You may want to read first item 77 in part 2 of this FAQ:
"Introduction to Fractal compression".


from Tal Kubo <kubo@zariski.harvard.edu>:

According to Barnsley's book 'Fractals Everywhere', this method is
based on a measure of deviation between a given image and its
approximation by an IFS code.  The Collage Theorem states that there is
a convergent process to minimize this deviation.  Unfortunately,
according to an article Barnsley wrote for BYTE a few years ago, this
convergence was rather slow, about 100 hours on a Cray, unless assisted by
a person.

Barnsley et al are not divulging any technical information beyond the
meager bit in 'Fractals Everywhere'.  The book explains the idea of IFS
codes at length, but is vague about the application of the Collage theorem
to specific compression problems.

There is reason to believe that Barnsley's company has
*no algorithm* which takes a given reasonable image and achieves
the compression ratios initially claimed for their fractal methods.
The 1000-to-1 compression advertised was achieved only for a 'rigged'
class of images, with human assistance. The best unaided
performance I've heard of is good lossy compression of about 80-1.

Steve Tate <srt@duke.cs.duke.edu> confirms:

Compression ratios (unzoomed) seem to range from 20:1 to 60:1...  The
quality is considerably worse than wavelets or JPEG on most of the
non-contrived images I have seen.

But Yuval Fisher <fisher@inls1.ucsd.edu> disagrees:

Their performance has improved dramatically beyond what they were
talking about in BYTE a few years ago.  Human assistance to the
compression is no longer needed and the compression time is
reasonable, although the more time and compute power you throw at the
compression, the smaller the resulting file for the same level of
quality.

Geoffrey A Stephenson <ketlux@ketlux.demon.co.uk> adds:

Iterated systems are shipping a general purpose compressor at about
300 Pounds in the UK that claims "640x480 24 bit colour compression of
about 1 min at 922k -> 10k on a 486/50 software only, decomp. to 8
bits in 3 secs, etc." At a recent multimedia conference in London they
handed out free demo disks that show the decomp. in action. The
package runs under both DOS anf WIN (DLLs provided for use in
applications). They also sell a board to speed up compression and
offer versions supporting full motion video (but not apparently at all
SVGA sizes like the static picture version). I have not yet got my
hands on a full version to test different types of pictures, but
friends have a and claim it looks good.


Thomas W. Colthurst <thomasc@athena.mit.edu> clarifies the distinction
between IFS and the Fractal Transform:

It is time, once and for all, to put to death the Barnsley myth that
IFSs are good for image compression.  They are not.  Various algorithms
have been proposed for this "inverse problem" ranging from the trendy
(genetic algorithms) to the deep (moment methods) to the ad hoc (the
hungry algorithm) to the absurd (the so-called "graduate student
algorithm", consisting of locking up a grad student in a tiny office
with a SGI workstation and not letting them out until they come up
with a good IFS for your image).  They are all useless for practical
image compression.

In fact, there are even good theoretical reasons for believing that
IFSs will never be useful for image compression.  For example, even
if you have an IFS for object A and an IFS for object B, there is no
way to combine these IFSs to get an IFS for object A union B or
object A intersect B.

Even Barnsley himself admits, in his latest book, that he doesn't use
IFS image compression.  Instead, he uses the so-called "fractal
transform," which is really just a variant of vector quantization
where you use the image itself, sampled at a higher scale, as the
VQ codebook.  To be fair, the fractal transform can be analyzed using
local IFSs, but local IFSs are immensely more complicated and general
than normal IFSs, to the point where one feels suspect even using the
word "IFS" to describe them.

It should be emphasized that the fractal transform is a real, working
method that performs about as well as other existing methods like VQ
or the discrete cosine transform. The fractal transform will probably
never beat vector quantization (VQ) as for size of the compressed
image, but does have the advantage that you don't need to carry your
codebook around.  The latest results have it slightly winning over
the discrete cosine transform; only time and more research will tell
if this advantage persists.  Just like VQ, the fractal transform
takes a while to compress, but is quick at decompression (Barnsley's
company has hardware to do this in realtime).

In short, IFSs are good for just about everything fractals are (and
more!), but are absolutely horrid for image compression.


Programs:

Check http://links.uwaterloo.ca/ for pointers to some fractal compression
programs and lots of papers on fractal compression.

The Waterloo BragZone (http://links.uwaterloo.ca/bragzone.base.html
or ftp://links.uwaterloo.ca/pub/BragZone/ ) compares the results of
various image compression schemes against a 32 element test suite.
Numerous rate-distortion graphs, data tables, and sample images are available.

A fractal image compression program is available by ftp in
ftp://inls.ucsd.edu/pub/young-fractal/ ; it contains source for
compression and decompression, source for X-windows decompression,
MSDOS executables and images. [Note from FAQ maintainer: Fisher's
program (see below) implements the same algorithm but is more general;
see http://inls.ucsd.edu/y/Fractals/ for the source code.]

A fractal image decompression program (note: decompression only) is
available in ftp://inls.ucsd.edu/pub/inls-ucsd/
In the same directory, fractal_paper.ps.Z is the paper "Fractal image
compression" by Yuval Fisher, Siggraph 92.  Reading this paper is
required to understand how the Young compression programs (see above) works.

A note from Yuval Fisher <yfisher@ucsd.edu>:

    Connect to http://inls.ucsd.edu/y/Fractals/ .  There is
    information there on my new book of contributed articles on
    fractal image compression, as well as the book's table of
    contents, some C code to encode and decode raw byte files of any
    size using a quadtree method, a manual explaining the use of the
    code, a fractal image compression bibliography (not guaranteed to
    be complete or close to it), some better executable code with
    sample encodings, and the SIGGRAPH '92 course notes on fractal
    image compression (these are based on appendix A of Chaos and
    Fractals by Peitgen et al., Springer Verlag). [The C code is also
    available in ftp://inls.ucsd.edu/pub/inls-ucsd/ ]

Another fractal compression program is available by ftp in
ftp://vision.auc.dk/pub/Limbo/. It is also based on quadtrees,
as yuvpak20 and frac_comp.

