We study theoretical properties of two inexact Hermitian/skew-Hermitian splitting (IHSS) iteration methods for the large sparse non-Hermitian positive definite system of linear equations. In the inner iteration processes, we employ the conjugate gradient (CG) method to solve the linear systems associated with the Hermitian part, and the Lanczos or conjugate gradient for normal equations (CGNE) method to solve the linear systems associated with the skew-Hermitian part, respectively, resulting in IHSS(CG, Lanczos) and IHSS(CG, CGNE) iteration methods, correspondingly. Theoretical analyses show that both IHSS(CG, Lanczos) and IHSS(CG, CGNE) converge unconditionally to the exact solution of the non-Hermitian positive definite linear system. Moreover, their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the Hermitian part, but are less dependent on the spectrum of the skew-Hermitian part, and are independent of the eigenvectors of the matrices involved. Optimal choices of the inner iteration steps in the IHSS(CG, Lanczos) and IHSS(CG, CGNE) iterations are discussed in detail by * Corresponding author.

Image denoising plays a important role in the areas of image processing. A real recorded image may be distorted by many expected or unexpected random factors, of which random noise is a unavoidable one.Multiplicative noise is naturally dependent on the image data, the recorded image g is the multiplication of original image u and noise n: g = un.(1)Here u, g and n are n 2 -by-1 vector corresponding to n-by-n image. whereis the data fitting term, u T V is the total variation (TV) regularization term [1], λ is the regularization parameter which measures the trade off between a good fit and a regularized solution.The main aim of this paper is to propose and study a convex objective function for multiplicative noise removal in images. we consider an auxiliary variable z = log u in the multiplicative noise removal model.The proposed unconstrained TV denoising problem is given bywhere α 1 and α 2 are positive regularization parameters. The main advantage of using the new data fitting in the new minimization method. We can interpret the total variation minimization scheme to denoise the multiplicative noise removed image z. The main advantage of the proposed method is that an exact TV norm is used in the noise removal process. Therefore the new method has the ability to preserve edges very well in the denoised image.An alternating minimization algorithm was proposed to solve (3). Starting from an initial guess w (0) , this method computes a sequence of iteratesWe remark that T (·) = S(R(·)) is non-expansive and asymptotically regular. Since the objective function J is coercive, the set of minimizers of J is non-empty. Therefore, the set of fixed points of T is non-empty. According to the Opial theorem [2], the sequence z converges to a fixed point of J , i.e., a minimizer of J .We present numerical results to demonstrate the performance of our proposed algorithm. The results are compared with those obtained by "AA" method proposed by Aubert and Aujol [1]. Relative error of the 1

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In this paper, we study the restoration of images corrupted by Gaussian plus salt-and-pepper noise, and propose a l 1-l 0 minimization approach where the l 1 term is from impulse noise and the l 0 term is from the assumption that the clean image patches admit a sparse representation over certain unknown dictionary. The main algorithm contains three phases. First, we identify the outlier candidates which are likely to be corrupted by salt-and-pepper impulse noise. Second, we recover the image via dictionary learning on the free-outlier pixels. Finally, an alternating minimization algorithm is employed to solve the proposed minimization energy function, leading to an further restoration based on the recovered image in the second step. Comparative experiments are carried out to show the leading performance of the proposed method.

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