2D Fourier Transform
Fourier Transform
∞ ∞
G(kx ,k y ) = F [ g( x, y )] =
Bioengineering 280A
Principles of Biomedical Imaging
Fall Quarter 2014
MRI Lecture 2
∫ ∫ 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
−∞ −∞
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TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
Phasor Diagram
Imaginary
∞
G(kx ) =
∫ g(x)exp(− j2πk x )dx
x
−∞
θ = −2 πkx x
Real
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θ = −2 πkx x
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kx = 1; x = 0
2πkx x = 0
€
€
x = 1/2
x = 3/ 4
2πkx x = π /2
2πkx x = π
2π k x x = 3π / 2
€
€
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
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1
Vector
sum
x
Vector
sum
25
x
Vector
sum
Vector
sum
-5
x
TT. Liu, BE280A, UCSD Fall 2014
-3
x
TT. Liu, BE280A, UCSD Fall 2014
Vector
sum
x
x
TT. Liu, BE280A, UCSD Fall 2014
13
Vector
sum
-1
x
Vector
sum
3
x
x
-13
Vector
sum
13
TT. Liu, BE280A, UCSD Fall 2014
2
Precession
Larmor Frequency
Analogous to motion of a gyroscope
dµ
= µ x γB
dt
B
Precesses at an angular frequency of
ω = γ Β
f = γ Β / (2 π)
This is known as the Larmor frequency.
dµ
Angular frequency in rad/sec
ω = γ Β
Frequency in cycles/sec or Hertz, Abbreviated Hz
For a 1.5 T system, the Larmor frequency is 63.86 MHz
which is 63.86 million cycles per second. For comparison,
KPBS-FM transmits at 89.5 MHz. µ
Note that the earth’s magnetic field is about 50 µΤ, so that
a 1.5T system is about 30,000 times stronger. TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
Gradients
MRI Gradients
Spins precess at the Larmor frequency, which is
proportional to the local magnetic field. In a constant
magnetic field Bz=B0, all the spins precess at the same
frequency (ignoring chemical shift). Gradient coils are used to add a spatial variation to Bz
such that Bz(x,y,z) = B0+Δ Bz(x,y,z) . Thus, spins at
different physical locations will precess at different
frequencies. http://www.magnet.fsu.edu/education/tutorials/magnetacademy/mri/fullarticle.html
TT. Liu, BE280A, UCSD Fall 2014
http://gulfnews.com/business/economy/german-factories-race-to-meet-big-surge-in-orders-1.638066
TT. Liu, BE280A, UCSD Fall 2014
3
Interpretation
Rotating Frame of Reference
Reference everything to the magnetic field at isocenter.
∆Bz(x)=Gxx
x
Spins Precess at γB0- γGxx
(slower)
Spins Precess at γB0
Spins Precess at
at γB0+ γGxx
(faster)
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
Spins
There is nothing that nuclear spins
will not do for you, as long as you
treat them as human beings. Erwin Hahn
kx=0; ky=0
kx=0; ky≠0
Fig 3.12 from Nishimura
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
4
Hanson 2009
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
Vector
sum
x
x
x
-13
Vector
sum
13
Hanson 2009
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
5
y
1
0
1
0
1
0
1
0
1
(a)
(b)
(d)
(e)
x
(c)
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
Gradient Fields
Gradient Fields
Define ∂Bz
∂B
∂B
x+ z y+ z z
∂x
∂y
∂z
= B0 + Gx x + Gy y + Gz z

G ≡ Gx iˆ + Gy ˆj + Gz kˆ
Bz (x, y,z) = B0 +
z
y
So that €
€
∂B
Gz = z > 0
∂z
TT. Liu, BE280A, UCSD Fall 2014
€

r ≡ xiˆ + yˆj + zkˆ
€
 
Gx x + Gy y + Gz z = G ⋅ r
€
Also, let the gradient fields be a function of time. Then
the z-directed magnetic field at each point in the
volume is given by : 


Bz ( r ,t) = B0 + G(t) ⋅ r
∂B
Gy = z > 0
∂y
TT. Liu, BE280A, UCSD Fall 2014
€
€
6
Time-Varying Gradient Fields
Phase
!
!
!
M (r,t) = M (r,0)eϕ ( r ,t )
In the presence of time-varying gradients the frequency
as a function of space and time is: Phase = angle of the magnetization phasor
Frequency = rate of change of angle (e.g. radians/sec)
Phase = time integral of frequency


