kCAIPI: Reduction of interleaved 3D acquisition into a set of 2D simultaneous multi-slice (SMS) reconstruction problems
Gilad Liberman1 and Benedikt A Poser1

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


The Fourier Transform (FT) of a vector of N=N1⋅N2 elements is decomposable into N1 FTs of N2-sized vectors followed by N2 FTs of N1-sized vectors, a fact utilized iteratively to produce the Fast FT algorithm. Put in MRI terminology, reconstructing N=k⋅M slices from k-undersampled kz-stacked trajectory can be achieved by FT, followed by solution of the M SMS problems of k slices. This can be used to reduce such 3D reconstruction problems into SMS problems, reducing memory and computational demands. The observation extends to CAIPI patterns. We term this approach kCAIPI.


3D acquisitions “partition encoding” along the second phase-encode (slice) axis are advantageous due to their greater SNR, and the shorter excitation TR reduces flip angles which alleviates RF power constraints. Moreover, 3D acquisitions provide greater flexibility in trajectory design to better exploit the the spatial encoding power of multi-coil receive arrays. 2D simultaneous multi-slice (SMS) schemes however enjoy much reduced computational demands, and independent, parallelizable reconstruction of single-shot data. SMS has improved 2D imaging by gaining back some of the SNR benefit, with minimal g-factor penalty when combined with CAIPIRINHA techniques 1,2. A common approach to reduce the computational demand of 3D reconstruction is the use of “stacked” trajectories, where the same (possibly non-Cartesian) 2D trajectory is repeated for different kz partitions on a Cartesian grid, followed by FT along kz and a per-slice in-plane reconstruction. We here note that an improvement analogous to that enabled by SMS-CAIPI for 2D towards 3D is possible for 3D towards 2D: using an interleaved multi-shot, multi-partition trajectory (Fig 3A) followed by FT along kz, the aliasing results in a reconstruction problem equivalent to that of SMS imaging 3. That is, retaining the SNR benefit of 3D as well as much of the freedom in trajectory design, while enjoying much reduced computational demands of a partitioned reconstruction.


The decomposability of the Fourier transform, i.e. the fact that the FT of a vector of length N=p⋅m can be calculated by combining p FTs of subvectors of length m thorough m FTs of the resulting vectors of length p (see Fig. 1), has first been utilized by Gauss 4 and laid the basic step of the Cooley-Tukey FFT algorithm 5. In MRI terminology, let z be a vector (of e.g. 12=4x3 “slices”) and k its iFT (value in each partition). The FT of k can be calculated by reordering k as below (each column is an undersampling of k), and FT along the rows: the result is a matrix where the values of each row come from the same 3 slices, analogously to SMS. Then (after multiplying by factors correcting for each column position in the z-axis, known in the FFT literature as “twiddle numbers”), FT along the row dimension, i.e. resolve the SMS bands. Similar observation led to the development of 2D CAIPIRINHA 6 limited to Cartesian patterns, and k-t SENSE 7 for dynamic acquisitions.


Cartesian data: The first inversion of an MP2-RAGE acquisition (0.7mm3 isovoxel, factor-3 subsampling, 7T Magnetom with 32ch head coil along PE, and fully-sampled along kz), was reduced along the z-axis to simulate a narrow slab of 5cm. For reference, as FT was then applied along z and 2 slices of half-slab apart were reconstructed (i) separately. The suggested approach was applied to reconstruct the 2 slices together, using (ii) the full (3x-PE-undersampled) data, (iii) by further undersampling of 2x along the z axis, and (iv) by further 2x undersampling along the z in a caipi fashion. All reconstructions were done using the pics module in BART 8 with l1 wavelet regularization 9. Spiral data: Stack-of-spiral data with 2 interleaves was acquired. The suggested approach was applied to reduce the volumetric data into separate 2-band CAIPIRINHA SMS reconstruction problems. These were reconstructed using an extension of the BART pics module that corrects for B0-field inhomogeneities, with wavelet regularization.


