Review of Partial Differential Equations in Image Processing: Edge Detection, Restoration, and Beyond

Abstract

Partial Differential Equations (PDEs) have become a cornerstone of modern image processing, offering a mathematically rigorous and physically inspired framework for solving a wide range of imaging problems. This review explores the development and applications of PDE-based methods, tracing their evolution from classical linear diffusion models to advanced nonlinear and variational formulations. PDEs provide a unified approach to fundamental tasks such as edge detection, denoising, deblurring, segmentation, and inpainting, enabling the preservation of critical structures while reducing noise and distortions. Notable contributions, including anisotropic diffusion (Perona–Malik) for selective smoothing and the Rudin–Osher–Fatemi (ROF) model for total variation minimization, highlight the effectiveness of PDEs in balancing clarity and detail preservation. Beyond restoration and edge enhancement, PDEs have been extended to higher-level tasks such as multiscale analysis, optical flow estimation, and texture reconstruction, demonstrating their versatility across diverse domains including medical imaging, remote sensing, cultural heritage preservation, and computer vision. While computational complexity and parameter sensitivity remain limitations, PDE-based methods retain unique strengths in interpretability and robustness, especially in scenarios with limited or noisy data. Furthermore, recent research emphasizes hybrid approaches that integrate PDE formulations with deep learning, combining the interpretability of PDEs with the adaptability of data-driven models. This review highlights the enduring significance of PDEs in advancing image processing theory and practice, while pointing toward future directions where mathematical rigor and modern computational techniques converge.