On solving double direction methods for convex constrained monotone nonlinear equations with image restoration

Vast applications of derivative-free methods to restore the blurred images in compressive sensing has become an important trend in recent years. This research, a double direction method for better image restoration is proposed. Besides this, two double direction algorithms to solve constrained monotone nonlinear equations are presented. The main idea employed in the first algorithm is to approximate the Jacobian matrix via acceleration parameter to propose an effective derivative-free method. The second algorithm involve hybridizing the scheme of the first algorithm with Picard–Mann hybrid iterative scheme. In addition, the step length is calculated using inexact line search technique. The proposed methods are proven to be globally convergent under some mild conditions . The numerical experiment, shown in this paper, depicts the efficiency of the proposed methods. Furthermore, the second method is successfully applied to handle the $$\ell _1$$ -norm regularization problem in image recovery which exhibits a better result than the existing method in the previous literature.