Abstract
In many domains of information processing, both vagueness, or imprecision, and bipolarity, encompassing positive and negative parts of information, are core features of the information to be modeled and processed. This led to the development of the concept of bipolar fuzzy sets, and of associated models and tools. Here we propose to extend these tools by defining algebraic relations between bipolar fuzzy sets, including intersection, inclusion, adjacency and RCC relations widely used in mereotopology, based on bipolar connectives (in a logical sense) and on mathematical morphology operators.
Keywords
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- 1.
Mereology is concerned with part-whole relations, while mereotopology adds topology and studies topological relations where regions (not points) are the primitive objects, useful for qualitative spatial reasoning, see e.g. [1] and the references therein.
- 2.
All proofs are quite straightforward, and omitted due to lack of space.
- 3.
Note that [0, 1] can be replaced by any poset or complete lattice, in the framework of L-fuzzy sets, and the proposed approach applies in this more general case.
- 4.
i.e.: \(\forall (a_1,a_2,a'_1,a'_2) \in {\mathcal L}^4, a_1 \preceq a'_1 \text { and } a_2 \preceq a'_2 \Rightarrow C(a_1,a_2) \preceq C(a'_1,a'_2)\).
- 5.
As detailed in [1], approaches for mereotopology differ depending on the interpretation of the connection and the properties of the considered regions (closed, open...).
References
Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds.): Handbook of Spatial Logic. Springer, Netherlands (2007)
Bloch, I.: Dilation and erosion of spatial bipolar fuzzy sets. In: Masulli, F., Mitra, S., Pasi, G. (eds.) WILF 2007. LNCS (LNAI), vol. 4578, pp. 385–393. Springer, Heidelberg (2007). doi:10.1007/978-3-540-73400-0_49
Bloch, I.: Bipolar fuzzy mathematical morphology for spatial reasoning. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) ISMM 2009. LNCS, vol. 5720, pp. 24–34. Springer, Heidelberg (2009). doi:10.1007/978-3-642-03613-2_3
Bloch, I.: Duality vs. adjunction for fuzzy mathematical morphology and general form of fuzzy erosions and dilations. Fuzzy Sets Syst. 160, 1858–1867 (2009)
Bloch, I.: Bipolar fuzzy spatial information: geometry, morphology, spatial reasoning. In: Jeansoulin, R., Papini, O., Prade, H., Schockaert, S. (eds.) Methods for Handling Imperfect Spatial Information, vol. 256, pp. 75–102. Springer, Heidelberg (2010)
Bloch, I.: Mathematical morphology on bipolar fuzzy sets: general algebraic framework. Int. J. Approx. Reason. 53, 1031–1061 (2012)
Bloch, I., Heijmans, H., Ronse, C.: Mathematical morphology. In: Aiello, M., Pratt-Hartman, I., van Benthem, J. (eds.) Handbook of Spatial Logics, chap. 13, pp. 857–947. Springer, Netherlands (2007)
Bloch, I., Maître, H., Anvari, M.: Fuzzy adjacency between image objects. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 5(6), 615–653 (1997)
Clementini, E., Felice, O.D.: Approximate topological relations. Int. J. Approx. Reason. 16, 173–204 (1997)
Cohn, A.G., Gotts, N.M.: The egg-yolk representation of regions with indeterminate boundaries. Geogr. Objects Indeterm. Bound. 2, 171–187 (1996)
Deschrijver, G., Cornelis, C., Kerre, E.: On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Trans. Fuzzy Syst. 12(1), 45–61 (2004)
Dubois, D., Prade, H.: Special issue on bipolar representations of information and preference. Int. J. Intell. Syst. 23(8–10), 999–1152 (2008)
Dubois, D., Gottwald, S., Hajek, P., Kacprzyk, J., Prade, H.: Terminology difficulties in fuzzy set theory - the case of “intuitionistic fuzzy sets”. Fuzzy Sets Syst. 156, 485–491 (2005)
Dubois, D., Prade, H.: An overview of the asymmetric bipolar representation of positive and negative information in possibility theory. Fuzzy Sets Syst. 160, 1355–1366 (2009)
Fargier, H., Wilson, N.: Algebraic structures for bipolar constraint-based reasoning. In: Mellouli, K. (ed.) ECSQARU 2007. LNCS (LNAI), vol. 4724, pp. 623–634. Springer, Heidelberg (2007). doi:10.1007/978-3-540-75256-1_55
Goguen, J.: L-fuzzy sets. J. Math. Anal. Appl. 18(1), 145–174 (1967)
Hazarika, S., Cohn, A.: A taxonomy for spatial vagueness: an alternative egg-yolk interpretation. In: Spatial Vagueness, Uncertainty and Granularity Symposium, Ogunquit, Maine, USA (2001)
Mélange, T., Nachtegael, M., Sussner, P., Kerre, E.: Basic properties of the interval-valued fuzzy morphological operators. In: IEEE World Congress on Computational Intelligence, WCCI 2010, Barcelona, Spain, pp. 822–829 (2010)
Randell, D., Cui, Z., Cohn, A.: A spatial logic based on regions and connection. In: Principles of Knowledge Representation and Reasoning, KR 1992, Kaufmann, San Mateo, CA, pp. 165–176 (1992)
Roy, A.J., Stell, J.G.: Spatial relations between indeterminate regions. Int. J. Approx. Reason. 27(3), 205–234 (2001)
Schockaert, S., De Cock, M., Cornelis, C., Kerre, E.E.: Fuzzy region connection calculus: an interpretation based on closeness. Int. J. Approx. Reason. 48(1), 332–347 (2008)
Schockaert, S., De Cock, M., Cornelis, C., Kerre, E.E.: Fuzzy region connection calculus: representing vague topological information. Int. J. Approx. Reason. 48(1), 314–331 (2008)
Schockaert, S., De Cock, M., Kerre, E.E.: Spatial reasoning in a fuzzy region connection calculus. Artif. Intell. 173(2), 258–298 (2009)
Sussner, P., Nachtegael, M., Mélange, T., Deschrijver, G., Esmi, E., Kerre, E.: Interval-valued and intuitionistic fuzzy mathematical morphologies as special cases of L-fuzzy mathematical morphology. J. Math. Imaging Vis. 43(1), 50–71 (2012)
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This work has been partly supported by the French ANR LOGIMA project.
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Bloch, I. (2017). Topological Relations Between Bipolar Fuzzy Sets Based on Mathematical Morphology. In: Angulo, J., Velasco-Forero, S., Meyer, F. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2017. Lecture Notes in Computer Science(), vol 10225. Springer, Cham. https://doi.org/10.1007/978-3-319-57240-6_4
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