# Single Image Super-Resolution Method Using CNN-Based Lightweight Neural Networks
## 3 Proposed Method
### Loss Function and Hyper-Parameters  
We set hyper-parameters as presented in Table 3. In order to find the optimal parameter \(\theta\) = {Filter weights, Biases}, we defined mean square error (MSE) as the loss function (6), where \(X_{HR}\), \(Y\), and \(N\) are the final output image of SR-LNN, the corresponding label image, and the batch size. Here, \(L(\theta)\) is minimized by Adam optimizer using the back-propagation. In particular, the number of epochs were set to 50 according to the Peak Signal-to-Nosie Ratio (PSNR) variations of the validation set (Set5) and the learning rates were empirically assigned to the intervals of epoch.  
Since it is important to set the optimal number of network parameters in the design of lightweight neural network, we investigated the relation between the number of parameters and PSNR according to the various filter sizes. As measured in Table 4, we implemented the most of convolution layers with 3x3 filter size, except for deconvolution layers to generate the interpolated feature map that accurately corresponds to the scaling factor:  
\[L(\theta)\ =\ \frac{1}{N}\sum_{i=0}^{N-1}\|X_{HR}\Big{(}X^{i}\Big{)}-Y^{i}\|_{2}^{2} \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Networks**} & **Filter Size** & **Num. of** & **PSNR** \\  & **(Width \(\times\) Height)** & **Parameters** & **(dB)** \\ \hline \multirow{4}{*}{SR-ILLNN} & 3 \(\times\) 3 & 439,393 & 31.41 \\  & 5 \(\times\) 5 & 1,153,121 & 31.38 \\ \cline{1-1}  & 7 \(\times\) 7 & 2,223,713 & 31.29 \\ \cline{1-1}  & 9 \(\times\) 9 & 3,651,169 & 28.44 \\ \hline \multirow{4}{*}{SR-SLNN} & 3 \(\times\) 3 & 262,977 & 31.29 \\ \cline{1-1}  & 5 \(\times\) 5 & 664,385 & 31.19 \\ \cline{1-1}  & 7 \(\times\) 7 & 1,266,497 & 31.16 \\ \cline{1-1}  & 9 \(\times\) 9 & 2,069,313 & 31.15 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Relation between the number of parameters and PSNR according to the various filter sizes.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Layer**} & **Kernel** & **Num. of** & **Padding** & **Output Feature Map** & **Num. of** \\
**Name** & **Size** & **Kernels** & **Size** & **(W \(\times\) H \(\times\) C)** & **Parameters** \\ \hline Conv1 & \(5\times 5\times 1\) & 64 & 2 & N \(\times\) M \(\times\) 64 & 1664 \\ Conv2 & \(3\times 3\times 64\) & 64 & 1 & N \(\times\) M \(\times\) 64 & 36,928 \\ Conv3 & \(3\times 3\times 128\) & 64 & 1 & N \(\times\) M \(\times\) 64 & 73,792 \\ Conv4 & \(3\times 3\times 192\) & 64 & 1 & N \(\times\) M \(\times\) 64 & 110,656 \\ Conv5 & \(1\times 1\times 64\) & 32 & 0 & N \(\times\) M \(\times\) 32 & 2080 \\ Deconv1 & \(4\times 4\times 32\) & 32 & 1 & 2N \(\times\) 2M \(\times\) 32 & 16,416 \\ Deconv2 & \(4\times 4\times 32\) & 32 & 1 & 4N \(\times\) 4M \(\times\) 32 & 16,416 \\ Conv6 & \(3\times 3\times 32\) & 16 & 1 & 4N \(\times\) 4M \(\times\) 16 & 4624 \\ Conv7 & \(5\times 5\times 16\) & 1 & 2 & 4N \(\times\) 4M \(\times\) 1 & 401 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Analysis of network parameters in SR-SLNN.  
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Optimizer**} & **Adam** \\ \hline Learning Rate & \(10^{-3}\) to \(10^{-5}\) \\ Activation function & ReLU \\ Padding Mode & Partial convolutional based padding [23] \\ Num. of epochs & 50 \\ Batch size & 128 \\ Initial weight & Xavier \\ \hline \hline \end{tabular}
\end{table}
Table 3: Hyper-parameters of the proposed methods.