Abstract
In the context of mathematical morphology based on structuring elements to define erosion and dilation, this paper generalizes the notion of a structuring element to a new setting called structuring neighborhood systems. While a structuring element is often defined as a subset of the space, a structuring neighborhood is a subset of the subsets of the space. This yields an extended definition of erosion; dilation can be obtained as well by a duality principle. With respect to the classical framework, this extension is sound in many ways. It is also strictly more expressive, for any structuring element can be represented as a structuring neighborhood but the converse is not true. A direct application of this framework is to generalize modal morpho-logic to a topological setting.
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
Notes
- 1.
While b(x) is often considered as a neighborhood of x according to a given topology on X, here our notion of neighborhood system refers to a set of subsets of X.
References
Aiguier, M., Atif, J., Bloch, I., Hudelot, C.: Belief revision, minimal change and relaxation: a general framework based on satisfaction systems, and applications to description logics. Artif. Intell. 256, 160–180 (2018)
Aiguier, M., Atif, J., Bloch, I., Pino Pérez, R.: Explanatory relations in arbitrary logics based on satisfaction systems, cutting and retraction. Int. J. Approx. Reason. 102, 1–20 (2018)
Aiguier, M., Bloch, I.: Dual logic concepts based on mathematical morphology in stratified institutions: applications to spatial reasoning. CoRR abs/1710.05661 (2017). http://arxiv.org/abs/1710.05661
Bloch, I., Heijmans, H., Ronse, C.: Mathematical morphology (Chap. 13). In: Aiello, M., Pratt-Hartman, I., van Benthem, J. (eds.) Handbook of Spatial Logics, pp. 857–947. Springer, Dordrecht (2007). https://doi.org/10.1007/978-1-4020-5587-4_14
Bloch, I., Lang, J.: Towards mathematical morpho-logics. In: 8th International Conference on Information Processing and Management of Uncertainty in Knowledge based Systems IPMU 2000, Madrid, Spain, vol. III, pp. 1405–1412 (2000)
Bloch, I., Lang, J., Pino Pérez, R., Uzcátegui, C.: Morphologic for knowledge dynamics: revision, fusion, abduction. Technical report arXiv:1802.05142, arXiv cs.AI, February 2018
Bloch, I., Pérez, R.P., Uzcategui, C.: A unified treatment of knowledge dynamics. In: International Conference on the Principles of Knowledge Representation and Reasoning, KR2004, Canada, pp. 329–337 (2004)
Bloch, I.: Modal logics based on mathematical morphology for qualitative spatial reasoning. J. Appl. Non-Classical Log. 12(3–4), 399–423 (2002)
Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press, Cambridge (1980)
Gorogiannis, N., Hunter, A.: Merging first-order knowledge using dilation operators. In: Hartmann, S., Kern-Isberner, G. (eds.) FoIKS 2008. LNCS, vol. 4932, pp. 132–150. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-77684-0_11
Heijmans, H.J.A.M., Ronse, C.: The algebraic basis of mathematical morphology - part I: dilations and erosions. Comput. Vis. Graph. Image Process. 50, 245–295 (1990)
Najman, L., Talbot, H. (eds.): Mathematical Morphology: From Theory to Applications. ISTE-Wiley, Hoboken (2010)
Pacuit, E.: Nijssen: Neighborhood Semantics for Modal Logic. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-319-67149-9
Pattinson, D.: Coalgebraic modal logic: soundness, completeness and decidability of local consequence. Theor. Comput. Sci. 309(1), 177–193 (2003)
Ronse, C.: Lattice-theoretical fixpoint theorems in morphological image filtering. J. Math. Imaging Vis. 4(1), 19–41 (1994)
Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Goy, A., Aiguier, M., Bloch, I. (2019). From Structuring Elements to Structuring Neighborhood Systems. In: Burgeth, B., Kleefeld, A., Naegel, B., Passat, N., Perret, B. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2019. Lecture Notes in Computer Science(), vol 11564. Springer, Cham. https://doi.org/10.1007/978-3-030-20867-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-20867-7_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-20866-0
Online ISBN: 978-3-030-20867-7
eBook Packages: Computer ScienceComputer Science (R0)