Two double direction methods for convex-constrained nonlinear monotone equations with image recovery of Spherical Parts

The recent emphasis on image restoration is largely due to its importance in engineering and scientific fields. This paper introduces two effective double-direction convex-constrained approaches to address large-scale monotone nonlinear equations to recover the imaging of spherical parts. The first method utilizes the difference between Broyden's update and its approximation using the Frobenius norm to derive an acceleration parameter, while the second approach incorporates a correction parameter through a Picard-Mann hybrid iterative procedure in its search direction. We prove the descent condition of the approaches and establish the global convergence and R-linear convergence rate of both methods under certain favorable conditions. Numerical simulations demonstrate that our approach significantly boosts the numerical performance of the proposed algorithms. We show that the test function used satisfied uniformly monotonicity conditions. Furthermore, the method has been successfully applied to restore blurred images of spherical parts, showcasing its practical relevance in mechanical engineering.