## 2 Materials and Methods
### Methodology
#### 2.3.2 Super-Resolution Generative Adversarial Network  
GANs consist of two networks: the generator and the discriminator. The generator's primary function is to transform a low-resolution (LR) input image into a synthesized high-resolution (HR) output image. Conversely, the discriminator takes two images as input--one LR image and one HR image--and generates a signal indicating the probability that the HR image is real. The discriminator's objective is to distinguish between real HR images and synthesized (or \"fake\") HR images. Concurrently, the generator aims to create high-quality HR images that deceive the discriminator and closely resemble real HR images. The structure of SRGAN is shown in Figure 7. The generator utilizes B residual blocks to extract image features and employs skip connections to enhance the flow of information across layers and mitigate the vanishing gradient problem associated with increasing network depth. Subsequently, a sub-pixel convolutional layer (PixelShuffler) is applied to upscale the spatial resolution of the input image. The generator was trained in the constructed LR-HR image pairs (\(I^{LR}\), \(I^{HR}\)), and the resolution of the generated images was magnified \(\times 4\). To distinguish between the real HR images and the generated SR images, the discriminator comprises eight convolutional layers, which use stride convolutions to reduce the image resolution and compute redundant information. Then, two dense layers and a Sigmoid activation function are used to scale the discriminator's output to the range of 0-1, representing whether the input image is real or generated. SRGAN introduces a significant innovation by designing a perceptual loss function. This loss aims to encourage the network to focus more on the semantic differences between reconstructed and original images during the learning process rather than on the differences in color and brightness at the pixel level. The perceptual loss was designed based on VGG-19 [34; 35], which can strengthen the extraction of low-frequency features and improve the effect of visual perception. The loss \(l^{SR}\) consists of two parts, including content loss \(l^{SR}_{X}\) and adversarial loss \(l^{SR}_{Gen}\):  
\[l^{SR}=\alpha_{X}l^{SR}_{X}+\alpha_{Gen}l^{SR}_{Gen}, \tag{2}\]  
where \(\alpha_{X}\) and \(\alpha_{Gen}\) are constant coefficients. The losses \(l^{SR}_{X}\) and \(l^{SR}_{Gen}\) are defined as Equations (3)-(5). There are two approaches to addressing the content loss:  
* The pixel-wise MSE loss is calculated as  
Figure 6: The examples of training dataset (LR-HR image pairs). (**a**) Gobi; (**b**) Lake; (**c**) Forest; (**d**) River. \[ \begin{split} I_{X}^{SR}=I_{MSE}^{SR}=\frac{1}{r^{2}WH}\sum_{x=1}^{rW} \sum_{y=1}^{H}\left(I_{x,y}^{HR}-G_{\theta_{G}}\left(I^{LR}\right)_{x,y}\right) ^{2},\end{split}  \tag{3}\]  
where \(r,W,H\) represent the downsampling factor and the dimensions of feature maps, respectively. * The VGG loss is defined based on the ReLU activation layers of the pre-trained 19-layer VGG network, where \(\phi_{i,j}\) represents the feature map obtained by the \(j\)-th convolution (after activation) before the \(i\)-th maxpooling layer within the VGG19 network, the VGG loss is the Euclidean distance between the feature representations of a reconstructed image \(G_{\theta_{G}}\left(I^{LR}\right)\) and the reference image \(I^{HR}\): \[ \begin{split} I_{X}^{SR}=I_{VGG_{ij}}^{SR}=\frac{1}{W_{i,j}H_{i,j }}\sum_{x=1}^{W_{i,j}}\sum_{y=1}^{H_{i,j}}\left(\phi_{i,j}\left(I^{HR}\right)_ {x,y}-\phi_{i,j}\left(G_{\theta_{G}}\left(I^{LR}\right)\right)_{x,y}\right) ^{2},\end{split} \] (4) where \(W_{i,j}\) and \(H_{i,j}\) describe the dimensions of the respective feature maps within the VGG network. \[ \begin{split} I_{Gen}^{SR}=\sum_{m=1}^{N}-logD_{\theta_{D}} \left(G_{\theta_{C}}\left(I^{LR}\right)\right),\end{split} \] (5)  
Adversarial loss, \(I_{Gen}^{SR}\), can enhance the texture details of the image, making the visual effect of the generated image more realistic.