| From 1960 s to now,image processing has been presented vigorous development,and there are almost no technical fields unrelated to digital image processing.During the processing of recording,storing and transferring,images may be degraded,such as blurring,noise corruption,quality degradation,and losing contrast and details.These degradations are usually due to the limitations of the imaging device and condition,such as out-of-focus of the camera,noise inside the charge-coupled device,corruption in the transmission channel and so on.Image restoration is trying to reconstruct or restore the degraded image by using some prior knowledge of the degradation phenomenon.Therefore,the restoration technology is to establish degradation model and use the opposite process to restore the original image.It is an important preprocessing step for mid-level and high-level image processing.Image restoration is generally ill-posed problems,and regularization methods are effective to overcome the ill-posed problems.The most classical regularization method is to employ Tikhonov regularization.This doctoral dissertation deeply focuses on investigating the iterative methods for solving standard Tikhonov regularization and the choice of the regularization matrix for general Tikhonov regularization model,and applies the obtained solution to the solution of image restoration problem.Our main works are as follows:1.Modified special HSS(MSHSS)method for image restoration is investigated.Based on the the augmented system derived by employing the Tikhonov regularization method,and by utilizing parameter accelerating technique and constructing a new splitting of the coefficient matrix we propose the modified special HSS(MSHSS)method,which further accelerates the convergence rate of the SHSS one.Next investigate analytically the convergence behavior of the MSHSS method and give the optimal parameters minimizing the spectral radius of the iteration matrix.Meanwhile,its SOR acceleration is also discussed.The presented numerical examples illustrate the efficiency of the MSHSS method and the SOR acceleration for the MSHSS method.2.Based on RHSS method,a special RHSS(SRHSS)iteration method for image restoration problem is given.Due to the special structures of the Hermitian part and the skew-Hermitian part of the coefficient matrix of the augmented system derived by employing the Tikhonov regularization method,the RHSS iterative method for solving non-Hermitian positive definite system is extended to solve the image restoration problem,and a special RHSS(SRHSS)iterative method is proposed.Theoretically analyze the convergence property of the SRHSS method and its optimal parameters.Numerical experiments are given to show that the SRHSS iteration method significantly outperforms the newly developed ones in iteration counts,computing times and image recovering quality.3.A class of nonstationary upper and lower triangular(MRULT)iteration methods for image restoration problem is studied.On the basis of the upper and lower triangular(ULT)split iteration method,the two types of the MRULT iteration methods are defined by introducing parameters into the two-step iteration sequence of the ULT iteration method.The parameters in the MRULT methods are determined by the minimize residual technique so as to minimize the residuals of each step of the MRULT iteration methods.The MRULT methods improve the convergence rates of the ULT ones.The corresponding convergence theory is established and the relationship of the optimal parameters is given.Numerical experiments are reported to show that the new methods have better numerical performance compared with some similar ones.4.Accelerated GNHSS(AGNHSS)iterative method and its preconditioner for weighted Toeplitz regularized least-squares problems from image restoration are studied.By introducing a parameter matrix P in the second step of the GNHSS iteration method the AGNHSS method is developed,accelerating the convergence rate of the GNHSS one.We obtain the convergence conditions and the quasi-optimal parameters.Besides,the convergence property of the AGNHSS method for solving the linear image restoration problem and its optimal parameters are analyzed.Meanwhile,the spectral properties of the preconditioned matrix are investigated.Numerical results show that the proposed methods are more effective than the similar ones.5.The selection of the regularization matrix in general Tikhonov regularization model is studied.And we propose a special modified regularization(SMTR)matrix including more useful information about images.The algorithm for determining the parameters in SMTR method is presented,which involves a few of the largest singular values of blur matrix and associated singular vectors.The computational complexity of the algorithm is small.Besides,the corresponding preconditioner is designed to accelerate the convergence of the CGLS method for solving Tikhonov regularization least-squares system.Numerical experiments are carried out to validate that the SMTR method and PSMTR preconditioner can get an appropriate solution with improved accuracy compared the related methods. |
