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Quasi-continuous images must be sampled prior to digital processing. The steps needed to render images and operators discrete are set out in full.

The theory of complete lattices is described in the language of set theory. The use of Hasse diagrams to represent finite lattices is described. Boolean lattices are defined and de Morgan's Laws are introduced. Regular closed sets are studied as an example of such lattices. Boolean functions and their representations are defined.

The algebraic properties of the morphological operators are important but they are not alone in this. Topological properties can be equally important and are explored in depth in this chapter. The basic elements of topology are summarized, followed by those of metric spaces, notably the Hausdorff metric. Compact sets are then defined and this leads to the hit-or-miss topology and the myope topology. Semi-continuity is explained and a proposition concerning basis representation is advanced.

In order to extract geometric information from images, suitable operators must be constructed. After a discussion of convexity and geodesic distance, the important notion of metric dilation is introduced, followed by that of distance transforms. Sections are then devoted to geodesic and conditional operators, granulometries, connectivity and skeletons. A final section considers discrete metric spaces.

It is shown that set operators can be extended to function operators. This makes it possible to transfer all the theory of binary morphology to grey-level images. For this, a function must be represented by a family of sets. The set operator is then applied to this family after which a transformed function is reconstructed for the family of transformed sets.

The mathematical morphology of grey-level images has many special features, which are the subject of this chapter. A general procedure for proceeding from binary to grey-level morphology was presented in an earlier chapter. Here, each aspect of the morphological analysis of grey-level images is examined in depth.

Opening and closings are examined in depth. The notion of structural opening, which constitutes a basis for the class of all openings (and likewise for closings) is introduced. A second important class of openings is constituted by the adjunctional openings. Annular openings, inf-overfilters and incidence structures are also explained.

A consistent algebraic framework is introduced with which the various translations encountered in the discussion of erosions and dilations can be unified. Adjunction is re-examined and the T-invariant operators are defined and discussed in detail. Self-dual and Boolean lattices are defined. Translation-invariant and polar morphology are studied. Additive and multiplicative structuring functions are considered in connection with grey-scale functions. T is examined in the non-Abelian situation, where the projection and lift operators are employed. Translation-rotation morphology is described.

The construction of morphological filters by iteration of an arbitrary increasing operator is described. The role of continuity is emphasized. It is shown that the finite window operators are suitable for this. All translation-invariant operators that use finite structuring elements belong to this family.

The basic concepts of mathematical morphology are introduced by considering two examples: The search for connected components in an image and the morphological analogue of the convolution operator. A brief history of the subject is included here.

The notion of translation-invariant operators is described and explained with reference to binary images. Erosion and dilation, opening and closing are discussed at length. Grey-scale morphology is introduced and the use of the umbra is presented. T-operators and their representation are described.