Nonlinear order statistics techniques for signal processing

In applications such as signal/image restoration and detection, linear techniques may be unable to deal with non-stationary or non-linearly distorted signals that are corrupted by non-additive, non-Gaussian noise. The class of non-linear order-statistics-based techniques that rely on rank-order information associated with observed discrete-time signals has been particularly successful at non-Gaussian impulsive noise suppression and in dealing with non-stationarities. This dissertation deals with two main aspects of such techniques.The first part of the work deals with performance characterization of order-statistics-based estimators when they form the basis of moving-window filters. Generally, performance of these filters is hard to evaluate due to their non-linear nature. We discuss an approximate approach to performance analysis which is based on a Taylor-series-like expansion of the statistical functionals that are used to represent these filters. We also show that the application of this approach to represent and analyze a class of adaptive order statistics filters leads to a framework for investigating new adaptive filters.The success of order-based techniques as robust processors in univariate signal processing leads one to seek their extensions for processing multivariate signals encountered in numerous applications such as color-image processing and band-pass communication systems. This task is made challenging by the fact that the concept of order associated with real-valued observations does not generalize naturally to higher dimensions. In the second part of this dissertation we describe some approaches that have been proposed to define a suitable notion of ordering in higher dimensions that result in different multivariate order statistics filters. Performance aspects of two of the definitions proposed for a multivariate median are discussed in some detail. One application of multivariate ordering techniques is in the area of robust multivariate regression. We use this application to investigate the possibility of exploiting multivariate ordering concepts for regression-based filtering.