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Fractal compression is quite controversial, with some people claiming it doesn't work well, and others claiming it works wonderfully. The basic idea behind fractal image compression is to express the image as an iterated function system (IFS). The image can then be displayed quickly and zooming will generate infinite levels of (synthetic) fractal detail. The problem is how to efficiently generate the IFS from the image.
Barnsley, who invented fractal image compression, has a patent on fractal compression techniques (4,941,193). Barnsley's company, Iterated Systems Inc, has a line of products including a Windows viewer, compressor, magnifier program, and hardware assist board.
Fractal compression is covered in detail in the comp.compression FAQ file(See "compression-faq"). Ftp: rtfm.mit.edu:/pub/usenet/comp.compression[18.70.0.209].
Two books describing fractal image compression are:
1. M. Barnsley, _Fractals Everywhere_, Academic Press Inc., 1988. ISBN 0-12-079062-9. This is an excellent text book on fractals. This is probably the best book for learning about the math underpinning fractals. It is also a good source for new fractal types.
2. M. Barnsley and L. Hurd, _Fractal Image Compression_, Jones and Bartlett. ISBN 0-86720-457-5. This book explores the science of the fractal transform in depth. The authors begin with a foundation in information theory and present the technical background for fractal image compression. In so doing, they explain the detailed workings of the fractal transform. Algorithms are illustrated using source code in C.
An introductory paper is:
1. A. E. Jacquin, Image Coding Based on a Fractal Theory of Iterated Contractive Image Transformation, _IEEE Transactions on Image Processing_, January 1992.