A High-Order Reaction-Diffusion Coupled System for Super-Resolution

Abstract

Super-resolution (SR) remains a critical challenge in imaging science, particularly in applications demanding fine-texture reconstruction and noise suppression. In this work, we introduce a novel nonvariational anisotropic diffusion model, formulated as a high-order nonlinear reaction-diffusion system, designed to enhance SR capabilities. Our approach utilizes a decomposition strategy based on the $H^{-1}$ norm, which effectively captures and preserves fine details and intricate textures in images. Theoretical analysis establishes the well-posedness of the model via the Schauder fixed-point theorem, ensuring mathematical rigor. To validate its performance, we implement a finite difference numerical scheme, testing the model against state-of-the-art SR techniques on challenging image datasets. Results demonstrate significant improvements in texture restoration, edge sharpness, and robustness to noise and degradation, surpassing competitive methods. This research lays the foundation for advanced PDE-based SR frameworks, offering significant potential for applications in biomedical imaging, satellite vision, and high-fidelity computational photography.