## 2 Methodology
### Generation of the Virtual Degraded Image
#### 2.1.2 Generation of the Point Spread Function
The discriminator \(D_{f}\) evaluates the authenticity of reconstructed frequency data by comparing it with the original data, thus encouraging \(G_{f}\) to generate frequency data indistinguishable from authentic data. After estimating and filling in the missing frequency components, we reconstruct the frequency domain data. We then apply the inverse discrete Fourier transform (IDFT) to convert the restored frequency domain data back into the spatial domain:  
\[\hat{f}(x,y)=\frac{1}{MN}\sum_{u=0}^{M-1}\sum_{v=0}^{N-1}\hat{F}(u,v)\cdot e^{ j2\pi(\frac{W}{M}+\frac{v2}{N})}, \tag{7}\]  
where \(\hat{F}(u,\ v)\) is the reconstructed frequency domain data, and \(\hat{f}(x,y)\) is the restored spatial domain image. In parallel, we employ a spatial domain network with generator \(G_{s}\) and discriminator \(D_{s}\). This network directly processes spatial images, correcting artifacts such as blur, noise, and aliasing. \(G_{s}\) applies anti-aliasing filtering to mitigate effects from under-sampling and other limitations, while \(D_{s}\) evaluates the restored images, guiding \(G_{s}\) to generate high-quality images closely resembling the originals. \(D_{s}\) evaluates input pairs of degraded and restored images by outputting a probability score that reflects how well the restored image matches real data. This evaluation leverages convolutional layers to extract and compare key features, including local textures, edge sharpness, and structural patterns. While metrics such as SSIM or PSNR are not directly measured using \(D_{s}\), its adversarial feedback during training effectively guides \(G_{s}\) to enhance visual fidelity and maintain consistency in restored images. Furthermore, \(D_{s}\)'s loss function, detailed in Appendix A.2,  
Figure 6: Representation of k-space regions showing lost or altered data for four different degraded sample images and the averaged loss across the entire dataset. ensures that \(G_{s}\) generates images indistinguishable from real ones by refining both local and global features. Both the frequency and spatial domain networks are trained simultaneously yet independently. Each pair of generator and discriminator works within its domain, minimizing discrepancies with the ground truth. Frequency domain training focuses on restoring structural integrity, addressing global features, while spatial domain training corrects visible artifacts and refines local details, enhancing textures and visual quality. The cGAN networks in the frequency domain (\(G_{f}\) and \(D_{f}\)) and spatial domain (\(G_{s}\) and \(D_{s}\)) share an identical architecture. The primary distinction lies in the input data and the domain-specific focus: a frequency domain network processes k-space representations, while a spatial domain network handles pixel-level images. This unified architecture ensures consistency while enabling domain-specific training for optimal restoration performance. The architectural details of the generator and discriminator components are described in Appendix A.1 and visually summarized in Figure A1. Additionally, the combined structure of a single cGAN network unit, comprising both the generator and the discriminator, is illustrated in Figure A2. By analyzing both domains, our model effectively addresses image degradation in UDC systems. Most deep learning methods primarily rely on spatial domain learning, which is intuitive and aligns with human visual perception. Spatial domain learning focuses on extracting features through convolutional operations, which model relationships between neighboring pixels. This approach is particularly effective at capturing local textures and details, as convolutional kernels operate within limited receptive fields [45; 46; 47]. However, this same reliance on limited receptive fields makes spatial domain learning less effective at capturing global contextual information.