## 2 Data and Methods
### Loss Function  
To address the issue of excessive image smoothing and lack of sensory realism caused by the use of the L1 loss function in the original model, we have adopted the Charbonnier loss function [23] as a replacement. [PERSON] et al. have demonstrated that compared to the L1 loss function, the Charbonnier loss function has a stronger capability to handle outliers, thereby being more effective in enhancing the performance of image super-resolution [24].  
\[\text{Charbonnier}_{\text{Loss}}=\sqrt{x^{2}+e^{2}} \tag{2}\]  
wherein, \(x\) denotes the interpolation between the predicted value and the actual value, and \(e\) is a small constant to prevent the non-existence of the derivative when \(x=0\). In the model proposed in this paper, \(e\) is set to \(10^{-3}\). Therefore, the total loss for the generator is as follows:  
\[L_{\text{G}}=L_{\text{percep}}+\lambda L_{\text{G}}^{Ra}+\eta L_{\text{charbonnier}} \tag{3}\]  
Perceptual loss \(L_{\text{percep}}\) is a type of loss function that constrains the features of network activations prior to activation layers. It is usually defined on the intermediate activation layers of a pre-trained deep network, which, however, may render the feature activations sparse. This phenomenon is particularly evident within deep learning frameworks. Furthermore, it could result in brightness inconsistencies between the super-resolution reconstructed images and the benchmark true images. To address these issues, this study has taken specific measures before the activation layers to ensure that the model can more accurately recover brightness and texture details. The definition of perceptual loss is as follows:  
\[L_{\text{percep}}=\frac{1}{WH\mathcal{C}}\sum_{x=1}^{W}\sum_{y=1}^{H}\sum_{z=1 }^{\mathcal{C}}\Bigl{(}\Phi(SR)_{xyz}-\Phi(HR)_{xyz}\Bigr{)}^{2} \tag{4}\]  
where \(W\), \(H\) and \(C\) respectively, represent the width, height, and number of channels of the feature map at the respective layer of the VGG network, and \(\Phi(SR)_{xyz}\) and \(\Phi(HR)_{xyz}\) denote the feature values at the coordinates \(((x,y,z))\) of the super-resolution image and high-resolution image, respectively. \(L_{\text{G}}^{Ra}\) is the adversarial loss, which is used to train the generator and discriminator in the generative adversarial network. The objective of the generator is to create high-resolution images that are realistic enough to \"fool\" the discriminator, while the discriminator attempts to distinguish between real and generated images. Since the adversarial loss occurs in tandem for the generator and the discriminator, the definition of adversarial losses for both is as follows:  
\[L_{\text{G}}^{Ra}=-\mathbb{E}_{x_{f}}\left[\log\Bigl{(}1-D_{Ra}\Bigl{(}x_{r}, x_{f}\Bigr{)}\Bigr{)}\right]-\mathbb{E}_{x_{f}}\Bigl{[}\log\Bigl{(}D_{Ra} \Bigl{(}x_{f},x_{r}\Bigr{)}\Bigr{)}\Bigr{]} \tag{5}\]  
\[L_{\text{D}}^{Ra}=-\mathbb{E}_{x_{f}}\left[\log\Bigl{(}D_{Ra}\Bigl{(}x_{r},x_{ f}\Bigr{)}\Bigr{)}\right]-\mathbb{E}_{x_{f}}\Bigl{[}\log\Bigl{(}1-D_{Ra} \Bigl{(}x_{f},x_{r}\Bigr{)}\Bigr{)}\Bigr{]} \tag{6}\]  
wherein, \(x_{f}=G(x_{i})\), with \(x_{i}\) represents the input low-resolution (LR) image, \(D_{Ra}\Bigl{(}x_{r},x_{f}\Bigr{)}=\sigma\Bigl{(}\mathcal{C}(x_{r})-\mathbb{E}_ {x_{f}}\Bigl{[}\mathcal{C}\Bigl{(}x_{f}\Bigr{)}\Bigr{]}\Bigr{)}\), and \(\mathbb{E}_{x_{f}}[\cdot]\) denote the operation of taking the average over all fake data in a mini-batch, \(\sigma\) is the sigmoid function [6], and C(x) represents the raw output of the discriminator.