Abstract
The basic operations of mathematical morphology, dilation and erosion, were introduced by Matheron and Serra. They were initially defined as Minkowski addition and subtraction on subsets of the Euclidean space, using translations, unions and intersections. Following Sternberg, they were generalized to the set of grey-level images with the help of umbras. Recently Serra and Matheron have generalized morphological operations to complete lattices, that is, sets in which the operations of supremum and infimum are well-defined. This generalization has proven useful by extending the scope of mathematical morphology to other structures. In this paper we show that it is also necessary for a mathematically coherent application of morphological operators to grey-level images. Indeed • In the continoous case, the definition of dilations on umbras is not exactly the same as for ordinary Euclidean sets; here the union must be replaced by a supremum operation similar to the one in the complete lattice of closed sets. Moreover, dilations and erosions can be defined directly with lattice-theoretic methods, without recourse to umbras. • In the digital case, when the set of grey-levels is bounded, the problem of grey-level overflow can be dealt with correctly only by taking into account the complete lattice structure of the set of grey-level images. Otherwise the properties of morphological operators are lost.
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