Abstract
We begin by defining the setup and the framework of connective segmentation. Then we start from a theorem based on connective criteria, established for the power set of an arbitrary set. As the power set is an example of a complete lattice, we formulate and prove an analogue of the theorem for general complete lattices.
Secondly, we consider partial partitions and partial connections. We recall the definitions, and quote a result that gives a characterization of (partial) connections. As a continuation of the work in the first part, we generalize this characterization to complete lattices as well.
Finally we link these two approaches by means of a commutative diagram, in two manners.
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
Braga-Neto, U., Goutsias, J.: Connectivity on Complete Lattices: New Results. Comput. Vis. Image. Underst. 85, 22–53 (2001)
Ore, O.: Galois connexions. Trans. Amer. Math. Soc. 55, 493–513 (1944)
Ronse, C.: Partial Partitions, Partial Connections and Connective Segmentation. J. Math. Imaging Vis. 32, 97–125 (2008)
Ronse, C.: Idempotent Block Splitting on Partial Partitions, I: Isotone Operators. Order. SpringerLink (2010)
Ronse, C., Serra, J.: Geodesy and Connectivity in Lattices. Fundamenta Informaticae 46, 349–395 (2001)
Ronse, C., Serra, J.: Algebraic foundations of morphology. In: Najman, L., Talbot, H. (eds.) Mathematical Morphology: From Theory to Applications. ISTE/J. Wiley & Sons, London (2010)
Serra, J.: Viscous Lattices. J. Math. Imaging Vis. 22, 269–282 (2005)
Serra, J.: A Lattice Approach to Image Segmentation. J. Math. Imaging Vis. 24, 83–130 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Alsaody, S., Serra, J. (2011). Connective Segmentation Generalized to Arbitrary Complete Lattices. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds) Mathematical Morphology and Its Applications to Image and Signal Processing. ISMM 2011. Lecture Notes in Computer Science, vol 6671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21569-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-21569-8_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21568-1
Online ISBN: 978-3-642-21569-8
eBook Packages: Computer ScienceComputer Science (R0)