## 2 Methodology
### Generation of the Virtual Degraded Image
#### 2.1.2 Generation of the Point Spread Function
The generator synthesizes images by using a combination of random noise and conditional information, aiming to generate images that closely mimic real data. The discriminator, in contrast, receives both real and generated data, along with the conditional information, working to distinguish between them. During training, the generator and discriminator engage in an adversarial process: the generator strives to produce realistic images, while the discriminator attempts to accurately classify images as real or generated. A distinctive feature of our approach is the integration of hybrid frequency-spatial domain learning within this adversarial framework. Unlike traditional cGANs, which operate primarily in the spatial domain, our model incorporates information from the frequency domain of degraded images. This is essential for capturing structural nuances that are often compromised in UDC systems due to panel-induced degradation. Our approach involves two separate but complementary cGAN networks: one for the frequency domain and the other for the spatial domain. Each network includes a generator and a discriminator, denoted as \(G_{f}\) and \(D_{f}\) for the frequency domain, and \(G_{s}\) and \(D_{s}\) for the spatial domain. Analyzing images in the frequency domain enables us to directly address the loss or alteration of specific frequency components, which often leads to image degradation issues such as blurring and aliasing. In the frequency domain, missing or degraded information appears as gaps or irregularities in k-space--the Fourier transform representation of spatial data. As explained in Equation (1), image degradation in a UDC system can be described using the convolution theorem. When the PSF of an imaging system is defined, it becomes possible to analyze how input signals are transformed into output signals. While degradation trends can be understood in the spatial domain for individual images, it is challenging to identify consistent patterns across multiple images. In contrast, the k-space domain provides clearer insights into where degradation occurs across diverse images. This clarity is due to the convolution theorem's representation in the k-space as a simple pixel-wise multiplication [40; 41]. In Figure 6, the left panels represent the Fourier domain information for four random sample images, each showing the lost or altered frequency components caused by degradation. While the overall patterns differ across the four images, it is evident that the loss consistently occurs within similar frequency bands. The rightmost panel displays the averaged k-space modulation data computed from all training images, providing a  
Figure 5: (**a**) Camera spectral sensitivity and (**b**) simulated 3 ch PSF fields. comprehensive visualization of the frequency regions that are systematically impacted by the UDC system. This analysis guided our model design, enabling it to learn in the Fourier domain by identifying the specific regions where k-space components are predominantly lost or altered. By doing so, the model gains an intuitive understanding of the degradation caused by the defined UDC system, thereby enhancing its ability to effectively restore degraded images. Building on these insights, we design a frequency domain network to predict and fill missing or altered values in k-space. First, we convert degraded images to the frequency domain using the discrete Fourier transform (DFT), which allows direct analysis and modification of frequency components. The DFT of an image \(f(x,y)\) is given by  
\[F(u,v)=\sum_{x=0}^{M-1}\sum_{y=0}^{N-1}f(x,y)\cdot e^{-j2\pi(\frac{(\frac{W}{M} +\frac{v2}{N})})}, \tag{6}\]  
where \(u\) and \(v\) are the frequency domain coordinates, and \(M\) and \(N\) are the dimensions of the image [42; 43; 44]. In the frequency domain, the generator \(G_{f}\) takes degraded frequency data as input, learning to predict and fill in the missing k-space components.