Gray-scale granulometries compatible with spatial scalings

Abstract In the field of mathematical morphology, granulometries form the major tool for the computation of size distributions for binary images. A granulometry can be defined as a one-parameter family of openings depending on a real parameter λ > 0 such that the opening becomes more and more active as λ increases. The granulometry is called an Euclidean granulometry if it is translation invariant and compatible with scalings. An important result due to Matheron states that one can build simple Euclidean granulometries by taking openings with convex compact structuring elements. In this paper we describe a general extension of binary Euclidean granulometries to gray-scale images using the notion of a spatial scaling (as opposed to umbral scaling). The main result of this paper is that one can build gray-scale Euclidean granulometries with one structuring function if and only if this function has a convex compact domain and is constant there (flat function).

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