Face super-resolution (FSR), also known as face hallucination, which is aimed at enhancing the resolution of low-resolution (LR) face images to generate high-resolution (HR) face images, is a domain-specific image super-resolution problem. Recently, FSR has received considerable attention and witnessed dazzling advances with the development of deep learning techniques. To date, few summaries of the studies on the deep learning-based FSR are available. In this survey, we present a comprehensive review of deep learning-based FSR methods in a systematic manner. First, we summarize the problem formulation of FSR and introduce popular assessment metrics and loss functions. Second, we elaborate on the facial characteristics and popular datasets used in FSR. Third, we roughly categorize existing methods according to the utilization of facial characteristics. In each category, we start with a general description of design principles, then present an overview of representative approaches, and then discuss the pros and cons among them. Fourth, we evaluate the performance of some state-of-the-art methods. Fifth, joint FSR and other tasks, and FSR-related applications are roughly introduced. Finally, we envision the prospects of further technological advancement in this field. A curated list of papers and resources to face super-resolution are available at url{https://github.com/junjun-jiang/Face-Hallucination-Benchmark}
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