Fuzzy Methods in Nonlinear Signal Processing: Part II—Applications - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 69

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Fuzzy Methods in Nonlinear Signal
Processing: Part II—Applications

Kenneth E. Barner, Yao Nie, and Yuzhong Shen

CONTENTS
3.1 Introduction ......................................................... 69

3.2 Affine Filter Applications ........................................... 70

3.3 Fuzzy Ordering and Fuzzy Median Filter Applications ............ 91

3.4 Conclusions .......................................................... 115

References .................................................................. 116

3.1 Introduction

This chapter builds on the theory of fuzzy methods in nonlinear signal pro-
cessing developed in Part I of this two-chapter set. The theory developed in
Part I shows that for the heavy-tailed Laplacian distribution, the maximum
likelihood estimation criteria yield the well-known median and weighted
median filters. Moreover, this development shows the importance of spatial
order and rank order in estimator, or filter, development. This importance
leads to the formalization of spatial and rank (SR) order theory and the more
general fuzzy SR order theory. The fuzzy generalization is particularly im-
portant in that it enables sample spread, or affinity, to be included in the SR
information.
Through use of the more general fuzzy ordering relation, well-established
filtering paradigms can be generalized to include sample affinity, or spread, in
the filtering process. The generalizations presented in the theory development
focus on two broad classes of fuzzy filters: affine filters, which are general-
izations of weighted sum filters that incorporate sample spread, and fuzzy
weighted median filters, which generalize the standard class of weighted
median filters by incorporating fuzzy samples. In both cases, there are well-
established optimization procedures and generalizations to multivariate data.