A First-order Dual Algorithm For Total Variation Image Restoration

In the process of collection and transmission,the image will inevitably produce degradation phenomenon.The image restoration technique, restoring the the maximum fidelity of the original image from degraded image.The algorithm based on Total Variation(TV) image restoration to maintain superiority in terms of image edge and denoising, this paper focuses on the performance of the total variation model of alternating direction algorithm, the primal dual hybrid gradient algorithm and forward-backward splitting algorithm for the total variation model. On the basis of these algorithm, we proposed some improved algorithms.In the field of image processing based on bounded variation, the total variation image restoration can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. Convex optimization problem can be a first-order or second-order method for solving methods, although second-order methods can be used to solve many convex programs, it may suffer from unbearably high computation cost when handling large scale problems. Furthermore,second-order methods have higher iteration complexity than first-order methods. Besides, instead of decomposing or calculating the inverse of the large scale matrix, first-order methods only need to calculate the function value and gradient information.Therefore, this paper focuses on solving the total variation image restoration problems, first-order methods are the better choice.First, for the established total variation image restoration model, we propose an linearized alternating direction algorithm to solving problems in image reconstruction with total variation regularization.The algorithm effectively combines an alternating direction technique to minimize the augmented Lagrangian function at each iteration and a Barzilai-Borwein(BB) stepsize rule. When the resulting subproblems do not have closed-form solutions, we propose to linearize these subproblems such that closed-form solutions of these linearized subproblems can be easily derived.Numerical results show that the linearized alternating direction algorithm can restore the degraded image with high quality, good convergence and stability.Second, we proposed a fast algorithm based on the primal dual hybrid gradient method to reduce the computational complexity.This algorithm adopts special projection operator, separation components of dual vector. In order to achieve different designs for different components of the weight, the gradient descent step size parameter understood as weighted coefficients, the weighting matrix is substituted with a weighting coefficient. Experiments on Total Variation(TV) image restoration indicate that optimization methods have the lower iteration complexity and higher computing efficiency than original primal dual hybrid gradient method.Forward Backward Splitting is a simple and effective method for solving the Total Variation(TV) image denoising model, it is often possible to re-formulate the problem using the duality theory into an equivalent dual problem. The efficiency of forward backward splitting is very sensitive to the choice of the stepsize. For this reason, much work has been devoted to studying adaptive stepsize strategies. In such a way, we propose an adaptive forward backward splitting iterative algorithms. At each iteration of the algorithms,forward backward splitting algorithms decouple the contributions of the otal Variation(TV) image denoising model in a gradient descent step determined by fidelity term and in a backward implicit step induced by regularization term.In order to accelerate the speed of convergence, we discuss Barzilai-Borwein stepsize methods, and how they can be adapted to forward backward splitting. Experimental results show that the adaptive orward backward splitting iterative algorithms automatically tune stepsize parameters in real time(as the algorithm runs) to achieve faster convergence than the original forward backward splitting algorithm.