Introduction

Partial magnetic resonance imaging (MRI) to accelerate scan time has been a hot research field ¹. One critical issue in this area is to recover the artifact-free image from a small subset of acquired data. Recently, approaches taking advantage of the low rankness of MRI data have been shown to provide promising reconstructions with low reconstruction errors ^2-5. However, these approaches need to arrange the MRI data into a large-scale low-rank Hankel matrix. This results in high computation demands and large memory consumption, leading to slow reconstruction time. In this work, we propose a new strategy aiming to alleviate this problem and meanwhile to provide better reconstructions. We proposed to simultaneously enforce the low rankness of a series of small-size Hankel matrices, avoiding constructing the large-scale low-rank Hankel matrix thereby enables much less memory cost and allows faster computation. Particularly, the proposed methods permit effective utilization of the low-rank property as the existing low-rank Hankel methods do. We validate the proposed approaches in accelerating 2D imaging and T2 mapping.

Method

Unlike the existing low-rank approaches needing to arrange large-scale Hankel matrices, we propose to simultaneously enforce the low rank of small-size Hankel matrices. The Hankel matrix size is significantly reduced allowing much less computation and memory consumption thus enables faster reconstruction time. Notably, the proposed method has lower reconstruction errors than other state-of-the-art approaches.
We proposed to solve the parallel MRI reconstruction problem as
$$\mathop{\min }\limits_{\bf{X}} \sum\limits_{m=1}^M {{{\left\| {{{{\bf{\tilde{H}}}}_{{\rm{vc}}}}{\bf{\tilde{W}}}{{{\bf{\tilde{F}}}}^{{\rm{1D}}}}{{\bf{P}}_m}{\bf{X}}} \right\|}_*}}+\sum\limits_{n=1}^N{{{\left\|{{{{\bf{\tilde{H}}}}_{{\rm{vc}}}}{\bf{\tilde{W}}}{{{\bf{\tilde{F}}}}^{1{\rm{D}}}}{{\bf{Q}}_n}{\bf{X}}} \right\|}_*}} +\frac{\lambda}{2}\left\|{{\bf{Y}}-{\bf{U}}{{\bf{F}}^{{\rm{2D}}}}{\bf{X}}}\right\|_F^2+\frac{{{\lambda_1}}}{2}\left\|{{\bf{X}}-{\bf{GX}}}\right\|_F^2,$$ where the $$${\bf{X}}\in{{\mathbb{C}}^{M\times{N}\times{J}}}$$$ denotes the desired multi-coil image, $$$M$$$, $$$N$$$, and $$$J$$$ are the number of the row, column, and coils, $$${\bf{P}}_m$$$ and $$${\bf{Q}}_n$$$ denote the operators that extract $$$m$$$-th row and $$$n$$$-th column of all coil from $$$\bf{X}$$$ for $$$m=1, \cdot, M$$$, and $$$n=1,\cdot,N$$$, i.e., define $$${\bf{x}}_{m,j}^{{\rm{row}}}$$$ as the vector of $$$m$$$-th row and the $$$j$$$-th coil, and $$${\bf{x}}_{n,j}^{{\rm{col}}}$$$ as the vector of $$$n$$$-th column and the $$$j$$$-th coil. We have, $$${{\bf{P}}_m}{\bf{X}} = \left[{\begin{array}{*{20}{c}}{{\bf{x}}_{m,1}^{{\rm{row}}}}& \cdots &{{\bf{x}}_{m,j}^{{\rm{row}}}}& \cdots &{{\bf{x}}_{m,J}^{{\rm{row}}}} \end{array}}\right]\in {{\mathbb{C}}^{N \times{J}}}$$$ and $$${{\bf{Q}}_n}{\bf{X}}=\left[ {\begin{array}{*{20}{c}} {{\bf{x}}_{n,1}^{{\rm{col}}}}& \cdots &{{\bf{x}}_{n,j}^{{\rm{col}}}}& \cdots &{{\bf{x}}_{n,J}^{{\rm{col}}}} \end{array}} \right]\in{\mathbb{C}^{M\times{J}}}$$$. The operator $$${{\bf{\tilde H}}_{vc}}$$$ is defined as $$${{\bf{\tilde H}}_{{\rm{vc}}}}{\bf{\tilde W}}{{\bf{\tilde F}}^{{\rm{1D}}}}{{\bf{P}}_m}{\bf{X}} = \left[ {\begin{array}{*{20}{c}} {{\bf{HW}}{{\bf{F}}^{{\rm{1D}}}}{\bf{x}}_{m,1}^{{\rm{row}}}}& \cdots &{{\bf{HW}}{{\bf{F}}^{{\rm{1D}}}}{\bf{x}}_{m,J}^{{\rm{row}}}}&{{\bf{HW}}{{\bf{F}}^{{\rm{1D}}}}{{\left( {{\bf{x}}_{m,1}^{{\rm{row}}}} \right)}^{\dagger} }}& \cdots &{{\bf{HW}}{{\bf{F}}^{{\rm{1D}}}}{{\left( {{\bf{x}}_{m,J}^{{\rm{row}}}} \right)}^{\dagger} }} \end{array}} \right]$$$, where $$${\dagger}$$$ denotes conjugating and reflecting around the zero-frequency, $$$\bf{H}$$$ denotes the operator converting a vector to a Hankel matrix, $$$\bf{F}^{\rm{1D}}$$$ the 1D Fourier transform, $$$\bf{W}$$$ the weighting matrix. $$$\bf{Y}$$$ is the acquired multi-coil k-space data with zero-filling in unacquired position, $$$\bf{F}^{\rm{2D}}$$$ the 2D Fourier transform, $$$\bf{U}$$$ the undersampling matrix, and $$$\bf{G}$$$ the SPIRiT operator matrix in the image domain.

