## 3 Results and Discussion  
Experiments on the proposed method were performed using TensorFlow code on a computer with a 3.40 GHz Intel (R) Core (TM) i7-6700 CPU, 16 GB of memory, and an NVIDIA GeForce RTX 2080 graphics card. (The code and dataset are available at [[https://github.com/hg-a/MFVDSR](https://github.com/hg-a/MFVDSR)]([https://github.com/hg-a/MFVDSR](https://github.com/hg-a/MFVDSR)).) The training dataset includes 215,316 sub-images created by data argument with 291 images that combine images from image data reported by [PERSON] et al. [9] and the Berkeley Segmentation Dataset [10]. Four datasets, namely, Set5, Set14, B100, and Urban100, were used as the test dataset for performance evaluation. The simulation results were compared with other super-resolution image reconstruction methods, such as A+ [11], SRCNN [1], and VDSR [2], by measuring the average peak signal-to-noise ratio (PSNR) for each test dataset. For the optimization of parameters in training, the Adam optimizer [12] was used for VDSR and the proposed method. In the proposed method, the \(k\) value was set to 2, 5, or 10, and different MFs \(\lambda_{i}\) were tested for various cases. Table 1 presents a comparison of PSNR among the proposed method, A+ [11], SRCNN [1], and VDSR [2]. The A+ method performs super-resolution reconstruction using sparse dictionaries [11]. Table 1 shows that the PSNR performance of the proposed method, having repetitive skip connections with MFs for layer groups, is better than those of A+, SRCNN, and VDSR. Among A+, SRCNN, and VDSR, the VDSR method shows the best PSNR performance for all datasets. Experiments were conducted with different combinations of \(k\), \(n\), and \(\lambda_{i}\) values to find the optimal combination that yields the best performance. The experimental results show that the proposed method shows the best PSNR performance when \(k=5\) and \(n=4\). Each MF \(\lambda_{i}\) (\(i=1,\cdots,n-1\)) value is initialized as 2.5 and optimized during the training process, while \(\lambda_{n}\) is fixed as 1.0. For the optimization of MF \(\lambda_{i}\) (\(i=1,\cdots,n-1\)) values in training, Adam optimizer [12] is used and the learning rate (step size) is set as 0.0001. For these datasets, the proposed method shows PSNR improvements of 0.03 \(\sim\) 0.16 dB over the VDSR method. For scales \(\times 2\), \(\times 3\), and \(\times 4\), the proposed method shows average PSNR improvements of 0.11, 0.08, and 0.06 dB, respectively, over the VDSR method. Table 2 compares the PSNRs of the proposed method using different combinations of \(k\), \(n\), and \(\lambda_{i}\). Fixed \(\lambda_{i}\) values are used in Case 1, Case 3, and Case 6. Meanwhile, in Case 2, Case 4, Case 5, and Case 7, some initial values are set for \(\lambda_{i}\), and these values are learned and updated during the training process. The results in Table 2 indicate that the cases where \(\lambda_{i}\) is set to an initial value and updated  
during the training process show better results than the cases where \(\lambda_{i}\) is fixed. Hence, we argue that treating the MFs, \(\lambda_{i}\), as parameters that are updated during the training process would result in improved performance with the proposed method. If we compare the results in terms of \(k\) and \(n\), Case 3, Case 4, and Case 5 with \(k=5\) and \(n=4\) show better PSNR performances than the other cases. In the case of \(k=5\) and \(n=4\), as shown in Figure 2, there are four skip connections from input data to the output of four layer groups. Figures 3 and 4 show the data distributions as histograms for data generated at the 60 th epoch of the training process in layers \(L_{6}\), \(L_{11}\), and \(L_{16}\), which are the first layers of layer groups \(\mathbf{G}_{2}\), \(\mathbf{G}_{3}\), and \(\mathbf{G}_{4}\), respectively. In the captions of Figures 3 and 4, \(\mu\) and \(\sigma\) denote the mean and standard deviation, respectively. Figure 3 shows a comparison of distributions for data generated before (as the input of) ReLU in VDSR and the proposed method, while Figure 4