## 3 Results and Discussion
### Image Restoration of Degraded Images  
The validation process in this study comprised two critical steps to confirm the reliability of the proposed approach. Initially, we validated both the standard cGAN and the hybrid frequency-spatial domain learning cGAN models using datasets not included in the training phase. The dataset encompassed a total of 3300 images, consisting of 3000 virtually augmented images and 300 experimentally acquired degraded images. Both datasets were divided into training and validation sets in an 8:2 ratio. Specifically, for training, we used 2400 virtual images and 240 real images, and for validation, we used 600 virtual images and 60 real images. This approach allowed us to train the models on both virtual and real data and to assess their performance comprehensively on validation sets that included both types of data. Both the cGAN and the hybrid frequency-spatial domain learning cGAN models were trained under identical conditions. The technical setup included Visual Studio Code IDE, Python 3.9.7, CUDA 11.8, and PyTorch 2.1.2, ensuring stable and efficient performance. Detailed information on hyperparameters is available in Appendix A.2. The image restoration results on the virtual degraded images are shown in Figure 9. The figure presents the original pristine, degraded, cGAN-restored, and hybrid domain learning cGAN-restored results, which are highlighted by green, red, yellow, and blue boxes, respectively. A qualitative analysis visually comparing the restored images to their pristine counterparts showed that both models produced visually appealing results. However, a closer examination of the enlarged sections of the images revealed that the images restored by the hybrid domain learning cGAN exhibited enhanced clarity in darker areas and reduced granular-patterned noise, in contrast to those restored by the standard cGAN. The average SSIM values are also summarized in Figure 9b on the virtual degraded images and the restored images by the two different models, where each case is displayed with different colors: blue for the hybrid domain learning cGAN, yellow for the standard cGAN, and red for the degraded images, respectively. Quantitatively, the hybrid domain learning cGAN achieved an average SSIM value of 0.9813, compared to 0.9604 for the standard cGAN, indicating approximately a 20.7% improvement in image quality. Similarly, Figure 9c presents the PSNR values for the same dataset and models, following the same color scheme. The degraded images showed an average PSNR of 27.112 dB, which was significantly improved to 33.420 dB by the standard cGAN. The hybrid domain learning cGAN achieved the highest average PSNR of 36.683 dB, further demonstrating its ability to enhance image quality. These results are consistent with the SSIM analysis, reaffirming the superior restoration capabilities of the hybrid domain learning approach compared to the standard cGAN. This evaluation verified the efficacy of hybrid frequency-spatial domain learning in image restoration. To provide a more comprehensive evaluation of the experimental results, we present additional summary statistics, including the minimum, maximum, and median values of SSIM and PSNR for each case. These statistics, summarized in Tables 1 and 2, offer deeper insights into the variability and consistency of the models' performance across the dataset. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**SSIM** & **Degraded Image** & **Standard cGAN** &
\begin{tabular}{c} **Hybrid Domain** \\ **Learning cGAN** \\ \end{tabular} \\ \hline Minimum & 0.5313 & 0.7347 & 0.8097 \\ Maximum & 0.9754 & 0.9903 & 0.9953 \\ Median & 0.8169 & 0.9603 & 0.9813 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary statistics of SSIM values for degraded images and images reconstructed by the standard cGAN and by the hybrid domain learning cGAN on virtual degraded images. The highest value in each row is underlined.