Inspired by the ROF model and the $L^1/TV$ image denoising model, we propose a combined model to remove Gaussian noise and salt-and-pepper noise simultaneously. This model combines the $L^1$ -data fidelity term, $L^2$ -data fidelity term and a fractional-order total variation regularization term, and is termed the $L^1{L}^2/{TV}^{\alpha }$ model. We have used the proximity algorithm to solve the proposed model. Through this method, the non-differentiable term is solved by using the fixed-point equations of the proximity operator. The numerical experiments show that the proposed model can effectively remove Gaussian noise and salt and pepper noise through implementation of the proximity algorithm. As we varied the fractional order $\alpha$ from 0.8 to 1.9 in increments of 0.1, we observed that different images correspond to different optimal values of α.

This paper proposes a model that eliminates combined additive and multiplicative noise by merging the Rudin, Osher, and Fatemi (ROF) and I-divergence data fidelity terms. Many important techniques for denoising and restoring medical images were included in this model. The addition of the I-divergence fidelity term increases the complexity of the model and difficulty of the solution in comparison with the ROF. To solve this model, we first proposed the generalized concept of the maximum common factor based on the inverse scale space algorithm. Different from general denoising algorithms, the inverse scale space method exploits the fact that $u$ starts at some value $c_0$ and gradually approaches the noisy image $f$ as time passes which better handles noise while preserving image details, resulting in sharper and more natural-looking images. Furthermore, a proof for the existence and uniqueness of the minimum solution of the model was provided. The experimental findings reveal that our proposed model has an excellent denoising effect on images destroyed by additive noise and multiplicative noise at the same time. Compared with general methods, numerical results demonstrate that the nonlinear inverse scale space method has better performance and faster running time on medical images especially including lesion images, with combined noises.