Heterogeneous morphological granulometries

The most basic class of binary granulometries is composed of unions of openings by structuring elements that are homogeneously scaled by a single parameter. These univariate granulometries have previously been extended to multivariate granulometries in which each structuring element is scaled by an individual parameter. This paper introduces the more general class of "lters in which each structuring element is scaled by a function of its sizing parameter, the result being multivariate heterogeneous granulometries. Owing to computational considerations, of particular importance are the univariate heterogeneous granulometries, for which scaling is by functions of a single variable. The basic morphological properties of heterogeneous granulometries are given, analytic and geometric relationships between multivariate and univariate heterogeneous pattern spectra are explored, and application to texture classi"cation is discussed. The homogeneous granulometric mixing theory, both the representation of granulometric moments and the asymptotic theory concerning the distributions of granulometric moments, is extended to heterogeneous scaling. ( 2000 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.