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 be a complete metric space. Suppose is a nonempty, compact subset of and let be given. Choose an iterated function system with contractivity factor, Suppose
where is the Hausdorff metric. Then
where A is the attractor of the IFS. Equivalently,
Informally, If is close to being stabilized by the IFS, then is also close to being the attractor of the IFS.
