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

The Fourier Transform (FT) of a vector of N=N_{1}⋅N_{2} elements is decomposable into N_{1} FTs of N_{2}-sized vectors followed by N_{2} FTs of N_{1}-sized vectors, a fact utilized iteratively to produce the Fast FT algorithm. Put in MRI terminology, reconstructing N=k⋅M slices from k-undersampled *k _{}*

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.

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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 *k*_{z} 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.