Advertisement

Nonlinear Dynamics

, Volume 52, Issue 1–2, pp 51–70 | Cite as

Generating an adaptive multiresolution image analysis with compact cupolets

  • Kourosh Zarringhalam
  • Kevin M. Short
Original Paper

Abstract

We present an efficient control scheme for stabilizing unstable periodic orbits of chaotic systems. The resulting orbits are called cupolets and have been proven to be useful in the representation of oscillatory or quasi periodic signals such as appear in music and image compression (Short et al., AES 118th Convention preprint 6446, May 2005; Short et al., AES 119th Convention preprint 6588, October 2005). In this paper we show that these cupolets can be used effectively to produce an adaptive basis for the space of real-valued functions of a discrete variable. From this basis, we construct a multiresolution analysis which allows for the approximation of signals at different resolution levels and apply it to image compression. This adaptive multiresolution analysis provides an interesting continuum between Fourier analysis and wavelet analysis.

Keywords

Controlling chaos Cupolets Image compression Signal processing Unstable periodic orbits 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Short, K.M., Garcia, R.A., Daniels, M., Curley, J., Glover, M.: An introduction to the koz scalable audio compression technology. AES 118th Convention preprint 6446 (May 2005) Google Scholar
  2. 2.
    Short, K.M., Garcia, R.A., Daniels, M.: Scalability in koz audio compression technology. AES 119th Convention preprint 6598 (October 2005) Google Scholar
  3. 3.
    Curry, J.H.: An algorithm for finding closed orbits. In: Global Theory of Dynamical Systems, vol. 819, pp. 111–120. Springer, New York (1979) CrossRefGoogle Scholar
  4. 4.
    Gregorio, S.D.: The study of periodic orbits of dynamical systems. The use of a computer. J. Stat. Phys. 38, 947–972 (1985) MATHCrossRefGoogle Scholar
  5. 5.
    Schwartz, I.B.: Estimating regions of existence of unstable periodic orbits using computer-based techniques. SIAM J. Numer. Anal. 20, 106–120 (1983) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Sparrow, C.: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer, New York (1982) MATHGoogle Scholar
  7. 7.
    Grebogi, C., Hammel, S.M., Yorke, J.A., Sauer, T.: Shadowing of physical trajectories in chaotic dynamics: Containment and refinement. Phys. Rev. Lett. 65, 1527–1530 (1990) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Coomes, B.A., Koçak, H., Palmer, K.J.: Shadowing orbits of ordinary differential equations. J. Comput. Appl. Math. 52, 35–43 (1994) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Coomes, B.A., Koçak, H., Palmer, K.J.: Long periodic shadowing. Numer. Algorithms 14, 55–78 (1997) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Parker, A.T.: Topics in chaotic secure communication. PhD thesis, University of New Hampshire (1999) Google Scholar
  11. 11.
    Short, K.M., Parker, A.T.: Security issues in chaotic communications. In: SIAM Conference on Dynamical Systems, Snowbird, UT, May 23–27 (1999) Google Scholar
  12. 12.
    Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hayes, S., Grebogi, C., Ott, E.: Communicating with chaos. Phys. Rev. Lett. 70, 3031–3034 (1993) CrossRefGoogle Scholar
  14. 14.
    Hayes, S., Grebogi, C., Ott, E., Mark, A.: Experimental control of chaos for communication. Phys. Rev. Lett. 73, 1781–1784 (1994) CrossRefGoogle Scholar
  15. 15.
    Beer, T.: Walsh transforms. Am. J. Phys. 49, 466–472 (1981) CrossRefGoogle Scholar
  16. 16.
    Petukhov, A.: Periodic discrete wavelets. Algebra Anal. 8, 151–183 (1996) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of New HampshireDurhamUSA

Personalised recommendations