## 3 Materials and Methods
### Spectral Normalization for Generator and Discriminator  
In stable GAN training methods, spectral normalization has recently been implemented in the discriminator, as introduced in the study in [41]. This technique constrains the Lipschitz constant of the discriminator by bounding the spectral norm of each layer. The spectral norm serves as an approximation of the Lipschitz constant for a composite function, which can be thought of as a sequence of convolutional layers and activation functions. According to the Lipschitz continuity theory, a composite function retains Lipschitz continuity if each individual function within it maintains this property. In the discriminator, the activation function is LeakyReLU, which inherently satisfies Lipschitz continuity and contributes to the stable training of the GAN network. Notably, unlike other normalization methods, spectral normalization does not necessitate additional hyperparameter tuning (in our experiments, setting the spectral norm of all weight layers to 1 yielded satisfactory results). Consequently, every convolutional layer must be designed to satisfy Lipschitz continuity. When considering an input \(h\) and a parameter matrix \(A\) for a convolutional layer, the spectral norm \(\sigma(A)\) is calculated as follows:  
\[\sigma(A):=\max_{h:h\
eq 0}\frac{||Ah||_{2}}{||h||_{2}}=\max_{||h||_{2}\leq 1 }||Ah||_{2}. \tag{6}\]  
This value is equivalent to the largest singular value of matrix \(A\), and it corresponds to the Lipschitz continuity constant, which is also equal to the spectral norm of the parameter matrix. By determining the spectral norm \(\sigma(W)\) of the parameter matrix \(W\) for the convolutional layer, we ensure that the maximum singular value of the normalized convolutional layer parameter matrix \(W_{\text{SN}}\) remains at 1. Consequently, for a linear layer, the norm can be expressed as \(g(h)=Wh\), and its Lipschitz constant is calculated as \(||g||_{\text{Lip}}=sup_{h}\sigma(\
abla g(h))=sup_{h}\sigma(W)=\sigma(W)\). The computation of \(W_{\text{SN}}\) is outlined as follows:  
\[W_{\text{SN}}(W)=\frac{W}{\sigma(W)}. \tag{7}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Input Image Size** & **Convolution Operation** & **Output Image Size** \\ \hline \(1\times 262\),144 & Reshape (BN) & \(64\times 64\times 64\) \\ \(64\times 64\times 64\) & Upsample:2 & \(64\times 128\times 128\) \\ \(64\times 128\times 128\) & Self-attention & \(64\times 128\times 128\) \\ \(64\times 128\times 128\) & Deconv2d:64 3 \(\times\) 3 & \(64\times 128\times 128\) \\  & stride = 1 (BN ReLU) & \\ \(64\times 128\times 128\) & Upsample:2 & \(64\times 256\times 256\) \\ \(64\times 256\times 256\) & Deconv2d:32 3 \(\times\) 3 & \(32\times 256\times 256\) \\  & stride = 1 (BN ReLU) & \\ \(32\times 256\times 256\) & Self-attention & \(32\times 256\times 256\) \\ \(32\times 256\times 256\) & Deconv2d:3 3\(\times\)3 & \(3\times 256\times 256\) \\  & stride = 1 (Tanh) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Detailed information of the proposed generator. Recent research has emphasized the significance of modulating generators for improving the performance of GANs. In particular, studies have shown that applying spectrum normalization to generators can yield several benefits [42]. By employing spectrum normalization, the risk of parameter escalation is minimized and potential gradient irregularities are avoided. Empirical evidence suggests that this technique encourages the generator to rely on a smaller number of discriminators per update, resulting in a significant reduction in computational training costs. Additionally, spectrum normalization has been observed to enhance training stability. To provide additional information, the DCGAN incorporates spectrum normalization in its discriminator's convolutional network, as shown in Table 2. The input consists of images with dimensions of 256 \(\times\) 256 \(\times\) 3. The convolutional kernel size is set to 3 \(\times\) 3, and the channel lengths progress as 16, 32, 64, 128, and 128.