Time-space fractional anisotropic diffusion equations for multiplicative noise removal

Abstract

In this paper, we propose a nonlinear time-space fractional diffusion model to remove the multiplicative gamma noise. This model incorporates Caputo time-fractional derivative into the existing space-fractional diffusion models. It leverages the memory effect of time-fractional derivatives to control the diffusion process, achieving a balance between edge preservation, texture retention and denoising effects. To establish the solvability of the proposed model, an auxiliary time-space fractional diffusion problem is first constructed, and the existence and uniqueness of its weak solution are proven using the Faedo-Galerkin method. Based on this, the existence and uniqueness of the weak solution for the proposed model are further confirmed via the Schauder fixed point theorem. Next, an explicit-implicit semi-discretization scheme is designed using Caputo fractional derivative with an order of $0<\beta \le 1$ in time, and the stability of the semi-implicit scheme ($\lambda =1$) is proven, ensuring that ${\Vert u^n\Vert }_2\le {\Vert f\Vert }_2$. For the fully discrete scheme, the more stable shifted Grünwald-Letnikov fractional derivative is used in space with an order of $1<\alpha <2$. Finally, to verify the effectiveness of the model, numerical experiments are conducted on images with different noise levels and features, and compared with the existing diffusion equation models. The results demonstrate that the proposed model exhibits superior denoising performance while preserving edges and textures.