Asymptotic granulometric mixing theorem: Morphological estimation of sizing parameters and mixture proportions

Abstract If a random set (binary image) is composed of randomly sized, disjoint translates arising as homothetics of a finite number of compact primitives and a granulometry is generated by a convex, compact set, then the granulometric moments of the random set can be expressed in terms of the granulometric moments of the generating primitives, their areas, and the radii of the homethetics. Moreover, if there is only a single generating primitive, then the granulometric moments of the random set are asymptotically normal relative to an increasing number of grains (translates) and methods exist to express their asymptotic moments. The present paper extends the asymptotic theory to multiple-primitive images. The extension is significant in two regards. First, the single-primitive asymptotic theory was based on the classical moment theory of Cramer; the new theory uses an approach based on analytic decomposition on the granulometric moments. Second, not only does the new theory demonstrate asymptotic normality, but it also permits method-of-moment estimation of the parameters of the distributions governing grain sizes and mixture proportions among grain types. In the present setting, granulometric method-of-moment estimation is based on asymptotic representation of expectations of granulometric moments in terms of grain-sizing model parameters, mixture proportions, and geometric constants. Method-of-moment estimation for sizing parameters and mixture proportions is achieved by replacing granulometric-moment expectations with moment estimations from image realizations and then solving the resulting system of equations for the model parameters. The theory is applied to both normal and gamma sizing distributions.

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