Physics Meets Pixels: PDE Models in Image Processing

Abstract

Partial Differential Equations (PDEs) have long been recognized as powerful tools for image processing and analysis, providing a framework to model and exploit structural and geometric properties inherent in visual data. Over the years, numerous PDE-based models have been developed and refined, inspired by natural analogies between physical phenomena and image spaces. These methods have proven highly effective in a wide range of applications, including denoising, deblurring, sharpening, inpainting, feature extraction, and others. This work provides a theoretical and computational exploration of both fundamental and innovative PDE models applied to image processing, accompanied by extensive numerical experimentation and objective and subjective analysis. Building upon well-established techniques, we introduce novel physical-based PDE models specifically designed for various image processing tasks. These models incorporate mathematical principles and approaches that, to the best of our knowledge, have not been previously applied in this domain, showcasing their potential to address challenges beyond the capabilities of traditional and existing PDE methods. By formulating and solving these mathematical models, we demonstrate their effectiveness in advancing image processing tasks while retaining a rigorous connection to their theoretical underpinnings. This work seeks to bridge foundational concepts and cutting-edge innovations, contributing to the evolution of PDE methodologies in digital image processing and related interdisciplinary fields.