2 triangles, example to show how fractal compression works

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.[1] Fractal algorithms convert these parts into mathematical data called "fractal codes" which are used to recreate the encoded image.

## Iterated function systems

 Main article: Iterated function system

Fractal image representation may be described mathematically as an iterated function system (IFS).[2]

### 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[2] 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.[3]

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.[4][5]

Fractal image compression has many similarities to vector quantization image compression.[6]

## 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.[7][8]

At common compression ratios, up to about 50:1, fractal compression provides similar results to DCT-based algorithms such as JPEG.[9] At high compression ratios fractal compression may offer superior quality. For satellite imagery, ratios of over 170:1[10] 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).[11]

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[12] 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.[13] Fractal interpolation maintains geometric detail very well compared to traditional interpolation methods like bilinear interpolation and bicubic interpolation.[14][15][16] 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.[17] 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.[18][19] All methods are based on the fractal transform using iterated function systems. Michael Barnsley and Alan Sloan formed Iterated Systems Inc.[20] 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[21] 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[22] 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,[23] 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.[24]

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.[25]

In 1996, Iterated Systems Inc. announced[26] an alliance with the Mitsubishi Corporation to market ClearVideo to their Japanese customers. The original ClearVideo 1.2 decoder driver is still supported[27] 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.[28] 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.

Research papers between 1997 and 2007 discussed possible solutions to improve fractal algorithms and encoding hardware.[29][30][31][32][33][34][35][36][37]

## Implementations

A library called Fiasco was created by Ullrich Hafner. In 2001, Fiasco was covered in the Linux Journal. [38] According to the 2000-04 Fiasco manual, Fiasco can be used for video compression. [39] The Netpbm library includes the Fiasco library. [40][41]

Femtosoft developed an implementation of fractal image compression in Object Pascal and Java. [42]

