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.

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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.

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