Collage theorem
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In mathematics, the collage theorem describes a constructive technique for approximating sets of points in Euclidean space (typically images) to any degree of precision with the attractor of an iterated function system. It is typically used in fractal compression.
Statement of the theorem
Let be a complete metric space. Let be given, and let be given. Choose an Iterated function system (IFS) with contractivity factor , so that
where is the Hausdorff metric. Then
where A is the attractor of the IFS. Equivalently,
See also
References
- Barnsley, Michael. (1988). Fractals Everywhere. Academic Press, Inc. ISBN 0-12-079062-9.
External links
- A description of the collage theorem and interactive Java applet at cut-the-knot.
- Notes on designing IFSs to approximate real images. [dead link]
- Expository Paper on Fractals and Collage theorem