2D FT Imaging MP/BME 574

Frequency Encoding Time (t) Temporal Frequency (f) FT Proportionality Position (x, or y) FT Proportionality Spatial Frequency (k)

2D Fast GRE Imaging GyGy RF GxGx TE Dephasing/ Rewinder Dephasing/ Rewinder Shinnar- LaRoux RF Phase Encode Asymmetric Readout GzGz 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 x k k Start Finish

3D FT y z k k k x T scan =N y N z 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: kzkz kyky

Case III FT k-space: Image Space: Methods: Sampling kzkz kyky

Case II Nyquist Case III Corner

Case II: Zero-filled FT k-space: Image Space: kzkz kyky

kzkz kyky 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:

Point Spread Functions Un-windowed: Radial Window:

RefCornersRadial

Angular Dependence w/o Zero-filling

Angular Dependence with Zero-filling

Experimental Results  = 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

Fain SB, et al., MRM 42 (1999) Start y z k Finish Overall Image Contrast High Detail Information Sampled Points k Step 2: Mapping to k-Space

Fain SB, et al., MRM 42 (1999) Step 2: Mapping to k-Space

Fain SB, et al., MRM 42 (1999) The Hankel Transform

Fain SB, et al., MRM 42 (1999) Step 3: Transform to Image Space

Fain SB, et al., MRM 42 (1999) Analysis: Spatial Resolution FWHM  2 FOV y  z  TR  11 Full Width at Half Maximum (FWHM) of the Point Spread Function is given by: where, FOV y and FOV z are the phase encoding Fields of View TR is the repetition time  1 is the time to peak enhancement of the bolus curve

Fain SB, et al., MRM 42 (1999) PSF Dependence on Acquisition Time

Fain SB, et al., MRM 42 (1999) 213 sec Z Y 10 sec 50 sec 90 sec Line Pairs/mm Acquisition Time (sec) PSF Dependence on Acquisition Time

Fain SB, et al., MRM 42 (1999) Experiment: FOV z 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 –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: seconds –20 cc Gd

Fain SB, et al., MRM 42 (1999) Left Carotid Artery Stenosis: Reconstruction at Multiple Time Points 33 sec22 sec11 sec44 sec Acquisition 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 sec22 sec33 sec44 sec Acquisition Time: X Z X Z

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

Fain SB, et al., MRM 42 (1999) Increased Scan Time

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

FI = fftshift(fft(fftshift(I))); for i = 1:192, FI_34(i,:) =FI(i,:); end I_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')

for i = 1:129, FI_Herm(i,:) =FI(i,:); end I_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')

count2 = 128; for i = 130:256, FI_Herm(i,:) =conj(FI(count2,:)); count2 = count2-1; end I_Herm2 = fftshift(ifft(fftshift(FI_Herm))); figure;subplot(2,2,1),imagesc(abs(I_Herm2));axis('image'); colorbar;colormap('gray');title('Magnitude') save phase_phantom subplot(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,:); end count2 = 128; for i = 130:256, FIp_Herm(i,:) =conj(FIp(count2,:)); count2 = count2-1; end Ip_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')