ABSTRACT
CONTENTS 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Maximum Likelihood Estimation and Spatial-Rank Ordering . . . . . . 38 2.3 Sample Affinity and Fuzzy Spatial-Rank Ordering . . . . . . . . . . . . . . . . . 46 2.4 Fuzzy Filter Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.5 Extensions to Multivariate Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Nonlinear signal processing methods, unlike linear methods, lack a unified and universal set of tools for analysis and design. Hundreds of nonlinear signal processing algorithms have been proposed. Most of the proposed methods, although well tailored for a given application, are not generally applicable. Although nonlinear signal processing is a dynamic, rapidly growing field, a large class of nonlinear signal processing algorithms can be studied with fundamentals that are well formulated. In this chapter, we approach the filtering problem from a maximum likelihood (ML) approach. It is shown that the ML optimization leads directly to the class of linear filters for signals with Gaussian statistics, and to the class of nonlinear weighted median filters for signals with double exponential, or Laplacian, distributions.
