| Discrete ill-posed problems occur frequently in the physical sciences. In this thesis, we present multilevel methods for a particular kind of discrete ill-posed problems, deblurring problems. Multigrid methods are well known as extremely efficient solvers for certain large-scale systems of equations, particularly those that result from the discretizations of partial differential equations and integral equations of the second kind. These have been extensively studied in recent years. However, for ill-posed problems, the classical multigrid approach is not immediately applicable. This work presents new wavelet-based multilevel methods for signal and image restoration problems as well as for blind deconvolution problems. In these methods, we use the orthogonal wavelet transform to define restriction and prolongation operators within a multigrid-type iteration. Specifically, the choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edge-preserving Tikhonov regularization is applied. Moreover, when solving a blind deconvolution problem by means of a Structured Total Least Norm formulation, we have again at each level a Structured Total Least Norm problem to solve. We present results that indicate the promise of these approaches for restoration of signals and images with edges as well as restoration of the blurring operator in the case of blind deconvolution problems. |
