2d ft imaging mp/bme 574. frequency encoding time (t) temporal frequency (f) ft proportionality...

2D FT Imaging

MP/BME 574

Frequency Encoding

T

yy

yy

dssGtk

generalIn

TGk

0

)()(

,

Time (t)

Temporal Frequency (f)

FT

Proportionality

Position (x, or y)

FT

Proportionality

Spatial Frequency (k)

2D Fast GRE Imaging

Gy

RF

Gx

TE

Dephasing/ Rewinder

Dephasing/ Rewinder

Shinnar-LaRoux RF

Phase Encode

Asymmetric Readout

Gz

TR = 6.6 msec

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

2D FT

y

xk

k

Start

Finish

22

,

)()(

)()(

0

0

yy

y

T

yy

t

xx

Nn

Nwhere

kynTG

dssGtk

dssGtk

3D FT

y

z

k

k

kx

Tscan =Ny Nz TR NEX

i.e. Time consuming!

Zero-padding/Sinc Interpolation

• Recall that the sampling theorem – Restoration of a compactly supported (band-

limited) function– Equivalent to convolution of the sampled

points with a sinc function

Case II

FT

k-space: Image Space:

kz

ky

Case III

FT


Methods: Sampling

kz

ky

Case II Nyquist Case III Corner

Case II: Zero-filled

FT

k-space: Image Space:kz

ky

kz

ky

Case III: Zero-Filled

FT


Methods: Sampling

Case II: Nyquist Zero-filled Case III: Corner Zero-filled

Apodization

• Rect windowing implies covolution with a truncated sinc function leading to Gibbs’ Ringing

• Desire to smooth the windowing function so as to diminish ringing.– Gaussian is one option discussed by Prof.

Holden– MRI often uses “Fermi” Filter:

;a)./beta))-)exp((abs(x+1./(1 = f

)()(),(1

1)(

2121 kHkHkkHe

kH

sep

k

),(),(),(

)(),(

212121

21 22

21

kkFkkHkkG

kHkkHkkkrradialradial

r

Point Spread Functions

Un-windowed: Radial Window:

),( 21 nnhsep ),( 21 nnhradial

Ref Corners Radial

),(),(),( 212121 nnfnnhnng sep ),(),(),( 212121 nnfnnhnng radial ),( 21 nnf

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

-5 15 35

Angle (degrees)

Re

solu

tio

n (

mm

/lp

)

Not windowed

Windowed

Cosine reference

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

-5 15 35

Angle (degrees)

Re

solu

tio

n (

mm

/lp

)

Not Windowed

Windowed

Cosine reference

Angular Dependence w/o Zero-filling

)cos(

)cos(max

max

xres

x

res

k

k

r

x

Angular Dependence with Zero-filling

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

-5 15 35

Angle (degrees)

Res

olut

ion

(mm

/lp)

Not windowed

Windowed

Cosine reference

Experimental Results

= 45º= 45º

0 Degrees

45 Degrees

Methods: Point response function

Summary

• Samples in 2D k-space represent 2D sinusoids at specific harmonics and at specific rotation angles

• Interpolation by zero-filling leads to:– Reduced partial volume artifact– Increased spatial resolution at specific angles

• Role of Apodization window– Increases SNR – Decreases ringing artifact– Choice effects the angular symmetry of the PSF

Point response function due to time-dependent contrast

• Example showing mapping on contrast-enhanced signal to model the point response function– Predict attainable resolution – Application to carotid artery MR angiography

Fain SB, Bernstein MA, Huston J III, Riederer SJ

Point Spread Function (PSF) Analysis

• Step 1: Measure enhancement curves in patients

• Step 2: Map enhancement curves to k-space

• Step 3: Transform result to image space to obtain the point

spread function

Fain SB, et al., MRM 42 (1999)

Step 1: Enhancement Model

Fitted Two Phase Gamma VariateC

on

tras

t E

nh

ance

men

t

Time (sec)0 10 20 30 40 50 60 70 80

0

20

40

60

80

100

120

140

160

Composite FitFirst Pass Fit

Residual FitMeasured Data

/)( knekttb


Start

y

z

k

Finish

Overall ImageContrast

High DetailInformation

SampledPoints

k

Step 2: Mapping to k-Space

)(2 tkkk

TRt

zy


Step 2: Mapping to k-Space

Spatial Frequency (cycles/mm)

Co

ntr

ast

En

han

ce

me

nt

(se

c)-1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.01

0.02

0.03

0.04

0.05

0.06

k-Space Weighting

Composite k-SpaceFirst Pass Only

Residual

Measured Data


The Hankel Transform

TR

kkM

tkkk

TRt

ekttb

zy

zy

kn

)(

)(

2

/


Step 3: Transform to Image Space

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.01

0

0.01

0.02

0.03

0.04

0.05

Analytical PSF for Fitted Curve

PS

F A

mp

litu

de

(mm

-sec

2 )-1

Radius (mm)

Composite PSF

First Pass PSF

Residual PSF

Image Contrast

Spatial Resolution


Analysis: Spatial Resolution

FWHM 2FOV y FOV z TR

1

Full Width at Half Maximum (FWHM) of the Point Spread Function is given by:

where,FOVy and FOVz are the phase encoding Fields of ViewTR is the repetition time1 is the time to peak enhancement of the bolus curve


PSF Dependence on Acquisition Time

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Radius (mm)

PS

F a

mp

litu

de

(mm

2 -sec

)-1

Dependence of PSF on Acquisition Time

Infinite ScanTacq = 230 secTacq = 90 secTacq = 50 secTacq = 10 sec

Tacq = Acquisition Time in seconds

10

50

90

230 sec, Maximum Spatial Resolution


213 sec1.2 1.6

2.0

2.6 3.2

4.2

5.2

Z

Y

10 sec50 sec

90 secLine Pairs/mm

Acquisition Time (sec)

