## 3 The Improved 3D Reconstruction Method of Satellite Images
### GAN-Based Image Enhancement  
According to the above introduction, the proposed method implements image enhancement on the satellite images by the GAN model. Here, we propose a modified generator network as shown in Figure 2, which is learned from the network designed in Ref. [21]. The generator network is mainly composed of the convolution layers, the batch-normalization layers, the residual blocks, the up-sampling block, and the down-sampling block. This generator network is trained to solve the minimization problem in Equation (5). We adopt Parametric-ReLU as the activation function and add a down-sampling block to  
Figure 1: The processing flow chart of the proposed reconstruction method. the generator network of the super-resolution GAN model to improve the quality of the output satellite images. Figure 3 shows the structure of the discriminator network, which is used to discriminate whether the samples generated by the generator network are high-quality satellite images. There are eight convolutional layers, seven batch-normalization layers, and two dense layers in the discriminator network. Here, we adopt Leaky-ReLU as the activation function, which is used to avoid maximum pooling [21]. The final activation function takes the form of the sigmoid function. We train this discriminator network to solve the maximization problem in Equation (5). The definition of the loss function directly affects the performance of the generator network. MSE is usually used to construct the loss function in image quality improvement research, but this method often lacks high-frequency details, which is inconsistent with our  
Figure 3: The architecture of the discriminator network in the proposed method. Figure 2: The architecture of the generator network in the proposed method. desire to make the image sharper. Therefore, the proposed method adopts a combination of MSE loss, VGG loss, adversarial loss, and regularization loss, as shown in Equation (6). \[l^{IE}=l^{IE}_{MSE}+6\times 10^{-3}l^{IE}_{VGG}+10^{-3}l^{IE}_{Adv}+2\times 10^{- 8}l^{IE}_{Reg}, \tag{6}\]  
where \(l^{IE}_{MSE}\) represents the MSE loss, whose structure is as below:  
\[l^{IE}_{MSE}=\frac{1}{WH}\sum_{w=1}^{W}\sum_{h=1}^{H}\biggl{(}l^{HQ}_{w,h}-G_{ \theta}\Bigl{(}l^{LQ}\Bigr{)}_{w,h}\biggr{)}^{2}, \tag{7}\]  
In Equation (7), \(W\) and \(H\) denote the width and the height of the high-quality satellite image. We define the MSE loss based on the Euclidean distance between the pixels of the reconstructed high-quality satellite image and the reference high-quality satellite image. The meaning of \(I^{HQ}\), \(I^{LQ}\), and \(G_{\theta}(\cdot)\) are the same as in Equation (4). In Equation (6), \(l^{IE}_{VGG}\) represents the VGG loss in Equation (6), whose structure is shown in Equation (8). \[l^{IE}_{VGG}=\frac{1}{W^{\prime}H^{\prime}}\sum_{w^{\prime}=1}^{W^{\prime}} \sum_{H^{\prime}=1}^{H^{\prime}}\biggl{(}\phi\Bigl{(}I^{HQ}\Bigr{)}_{w,h}- \phi\Bigl{(}G_{\theta}\Bigl{(}I^{LQ}\Bigr{)}\Bigr{)}_{w,h}\biggr{)}^{2}, \tag{8}\]  
where \(W^{\prime}\) and \(H^{\prime}\) describe the dimensions of the feature maps in the VGG network, and \(\phi(\cdot)\) stands for the obtained feature map. In fact, the VGG loss is defined based on the Euclidean distance between the reconstructed satellite image features and the reference satellite image features. As mentioned in Section 2.2, the generator network is important for the entire training process. In order to generate high-quality satellite images, we add adversarial loss to the loss function, as shown in Equation (9). \[l^{IE}_{Adv}=-\sum_{n=1}^{N}\log D_{\rho}\Bigl{(}G_{\theta}\Bigl{(}I^{LQ} \Bigr{)}\Bigr{)}, \tag{9}\]  
Here, we define the adversarial loss based on the sum of the probabilities that the high-quality satellite image generated by the generator network \(G_{\theta}(\cdot)\) will be judged true by the discriminator network \(D_{\rho}(\cdot)\). We adopt the logarithmic form in Equation (9) to facilitate the solution later.