## 2 Materials and Methods
### Evaluation Metrics of GAN Algorithms  
GAN algorithms are evaluated using a variety of metrics, including the Inception Score (IS), the Frechet Inception Distance (FID), the Structural Similarity Index (SSIM), and the Frechet Distance (FD) [25; 26]. The IS measures the quality of the generated images, while the FID and the SSIM measure the similarity between the real and generated images. The FD measures the similarity between the distributions of real and generated images. Additionally, GAN algorithms can be evaluated using user studies or metrics such as the Human Visual System (HVS) or classification accuracy. The Inception Score (IS) is a metric that measures the quality of generative models, specifically synthetic images output by generative adversarial network models. It is based on a convolutional network and is calculated by taking the mean of the KL-divergence [27] between the predicted label distributions for a set of generated images and the label distributions for the ground truth. The quantitative evaluation of IS is calculated with the equation  
\[I=\exp(E_{x}D_{KL}(p(y|x)||p(y))) \tag{4}\]  
where x denotes one generated sample and y is the label predicted by the model. The Frechet Inception Distance (FID) is a metric used to measure the similarity between two sets of images. The FID score is calculated by comparing the statistics of the feature representations of the two image sets. The lower the FID score, the more similar the two image sets are considered to be. Frechet distance is used to compute the distance between two \"multivariate\" normal distributions. For a \"univariate\" normal distribution, the Frechet Distance is given as,  
\[\mathrm{d}(X,Y)=\left(\mu_{X}-\mu_{Y}\right)^{2}+\left(\sigma_{X}-\sigma_{Y} \right)^{2} \tag{5}\]  
where \(\mu\) and \(\sigma\) are the mean and standard deviation of the normal distributions, and X and Y are two normal distributions. In common, the FID metric is the squared Wasserstein metric between two multidimensional Gaussian distributions: N(\(\mu\), \(\Sigma\)), the distribution of some neural network features of the images generated by the GAN, and N(uw, \(\Sigma w\)), the distribution of the same neural network features from the \"world\" or real images used to train the GAN:  
\[\mathrm{FID=}\parallel\mu_{X}-\mu_{Y}\parallel ^{2}-\mathrm{Tr}\left(\sum_{X}+\sum_{Y}-2\sqrt{\sum_{X}\sum_{Y}}\right) \tag{6}\]  
where X and Y are the real and fake embeddings assumed to be two multivariate normal distributions. \(\mu_{x}\) and \(\mu_{y}\) are the magnitudes of the vectors X and Y. Tr is the trace of the matrix and \(\sum\)X and \(\sum\)Y are the covariance matrix of the vectors. The Structural Similarity Index (SSIM) is an index designed to assess visual differences between people instead of numerical errors. This is because human vision specializes in extracting structural information from images, so distortion of structural information has a large impact on perception. Using this approach, image quality is evaluated based on three variables: luminance, contrast, and structure. These three methods are based on the fact that they have the greatest impact on the human visual system [28]:  
\[\mathrm{I(x,y)}=\frac{2\mu_{x}\mu_{y}+C_{1}}{\mu_{x}^{2}+\mu_{y}^{2}+C_{1}} \tag{7}\]  
\[\mathrm{c(x,y)}=\frac{2\sigma_{x}\sigma_{y}+C_{2}}{\sigma_{x}^{2}+\sigma_{y}^ {2}+C_{2}} \tag{8}\]  
\[\mathrm{s(x,y)}=\frac{\sigma_{xy}+C_{3}}{\sigma_{x}\sigma_{y}+C_{3}} \tag{9}\]  
where \(\mathrm{I(x,y)}\), \(\mathrm{c(x,y)}\), and \(\mathrm{s(x,y)}\) represent brightness, contrast, and the structure of an image, respectively. \(C_{1}\), \(C_{2}\), and \(C_{3}\) are weighting constants to prevent denominators from being zero. \(C_{1}\), \(C_{2}\), and \(C_{3}\) are weighting constants to prevent denominators from being zero. It is in general expressed as \(C_{1}=\left(K_{1}L\right)^{2}\) and \(C_{2}=\left(K_{2}L\right)^{2}\), and the nominal values are given by \(K_{1}=0.01\), \(K_{2}=0.03\), and L = 255, respectively.