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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.