PSF Dependence on Acquisition Time


Experiment: FOVz Reduction

13 cm X 6.4 cm

13 cm X 4.0 cm

Z

Y


Carotid and Vertebral Arteries: Acquisition Parameters

– FOV: 22 cm (S/I) X 15 cm (R/L) X 6 cm (A/P)– Matrix: 256 X 168 X 40-44– Acquired Voxel: 0.9 mm X 0.9 mm X 1.4 mm

– 2X Zerofilling in all three directions– TR/TE 6.6 msec/1.4 msec– Acquisition Time: 44-51 seconds– 20 cc Gd


Left Carotid Artery Stenosis: Reconstruction at Multiple Time Points

33 sec22 sec11 sec 44 secAcquisition Time:

X

Z

X

Z

Coronal MIP, Full Data Set:

MIP Reprojec-

tions


Right Carotid Artery Stenosis: Reconstruction at Multiple Time Points

11 sec 22 sec 33 sec 44 secAcquisition Time:

X

Z

X

Z


Decreased FOV

1.0 mm 1.2 mm 1.6 mm 2.0 mm 2.6 mm

15 cm X 6.0 cm

20 cm X 6.0 cm

FOVy

13 cm X 6.4 cm

13 cm X 4.0 cm

FOVz

Z

Y

Z

Y


Increased Scan Time

Z

Y

10 sec50 sec

90 sec213 sec1.2

1.6

2.0

2.6

3.2

4.2

5.2

Partial k-Space Acquisition

• Means of accelerating image acquisition at the expense of minor artifacts– ¾ k-space– ½ k-space -> Hermetian symmetry

• Phase in the image space complicates matters– In practice, MR images have non-zero phase due to

magnetic field variations• Susceptibility• General field inhomogeneity

– “Homodyne” reconstruction required • Low spatial frequency estimation of the phase

2D FT

y

xk

k

Start

Finish

22

,

)()(

)()(

0

0

yy

y

T

yy

t

xx

Nn

Nwhere

kynTG

dssGtk

dssGtk

2D FT

y

xk

k

Start

Finish

22

,

)()(

)()(

0

0

yy

y

T

yy

t

xx

Nn

Nwhere

kynTG

dssGtk

dssGtk

2

N

4

N- where yy n

FI = fftshift(fft(fftshift(I)));for i = 1:192,FI_34(i,:) =FI(i,:);endI_34 = fftshift(ifft(fftshift(FI_34)));figure;subplot(2,2,1),imagesc(abs(I_34));axis('image');colorbar;colormap('gray');title('Magnitude')subplot(2,2,2),imagesc(angle(I_34));axis('image');colorbar;colormap('gray');title('Phase')subplot(2,2,3),imagesc(abs(I-I_34));axis('image');colorbar;colormap('gray');title('Error')gtext('Three-quarter k-space')

2D FT

y

xk

k

Start

Finish

22

,

)()(

)()(

0

0

yy

y

T

yy

t

xx

Nn

Nwhere

kynTG

dssGtk

dssGtk

2

N0 where yn

for i = 1:129,FI_Herm(i,:) =FI(i,:);endI_Herm = fftshift(ifft(fftshift(FI_Herm)));figure;subplot(2,2,1),imagesc(abs(I_Herm2));axis('image');colorbar;colormap('gray');title('Magnitude')figure;subplot(2,2,1),imagesc(abs(I_Herm));axis('image');colorbar;colormap('gray');title('Magnitude')subplot(2,2,2),imagesc(angle(I_Herm));axis('image');colorbar;colormap('gray');title('Phase')subplot(2,2,3),imagesc(abs(I-I_Herm));axis('image');colorbar;colormap('gray');title('Error')gtext('One-half k-space')

2D FT

y

xk

k

Start

Finish

22

,

)()(

)()(

0

0

yy

y

T

yy

t

xx

Nn

Nwhere

kynTG

dssGtk

dssGtk

2

N0

;12

N- where

y

''

y

n

n

count2 = 128;for i = 130:256,FI_Herm(i,:) =conj(FI(count2,:));count2 = count2-1;endI_Herm2 = fftshift(ifft(fftshift(FI_Herm)));figure;subplot(2,2,1),imagesc(abs(I_Herm2));axis('image');colorbar;colormap('gray');title('Magnitude')save phase_phantomsubplot(2,2,2),imagesc(angle(I_Herm2));axis('image');colorbar;colormap('gray');title('Phase')subplot(2,2,3),imagesc(abs(I-I_Herm2));axis('image');colorbar;colormap('gray');title('Error')gtext('Hermetian k-space')

FIp = fftshift(fft(fftshift(IIII)));FIp_Herm = zeros(256);for i = 1:129,FIp_Herm(i,:) =FIp(i,:);endcount2 = 128;for i = 130:256,FIp_Herm(i,:) =conj(FIp(count2,:));count2 = count2-1;endIp_Herm = fftshift(ifft(fftshift(FIp_Herm)));figure;subplot(2,2,1),imagesc(abs(Ip_Herm));axis('image');colorbar;colormap('gray');title('Magnitude')subplot(2,2,2),imagesc(angle(Ip_Herm));axis('image');colorbar;colormap('gray');title('Phase')subplot(2,2,3),imagesc(abs(I-Ip_Herm));axis('image');colorbar;colormap('gray');title('Error')gtext('Attempt at Hermetian k-space for Image with Phase')

2d ft imaging mp/bme 574. frequency encoding time (t) temporal frequency (f) ft proportionality...

Documents

image space slide

sampling slide

filling slide

psf slide

finish slide

corner slide

msec slide

refcornersradial slide