Reconstruction of Non-Cartesian MRI Data

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1 G Practical Magnetic Resonance Imaging II Sacler Institute of Biomedical Sciences New Yor Universit School of Medicine Reconstruction of Non-Cartesian MRI Data Ricardo Otazo PhD

2 Non-Cartesian MRI -space trajector does not fall on a Cartesian grid Radial EPI Spiral Faster more motion robust than Cartesian MRI But reconstruction is more complicated

3 Reconstruction of non-cartesian MRI data Direct FFT won t wor Radial MRI Bacprojection reconstruction lie in CT In general Compute the inverse DFT according to the trajector slow Regridding: resample the non-cartesian MRI data into a Cartesian grid and appl inverse FFT fast

4 Regridding idea Convolve with a -space ernel Evaluate the convolution at the Cartesian grid Wh would this wor? The image support is finite then each point in -space can be estimated b convolution with an infinite sinc

5 Mathematical description of regridding Non-Cartesian sampling function: = i i i S δ Sampled data: S M Convolution with the regridding ernel and the resampling on the Cartesian grid: [ ] = III C S M M ˆ After appling the inverse Fourier transform: [ ] = FOV FOV III c s m m ˆ

6 Effect of regridding operations Original signal Blurring + side lobes Apodization Replication

7 Simple regridding 5-point triangular ernel Radial -space 0000 grid Spiral -space 1818 grid

8 Regridding design considerations Non-Cartesian sampling trajector Sidelobes Densit Convolution ernel Apodization Aliasing Grid densit Aliasing Apodization

9 Sampling densit compensation Non-Cartesian trajectories perform a variable-densit sampling of -space Radial imaging: the central point is acquired N times Non-uniform -space weighting

10 Sampling densit compensation Pre-compensation ideal Sampling densit ρ must be pre-computed Using geometr Assign an area to each -space sample numerical method E.g. Voronoi diagram = III C S M M ˆ ρ 0 -W/ W/ 1/N 1 1/ρ For radial MRI:

11 Sampling densit compensation Post-compensation [ ] = III C S M M 1 ˆ ρ Find ρ b regridding M =1 [ ] = III C S M ρ

12 Sampling densit compensation Radial Spiral Without densit compensation With densit compensation Aliasing

13 Convolution ernel The ideal ernel would be an infinite sinc impractical Windowed sinc Aliasing

14 Convolution ernel Kaiser-Bessel function Best ernel b consensus Inverse Fourier transform = W rect W b I W C I 0 : zero-order modified Bessel function of the first ind W: width of the ernel b: scaling parameter sin b W b W c = π π

15 Oversampling the Cartesian grid Removes aliasing Reduces apodization

16 Oversampling the Cartesian grid X grid Crop in the image domain

17 Deapodization Divide the reconstructed image b the inverse Fourier transform of the regridding ernel Without deapodization Without deapodization With deapodization With deapodization

18 Wh the Kaiser-Bessel ernel is preferred? Less oversampling Triangular Kaiser-Bessel 1.5X grid 1.5X grid

19 Summar of regridding reconstruction Compute the non-cartesian -space sampling pattern Choose the regridding ernel e.g. Kaiser-Bessel Densit pre-compensation if possible Convolve the pre-compensated -space data with the regridding ernel and evaluate the convolution at the Cartesian grid oversampled Appl inverse FFT Appl the de-apodization function Appl densit post-compensation optional Remove the oversampling b cropping the image

20 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 b the MR communit J Fessler Universit of Michigan