Abstract
This paper investigates algebraic and continuity properties of increasing set operators underlying dynamic systems. We recall algebraic properties of increasing operators on complete lattices and some topologies used for the study of continuity properties of lattice operators. We apply these notions to several operators induced by differential equation or differential inclusion. We focus especially on the operators associating with any closed subset its reachable set, its exit tube, its viability kernel or its invariance kernel. At the end, we show that morphological operators used in image processing are particular cases of operators induced by constant differential inclusion.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
J.-P. Aubin. Viability theory. Birkhauser, Systems and Control: Foundations and Applications, 1991.
J.-P. Aubin and A. Cellina. Differential Inclusions (Set-valued maps and viability theory). Springer-Verlag, 1984.
J.-P. Aubin and H. Frankowska. Set-Valued Analysis. Birkhauser, 1990. Systems and Control: Foundations and Applications.
G. Birkhoff. Lattice Theory. Am. Math. Soc. Colloq. Publ., vol. 25, 1983.
H. Frankowska. Control of nonlinear systems and differential inclusions. Birkhäuser, to appear.
H.J.A.M. Heijmans. Morphological Image Operators. Academic Press, Boston, 1994.
H.J.A.M. Heijmans and C. Ronse. The Algebraic Basis of Mathematical Morphology: I. Dilatations and Erosions. Computer Vision, Graphics, and Image Processing, 50:245–295, 1990.
H.J.A.M. Heijmans and J. Serra. Convergence, continuity and iteration in mathematical morphology. Journal of Visual Communication and Image Representation, 3(No. 1):84–102, March 1992.
G. Matheron. Random Sets and Integral Geometry. John Wiley and Sons, New York, 1975.
G. Matheron. Dilations on topological spaces. In J. Serra, editor, Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances. Academic Press, London, 1988.
G. Matheron. Filters and lattices. In J. Serra, editor, Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances. Academic Press, London, 1988.
J. Mattioli, L. Doyen and L. Najman Lattice operators underlying dynamic systems Set-Valued Analysis. To appear.
M. Quincampoix. Enveloppe d’invariance pour les inclusions différentielles lipschitziennes: applications aux problèmes de cibles. C.R. Acad. Sci. Paris, Tome 314:343–347, 1992.
C. Ronse and H.J.A.M. Heijmans. The Algebraic Basis of Mathematical Morphology: II. Openings and Closings. Computer Vision, Graphics, and Image Processing, 54(l):74–97, july 1991.
J. Serra. Image Analysis and Mathematical Morphology. Academic Press, London, 1982.
J. Serra, editor. Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances. Academic Press, London, 1988.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Kluwer Academic Publishers
About this chapter
Cite this chapter
Mattioli, J., Doyen, L., Najman, L. (1996). Lattice Operators Underlying Dynamic Systems. In: Maragos, P., Schafer, R.W., Butt, M.A. (eds) Mathematical Morphology and its Applications to Image and Signal Processing. Computational Imaging and Vision, vol 5. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0469-2_4
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0469-2_4
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-8063-4
Online ISBN: 978-1-4613-0469-2
eBook Packages: Springer Book Archive