# Various Generative Adversarial Networks Model for Synthetic Prohibitory Sign Image Generation
## 2 Materials and Methods
### SSIM and MSE  
The structural similarity (SSIM) index is a good indicator of perceived image quality. The SSIM assessment approach distinguishes the brightness and contrast of the required image detail and incorporates structural information for image quality evaluation [41; 42]. The structural similarity measurement is split into three parts: the luminance function \(l(x,y)\), the contrast function \(c(x,y)\), and the structure comparison function \(s(x,y)\)[43]. These three factors will become indicators of how similar the structure is. The mean value is an estimate of brightness, the standard deviation is used as a contrast estimate, and the  
Figure 2: Schematic of the Wasserstein Generative Adversarial Network (WGAN).  
total variation number is used as a structural resemblance measure. The SSIM functions Formulae (8)-(11) areas follows [44; 45].  
\[l(x,y)=\frac{\left(2\mu_{x}\mu_{y}+C_{1}\right)}{\left(\mu_{x}^{2}+\mu_{y}^{2}+C _{1}\right)} \tag{8}\]  
\[c(x,y)=\frac{\left(2\sigma_{xy}+C_{2}\right)}{\left(\sigma_{x}+\sigma_{y}+C_{ 2}\right)} \tag{9}\]  
\[s(x,y)=\frac{\left(\sigma_{xy}+C_{3}\right)}{\left(\sigma_{x}\sigma_{y}+C_{3} \right)} \tag{10}\]  
\[SSIM(x,y)=\frac{\left(2\mu_{x}\mu_{y}+C_{1}\right)\left(2\sigma_{xy}+C_{2} \right)}{\left(\mu_{x}^{2}+\mu_{y}^{2}+C_{1}\right)\left(\sigma_{x}^{2}+\sigma _{y}^{2}+C_{2}\right)} \tag{11}\]  
where \(\mu_{x}\) is the average of \(x\), \(\mu_{y}\) is the average of \(y\), \(\sigma_{x}^{2}\) is the variance of \(x\), \(\sigma_{y}^{2}\) is the variance of \(y\), and \(\sigma_{xy}\) is the covariance of \(x\) and \(y\). The input of SSIM [46] is a pair image, one an undistorted image, and the other a distorted image. The structural similarity between both images can be observed as an image quality indicator of the distorted image. Contrasted with traditional image quality measurement indicators, such as Peak Signal-to-Noise Ratio (PSNR), and Mean Squared Error (MSE) [47], the structural similarity is more in line with the human eye for image quality in terms of image quality measurement judgment. The relation between SSIM and more conventional quality metrics in a vector field of the image components can be demonstrated geometrically. The components of these images might be pixels or other derived elements, for example, linear coefficients. [48].  
Mean Square Error (MSE) is adopted to determine the discrepancy between estimated values, and the original values of the quantity being estimated are the square of the difference of pixels. The error is the amount by which the value implied by the estimator differs from the quantity to be estimated shown in Formula (12) [49].  
\[MSE=\frac{\sum_{i=1}^{n}(P_{i}-Q_{i})^{2}}{n} \tag{12}\]  
where \(P_{i}\) represents observed value, \(n\) is the number of data points, and \(Q_{i}\) represents predicted value. In our works, synthetic images generated by DCGAN, LSGAN, and WGAN are evaluated using SSIM and MSE. However, the value of SSIM is between \(-1\) and \(1\), the higher is better. In contrast, smaller MSE values suggest a more favorable result.