Contents I. Introduction
It has been recognized that multiresolution signal decomposition schemes provide convenient and effective ways to process information [1]. Most of the modern multiresolution decomposition schemes are based on the theories of pyramid and wavelet, using the convolution and time-frequency-domain transformations. However, the linear filtering approaches to multiresolution signal decomposition have not been theoretically justified. In particular, the operators used for generating various levels of signal components in a pyramid must crucially depend on an application. Therefore, in recent years, a number of researchers have proposed nonlinear multiresolution signal decomposition schemes based on morphological operators. However, until Goutsias and Heijmans presented a set of fundamental theories named morphological pyramid (MP) and morphological wavelet (MW), which are derived from traditional wavelet and pyramid theories, there was not a unified standpoint and framework for nonlinear pyramids, filter banks and wavelets, including MPs and wavelets construction [2] [3]. MP and MW inherit the multidimension and multilevel analysis of wavelet and pyramid, whilst they do not require the time-frequency-domain analysis. Furthermore, MP and MW extend the original wavelet and pyramid from the linear domain, which is based on the convolution and time-frequency-domain transformation, to the nonlinear domain. The theories presented by Goutsias and Heijmans can be regarded as a framework for construction of the MW and pyramid. Based on this framework, some schemes have been developed for specific applications of image processing [4]–[7]. However the details of the operators proposed in this framework have not been investigated comprehensively. On the other hand, the study of this aspect has not been attempted for signal processing.
