An Adaptive Double-Inertial Algorithm for Variational Inequality and Split Common Fixed Point Problems with Applications to Image Restoration

In this paper, we introduce a new adaptive double-inertial viscosity algorithm to solve a variational inequality and split common fixed point problem in Hilbert spaces. In our proposed method, we combine two inertial terms with a projected variational inequalities with a self-adaptive step size, an averaged demicontractive mapping and a viscosity approximation. This combination allows the algorithm to use information from previous iterates in each step. Under suitable assumptions on mappings used and sequences of the parameters, we prove the proposed sequence converges strongly to a solution of the considered problem. Several numerical examples are presented to study the convergence behavior of the proposed method. The numerical results show that the proposed algorithm converges faster for different initial points and performs better than the comparable existing algorithm in the tested cases. As an application, the proposed method is used for image restoration. The restoration quality is evaluated using PSNR, SSIM, and MSE, together with CPU time and the number of iterations. The proposed method gives the highest PSNR and SSIM values and the lowest MSE for all considered images. The visual results also show that the proposed method can remove degradation while preserving important image structures. The proposed algorithm is also assessed through a numerical example, and its performance is compared graphically with several existing methods from the literature. These results show that the proposed algorithm is effective both for the theoretical problem and for the considered image-restoration application.