Abstract
Mathematical morphology is based on set-theoretic notions such as inclusion and intersection. In this paper it is shown how such notions can be extended to complete lattices. This results in so-called dominance and incidence structures. Such structures can be regarded as the geometric counterpart of adjunctions, which play an important role in morphology. This paper discusses the theoretical foundations, some applications from mathematical morphology, and an application to the Buffon-Sylvester problem in stochastic geometry.
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.
References
R. V. Ambartzumian. Combinatorial Integral Geometry, with applications to mathematical stereology. John Wiley and Sons, Chichester, 1982.
A. J. Baddeley. Appendix A. In R. V. Ambartzumian,Combinatorial Integral Geometry, pages 194–214. John Wiley and Sons, New York and Chichester, 1982.
A. J. Baddeley and H. J. A. M. Heijmans. Incidence and lattice calculus with applications to stochastic geometry and image analysis. Research report BS-R9213, CWI, 1992.
L. Dorst and R. van den Boomgaard. Morphological signal processing and the slope transform. Signal Processing, 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 – part I: Dilations and erosions. Computer Vision, Graphics and Image Processing,50:245–295, 1990.
P. Maragos. Morphological systems: slope transforms and max-min difference and differential equations. Signal Processing, to appear.
L. A. Santaló. Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and Its Applications, vol. 1. Addison-Wesley, 1976.
J. Serra. Image Analysis and Mathematical Morphology. Academic Press, London, 1982.
J. J. Sylvester. On a funicular solution of buffon’s `problem of the needle’ in its most general form. Acta Mathematica,14:185–205, 1890.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Heijmans, H.J.A.M., Baddeley, A.J. (1994). Dominance and Incidence Structures with Applications to Stochastic Geometry and Mathematical Morphology. In: Serra, J., Soille, P. (eds) Mathematical Morphology and Its Applications to Image Processing. Computational Imaging and Vision, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1040-2_22
Download citation
DOI: https://doi.org/10.1007/978-94-011-1040-2_22
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4453-0
Online ISBN: 978-94-011-1040-2
eBook Packages: Springer Book Archive