Fractal compression

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Fractal compression is a lossy image compression method using fractals. The method is best suited for textures and natural images, relying on the fact that parts of an image often resemble other parts of the same image.[citation needed] Fractal algorithms convert these parts into mathematical data called "fractal codes" which are used to recreate the encoded image. Fractal compression differs from pixel-based compression schemes such as JPEG, GIF and MPEG since no pixels are saved. Once an image has been converted into fractal code, the image can be recreated to fill any screen size without the loss of sharpness that occurs in conventional compression schemes.


Iterated Function Systems

Fractal image representation can be described mathematically as an iterated function system (IFS).

For Binary Images

We begin with the representation of a binary image, where the image may be thought of as a subset of \mathbb{R}^2. An IFS is a set of contraction mappings ƒ1,...,ƒN,

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

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 initial set A0, the iteration Ak+1 = H(Ak) converges to S.

The set S is self-similar because H(S) = S implies that S is a union of mapped copies of itself:

So we see the IFS is a fractal representation of S.

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