Aliased Coil Compression
Gilad Liberman1 and Benedikt A Poser1

1Faculty of Psychology and Neuroscience, Maastricht University, Maastricht, Netherlands


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.


Under ideal conditions, parallel-acquired data can be summarized by the sample and a single set of sensitivity maps. When full sampling is given, the image can be reconstructed from the separate channels’ images by combining them by voxelwise dot-product with the conjugated sensitivity maps. The redundancy can also be exploited by compressing the channels with e.g. SVD, (i.e. SCC, Software coil-compression1), resulting in reduction of the noise and reduced computation with minimal signal loss. A more efficient compression can be achieved by Geometrical coil-compression2 (GCC), ESPIRiT coil compression3 (ECC), with space-localized coil compression matrices along fully-sampled axes. Under sub-sampling along one axes, these methods use a single compression matrix along that axis, resulting in a trade-off between compression and signal loss, which is especially problematic in UHF, where sensitivity maps are faster-varying spatially. Reconstructing k-subsampled data deals with separating an image from its Nyquist ghosts, i.e. FOV/k shifted versions of the image. This is analogous to a SMS problem, where each band is an FOV/k portion of the image. The g-factor maps associated with such problem are defined by the difference between the sensitivity maps of the image and a shifted version of themselves. Thus, a k-subsampled image can be optimally coil-compressed to exactly k virtual coils, where one of the is the uniform ‘1’s coil, and the others are the results of localized dot-multiplication of the sensitivity maps with its conjugated shifted replica.

Theory and Methods

Image-domain approach:

Let SZF 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 SZF, on all channels, and IA be the single aliased image resulting from voxelwise dot-multiplication of A with the conjugate of M. Thus, IA is of the size of the required image, exactly k times bigger than the single-channel subsampled data. IA will serve as the signal for the image reconstruction. This process is depicted in Figure 1. Let MACC,i , for i=1..k be the result of voxelwise dot-multiplication of M with its ((i-1)*FOV/k) shifted version. Thus MACC,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 MACC. 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 Scond be the raw signal condensed, of NPE/k rows (all channels). Applying FT on Scond will result in Icond, 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 Icond,CC, and its iFT SACC. SACC may be reconstructed using Slice-GRAPPA4 or RO-SENSE-GRAPPA5, 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.


Figure 3 shows reconstructed images using the proposed approach and using SCC and GCC. For SENSE reconstruction, ACC preserved the image quality obtained without compression, however enjoying reduced memory use and computational time. For L1 regularized CG-SENSE, using SCC and GCC did not result in reduced noise in our experiments. GRAPPA reconstruction did reproduce the effect. Figure 4 shows ACC for GRAPPA; This approach exhibited some degree of N/3 banding is which is attributable to the low number of 24 auto-calibration lines, resulting in a 3x8 matrix after the reshaping operation.

Discussion and Conclusion

The ACC scheme for uniformly subsampled acquisition patterns is optimal in compression factor and no loss of signal. It trivially generalizes to multi-map and multi-band, as well as to CAIPI and 2D CAIPIRINHA patterns. ACC is particularly promising for image-domain reconstruction techniques and where memory resources are limited.


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.


1. Buehrer M, Pruessmann KP, Boesiger P, Kozerke S. Array compression for MRI with large coil arrays. Magn. Reson. Med. 2007;57:1131–1139. doi: 10.1002/mrm.21237.

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. Blippedā€Controlled 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.

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