Image processing is a fundamental field that plays a crucial role in various applications such as medical imaging, remote sensing, and computer vision. Mathematical methods form the foundation of image processing algorithms, providing the framework for understanding image properties and designing efficient processing techniques. This survey provides a comprehensive overview of the mathematical methods used in image processing, including digital image representation, image enhancement, restoration, and segmentation. Advanced mathematical methods such as sparse representations, variational methods, Markov random fields, and deep learning techniques are also discussed, highlighting their significance in advancing the field of image processing. The applications of mathematical methods in various domains, including medical imaging, satellite image analysis, forensic image analysis, and industrial image processing, are explored. Finally, the survey discusses the challenges faced by researchers in image processing, such as computational complexity, big data management, interdisciplinary research trends, and emerging technologies, and suggests future directions for research in this area.