Encoding Images by Simple Transformations

  • Heinz-Otto Peitgen
  • Hartmut Jürgens
  • Dietmar Saupe


So far, we have discussed two extreme ends of fractal geometry. We have explored fractal monsters, such as the Cantor set, the Koch curve, and the Sierpinski gasket; and we have argued that there are many fractals in natural structures and patterns, such as coastlines, blood vessel systems, and cauliflowers. We have discussed the common features, such as self-similarity, scaling properties, and fractal dimensions shared by those natural structures and the monsters; but we have not yet seen that they are close relatives in the sense that maybe a cauliflower is just a ‘mutant’ of a Sierpinski gasket, and a fern is just a Koch curve ‘let loose’. Or phrased as a question, is there a framework in which a natural structure, such as a cauliflower, and an artificial structure, such as a Sierpinski gasket, are just examples of one unifying approach; and if so, what is it? Believe it or not, there is such a theory, and this chapter is devoted to it. It goes back to Mandelbrot’s book, The Fractal Geometry of Nature, and a beautiful paper by the Australian mathematician Hutchinson.2 Barnsley and Berger have extended these ideas and advocated the point of view that they are very promising for the encoding of images.3


Fractal Geometry Hausdorff Distance Initial Image Encode Image Final 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.
    Michael F. Barnsley, Fractals Everywhere, Academic Press, 1988.Google Scholar
  2. 2.
    J. Hutchinson, Fractals and self-similarity Indiana Journal of Mathematics 30 (1981) 713–747. Some of the ideas can already be found in R. F. Williams, Compositions of contractions, Bol. Soc. Brasil. Mat. 2 (1971) 55–59.Google Scholar
  3. 3.
    M. F. Barnsley, V. Ervin, D. Hardin, and J. Lancaster, Solution of an inverse problem for fractals and other sets, Proceedings of the National Academy of Sciences 83 (1986) 1975–1977.Google Scholar
  4. M. Berger, Encoding images through transition probablities, Math. Comp. Modelling 11 (1988) 575–577.Google Scholar
  5. A survey article is: E. R. Vrscay, Iterated function systems: Theory,applications and the inverse problem, in: Proceedings of the NATO Advanced Study Institute on Fractal Geometry, July 1989. Kluwer Academic Publishers, 1991.Google Scholar
  6. A very promising approach seems to be presented in the recent paper A. E. Jacquin, Image coding based on a fractal theory of iterated contractive image transformations, to appear in WEF. Transactions on Signal Processing, March 1992.Google Scholar
  7. 4.
    A similar metaphor has been ussed by Barnsley in his popularization of iterated function systems (IFS), which is the mathematical notation for MRCMs.Google Scholar
  8. 5.
    Almost any image can be used for this purpose. Images with certain symmetries provide some exceptions. We will study these in detail further below.Google Scholar
  9. 6.
    Being more mathematically technical, we allow A to be any compact set in the plane. Compactness means, that A is bounded and that A contains all its limit points, i.e. for any sequence of points from A with a cluster point, we have that the cluster point also belongs to A. The open unit disk of all points in the plane with a distance less than 1 from the origin is not a compact set, but the closed unit disk of all points with a distance not exceeding 1 is compact.Google Scholar
  10. 13.
    The unit sets are defined to be the sets of points with a distance not greater than 1 from the origin. Thus, they depend on the metric used. For example, the unit set for the Euclidean metric is a disk, while it is a square for the maximum metric (see fgures 532 and 534).Google Scholar
  11. 15.
    The computational problem evaluating the Hausdorff distance for digitized images is addressed in R. Shonkwiller, An image algorithm for computing the Hausdorff distance efficiently in linear time Info. Proc. Lett. 30 (1989) 87–89.Google Scholar
  12. 17.
    More precisely, the fern without the stem is self-affine, not self-similar, because the transformations which produce the leaves are only approximate similitudes.Google Scholar
  13. Comp. Modelling 11 (1988) 575–577. R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions,Trans. Amer. Math. Soc. 309 (1988) 811–829. G. Edgar, Measures,Topology and Fractal Geometry, Springer-Verlag, New York, 1990. The first ideas in this regard seem to be in T. Bedford, Dynamics and dimension for fractal recurrent sets, J. London Math. Soc. 33 (1986) 89–100.Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Heinz-Otto Peitgen
    • 1
    • 2
  • Hartmut Jürgens
    • 1
  • Dietmar Saupe
    • 1
  1. 1.Institut für Dynamische SystemeUniversität BremenBremen 33Federal Republic of Germany
  2. 2.Department of MathematicsFlorida Atlantic UniversityBoca RatonUSA

Personalised recommendations