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.

References

Barnsley, Michael. (1988). Fractals Everywhere. Academic Press, Inc. ISBN 0-12-079062-9.

External links

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Collage theorem

Statement of the theorem

See also

References

External links