Collage theorem

In mathematics, the collage theorem characterises an iterated function system whose attractor is close, relative to the Hausdorff metric, to a given set. The IFS described is composed of contractions whose images, as a collage or union when mapping the given set, are arbitrarily close to the given set. It is typically used in fractal compression.

Statement

Let be a complete metric space. Suppose is a nonempty, compact subset of and let be given. Choose an iterated function system (IFS) with contractivity factor where (the contractivity factor of the IFS is the maximum of the contractivity factors of the maps ). Suppose

where is the Hausdorff metric. Then

where A is the attractor of the IFS. Equivalently,

, for all nonempty, compact subsets L of .

Informally, If is close to being stabilized by the IFS, then is also close to being the attractor of the IFS.

See also

References

  • Barnsley, Michael. (1988). Fractals Everywhere. Academic Press, Inc. ISBN 0-12-079062-9.


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