Gilad Liberman^{1} and Benedikt A Poser^{1}

Cartesian sub-sampling patterns play a major role in routine MRI, usually reconstructed using GRAPPA or SENSE and image based regularization. Coil compression is commonly applied to reduce computational load and noise. Software coil compression achieves only mediocre compression factors without compromising signal. Geometrical/ESPIRiT coil-compression use fully-sampled axes, when availables, to improve compression factors without reducing signal or reconstruction level. In this work we present Aliased Coil Compression for Cartesian subsampling patterns, achieving optimal compression without any signal loss. The method is especially useful for alleviating fast image-domain regularization (compressed sensing or deep learning) for available sequences.

*Image-domain approach*:

Let *S*_{ZF} be the zero-filled subsampled data, and *M* be the sensitivity maps obtained e.g. in a separate acquisition. Let *A*, the multi-channel aliased image be the result of applying Fourier transform on *S*_{ZF}, on all channels, and *I*_{A} be the single aliased image resulting from voxelwise dot-multiplication of *A* with the conjugate of *M*. Thus, *I*_{A} is of the size of the required image, exactly k times bigger than the single-channel subsampled data. *I*_{A} will serve as the signal for the image reconstruction. This process is depicted in Figure 1. Let *M*_{ACC,i} , for i=1..k be the result of voxelwise dot-multiplication of *M* with its ((i-1)*FOV/k) shifted version. Thus *M*_{ACC,1} is a uniform ‘1’s map. Figure 2 illustrates the corresponding image-to-signal operator which is composed of the following steps: i) Multiply the image by (each of) the ACC maps *M*_{ACC}. ii) Shift each mapped channel by the appropriate FOV/k shift. iii) Sum along the map direction to get the “aliased signal”. Thus, we have created an optimal coil compression for the k-subsampled case, with 2 beneficial properties: 1) The data is compressed into data that is exactly of the full-resolution image, i.e. the size of k sub-sampled channels. 2) The compression is done voxelwise, without trade-off between preserved signal and noise reduction.

*k-domain approach*:

Let *S*_{cond} be the raw signal condensed, of *N*_{PE}/k rows (all channels). Applying FT on Scond will result in *I*_{cond}, an image were k partitions of the image are overlaid over each other. Similarly to the image-domain method, dot-multiplication with the conjugate of each of the k partitions of size FOV/k of *M* results in the compressed image *I*_{cond,CC}, and its iFT *S*_{ACC}. *S*_{ACC }may be reconstructed using Slice-GRAPPA^{4} or RO-SENSE-GRAPPA^{5}, where the calibration of each ”slice” is a FOV/K (in image-space) partition a full calibration data.

The approach is demonstrated on 2D slices from a MP2RAGE acquired at 7T with 3-fold in-plane sub-sampling with 24 ACS lines.

NB: This abstract stems from a similar observation as our submission entitled 'kCAIPI: Reduction of interleaved 3D acquisition into a set of 2D simultaneous multi-slice (SMS) reconstruction problems'. Their use and meaning diverges, therefore combining them into one would result in a very convoluted message. They should, however, considered as being part of a single conceptual work.

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2. Zhang T, Pauly JM, Vasanawala SS, Lustig M. Coil compression for accelerated imaging with Cartesian sampling. Magn. Reson. Med. 2013;69:571–582. doi: 10.1002/mrm.24267.

3. Bahri, D., Uecker, M., & Lustig, M. (2013). ESPIRIT-Based Coil Compression for Cartesian Sampling. In Proceedings of the 21st Annual Meeting of ISMRM, Salt Lake City, Utah, USA.

4. Setsompop, K., Gagoski, B.A. , Polimeni, J., Witzel, T., Wedeen, V.J. & Wald, L.L. BlippedāControlled Aliasing in Parallel Imaging (blipped-CAIPI) for simultaneous multi-slice EPI with reduced g-factor penaltyMagn. Reson. Med., 67 (2012), pp. 1210-1224

5. Moeller, S., Vu, A. T., Auerbach, E., Ugurbil, K., & Yacoub, E. RO extended FOV SENSE/GRAPPA for multiband imaging with FOV shift. In Proceedings of the 22nd Annual Meeting of ISMRM (2014), Milan, Italy (p. 4396).

Figure 1: Producing the "*Aliased Signal*" used as that data to be fitted by the reconstruction algorithm.

Figure 2: Visualisaion of the image-to-aliased-signal operator.

Figure 3: Results of reconstruction with SENSE and GRAPPA using the proposed coil compression, no coil compression, SCC and GCC, on 4 slices. Coil compression resulted in noise reduction only when using GRAPPA, while SENSE results depended on the regularization terms. ACC resulted in the highest compression and accelerated computation.

Figure 4: Magnitude and phase images obtained by GRAPPA-ACC reconstruction of the 3-fold undersampled data.