Properties of multiscale morphological filters, namely, the morphology decomposition theorem

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Abstract
Sieves decompose 1D bounded functions, e.g., f to a set of increasing scale granule functions {dm, m equals 1 ...R}, that represent the information in a manner that is analogous to the pyramid of wavelets obtained by linear decomposition. Sieves based on sequences of increasing scale open-closings with flat structuring elements (M and N filters) map f to {d} and the inverse process maps {d} to f. Experiments show that a more general inverse exists such that {d} maps to f and back to {d}, where the granule functions {d}, are a subset of {d} in which granules may have changed amplitudes, that may include zero but not a change of sign. An analytical proof of this inverse is presented. This key property could prove important for feature recognition and opens the way for an analysis of the noise resistance of these sieves. The resulting theorems neither apply to parallel open-closing filters nor to median based sieves, although root median sieves do `nearly' invert and offer better statistical properties.

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