp
TT. Liu, BE280A, UCSD Fall 2015
Sampling
€
Wkx€
Wky
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.
ky
y
FOV/4
1/FOV
4/FOV
FOV
TT. Liu, BE280A, UCSD Fall 2015
ExampleConsider the k-space trajectory shown below. ADC samples are acquired at the points shownwith
€
Δt = 10 µsec. The desired FOV (both x and y) is 10 cm and the desired resolution (both xand y) is 2.5 cm. Draw the gradient waveforms required to achieve the k-space trajectory. Labelthe waveform with the gradient amplitudes required to achieve the desired FOV and resolution.Also, make sure to label the time axis correctly.
PollEv.com/be280a
Assume γ 2π( ) = 4000Hz /G
TT. Liu, BE280A, UCSD Fall 2015
Gibbs Artifact
256x256 image 256x128 image
Images from http://www.mritutor.org/mritutor/gibbs.htm
* =
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TT. Liu, BE280A, UCSD Fall 2015
Apodization
Images from http://www.mritutor.org/mritutor/gibbs.htm
* =
rect(kx)
h(kx )=1/2(1+cos(2πkx)Hanning Window
sinc(x)
0.5sinc(x)+0.25sinc(x-1)+0.25sinc(x+1)
TT. Liu, BE280A, UCSD Fall 2015
Aliasing and Bandwidth
x
LPF ADC
ADC€
cosω0t
€
sinω0t
RF Signal
I
QLPF
x
*
xf
t
x
t
FOV 2FOV/3
Temporal filtering in the readout direction limits the readout FOV. So there should never be aliasing in the readout direction.
TT. Liu, BE280A, UCSD Fall 2015
Aliasing and Bandwidth
Slower
Fasterx
f
Lowpass filterin the readout direction toprevent aliasing.
readout
FOVx
B=γGxrFOVx
TT. Liu, BE280A, UCSD Fall 2015 GE Medical Systems 2003