3 - Spatial-Rank Order Selection Filters

Abstract

This chapter explores the joint use of spatial and rank (SR) ordering information in a selection filter framework and applies the methods developed to several common image processing problems. The chapter begins with a theoretical discussion of maximum likelihood (ML) estimation that motivates the use of rank order in the processing of signals with heavy-tailed distributions. The ML development leads to several rank order selection filters, which are then extended to the general class of selection filters. The chapter then introduces a general framework for relating the spatial and rank orderings of samples. It also presents a discussion of broad class of SR selection filters, beginning with permutation filters, which utilize the full SR ordering information. The factorial growth in the number of SR ordering limits the size of permutation filter window that can be utilized in practice. To efficiently utilize partial SR information in the filtering process, M permutation and colored permutation filters are developed. These methods utilize the rank order information of specific spatial samples and allow ordering equivalences to be established in order to efficiently utilize the most important SR information in a given application. This chapter extends the SR selection filtering framework to include augmented observation sets that may include functions of the observed samples.