Skip to main content
SpringerLink
Log in
Book cover

International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems

IPMU 2016: Information Processing and Management of Uncertainty in Knowledge-Based Systems pp 459–469Cite as

Adjoint Fuzzy Partition and Generalized Sampling Theorem

  • Conference paper
  • First Online:

  • 822 Accesses

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 611))

Abstract

A new notion of adjoint fuzzy partition is introduced and the reconstruction of a function from its F-transform components is analyzed. An analogy with the Nyquist-Shannon-Kotelnikov sampling theorem is discussed.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    The extended version of this contribution together with the application to the problem of function “de-noising” was submitted to [11].

  2. 2.

    For simplicity of representation, we assume that \(t_0=0\).

  3. 3.

    We distinguish between a generating function of an adjoint partition (in this paper, denoted by b) and a generating function of a fuzzy partition (in this paper, denoted by a). The latter is characterized in Definition 1, while the former is associated with an adjoint partition and can have values outside the interval [0, 1].

  4. 4.

    Strictly speaking, the Dirac’s delta is not a function, but a generalized function or a linear functional. Therefore, it makes sense to use it only if it appears inside an integral. In our paper, we always follow this restriction.

References

  1. Crouzet, J.-F.: Fuzzy projection versus inverse fuzzy transform as ampling/interpolation schemes. Fuzzy Sets Syst. 193, 108–121 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Holčapek, M., Tichý, T.: A smoothing filter based on fuzzy transform. Fuzzy Sets Syst. 180, 69–97 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Holčapek, M., Perfilieva, I., Novák, V., Kreinovich, V.: Necessary and sufficient conditions for generalized uniform fuzzy partitions. Fuzzy Sets Syst. 277, 97–121 (2015)

    Article  MathSciNet  Google Scholar 

  4. Kotelnikov, V.A.: On the carrying capacity of the ether and wire in telecommunications. In: Material for the First All-Union Conference on Questions of Communication, Izd. Red. Upr. Svyazi RKKA, Moscow (1933). (in Russian)

    Google Scholar 

  5. Novák, V., Štěpnička, M., Dvořák, A., Perfilieva, I., Pavliska, V., Vavřícková, L.: Analysis of seasonal time series using fuzzy approach. Int. J. General Syst. 39(3), 305–328 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Nyquist, H.: Certain topics in telegraph transmission theory. Trans. AIEE 47, 617–644 (1928)

    Google Scholar 

  7. Patanè, G.: Fuzzy transform and least-squares approximation: analogies differences and generalizations. Fuzzy Sets Syst. 180, 41–54 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Perfilieva, I.: Fuzzy transforms: theory and applications. Fuzzy Sets Syst. 157, 993–1023 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Perfilieva, I.: Fuzzy transforms: a challenge to conventional transforms. In: Hawkes, P.W. (ed.) Advances in Images and Electron Physics, vol. 147, pp. 137–196. Elsevier Academic Press, San Diego (2007)

    Google Scholar 

  10. Perfilieva, I.: F-transform. In: Kacprzyk, J., Pedrycz, W. (eds.) Springer Handbook of Computational Intelligence, pp. 113–130. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  11. Perfilieva, I., Holčapek, M., Kreinovich, V.: A new reconstruction from the F-Transform components. Fuzzy Sets Syst. 288, 3–25 (2016)

    Article  MathSciNet  Google Scholar 

  12. Shannon, C.E.: Communication in the presence of noise. Proc. Inst. Radio Eng. 37(1), 10–21 (1949)

    MathSciNet  Google Scholar 

  13. Stefanini, L.: F-transform with parametric generalized fuzzy partitions. Fuzzy Sets Syst. 180, 98–120 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Takagi, T., Sugeno, M.: Fuzzy identification of systems and its application to modeling and control. IEEE Trans. Syst. Man Cybern. 15, 116–132 (1985)

    Article  MATH  Google Scholar 

  15. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgement

This work was partially supported by the project LQ1602 IT4Innovations excellence in science.

Author information

Authors and Affiliations

  1. Centre of Excellence IT4Innovations, Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. Dubna 22, 701 03, Ostrava 1, Czech Republic

    Irina Perfilieva & Michal Holčapek

  2. Department of Computer Science, University of Texas at El Paso, 500 W. University, El Paso, TX, 79968, USA

    Vladik Kreinovich

Authors
  1. Irina Perfilieva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michal Holčapek
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vladik Kreinovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Irina Perfilieva .

Editor information

Editors and Affiliations

  1. INESC-ID,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Joao Paulo Carvalho

  2. LIP 6, Université Pierre et Marie Curie, Paris, France

    Marie-Jeanne Lesot

  3. School of Industrial Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    Uzay Kaymak

  4. IDMEC,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Susana Vieira

  5. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France

    Bernadette Bouchon-Meunier

  6. Iona College, Machine Intelligence Institute, New Rochelle, New York, USA

    Ronald R. Yager

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Perfilieva, I., Holčapek, M., Kreinovich, V. (2016). Adjoint Fuzzy Partition and Generalized Sampling Theorem. In: Carvalho, J., Lesot, MJ., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, vol 611. Springer, Cham. https://doi.org/10.1007/978-3-319-40581-0_37

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-40581-0_37

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-40580-3

  • Online ISBN: 978-3-319-40581-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation