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Memory-friendly and Robust Deep Learning Architecture for Accelerated MRI
Zi Wang1, Chen Qian1, Di Guo2, Hongwei Sun3, Rushuai Li4, and Xiaobo Qu1
1Department of Electronic Science, Biomedical Intelligent Cloud R&D Center, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, National Institute for Data Science in Health and Medicine, Xiamen University, Xiamen, China, 2School of Computer and Information Engineering, Xiamen University of Technology, Xiamen, China, 3United Imaging Research Institute of Intelligent Imaging, Beijing, China, 4Department of Nuclear Medicine, Nanjing First Hospital, Nanjing Medical University, Nanjing, China

### Synopsis

Deep learning has shown astonishing performance in accelerated MRI. Most methods adopt the convolutional neural network and perform 2D convolution since many MR images or their corresponding k-space are in 2D. In this work, we try a different approach that explores the memory-friendly 1D convolution, making the deep network easier to be trained and generalized. Furthermore, a one-dimensional deep learning architecture (ODL) is proposed for MRI reconstruction. Results demonstrate that, the proposed ODL provides improved reconstructions than state-of-the-art methods and shows nice robustness to some mismatches between the training and test data.

### Purpose

MRI plays an indispensable role in modern medical diagnosis[1]. Parallel imaging and sparse sampling are employed to shorten the scan time[2, 3], and image reconstruction is required. Recently, deep learning has shown astonishing potential with the powerful deep convolutional neural network (CNN)[4-7], but it is still limited by the lack of training subjects and robustness. As MR images or k-space are usually in 2D, most previous deep learning methods adopt 2D convolution. Here, we argue that there exists another effective scheme without considering spatial location information during the analysis, i.e., 1D CNN, which may address mentioned limitations. In this work, for the MRI reconstruction under the commonly used 1D phase encoding (PE) undersampling[2-3, 8-9], we propose a 1D learning scheme, and a memory-friendly deep learning architecture with nice robustness.

### Method

In the 1D PE undersampling, the k-space is fully- or non-sampled along the frequency encoding (FE). Thus, taking the 1D inverse Fourier transform (IFT) of the undersampled k-space along the FE can decouple the 2D k-space into $m$ independent 1D hybrid signals[10, 11], where $m (m=1,...,M)$ is the length of the FE direction. Decoupling allows us to reduce the dimension of reconstructed target signals from 2D to 1D, making the network training easier. Meanwhile, given the same number of available training subjects, decoupling provides abundant 1D hybrid signals as training samples, leading to the data augmentation of more than two orders of magnitude. Therefore, we propose a 1D learning scheme to promote better network training and generalization. Figure 1 shows the flowchart of the 1D reconstructions from the 1D PE undersampled k-space $\mathbf{z}_{m}$.
Furthermore, we propose a One-dimensional Deep Learning architecture (ODL). Its unrolled reconstruction incorporates both k-space and image-domain priors. As shown in Figure 2, the $k^{th}(k=1,...,K)$ network phase is designed as
$$(k-space\ learning\ module):\ \mathbf{r}_{m}^{(k)}=\lambda^{(k)}[{{\mathcal{N}}_{1}}(\mathbf{e}_{m}^{(k-1)})],\ \tag {1}$$
$$(Data\ consistency\ module):\ \mathbf{d}_{m}^{(k)}=\mathbf{e}_{m}^{(k-1)}-{{\gamma }^{(k)}}[{{\mathcal{U}}^{\text{*}}}(\mathcal{U}\mathbf{e}_{m}^{(k-1)}-{{\mathbf{z}}_{m}})+2\mathbf{r}_{m}^{(k)}],\ \tag {2}$$
$$(Image-domain\ learning\ module):\ \mathbf{e}_{m}^{(k)}={{\mathcal{F}}_{PE}}{{\mathcal{N}}_{3}}[soft({{\mathcal{N}}_{2}}(\mathcal{F}_{PE}^{\text{*}}\mathbf{d}_{m}^{(k)});{{\theta }^{(k)}})],\ \tag {3}$$
where $e_{m}$ is the 1D hybrid data to be reconstructed, $\mathcal{U}$ is the undersampling operator with zero-filling, $\mathcal{F}_{PE}$ is the 1D FT along the PE, the superscript $*$ represents the inverse operation. ${\lambda }^{(k)}$ and ${\gamma }^{(k)}$ are learnable network parameters initialized to 0.001 and 1, respectively. ${\mathcal{N}}_{1}$, ${\mathcal{N}}_{2}$, and ${\mathcal{N}}_{3}$ are multi-layer 1D CNNs, whose number of layers are 6, 3, and 3, respectively. Each convolutional layer contains 48 1D convolution filers of size 3, followed by the batch normalization[12] and ReLU. $soft(x;\rho )=\max \left\{ \left| x \right|-\rho \right\}\cdot {x}/{\left| x \right|}\;$ is the element-wise soft-thresholding, ${\theta }^{(k)}$ is the learnable threshold initialized to 0.001. When $k=1$, the initialized network input $\mathbf{e}_{m}^{(0)}={{\mathsf{\mathcal{U}}}^{\text{*}}}{{\mathbf{z}}_{m}}$ is the zero-filled 1D hybrid data with strong artifacts. The overall number of network phase in our implementation is 10, i.e. $K=10$. We employ the proposed 1D learning scheme to train our network by minimizing the loss function (mean square error).
In the reconstruction stage, the 1D IFT along the FE is first performed on the undersampled k-space to obtain $\mathbf{Z}=\mathsf{\mathcal{F}}_{FE}^{\text{*}}\mathbf{Y}$, all rows of $\mathbf{Z}$ form a batch that is then reconstructed in parallel and stitched back together to yield the reconstructed hybrid data $\mathbf{\hat{E}}$. After performing the 1D IFT along the PE, we can obtain the final reconstructed image $\mathbf{\hat{S}}=\mathsf{\mathcal{F}}_{PE}^{\text{*}}\mathbf{\hat{E}}$, as shown in Figure 1.

