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International Conference on Analysis of Images, Social Networks and Texts

AIST 2015: Analysis of Images, Social Networks and Texts pp 57–70Cite as

Fréchet Filters for Color and Hyperspectral Images Filtering

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Part of the book series: Communications in Computer and Information Science ((CCIS,volume 542))

Abstract

Median filtering has been widely used in scalar-valued image processing as an edge preserving operation. The basic idea is that the pixel value is replaced by the median of the pixels contained in a window around it. In this paper, we extend the notion of the Fréchet vector median to the general Fréchet vector median, which minimizes the Fréchet cost function (FCF) in the form of an aggregation function instead of the ordinary sum. Moreover, we propose to use an aggregation distance instead of the classical one. We use the generalized Fréchet median for constructing new nonlinear filters based on an arbitrary pair of aggregation operators that can be changed independently. For each pair of parameters, we get the unique class of new nonlinear filters.

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References

  1. Frèchet, M.: Les elements aleatoires de nature quelconque dans un espace distancie. Ann. Inst. Henri Poincare 10(3), 215–310 (1948)

    MATH  Google Scholar 

  2. Bajaj, C.: Proving geometric algorithms nonsolvability: an application of factoring polynomials. J. Symbolic Comput. 2, 99–102 (1986)

    Article  MathSciNet  Google Scholar 

  3. Bajaj, C.: The algebraic degree of geometric optimization problems. Discrete Comput. Geom. 3, 177–191 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chandrasekaran, R., Tamir, F.: Algebraic optimization: the fermat-weber problem. Math. Program. 46, 219–224 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Astola, J., Haavisto, P., Neuvo, Y.: Vector median filters. Proc. IEEE 78, 678–689 (1990)

    Article  Google Scholar 

  6. Tang, K., Astola, J., Neuvo, Y.: Nonlinear multivariate image filtering techniques. IEEE Trans. Image Process. 4, 788–798 (1996)

    Article  Google Scholar 

  7. Mayor, G., Trillas, E.: On the representation of some aggregation functions. Proc. ISMVL 20, 111–114 (1986)

    Google Scholar 

  8. Ovchinnikov, S.: On robust aggregation procedures. In: Bouchon-Meunier, B. (ed.) Aggregation Operators for Fusion under Fuzziness, pp. 3–10. Physica, Heidelberg (1998)

    Google Scholar 

  9. Kolmogorov, A.: Sur la notion de la moyenne. Atti Accad. Naz. Lincei 12, 388–391 (1930)

    MATH  Google Scholar 

  10. Labunets V. G.: Filters based on aggregation operators Part 1. Aggregation operators. In: 24th International Crimean Conference Microwave and Telecommunication Technology (CriMiCo2014), Sevastopol, Crimea, Russia, vol. 24, pp. 1239–1240 (2014)

    Google Scholar 

  11. Labunets, V.G., Gainanov, D.N., Ostheimer, E.: Filters based on aggregation operators. Part 2. The kolmogorov filters. In: 24th International Crimean Conference Microwave and Telecommunication Technology (CriMiCo2014), Sevastopol, Crimea, Russia, vol. 24, pp. 1241–1242 (2014)

    Google Scholar 

  12. Labunets, V.G., Gainanov, D.N., Tarasov A.D., Ostheimer E.: Filters based on aggregation operators. Part 3. The heron filters. In: 24th International Crimean Conference “Microwave and Telecommunication Technology” (CriMiCo2014), Sevastopol, Crimea, Russia, vol. 24, pp. 1243–1244 (2014)

    Google Scholar 

  13. Labunets, V.G., Gainanov, D.N., Arslanova, R.A., Ostheimer, E.: Filters based on aggregation operators. Part 4. Generalized vector median filters. In: 24th International Crimean Conference Microwave and Telecommunication Technology (CriMiCo2014), Sevastopol, Crimea, Russia, vol. 24, pp. 1245–1246 (2014)

    Google Scholar 

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Acknowledgements

This work was supported by grants the RFBR Nos.13-07-12168 and 13-07-00785.

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Authors and Affiliations

  1. Capricat LLC, 1340 S., Ocean Blvd., Suite 209, Pompano Beach, FL, 33062, USA

    Ekaterina Ostheimer

  2. Ural Federal University, Pr. Mira, 19, Yekaterinburg, 620002, Russian Federation

    Valeriy Labunets, Denis Komarov & Tat’yana Fedorova

Authors
  1. Ekaterina Ostheimer
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  2. Valeriy Labunets
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  3. Denis Komarov
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  4. Tat’yana Fedorova
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Corresponding author

Correspondence to Ekaterina Ostheimer .

Editor information

Editors and Affiliations

  1. Krasovsky Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Mikhail Yu. Khachay

  2. Wolverhampton, United Kingdom

    Natalia Konstantinova

  3. Technische Universität Darmstadt, Darmstadt, Germany

    Alexander Panchenko

  4. National Research University Higher School of Economics, Moscow, Russia

    Dmitry Ignatov

  5. Ural Federal University, Yekaterinbug, Russia

    Valeri G. Labunets

Appendix. Figures

Appendix. Figures

Fig. 3.
figure 3

Original (a) and noised (b) images; noise: Salt-Pepper; denoised images (c)–(f)

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Fig. 4.
figure 4

Original (a) and noised (b) images; noise: Gaussian PDF; denoised images (c)–(f)

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Fig. 5.
figure 5

Original (a) and noised (b) images; noise: Laplasian PDF; denoised images (c)–(f)

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© 2015 Springer International Publishing Switzerland

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Ostheimer, E., Labunets, V., Komarov, D., Fedorova, T. (2015). Fréchet Filters for Color and Hyperspectral Images Filtering. In: Khachay, M., Konstantinova, N., Panchenko, A., Ignatov, D., Labunets, V. (eds) Analysis of Images, Social Networks and Texts. AIST 2015. Communications in Computer and Information Science, vol 542. Springer, Cham. https://doi.org/10.1007/978-3-319-26123-2_6

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  • DOI: https://doi.org/10.1007/978-3-319-26123-2_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26122-5

  • Online ISBN: 978-3-319-26123-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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