MULTISCALE NONLINEAR DECOMPOSITION - THE SIEVE DECOMPOSITION THEOREM

Sieves decompose one dimensional bounded functions, e.g., f to a seque nce of increasing scale granule functions, (d(m))(m=1)(R) that represe nt the information in a manner that is analogous to the pyramid of wav elets obtained by linear decomposition. Sieves based on sequences of i ncreasing scale open-closings with flat structuring elements (M and N filters) map f to {d} and the recomposition. consisting of adding up a ll 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 ca p}, are obtained from {(d) over cap} by applying any operator alpha co nsisting of changing the amplitudes of some granules, including zero, without changing their signs. in other words, the set of granule funct ion vectors produced by the decomposition is closed under the operatio n alpha. An analytical proof of this property is presented. This prope rty means that filters are useful in the context of feature recognitio n and, in addition, opens the way for an analysis of the noise resista nce of sieves.