An adaptive hybrid conjugate gradient algorithm for the optimization of image restoration model

Image restoration is a fundamental problem in image processing aimed at recovering high-quality images from degraded observations corrupted by blur and noise. This task is commonly formulated as a constrained or unconstrained optimization problem consisting of a data fidelity term and a regularization term to preserve important image features such as edges and textures. Due to the large-scale and ill-conditioned nature of image restoration problems, efficient iterative optimization techniques are required.
Conjugate gradient (CG) methods have been widely applied to image restoration because of their low memory requirements and fast convergence properties in solving large sparse linear and nonlinear optimization problems. Several CG variants have been investigated in the literature and successfully integrated into image restoration frameworks. However, the performance of classical CG methods strongly depends on the choice of search direction and step size parameters. In particular, inappropriate line search strategies may lead to slow convergence or instability when applied to nonlinear and nonconvex image restoration problems. To address these limitations, this paper proposes an adaptive hybrid conjugate gradient algorithm that combines the Fletcher–Reeves and Polak–Ribiere schemes through an adaptive weighting strategy. The proposed method exploits the numerical stability of the Fletcher–Reeves approach and the fast convergence behavior of the Polak–Ribiere method, thereby achieving a balanced and robust optimization performance.
Furthermore, the strong Wolfe line search conditions are adopted to guarantee sufficient descent and curvature properties at each iteration, ensuring global convergence of the proposed algorithm. A theoretical convergence analysis is presented under standard assumptions. The experimental results on blurred and noisy images demonstrate that the proposed adaptive hybrid conjugate gradient method outperforms conventional CG algorithms in terms of both convergence speed and restoration quality, as measured by Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM).