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

k-space: Image Space:

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

k-space: Image Space:

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

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

Start

y

z

k

Finish

Overall ImageContrast

High DetailInformation

SampledPoints

k

Step 2: Mapping to k-Space

)(2 tkkk

TRt

zy

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

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

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

The Hankel Transform

TR

kkM

tkkk

TRt

ekttb

zy

zy

kn

)(

)(

2

/

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

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

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

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

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

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

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

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

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

Experiment: FOVz Reduction

13 cm X 6.4 cm

13 cm X 4.0 cm

Z

Y

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

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

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

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

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

Right Carotid Artery Stenosis: Reconstruction at Multiple Time Points

11 sec 22 sec 33 sec 44 secAcquisition Time:

X

Z

X

Z

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

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

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

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')