Juan Zou^{1}, Cheng Li^{1}, Ruoyou Wu^{1}, Zhenzhen Xue^{1}, Xin Liu^{1}, Hairong Zheng^{1}, and Shanshan Wang^{1}

^{1}Paul C. Lauterbur Research Center for Biomedical Imaging, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, shenzhen city, China

Fast data acquisition and high-quality image reconstruction are vital for dynamic MRI, which can capture both anatomical and temporal information. High-resolution acquisition approaches in k-space and super-resolution approaches after reconstruction have been frequently reported. However, these methods may get details lost at high acceleration factors. To address this issue, we propose a multi-scale detail preserving reconstruction method for dynamic MR images. The residuals of multi-scale intermediate images in the iterative procedure are explored and the temporal and spatial dependencies between frames are considered. Promising results are achieved by the proposed method at the high acceleration factor of 11.

High-resolution acquisition approaches in k-space and super-resolution approaches after reconstruction in the image domain have been reported [1-3]. These methods either have limited performance at certain high acceleration factors or suffer from the error accumulation of a two-step structure. In order to preserve the important details, we propose a multi-scale detail preserving reconstruction method for dynamic MR image reconstruction. Specifically, the residuals of multi-scale intermediate images in the iterative procedure are explored and the temporal and spatial dependencies between frames are considered. We employ multi-scale super-resolution and bidirectional convLSTM as the sparse regularization term in each reconstruction step to explore its prior knowledge. This method can refine image details that are neglected by existing methods, and achieve superior visual and quantified performance at high acceleration factors, such as 11.

$$y=Ax+e\quad\quad\quad\quad\quad (1)$$

$$arg\min_xR(x)+ \frac{1}{2}‖Ax-y‖_2^2\quad\quad\quad\quad (2)$$

$$arg\min_x\frac{1}{2}‖Ax-y‖_2^2+R(s)+δ‖s-z‖_2^2+ε‖z-x‖_2^2\quad\quad\quad\quad (3)$$

\begin{cases}\displaystyle s_n=arg\min_sδ‖s-z_{n-1}‖_2^2+R(s)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (4a)\\\displaystyle z_n=arg\min_zδ‖z-s_n‖_2^2+∇F(s_n)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (4b) \\\displaystyle x_n=arg\min_s\frac{α}{2}‖Ax-y‖_2^2+ε‖x-z_n‖_2^2+r(x_n )\quad\quad\quad\quad (4c) \\\end{cases}

\begin{cases}s_n=Prox(z_{n-1})\quad\quad\quad\quad\quad\quad\quad\quad (5a)\\z_n=s_n-ωA^H (As_n-y)\quad\quad\quad\quad\quad (5b)\\x_n=z_n+∆x_n\quad\quad\quad\quad\quad\quad\quad\quad\quad (5c)\\\end{cases}

In Eq. 1, $$$x∈\mathbb{C}^N$$$ represents 2D cardiac high-quality sequences, $$$A∈\mathbb{C}^{M×N}$$$ is the under-sampling operator, $$$y∈\mathbb{C}^M (M≪N)$$$ is the under-sampled k-space data, and $$$e∈\mathbb{C}^M$$$ is the noise term. In Eq. 2, $$$R(x)$$$ represents the regularization of $$$x$$$, $$$\frac{1}{2}‖Ax-y‖_2^2$$$ is data fidelity term, and $$$δ$$$ and $$$ε$$$ are two penalty parameters. In Eq. 3, $$$R(s)$$$ represents the regularization of $$$s$$$. In Eq. 4, we define $$$F(x)$$$ as the data fidelity term $$$\frac{1}{2}‖Ax-y‖_2^2$$$, and we can update $$$z_n$$$ from $$$s_n$$$ by minimizing $$$F(x)$$$. Since $$$z_n$$$ is a linear combination of $$$s_n$$$ and the original measurement $$$y$$$, we can transfer Eq. 4b to minimize $$$‖z-s_n‖_2^2$$$ and the correction term $$$∇F(s_n)$$$. In order to update $$$x_n$$$ from $$$z_n$$$, we expect the reconstruction values in iteration steps $$$n$$$ and $$$n-1$$$ to be equal by introducing a correction term $$$r(x_n )$$$ between $$$x_n$$$ and $$$x_{n-1}$$$.

We unroll the iterations into a deep neural network. The three procedures in Eq. 5 correspond to three modules in the network as shown in Fig. 1. They are named as the reconstruction layer $$$s_n$$$, the data consistency layer $$$z_n$$$, and the details preserving layer $$$x_n$$$. All parameters $$$(θ_1,ω,θ_2)$$$ in Eq. 6-8 are learnable network parameters.

\begin{cases}s_n=C_{θ_1}(z_{n-1})+z_{n-1}\quad\quad\quad\quad\quad (6)\\z_n=s_n-ωA^H(As_n-y)\quad\quad\quad\quad (7)\\x_n=z_n+C_{θ_2} (x_{n-1})\quad\quad\quad\quad\quad\quad (8)\\\end{cases}

where $$$C_{θ_1}$$$ represents the five 3D convolution layers in the reconstruction module and $$$θ_1$$$ refers to the learnable parameters. $$$ω$$$ is the update step size in the data consistency modules. $$$x_n$$$ consists of two parts. One is a cascade of seven 2D convolution layers and the ReLU activation function. The other is bidirectional convLSTM between $$$ x_n$$$ and $$$x_{n-1}$$$. $$$θ_2$$$ represents the learnable parameters in $$$C_{θ_2}$$$ .

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5. Shanshan Wang, Ziwen Ke, Huitao Cheng, Sen Jia, Leslie Ying, Hairong Zheng, Dong Liang. DIMENSION: Dynamic MR imaging with both k‐space and spatial prior knowledge obtained via multi‐supervised network training. NMR Biomed, 2019: E4131.

6. Chen Qin, Jo Schlemper, Jose Caballero, Anthony N. Price, Joseph V. Hajnal, Daniel Rueckert. Convolutional recurrent neural networks for dynamic MR image reconstruction. IEEE Transactions on Medical Imaging, 2018, 38, 1, 280–290.

Figure
1. Overall architecture of the proposed detail-preserving multi-scale deep
learning reconstruction network.

Figure 2. Reconstruction results at the
acceleration factor of 11. From left to right, the first row plots the ground
truth, zero-filling image and the reconstruction results of different methods,
whereas the second row gives the undersampling mask and the error maps of
different methods.

Table
1. The average PSNR and SSIM values
achieved by D5C5, DIMENSION, and Our Method on the test dataset at the acceleration
factor of 11

DOI: https://doi.org/10.58530/2022/4166