k-space Data Pre-processing
for Artifact Reduction in MRI
SK Patch
UW-Milwaukee
thanks
KF King, L Estkowski, S Rand for comments on presentation
A Gaddipatti and M Hartley for collaboration on Propeller productization.
660Hz
523.2Hz
392Hz
G
E
C
temporal
frequency
time
f x   F Ff x 
Ff k 
1
apodized
log of k-space magnitude data. reconstructed image.
checkerboard pattern 
strong k-space signal along
axes
Heisenberg, Riemann &
Lebesgue
Heisenberg Functions cannot be space- and band-limited.
f ( x)  0
implies
Ff k   0
for x  D
as k  
Riemann-Lebesgue k-space data decays with frequency
Ff k   0 as k  
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.
non-Cartesian sampling
requires gridding 
additional errors
spirals – fast acquisition
From Handbook of MRI Pulse
Sequences.
Propeller –
redundant
data permits motion
correction.
CT vs. MRI
CT errors
high-frequency &
localized
MR errors
low-frequency &
global
high-order
naive k-space
interp overshoots
gridding
corrected
low-orderforinterp
gridding
smooths
errors
linear interpolation =
convolve w/“tent”
function
“gridding” = convolve
w/kernel (typically
smooth, w/small support)
convolution – “shift & sum”
 f  e x    f  y  ex  y  dy
 sin y sinc 16x  y  dy




sin
y
sinc
16




sin
y
sinc
16
0

y
dy

0
2  y dy 



16
convolution –
properties
 f  e x    f  y  ex  y  dy
F fe k   Ff  Fek 
Avoid Aliasing Artifacts
 fe x
2x Field-of-View
F fe k 
sinc interp in
k-space
Ff  Fek 
Avoid Aliasing Artifacts
Propeller k-space data interpolated onto
4x fine grid
convolution –
properties
 f  e x    f  y  ex  y  dy
F Ff Fex    f  ex 
1
Image Space Upsampling
F Ff Fex 
1
Ff Fek 
 f  ex
Image Space Upsampling
image from a phase
corrected Propeller blade
with ETL=36 and readout
length=320.
sinc-interpolated up to
64x512.
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.
Low-frequency Gridding Errors
no interpolation-no shading;
interpolation onto Dk/4 lattice  4xFOV
linear interpolation
“tent” function
against which
k-space data is
convolved
cubic interp
linear interp
k-space data sampled
at ‘X’s and linearly
interpolated onto ‘’s.
no interpolation
no shading
cubic interp
linear interp
high-order
w/o gridding
interp
deconvolution
overshoots
after gridding deconv
Cartesian sampling
suited to sinc-interpolation
F Ff Fex 
1
Ff Fek 
 f  ex
Radial sampling
(PR, spiral, Propeller)
suited to jinc-interpolation
multiply image
Propeller – Phase Correct
Redundant data must agree,
remove phase from each
blade image
Propeller – Phase Correct one blade
CORRECTED
RAW
Propeller - Motion Correct
2 scans
– sans
motion
sans motion
correction
w/motion
correction
artifacts due to
blade #1 errors
Propeller – Blade Correlation
interpret - data
throw out bad – or difficult to interpolate
- data
blade weights
blade #1
rotations in degrees
1
blade #
23
shifts in pixels
Fourier Transform Properties
shift image  phase roll across data
F bx  Dxk   F bk e
2ik Dx
b is blade image, r is reference image


F Fb Fr x   b * r x 
-1
max at Dx


F Fb Fr x   b * r x 
-1
No correction,
with correction
shifts in pixels
Fourier Transform Properties
F  f R x k   Ff R k 
rotate imagerotate data
“holes” in k-space
no correction
correlation
correction only
motion
correction only
full corrections
Backup Slides
Simulations 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 fieldof-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 ‘’