Ziwen Ke^{1,2}, Shanshan Wang^{2}, Huitao Cheng^{1,2}, Leslie Ying^{3}, Xin Liu^{2}, Hairong Zheng^{2}, and Dong Liang^{1,2}

Dynamic MR image reconstruction from incomplete k-space data is an important technique for reducing its scan time. Deep learning has shown great potential in assisting this task. Nevertheless, most frameworks only adopt a final loss for network training and the intermediate results generated during the network forward pass haven't been considered for the network training. This work proposes a multi-supervised learning strategy, which constrains the frequency domain information and reconstruction results at different levels. Improved reconstruction results have been achieved with the proposed strategy.

**Introduction**

**Theory and method**

In this work, we propose a novel multi-supervised learning based on a cascaded cross-domain neural network. The network architecture and the multi-supervised learning strategy are shown in Figure 1. The cross-domain network consists of two parts: the first part is frequency domain network for k-space completion termed as FDN; the second part is spatial domain network term as SDN, which is used to extract high-level features of images. The two are connected by Fourier inversion (see Inverse Fast Fourier Transform (IFFT) in Figure 1). The multi-supervised learning consists of two parts: progressive loss and primary loss. In the ordinary deep learning methods, there is only primary loss, while the proposed multi-supervised learning introduces progressive loss. The primary loss is the mean squared error (MSE) between the reconstructions and corresponding fully sampled images. The progressive loss is auxiliary loss include k-space loss and spatial loss. Let $$$k_u, k_c, k_f, x$$$ be the undersampled k-space, completed k-space, fully sampled k-space and fully sampled image respectively. Let $$$b_1,b_2,b_3,b_4$$$ be the output of block I1, block I2, block I3 and block I4 respectively. The total loss of the network training can be expressed as the formula: $$total\ loss=||x-b_4||^2_2+\sum_{i=1}^3\lambda_i||x-b_i||^2_2+\lambda_k||k_f-k_c||_2^2$$

(1) The first term is primary loss, the second term is spatial loss and the third term is k-space, where $$$\lambda_i$$$ and $$$\lambda_k$$$ are the weights of spatial loss and k-space loss respectively.

**Experiment**

**Conclusion **

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Figure
1. The multi-supervised learning
strategy in the cross-domain neural networks.

Figure
2. The comparison of cardiac MR
reconstructions from different methods (k-t FOCUSS, k-t SLR, D5C5, CDN and the
proposed method). (a) ground truth, (b) mask, (c) zero-filling image, (d) k-t
FOCUSS reconstruction, (e) k-t SLR reconstruction, (f) D5C5 reconstruction, (g)
CND reconstruction and (h) the proposed method reconstruction; (i), (j), (k),
(l) and (m) their corresponding error maps with display ranges [0, 0.07].

Table 1. The MSE, PSNR and SSIM of
zero-filling, k-t FOCUSS, k-t SLR, D5C5 and the proposed method.