Journal of Mathematical Imaging and Vision

, Volume 14, Issue 1, pp 73–81 | Cite as

Images and Benford's Law

  • Jean-Michel Jolion


Benford's law had been proposed in the past as a way to modelize the probability distribution of the first digit in a set of natural numbers. We show in this paper that the magnitude of the gradient of an image obeys this law. We show, experimentally, that this also applies for the laplacian pyramid code. This yields to the field of entropy based coding which takes advantage of the a priori information about the probability of any symbol in the signal.

image's distribution Benford's law entropy based coding 


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  1. 1.
    F. Benford, “The law of anomalous numbers,” Proc. Amer. Phil. Soc., Vol. 78, pp. 551–572, 1938.Google Scholar
  2. 2.
    P.J. Burt and E.H. Adelson, “The Laplacian pyramid as a compact image code,” IEEE Trans. on Comm., Vol. 31, No. 4, pp. 532–540–572, 1983.Google Scholar
  3. 3.
    R. Deriche, “Optimal edge detection using recursive filtering,” in Proc. 1st Int. Conf. on Computer Vision, London, June 8–11, 1987, pp. 501–505.Google Scholar
  4. 4.
    T.P. Hill, “A statistical derivation for the significant-digit law,” Statistical Science, Vol. 10, pp. 354–363, 1996.Google Scholar
  5. 5.
    M. Kunt, Techniques modernes de traitement numérique des signaux, Presses polytechniques et universitaires romandes, collection électricité, Vol. 1, 397–400, 1991.Google Scholar
  6. 6.
    S. Newcomb, “Note on the frequency of the use of digits in natural numbers,” Amer. Journal Math., Vol. 4, pp. 39–40, 1881.Google Scholar
  7. 7.
    H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press: Cambridge, pp. 472–475, 1987.Google Scholar
  8. 8.
    E.P. Simoncelli, “Modeling the joint statistics of images in the wavelet domain,” in Proc. of SPIE 44th Annual Meeting, Denver, Colorado, USA, July 1999, Vol. 3813.Google Scholar
  9. 9.
    A. van der Schaaf and J.H. van Hateren, “Modeling the power spectra of natural images: Statistics and information,” Vision Research, Vol. 36, No. 17, pp. 2759–2770, 1996.Google Scholar
  10. 10.
    H. Voorhees and T. Poggio, “Detecting blobs as textons in natural images,” in Proc. of Image Understanding Workshop, Los Angeles, 1987, pp. 892–899.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Jean-Michel Jolion
    • 1
  1. 1.Laboratoire Reconnaissance de Formes et VisionINSA LyonFrance

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