A Rigorous Stability Analysis of a Fractional-Order Anisotropic Diffusion Model for Edge-Preserving Biomedical Imaging

Classical Perona-Malik models for image processing often suffer from mathematical ill-posedness and can introduce oversmoothing artifacts. This paper addresses these limitations by providing a rigorous theoretical analysis of a fractional-order (FO) anisotropic diffusion model. The primary contribution is the establishment of the model’s well-posedness and stability properties, which have been underexplored. Using spectral decomposition and Lyapunov-energy methods, we prove the existence, uniqueness, and global Mittag-Leffler stability (MLS) of solutions under Neumann boundary conditions. This form of stability guarantees a predictable, algebraic decay to equilibrium, which is crucial for robust performance. Numerical simulations validate the theoretical framework and include a comparative analysis against the classical integer-order model. The results, evaluated with quantitative metrics, demonstrate the FO model’s superior capability in preserving critical edge information while mitigating oversmoothing. By bridging rigorous mathematical theory with practical performance, this study provides a sound foundation for applying FO diffusion in biomedical imaging tasks, such as MRI denoising, where precision and stability are paramount.