## 3 Results
### Evaluation Metrics  
To perform a quantitative evaluation of the generated signatures, we employed both the structural similarity index measure (SSIM) and the root mean square error (RMSE) as evaluation metrics. [PERSON] et al. [22] also assessed the performance of their GAN-based signature generation model by calculating the SSIM between the original signature input to the model and the generated signature. SSIM is a method for assessing the similarity between two images. Unlike conventional methods that simply compare pixel-wise differences, SSIM measures the degree of the preservation of structural information. This can be calculated using Equation (15), with values ranging from \(-1\) to \(1\). A value of \(1\) indicates that the two images are identical, while a value of \(0\) signifies that they are completely different. A value of \(-1\) indicates that the structures are identical but the pixel values are inverted:  
\[\text{SSIM}(x,y)=\frac{(2\mu_{x}\mu_{y}+C_{1})(2\sigma_{xy}+C_{2})}{(\mu_{x}^{ 2}+\mu_{y}^{2}+C_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2})} \tag{15}\]  
In Equation (15), \(x\) and \(y\) represent the two images being compared. \(\mu_{x}\) and \(\mu_{y}\) denote the average brightness of each image, while \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\) represent the variance of each image, and \(\sigma_{xy}\) is the covariance between the two images. \(C_{1}\) and \(C_{2}\) are constants for stability, calculated as follows: generally, \(K_{1}\) and \(K_{2}\) are set to \(0.01\) and \(0.03\), respectively, and \(L\) represents the dynamic range of the pixel values. \[C_{1}=(K_{1}L)^{2},\quad C_{2}=(K_{2}L)^{2} \tag{16}\]  
[PERSON] et al. [32] utilized RMSE to assess how accurately the model predicts output values given specific inputs in nonlinear systems. RMSE is the square root of the mean of the squared differences between the pixel values of two images, where a smaller value indicates higher similarity between the images. RMSE is the square root of the mean of the squared differences between the pixel values of two images, where a smaller value indicates higher similarity between the images. \[\text{RMSE}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_{i}-y_{i})^{2}} \tag{17}\]  
In this equation, \(x_{i}\) and \(y_{i}\) represent the \(i^{th}\) pixel values of the original and generated signatures, respectively, and \(N\) is the total number of pixels. RMSE quantifies the pixel-wise differences between the two signatures, providing insight into how visually similar the generated signature is to the original. In this study, we computed both the SSIM and RMSE after applying a preprocessing step to the comparison signature pairs. This preprocessing involved a high-frequency synthesis process based on the wavelet transform used in the auxiliary classification process. In Table 2, the rows labeled \"Org-Forg\" in the Model category represent the measurement results for the original-forged signature pairs included in the CEDAR dataset. The Train Set Case shows the maximum, minimum, and average SSIM values, as well as the average RMSE value for all original-forged signature pairs used in training. The Highest Class Case presents the maximum, minimum, and average SSIM values and the average RMSE value only for the class with the highest SSIM. The class with the highest SSIM among the original-forged pairs used in training was Class 46. For the original-forged signature pairs used in training, the highest SSIM observed was \(0.9976\), the lowest was \(0.9749\), the average was \(0.9915\), and the average RMSE was \(0.1658\). When only considering Class 46, which had the highest SSIM among the pairs, the highest SSIM was \(0.9976\), the average was \(0.9955\), the lowest was \(0.99\), and the average RMSE was \(0.1427\). In Table 2, the model labeled \"Aux\" represents the model with the auxiliary classification process, while \"Non-Aux\" refers to the model without this process.