# RETRACTED: Image Super-Resolution Algorithm Based on Dual-Channel Convolutional Neural Networks
## 3 Dual-Channel Convolutional Neural Networks
### Loss Function and Evaluation Standard  
(1) Loss Function  
By minimizing the loss cost between the super-resolution image and the real high-resolution image, the network constantly adjusts the network parameters \(\Theta=\{w_{i},\,b_{i}\}\). For a group of real high-resolution images \(X_{j}\) and a group of super-resolution images, \(F^{j}(Y;\Theta)\), is reconstructed by the network. This paper uses MSE as the cost function:  
\[L(\Theta)=\frac{1}{n}\sum_{i=1}^{n}\|F^{j}(Y;\Theta)-X_{j}\| \tag{8}\]  
where \(n\) represents the number of training samples. Because the weights of the dual-channel network are not shared, they are converted to a dual-channel cost function problem:  
\[L_{EDC}=\min(L_{d}(\Theta)+L_{s}(\Theta)) \tag{9}\]  
Here, \(L_{d}(\Theta)\) and \(L_{s}(\Theta)\) are the loss costs of the deep channel and shallow channel respectively. The network uses the Adam optimization method and back-propagation algorithm [38] to minimize MSE in order to adjust the network parameters, and the update process of the network weights is as in Equation (10):  
\[\mathbf{A}+1=0.9\times\Delta_{k}-\eta\times\frac{\partial L}{\partial W_{k}^{l}},W_{k+1}^{l}=W_{k}^{l}+\Delta_{k+1} \tag{10}\]  
\(\Delta_{k}\) represents the updating value of the last weight, \(l\) represents the number of layers of the network, and \(k\) represents the number of iterations from the network; \(\eta\) is the learning rate, \(\mathbf{W^{l}_{k}}\) represents the weight of the \(k\)th iteration in level \(l\), \(\frac{\partial L}{\partial W_{k}^{l}}\) represents the corresponding weight of the cost function and derivation of the derivative. The weights are randomly initialized according to a Gaussian distribution with mean value of zero and variance of 0.001. The model can automatically adjust the learning rate in the range of training, making the learning of the parameters more stable.  
(2) Evaluation Standards  
In this paper, the difference between the generated image quality and the quality of the original high-resolution image is measured by means of two common evaluation indexes, namely the Peak Signal-to-Noise Ratio (PSNR) and the Structural Similarity Index (SSIM) [29; 30].  
PSNR is used as an objective evaluation index of image quality, which is measured by calculating the error between corresponding pixels. The PSNR's unit is decibel (dB) [16]. The larger the value, the smaller the image distortion. The calculating equation is Equation (11):  
\[PSNR=10\log_{10}\left(\frac{\left(2^{n}-1\right)^{2}}{MSE}\right) \tag{11}\]  
Here, MSE is the direct Mean Squared Error of the original image and the super-resolution image, \((2^{n}-1)^{2}\) is the signal maximum square, and \(n\) is the number of bits per sampling value.  
The SSIM measures image similarity in terms of three aspects: brightness, contrast ratio, and structure. The range of SSIM is [0,1], and its value is closer to one. The distortion effect is smaller. The calculation equations are as follows:  
\[SSIM(X,Y)=l(X,Y)\cdot c(X,Y)\cdot s(X,Y) \tag{12}\]  
\[l(X,Y)=\frac{2\mu_{X}\mu_{Y}+C_{1}}{\mu_{X}^{2}+\mu_{Y}^{2}+C_{1}} \tag{13}\]  
\[c(X,Y)=\frac{2\sigma_{X}\sigma_{Y}+C_{2}}{\sigma_{X}^{2}+\sigma_{Y}^{2}+C_{2}} \tag{14}\]  
\[s(X,Y)=\frac{2\sigma_{XY}+C_{3}}{\sigma_{X}\sigma_{Y}+C_{3}} \tag{15}\]  
\(X\) is the super-resolution image of the LR image obtained through network training, \(Y\) is the original HR image. The variances of \(\mu_{X}\) and \(\mu_{Y}\) are represented by \(X\) and \(Y\), respectively, while \(\sigma_{X}\) and \(\sigma_{Y}\) represent the variances of the super-resolution image and of the original high-resolution image, respectively, and \(\sigma_{XY}\) represents the covariance of the super-resolution image and the original high-resolution image. \(C_{1}\), \(C_{2}\), \(C_{3}\) are constant terms. In order to avoid a zero in the denominator, the usual practice is to take \(C_{1}\), \(C_{2}\), \(C_{2}\) = \((K_{2}\times L)^{2}\), \(C_{3}\) = \(C_{2}\)/2 and, generally, \(K_{1}\) = 0.01, \(K_{2}\) = 0.03, \(L\) = 255.