Adaptive and Accelerated Golden-Ratio Primal-Dual Algorithms for Total Variation Image Denoising

Abstract Total variation (TV) denoising is a widely used variational approach for recovering images from noisy observations while preserving sharp edges. The resulting Rudin--Osher--Fatemi (ROF) model is convex but nonsmooth and can be efficiently treated by primal--dual first-order methods. This paper specializes adaptive and accelerated golden-ratio primal--dual algorithms to the ROF TV-denoising problem. First, an extended golden-ratio primal--dual algorithm (E-GRPDA) with a non-increasing adaptive stepsize rule is considered. The resulting scheme avoids explicit estimation of the discrete gradient operator norm and backtracking while retaining simple matrix-free proximal updates. We then exploit the strong convexity of the quadratic fidelity term to specialize a parameter-free accelerated golden-ratio primal--dual algorithm (PF-AGRPDA), for which an $O(1/N^2)$ convergence rate is obtained for the corresponding weighted ergodic quantities. Numerical experiments on the Cameraman, House, and Peppers images at two Gaussian-noise levels compare the primal--dual hybrid gradient (PDHG) method, E-GRPDA, and PF-AGRPDA. PDHG gives the fastest decrease of the reported nonergodic objective residual, whereas PF-AGRPDA provides competitive reconstruction quality and achieves the highest final PSNR and the lowest relative $L_2$ error in the reported tests. The results demonstrate the practical viability of the adaptive and accelerated golden-ratio frameworks while highlighting the distinction between theoretical ergodic acceleration and finite-iteration computational performance.