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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.^{[citation needed]} Fractal algorithms convert these parts into mathematical data called "fractal codes" which are used to recreate the encoded image.
Fractal image representation can 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 . An IFS is a set of contraction mappings ƒ_{1},...,ƒ_{N},
According to these mapping functions, the IFS describes a twodimensional 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 A_{0}, the iteration A_{k+1} = H(A_{k}) converges to S.
The set S is selfsimilar 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.
IFS representation can be extended to a grayscale image by considering the image's graph as a subset of . For a grayscale image u(x,y), consider the set S = {(x,y,u(x,y))}. Then similar to the binary case, S is described by an IFS using a set of contraction mappings ƒ_{1},...,ƒ_{N}, but in ,
A challenging problem of ongoing research in fractal image representation is how to choose the ƒ_{1},...,ƒ_{N} such that its fixed point approximates the input image, and how to do this efficiently. A simple approach^{[1]} for doing so is the following:
In the second step, it is important to find a similar block so that the IFS accurately represents the input image, so a sufficient number of candidate blocks for D_{i} need to be considered. On the other hand, a large search considering many blocks is computationally costly. This bottleneck of searching for similar blocks is why fractal encoding is much slower than for example DCT and wavelet based image representations.
With fractal compression, encoding is extremely computationally expensive because of the search used to find the selfsimilarities. Decoding, however is quite fast. While this asymmetry has so far made it impractical for real time applications, when video is archived for distribution from disk storage or file downloads fractal compression becomes more competitive.^{[2]}^{[3]}
At common compression ratios, up to about 50:1, Fractal compression provides similar results to DCTbased algorithms such as JPEG. ^{[4]} At high compression ratios fractal compression may offer superior quality. For satellite imagery, ratios of over 170:1^{[5]} have been achieved with acceptable results. Fractal video compression ratios of 25:1244:1 have been achieved in reasonable compression times (2.4 to 66 sec/frame).^{[6]}
Compression efficiency increases with higher image complexity and color depth, compared to simple grayscale images.
An inherent feature of fractal compression is that images become resolution independent^{[7]} after being converted to fractal code. This is because the iterated function systems in the compressed file scale indefinitely. This indefinite scaling property of a fractal is known as "fractal scaling".
The resolution independence of a fractalencoded image can be used to increase the display resolution of an image. This process is also known as "fractal interpolation". In fractal interpolation, an image is encoded into fractal codes via fractal compression, and subsequently decompressed at a higher resolution. The result is an upsampled image in which iterated function systems have been used as the interpolant.^{[8]} Fractal interpolation maintains geometric detail very well compared to traditional interpolation methods like bilinear interpolation and bicubic interpolation.^{[9]}^{[10]}^{[11]}
Michael Barnsley led development of fractal compression in 1987, and was granted several patents on the technology.^{[12]} The most widely known practical fractal compression algorithm was invented by Barnsley and Alan Sloan. Barnsley's graduate student Arnaud Jacquin implemented the first automatic algorithm in software in 1992.^{[13]}^{[14]} All methods are based on the fractal transform using iterated function systems. Michael Barnsley and Alan Sloan formed Iterated Systems Inc.^{[15]} in 1987 which was granted over 20 additional patents related to fractal compression.
A major breakthrough for Iterated Systems Inc. was the automatic fractal transform process which eliminated the need for human intervention during compression as was the case in early experimentation with fractal compression technology. In 1992 Iterated Systems Inc. received a $2.1 million government grant^{[16]} to develop a prototype digital image storage and decompression chip using fractal transform image compression technology.
Fractal image compression has been used in a number of commercial applications: onOne Software, developed under license from Iterated Systems Inc., Genuine Fractals 5^{[17]} which is a Photoshop plugin capable of saving files in compressed FIF (Fractal Image Format). To date the most successful use of still fractal image compression is by Microsoft in its Encarta multimedia encyclopedia,^{[18]} also under license.
Iterated Systems Inc. supplied a shareware encoder (Fractal Imager), a stand alone decoder, a Netscape plugin decoder and a development package for use under Windows. As waveletbased methods of image compression improved and were more easily licensed by commercial software vendors the adoption of the Fractal Image Format failed to evolve.^{[citation needed]} The redistribution of the "decompressor DLL" provided by the ColorBox III SDK was governed by restrictive perdisk or yearbyyear licensing regimes for proprietary software vendors and by a discretionary scheme that entailed the promotion of the Iterated Systems products for certain classes of other users.^{[19]}
During the 1990s Iterated Systems Inc. and its partners expended considerable resources to bring fractal compression to video. While compression results were promising, computer hardware of that time lacked the processing power for fractal video compression to be practical beyond a few select usages. Up to 15 hours were required to compress a single minute of video.
ClearVideo — also known as RealVideo (Fractal) — and SoftVideo were early fractal video compression products. ClearFusion was Iterated's freely distributed streaming video plugin for web browsers. In 1994 SoftVideo was licensed to Spectrum Holobyte for use in its CDROM games including Falcon Gold and Star Trek: The Next Generation A Final Unity.^{[20]}
In 1996 Iterated Systems Inc. announced^{[21]} an alliance with the Mitsubishi Corporation to market ClearVideo to their Japanese customers. The original ClearVideo 1.2 decoder driver is still supported^{[22]} by Microsoft in Windows Media Player although the encoder is no longer supported.
Numerous research papers have been published during the past few years discussing possible solutions to improve fractal algorithms and encoding hardware.^{[23]}^{[24]}^{[25]}^{[26]}^{[27]}^{[28]}^{[29]}^{[30]}^{[31]}
A library called Fiasco was created by Ullrich Hafner and described in Linux Journal.^{[32]}
The Netpbm library includes a Fiasco library. ^{[33]} ^{[34]}
There is a video library for fractal compression. ^{[35]}
There is another example implementation from Femtosoft. ^{[36]}
