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<p>A unifying theory for many concepts and operations encountered in or related to morphological image and signal analysis is presented. The unification requires a set-theoretic methodology, where signals are modeled as sets, systems (signal transformations) are viewed as set mappings, and translational-invariant systems are uniquely characterized by special collections of input signals. This approach leads to a general representation theory, in which any translation-invariant, increasing, upper semicontinuous system can be presented exactly as a minimal nonlinear superposition of morphological erosions or dilations. The theory is used to analyze some special cases of image/signal analysis systems, such as morphological filters, median and order-statistic filters, linear filters, and shape recognition transforms. Although the developed theory is algebraic, its prototype operations are well suited for shape analysis; hence, the results also apply to systems that extract information about the geometrical structure of signals.</p>
set theory; picture processing; representation theory; morphological image; signal processing; signal transformations; set mappings; semicontinuous system; minimal nonlinear superposition; morphological filters; order-statistic filters; linear filters; shape recognition transforms; shape analysis; geometrical structure; pattern recognition; picture processing; set theory; signal processing

P. Maragos, "A Representation Theory for Morphological Image and Signal Processing," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 11, no. , pp. 586-599, 1989.
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