Exploring Partial Differential Equations for Image Processing: Edge Detection to Restoration

Abstract

Partial Differential Equations (PDEs) have become a cornerstone of modern image processing, providing a powerful mathematical framework to analyze, enhance, and restore digital images. By modeling an image as a continuous function, PDE-based methods effectively capture structural features such as edges and textures while minimizing the impact of noise and distortions. In edge detection, PDEs surpass traditional gradient-based operators by incorporating geometric and contextual information, which improves the accuracy of boundary identification in noisy or complex images. Likewise, PDE-based diffusion models, inspired by physical processes such as heat conduction, have been widely used for image restoration tasks including denoising, deblurring, and inpainting. Advanced nonlinear PDEs further refine these processes by adapting to local image characteristics, preserving sharp details while suppressing unwanted artifacts. This dual capability—detecting meaningful structures and restoring degraded information—highlights the versatility of PDEs in both theoretical and applied contexts. From medical imaging to computer vision and remote sensing, PDE-driven techniques continue to play a pivotal role in ensuring image clarity, reliability, and interpretability. This paper explores the applications of PDEs in image processing, tracing their role from edge detection to image restoration, and emphasizing their enduring significance in advancing digital imaging technologies.