Inertial Forward–Backward–Forward Algorithm with Moving Point Projection for Monotone Inclusions and Image Restoration

This paper introduces a novel inertial forward–backward–forward algorithm driven by a newly conceptualized moving point projection technique for solving monotone inclusion problems in real Hilbert spaces. By leveraging the properties of a Lipschitz continuous, monotone operator and a maximally monotone operator alongside this innovative projection strategy, we dynamically construct a sequence of nonempty, closed, and convex sets that contain the zeros of the sum of the two operators. This geometric construction ensures that the resulting sequence is well defined and guarantees its weak convergence to a solution. Furthermore, to validate the practical efficacy of the proposed theoretical framework, we evaluate our method on image restoration problems. Numerical experiments measuring the improvement in signal-to-noise ratio (ISNR) and the structural similarity index measure (SSIM) confirm that the proposed algorithm is highly efficient and significantly outperforms existing state-of-the-art methods.