Mathematical morphology is a theory of image transformations and image functionals which is based on set-theoretical, geometrical, and topological concepts. The methodology is particularly useful for the analysis of the geometrical struct~re in an i'.11age. The main goal of this paper is to give an impression of the underlying philosophy and the mathematical theones which are relevant to this field. The following topics are discussed: introduction to mathematical morphology; generalization to

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... mplete lattices; morphological filters and their construction by iteration; geometrical aspects of morphology (e.g., convexity, distance, geodesic operators, granulometries, metric dilations, distance transform, cost functions); and extension of binary operators to grey-scale images. In the final section we point out a few other subjects which are not discussed in this paper but which may also be of interest. A general account on mathematical morophology can be found in the two books by Serra [40] , [41] and in a monograph by Matheron [28]; the latter contains a comprehensive discussion on random sets and integral geometry. Actually, it is this probabilistic branch which has made morphology into such a powerful methodology, and it is somewhat unfortunate that this aspect has been given so little attention in the recent literature. Furthermore, the interested reader may refer to a monograph by Heijmans [ 17] dealing with various mathematical aspects of morphology. Some other elementary references are the books by Dougherty and Giardina [6], [7] and the tutorial paper by Haralick, Sternberg, and Zhuang [10] . Besides this introduction this paper comprises six sections, dealing with various aspects. Section 2 acquaints the reader with the morphological approach in image processing and discusses some basic morphological operators for binary images which are invariant under translations. In §3 we discuss an extension to the framework of complete lattices. Such an abstract theory also allows the construction of operators which are invariant under transformation groups other than translations. Section 4 deals with morphological filters; these are operators which preserve the partial ordering structure and which are idempotent. We explain how to construct filters by iteration of operators which are not idempotent but do have certain continuity properties. Geometrical aspects of morphology are discussed in §5. Despite the patchy contents of that section we hope it gives the reader an intuition for the kind of problems which emerge. Then, in §6 we discuss some extensions of the binary theory to grey-scale images, and finally, in §7, we mention some problems which have not found a place elsewhere in this paper. What is mathematical morphology? 2.1. Hit-or-miss operator. A convenient way to model binary (=black and white) images, both continuous and discrete, is by means of sets. Unless stated otherwise we assume that E =Rd or 'llf By P(E) we denote the power set of E. The key principle underlying mathematical morphology is to gain topological or geometrical information about a binary ima.g~ X s; E by probing it with another small set A, called a structuring element, at every pos1t10n h E E. By "probing" we mean testing whether the set Ah hits X (i.e., Ahn X i= 0), misses X (i.e., Ah n X = 0), or lies entirely inside X (i.e., Ah s; X), Here Ah denotes the translate of A along the vector h: Ah ={a+ hla E A}. The hit-or-miss operator is a mapping on the space of binary images 'P(E) which is based on this intuitive idea. Let A, B c E be two structuring elements such that An B = 0, and define -(2.1) X@ (A, B) = {h E EIAh s; X and Bh s;; X'l H~re ~c denotes the complement of X, or, in image processing terminology, the background ?f the 1~ag~ X. See Fig. 1 for an example. It is obvious that A and B must have an empty mtersect1~n m or?er to obtain nontrivial results. If not, the resulting set will be empty. The h1.t-o~-m1ss operator is an easy example of a set operator (i.e., a mapping i/I : °P( £) -+ P(E)) which JS translation invariant, that is, (2.2) i/l(X") = [i/l(X)]h for X E 'P(E) and h E £.