Euclidean gray-scale granulometries: Representation and umbra inducement
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A basic filter theory within mathematical morphology involves the binary granulometric theory of G. Matheron. This theory results in a size-distribution analysis that has proved beneficial in the characterization of random-grain image models. Of central importance are the Euclidean granulometries, which admit a particularly elegant representation. The present paper extends the Matheron binary Euclidean granulometric theory to the gray scale. In doing so, it draws on the lattice formulations of openings and abstract granulometric processes. Given these, the concept of gray-scale Euclidean granulometry is defined in terms of a signal scalar multiplication that is compatible with the umbra transform. Once the definition has been extended, the fundamental Matheron representation in terms of unit generators is generalized to the gray scale. Attention is then paid to the inducements of gray-scale-to-binary and binary-to-gray-scale operators that result naturally from the umbra methodology. It is demonstrated that this inducement results in the inducement of Euclidean granulometries in both directions. 1992 Kluwer Academic Publishers.