Fractal image compression

Fractal compression is a lossy compression method for digital images, based on 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. Fractal algorithms convert these parts into mathematical data called "fractal codes" which are used to recreate the encoded image.

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

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

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 A_k+1 = H(A_k) converges to S.

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

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