Simultaneous multi-slice (SMS) acquisition has recently gained attention in clinical and research applications^1-3. However, since the spatial variation of coil sensitivity along the slice direction is typically insufficient and thus SMS reconstruction including SENSE/GRAPPA and slice-GRAPPA is potentially ill-conditioned, it is challenging to separate the aliased slices in the presence of noise with increasing multi-band factors (MB)^3,4. In this work, we propose a novel, SMS reconstruction method that exploits Hankel subspace learning (SMS-HSL) for aliasing separation in the slice direction^5-8, in which SMS signals are projected onto an individual subspace specific to each slice by incorporating the proposed SMS model into a constrained optimization with low rank and magnitude priors. Simulation and experiments were performed at high MB factors to demonstrate the effectiveness of the proposed SMS-HSL over conventional SMS methods.

1) Aliasing Separation with Hankel Subspace Learning: The hypothesis of this work is that the null space of k-space specific to each slice is spanned by the coil sensitivities and different coil sensitivities will have different null spaces. In multi-band excitation (MB≥2), the aliased SMS signal is modeled as a superposition of multiple slices:

$$ \bf\mathcal{H}\left ( y \right ) = \mathcal{H}\left ( x_s \right ) + \mathcal{H}\left ( x_s^c \right )+ N $$

$$ \bf\mathcal{H}\left ( x_s^c \right ) = \sum_{m=1}^{N_s} \mathcal{H}\left ( x_m \right ) - \mathcal{H}\left ( x_s \right ) $$

where $$$\bf\mathcal{H}$$$ is the Hankel operator, $$$ \bf{y} $$$ is the measured, aliased SMS signals, $$$ \bf{x_s} $$$ and $$$ \bf{x}_s^c $$$ are the k-space for the s^th slice of interest and its complementary signals, and $$$ \bf{N} $$$ is Hankel-structured noise matrix. The aliased Hankel matrix and slice-specific Hankel matrices are projected onto the null space spanned by the complementary null space $$$\bf\mathcal{N}_s^c$$$:

$$ \bf\mathcal{H}\left ( y \right ) \mathcal{N}_s^c = \mathcal{H}\left ( x_s \right ) \mathcal{N}_s^c $$

where $$$\bf\mathcal{N}_s^c$$$ is the null space of $$$\bf\mathcal{H}\left ( x_s^c \right ) $$$, which is spanned by the right singular vectors in the SVD with their singular values below a certain threshold⁸, and $$$\bf\mathcal{H}\left ( x_s^c \right ) \mathcal{N}_s^c $$$ and $$$ \bf{N}\mathcal{N}_s^c $$$ are then assumed to be negligible after the null space projection. Applying the pseudo-inverse of the complementary null space, aliasing separation can be achieved by:

$$ \bf\hat{x}_s = \underset{x_s}{min} \left \| \left ( \mathcal{H}\left ( y \right ) - \mathcal{H}\left ( x_s \right ) \right ) \mathcal{N}_s^c \right \|_ F^2 $$

$$ \bf=\mathcal{H}^\dagger \left ( \mathcal{H}\left ( y \right ) \mathcal{N}_s^c \mathcal{N}_s^{c \ \dagger} \right ) $$

2) SMS-HSL with Low Rank and Magnitude Priors: It is well known that the linear dependency of k-space over all coils promotes a low rank property and thus the Hankel-structured matrix is highly rank-deficient. Additionally, although the phase of k-space in the reference is typically deviant from that in the corresponding SMS estimate due to time-drifting field inhomogeneities, the magnitude of k-space in between reference and its SMS estimate remain similar unless there is large discrepancy between the reference and the SMS acquisitions. Given the consideration above, we propose a new optimization framework with low rank and magnitude priors:

$$ \bf\hat{x}_s = \underset{x_s}{min} \left \| \left ( \mathcal{H}\left ( y \right ) - \mathcal{H}\left ( x_s \right ) \right ) \mathcal{N}_s^c \right \|_ F^2 + \lambda_L \left \| \mathcal{H}\left ( x_s \right ) \right \| $$

$$\bf{s.t. \ \ r_s=\left | Dx_s \right |}$$

where $$$ \left \| \cdot \right \|_*$$$ is the nuclear norm, $$$ \bf\lambda_L $$$ is the regularization parameter, $$$ \bf{r_s} $$$ is the magnitude of the reference for the s^th slice, and $$$ \bf{D} $$$ is a mask matrix specified by the reference. This is solved using phase retrieval and variable splitting methods under the framework of alternating direction approach.Simulations and experiments were performed to evaluate the performance of SMS-HSL using both the retrospectively emulated and prospectively acquired SMS data like CAIPIRINHA. For comparison, images were reconstructed using slice-GRAPPA. Fig. 1 shows the correlation maps between null spaces in the dual-band excitation and its corresponding SMS-HSL images changing the slice distances and FOV shifts. As expected, the correlation is decreased as both the slice distance and the FOV shift are increased, thus well separating the multiple slices while artifacts and noise are reduced. Fig. 2 represents brain images at MB=6. The SMS-HSL results in robust reconstruction though slightly amplified noise is visible. Fig. 3 demonstrates the effectiveness of SMS-HSL in suppressing artifacts and noise using actual SMS acquisition at MB=4.We proposed a novel SMS reconstruction, and then successfully demonstrated the superior performance of the proposed SMS-HSL over existing method, exhibiting advantages with low rank and magnitude priors. It is expected that SMS-HSL would enable more rapid SMS imaging and widen its applications.