## 3 Method or Methodology
### Architecture of DMFN  
Similar to SRFBN [21], our DMFN can be unfolded into several iterations because of the feedback manner, and the iteration t is set from 1 to T. The back-projection units feed back their output results to each other iteratively in a dual mutual-feedback manner. As shown in Figure 3, two convolutional layers are firstly used to obtain shallow features, which are then up-sampled. Then, the shallow features and the up-sampled shallow features are learned by the dual mutual-feedback component. In the dual mutual-feedback component, the HR features generated by the up-projection unit are fed back into the down-projection unit, and the LR features generated by the down-projection unit are fed back into the up-projection unit in the next iteration, which forms a dual mutual-feedback structure. Then, the outputs of dual mutual feedback from all iterations are concatenated for image reconstruction by fusing them with the bicubic interpolated results. Finally, we down-sampled the SR results to LR images as the inverse process of SR in the unity feedback component. Then, we compared it with the input LR to calculate the unity feedback loss.  
We define \(L_{in}\) and \(H_{in}\) as the shallow features learned by the first component, which can be obtained by  
\[\left\{\begin{array}{l}L_{in}=f_{c}(LR)\\ H_{in}=f_{up}(L_{in})\end{array}\right\}, \tag{1}\]  
where \(f_{c}\) contains two convolutional layers to obtain shallow LR features. \(f_{up}\) is a deconvolutional upsampling operation.  
In the dual mutual-feedback component of the \(t\)-th iteration, we use \(L_{out}^{t}\) to represent the LR features generated by the down-projection unit, and \(H_{out}^{t}\) to represent the HR features generated by the up-projection unit. The functions are as follows:  
\[L_{out}^{t}=\left\{\begin{array}{l}f_{d-p}(H_{in})\\ f_{d-p}([H_{in},H_{out}^{t-1}])\end{array}\right.\begin{array}{l}t=1\\ t\geq 2\end{array}\right\}, \tag{2}\]  
\[H_{out}^{t}=\left\{\begin{array}{l}f_{u-p}(L_{in})\\ f_{u-p}([L_{in},L_{out}^{t-1}])\end{array}\right.\begin{array}{l}t=1\\ t\geq 2\end{array}\right\}, \tag{3}\]  
where \(f_{d-p}\) are the operations of the down-projection unit, which contains some features from the up-projection unit because of themutual learning between the two back-projection units. \(f_{u-p}\) is the operations of the up-projection unit, which also contains some features from the down-projection unit. [] is the concat function.  
For reconstruction, we up-scale the LR features generated by the down-projection unit, which are then fused with the HR features generated by the up-projection unit. We define the final HR feature results of the \(t\)-th iteration as follows:  
\[H_{rb}^{t}=H_{out}^{t}+f_{up}(L_{out}^{t}). \tag{4}\]  
Since the final HR features of all iterations are fused and then added to the bicubic interpolated result of LR input, the final SR result is as follows:  
Figure 3: The unfolded DMFN.  
\[SR=f_{cm}([H_{up}^{1},H_{up}^{2},...,H_{rb}^{T}])+f_{\rm BC}(LR), \tag{5}\]  
where \(f_{cm}\) is a conv-3 compression layer, and \(f_{BC}\) represents the bicubic up-sample function.  
Finally, we down sample the SR result to the LR image named \(LR^{\prime}\) by the down-sampling function \(f_{down}\), which contains two convolutional layers for down sampling and channel transformation. The unity feedback loss calculated by \(LR^{\prime}\) and LR is used to narrow the search domain of the SR function.  
\[LR^{\prime}=f_{down}(SR), \tag{6}\]