A fractional-time PDE-constrained parameter identification for inverse image noise removal problem

Abstract

This study introduces a novel PDE-constrained optimization approach tailored to determine the optimal fractional-time order $\alpha$ in diffusion PDEs for image denoising. By incorporating a time-fractional derivative, our framework effectively enhances image clarity and reduces virtual artifacts. The Accelerated Primal–Dual algorithm is utilized to improve the efficiency of the model. We conduct a comprehensive evaluation of the denoising performance of this PDE-constrained method through various numerical experiments, considering different images and noise levels across a wide range of noise intensities. Furthermore, the robustness of the model is tested under high noise conditions, and a detailed analysis of the behavior of the fractional-time derivative is provided. The experimental results demonstrate the model’s effectiveness and resilience in noise reduction, supported by both visual inspections and quantitative metrics. Compared to several state-of-the-art techniques, our approach delivers superior image denoising, producing images that are significantly cleaner, exhibit a natural appearance, and show a marked reduction in undesirable artifacts.