Abstract
Today, the theoretical framework of mathematical morphology is phrased in terms of complete lattices and operators defined on them. The characterization of a particular class of operators, such as erosions or openings, depends almost entirely upon the choice of the underlying partial ordering. This is not so strange if one realizes that the partial ordering formalizes the notions of foreground and background of an image. The duality principle for partially ordered sets, which says that the opposite of a partial ordering is also a partial ordering, gives rise to the fact that all morphological operators occur in pairs, e.g., dilation and erosion, opening and closing, etc. This phenomenon often prohibits the construction of tools that treat foreground and background of signals in exactly the same way. In this paper we discuss an alternative framework for morphological image processing that gives rise to image operators which are intrinsically self-dual. As one might expect, this alternative framework is entirely based upon the definition of a new self-dual partial ordering.
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References
G. Birkhoff, “Lattice Theory,” 3rd ed., Vol. 25 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 1967.
P. Dubreil, and Dubreil-Jacotin, M.L. Lecons d'Alebre Moderne, Dunod, Paris, 1961.
L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford, 1963.
H.J.A.M. Heijmans, Morphological Image Operators, Academic Press, Boston, 1994.
H.J.A.M. Heijmans, “Self-dual morphological operators and filters,” Journal of Mathematical Imaging and Vision Vol. 6, No. 1, pp. 15-36, 1996.
R. Keshet (Kresch), “Mathematical morphology on complete semilattices and its applications to image processing,” Fundamenta Informaticae, Vol. 41, No. 1-2, pp. 33-56, 2000.
R. Kresch, “Extensions of morphological operations to complete semilattices and its applications to image and video processing,” In Mathematical Morphology and its Application to Image and Signal Processing, H.J.A.M. Heijmans and J.B.T.M. Roerdink, Eds. Kluwer Academic Publishers, Dordrecht, pp. 35-42, 1998.
A.J.H. Mehnert and P.T. Jackway, “Folding induced self-dual filters,” In Mathematical Morphology and its Application to Image and Signal Processing, J. Goutsias, L. Vincent, and D.S. Bloomberg, Eds. Kluwer Academic Publishers, Boston, pp. 99-108, 2000.
F. Meyer, “The levelings” In Mathematical Morphology and its Applications to Image and Signal Processing, H.J.A.M. Heijmans and J.B.T.M. Roerdink, Eds. Kluwer Academic Publishers, Dordrecht, pp. 199-207, 1998.
J. Serra, Image Analysis and Mathematical Morphology, Academic Press, London, 1982.
J. Serra, (Ed.) Image Analysis and Mathematical Morphology. II: Theoretical Advances. Academic Press, London, 1988.
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Heijmans, H.J., Keshet, R. Inf-Semilattice Approach to Self-Dual Morphology. Journal of Mathematical Imaging and Vision 17, 55–80 (2002). https://doi.org/10.1023/A:1020726725590
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DOI: https://doi.org/10.1023/A:1020726725590