# 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. Let $L \in H(\mathbb{X})$ be given, 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 $0 \leq s < 1$, so that

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

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,

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