Collage theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

Let be a complete metric space. Suppose is a nonempty, compact subset of and let be given. Choose an iterated function system (IFS) with contractivity factor , (The contractivity factor of the IFS is the maximum of the contractivity factors of the maps .) Suppose

where is the Hausdorff metric. Then

where A is the attractor of the IFS. Equivalently,

, for all nonempty, compact subsets L of .

Informally, If is close to being stabilized by the IFS, then is also close to being the attractor of the IFS.

See also[edit]

References[edit]

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

External links[edit]