An Application of the Classical Conjugate Gradient Method in Image Restoration Problem

Abstract

Optimization was a collection of mathematical methods applied across various fields to find the most effective solution to a problem. One of the optimization methods was the conjugate gradient method, selected due to its low memory demands and capacity to make use of the optimization problem’s structure. In this study, the classical conjugate gradient method was focused. The aim was to investigate the performance of classical conjugate gradient methods and assess their applicability and effectiveness in an image restoration problem. Three methods, namely the Mouiyad, Mustafa and Rivaie (MMR), Nurul Hajar, Mustafa and Rivaie (NHMR), and Saleh, Fadhilah, and Aidya Hanum Aizam (SFA) methods, were evaluated on ten selected test functions using number of iterations (NOI) and computational time (CPU). The methods were subsequently applied to image restoration problems in which test images were degraded by 20% to 90% salt-and-pepper noise. Restoration quality was evaluated using peak signal-to-noise ratio (PSNR), relative error (Relerr), and computational time (CPU). Comparative analysis indicated that MMR achieved the lowest NOI and CPU time across a wide range of test problems, while SFA displayed moderate performance and NHMR showed lower efficiency. For image restoration, the MMR method achieved the highest overall restoration quality, while SFA and NHMR showed moderate and lower restoration performance, respectively. Overall, the findings indicated that the MMR method demonstrated stronger performance and flexibility than NHMR and SFA in both the benchmark test problems and image restoration application. The results also indicated that a method with superior convergence characteristics on the test problems tended to show stronger restoration performance when applied to image restoration.