Analysis of threshold decomposition and continuity for median-type filters and its application to nonlinear filter design

Abstract

In this thesis, the concept of threshold decomposition is revisited and some useful properties of threshold decomposable operators are developed in the sequel. These properties can be used for illuminating some natures of threshold decomposable (TD) filters and for finding out the relations of TD filters to other existing filters. With the help of these properties and ordering output tables listing all possible rank orderings of input samples and corresponding outputs, it is derived that a rank based nonlinear operator is continuous if and only if it is threshold decomposable. A general representation of the TD filter is also derived. This representation in which the TD filter can be expressed as a weighted sum of minimum (or maximum) subfiltered outputs, leads to the class of K- th order TD filters. The proposed representation provides a tradeoff between computational complexity and filtering performance. This kind of flexibility cannot be offered when the other conventional representations are used. A practically efficient subclass of threshold decomposable filters, called the FIR-L hybrid (FLH) filter, is introduced. The FLH filter has both characteristics of FIR and L filters by simply combining both the structures in parallel. The structure of an FLH filter is much simpler than those of other unified filters. The relationship between FLH and C filters is investigated and it is proven that the threshold decomposable (or equivalently continuous) subclass of C filters is identical with the class of FLH filters. Some undesirable phenomena of C filters which are not continuous are investigated. In addition, threshold decomposability and continuity of FIR-WOS hybrid (FWH) filters are studied and it turned out that the FWH filter is continuous but not threshold decomposable. The efficiency of FLH filters over other existing filters is proven through simulations in which their filtering performances and computational complexities are compared to the existing ones.