## 1 Introduction  
In recent years, smartphones have become an integral part of our lives, serving crucial roles as communication devices, media consumption tools, and gateways to the virtual world. The pursuit of larger and more immersive displays has catalyzed significant advancements in smartphone design. For instance, bezel sizes have decreased, and innovative solutions, such as notch and punch-hole designs, have been developed to accommodate front-facing cameras. Among these advancements, the under-display camera (UDC) system [1, 2] stands out as a groundbreaking technology aimed at achieving full-screen displays. This is accomplished by reducing pixel density to enhance light transmittance and integrating the front-facing camera beneath the display panel, resulting in an aesthetically pleasing effect in which the camera is virtually invisible to the naked eye. However, UDC systems face challenges related to poor light transmittance and diffraction, leading to degraded images with undesirable effects, such as noise, ghost, haze, fog, and blur [3; 4; 5], as illustrated in Figure 1. These degradation phenomena occur due to the UDC panel positioned in front of the camera lens. They can be mathematically described using the convolution theorem as follows:  
\[y=k*x, \tag{1}\]  
where \(*\) denotes the convolution operation, \(y\) represents the degraded image result, \(k\) denotes the point spread function (PSF) of the UDC system, and \(x\) represents the original clear image. The PSF characterizes how a focused optical imaging system responds to a point source [6; 7; 8; 9]. To address the challenges arising from poor light transmittance and diffraction in UDC systems, research has focused on the structural optimization of the display panel [10; 11; 12]. Additionally, a computational approach leveraging artificial intelligence (AI) techniques has been explored to enhance image quality in UDC systems, complementing structural optimization efforts. To mitigate undesired artifacts in degraded images, numerous deep neural networks, predominantly based on U-Net architecture, have been proposed [13; 14; 15; 16; 17; 18; 19]. These studies have primarily utilized a U-Net-based structure to analyze the mapping relationship between original and degraded images within the spatial domain. In this study, we propose a hybrid frequency-spatial domain learning approach to realize a more effective image restoration process in UDC systems. By training an AI model not only in the spatial (image) domain but also in the frequency (Fourier) domain, we address the technical challenges inherent in UDC systems. The integration of frequency domain learning enables the model to directly analyze and restore global patterns and structural degradations, such as those introduced by diffraction. In addition, spatial domain learning focuses on refining local textures and correcting visible artifacts. This hybrid domain approach allows the model to learn a more intuitive correlation between degraded and original images, significantly enhancing its predictive accuracy and reliability compared to traditional spatial domain-only methods. For training in most image restoration tasks, a large amount of training data consisting of degraded images and their corresponding original images is essential, and UDC scenarios are no exception. However, obtaining degraded images and original images simultaneously in UDC systems is not straightforward. In outdoor shooting, unless images are captured simultaneously, the subjects may change, making it impossible to obtain exactly corresponding data. Even if an optical system capable of simultaneous shooting is  
Figure 1: Conceptual pixel layouts of (**a**) a typical punch-hole camera system and (**b**) a UDC system, respectively, and their respective image quality. implemented, collecting such data directly is time-consuming and labor-intensive. This challenge is further compounded in UDC systems, where degradation is often specific to the unique optical properties of the display panel.