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 of the theorem

Let $\mathbb{X}$ be a complete metric space. Suppose $L$ is a nonempty, compact subset of $\mathbb{X}$ and let $\epsilon \geq 0$ be given. Choose an iterated function system (IFS) $\{ \mathbb{X} ; w_1, w_2, \dots, w_N\}$ with contractivity factor $0 \leq s < 1$, (The contractivity factor of the IFS is the maximum of the contractivity factors of the maps $w_i$.) Suppose

$h\left( L, \bigcup_{n=1}^N w_n (L) \right) \leq \varepsilon,$

where $h(d)$ is the Hausdorff metric. Then

$h(L,A) \leq \frac{\varepsilon}{1-s}$

where A is the attractor of the IFS. Equivalently,

$h(L,A) \leq (1-s)^{-1} h\left(L,\cup_{n=1}^N w_n(L)\right) \quad$, for all nonempty, compact subsets L of $\mathbb{X}$.

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

See also

• Michael Barnsley

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