Continuous Evolution of Fractal Transforms and Nonlocal PDE Imaging

  • Edward R. Vrscay
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4141)


Traditional fractal image coding seeks to approximate an image function u as a union of spatially-contracted and greyscale-modified copies of itself, i.e., uTu, where T is a contractive fractal transform operator on an appropriate space of functions. Consequently u is well approximated by \(\bar u\), the unique fixed point of T, which can then be constructed by the discrete iteration procedure u n + 1 = T n .

In a previous work, we showed that the evolution equation y t = Oyy produces a continuous evolution y(x,t) to \(\bar y\), the fixed point of a contractive operator O. This method was applied to the discrete fractal transform operator, in which case the evolution equation takes the form of a nonlocal differential equation under which regions of the image are modified according to information from other regions.

In this paper we extend the scope of this evolution equation by introducing additional operators, e.g., diffusion or curvature operators, that “compete” with the fractal transform operator. As a result, the asymptotic limiting function y  ∞  is a modification of the fixed point \(\bar u\) of the original fractal transform. The modification can be viewed as a replacement of traditional postprocessing methods that are employed to “touch up” the attractor function \(\bar{u}\).


Continuous Evolution Image Function Image Denoising Lena Image Fractal Image 
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.
    Alexander, S.K.: Multiscale methods in image modelling and image processing, Ph.D. Thesis, Department of Applied Mathematics, University of Waterloo (2005)Google Scholar
  2. 2.
    Alexander, S.K., Vrscay, E.R., Tsurumi, S.: An examination of the statistical properties of domain-range block matching in fractal image coding (preprint)Google Scholar
  3. 3.
    Barnsley, M.F.: Fractals Everywhere. Academic Press, New York (1988)MATHGoogle Scholar
  4. 4.
    Barnsley, M.F., Demko, S.: Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London A399, 243–275 (1985)MathSciNetGoogle Scholar
  5. 5.
    Barnsley, M.F., Hurd, L.P.: Fractal Image Compression. A.K. Peters, Wellesley (1993)MATHGoogle Scholar
  6. 6.
    Bona, J., Vrscay, E.R.: Continuous evolution of functions and measures toward fixed points of contraction mappings. In: Levy-Vehel, J., Lutton, E. (eds.) Fractals in Engineering: New Trends in Theory and Applications, pp. 237–253. Springer, London (2005)Google Scholar
  7. 7.
    Buades, A., Coll, B., Morel, J.-M.: A nonlocal algorithm for image denoising. CVPR (2), 60–65 (2005)Google Scholar
  8. 8.
    Falconer, K.: The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985)CrossRefMATHGoogle Scholar
  9. 9.
    Fisher, Y.: Fractal Image Compression. Springer, New York (1995)Google Scholar
  10. 10.
    Forte, B., Vrscay, E.R.: Theory of generalized fractal transforms. In: Fisher, Y. (ed.) Fractal Image Encoding and Analysis. NATO ASI Series F, vol. 159. Springer, New York (1998)Google Scholar
  11. 11.
    Ghazel, M., Freeman, G., Vrscay, E.R.: Fractal image denoising. IEEE Transactions on Image Processing 12(12), 1560–1578 (2003)CrossRefGoogle Scholar
  12. 12.
    Hutchinson, J.: Fractals and self-similarity. Indiana Univ. J. Math. 30, 713–747 (1981)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Jacquin, A.: Image coding based on a fractal theory of iterated contractive image transformations. IEEE Trans. Image Proc. 1, 18–30 (1992)CrossRefGoogle Scholar
  14. 14.
    Lu, N.: Fractal Imaging. Academic Press, New York (1997)MATHGoogle Scholar
  15. 15.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. PAMI 12, 629–639 (1990)Google Scholar
  16. 16.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)CrossRefMATHGoogle Scholar
  17. 17.
    Sapiro, G.: Geometric partial differential equations and image analysis. Cambridge University Press, New York (2001)CrossRefMATHGoogle Scholar
  18. 18.
    Youla, D., Webb, H.: Image restoration by the method of convex projections:Part 1-Theory. IEEE Transactions on Medical Imaging MI-1(2), 81–94 (1982)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Edward R. Vrscay
    • 1
  1. 1.Department of Applied Mathematics, Faculty of MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations