Pseudo-Cartesian k-Space Interpolation Using Artificial Neural Networks
Nikolai J Mickevicius1, Eric S Paulson1, and Andrew S Nencka2

1Radiation Oncology, Medical College of Wisconsin, Milwaukee, WI, United States, 2Radiology, Medical College of Wisconsin, Milwaukee, WI, United States


This work aims to extend the RAKI method for artificial intelligence-based k-space interpolation to non-Cartesian acquisitions. It was tested in radial acquisitions up to acceleration factors of 7. This method performs similarly well, or better than total-variation regularized sensitivity encoding.


Non-Cartesian k-space trajectories typically allow for higher acceleration factors relative to Cartesian acquisitions due to the spreading of aliasing energy throughout the entire image. Parallel imaging[1], compressed sensing[2], and regularized inversion[3] methods have been developed over the years to accelerate non-Cartesian data acquisition and thus reduce scan time. More recently, the use of artificial intelligence has been incorporated into reconstruction pipelines with promising results. The RAKI method was presented as an alternative to GRAPPA by using a simple 3-layer neural network to perform a non-linear k-space interpolation[4]. The application of RAKI, in its initial presentation, was limited to uniformly-accelerated Cartesian acquisitions due to the use of regularly-shaped convolution kernels. In this work, we extend the capabilities of RAKI to interpolate skipped Cartesian k-space samples from undersampled GROG-gridded[5] non-Cartesian data.


The Pseudo-Cartesian Robust Artificial-neural-network for K-space Interpolation (PC-RAKI) method was implemented as a regression in Keras with a TensorFlow backend. For each non-acquired Cartesian grid location, a kernel selecting two nearby source points was determined. A separate neural network was trained for each kernel shape to reconstruct missing target points. Each network consisted of three layers. The input to the network was the concatenated real/imaginary k-space values for each coil and source point. The output size for each layer was as follows: #1: 128, #2: 128, #3: 2*Ncoils. The sizes for layers 1 and 2 were chosen heuristically. The multiplicative factor of two in the final layer accounts for the real/imaginary components of the target point. Rectified linear unit (ReLU) activations were applied in the 1st and 2nd layers. The PC-RAKI method is summarized graphically in Figure 1. The networks were trained using 2000 randomly selected target points in the NUFFT-gridded[6] Cartesian k-space from the fully sampled radial acquisition. Note: like standard RAKI, this is a database-free, scan-specific deep learning approach.

A parallel imaging-based, nearest-neighbor gridding (via GROG) of non-Cartesian k-space samples was performed with golden-angle radial data acquired on an Elekta 1.5T MR-Linac. Eight receive coils were used. The abdomen of a free-breathing, consenting healthy volunteer was scanned using a spoiled gradient echo 3D stack-of-stars acquisition. An in-plane matrix size of 256x256 with 40 slice partitions was prescribed. The data were retrospectively undersampled to 144, 89, and 55 spokes corresponding to acceleration factors of R=2.8, R=4.5, and R=7.3, respectively. To interpolate the skipped points for each acceleration factor, 25, 69, and 156 kernels (i.e. trained networks) were required, respectively. For a central slice of the acquisition, PC-RAKI was compared with a reference image (610 spokes), NUFFT, GROG-gridding, and total variation (TV)-regularized CG-SENSE[7].


The images from the 144, 89, and 55 spoke reconstructions can be seen in Figures 2-4. PC-RAKI is able to largely remove residual undersampling artifacts, even at an acceleration factor of 7. Some streaking artifacts still remain in the PC-RAKI images, but this is likely due to the poor kernel geometry employed for ease of initial implementation. On a single 2.8 GHz CPU, each network took approximately one second to train.


This initial PC-RAKI implementation offers promising results compared with the standard CG-SENSE algorithm in reconstructing accelerated non-Cartesian data. Future implementations will incorporate more source points with more optimal kernel shapes such as those described in Seiberlich et. al. (2008)[8]. This should allow better performance at higher acceleration factors. Also, an extension of this method to simultaneous multislice acquisitions will be made. The use of GPUs will of course be incorporated for improved computation times in future work.


PC-RAKI shows promise for removing undersampling artifacts from accelerated radial acquisitions. It may be useful in reducing scan time for free-breathing abdominal exams pending developments aimed to reduce computation time of the scan-specific neural networks.


No acknowledgement found.


[1] Pruessmann KP, Weiger M, Börnert P, Boesiger P. Advances in sensitivity encoding with arbitrary k-space trajectories. Magn Reson Med 2001;46:638–51. doi:10.1002/mrm.1241.

[2] Lustig M, Donoho D, Pauly JM. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magn Reson Med 2007;58:1182–95. doi:10.1002/mrm.21391.

[3] Uecker M, Zhang S, Frahm J. Nonlinear inverse reconstruction for real-time MRI of the human heart using undersampled radial FLASH. Magn Reson Med 2010;63:1456–62. doi:10.1002/mrm.22453.

[4] Akçakaya M, Moeller S, Weingärtner S, U─čurbil K. Scan-specific robust artificial-neural-networks for k-space interpolation (RAKI) reconstruction: Database-free deep learning for fast imaging. Magn Reson Med 2018. doi:10.1002/mrm.27420.

[5] Seiberlich N, Breuer FA, Blaimer M, Barkauskas K, Jakob PM, Griswold MA. Non-Cartesian data reconstruction using GRAPPA operator gridding (GROG). Magn Reson Med 2007;58:1257–65. doi:10.1002/mrm.21435.

[6] Fessler JA, Sutton BP. Nonuniform fast fourier transforms using min-max interpolation. IEEE Trans Signal Process 2003;51:560–74. doi:10.1109/TSP.2002.807005.

[7] Block KT, Uecker M, Frahm J. Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint. Magn Reson Med 2007;57:1086–98. doi:10.1002/mrm.21236.

[8] Seiberlich N, Breuer F, Heidemann R, Blaimer M, Griswold M, Jakob P. Reconstruction of undersampled non-Cartesian data sets using pseudo-Cartesian GRAPPA in conjunction with GROG. Magn Reson Med 2008;59:1127–37. doi:10.1002/mrm.21602.


The radial data are interpolated onto a Cartesian grid via GROG. The kernel shapes necessary to interpolate missing data points are determined. For each kernel shape, a 3-layer Neural network is trained using Cartesian calibration data. The weights are then applied to the undersampled data to create the interpolated data output.

Results for R=2.8 (144 spokes)

Results for R=4.5 (89 spokes)

Results for R=7.3 (55 spokes)

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)