| In this thesis, we study two aspects in image processing. Part I is about image denoising and deblurring under impulse noise, and Part II is about framelet-based methods for image reconstruction.; In Part I of the thesis, we study the problems of image denoising and de-blurring under impulse noise. We consider two-phase methods for solving these problems. In the first phase, efficient detectors are applied to detect the outliers. In the second phase, variational methods utilizing the outputs of the first phase are performed. For denoising, we prove that the functionals to be minimized in the second phase have many good properties such as maximum principle, Lipschitz continuity and etc. Based on the results, we propose conjugate gradient methods and quasi-Newton methods to minimize the functional efficiently. For deblurring, we propose a two-phase method combining the median-type filters and a variational method with Mumford-Shah regularization term. The experiments show that the two-phase methods give much better results than both the median-type filters and full variational methods.; Part II of the thesis focuses on framelet-based methods for image reconstruction. In particular, we consider framelet-based methods for chopped and nodded image reconstruction and image inpainting. By interpreting both the problems as recovery of missing data, framelet, a generalization of wavelet, is applied to solve the problems. We incorporate sophisticated thresholding schemes into the algorithm, hence the regularities of the restored images can be guaranteed. By the theory of convex analysis, we prove the convergence of the framelet-based methods. We find that the limits of the framelet-based methods satisfy some minimization properties, hence connections with variational methods are established. |
