Rank Minimization And Variable Transformation Based Methods For Image Restoration

During the formation and transformation process,images are inevitably degraded by noise and blur caused by intrinsic properties of the electronic imaging system or unavoidable effects from external environments.By utilizing the information of contaminated images,the aim of image restoration is to construct efficient methods to obtain recovered images which are as close as possible to the original ones.Image restoration problem arises in many digital imaging applications,it is a basic problem in image processing.Therefore,the study of image restoration is of greatly theoretical and practical significance.In this paper,the restoration for images contaminated by both additive Gaussian noise and blur,as well as multiplicative noise removal problems are considered.Image restoration methods based on matrix rank minimization and variable transformation are constructed.In recent decades,many efficient methods for image restoration have been proposed.These methods include filtering method,regularization method,sparse representation and non-local methods etc.Restoration methods based on rank minimization are important ones which have been extensively developed in recent years.Rank minimization problem is a constrained optimization problem which aims at getting a low-rank matrix under certain constraints.Because of the difficulty in solving the rank minimization problem,its solution is always approximated by the solution of the nuclear norm minimization problem.A matrix's nuclear norm is the sum of all singular values of the matrix and it is used as an approximation of the matrix's rank.In this paper,we propose a new method for rank minimization problem by utilizing matrix's rank as the regularization term in the energy function.We also prove that the proposed model can be solved by hardthresholding operation on the observed matrix's singular values.By utilizing image selfsimilarity and image block matching scheme,we apply the proposed rank minimization method to remove white Gaussian additive noise in images.Gamma multiplicative noise is also removed in logarithm domain.Numerical results illustrate that the proposed rank minimization method can remove noises in images efficiently.Meanwhile,edges and details in images are also preserved.An new image restoration model is also proposed based on weighted nuclear norm minimization for restoring images contaminated by both blur and Gaussian additive noise.Weighted nuclear norm is the weighted sum of all singular values of a matrix which is more efficient than nuclear norm minimization for image noise removal.For each image block in the observed image,its block matching matrix can be formed by its similar blocks obtained through matching process in the image.The block matching matrices are used to establish the energy function of the proposed model.The regularization term is the sum of the block matching matrices' weighted nuclear norm.We further design an alternating iterative algorithm to solve the proposed model.Convergence of the algorithm is also analyzed and proved.Ill-conditioned problem does not appear in the proposed algorithm for the proposed image restoration model while such problem often arises in most of the nuclear norm based minimization problems.The computational efficiency is greatly improved.Numerical results show that the image recovered quality by the proposed method is good.Non-Gaussian noise removal problem by variable transformation is also considered.By transforming non-Gaussian noises into noises with Gaussian or near Gaussian distribution,the existing efficient Gaussian additive noise removal methods can be applied to remove these transformed noises.The inverse transformation are then used to get the final recovered images.Box-Cox transformation is an important data transformation which is often used in mathematical statistics to transform non-Gaussian distribution data into Gaussian distribution data.The selection of the parameter in Box-Cox transformation is very important since it determines if the transformed data follows Gaussian distribution.In this paper,we devise a method for multiplicative noise removal based on Box-Cox transformation.Firstly,we design a maximum likelihood method to determine the optimal transform parameter in Box-Cox transformation for multiplicative denoising problem.Then we convert the Gamma multiplicative noise removal problem into a Gaussian additive noise removal problem by using Box-Cox transformation and block matching three dimensional filtering method(BM3D)is applied to remove the transformed noise.BM3 D is an effective Gaussian additive noise removal method.The final multiplicative denoised image is obtained from the additive denoised image by using an unbiased inverse Box-Cox transformation.Both theoretical analysis and experimental results demonstrate that the variable transformation based methods can remove non-Gaussian noises in images efficiently.Box-Cox transformation based multiplicative noise removal method is also very efficient.