A Restart Fast ADMM Method For Second-Order Variational Image Restoration Models

Objective Variational models have been very successful used in image denoising,image segmentation and image restoration.It is still one of the most active research areas in image processing and computer vision.Among them,variational model of image restoration has a fundamental position.Variational model of image restoration based on the second-order derivative can maintain the image edge and smooth features.But its regular terms are generally non-linear,non-smooth,or even non-convex.These features not only lead to the difficulty of numerical algorithm design,but also the low computational efficiency of tis numerical method.At the same time,these features restrict the design of its fast algorithm.It forms one of the main bottlenecks in the engineering application of variational image processing and analysis models.In order to solve convex optimization problem,Alternating Direction Methods of Multipliers decompose the global optimization problem into several local subproblems that are easy to be solved.The optimal solution can be obtained by means of alternating iterative optimization of subproblems.The iteration cost of ADMM method is lower,but the convergence speed is slower.As the size of the problem grows,the algorithm that solves the optimization problem should maintain efficient performance.The key to design of acceleration method is to design optimal inertial parameters.However,the variational image processing models are often locally strongly convex or completely non-convex,which makes it difficult or time-consuming to estimate the optimal inertial parameters.Its inertial acceleration algorithms can cause ripples and fail to achieve the expected acceleration effect.Based on alternating direction methods of multipliers method,combined with Nesterov's inertial acceleration method and restart idea to avoid ripples,the corresponding restart fast ADMM algorithms are designed.In calculation process,restart fast ADMM algorithm adaptively adjust step size to eliminate ripples and improve computational efficiency according to size of the combined residual.The research of improved monotonic algorithm,backtracking algorithm and restart algorithm can avoid ripples phenomenon and keep algorithm convergence rate.The purpose of this paper is to use fast ADMM method as framework to explore possibility of the restart fast algorithm in second-order variational models.Main task is trying to design restart fast algorithms of Total Variation(TV)? total Laplacian(TL)model and Euler's elastic(EE)model.Firstly,for TV,TL and EE image restoration variational models,the original optimization problem was transformed into an alternate optimization of multiple subproblems by introducing auxiliary variables,Lagrange multiplier and penalty parameters through variable splitting.Then,the subproblems were solved by fast Fourier transform and soft threshold formula respectively.Secondly,based on alternating direction methods of multipliers method,combined with Nesterov's inertial acceleration method and restart idea to avoid ripples,the corresponding restart fast ADMM algorithms are designed.Finally,the original ADMM algorithm,FAST ADMM algorithm and restart fast ADMM algorithm are applied to the image restoration variational model,and the iteration times,running time,PSNR and computational efficiency of the three algorithms are compared under different models,different images,different noises and different parameters.Numerical experiments show that restart strategy can improve computational efficiency of original ADMM greatly as well as algorithm robustness on penalty parameters.It is obvious from the energy change curve and the convergence curve that the fast ADMM algorithm produced ripples and the restart fast ADMM algorithm decreased monotonously.However,ADMM algorithm and its restart fast algorithm lack sufficient theoretical support for the design of nonlinear,non-smooth,and non-convex variational models with high-order derivatives.So current theoretical research is limited to the optimization problem of objective function composed of two functions.Deterministic theoretical results focus on the optimization problems of smooth,strong convex and containing a linear constraint.For fast algorithm research of non-smooth and non-convex variational models of high-order derivatives in computer vision,the work of this paper is limited to tentative algorithm design and numerical verification.