Skip to main content

Computing Self-Similar Contractive Functions for the IFS Inverse Problem Through the Cuckoo Search Algorithm

  • Conference paper
  • First Online:
Harmony Search Algorithm (ICHSA 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 514))

Included in the following conference series:

  • 758 Accesses

Abstract

One of the most powerful and popular methods to generate fractal images is the so-called iterated function systems (IFS). Given a finite system of contractive maps \(\{w_i\}_{i=1,\dots ,n}\) on the compact metric space \(\mathbb {R}^2\), this system has a unique non-empty compact fixed set \(\mathcal {A}\), called the attractor of the IFS. The graphical representation of this attractor is a self-similar fractal image. The opposite is also true: each self-similar fractal image in \(\mathbb {R}^2\) can be mathematically represented as the only attractor of an IFS. Obtaining the parameters of the IFS system (called the IFS inverse problem) is a very difficult issue. A good strategy to address it consists of solving firstly the sub-problem of computing a suitable set of self-similar contractive functions to be further applied to obtain the optimal IFS for the inverse problem. In this paper we address this sub-problem by using a powerful metaheuristic technique called cuckoo search algorithm. Our experimental results show that the method performs quite well for several self-similar fractal images.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Barnsley, M.F.: Fractals Everywhere, 2nd edn. Academic Press, San Diego (1993)

    MATH  Google Scholar 

  2. Elton, J.H.: An ergodic theorem for iterated maps. Ergodic Theor. Dynam. Syst. 7, 481–488 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester (2003)

    Book  MATH  Google Scholar 

  4. Gálvez, A.: IFS Matlab generator: a computer tool for displaying IFS fractals. In: Proceedings of ICCSA 2009, pp. 132–142. IEEE CS Press, Los Alamitos (2009)

    Google Scholar 

  5. Gálvez, A., Iglesias, A.: Cuckoo search with Lévy flights for weighted Bayesian energy functional optimization in global-support curve data fitting. Sci. World J. 2014, 11 page (2014). Article ID 138760

    Google Scholar 

  6. Gálvez, A., Iglesias, A., Takato, S.: Matlab-based KETpic add-on for generating and rendering IFS fractals. CCIS 56, 334–341 (2009)

    Google Scholar 

  7. Gálvez, A., Iglesias, A., Takato, S.: KETpic Matlab binding for efficient handling of fractal images. Int. J. Future Gener. Comm. Netw. 3(2), 1–14 (2010)

    Google Scholar 

  8. Gálvez, A., Kitahara, K., Kaneko, M.: IFSGen4 : interactive graphical user interface for generation and visualization of iterated function systems in. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 554–561. Springer, Heidelberg (2014). doi:10.1007/978-3-662-44199-2_84

    Google Scholar 

  9. Graf, S.: Barnsley’s scheme for the fractal encoding of images. J. Complex. 8, 72–78 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gutiérrez, J.M., Iglesias, A.: A mathematica package for the analysis and control of chaos in nonlinear systems. Comput. Phys. 12(6), 608–619 (1998)

    Article  Google Scholar 

  11. Gutiérrez, J.M., Iglesias, A., Rodríguez, M.A.: A multifractal analysis of IFSP invariant measures with application to fractal image generation. Fractals 4(1), 17–27 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gutiérrez, J.M., Iglesias, A., Rodríguez, M.A., Burgos, J.D., Moreno, P.A.: Analyzing the multifractal structure of DNA nucleotide sequences. In: Chaos and Noise in Biology and Medicine, vol. 7, pp. 315–319. World Scientific, Singapore (1998)

    Google Scholar 

  13. Gutiérrez, J.M., Iglesias, A., Rodríguez, M.A., Rodríguez, V.J.: Generating and rendering fractal images. Math. J. 7(1), 6–13 (1997)

    Google Scholar 

  14. Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Iglesias, A., Gálvez, A.: Cuckoo search with Lévy flights for reconstruction of outline curves of computer fonts with rational Bézier curves. In: Proceedings of Congress on Evolutionary Computation-CEC 2016. IEEE CS Press, Los Alamitos (2016)

    Google Scholar 

  16. Yang, X.-S.: Nature-Inspired Metaheuristic Algorithms, 2nd edn. Luniver Press, Frome (2010)

    Google Scholar 

  17. Yang, X.S., Deb, S.: Cuckoo search via Lévy flights. In: Proceedings World Congress on Nature and Biologically Inspired Computing (NaBIC), pp. 210–214. IEEE Press, New York (2009)

    Google Scholar 

  18. Yang, X.S., Deb, S.: Engineering optimization by cuckoo search. Int. J. Math. Model. Numer. Optim. 1(4), 330–343 (2010)

    MATH  Google Scholar 

Download references

Acknowledgements

This research has been kindly supported by the Computer Science National Program of the Spanish Ministry of Economy and Competitiveness, Project Ref. #TIN2012-30768, Toho University, and the University of Cantabria.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andrés Iglesias .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Nature Singapore Pte Ltd.

About this paper

Cite this paper

Quirce, J., Gálvez, A., Iglesias, A. (2017). Computing Self-Similar Contractive Functions for the IFS Inverse Problem Through the Cuckoo Search Algorithm. In: Del Ser, J. (eds) Harmony Search Algorithm. ICHSA 2017. Advances in Intelligent Systems and Computing, vol 514. Springer, Singapore. https://doi.org/10.1007/978-981-10-3728-3_33

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-3728-3_33

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-3727-6

  • Online ISBN: 978-981-10-3728-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics