1D Fourier Transform 2D Plane Waves · 2 * decay 2 € exp−t/T* v ((r )) The overall decay has...

1

TT Liu, SOMI276A, UCSD Winter 2006

SOMI 276AFMRI in Cognitive Science: Foundations

Winter Quarter 2006MRI:

Images and Artifacts


k-spaceImage space k-space

x

y

kx

ky

Fourier Transform


1D Fourier Transform

KPBS

KIFM

KIOZ

Fourier

Transform


2D Plane Waves

cos(2πkxx) cos(2πkyy) cos(2πkxx +2πkyy)

1/kx

1/ky€

1kx2 + ky

2

€

e j2π (kxx+kyy ) = cos 2π (kxx + kyy)( ) + j sin 2π (kxx + kyy)( )


Figure 2.5 from Prince and Link



2


Examples


Examples


Examples


Examples


Examples



€

Fourier Transform

G(kx,ky ) = g(x,y)−∞

∞

∫ e− j2π kxx+kyy( )dxdy−∞

∞

∫

Inverse Fourier Transform

g(x,y) = G(kx,ky )−∞

∞

∫ e j 2π kxx+kyy( )dkxdky−∞

∞

∫

3


Phasor Diagram

€

Recall that a complex number has the form

z = a + jb = z exp( jθ) = z cosθ + j sinθ( )

where z = a 2 + b2 and θ = tan−1 b /a( )

e− j2πkxx = cos 2πkx x( )− j sin 2πkx x( )

Real

Imaginary

€

θ = −2πkx x


Interpretation

∆x 2∆x-∆x-2∆x 0

€

exp − j2π 18Δx

x

€

exp − j2π 28Δx

x

€

exp − j2π 08Δx

x


Interpretation

Fig 3.12 from Nishimura

kx=0; ky=0 kx=0; ky≠0


Simplified Drawing of Basic Instrumentation.Body lies on table encompassed by

coils for static field Bo, gradient fields (two of three shown),

and radiofrequency field B1.

MRI System

Image, caption: copyright Nishimura, Fig. 3.15


Gradient Fields

€

Bz(x,y,z) = B0 +∂Bz

∂xx +

∂Bz

∂yy +

∂Bz

∂zz

= B0 +Gxx +Gyy +Gzzz

€

Gz =∂Bz

∂z> 0

€

Gy =∂Bz

∂y> 0

y


Interpretation

∆Bz(x)=Gxx

Spins Precess atat γB0+ γGxx(faster)

Spins Precess at γB0- γGxx(slower)

x

Spins Precess at γB0

4


Phase with time-varying gradient


Interpretation

∆x 2∆x-∆x-2∆x 0

∆Bz(x)=Gxx

€

exp − j2π 18Δx

x

€

exp − j2π 28Δx

x

€

exp − j2π 08Δx

x

FasterSlower


K-space trajectoryGx(t)

t

€

kx (t) =γ2π

Gx (τ )dτ0

t∫

t1 t2

kx

ky

€

kx (t1)

€

kx (t2)



tt1 t2

ky

€

kx (t1)

€

kx (t2)

Gy(t)

t3 t4kx

€

ky (t4 )

€

ky (t3)



tt1 t2

ky

Gy(t)

kx


Spin-WarpGx(t)

t1

ky

Gy(t)

kx

5


Spin-WarpGx(t)

t1 ky

Gy(t)

kx


Spin-Warp Pulse Sequence

Gx(t)

ky

kx

Gy(t)

RF


Sampling in k-space


Aliasing


Aliasing

Period =1/kx


Aliasing

Period =1/kx

6


K-space trajectories

kx

ky ky

kx

EPI Spiral

Credit: Larry Frank TT Liu, SOMI276A, UCSD Winter 2006

Echoplanar Imaging

GE Medical Systems 2003


Ideal WorldGx

ADC

kx

kyky

kx


+ =1

1

1

-1

1

-1

Image fromEven lines

Image fromOdd lines


Non-Ideal WorldGx

ADC

kx

kyky

kx


+ =

Image fromEven lines

Image fromOdd lines

7


Nyquist Ghosts



Field InhomogeneitiesIn the ideal situation, the static magnetic field is totally uniformand the reconstructed object is determined solely by the appliedgradient fields. In reality, the magnet is not perfect and will notbe totally uniform. Part of this can be addressed by additionalcoils called “shim” coils, and the process of making the fieldmore uniform is called “shimming”. In the old days this wasdone manually, but modern magnets can do this automatically.

In addition to magnet imperfections, most biological samplesare inhomogeneous and this will lead to inhomogeneity in thefield. This is because, each tissue has different magneticproperties and will distort the field.


Field Inhomogeneities


FasterSlower

Precesses slower becauseof chemical shift or localinhomogeneities


For EPI scans, distortion occurs mostly in the phase-encode direction, since data are acquired more slowly inthis directon. For spiral scans, the picture is morecomplicated.

GE Medical Systems 2003

Phase Encode

8

TT Liu, SOMI276A, UCSD Winter 2006 TT Liu, SOMI276A, UCSD Winter 2006


Top down vs. bottom up


Field Map Correction


Distortions can be reduced by moving more quicklythrough k-space. This can be achieved with interleavedEPI or Spiral scans, albeit with a loss of temporalresolution.

On modern imaging systems, parallel imaging offersanother way of reducing the acquisition time, albeit witha loss of signal-to-noise.


InterleavedSpirals

1

2

3

4

9


Spiral SENSE

1

2

3

4


Signal Dropouts

Field inhomogeneities also cause the spins to dephase with timeand thus for the signal to decrease more rapidly. To first order thiscan be modeled as an additional decay term.


T2* decay

€

exp −t /T2* v r ( )( )

The overall decay has the form.

€

1T2* =

1T2

+1′ T 2

where

Due to random motions of spins.Not reversible.

Due to staticinhomogeneities. Reversiblewith a spin-echo sequence.


T2* decay

Gradient echo sequences exhibit T2* decay.

Gx(t)

Gy(t)

RF

Gz(t)Slice select gradient

Slice refocusing gradient

ADC

TE = echo time

Gradient echo hasexp(-TE/T2

*)

weighting


Spin EchoDiscovered by Erwin Hahn in 1950.

There is nothing that nuclear spins will not do for you, aslong as you treat them as human beings. Erwin Hahn

Image: Larry Frank

τ τ180º

The spin-echo can refocus the dephasing of spins dueto static inhomogeneities. However, there will still beT2 dephasing due to random motion of spins.


Spin-echo TE = 35 ms Gradient Echo TE = 14ms

1D Fourier Transform 2D Plane Waves · 2 * decay 2 € exp−t/T* v ((r )) The overall decay has...

Documents

Transcript of 1D Fourier Transform 2D Plane Waves · 2 * decay 2 € exp−t/T* v ((r )) The overall decay has...