Nonlinear image restoration techniques

In the image formation process, a realistic imaging system introduces both linear and nonlinear degradations, as well as noise. The linear transformation may depict the properties of the optical medium, or the characteristics of the imaging device. The nonlinear degradation is introduced either through the imaging system in the recording process, or through a transformation of the original model. The existence of the nonlinearity complicates the restoration process in various ways. First the solution of the problem has to assume nonlinear form requiring expensive and complicated computations. Furthermore, the evaluation of the statistics of the original image becomes difficult. Another process that complicates the image restoration problem is the noise that can assume short, medium or long-tailed form, depending on the problem under consideration.; The nonlinear-image restoration problem, as well as the parameters that affect the restoration process, are considered in this thesis. More specifically, the contribution of this research project is threefold. First, the nonlinear restoration problem is rigorously studied. In the case of short and/or medium-tailed noise corruption, two non-adaptive approaches are introduced. The minimum mean square error (MMSE) approach results in a direct algorithm, which is implemented in the Fourier transform (FT) domain. Alternatively, the least-squares error (LSE) approach leads to an iterative algorithm which is efficiently implemented in a mixed, image-space and FT domain. In contrast to the MMSE, the nonlinear LSE algorithm requires only deterministic information regarding the imaging system. Furthermore, modifications that drastically improve the convergence rate are proposed.; The restoration algorithms result in a certain degradation of the detailed structure. Moreover, the presence of long-tailed noise drastically degrades their performance. The second major concern of this study is the development of a rigorous approach for the incorporation of adaptivity in direct algorithms. The formulation, and the joint optimization of a combined criterion is proposed. This approach introduces a local trade-off between the restored-image resolution and the noise preservation. Moreover, it introduces flexibility with respect to the type of noise that is locally present.; A third major contribution of this thesis is the application of robust M-estimators in iterative restoration algorithms. M-estimators have been solely utilized in recursive algorithms. Nevertheless, their nature often requires iterative implementation. Through the incorporation of the M-estimation concept in the nonlinear LSE criterion, the evaluation of the M-estimator is embedded in the overall iterative scheme, which is designed for the minimization of the corresponding problem. Important concepts of the algorithms developed are thoroughly studied throughout the context of this thesis.