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

eque

ncy

392Hz

G

660Hz

E

523.2Hz

C

pitc

h/fr

eque

ncy

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 ‘’