# Fractal compression

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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.

## Iterated function systems

Fractal image representation may be described mathematically as an iterated function system (IFS).[1]

### For binary images

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,...,ƒN,

${\displaystyle f_{i}:\mathbb {R} ^{2}\to \mathbb {R} ^{2}.}$

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

${\displaystyle H(A)=\bigcup _{i=1}^{N}f_{i}(A),\quad A\subset \mathbb {R} ^{2}.}$

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:

${\displaystyle S=f_{1}(S)\cup f_{2}(S)\cup \cdots \cup f_{N}(S)}$

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

### Extension to grayscale

IFS representation can be extended to a grayscale image by considering the image's graph as a subset of ${\displaystyle \mathbb {R} ^{3}}$. 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 ƒ1,...,ƒN, but in ${\displaystyle \mathbb {R} ^{3}}$,

${\displaystyle f_{i}:\mathbb {R} ^{3}\to \mathbb {R} ^{3}.}$

### Encoding

A challenging problem of ongoing research in fractal image representation is how to choose the ƒ1,...,ƒN 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 partitioned iterated function system (PIFS):

1. Partition the image domain into range blocks Ri of size s×s.
2. For each Ri, search the image to find a block Di of size 2s×2s that is very similar to Ri.
3. Select the mapping functions such that H(Di) = Ri for each i.

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 Di 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 PIFS fractal encoding is much slower than for example DCT and wavelet based image representation.

The initial square partitioning and brute-force search algorithm presented by Jacquin provides a starting point for further research and extensions in many possible directions -- different ways of partitioning the image into range blocks of various sizes and shapes; fast techniques for quickly finding a close-enough matching domain block for each range block rather than brute-force searching, such as fast motion estimation algorithms; different ways of encoding the mapping from the domain block to the range block; etc.[2]

Other researchers attempt to find algorithms to automatically encode an arbitrary image as RIFS (recurrent iterated function systems) or global IFS, rather than PIFS; and algorithms for fractal video compression including motion compensation and three dimensional iterated function systems.[3][4]

Fractal image compression has many similarities to vector quantization image compression.[5]

## Features

With fractal compression, encoding is extremely computationally expensive because of the search used to find the self-similarities. 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.[6][7]

At common compression ratios, up to about 50:1, Fractal compression provides similar results to DCT-based algorithms such as JPEG.[8] At high compression ratios fractal compression may offer superior quality. For satellite imagery, ratios of over 170:1[9] have been achieved with acceptable results. Fractal video compression ratios of 25:1–244:1 have been achieved in reasonable compression times (2.4 to 66 sec/frame).[10]

Compression efficiency increases with higher image complexity and color depth, compared to simple grayscale images.

### Resolution independence and fractal scaling

An inherent feature of fractal compression is that images become resolution independent[11] 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".

### Fractal interpolation

The resolution independence of a fractal-encoded 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 up-sampled image in which iterated function systems have been used as the interpolant.[12] Fractal interpolation maintains geometric detail very well compared to traditional interpolation methods like bilinear interpolation and bicubic interpolation.[13][14][15] Since the interpolation cannot reverse Shannon entropy however, it ends up sharpening the image by adding random instead of meaningful detail. One cannot, for example, enlarge an image of a crowd where each person's face is one or two pixels and hope to identify them.

## History

Michael Barnsley led development of fractal compression in 1987, and was granted several patents on the technology.[16] 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.[17][18] All methods are based on the fractal transform using iterated function systems. Michael Barnsley and Alan Sloan formed Iterated Systems Inc.[19] 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 US\$2.1 million government grant[20] 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[21] 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,[22] also under license.

Iterated Systems Inc. supplied a shareware encoder (Fractal Imager), a stand-alone decoder, a Netscape plug-in decoder and a development package for use under Windows. As wavelet-based 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 per-disk or year-by-year 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.[23]

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 CD-ROM games including Falcon Gold and Star Trek: The Next Generation A Final Unity.[24]

In 1996, Iterated Systems Inc. announced[25] an alliance with the Mitsubishi Corporation to market ClearVideo to their Japanese customers. The original ClearVideo 1.2 decoder driver is still supported[26] by Microsoft in Windows Media Player although the encoder is no longer supported.

Two firms, Total Multimedia Inc. and Dimension, both claim to own or have the exclusive licence to Iterated's video technology, but neither has yet released a working product. The technology basis appears to be Dimension's U.S. patents 8639053 and 8351509, which have been considerably analyzed.[27] In summary, it is a simple quadtree block-copying system with neither the bandwidth efficiency nor PSNR quality of traditional DCT-based codecs. In January 2016, TMMI announced that it was abandoning fractal-based technology altogether.

