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Parameter Identification of 1D Recurrent Fractal Interpolation Functions with Applications to Imaging and Signal Processing

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Abstract

Recurrent fractal interpolation functions are very useful in modelling irregular (non-smooth) data. Two methods that use bounding volumes and one that uses the concept of box-counting dimension are introduced for the identification of the vertical scaling factors of such functions. The first two minimize the area of the symmetric difference between the bounding volumes of the data points and their transformed images, while the latter aims at achieving the same box-counting dimension between the original and the reconstructed data. Comparative results with existing methods in imaging applications are given, indicating that the proposed ones are competitive alternatives for both low and high compression ratios.

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References

  1. Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Barnsley, M.F., Elton, J.H., Hardin, D.P.: Recurrent iterated function systems. Constr. Approx. 5, 3–31 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brinks, R.: A hybrid algorithm for the solution of the inverse problem in fractal interpolation. Fractals 13(3), 215–226 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

    MATH  Google Scholar 

  6. Gantmacher, F.R.: Matrix Theory, vol. 2. Chelsea, New York (2000)

    Google Scholar 

  7. Levkovich-Maslyuk, L.I.: Wavelet-based determination of generating matrices for fractal interpolation functions. Regul. Chaotic Dyn. 3(2), 20–29 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Manousopoulos, P., Drakopoulos, V., Theoharis, T., Stavrou, P.: Effective representation of 2D and 3D data using fractal interpolation. In: Proceedings of International Conference on Cyberworlds, pp. 457–464. IEEE Comput. Soc., Los Alamitos (2007)

    Google Scholar 

  9. Manousopoulos, P., Drakopoulos, V., Theoharis, T.: Curve fitting by fractal interpolation. Trans. Comput. Sci. 1, 85–103 (2008)

    Article  MathSciNet  Google Scholar 

  10. Manousopoulos, P., Drakopoulos, V., Theoharis, T.: Parameter identification of 1D fractal interpolation functions using bounding volumes. J. Comput. Appl. Math. 233(4), 1063–1082 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Massopust, P.R.: Interpolation and Approximation with Splines and Fractals. Oxford University Press, London (2009)

    Google Scholar 

  12. Mazel, D.S., Hayes, M.H.: Using iterated function systems to model discrete sequences. IEEE Trans. Signal Process. 40, 1724–1734 (1992)

    Article  MATH  Google Scholar 

  13. Navascues, M.A., Sebastian, M.V.: Fitting curves by fractal interpolation: An application to the quantification of cognitive brain processes. In: Novak, M.M. (ed.) Thinking in Patterns: Fractal and Related Phenomena in Nature, pp. 143–154. World Scientific, Singapore (2004)

    Chapter  Google Scholar 

  14. Navascues, M.A., Sebastian, M.V.: Spectral and affine fractal methods in signal processing. Int. Math. Forum 1(29), 1405–1422 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Penn, A.I., Loew, M.H.: Estimating fractal dimension with fractal interpolation function models. IEEE Trans. Med. Imaging 16(6), 930–937 (1997)

    Article  Google Scholar 

  16. Penn, A.I., Bolinger, L., Schnall, M.D., Loew, M.H.: Discrimination of MR images of breast masses with fractal interpolation function models. Acad. Radiol. 6(3), 156–163 (1999)

    Article  Google Scholar 

  17. Saupe, D.: Efficient computation of julia sets and their fractal dimension. Physica D 28(3), 358–370 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. Uemura, S., Haseyama, M., Kitajima, H.: Efficient contour shape description by using fractal interpolation functions. In: IEEE Proc. ICIP, vol. 1, pp. 485–488 (2002)

    Google Scholar 

  19. Zhao, C., Shi, W., Deng, Y.: A new Hausdorff distance for image matching. Pattern Recognit. Lett. 26, 581–586 (2005)

    Article  Google Scholar 

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Correspondence to Polychronis Manousopoulos.

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Manousopoulos, P., Drakopoulos, V. & Theoharis, T. Parameter Identification of 1D Recurrent Fractal Interpolation Functions with Applications to Imaging and Signal Processing. J Math Imaging Vis 40, 162–170 (2011). https://doi.org/10.1007/s10851-010-0253-z

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