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.
External links
- A description of the collage theorem and interactive Java applet at cut-the-knot.
- Notes on designing IFSs to approximate real images.
- Expository Paper on Fractals and Collage theorem