ω ( r ,t ) = γBz ( r ,t)


= γB0 + γG(t) ⋅ r

= ω 0 + Δω ( r ,t)
t


ϕ ( r ,t ) = − ∫ 0 ω ( r , τ )dτ

= −ω 0 t + Δϕ ( r ,t )
MY
MX
!
θ = 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
TT. Liu, BE280A, UCSD Fall 2014
TT. Liu, BE280A, UCSD Fall 2014
€
Phase with constant gradient
t1
!
!
Δϕ ( r,t1 ) = − ∫ Δω (r, τ )dτ
t3
!
!
Δϕ ( r,t3 ) = − ∫ Δω (r, τ )dτ
0
= −γ Gx xt1
0
t2
!
!
Δϕ ( r,t2 ) = − ∫ Δω (r, τ )dτ
0
!
= −Δω (r )t2
= −γ Gx xt3
Phase with time-varying gradient
t1
!
!
Δϕ ( r,t1 ) = − ∫ Δω (r, τ )dτ
0
= −γ Gx xt1
0
t1
!
= − ∫ Δω (r, τ )dτ + 0
0
= − γ Gx xt2
TT. Liu, BE280A, UCSD Fall 2014
t3
!
!
Δϕ ( r,t3 ) = − ∫ Δω (r, τ )dτ
0
t2
!
!
Δϕ ( r,t2 ) = − ∫ Δω (r, τ )dτ
= −γ Gx x (t1 + (t3 − t2 ))
TT. Liu, BE280A, UCSD Fall 2014
= −γ Gx xt1
7
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 )e− jω 0 t
(
)


exp(− jγ ∫ G(τ ) ⋅ r dτ )
t

∫ M(r,t)dV
= ∫ ∫ ∫ M(x, y,z,0)e
sr (t) =
V
x
y

−t /T2 ( r ) − jω 0 t
e
z
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
o
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
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
∫ ∫
x
Where
y
y
(
γ
2π
γ
ky (t) =
2π
TT. Liu, BE280A, UCSD Fall 2014
y
)
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
€
8
K-space
Recap
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 )
•  Frequency = rate of change of phase.
•  Higher magnetic field -> higher Larmor frequency ->
phase changes more rapidly with time.
•  With a constant gradient Gx, spins at different x locations
precess at different frequencies -> spins at greater x-values
change phase more rapidly.
•  With a constant gradient, distribution of phases across x
locations changes with time. (phase modulation)
•  More rapid change of phase with x -> higher spatial
frequency kx
TT. Liu, BE280A, UCSD Fall 2014
x
evaluated at the spatial frequencies:
€
€
€
kx (t 2 )
0
y
∫
t
∫ G (τ )dτ
x
TT. Liu, BE280A, UCSD Fall 2014
t
∫ G (τ )dτ
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
kx
0
x
Units
ky
γ
2π
€
0
TT. Liu, BE280A, UCSD Fall 2014
t2
kx (t1 )
t
∫ G (τ )dτ
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.
t
t1
kx (t) =
γ
2π
γ
ky (t) =
2π
kx (t) =
K-space trajectory
Gx(t)
y
= [Hz /Gauss][Gauss /cm][sec]
= [1/cm]
TT. Liu, BE280A, UCSD Fall 2014
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Gx(t) = 1 Gauss/cm
Example
K-space trajectory
Gx(t)
ky
t
t2 = 0.235ms
t
Gy(t)
ky
t1
ky (t 4 )
t2
ky (t 3 )
€
kx
kx (t1 )
€
€
TT. Liu, BE280A, UCSD Fall 2014
kx
t4
kx (t€1 )
kx (t 2 )
γ t
∫ Gx (τ )dτ
2π 0
= 4257Hz /G ⋅1G /cm ⋅ 0.235 ×10−3 s
=1 cm −1
kx (t2 ) =
t3
1 cm
€
kx (t 2 )
€
TT. Liu, BE280A, UCSD Fall 2014
€
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