Abstract
In many cases, the mere distinction between convex and nonconvex sets is too coarse. From the simple notion of a metric it is possible to generalize the very notion of Euclidean convexity and to go into a nonconvex domain. After a brief discussion on the basic properties of metric convexity it is indicated how its application in mathematical morphology can give rise to a number of mathematically interesting results and computationally efficient algorithms.
A part of this research was carried out during the first author’s visit to CWI This visit was supported by the Netherlands Organisation for Scientific Research NWO.
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
L.M. Blumenthal, Theory and Applications of Distance Geometry, Clarendon Press, 1953.
H. Busemann, The Geometry of Geodesics, Academic Press, New York, 1955.
P.K. Ghosh, “An algebra of polygons through the notion of negative shapes”, CVGIP: Image Understanding, Vol.54, No.1, 1991, 119–144.
P.K. Ghosh, “A unified computational framework for Minkowski Operations”, Computers and Graphics, Vol.17, No.4, 1993, 357–378.
P.K. Ghosh, “The Indecomposability Problem in Binary Morphology”, to be published in the Journal of Mathematical Imaging and Vision.
H. J.A.M. Heijmans, Morphological Image Operators, Academic Press, 1994.
H. J.A.M. Heijmans, P.K. Ghosh, “Metric convexity with applications to mathematical morphology”, under preparation.
J. Mattioli, Problèmes Inverses et Relations Différentielles en Morphologie Mathématique, PhD thesis, Université Paris Dauphine, Paris, 1993.
W. Rinow, Die Innere Geometrie der Metrischen Raume, Springer-Verlag, Berlin, 1961.
M. Schmitt, and J. Mattioli, “Strong and weak convex hulls in non-Euclidean metric: theory and application”, Pattern Recognition Letters, 15, 1994, 943–947.
J. Serra, Image Analysis and Mathematical Morphology, Academic Press, 1982.
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
Ghosh, P.K., Heijmans, H.J.A.M. (1996). Metric Convexity in the Context of Mathematical Morphology. 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_2
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0469-2_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-8063-4
Online ISBN: 978-1-4613-0469-2
eBook Packages: Springer Book Archive