## 1 Introduction  
Single image super-resolution (SISR) is a method to reconstruct a super-resolution image from a single low-resolution image [1; 2]. The reconstruction of a super-resolution image is generally difficult because of various issues, such as blur and noise. Image processing methods such as interpolation were developed for this purpose before the advent of deep learning. Many applications are based on deep learning in the image processing and computer vision field [1; 2; 3; 4]. The first solution for super-resolution reconstruction from a low-resolution image using deep learning with convolutional neural networks (CNNs) was the super-resolution convolutional neural network (SRCNN) method [1]. However, in the SRCNN method, learning was not performed well in deep layers. The very deep super-resolution (VDSR) method [2] is more efficient for learning in deep layers and achieves better performance than SRCNN for super-resolution reconstruction. Although VDSR has layers much deeper than those in SRCNN, it is efficient because it focuses on generating only residual (high-frequency) information by connecting the input data to the output of the last layer with a skip connection. However, in the VDSR method, the gradient information vanishes, owing to repeated rectified linear unit (ReLU) operations. It was observed that the number of hidden data units with vanishing gradients increases as the training proceeds with many iterations [5]. To resolve the problem of gradient vanishing, batch normalization [6] can be applied, but it may cause data distortion and other negative effects for reconstructing super-resolution images. The super-resolution image reconstruction performance of the VDSR method [2] is significantly better than that of SRCNN because it uses deep layers and a skip connection. The skip connection is applied only once between the input data and the output of the last layer in the existing VDSR method [2]. In the existing VDSR method, it is difficult to maintain the characteristics of the input data entering the neural network. In each layer, a ReLU operation is performed after a convolution operation. The ReLU operation is defined as  
\[ReLU(z)=\max(0,z) \tag{1}\]  
where negative values are clipped to zero. In VDSR, the data entering network layers would be residual-type with high-frequency components. Approximately half of the output values of the convolution operation are negative in each layer. Because negative values are clipped to zero, approximately half of the output values of ReLU are zero. The remaining positive values after ReLU operation are redistributed into positive and negative values through the convolution operation in the next layer. As the epoch with many iterations proceeds, the percentage of zero and small (close to zero) values increases in the output data of ReLU in each layer, which causes serious extinction of the information used in the learning for reconstruction. This would be the reason for the problem of gradient vanishing after repeated ReLU operations for residual-type data. Figure 1 shows the network structure of VDSR. In the figure, \(\mathbf{G}\) denotes the whole layer group, which is composed of \(l\) layers (\(l=20\) in Figure 1). The layer group \(\mathbf{G}\) performs whole operations on residual data. Each layer block \(L_{i}\) (\(i=1,\cdots,l\)) performs a convolution operation and a ReLU activation function operation. In VDSR, the multiplication factor (MF) \(\lambda\) for the skip connection is fixed as 1.0. Let \(\mathcal{F}(\cdot)\) represent the function of the layer group \(\mathbf{G}\). When a low-resolution image is the input data \(\mathbf{x}\) for the neural network, the operation result of the layer group \(\mathbf{G}\) in VDSR can be expressed as \(\mathcal{F}(\mathbf{x})\). The output data \(\mathbf{y}\), which is the super-resolution image, can be expressed as follows:  
\[\mathbf{y}=\mathcal{F}(\mathbf{x})+\lambda\mathbf{x}. \tag{2}\]  
Skip connections have been used in other residual-type networks [7; 8]. [PERSON] et al.