An effective double direction method for solving convex-constrained nonlinear equations with image restoration

The restoration of medical imaging has recently gained significant attention due to its critical role in the medical sciences. This research introduces two efficient double-direction approaches for solving large-scale monotone nonlinear equations and addressing medical image restoration problems. The acceleration parameter in the first algorithm was derived by approximating the Jacobian matrix with a diagonal matrix, while the second algorithm integrates a correction parameter using the Picard-Mann hybrid iterative procedure. The study established the descent condition and demonstrated both the global convergence and R-linear convergence rate of the proposed method under favorable conditions. Numerical experiments showed that the procedure significantly enhances the numerical performance of the proposed methods. Additionally, one of the techniques was successfully applied to restore blurred medical images, highlighting its practical applicability in the field of medical imaging.