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 $\mathbb$ be a complete metric space.
Suppose $L$ is a nonempty, compact subset of $\mathbb$ and let $\epsilon >0$ be given.
Choose an iterated function system (IFS) $\$ with contractivity factor $s,$ where $0 \leq s < 1$ (the contractivity factor $s$ of the IFS is the maximum of the contractivity factors of the maps $w_i$). Suppose

:$h\left( L, \bigcup_^N w_n (L) \right) \leq \varepsilon,$

where $h(\cdot,\cdot)$ is the Hausdorff metric. Then

:$h(L,A) \leq \frac$

where ''A'' is the attractor of the IFS. Equivalently,

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

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
* Barnsley fern

References

*

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

Fractals
Theorems in geometry

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