## 4 Conclusions  
In this study, we addressed a novel image restoration method for the UDC system of smartphones by introducing hybrid frequency-spatial domain learning into a deep neural network. This method, which incorporates additional information from the Fourier domain during the training process, has demonstrated substantial improvement in image quality. By analyzing images in the frequency domain, the model directly addresses the loss or alteration of specific frequency components, which are often responsible for image degradation phenomena such as blurring, aliasing, and other artifacts. The spatial domain learning complements this by correcting visible artifacts by addressing distortions and degradations observable in the spatial domain, such as noise and blur. The effectiveness of this approach is highlighted by the increase in the SSIM index from 0.8130 to 0.9813 for the virtual validation datasets, compared to the standard cGAN's increase from 0.8130 to 0.9604. Similarly, the PSNR values improved significantly, with the hybrid domain learning cGAN achieving an average PSNR of 36.683 dB compared to 33.420 dB for the standard cGAN. Moreover, the model has shown its capability to restore experimentally obtained degraded images, achieving an SSIM index increase from 0.8047 to 0.9608 and a PSNR increase from 26.383 dB to 36.046 dB. In contrast, the standard cGAN, while also effective, exhibited slightly lower performance compared to the hybrid domain learning cGAN. The consistent outperformance of the hybrid frequency-spatial domain learning cGAN over the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**PSNR** & **Degraded Image** & **Standard cGAN** &
\begin{tabular}{c} **Hybrid Domain** \\ **Learning cGAN** \\ \end{tabular} \\ \hline Minimum & 18.163 & 18.712 & 20.625 \\ Maximum & 30.9682 & 39.188 & 44.692 \\ Median & 26.704 & 33.061 & 37.558 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Summary statistics of PSNR values for degraded images and images reconstructed by the standard cGAN and by the hybrid domain learning cGAN on actual degraded datasets. The highest value in each row is underlined. \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 & 0.6404 & 0.8502 & 0.8702 \\ Maximum & 0.8972 & 0.9752 & 0.9865 \\ Median & 0.8099 & 0.9416 & 0.9714 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary statistics of SSIM values for degraded images, images reconstructed by the standard cGAN, and images reconstructed by the hybrid domain learning cGAN on actual degraded datasets. The highest value in each row is underlined. standard cGAN, in both SSIM and PSNR metrics, demonstrates the value and effectiveness of the hybrid domain learning approach in image restoration. In addition to developing the restoration model, we generated a dataset of virtual degraded images using simulations based on the ASM, facilitating efficient training. This data augmentation strategy not only allows for the expansion of the dataset to cover a broad range of UDC imaging system scenarios but also reduces the need for laborious manual data acquisition. Finally, the computational complexity of the proposed model was analyzed. The proposed model's computational complexity for processing a 256 \(\times\) 256 resolution image is calculated as 38.5 GFLOPs, as detailed in this study. When scaling this to the resolution of a standard smartphone front-facing camera, assumed to be 2560 \(\times\) 1440 (commonly referred to as QHD resolution), the computational complexity increases by approximately 56 times, reaching nearly 2.17 TFLOPs per frame. Based on the hardware specifications of the current commercial UDC products, the processing time is estimated to be approximately 0.85 s per frame, suggesting that the proposed model could be integrated into commercial UDC products.