Abstract
In mathematical morphology, Matheron has described the complete lattice structure of several classes of increasing (isotone) operators, such as filters. These results can be interpreted in terms of fixpoints of certain types of transformations of the lattice of increasing operators. Moreover, Heijmans and Serra have given conditions for the construction of such operators by convergence of an iteration of increasing operators. We recall some known lattice-theoretical fixpoint ideas originating from Tarski's 1955 paper. A slight elaboration on the underlying methodology leads to the aforementioned results and some others as a consequence of the general theory.
Similar content being viewed by others
References
J. Serra,Image Analysis and Mathematical Morphology, Academic Press: London, 1982.
H.J.A.M. Heijmans and C. Ronse, “The algebraic basis of mathematical morphology I: Dilations and erosions,”Comput. Vis. Graph., Image Process., vol. 50, pp. 245–295, 1990.
C. Ronse and H.J.A.M. Heijmans, “The algebraic basis of mathematical morphology II: Openings and closings,”Comput. Vis., Graph., Image Process.: Image Understanding, vol. 54, pp. 74–97, 1991.
J. Serra, ed.,Image Analysis and Mathematical Morphology, vol. 2: Theoretical Advances, Academic Press: London, 1988.
J.E. Goodman, “On the largest convex polygon contained in a non-convexn-gon, or how to peel a potato,”Geom. Dedicata, vol. 11, pp. 99–106, 1986.
S.J. Wilson, “Convergence of iterated median rule,”Comput. Vis., Graph., Image Process., vol. 47, pp. 105–110, 1989.
H.J.A.M. Heijmans, “Iteration of morphological transformations,”CWI Quarterly, vol. 2, no. 1, pp. 19–36, 1989.
H.J.A.M. Heijmans, “Morphological filtering and iteration,”Proc. Soc. Photo-Opt. Instrum. Eng., vol. 1360, pp. 166–175, 1990.
H.J.A.M. Heijmans and J. Serra, “Convergence, continuity and iteration in mathematical morphology,”J. Vis. Commun. Image Rep., vol. 3, pp. 84–102, 1992.
A. Tarski, “A lattice-theoretical fixpoint theorem and its applications,”Pacific J. Math., vol. 5, pp. 285–309, 1955.
J. van Leeuwen, ed.,Handbook of Theoretical Computer Science, vol. B: Formal Models and Semantics, Elsevier: Amsterdam, 1990.
A.C. Davis, “A characterization of complete lattices,”Pacific J. Math., vol. 5, pp. 311–319, 1955.
G. Birkhoff,Lattice Theory, American Mathematical Society Colloquium Publications, vol. 25, 3rd ed., American Mathematical Society: Providence, RI, 1984.
J. Serra, “Itérations et convergence,” Centre de Morphologie Mathématique, Fontainebleau, France, Report N-5/89/MM, 1989.
R. Lalement,Logique, Réduction, Résolution, Etudes et Recherches en Informatique, Masson: Paris, 1990.
M.E. Munroe,Measure and Integration, Addison-Wesley: Reading, MA, 1971.
W. Rudin,Real and Complex Analysis, McGraw-Hill: New York, 1970.
F. Maisonneuve, “Ordinaux transfinis et sur-(ou sous-) potentes,” Centre de Morphologie Mathématique, Fontainebleau, France, Report N780, 1982.
E.M. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press: Princeton, NJ, 1971.
G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott,A Compendium of Continuous Lattices, Springer-Verlag: Berlin, 1980.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ronse, C. Lattice-theoretical fixpoint theorems in morphological image filtering. J Math Imaging Vis 4, 19–41 (1994). https://doi.org/10.1007/BF01250002
Issue Date:
DOI: https://doi.org/10.1007/BF01250002