Abstract
Although mathematical morphology and formal concept analysis are two lattice-based data analysis theories, they are still developed in two disconnected research communities. The aim of this paper is to contribute to fill this gap, beyond the classical relationship between the Galois connections defined by the derivation operators and the adjunctions underlying the algebraic mathematical morphology framework. In particular we define mathematical morphology operators over concept lattices, based on distances, valuations, or neighborhood relations in concept lattices. Their properties are also discussed. These operators provide new tools for reasoning over concept lattices.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Atif, J., Hudelot, C., Bloch, I.: Abduction in description logics using formal concept analysis and mathematical morphology: Application to image interpretation. In: 8th International Conference on Concept Lattices and Their Applications (CLA 2011), Nancy, Paris, pp. 405–408 (October 2011)
Baader, F.: Computing a minimal representation of the subsumption lattice of all conjunctions of concepts defined in a terminology. In: 1st International KRUSE Symposium on Knowledge Retrieval, Use and Storage for Efficiency, pp. 168–178 (1995)
Birkhoff, G.: Lattice theory, 3rd edn., vol. 25. American Mathematical Society (1979)
Bloch, I.: On Links between Mathematical Morphology and Rough Sets. Pattern Recognition 33(9), 1487–1496 (2000)
Bloch, I., Heijmans, H., Ronse, C.: Mathematical Morphology. In: Aiello, M., Pratt-Hartman, I., van Benthem, J. (eds.) Handbook of Spatial Logics, ch. 13, pp. 857–947. Springer (2007)
Bloch, I., Maître, H.: Fuzzy Mathematical Morphologies: A Comparative Study. Pattern Recognition 28(9), 1341–1387 (1995)
Bloch, I., Pino-Pérez, R., Uzcategui, C.: A Unified Treatment of Knowledge Dynamics. In: International Conference on the Principles of Knowledge Representation and Reasoning, KR 2004, Canada, pp. 329–337 (2004)
Ferré, S., Ridoux, O.: A logical generalization of formal concept analysis. In: Ganter, B., Mineau, G.W. (eds.) ICCS 2000. LNCS (LNAI), vol. 1867, pp. 371–384. Springer, Heidelberg (2000)
Ganter, B., Wille, R., Franzke, C.: Formal concept analysis: Mathematical foundations. Springer-Verlag New York, Inc. (1997)
Heijmans, H.J.A.M.: Morphological Image Operators. Academic Press, Boston (1994)
Heijmans, H.J.A.M., Ronse, C.: The Algebraic Basis of Mathematical Morphology – Part I: Dilations and Erosions. Computer Vision, Graphics and Image Processing 50, 245–295 (1990)
Leclerc, B.: Lattice valuations, medians and majorities. Discrete Mathematics 111(1), 345–356 (1993)
Monjardet, B.: Metrics on partially ordered sets–a survey. Discrete Mathematics 35(1), 173–184 (1981)
Orum, C., Joslyn, C.: Valuations and metrics on partially ordered sets, arXiv preprint arXiv:0903.2679 (2009)
Ronse, C., Heijmans, H.J.A.M.: The Algebraic Basis of Mathematical Morphology – Part II: Openings and Closings. Computer Vision, Graphics and Image Processing 54, 74–97 (1991)
Ronse, C.: Adjunctions on the lattices of partitions and of partial partitions. Applicable Algebra in Engineering, Communication and Computing 21(5), 343–396 (2010)
Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, New-York (1982)
Serra, J. (ed.): Image Analysis and Mathematical Morphology, Part II: Theoretical Advances. Academic Press, London (1988)
Simovici, D.: Betweenness, metrics and entropies in lattices. In: 38th IEEE International Symposium on Multiple Valued Logic, ISMVL, pp. 26–31 (2008)
Stern, M.: Semimodular lattices: Theory and applications. Cambridge University Press (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Atif, J., Bloch, I., Distel, F., Hudelot, C. (2013). Mathematical Morphology Operators over Concept Lattices. In: Cellier, P., Distel, F., Ganter, B. (eds) Formal Concept Analysis. ICFCA 2013. Lecture Notes in Computer Science(), vol 7880. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38317-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-38317-5_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38316-8
Online ISBN: 978-3-642-38317-5
eBook Packages: Computer ScienceComputer Science (R0)