## Notes

1. ^ May, Mike (1996). "Fractal Image Compression". American Scientist. 84 (5): 442–451. Bibcode:1996AmSci..84..442M. JSTOR 29775747. ProQuest 215266230.
2. ^ a b Fischer, Yuval (1992-08-12). Przemyslaw Prusinkiewicz (ed.). SIGGRAPH'92 course notes - Fractal Image Compression (PDF). SIGGRAPH. Vol. Fractals - From Folk Art to Hyperreality. ACM SIGGRAPH.
3. ^ Saupe, Dietmar; Hamzaoui, Raouf (November 1994). "A review of the fractal image compression literature". ACM SIGGRAPH Computer Graphics. 28 (4): 268–276. doi:10.1145/193234.193246. S2CID 10489445.
4. ^ Lacroix, Bruno (1999). Fractal image compression (Thesis). doi:10.22215/etd/1999-04159. OCLC 1103597126. ProQuest 304520711.
5. ^ Fisher, Yuval (2012). Fractal Image Compression: Theory and Application. Springer Science & Business Media. p. 300. ISBN 978-1-4612-2472-3.
6. ^ Henry Xiao. "Fractal Compression". 2004.
7. ^ 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
8. ^ Steve Heath (23 August 1999). Multimedia and communications technology. Focal Press. pp. 120–123. ISBN 978-0-240-51529-8. Focal Press link
9. ^ Sayood, Khalid (2006). Introduction to Data Compression. Elsevier. pp. 560–569. ISBN 978-0-12-620862-7.
10. ^ Wee Meng Woon; Anthony Tung Shuen Ho; Tao Yu; Siu Chung Tam; Siong Chai Tan; Lian Teck Yap (2000). "Achieving high data compression of self-similar satellite images using fractal". IGARSS 2000. IEEE 2000 International Geoscience and Remote Sensing Symposium. Taking the Pulse of the Planet: The Role of Remote Sensing in Managing the Environment. Proceedings (Cat. No.00CH37120). Vol. 2. pp. 609–611. doi:10.1109/IGARSS.2000.861646. ISBN 0-7803-6359-0. S2CID 14516581.
11. ^ Fisher, Y. (July 1995). Fractal encoding of video sequences. Fractal image encoding and analysis. Trondheim. INIST:1572685.
12. ^ Walking, Talking Web Archived 2008-01-06 at the Wayback Machine Byte Magazine article on fractal compression/resolution independence
13. ^ He, Chuan-jiang; Li, Gao-ping; Shen, Xiao-na (May 2007). "Interpolation decoding method with variable parameters for fractal image compression". Chaos, Solitons & Fractals. 32 (4): 1429–1439. Bibcode:2007CSF....32.1429H. doi:10.1016/j.chaos.2005.11.058.
14. ^ Navascués, M. A.; Sebastián, M. V. (2006). "Smooth fractal interpolation". Journal of Inequalities and Applications. 2006: 1–20. doi:10.1155/JIA/2006/78734. S2CID 20352406.
15. ^ Uemura, Satoshi; Haseyama, Miki; Kitajima, Hideo (28 January 2003). "EFIFを用いた自己アフィンフラクタル図形の拡大処理に関する考察" [A Note on Expansion Technique for Self-Affine Fractal Objects Using Extended Fractal Interpolation Functions]. IEICE Technical Report (in Japanese). 102 (630): 95–100. doi:10.11485/itetr.27.9.0_95. NAID 110003171506.
16. ^ Kuroda, Hideo; Hu, Xiaotong; Fujimura, Makoto (1 February 2003). "フラクタル画像符号化におけるスケーリングファクタに関する考察" [Studies on Scaling Factor for Fractal Image Coding]. The Transactions of the Institute of Electronics, Information and Communication Engineers (in Japanese). 86 (2): 359–363. NAID 110003170896.
17. ^ U.S. Patent 4,941,193 – Barnsley and Sloan's first iterated function system patent, filed in October 1987
18. ^
19. ^ Jacquin, A.E. (1992). "Image coding based on a fractal theory of iterated contractive image transformations". IEEE Transactions on Image Processing. 1 (1): 18–30. Bibcode:1992ITIP....1...18J. CiteSeerX 10.1.1.456.1530. doi:10.1109/83.128028. PMID 18296137.
20. ^ Iterated Systems Inc. changed its name to MediaBin Inc. Inc. in 2001 and in turn was bought out by Interwoven, Inc. in 2003)
21. ^
22. ^ Genuine Fractals Product Review
23. ^ "MAW 1998: Theme Essay". www.mathaware.org. Archived from the original on 31 August 2016. Retrieved 18 April 2018.
24. ^ Aitken, William (May 1994). "The big squeeze". Personal Computer World.
26. ^ "Mitsubishi Corporation Inks Agreement With Iterated Systems" (Press release). Iterated Systems. 29 October 1996. Archived from the original on 8 July 2012.
27. ^ "Windows Media Player for Windows XP Supported Codecs". 31 October 2003. Archived from the original on 28 October 2004.
28. ^ "April - 2014 - Due Diligence Study of Fractal Video Technology". paulschlessinger.wordpress.com. Retrieved 18 April 2018.
29. ^ Kominek, John (1 June 1997). "Advances in fractal compression for multimedia applications". Multimedia Systems. 5 (4): 255–270. CiteSeerX 10.1.1.47.3709. doi:10.1007/s005300050059. S2CID 6016583.
30. ^ Harada, Masaki; Kimoto, Tadahiko; Fujii, Toshiaki; Tanimoto, Masayuki (2000). "Fast calculation of IFS parameters for fractal image coding". In Ngan, King N; Sikora, Thomas; Sun, Ming-Ting (eds.). Visual Communications and Image Processing 2000. Vol. 4067. p. 457–464. Bibcode:2000SPIE.4067..457H. doi:10.1117/12.386580. S2CID 30148845. INIST:1380599.
31. ^ Rajkumar, Wathap Sapankumar; Kulkarni, M.V.; Dhore, M.L.; Mali, S.N. (2006). "Fractal image compression performance synthesis through HV partitioning". 2006 International Conference on Advanced Computing and Communications. pp. 636–637. doi:10.1109/ADCOM.2006.4289976. ISBN 978-1-4244-0715-6. S2CID 15370862.
32. ^ Simple and Fast Fractal Image Compression Circuits, Signals, and Systems - 2003
33. ^ Wu, Ming-Sheng; Jeng, Jyh-Horng; Hsieh, Jer-Guang (June 2007). "Schema genetic algorithm for fractal image compression". Engineering Applications of Artificial Intelligence. 20 (4): 531–538. doi:10.1016/j.engappai.2006.08.005.
34. ^ Wu, Xianwei; Jackson, David Jeff; Chen, Hui-Chuan (September 2005). "A fast fractal image encoding method based on intelligent search of standard deviation". Computers & Electrical Engineering. 31 (6): 402–421. doi:10.1016/j.compeleceng.2005.02.003.
35. ^ Wu, Xianwei; Jackson, David Jeff; Chen, Hui-Chuan (2005). "Novel fractal image-encoding algorithm based on a full-binary-tree searchless iterated function system". Optical Engineering. 44 (10): 107002. Bibcode:2005OptEn..44j7002W. doi:10.1117/1.2076828.
36. ^ Truong, Trieu-Kien; Jeng, Jyh H. (2000). "Fast classification method for fractal image compression". In Schmalz, Mark S (ed.). Mathematics and Applications of Data/Image Coding, Compression, and Encryption III. Mathematics and Applications of Data/Image Coding. Vol. 4122. pp. 190–193. Bibcode:2000SPIE.4122..190T. doi:10.1117/12.409247. S2CID 120032052.
37. ^ Erra, Ugo (2005). "Toward Real Time Fractal Image Compression Using Graphics Hardware". Advances in Visual Computing. Lecture Notes in Computer Science. Vol. 3804. pp. 723–728. doi:10.1007/11595755_92. hdl:11563/14075. ISBN 978-3-540-30750-1.
38. ^ Hafner, Ullrich (2001). "FIASCO - An Open-Source Fractal Image and Sequence Codec". Linux Journal (81). Retrieved February 19, 2013.
39. ^ "Manpage of fiasco". castor.am.gdynia.pl. Archived from the original on 9 March 2012. Retrieved 18 April 2018.
40. ^ "Pnmtofiasco User Manual". netpbm.sourceforge.net. Retrieved 18 April 2018.
41. ^ "Fiascotopnm User Manual". netpbm.sourceforge.net. Retrieved 18 April 2018.
42. ^ "Femtosoft Technologies - Fractal Imaging Software". Archived from the original on 2010-10-23. Retrieved 2010-07-10.