Proximity algorithms for the $ {\mathit{L}}^{1}{\mathit{L}}^{2}/{\mathit{T}\mathit{V}}^{\mathit{\alpha }} $ image denoising model

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.