## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dataset/Scale** & **A+** & **SRCNN** & **VDSR** & **Proposed** \\ \hline \multirow{3}{*}{Set5} & \(\times 2\) & 36.54 & 36.66 & 37.16 & 37.30 \\  & \(\times 3\) & 32.58 & 32.75 & 33.26 & 33.34 \\  & \(\times 4\) & 30.28 & 30.48 & 30.92 & 31.03 \\ \hline \multirow{3}{*}{Set14} & \(\times 2\) & 32.28 & 32.42 & 32.69 & 32.85 \\  & \(\times 3\) & 29.13 & 29.28 & 29.52 & 29.62 \\  & \(\times 4\) & 27.32 & 27.49 & 27.79 & 27.82 \\ \hline \multirow{3}{*}{B100} & \(\times 2\) & 31.21 & 31.36 & 31.69 & 31.75 \\  & \(\times 3\) & 28.29 & 28.41 & 28.64 & 28.68 \\  & \(\times 4\) & 26.82 & 26.90 & 27.12 & 27.15 \\ \hline \multirow{3}{*}{Urban100} & \(\times 2\) & 29.20 & 29.50 & 30.29 & 30.36 \\  & \(\times 3\) & 26.03 & 26.24 & 26.68 & 26.78 \\  & \(\times 4\) & 24.32 & 24.52 & 24.84 & 24.91 \\ \hline \multirow{3}{*}{Average} & \(\times 2\) & 32.31 & 32.49 & 32.96 & 33.07 \\  & \(\times 3\) & 29.01 & 29.17 & 29.53 & 29.61 \\  & \(\times 4\) & 27.19 & 27.35 & 27.67 & 27.73 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of peak signal-to-noise ratio (PSNR) among the proposed method, A+ [11], SRCNN [1], and VDSR [2]. (The results of A+ and SRCNN are taken from [2].) For the proposed method, \(k=5\), \(n=4\), and \(\lambda_{i}=2.5\) (initial). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Dataset/Scale** & **Case 1** & **Case 2** & **Case 3** & **Case 4** & **Case 5** & **Case 6** & **Case 7** \\ \hline \multirow{3}{*}{Set5} & \(\times 2\) & 37.24 & 37.26 & 37.19 & 37.23 & 37.30 & 37.17 & 37.21 \\  & \(\times 3\) & 33.27 & 33.32 & 33.30 & 33.27 & 33.34 & 33.20 & 33.20 \\  & \(\times 4\) & 30.99 & 30.95 & 30.87 & 30.97 & 31.03 & 30.67 & 30.9 shows a comparison of distributions for data generated after (as the output of) ReLU in VDSR and the proposed method. The comparison of Figures 3 and 4 shows that negative values are clipped to zero, and the standard deviation \(\sigma\) significantly decreases after the ReLU operation for each case. In Figure 4a-c, the repeated ReLU operations in VDSR force the data distribution to be concentrated in a very narrow range, and the \(\sigma\) values are small. Meanwhile, in Figure 4d-f, the repetitive skip connections from the input data to the layer groups in the proposed method result in data distributions in wider ranges, and the \(\sigma\) values are much larger than in the VDSR method. This effect increases the number of contributing data units for learning and maintains the features of the input image for super-resolution reconstruction more robustly. Therefore, the proposed method shows better super-resolution reconstruction performance than the VDSR method, even though it has the same number of layers. Table 3 presents the comparison of training and test running time (sec) for SRCNN, VDSR, and the proposed method. The running time for training is very similar for VDSR and the proposed method. Since the network structure of SRCNN is relatively simple, the running times (per epoch and per 60 epochs) for training in SRCNN look smaller than VDSR and the proposed method. The running times for testing per image are very close for SRCNN, VDSR, and the proposed method. Note that the loss value for training (or test PSNR value) converges at around 60 epochs for VDSR and the proposed method, whereas it does not converge at 60 epochs for SRCNN. For the training of SRCNN, 24,800 sub-images are used [1]. In experimental results in [1], the number of backpropagations should be at least \(5\times 10^{8}\) (more than 20,000 epochs) for convergence in training for SRCNN. In our experiments for VDSR and the proposed method, 215,316 sub-images are used. To achieve more than \(5\times 10^{8}\) backpropagations with 215,316 sub-images in training, it requires at least 2300 epochs in training. This means that it would require at least \(5\times 10^{5}\) s for the convergence of training in SRCNN. Hence, SRCNN would require much more time for training than VDSR and the proposed method.