Dual Splitting Method For TV Regularization

In this paper,we study the image denoising issue where the images are denoised by the impulsive noise.In general,the traditional Tikhonov regularization method uses quadratic data fidelity which can remove the Gaussian noise in the images well.However,the traditional Tikhonov regularization is not suitable for the situation where the noise is impulse noise.In order to remove the impulse noise efficiently,we investigate the TV(total variation)regularization coupled with l1 data fidelity.Compared with traditional quadratic smooth regularization,the crucial drawback is its poor stability.In order to improve the stability,we add a smooth l2 regularization term on the TV regularization,which can improve the stability of the functional.The proposed new optimization method can not only remove the impulsive noise and preserve the edge of the image but also improve the stability.We proves the existence,stability,convergence as well as convergence rate of the regularized solution of the TV regularization,which shows that this method is a regularization method.Since the l1 fidelity and the TV regularization term are non-differentiable,the traditional gradient algorithm is no longer suitable for this case.Hence,we adapt a dual method to transform new functional to a constraint smooth functional such that the projected gradient algorithl can be utilized to compute the minimizer.Then we utilize the linear constraint equation and the augmented Lagrange functional to simplify the computation and we prove convergence of the proposed algorithm.Finally,the numerical examples present the application of the proposed method on compressive sensing and image restoration,the results testify the efficiency of the proposed method.In addition,we compare the splitting method with ADM(altermating direction method)and TNIP(Newton break point method),it is shown that the convergence rate of the splitting algorithm is better than that of ADM and TNIR The results show that the performance of the splitting method is obviously better than that of the other two methods.