On the Compatibility of Granulometry with Wavelet Analysis

In this paper, regular closed Jordan measurable sets are proposed as models of digital images, and it is shown, that these are precisely the sets that are well approximated by their family of granulometric openings in measure theoretic sense. Moreover, compatibility of mathematical morphology and wavelet analysis is established in the following sense: two sets, that are indistinguishable by wavelets (i.e. their symmetric difference is a Lebesgue zero set) have undistinguishable granulometric openings, and a finite resolution wavelet approximation to a Jordan measurable set is sufficient to determine its granulometric openings up to a specified margin of error.