Numerical Linear Algebra And Sparse Optimization For Image Restoration Problems

During the formation, recording, processing and transmitting, imperfect imagingsystems, recording equipments and transmission medium will result in the degradationof images. For examples, effects of atmospheric turbulence, noise caused by sensors andchanges of environment condition, motion blurs caused by the relative motion betweenthe camera and the entire scene. Image restoration is the reconstruction of the original im-age from a degraded observation with a priori knowledge of images to improve the qualityof images. With rising requirements of image resolution and image quality, and real-timerequirements in practice, the computational burdens increase significantly. This providesgrand challenges and also a promising future for the development of image restorationtechniques. The contents of this thesis are organized into regularization methods studyparts and models study parts, among which, the following first four parts are attributedto the first kind of study parts while the latter successive two parts belong to the secondkind. They are illustrated in detail as follows:Based on the matrix splitting of a Toeplitz-plus-Hankel matrix, an efficient splittingiteration method for Toeplitz-plus-Hankel systems has been developed. The convergenceproperties and the computational cost of the proposed splitting iteration method are dis-cussed. Moreover, we have studied the problem of choosing optimal or quasi-optimalparameters of the proposed splitting iteration method. The performance of this splittingiteration method is illustrated by numerical experiments.As alternatives to classic boundary conditions, new mean boundary conditions havebeen recently introduced for image restoration problems. We have proposed an efficien-t scheme of computing optimal Kronecker product approximations of blurring matriceswith new mean boundary conditions. Based on Kronecker product approximations, trun-cated singular value decomposition (TSVD) type regularization method is developed forimage restoration problems with new mean boundary conditions.Landweber method is one of classical iterative regularization methods for solvinglinear discrete ill-posed problems. However, the slow convergence limits its availabilityfor widespread applications. We have presented the vector extrapolation based Landwebermethod which exhibits a fast and stable convergence behavior. Moreover, the restarted version of the vector extrapolation based Landweber method is considered for practicalconsiderations. Numerical results are given to illustrate the performance of the vectorextrapolation based Landweber method.Motivated by the excellent performance of the vector extrapolation enhanced TSVDmethod on small-to-medium sized problems, we have proposed an efficient hybrid methodfor large-scale linear discrete ill-posed problems, which applies the vector extrapolationenhanced TSVD method to small-to-medium sized problems generated by Krylov sub-space methods. Numerical experiments are reported to illustrate the performance of theproposed hybrid method for large-scale linear discrete ill-posed problems.In practice, one is often faced with imprecise knowledge of the point spread function-s. We studied the functional model by minimizing two variables: the restored image andthe noise on the data-fitting term, the magnitude of the noise in the point spread function,and the total variation regularization term. By making use of the structure of the func-tional, an efficient alternating minimization scheme is developed to solve the proposedmodel. The existence of minimizers of the proposed model and the convergence of theproposed scheme are established. Numerical examples are also presented to demonstratethe effectiveness of the proposed model and the efficiency of the numerical scheme.Unmixing and deblurring are key issues of hyperspectral imaging. We have proposedan unmixing and deblurring model of hyperspactral data and developed an alternatingdirection method to solve the proposed model. Extensive numerical results are reportedto demonstrate the effectiveness of the proposed model and the efficiency of the proposedscheme.