Models And Algorithms For Image Restoration And Low Rank Tensor Approximation

With the rapid development of computer technology,digital image processing has wide applications in communication,medicine,aeronautics and astronautics,etc.Image restoration is an important field of digital image processing,which refers to recoverying an image from the degradation image.It is basic tasks in imaging and vision science.Moreover,with the increase of the dimension of data,tensor plays an important role in daily life,which is the high order array.Tensor completion is to recover a tensor with partial observed data,which has wide applications in machine learning,artificial intelligence,image processing,etc.The main focus of this thesis are two parts,i.e.,image restoration and tensor completion.First,for the problem of image restoration of observed images corrupted by blur and impulse noise,the widely-used L1TV model may deviate from both the data-acquisition model and the prior model,especially for high noise levels.In order to seek a solution of high recovery quality beyond the reach of the LI TV model,we propose an adaptive correction procedure for LI TV image deblurring under impulse noise.Then,a proximal alternating direction method of multipliers(ADMM)is presented to solve the corrected L1TV model and its convergence is also established under very mild conditions.It is verified by numerical experiments that our proposed approach outperforms the LI TV model in terms of signal-to-noise ratio(SNR)values and visual quality,especially for high noise levels:it can handle salt-and-pepper noise as high as 90%and random-valued noise as high as 70%.In addition,a comparison with a state-of-the-art method,the two-phase method,demonstrates the superiority of the proposed approach.Second,we propose and study a nonconvex data fitting term and a total variation regularization term for image restoration with impulse noise removal.The proposed model is different from existing image restoration models where the data fitting term is based on l1 or l2-norm,and the regularization term is based on total variation,l1-norm or some nonconvex functions.Theoretically,we analyze the properties of minimizers of the proposed objective function with the nonconvex data fitting term and the total variation regularization term.We show that minimizers can preserve piecewise constant regions and/or match with the data points perfectly.This property is particularly use-ful for impulse noise removal.The proposed image restoration model can be solved by the proximal linearized minimization algorithm,and the global convergence of the iter-ative algorithm can also be established according to Kurdyka-Lojasiewicz property.The performance of the proposed model is tested for image restoration with salt-and-pepper impulse noise or random-valued impulse noise.We demonstrate that the restored images by the proposed Nonconvex-TV model are better(in terms of PSNR and visual quality)than those by the other existing data fitting plus regularization models including l1 plus total variation(L1TV),and l1 plus nonconvex(L1Nonconvex)methods.Finally,we study the tensor completion problem on recovery of the multilinear data under limited sampling.A popular convex relaxation of this problem is to minimize the nuclear norm of the more square matrix produced by matricizing a tensor.However,it may fail to produce a high accurate solution under low sample ratio.In order to get a recovery with high accuracy,we propose an adaptive correction approach for tensor completion.First,a corrected model for matrix completion with bound constraint is pro-posed and its error bound is established.Then,we extend it to tensor completion with bound constraint and propose a corrected model for tensor completion.The adaptive correction approach consists of solving a series of corrected models with an initial esti-mator where the initial estimator used for the next step is computed from the value of the current solution.Moreover,the error bound of the corrected model for tensor completion is also established.A convergent 3-block ADMM is applied to solve the dual problem of the corrected model.Numerical experiments on both random data and real world data validate the efficiency of our proposed correction approach.