A Parallel Moving Point Projection Algorithm for Generalized Split Convex Feasibility Problems with Applications to Image Restoration

The split convex feasibility problem serves as a unified mathematical framework for solving inverse problems, medical image reconstruction, and signal recovery across multiple Hilbert spaces. Traditional iterative schemes, such as hybrid and shrinking projection algorithms, typically enforce strong convergence by projecting the initial point onto the intersection of successive half-spaces. However, this strategy introduces a severe computational bottleneck due to the necessity of solving constrained quadratic subproblems at every iteration. In this paper, we propose a novel Parallel Moving Point Projection (PMPP) algorithm designed to overcome this limitation. The proposed framework integrates a parallel projection architecture across multiple bounded linear operators with a dynamic moving point projection mechanism, where projections are performed onto the intersection of two adaptively updated half-spaces, which can be computed in explicit closed form. Under mild and standard assumptions on the control parameters, we establish a strong convergence theorem for the generated sequence in real Hilbert spaces without requiring any compact embedding. Furthermore, to validate the theoretical framework and computational efficiency of our approach, we perform extensive numerical experiments on both synthetic benchmark problems in finite-dimensional Euclidean spaces and color image restoration tasks corrupted by motion blur, average blur, and additive Gaussian noise. The experimental results demonstrate that the proposed PMPP algorithm achieves competitive execution times, lower iteration counts in some cases, and higher restoration quality in terms of signal-to-noise ratio (SNR), improvement in signal-to-noise ratio (ISNR), and structural similarity index measure (SSIM) compared to prominent existing parallel hybrid projection methods.