Ricardo Otazo, PhD [email protected]
G16.4428 – Practical Magnetic Resonance Imaging II Sackler Institute of Biomedical Sciences New York University School of Medicine
Reconstruction of Non-Cartesian MRI Data
Non-Cartesian MRI • k-space trajectory does not fall on a Cartesian grid
Radial Spiral EPI
• Faster, more motion robust than Cartesian MRI
• But, reconstruction is more complicated …
Reconstruction of non-Cartesian MRI data • Direct FFT won’t work
• Radial MRI
– Backprojection reconstruction, like in CT
• In general – Compute the inverse DFT according to the trajectory
(slow) – Regridding: resample the non-Cartesian MRI data into
a Cartesian grid and apply inverse FFT (fast)
Regridding idea • Convolve with a k-space kernel • Evaluate the convolution at the Cartesian grid
Why would this work? The image support is finite, then each point in k-space can be estimated by convolution with an infinite sinc
Mathematical description of regridding • Non-Cartesian sampling function: ( )∑ −−=
iiyyixxyx kkkkkkS ,, ,),( δ
• Sampled data: ),(),( yxyx kkSkkM
• Convolution with the regridding kernel and the resampling on the Cartesian grid:
( )[ ]
∆∆×∗=
y
y
x
xyxyxyxyx k
kk
kIIIkkCkkSkkMkkM ,),(),(),(),(ˆ
• After applying the inverse Fourier transform:
( )[ ]
×∗=
yx FOVy
FOVxIIIyxcyxsyxmyxm ,),(),(),(),(ˆ
Effect of regridding operations
Original signal
Blurring + side lobes
Apodization
Replication
Simple regridding • 5-point triangular kernel
Radial k-space 200x200 grid
Spiral k-space 128x128 grid
Regridding design considerations • Non-Cartesian sampling trajectory
– Sidelobes – Density
• Convolution kernel
– Apodization – Aliasing
• Grid density – Aliasing – Apodization
Sampling density compensation • Non-Cartesian trajectories perform a variable-density
sampling of k-space – Radial imaging: the central point is acquired N times
• Non-uniform k-space weighting
Sampling density compensation • Pre-compensation (ideal)
– Sampling density (ρ) must be pre-computed
– Using geometry
– Assign an area to each k-space sample (numerical method) • E.g. Voronoi diagram
∆∆×
∗
=
y
y
x
xyxyx
yx
yxyx k
kk
kIIIkkCkkSkkkkM
kkM ,),(),(),(),(
),(ˆρ
k 0 -W/2 W/2
1/N
1
1/ρ(k)
For radial MRI:
Sampling density compensation • Post-compensation
( )[ ]
∆∆×∗=
y
y
x
xyxyxyx
yxyx k
kk
kIIIkkCkkSkkMkk
kkM ,),(),(),(),(
1),(ˆρ
• Find ρ by regridding M(kx,ky)=1
( )[ ]
∆∆×∗=
y
y
x
xyxyxyxyx k
kk
kIIIkkCkkSkkMkk ,),(),(),(),(ρ
Sampling density compensation Radial Spiral
Without density
compensation
With density
compensation
Aliasing
Convolution kernel • The ideal kernel would be an infinite sinc (impractical)
• Windowed sinc
Aliasing
Convolution kernel • Kaiser-Bessel function
– Best kernel (by consensus)
– Inverse Fourier transform
−=
Wkrect
WkbI
WkC 2211)(
2
0
I0: zero-order modified Bessel function of the first kind W: width of the kernel b: scaling parameter
( )2222
2222sin)(bxW
bxWxc−
−=
ππ
Oversampling the Cartesian grid • Removes aliasing • Reduces apodization
Oversampling the Cartesian grid 2X grid
Crop in the image
domain
Deapodization • Divide the reconstructed image by the inverse Fourier
transform of the regridding kernel
With
out d
eapo
diza
tion
With
dea
podi
zatio
n
Without deapodization With deapodization
Why the Kaiser-Bessel kernel is preferred? • Less oversampling
Triangular Kaiser-Bessel
1.5X grid
1.25X grid
Summary of regridding reconstruction • Compute the non-Cartesian k-space sampling pattern • Choose the regridding kernel (e.g. Kaiser-Bessel) • Density pre-compensation (if possible) • Convolve the pre-compensated k-space data with the
regridding kernel and evaluate the convolution at the Cartesian grid (oversampled)
• Apply inverse FFT • Apply the de-apodization function • Apply density post-compensation (optional) • Remove the oversampling by cropping the image
Non-Uniform FFT (NUFFT) • Generalized version of the regridding algorithm • Similar idea, but fast implementation • Forward and inverse implementation
– Interesting for iterative algorithms
• Popular implementation used by the MR community – J Fessler, University of Michigan
