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International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing

ISMM 2017: Mathematical Morphology and Its Applications to Signal and Image Processing pp 40–51Cite as

Topological Relations Between Bipolar Fuzzy Sets Based on Mathematical Morphology

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Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10225))

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.

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Notes

  1. 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. 2.

    All proofs are quite straightforward, and omitted due to lack of space.

  3. 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. 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. 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...).

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Acknowledgments

This work has been partly supported by the French ANR LOGIMA project.

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Authors and Affiliations

  1. LTCI, Télécom ParisTech, Université Paris-Saclay, Paris, France

    Isabelle Bloch

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  1. Isabelle Bloch
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Correspondence to Isabelle Bloch .

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Editors and Affiliations

  1. MINES ParisTech, PSL-Research University, Fontainebleau, France

    Jesús Angulo

  2. MINES ParisTech, PSL-Research University , Fontainebleau, France

    Santiago Velasco-Forero

  3. MINES ParisTech, PSL-Research University, Fontainebleau, France

    Fernand Meyer

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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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  • DOI: https://doi.org/10.1007/978-3-319-57240-6_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57239-0

  • Online ISBN: 978-3-319-57240-6

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