Nonlinear filter design: methodologies and challenges

Linear filtering techniques have serious limitations in dealing with signals that have been created or processed by a system exhibiting some degree of nonlinearity, or, in general, situations where the relevance of information cannot be specified in the frequency domain. In image processing, many of these characteristics are often present, and it is no wonder that image processing is the field where nonlinear filtering techniques have first shown clear superiority over linear filters. Since nonlinear filters are all of those filters that are not linear, there is a large variety of different filters in use, and no common theory can exist. This makes filter design challenging, and optimization is meaningful only after restricting the class. It can be done in several conceptually different ways and, in this paper, we consider these techniques and the optimization methods that go together with the particular restriction. We review polynomial and rational filter classes and the optimization of stack filters under structural constraints and statistical constraints.