Introduction

Structured low-rank (SLR) methods have been used in a variety of MR image reconstruction problems, and offer great flexibility to exploit the intrinsic linear dependency in single channel images¹, or across multiple channels/shots/contrasts of MR data^2-6. SLR methods formulate this linear dependency in MR data as the low-rank property of a structured Hankel matrix generated from k-space data. These methods have also been extended to multi-dimensional structured tensor problems^6,7. While conventional SLR methods enforce the low-rank constraint on the entire structured matrix, we recently proposed the Locally Structured Low-rank method (LSLR), which relaxes the low-rank constraint by enforcing it on submatrices of the Hankel structured matrix. Here we extend the LSLR method to structured low-rank tensor reconstruction, and demonstrate its benefit over conventional SLR approaches on a calibration-less, multi-echo joint reconstruction of retrospectively under-sampled GRE data.

Methods

The LSLR method is demonstrated here for a calibration-less multi-echo joint tensor reconstruction using a non-convex formulation:
$$\widehat{X}=argmin\left \| EX-Y \right \|^{2}\\s.t. rank(\Gamma _iH_1X)=r_1,\forall {\Gamma _i}\in\Omega _1 \\ ~~~~~ rank(\Gamma_jH_2X)=r_2,\forall {\Gamma _j}\in\Omega _2$$
Where $$$X\in\mathbb{C} ^{n^2 \times N_c \times N_e}$$$ corresponds to the k-space data of a $$$n \times n$$$ image with $$$N_c$$$ channels and $$$N_e$$$ echoes. $$$Y\in\mathbb{C} ^{N_k\times N_c\times N_e}$$$denotes the sampled data and E represents the sampling operator. We define an underlying Hankel transform operator for multi-coil data with a $$$n \times n$$$ patch size as $$$H:\mathbb{C} ^{n^2 \times N_c}\rightarrow \mathbb{C} ^{l\times d^2 N_c}$$$ where $$$l=(n-d+1)^2$$$, and a low-rank tensor $$$\mathbb{C} ^{l\times d^2 N_c \times N_e}$$$ can be constructed using an additional echo dimension. The operator $$$H_1:\mathbb{C} ^{n^2 \times N_c \times N_e}\rightarrow \mathbb{C} ^{l\times d^2 N_c N_e}$$$ concatenates the Hankel matrix representation of each echo across the column (channel) dimension, and $$$H_2:\mathbb{C} ^{n^2\times N_c \times N_e}\rightarrow \mathbb{C} ^{lN_e\times d^2 N_c}$$$concatenates them along the row (patch) dimension, so $$$H_1X$$$ and $$$H_2X$$$ correspond to the one-mode and two-mode unfolding matrices of the low-rank tensor as defined in⁶. $$$r_1$$$ and $$$r_2$$$ are two rank parameters, and $$$\Gamma _i$$$ and $$$\Gamma _j$$$ are submatrix selection operators (see Fig.1). $$$\Gamma _i:\mathbb{C} ^{l \times d^2N_cN_e}\rightarrow \mathbb{C} ^{s \times d^2N_cN_e}$$$selects a submatrix of $$$s$$$ consecutive rows from $$$H_1X$$$ . Similarly, $$$\Gamma _j:\mathbb{C} ^{l N_e \times d^2N_c}\rightarrow \mathbb{C} ^{s \times d^2N_c}$$$selects a submatrix of $$$s/N_e$$$ consecutive rows from the Hankel matrix of each echo (same relative position) and horizontally concatenate them together. $$$\Omega _1$$$ is the partition consisting of all possible $$$\Gamma _i$$$-selecting non-overlapping submatrices of $$$H_1X$$$, and similarly for $$$\Omega _2$$$ with $$$\Gamma _j$$$ and $$$H_2X$$$. As the Hankel structured matrix $$$H_*X$$$ is assumed to be low-rank in conventional SLR reconstructions, each submatrix should also be low-rank. The optimization was solved by ADMM^8. To avoid boundary artifacts, the submatrices were randomly shifted every iteration⁹.
A GRE dataset with 8 compressed channels and 2 echoes was retrospectively under-sampled. Three different under-sampling patterns were explored (4 extra central lines were always sampled and complementary sampling was used for two echoes): 1) “Uniform sampling”: uniformly sampled readout lines for each echo (R=4); 2) “PF sampling”: an additional 3/4 PF sampling was applied on each echo¹⁰; 3) “Random sampling”: randomly sampled readout lines for each echo (R=4). A single echo reconstruction and three separate multi-echo joint reconstructions were performed by enforcing the low rank property on the one-mode unfolding matrix only, on the two-mode unfolding matrix only and on both unfolding matrices simultaneously. The virtual conjugate coils approach (VC)¹¹ was also incorporated for PF and randomly sampled data. The kernel size of Hankel transform was $$$5 \times 5$$$ . A maximum of 1500 iterations was used for the uniform and PF sampling, and 2500 iterations for the random sampling due to slower convergence. Rank parameters were optimized for each SLR reconstruction. The number of submatrices $$$m$$$ used for LSLR is specified as in LSLR($$$m$$$). Normalized Root Mean Square Error (NRMSE) was calculated by using the fully-sampled data as reference.

Results

Figure 2 shows the single echo reconstruction for the first echo by SLR and LSLR. We compare SLR and LSLR across a variety of rank parameters, where all LSLR(2) reconstructions converge to a lower NRMSE than SLR at the same rank, at the cost of slower convergence. LSLR(4) reconstructions converge to a comparable NRMSE to LSLR(2) but requiring more iterations, except at r=80 where it had not yet converged at the maximum iteration. Figure 3 shows the multi-echo joint reconstruction results of the uniform sampled data, highlighting improved NRMSE for LSLR compared to SLR for each unfolding and in the tensor formulation ("both"). Figures 4 and 5 show the tensor reconstruction results of the PF sampled data and randomly sampled data respectively, indicating reduced NRMSE compared to conventional SLR constraint in both cases.

Conclusion

In contrast to conventional SLR reconstruction, the proposed LSLR enforces low-rank constraints on submatrices of the structured low-rank matrix or unfolding matrices of the low-rank tensor. This simple modification leads to a robust improvement over conventional SLR reconstruction, which is validated across a variety of low-rank reconstruction formulations. LSLR(2) is a good choice balancing reconstruction performance and computational efficiency. Although the optimal number of submatrices is data-dependent, a relatively small number (e.g. 2/4/6) is sufficient to achieve consistent improvements in our preliminary experience.

Acknowledgements

The Wellcome Centre for Integrative Neuroimaging is supported by core funding from the Wellcome Trust (203139/Z/16/Z). MC and WW are supported by the Royal Academy of Engineering (RF201617\16\23, RF201819\18\92). MC also receives research support from Engineering and Physical Sciences Research Council (EP/T013133/1).

References

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