Abstract
This paper develops an abstract theory for mathematical morphology on complete lattices. It differs from previous work in this direction in the sense that it does not merely assume the existence of a binary operation on the underlying lattice. Rather, the starting point is the recognition of the fact that, in general, objects are only known through information resulting from a given collection of measurements, called evaluations. Such an abstract approach leads in a natural way to the concept of convolution lattice, where ‘convolution’ has to be understood in the sense of an abstract Minkowski addition. The paper contains various examples. Two applications are treated in great detail, the morphological slope transform and random set theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Birkhoff, “Lattice theory,” American Mathematical Society Colloquium Publications, American Mathematical Society: Providence, RI, 3rd ed., Vol. 25, 1967.
L. Dorst and R. van den Boomgaard, “Morphological signal processing and the slope transform,” Signal Processing, Vol. 38, pp. 79–98, 1994.
S. Doss, “Sur la moyenne d'un élément aléatoire dans un espace distancié,” Bull. Sci. Math., Vol. 73, pp. 48–72, 1949.
H.J.A.M. Heijmans, Morphological Image Operators, Academic Press: Boston, 1994
H.J.A.M. Heijmans and P. Maragos, “Lattice calculus of the morphological slope transform,” Signal Processing, Vol. 59, pp. 17–42, 1997.
H.J.A.M. Heijmans and C. Ronse, “The algebraic basis of mathematical morphology—Part I: Dilations and erosions,” Computer Visions, Graphics and Image Processing, Vol. 50, pp. 245–295, 1990.
J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer-Verlag: Berlin, 1993.
D.G. Kendall, “Foundations of a theory of random sets,” in Stochastic Geometry, E.F. Harding and D.G. Kendall (Eds.), John Wiley & Sons: Chichester, 1974, pp. 322–376.
P. Maragos, “Morphological systems: Slope transforms and max-min difference and differential equations,” Signal Processing, Vol. 38, pp. 57–77, 1994.
P. Maragos, “Slope transforms: Theory and application to nonlinear signal processing,” IEEE Transactions on Signal Processing, Vol. 43, No.4, pp. 864–877, 1995.
G. Matheron, Random Sets and Integral Geometry, John Wiley & Sons: New York, 1975.
I.S. Molchanov, “Limit theorems for unions of random closed sets,” Lecture Notes in Mathematics, Springer-Verlag: Berlin, 1993, Vol. 1561.
R.T. Rockafellar, Convex Analysis, Princeton University Press: Princeton, 1972.
C. Ronse, “Why mathematical morphology needs complete lattices,” Signal Processing, Vol. 21, pp. 129–154, 1990.
C. Ronse and H.J.A.M. Heijmans, “The algebraic basis of mathematical morphology—Part II: Openings and closings,” Computer Vision, Graphics and Image Processing: Image Understanding, Vol. 54, pp. 74–97, 1991.
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press: Cambridge, 1993.
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.
D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields, Wiley: Chichester, 1994.
R. Vitale, “An alternate formulation of mean value for random geometric figures,” Journal of Microscopy, Vol. 151, pp. 197–204, 1988.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Heijmans, H.J., Molchanov, I.S. Morphology on Convolution Lattices with Applications to the Slope Transform and Random Set Theory. Journal of Mathematical Imaging and Vision 8, 199–214 (1998). https://doi.org/10.1023/A:1008226416181
Issue Date:
DOI: https://doi.org/10.1023/A:1008226416181