Abstract
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 will make you see everything differently. There is a danger in reading further. You risk the loss of your childhood vision of clouds, forests, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpets, bricks, and much else besides. Never again will your interpretation of these things be quite the same.
Michael F. Barnsley1
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References
Michael F. Barnsley, Fractals Everywhere, Academic Press, 1988.
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.
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.
M. Berger, Encoding images through transition probablities, Math. Comp. Modelling 11 (1988) 575–577.
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.
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.
A similar metaphor has been ussed by Barnsley in his popularization of iterated function systems (IFS), which is the mathematical notation for MRCMs.
Almost any image can be used for this purpose. Images with certain symmetries provide some exceptions. We will study these in detail further below.
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.
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).
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.
More precisely, the fern without the stem is self-affine, not self-similar, because the transformations which produce the leaves are only approximate similitudes.
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.
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© 1992 Springer Science+Business Media New York
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Peitgen, HO., Jürgens, H., Saupe, D. (1992). Encoding Images by Simple Transformations. In: Fractals for the Classroom. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2172-0_5
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DOI: https://doi.org/10.1007/978-1-4757-2172-0_5
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