A Representation Theory for Morphological Image and Signal Processing

This paper presents a unifying theory for many concepts and operations encountered in or related to morphological image and signal analysis. This unification requires a set-theoretic methodology, where signals are modeled as sets, systems (signal transformations) are viewed as set mappings, and translation-invariant systems are uniquely characterized by special collections of input signals. This approach leads to a general representation theory, in which any translation-in- variant, increasing, upper semicontinuous system can be represented exactly as a minimal nonlinear superposition of morphological erosions or dilations. In this representation, many similarities and a few differences are observed between systems processing binary or multi- level signals, and continuous-domain or discrete-domain signals. The theory is used to analyze some special cases of image/signal analysis systems, such as morphological filters, median and order-statistic fil- ters, linear filters, and shape recognition transforms. Although the de- veloped theory is algebraic, its prototype operations are well suitable for shape analysis; hence, the results of this study also apply to systems that extract information about the geometrical structure of signals. Zndex Terms-Imagelsignal processing, mathematical morphology, nonlinearllinear filtering, shape analysis, systems representation.