Fractal Modulation and Other Applications from a Theory of the Statistics of Dimension

  • J. M. Blackledge
  • S. Mikhailov
  • M. J. Turner
Conference paper
Part of the The IMA Volumes in Mathematics and its Application book series (IMA, volume 132)


This paper considers a model and the applications of a non-stationary stochastic field u(x,t) using a fractional partial differential equation of (time dependent) order q(t) given by
$$ \left[ {\frac{{{\partial ^2}}}{{\partial {x^2}}} - {r^{q(t)}}\frac{{{\partial ^{q(t)}}}}{{\partial {t^{q(t)}}}}} \right]u(x,t) = - F(x,t),{\kern 1pt} - \infty < q(t) < \infty ,{\kern 1pt} \forall t $$
where r is a constant and F and q are stochastic functions. Numerical algorithms for computing this solution are introduced and results presented to illustrate its characteristics which depend on the random behaviour of q(t). An important aspect of this approach is concerned with the effect of changing the statistics [i.e. the Probability Density Function (PDF) of q(t)] used to ‘drive’ the solution. Since q is a dimension (the ‘Fourier dimension’ which is related to the fractal dimension), the introduction of a PDF associated with q(t) leads directly to the notion of the ‘statistics of dimension’. We address the inverse problem in which a discrete solution is found for q(t) from a known stochastic field u(x,t) when x→0. The application of fractal modulation, where q(t) is assigned just two states for all t as well as other applications are considered.


Fractal Dimension Probability Density Function Discrete Fourier Transform Fractal Modulation Power Spectral Density Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. Bak, How Nature Works: The Science of Self-Organized Criticality, Oxford University Press, 1997. ISBN: 038798738X.Google Scholar
  2. [2]
    M. Bertero, P. Boccaci, and P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics Publishing, Bristol, April 1998. ISBN: 0750304359.MATHCrossRefGoogle Scholar
  3. [3]
    J. Blackledge, Quantitative Coherent Imaging: Theory, Methods and Some Applications (Techniques of Physics), Academic Press, 1989.Google Scholar
  4. [4]
    J. Blackledge, B. Foxon, and S. Mikhailov, Fractal dimension segmentation, in Proceedings of the First IMA Conference on Image Processing, J. Blackledge, ed., Oxford University Press, 1997, pp. 249–289.Google Scholar
  5. [5]
    J. Blackledge, M. London, S. Mikhailov, and R. Smith, On the statistics of dimension: Fractal modulation and quantum fractional dynamics, in Image Processing II: Mathematical Methods, Algorithms and Applications, J. Blackledge and M. Turner, eds., Institute of Mathematics and its Applications, Horwood Publishing, 2000, pp. 184–227. ISBN:1898563616.Google Scholar
  6. [6]
    M. Buchanan, One law to rule them all, New Scientist, (1997), pp. 30–35.Google Scholar
  7. [7]
    M. Buchanan, Ubiquity: the science of history… or why the world is simpler than we think., Weidenfeld and Nicolson, 2000. ISBN: 0297643762.Google Scholar
  8. [8]
    B. Buck, ed., Maximum Entropy in Action: A Collection of Expository Essays, Oxford University Press, July 1991. ISBN: 0198539630.Google Scholar
  9. [9]
    D. S. Ebert, F. K. Musgrave, D. Peachey, S. Worley, and K. Perlin, Texturing and Modeling, Morgan Kaufmann Publishers, July 1998. ISBN: 0122287304.Google Scholar
  10. [10]
    A. Evans, The Fourier dimension and the fractal dimensions, Chaos, Solitons and Fractals, 9 (1998), pp. 1977–1982.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    R. Hilfer, Scaling theory and the classification of phase transitions, Modern Physics Letters B, 6 (1992), pp. 773–784.MathSciNetCrossRefGoogle Scholar
  12. [12]
    F.S.J. Klafter and G. Zumofen, Beyond brownian motion, Physics Today (1996), pp. 33–39.Google Scholar
  13. [13]
    M. London, A. Evans, and M. Turner, Conditional entropy and randomness in financial time series, Quantitative Finance, 1 (2001), pp. 414–426. Institute of Physics and IOP Publishing Limited.CrossRefGoogle Scholar
  14. [14]
    S. Mikhailov, Fractal Modulation and Encryption, PhD thesis, De Montfort University, 1999.Google Scholar
  15. [15]
    H.-O. Peitgen and D. Saupe, eds., The Science of Fractal Images, Springer, 1988.MATHGoogle Scholar
  16. [16]
    M. Shlesinger, M. Zaslavsky, and J. Klafter, Strange kinetics, Nature, 363 (1993), pp. 31–37.CrossRefGoogle Scholar
  17. [17]
    M.F. Shlesinger, G.M. Zaslavskii, and U. Frisch, eds., Levy Flights and Related Topics in Physics: Proceedings of the International Workshop 27–30 June, Nice, France, January 1996, Springer. Lecture Notes in Physics ISBN: 3540592229.Google Scholar
  18. [18]
    F. Tatom, The Application of Fractional Calculus to the Simulation of Stochastic Processes, Engineering Analysis Inc., Huntsville, Alabama, 1989. AIAA-89/0792.Google Scholar
  19. [19]
    M. Turner, J. Blackledge, and P. Andrews, Fractal Geometry in Digital Imaging, Academic Press, 1998. ISBN: 0127039708.Google Scholar
  20. [20]
    P. Wilmott, S. Howison, and J.D. (Contributor), The Mathematics of Financial Derivatives: A Student Introduction, Cambridge University Press, August 1995. ISBN: 0521497892.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 2002

Authors and Affiliations

  • J. M. Blackledge
    • 1
  • S. Mikhailov
    • 1
  • M. J. Turner
    • 1
  1. 1.IMSS, SERCDe Montfort UniversityLeicesterUK

Personalised recommendations