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 of the theorem
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 , (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.
- 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
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