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.

Let ${\displaystyle \mathbb {X} }$ be a complete metric space. Suppose ${\displaystyle L}$ is a nonempty, compact subset of ${\displaystyle \mathbb {X} }$ and let ${\displaystyle \varepsilon >0}$ be given. Choose an iterated function system (IFS) ${\displaystyle \{\mathbb {X} ;w_{1},w_{2},\dots ,w_{N}\}}$ with contractivity factor ${\displaystyle s}$, where ${\displaystyle 0\leq s<1}$ (the contractivity factor ${\displaystyle s}$ of the IFS is the maximum of the contractivity factors of the maps ${\displaystyle w_{i}}$). Suppose

${\displaystyle h\left(L,\bigcup _{n=1}^{N}w_{n}(L)\right)\leq \varepsilon ,}$

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

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

where ${\displaystyle A}$ is the attractor of the IFS. Equivalently,

${\displaystyle h(L,A)\leq (1-s)^{-1}h\left(L,\bigcup _{n=1}^{N}w_{n}(L)\right)}$

for all nonempty, compact subsets ${\displaystyle L}$ of ${\displaystyle \mathbb {X} }$.

Informally, if ${\displaystyle L}$ is close to being stabilized by the IFS, then ${\displaystyle L}$ is also close to being the attractor of the IFS.