Fast parallel image reconstructions with non-cartesian trajectories (2003)

Synopsis: A fast reconstruction approach for partially parallel acqusition (PPA) techniques with non-Cartesian sampling trajectories is presented in this abstract. In our approach, the regridding of the non-Cartesian sampled lines is done with a PPA reconstruction. Necessary reconstruction parameters for a shift onto a cartesian grid are derived by a SMASH-like procedure. Instead of calculating these parameters for every required shift, this calculation is performed for only a few shifts. Reconstruction parameters for intermediate shifts are interpolated resulting in dramatically reduced reconstruction times. Introduction: Non-Cartesian sampling present unique problems for parallel imaging reconstructions. In general, large linear systems are solved to reconstruct images from undersampled data (1). An alternative was proposed by Yeh et al (2), which uses non-integer shifts for a SMASH like reconstruction. This method has been extended so that more than 1 point is used in each reconstruction, however this results in a drastic increase in reconstruction time, beyond that of conjugate-gradient methods (3). In this work continuously variable density (VD) trajectories are used, similar to the VD-sampling proposed in (4), using a non-Cartesian sampling trajectory. The distance between adjacent lines in this acquisition grows continously from a minimal to a maximal value. In our approach, we use a 2 step approach. First, each line is reconstructed onto a cartesian grid position using reconstruction parameters determined by a SMASH (5) like procedure. However, instead of using a separate inversion for each required shift, the reconstruction parameters for a small number of shifts are calculated allowing the weights for intermediate shifts to be determined by interpolation, resulting in increased reconstruction speed. Missing lines are then reconstructed in a second step using integer shifts from the regridded data. Methods: For the simulations shown here, the distance between adjacent lines in the non-Cartesian VD-sampling scheme varies continiously from 0.5 to 3.0 ∆ky. Asuming a cartesian grid from -128 to +127 ∆ky, 256 phase encoding steps are necessary for the acquisition. The same range of ky can be covered with 128 VD-lines on a non-Cartesian grid, as shown in Fig. 1 in the middle. In the first step, all acquired VD-lines are regridded onto their nearest Cartesian grid position using a SMASH reconstruction with fractional harmonics. For the reconstruction, only 11 sets of reconstruction parameters were calculated, covering shifts from -∆ky to +∆ky in steps of 0.1 (see Fig.1 right, gray points). The reconstruction parameters vary slowly over the fractional harmonic values, so that all intermediate shifts necessary for the whole reconstruction can be easily interpolated from these 11 calculated sets. In a second step missing Cartesian lines in the outer part of k-space were reconstructed with the ± 1.0 ∆ky reconstruction parameters using the regridded Cartesian data lines from adjacent positions. Results: In order to demonstrate the feasibility of our approach, results from simulations are shown in Fig. 2. The direct Fourier Transformation (dFT) of a nonCartesian VD-data set with 128x256 matrix size is shown in Fig. 2a. Besides the ringing artifacts due to the low resolution, foldover artifacts are clearly visible as a result of the non uniform FOV. Application of our approach to the same data set leads to significantly reduced artifacts, see Fig. 2b. Applying the procedure in +ky and –ky direction allows one to reconstruct 256 cartesian lines from 128 VD-lines, resulting in reduced ringing artifacts. Discussion: In this work we have demonstrated that it is feasible to use interpolated reconstruction parameters for parallel imaging using a 1D non-Cartesian VD-sampling trajectory. The flexible implementation of our approach allows one to use other regenerative k-space based PPA methods, such as AUTO-SMASH (6) or VDAUTO-SMASH (7) to determine the reconstruction parameters. This technique could easliy be extended to 2D non-Cartesian trajectories (e.g. spiral) using multidimensional interpolation as well. References: (1) Yeh E, et al. ISMRM 2002 #2390 (2) Yeh E, et al. ISMRM 2001 #1796 (3) Pruessmann K, et al. Magn Reson Med 2001;46:638-651. (4) Kyriakos W, et al. Magn Reson Med 2000;44:301-308. (5) Sodickson D, et al. Magn Reson Med 1997;38:591-603. (6) Jakob P, et al. MAGMA 1998;7:42-54. (7) Heidemann R, et al. Magn Reson Med 2001;45:1066-1074. a b