We also applied the memory-efficient low-rank Hankel matrix approach to T2 mapping reconstruction. Denote the desired T2 mapping image $$${\bf{X}}\in{\mathbb{C}^{M\times{N}\times{L}\times{J}}}$$$, $$$L$$$ and $$$J$$$ are the number of echoes and coils, $$$M$$$ and $$$N$$$ are the number points in the readout and phase encoding dimensions, respectively. We reconstructed the y-t plane image $$${\bf{X}}_m^{{\rm{y-t}}} \in {\mathbb{C}^{N\times{L}\times{J}}}$$$ along the readout dimension one by one with the reconstruction problem: $$\mathop {\min }\limits_{{\bf{X}}_m^{{\rm{y-t}}}} \sum\limits_{n=1}^N {{{\left\| {{\bf{\tilde H}}{{\bf{P}}_n}{\bf{X}}_m^{{\rm{y-t}}}} \right\|}_*}}+{\lambda _2}\sum\limits_{l=1}^L{{{\left\| {{{{\bf{\tilde{H}}}}_{{\rm{vc}}}}{\bf{\tilde W}}{{{\bf{\tilde F}}}^{1{\rm{D}}}}{{\bf{Q}}_l}{\bf{X}}_m^{{\rm{y-t}}}} \right\|}_*}}+\frac{\lambda }{2}\left\| {{\bf{Y}}_m^{{\rm{y-t}}}-{\bf{U}}{{\bf{F}}^{{\rm{1D}}}}{\bf{X}}_m^{{\rm{y-t}}}} \right\|_F^2,$$
where $$${\bf{Y}}_{m}^{\rm{y-t}}$$$ denotes the acquired k-space data of $$$m$$$-th y-t plane (inverse Fourier transform has been done in the readout dimension).

Results

1. Accelerate 2D imaging
As shown in Fig. 1, the proposed approach produces an image with the lowest error. The errors inside the skull are uniform and very small compared to the competing results. Besides, the computational time of $$$\ell_1$$$-SPIRiT ⁶, AC-LORAKS ⁷, STDLR-SPIRiT ⁵, and the proposed method are 22.5 seconds, 7.4 seconds, 1090.3 seconds, and 80.3seconds, respectively. Overall, the results demonstrate the promising performance and relatively fast reconstruction speed of the proposed method, e.g., a more than 12 times acceleration was obtained by the proposed when compared to the structured low-rank method, STDLR-SPIRiT.

2. Accelerate T2 mapping
We performed reconstruction on brain T2 mapping (16 echoes) using 1D Cartesian with partial Fourier sampling pattern. As shown in Fig. 2, the proposed approach permits the lowest RLNEs for all 16 images than the state-of-the-art approaches. The estimated T2 map of the proposed method also enables the lowest error (Fig. 3), indicating the proposed approach can provide more accurate quantitative results than other state-of-the-art methods. Besides, the proposed method reconstructed all echo images with the fastest speed using only 197 seconds, while ALOHA ⁹ used 309 seconds and MORASA ⁸ cost 1200 seconds.

Conclusion

We present a new strategy for enforcing the low-rankness of parallel MRI data to achieve lower computations and memory consumption and also provide promising results. We validated the idea in both spatial encoding MRI and T2 mapping problems. In both cases, the proposed approaches outperform the state-of-the-art methods in terms of lower errors and reduced computational time significantly compared to a majority of low-rank Hankel approaches.