Multiscale nonlinear decomposition: The sieve decomposition theorem

Sieves decompose one dimensional bounded functions, e.g., f to a sequence of increasing scale granule functions, (d(m))(m=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 recomposition. consisting of adding up all the granule functions, maps {d} to f. Experiments show that a more general property exists such that {(d) over cap} maps to (f) over cap and back to {<(d) over cap>}, where the granule functions {(d) over cap}, are obtained from {(d) over cap} by applying any operator alpha consisting of changing the amplitudes of some granules, including zero, without changing their signs. in other words, the set of granule function vectors produced by the decomposition is closed under the operation alpha. An analytical proof of this property is presented. This property means that filters are useful in the context of feature recognition and, in addition, opens the way for an analysis of the noise resistance of sieves.