A New Approach to Image Processing by Nonlinear Estimation

The inherent nonlinear aspect of many practical systems and observation models is explicitly suggestive of the importance and necessity of considering the nonlinear behavior of such a class of systems. The theoretical part of nonlinear systems and in particular nonlinear estimation and filtering has been developed through the decades of the sixties and beginning part of the seventies. The main obstacles in exact practical implementation of the theoretical results still remain unchallenged. But based on the possibility of realizing an adequate approximation criterion to feasibly utilize the scattered theory, this work has concentrated on least square approximation techniques as well as the Gauss-Hermite numerical integration method. A concise theoretical background of nonlinear filters is provided in the first chapter, briefly encompassing continuous estimation and elaborating more extensively on the discrete case. By applying Bayes\u27 law, the probability density function for the discrete case is derived at the end of the chapter; this conditional density function is the pillar of the remaining calculating efforts in the forthcoming chapters. The approximation methods discussed in Chapter 2 are either the ones deployed later on, or those that represent themselves as good candidates for the same tasks. The topics of interest include least square approximation techniques which are used to develop a two dimensional interpolation formula. This formula as well as the Gauss-Hermite quadrature formula are the ones of immediate concern. A few other quadrature formulas are also referred to as well. These formulas are similarly applicable to the approximation tasks undertaken by the Gauss-Hermite formula, where some modifications to the results are required. Chapter 3 includes mathematical models of the image processors; these models take into account the noise sources which are prominent in the process. Also, nonlinear aspects of image processing, and image restoration in particular, are discussed and implemented correspondingly. The remainder of the chapter consists of steps to obtain the probability density function of the image, based on the depicted nonlinear model. In Chapter 4, by the use of autoregressive models and their transformation to state space models, the conditional probability density function of the image is calculated. Then least square approximation and Gauss-Hermite quadrature techniques are applied. This approach makes it feasible to calculate the density function by numerical methods..