### Results

The knee[13] and brain (United Imaging collected) datasets are used in our experiments. The proposed ODL is compared with the state-of-the-art 2D CNN-based methods including IUNET[4], HDSLR[5], and DOTA[6]. GRAPPA[14] is used as the reconstruction baseline. To quantitatively evaluate the reconstruction performance, peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM)[15] are utilized. The higher PSNR and SSIM indicate less image distortions and better detail preservation in reconstructions, respectively.
(i) Matched reconstruction: Figures 3 demonstrates that, for any number of training subjects NTS, the proposed ODL consistently outperforms other CNN-based methods in terms of PSNR and SSIM. Notably, the proposed ODL can already provide better reconstructions than the baseline GRAPPA when training subjects are very limited (NTS=2 or 1). These results imply that, the proposed method is suitable for the case of limited training subjects. Representative images in Figure 4 show that, ODL outperforms other methods both visually and quantitatively in two datasets, indicating the excellent ability of the artifact suppression and detail preservation.
(ii) Mismatched reconstruction: Mismatched reconstruction refers to utilizing a trained network to reconstruct test datasets which are different from the acquisition specification of training datasets. Here, we focus on the mismatch of the knee plane orientation (Train/reconstruct: Coronal/sagittal) and brain contrast weighting (Train/reconstruct: T2w/T1w). Figure 5 shows that, the proposed ODL not only outperforms other deep learning methods, but also is the only one with lower reconstruction errors than the baseline GRAPPA, indicating the nice robustness to some mismatches between the training and test data.
(iii) By virtue of 1D CNN, the proposed ODL has few trainable parameters (664350), about 3% of IUNET, 2% of DOTA, and 26% of HDSLR, thus it is memory-friendly.

### Conclusion

We believe that the success of the 1D learning scheme and memory-friendly 1D deep learning architecture with nice robustness in fast MRI is of great significance for clinical applications, since the available training subjects and the machine memory are relatively limited.

### Acknowledgements

See more details in the full-length paper: https://csrc.xmu.edu.cn/. This work was supported in part by the National Natural Science Foundation of China under grants 62122064, 61971361, 61871341, and 61811530021, the National Key R&D Program of China under grant 2017YFC0108703, and the Xiamen University Nanqiang Outstanding Talents Program. The authors thank Xinlin Zhang and Jian Wu for assisting in data processing and helpful discussions. The authors thank Weiping He, Shaorong Fang, and Tianfu Wu from Information and Network Center of Xiamen University for the help with the GPU computing. The authors also thank Drs. Michael Lustig, Jong Chul Ye, Taejoon Eo, and Mathews Jacob for sharing their codes online.

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

### References

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### Figures

Figure 1. Flowchart of the 1D reconstructions from the 1D PE undersampled k-space.

Figure 2. The proposed ODL for accelerated MRI. (a) The recursive ODL architecture that alternates between the k-space learning (KL) module, data consistency (DC) module, and image-domain learning (IL) module. (b)-(d) The detailed structure of the KL module, DC module, and IL module, respectively. Note: The variables marked red are learnable. “BN” is the batch normalization.

Figure 3. Quantitative comparison of matched reconstructions using different number of training subjects NTS of a knee and a brain dataset. (a)-(b) are the mean of PSNR and SSIM of the reconstructions of the knee dataset, respectively. (c)-(d) are the mean of PSNR and SSIM of the reconstructions of the brain dataset, respectively. Note: The Cartesian undersampling pattern with AF=4 is used. The means are computed over all test data.

Figure 4. Matched reconstruction results of different methods. (a) (or (g)) is the fully sampled image and the Cartesian undersampling pattern with AF=4. (b)-(f) (or (h)-(l)) are reconstructed results and the corresponding error maps. Note: PSNR(dB)/SSIM are listed for each reconstruction. The deep learning methods are trained using all available training subjects.

Figure 5. Mismatched reconstruction results of different methods under the Cartesian undersampling patterns with AF=4. (a) (or (g)) is the fully sampled image. (b)-(f) (or (h)-(l)) are reconstruction error maps. Note: (b)-(d) are trained by the coronal knee dataset to reconstruct the sagittal knee dataset. (h-l) are trained by the T2w brain dataset to reconstruct the T1w brain dataset.

Proc. Intl. Soc. Mag. Reson. Med. 30 (2022)
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