The BV Solution for a Nonlocal Diffusion Equation and Its Application on Multiplicative Noise Removal

Multiplicative Gamma denoising is always a critical challenge in image processing. The multiplicative mechanism and the heavy-tailed distribution of the Gamma noise significantly complicate its removal. To address these two characteristics, this paper proposes a novel gray value indicator function, which is inserted into a new nonlocal diffusion equation model that combines the strengths of the Perona–Malik model and the regularized Perona–Malik model. Furthermore, the concept of generalized Young measure is used to prove the existence for BV solutions and the maximum principle of the proposed model when the initial value $u_0\in BV\left(\Omega \right)\cap L^{\infty }\left(\Omega \right)$. We employ the explicit finite difference method to implement our model, and comparative experiments are conducted.