Editing Fractal compression (section)

===For binary images===
We begin with the representation of a [[binary image]], where the image may be thought of as a subset of <math>\mathbb{R}^2</math>. An IFS is a set of [[contraction mapping]]s ''ƒ''<sub>1</sub>,...,''ƒ<sub>N</sub>'',

:<math>f_i:\mathbb{R}^2\to \mathbb{R}^2.</math>

According to these mapping functions, the IFS describes a two-dimensional set ''S'' as the fixed point of the [[Hutchinson operator]]

:<math>H(A)=\bigcup_{i=1}^N f_i(A), \quad A \subset \mathbb{R}^2.</math>

That is, ''H'' is an operator mapping sets to sets, and ''S'' is the unique set satisfying ''H''(''S'') = ''S''. The idea is to construct the IFS such that this set ''S'' is the input binary image. The set ''S'' can be recovered from the IFS by [[fixed point iteration]]: for any nonempty [[compact space|compact]] initial set ''A''<sub>0</sub>, the iteration ''A''<sub>''k''+1</sub>&nbsp;= ''H''(''A<sub>k</sub>'') converges to ''S''.

The set ''S'' is self-similar because ''H''(''S'') = ''S'' implies that ''S'' is a union of mapped copies of itself:
:<math>S=f_1(S)\cup f_2(S) \cup\cdots\cup f_N(S)</math>
So we see the IFS is a fractal representation of ''S''.