Figure 2 shows the results on data with fold undersampling along PE, using ‘single-slice’ reconstruction (i.e. after FT the full kz dimension), as a MB reconstruction, (i.e. after FT the odd and even field of kz), and by adding 2-subsampling along kz: by FT the subsampled data, and by using a CAIPI pattern. Comparing the reconstructions of the 6x accelerated data reveals that the CAIPI pattern reduces artifacts (red arrow) due to the better use of the coil geometry. Figure 3 shows reconstruction of the 2-shot interleaved stack-of-spirals data, using kCAIPI, i.e. reducing the 3D reconstruction into 2-band SMS reconstructions. The through-plane slices taken after reconstructing the full volume indicate the successful reconstruction.

Discussion and Conclusion

Using the FT directly on undersampled/interleaved data results in a reconstruction problem that is equivalent to the corresponding SMS problem. This observation may ease the application of newly developed reconstruction techniques for 2D and SMS (e.g. deep learning, dictionary and sparse coding learning, low rank regularization, etc..) to 3D acquired data, pulling benefits from both worlds.

Using the FT directly on undersampled/interleaved data results in a reconstruction problem that is equivalent to the corresponding SMS reconstruction. This observation may ease the application of newly developed reconstruction techniques for 2D and SMS (e.g. deep learning, dictionary and sparse coding learning, low rank regularization, etc..) to 3D acquired data, pulling benefits from both worlds.


NB: This abstract stems from a similar observation as our submission on ' Aliased Coil Compression'. Their use and meaning diverges, so combining into one would result in a very convoluted message. They should, however, considered as being part of a single conceptual work.


1. Breuer FA, Blaimer M, Heidemann RM, Mueller MF, Griswold MA, Jakob PM. Controlled aliasing in parallel imaging results in higher acceleration (CAIPIRINHA) for multi-slice imaging. Magn. Reson. Med. 2005;53:684–691. doi: 10.1002/mrm.20401.

2. Setsompop K, Gagoski BA, Polimeni JR, Witzel T, Wedeen VJ, Wald LL. Blipped-controlled aliasing in parallel imaging for simultaneous multislice echo planar imaging with reduced g-factor penalty. Magn. Reson. Med. 2012;67:1210–1224. doi: 10.1002/mrm.23097.

3. Zahneisen B, Ernst T, Poser BA. SENSE and simultaneous multislice imaging. Magn. Reson. Med. 2015;74:1356–1362. doi: 10.1002/mrm.25519.

4. Gauss CF. Theoria interpolationis methodo nova tractata Werke band 3, 265–327.

5. Cooley JW, Tukey JW. An algorithm for the machine calculation of complex Fourier series. Math. Comput. 1965;19:297–297. doi: 10.1090/S0025-5718-1965-0178586-1.

6. Breuer FA, Blaimer M, Mueller MF, Seiberlich N, Heidemann RM, Griswold MA, Jakob PM. Controlled aliasing in volumetric parallel imaging (2D CAIPIRINHA). Magn. Reson. Med. 2006;55:549–556. doi: 10.1002/mrm.20787.

7. Tsao J, Boesiger P, Pruessmann KP. k-t BLAST and k-t SENSE: dynamic MRI with high frame rate exploiting spatiotemporal correlations. Magn. Reson. Med. 2003;50:1031–1042. doi: 10.1002/mrm.10611.

8. Uecker M, Tamir J, Ong F, Holme C, Lustig M. BART: version 0.4.01. 2017.

9. Lustig M, Donoho D, Pauly JM. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magn. Reson. Med. 2007;58:1182–1195. doi: 10.1002/mrm.21391.


Figure 1: Decomposing the FT into subproblems. The latter part can be done using e.g. coil sensitivities, rather than using fully acquired data.

Figure 2: Reconstruction of fully sampled kz in (A) a single-slice manner, (B) as an SMS reconstruction, and reconstruction of further 2x subsampling along kz reduced using kCAIPI into SMS problem, (C) without and (D) with interleaved pattern.

Figure 3: Left: 2-shot interleaved stack-of-spirals trajectory. MIddle and Right: Images that results from the kCAIPI reconstructions, consisting of a series of 2-band SMS reconstructions to reduce memory and computational requirements.

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