Xinlin Zhang^{1}, Hengfa Lu^{2}, Zi Wang^{1}, Xi Peng^{3}, Feng Huang^{4}, Qin Xu^{4}, Di Guo^{5}, and Xiaobo Qu^{1}

^{1}Department of Electronic Science, School of Electronic Science and Engineering (National Model Microelectronics College), National Institute for Data Science in Health and Medicine, Xiamen University, Xiamen, China, ^{2}College of Optical Science and Engineering, Zhejiang University, Hangzhou, China, ^{3}Department of Radiology, Mayo Clinic, Rochester, MN, United States, ^{4}Neusoft Medical System, Shanghai, China, ^{5}School of Computer and Information Engineering, Fujian Provincial University Key Laboratory of Internet of Things Application Technology, Xiamen University of Technology, Xiamen, China

Being the state-of-the-art parallel magnetic resonance imaging methods other than the deep learning approaches, the low-rank Hankel approaches embrace the advantage of holding low reconstruction errors. However, they demand intensive computations and high memory consumptions, thereby result in long reconstruction time. We proposed a new strategy for exploiting the low rankness and applied it to accelerate 2D imaging and T2 mapping. It is shown that the proposed method outperforms the state-of-the-art approaches in terms of lower reconstruction errors and more accurate mapping estimations. Besides, the proposed method required much less computation and memory consumption.

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).

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

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

This work was supported in part by the National Key R&D Program of China (2017YFC0108703), National Natural Science Foundation of China (61971361, 61871341, 61811530021, U1632274), Natural Science Foundation of Fujian Province of China (2018J06018), Fundamental Research Funds for the Central Universities (20720180056), and Xiamen University Nanqiang Outstanding Talents Program.

The correspondence should be sent to Dr. Xiaobo Qu (Email: quxiaobo@xmu.edu.cn).

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Figure
1. Reconstruction results of different methods for 2D imaging. Note: The relative $$$\ell_2$$$ norm error
(RLNE) of $$$\ell_1$$$-SPIRiT, AC-LORAKS, STDLR-SPIRiT, and proposed method
are 0.0866, 0.0759, 0.0735, and 0.0614, respectively.

Figure 2. The reconstruction RLNE versus
different echo images in T2 mapping.

Figure
3. The T2 map results reconstructed by different methods. Note: 1D Cartesian with
partial Fourier sampling pattern (R=6) were adopted in this experiment. The T2
map RLNEs of ALOHA, MORASA, and the proposed method are 0.1370, 0.1309, and
0.1113.