| Image restoration is a basic problem of image processing. Its objective is to restore an image, blurred by a smoothing operator or contaminated by additive noises, from its deterioration. Image restoration has been widely applied to medical imaging and astronomy.;Regularization is a primary method used in mage restoration. The variation functional of a regularized method contains two terms: a fidelity term and a regularization term. Therefore, characteristics of a regularized method are determined by selections of the fidelity and the regularizer. In past decades, a great amount of studies of regularized methods have focused on selection of the fidelity as an Lp norm ( p = 1, 2). In this thesis, we concentrate on selection of other possible fidelities with a total variation regularizer.;We characterize a general class of fidelities, which includes existing fidelities in the literatures as special examples. We prove the well-posedness of the resulting regularized minimization problem, composed of a fidelity from the general class of fidelities and the total variation regularization. We also show convergence of minimizers of the regularized minimization problem as errors in data and the blurring operator tend to zero.;We specify a new subclass of fidelities: content-driven fidelities. Content-driven fidelities provide spatial varying measurements of the recovered image to the observed image, which can accurately adapt to local image contents. They interpolate strengths of a variety of existing fidelities under different image contents. Four gradient-based algorithms, including the steepest descent algorithm (SD), heavy ball algorithm (HB), steepest descent algorithm with two point step size (SD-TPSS), and conjugate gradient algorithm (CG), are applied to solve the content-driven minimization composed of a content-driven fidelity and the total variation regularization. We also show convergence of the steepest gradient algorithm and the conjugate gradient algorithm based on the content-driven estimation.;We use numerical studies to assess properties of the L p-norm estimations, composed of Lp-norm fidelities and the total variation regularizer, for 1 ≤ p ≤ 2 under different image contents. We observe the relationship of the performance of the Lp-norm estimation, the selection of p, the noise levels, and the presence of outliers. Finally, practical effectiveness of the content-driven approach is shown on synthetically generated data. The superiorities of the content-driven estimation over the Lp-norm estimation (1 ≤ p ≤ 2) and other existing estimations are numerically demonstrated. |