The source code for the program published in the Oct 93 issue of
Byte is in ftp://ftp.uu.net/published/byte/93oct/fractal.exe. This is
a self-extractible arc file (must be run on MSDOS for extraction).
The source code is for a TARGA video board. [Note from FAQ maintainer:
this code is taken from Barnsley's book "Fractal Image Compression";
it implements the brute force method and is thus very slow.]

Iterated Systems have released a beta version of their fractal imager.
It will let you view a number of formats including JPG and do
conversions to their fractal format.  The program can be downloaded
from http://www.iterated.com

"The Data Compression Book" (see [NEL 1996] in item 7 above) contains
a chapter on fractal compression; it includes source code for a simple
fractal compression program. The source is also available at
http://www.teaser.fr/~jlgailly/frac-1.0.tar.gz

Several fractal compression programs, including a volume coder, are available
at http://www.eecs.wsu.edu/~wcochran/

Several papers on fractal image compression are available on
ftp.informatik.uni-freiburg.de in directory /documents/papers/fractal . A
biliography is in ftp://schmance.uwaterloo.ca/pub/Fractal/

References:
  A. Jacquin, 'Fractal image coding based on a theory of iterated
    contractive image transformations', Proc. SPIE Visual Communications
    and Image Processing, 1990, pages 227-239.  (The best paper that explains
    the concept in a simple way.)

  A. Jacquin, "A Fractal Theory of Iterated Markov Operators with
    Applications to Digital Image Coding", PhD Thesis, Georgia Tech, 1989.
  It can be obtained from university microfilms for $35, phone 1-800-521-0600.

  M. Barnsley, L. Anson, "Graphics Compression Technology, SunWorld,
    October 1991, pp. 42-52.
  M.F. Barnsley, A. Jacquin, F. Malassenet, L. Reuter & A.D. Sloan,
    'Harnessing chaos for image synthesis', Computer Graphics,
    vol 22 no 4 pp 131-140, 1988.
  M.F. Barnsley, A.E. Jacquin, 'Application of recurrent iterated
    function systems to images', Visual Comm. and Image Processing,
    vol SPIE-1001, 1988.
  A. Jacquin, "Image Coding Based on a Fractal Theory of Iterated Contractive
    Image Transformations" p.18, January 1992 (Vol 1 Issue 1) of IEEE Trans
    on Image Processing.
  A.E. Jacquin, 'A novel fractal block-coding technique for digital
    images', Proc. ICASSP 1990.
  G.E. Oien, S. Lepsoy & T.A. Ramstad, 'An inner product space
    approach to image coding by contractive transformations',
    Proc. ICASSP 1991, pp 2773-2776.
  D.S. Mazel, Fractal Modeling of Time-Series Data, PhD Thesis,
    Georgia Tech, 1991.    (One dimensional, not pictures)
  S. A. Hollatz, "Digital image compression with two-dimensional affine
    fractal interpolation functions", Department of Mathematics and
    Statistics, University of Minnesota-Duluth, Technical Report 91-2.
    (a nuts-and-bolts how-to-do-it paper on the technique)
  Stark, J., "Iterated function systems as neural networks",
    Neural Networks, Vol 4, pp 679-690, Pergamon Press, 1991.
  Monro D M and Dudbridge F, "Fractal block coding of images",
    Electronics Letters 28(11):1053-1054 (1992)
  Beaumont J M, "Image data compression using fractal techniques",
    British Telecom Technological Journal 9(4):93-108 (1991)
  Fisher Y, "Fractal image compression", Siggraph 92
  Graf S, "Barnsley's Scheme for the Fractal Encoding of Images",
    Journal Of Complexity, V8, 72-78 (1992).
  Monro D.M. 'A hybrid fractal transform', Proc ICASSP 93, pp. V: 169-72
  Monro D.M. & Dudbridge F. 'Fractal approximation of image blocks',
    Proc ICASSP 92, pp. III: 485-488
  Monro D.M., Wilson D., Nicholls J.A. 'High speed image coding with the Bath
    Fractal Transform', IEEE International Symposium on Multimedia Technologies
    Southampton, April 1993
  Jacobs, E.W., Y. Fisher and R.D. Boss.  "Image Compression:  A study
    of the Iterated Transform Method."  _Signal Processing 29_  (1992) 25-263
  Vrscay, Edward R.  "Iterated Function Systems:  Theory, Applications,
    and the Inverse Problem."  _Fractal Geometry and Analysis_,
    J. Belair and S. Dubuc (eds.)  Kluwer Academic, 1991.  405-468.

Books:
    Fractal Image Compression: Theory and Application, Yuval Fisher (ed.),
    Springer Verlag, New York, 1995.
    To order the book, call 1-800-SPRINGER and ask for the book with
    ISBN number 0-387-94211-4 or check http://www.springer-ny.com/

    Fractal Image Compression
    Michael F. Barnsley and Lyman P. Hurd
    ISBN 0-86720-457-5, ca. 250 pp., $49.95
    Copies can be ordered directly from the publisher by sending a message
    to kpeters@math.harvard.edu with name, address and a Mastercard or
    Visa card number with expiration date.

Barnsley's company is:

Iterated Systems, Inc.
5550A Peachtree Parkway, Suite 650
Norcross, GA  30092
tel: 404-840-0310 or 1-800-4FRACTL
fax: 404-840-0806
In UK: Phone (0734) 880261, Fax (0734) 880360