Fundamenta Informaticae 41 (2000) 1-31 1 !OS Press Fundamenta Morphologicae Mathematicae * John Goutsias t Center for Imaging Science and Department of Electrical and Computer· Engineering The Johns Hopkins University Baltimore, MD 21218, U. S. A.
[email protected] Henk J .A.M. Heijmans Centre for Mathematics and Computer Science P.O. Box 94079 1090 GB Amsterdam, The Netherlands
[email protected] Abstract. Mathematical morphology is a geometric approach in image processing and anal ysis with a strong mathematical f!avor. Originally, it was developed as a powerful tool for shape analysis in binary and, later, grey-scale images. But it was soon recognized that the underlying ideas could be extended naturally to a much wider class of mathematical ob jects, namely complete lattices. This paper presents, in a bird's eye view, the foundations of mathematical morphology, or more precisely, the theory of morphological operators on complete lattices. Keywords: adjunction, alternating sequential filter, Boolean function, closing, complete lattice, connectivity, connected operator, dilation, erosion, flat operator, fuzzy set, grain operator, granulometry, grey-scale image, idempotence, mathematical morphology, morpho logical filter, morphological operator, multivalued image, opening, thresholding. 1. Introduction In 1975, a seminal book by Georges Matheron, entitled Random Sets and Integral Geometry, appeared in the literature [20]. This book laid down the foundations of a novel technique for • This work was supported in part by NATO Collaborative Research Grant CRG.971503. The work of the first author was also supported by the Office of Naval Research, Mathematical, Computer, and Information Sciences Division, under ONR Grant N00014-90-1345, and by the National Science Foundation, under NSF Award #9729576.