Fuzzy Methods in Nonlinear Signal Processing: Part I—Theory - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 38

correlations between neighboring samples and (2) rank order can be used to
isolate outliers and ensure robust behavior. Although the exploitation of SR
ordering information in nonlinear filtering algorithms has yielded good re-
sults, traditional ordering information is based on a crisp relationship. Such
crisp relations yield no information on important quantities such as sample
spread or diversity.
The simple relaxation of the ordering relation from a crisp (binary) operator
to a more general affinity (real-valued) operator leads to the concept of fuzzy
SR orderings. Thus, fuzzy SR orderings not only relate spatial and rank order-
ings, but also contain information on sample spread (affinity). Powerful fuzzy
nonlinear filtering algorithms can be realized by embedding fuzzy SR order-
ing information into the filter structure. Such filters can be simply realized as
(1) generalizations of existing nonlinear filters that employ fuzzy, rather than
crisp, ordering relations; (2) generalizations of linear filters that embed fuzzy
SR ordering information into the traditional weighted sum filter structure; or
(3) new filter structures specifically designed to exploit fuzzy SR information.
Fuzzy SR methods in signal processing is a broad topic covered here in
two chapters. This first chapter theoretically motivates the use of fuzzy SR
ordering information, details the theory of fuzzy ordering and fuzzy order
statistics, and develops several classes of fuzzy nonlinear filters based on
these concepts. Specifically, the classes of affine filters and fuzzy weighted
median filters are developed. In the next chapter these methods are applied
to a wide range of signal processing and communications problems.
The topics covered in this chapter are organized as follows: Section 2.2
begins with a theoretical discussion of ML estimation and formally develops
the concept of SR ordering. The crisp ordering relation is relaxed in Section 2.3,
which develops the concepts of sample affinity and the resulting fuzzy SR
ordering. Fuzzy filter generalizations are developed in Section 2.4, where we
focus on the fuzzy weighted median and affine filter classes. In the case of
affine filters, the discussion is limited to the two important median affine and
center affine filter subclasses. Extensions to multivariate data are covered
in Section 2.5 and conclusions are drawn in Section 2.6. The next chapter
presents the results of applying the filters developed here to several important
signal processing and communications problems, including robust frequency
selective filtering, Synthetic Aperture Radar image filtering, time-frequency
domain filtering, multiresolution signal representations, surface smoothing,
image smoothing, zooming, and deblocking, and multiuser detection.

2.2 Maximum Likelihood Estimation and Spatial-Rank Ordering

2.2.1 ML Estimation

To motivate the development of theoretically sound signal processing meth-
ods, consider first the modeling of observation samples. In all but trivial cases,