Junhyeok Lee^{1}, Junghwa Kang^{1}, Se-hong Oh^{1}, and Dong Hye Ye^{2}

^{1}Biomedical Engineering, Hankuk University of Foreign Studies, Yongin-si, Gyeonggi-do, Korea, Republic of, ^{2}Department of Electrical and Computer Engineering, Marquette University, Milwaukee, WI, United States

We performed parallel MRI reconstruction from under-sampled k-space data using the Multi-Domain Neumann Network with Sensitivity Maps. The Neumann network solves the inverse problem with recursive neural networks taking account into the forward model. We adapt the Neumann network with the coil sensitivity estimation and k-space data regurlaization to take account into MR physical models. Our proposed method shows uppressed image artifacts and enhanced spatial resolution compared with GRAPPA, U-Net and Neumann network.

Where $$$\tilde{x}^{j}$$$ is the output of the $$$j$$$-th iteration and $$$y$$$ is multi-coil under-sampled k-space. $$$F$$$ and $$$R$$$ denote forward operator (e.g., Fourier Transform (FT)), and regularization, respectively. $$$\lambda$$$ represents the balance parameter between data consistency and regularization term. The final output $$$\hat{x}$$$ of Neumann network is the summation of intermediate reconstructed results.

We adapt the Neumann network to utilize k-space data regularization and coil sensitivity map based on MR physics. First, we introduce the new forward model $$$A$$$ that takes account into coil sensitivity maps.$$A=M\circ F\circ S\quad (\text{Eq. 4})$$

Where $$$A$$$ is a forward operator consisting of sensitivity map multiplication $$$S$$$, and under-sampling mask operator $$$M$$$. We now plug the new forward model $$$A$$$ in Eq. 1 and 2 as following.$$\tilde{x}^{0} = \lambda A^{-1}y \quad (\text{Eq. 5})$$$$\tilde{x}^{j+1} = (I - \lambda A^{-1}A)\tilde{x}^{j} - \lambda R(\tilde{x}^{j};A), \quad \forall j = 1, ..., N \quad (\text{Eq. 6})$$

It is worth noting that $$$R$$$ depends on both image $$$\tilde{x}^{j}$$$ and forward model $$$A$$$ for multi-domain regularization.

We also report quantitative evaluation scores for parallel MRI reconstruction with acceleration factor 4 and 8 in Table 1. Our proposed MDNNSM produces significantly lower NMSE and higher PSNR and SSIM than other reconstruction methods including the original Neumann network.

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DOI: https://doi.org/10.58530/2022/2345