k-space Data Pre-processing for Artifact Reduction in MRI
SK Patch
UW-Milwaukee
thanksKF King, L Estkowski, S Rand for comments on presentation
A Gaddipatti and M Hartley for collaboration on Propeller productization.
pitc
h/fr
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ncy
392Hz
G
660Hz
E
523.2Hz
C
pitc
h/fr
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time temporal frequency
log of k-space magnitude data.
apodized
kfF
reconstructed image.
checkerboard pattern strong k-space signal along axes
xfxf FF 1
Heisenberg, Riemann & Lebesgue
Heisenberg Functions cannot be space- and band-limited.
Dx
xf
for
0)(
k
kf
as
0Fimplies
Riemann-Lebesgue k-space data decays with frequency
kaskf 0F
Cartesian sampling
reconstruct directly with Fast Fourier Transform (FFT)
Ringing near the edge of a disc. Solid line for k-space data sampled on 512x512; dashed for 128x128; dashed-dot on 64x64 grid.
spirals – fast acquisition From Handbook of MRI Pulse Sequences.
non-Cartesian sampling
requires gridding additional errors
Propeller – redundant data permits motion correction.
CT errors high-frequency &
localized
MR errors low-frequency &
global
CT vs. MRI
high-order interp overshoots low-order interp smoothsnaive k-space gridding corrected for gridding errors
linear interpolation = convolve w/“tent” function
“gridding” = convolve w/kernel (typically smooth, w/small support)
convolution – “shift & sum”
dyyxeyfxef
0016sincsin dyyy 16216sincsin dyyy
dyyxy 16sincsin
convolution – properties
dyyxeyfxef
kefkfe FFF
2x Field-of-View
xfe
Avoid Aliasing Artifacts
kef FF
sinc interp in k-space kfeF
Avoid Aliasing Artifacts
Propeller k-space data interpolated onto 4x fine grid
xef
sinc interp
xefxef FFF 1
kef FF
xef FFF 1
convolution – properties
dyyxeyfxef
Image Space Upsampling
image from a phase corrected Propeller blade with ETL=36 and readout length=320.
sinc-interpolated up to 64x512.
Image Space Upsampling
Ringing near the edge of a disc. Solid line for k-space data sampled on 512x512; dashed for 128x128; dashed-dot on 64x64 grid.
Reprinted with permission from Handbook of MRI Pulse Sequences. Elsevier, 2004.
Tukey window function in k-space PSF in image space.
k-space apodization
Low-frequency Gridding Errors
linear interpolation “tent” function against which k-space data is convolved
no interpolation-no shading; interpolation onto k/4 lattice 4xFOV
cubic interp
linear interp
k-space data sampled at ‘X’s and linearly interpolated onto ‘’s. cubic interp
linear interp
no interpolation no shading
high-order interp overshootsw/o gridding deconvolution after gridding deconv
xef
sinc interp
kef FF
xef FFF 1
Cartesian sampling
suited to sinc-interpolation
Radial sampling
(PR, spiral, Propeller)
suited to jinc-interpolation
64 256
“fast” conv
kernelperfect jinc
kernel
multiply image
Propeller – Phase Correct
Redundant data must agree, remove phase from each
blade image
RAW
Propeller – Phase Correct one blade
CORRECTED
Propeller - Motion Correct
2 scans – sans motion
sans motion correction
w/motion correction
artifacts due to
blade #1 errors
1 blade # 23
shifts in pixels
rotations in degrees
blade weights
Propeller – Blade Correlation
throw out bad – or difficult to interpret - data
blade #1
Propeller – Blade Correlation
throw out bad – or difficult to interpolate - data
Fourier Transform Properties
shift image phase roll across data
xkkkxx iebb 2 F F
xrbrb * FFF -1 x
b is blade image, r is reference image
xrbrb * FFF -1 x
max at x
No correction, with correction
shifts in pixels
rotate imagerotate data
kkx RfRf F F
Fourier Transform Properties
“holes” in k-space
no correction
correlation correction only
motion correction only
full corrections
Backup SlidesSimulations show Cartesian acquisitions are robust to field inhomogeneity. (top left) Field inhomogeneity translates and distorts k-space sampling more coherently than in spiral scans. (top right) magnitude image suffers fewer artifacts than spiral, despite (bottom left) severe phase roll. (bottom right) Image distortion displayed in difference image between magnitude images with and without field inhomogeneity. k-space stretching decreases the field-of-view (FOV), essentially stretching the imaging object.
Backup Slides
Propeller blades sample at points denoted with ‘o’ and are upsampled via sinc interpolation to the points denoted with ‘’
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