Numerous research papers have been published during the past few years discussing possible solutions to improve fractal algorithms and encoding hardware.[28][29][30][31][32][33][34][35][36]

## Open Source

A library called Fiasco was created by Ullrich Hafner and described in Linux Journal.[37]

The Netpbm library includes a Fiasco library.[38][39]

There is a video library for fractal compression.[40]

There is another example implementation from Femtosoft.[41]

## Notes

1. ^ a b Fischer, Yuval (1992-08-12). Przemyslaw Prusinkiewicz, ed. SIGGRAPH'92 course notes - Fractal Image Compression (PDF). SIGGRAPH. Fractals - From Folk Art to Hyperreality. ACM SIGGRAPH.
2. ^ Dietmar Saupe, Raouf Hamzaoui. "A Review of the Fractal Image Compression Literature". 1994. doi: 10.1145/193234.193246
3. ^ Bruno Lacroix. "Fractal Image Compression". 1998.
4. ^ Yuval Fisher. "Fractal Image Compression: Theory and Application". 2012. p. 300
5. ^ Henry Xiao. "Fractal Compression". 2004.
6. ^ John R. Jensen, "Remote Sensing Textbooks", Image Compression Alternatives and Media Storage Considerations (reference to compression/decompression time), University of South Carolina, archived from the original on 2008-03-03
7. ^ Steve Heath (23 August 1999). Multimedia and communications technology. Focal Press. pp. 120–123. ISBN 978-0-240-51529-8. Focal Press link
8. ^ Sayood, Khalid (2006). Introduction to Data Compression, Third Edition. Morgan Kaufmann Publishers. pp. 560–569. ISBN 978-0-12-620862-7.
9. ^ Wee Meng Woon; Anthony Tung Shuen Ho; Tao Yu; Siu Chung Tam; Siong Chai Tan; Lian Teck Yap, "Achieving high data compression of self-similar satellite images using fractal" (PDF), Geoscience and Remote Sensing Symposium paper, IGARSS 2000, Achieving high data compression of self-similar satellite images using fractal
10. ^ Fractal encoding of video sequences
11. ^ Walking, Talking Web Archived 2008-01-06 at the Wayback Machine. Byte Magazine article on fractal compression/resolution independence
12. ^ Interpolation decoding method with variable parameters for fractal image compression College of Mathematics and Physics, Chongqing University, China
13. ^ Smooth fractal interpolation Departamento de Matemáticas, Universidad de Zaragoza, Campus Plaza de San Francisco, Zaragoza, Spain
14. ^ A Note on Expansion Technique for Self-Affine Fractal Objects Using Extended Fractal Interpolation Functions Archived 2011-01-01 at the Wayback Machine. Hokkaido Univ., Graduate School of Engineering, JPN
15. ^ Studies on Scaling Factor for Fractal Image Coding Archived 2008-01-27 at the Wayback Machine. Nagasaki University, Faculty of Engineering
16. ^ U.S. Patent 4,941,193 – Barnsley and Sloan's first iterated function system patent, filed in October 1987
17. ^
18. ^ Arnaud E. Jacquin. Image Coding Based on a Fractal Theory of Iterated Contractive Image Transformations. IEEE Transactions on Image Processing, 1(1), 1992.
19. ^ Iterated Systems Inc. changed its name to MediaBin Inc. Inc. Archived 2008-01-31 at the Wayback Machine. in 2001 and in turn was bought out by Interwoven, Inc. in 2003)
20. ^
21. ^
22. ^ Mathematics Awareness Week - April 1998 reference to Microsoft's Encarta fractal image compression
23. ^ Aitken, William (May 1994). "The big squeeze". Personal Computer World.
25. ^ Mitsubishi Corporation ClearVideo press release
26. ^ Microsoft ClearVideo support
27. ^ Due Diligence Study of Fractal Video Technology
28. ^ Advances in fractal compression for multimedia applications
29. ^ Fast calculation of IFS parameters for fractal image coding
30. ^ Fractal image compression performance synthesis through HV partitioning
31. ^ Simple and Fast Fractal Image Compression Circuits, Signals, and Systems - 2003
32. ^ Schema genetic algorithm for fractal image compression Department of Electrical Engineering, National Sun Yet-Sen University, Kaohsiung, Taiwan
33. ^ A fast fractal image encoding method based on intelligent search of standard deviation Department of Electrical and Computer Engineering, The University of Alabama
34. ^ Novel fractal image-encoding algorithm based on a full-binary-tree searchless iterated function system Department of Electrical and Computer Engineering, The University of Alabama
35. ^ Fast classification method for fractal image compression Proc. SPIE Vol. 4122, p. 190-193, Mathematics and Applications of Data/Image Coding, Compression, and Encryption III, Mark S. Schmalz; Ed
36. ^ Toward Real Time Fractal Image Compression Using Graphics Hardware Dipartimento di Informatica e Applicazioni, Università degli Studi di Salerno
37. ^ Hafner, Ullrich (2001). "FIASCO - An Open-Source Fractal Image and Sequence Codec". Linux Journal. linuxjournal.com (81). Retrieved February 19, 2013.
38. ^ http://netpbm.sourceforge.net/doc/pnmtofiasco.html
39. ^ http://netpbm.sourceforge.net/doc/fiascotopnm.html
40. ^ http://castor.am.gdynia.pl/cgi-bin/man/man2html?3+fiasco_decoder_get_frame
41. ^ "Archived copy". Archived from the original on 2010-10-23. Retrieved 2010-07-10.