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

Cardiac magnetic resonance (MR) imaging provides a powerful imaging tool for clinical diagnosis. However, due to the constraints of magnetic resonance (MR) physics and reconstruction algorithms, dynamic MR imaging takes a long time to scan. Recently, deep learning has achieved preliminary success in MR reconstruction. Compared with the classical iterative optimization algorithms, the deep learning based methods can get improved reconstruction results in shorter time. However, most current deep convolutional neural network (CNN) based methods use mean square error (MSE) as the loss function, which might be a reason for image smooth in the reconstruction. In this work, we propose to employ edge-enhanced constraint for loss function and explore different types of total variation on network training. Encouraging performances have been achieved.

**Theory and method**

The DC-CNN [1] model is selected as our network framework as shown in Figure 1. To explore the effects of TV constraints on dynamic MR reconstruction, we configure five CNN models as shown in Table 1. Specifically, the Model 0 is the original DC-CNN model, whose loss function is MSE.The isotropic TV constraint is introduced into the loss function of the Model 1, where the isotropic TV constraint is defined in Eq. (1). The Model 2 introduces anisotropic TV constraint (as shown in Eq. (2)). And the Model 3 and Model 4 respectively introduce the HDTV (degree=2 and degree=3), whose definitions are shown in Eq. (3) and (4). The derivation and symbols of the formula can be referred in [6].

$${\rm TV}_{iso}(f)=\int_{\Omega} \sqrt{(\frac{\partial f(r)}{\partial x})^2+(\frac{\partial f(r)}{\partial y})^2}dr\ \ \ \ \ (1)$$

$${\rm TV}_{aniso}(f)=\int_{\Omega} |\frac{\partial f(r)}{\partial x}|+|\frac{\partial f(r)}{\partial y}|dr\ \ \ \ \ (2)$$

$${\rm HDTV}_{2}(f)=\int_{\Omega}\sqrt{(3|f_{xx}|^2+3|f_{yy}|^2+4|f_{xy}|^2+2\mathcal{R}(f_{xx}f_{yy}))/8}dr\ \ \ \ \ (3)$$

$${\rm HDTV}_{3}(f)=\int_{\Omega}\sqrt{5(|f_{xxx}|^2+|f_{yyy}|^2)+6\mathcal{R}(f_{xxx}f_{xyy}+f_{yyy}f_{xxy})+9(|f_{xxy}|^2+|f_{xyy}|^2)}dr/4\sqrt(2)\ \ \ \ \ (4)$$

The loss functions can be defined as the following paradigm:

$${\rm loss\ \ function}={\rm MSE}(f, \hat{f})+\lambda{\rm TV}(f)\ \ \ \ (5)$$

where $$$f$$$ is the reconstructed image and $$$\hat{f}$$$ is the ground truth. $$$\lambda$$$ is a hyper-parameter and we set $$$\lambda=10^{-8}$$$ here.

**Experiment**** **

**Results and discussion**** **

**Conclusion**** **

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Figure
1. The DC-CNN model we
selected for cardiac MR reconstruction.

Table
1. The
five models we configure to explore the effects of different TV constraints on
dynamic MR reconstruction.

Figure
2．Cardiac
MR reconstruction results of different methods (k-t FOCUSS, L+S,
DC-CNN and the TV-based methods). (a) ground truth, (b) mask, (c) zero-filling image (d-f, j-m) reconstruction of k-t FOCUSS, L+S, D5C5,
and the TV-based methods, respectively; (g-i, n-q) their corresponding error
maps with display ranges of [0, 0.04].

Table 2. Evaluation metrics of the
reconstruction results generated by zero-filling, k-t FOCUSS, L+S,
DC-CNN and the TV-based methods.