## 1 Introduction  
Face super-resolution (FSR), also known as face hallucination [1], is a technology for enhancing low-resolution (LR) face images into high-resolution (HR) ones. Due to low-cost imaging equipment and limited imaging conditions, face images often have a lower spatial resolution, which severely degrades the performance of most practical downstream applications such as face analysis and face recognition. Therefore, FSR has become increasingly popular in the computer vision and image processing fields, making it an important scientific tool [2]. Unlike general image super-resolution, FSR focuses on recovering pivotal facial structures. Although facial structures only occupy a small portion of the face, they are crucial in distinguishing different faces. The first FSR method, proposed by [PERSON] and [PERSON] [1], triggered the upsurge of traditional FSR methods. Over time, various traditional techniques for FSR have been developed, including the interpolation approach [3], PCA [4], convex optimization [5], Bayesian approach [6], kernel regression [7], and manifold learning [8]. Traditional methods are limited by their shallow structure and representation abilities, making them incompetent in producing plausible facial images. Alongside the rise of deep learning techniques [2, 9, 10], deep convolution neural networks (CNNs) [11, 12, 13, 14, 15] have achieved remarkable advancements in improving face image quality. The first CNN-based FSR method was introduced by [PERSON] et al. [11], greatly improving FSR performance compared with traditional FSR methods. To further explore facial information, [PERSON] et al. [12] utilized reinforcement learning to capture the interdependency among facial parts. [PERSON] et al. [13] introduced super-identity loss to assist the network in generating more accurately identified super-resolution face images. In contrast to the FSR methods above, which recover face images directly, [PERSON] et al. [14] used the wavelet transform to project face images into wavelet spaces to capture rich contextual information, while [PERSON] et al. [15] applied the Fourier transform to obtain an image-size receptive field for capturing global facial structure. Inspired by the great success of generative adversarial networks (GANs) in the image processing field [16; 17; 18], [PERSON] et al. [19] introduced a collaborative suppression and replenishment framework based on GANs. [PERSON] et al. [20] proposed a conditional generative model based on the diffusion model, which replaces the U-Net in super-resolution to capture complex details and fine textures. Addressing the fact that GAN-based methods require greater computational resources, PCA-SRGAN [21] uses Principal Component Analysis decomposition, while SPGAN [22] employs supervised pixel-wise loss to ease the GAN training process. Due to the highly structured nature of the human faces, many FSR methods leverage facial priors, e.g., face landmarks and parsing maps, to enhance the reconstruction performance. [PERSON] et al. [23] utilized facial parsing maps to guide the end-to-end FSR convolution network. Meanwhile, [PERSON] et al. [24] combined GANs with a well-designed heatmap loss to constrain the face structure between HR and super-resolved (SR) face images. To better capture sharp facial structures in face images with large pose variations, [PERSON] et al. [25] introduced 3D facial priors instead of the commonly used 2D ones. Considering the challenge of estimating priors from degraded LR face images directly, DIC [26] used an iterative process in which FSR and prior estimation were performed repeatedly to enhance FSR performance. Recently, the attention mechanism has emerged as a new trend in computer vision tasks [27; 28; 29; 30; 31]. [PERSON] et al. [32] devised a face attention unit to capture facial structure information, while [PERSON] et al. [33] designed an external-internal split attention group to reconstruct clear facial images.