Image compression

MANAGING MEGABYTES

A Better Way to
Compress Images
Mathematics is providing a novel technique for achieving
compression ratios of 10, 000 to 1-and higher

Michael F. Barnsley and Alan D.
Sloan
THE NATURAL WORLD is filled Georgia Institute of Technology is devel- geometry provides just such a
with intricate detail. Consider the oping the system, with funding provided collection of shapes. For a hint of this,
geometry on the back of your hand: the by the Defense Advanced Research Proj- glance at the pictures in The Fractal
pores, the fine lines, and the color ects Agency (DARPA) and the Georgia Geometry of Nature by Benoit
variations. A camera can capture that Tech Research Corporation (GTRC). Our Mandelbrot, who coined the term fractal
detail and, at your leisure, you can study description is necessarily simplified, but to describe objects that are very
the photo to see things you never noticed it will show you how a fractal image- "fractured" (see references for additional
before. Can personal computers be made compression scheme operates and how to books and articles). Some elementary
to carry out similar functions of image use it to create exciting images. fractal images accompany this article.
storage and analysis? If so, then image Using fractals to simulate landscapes
compression will certainly play a central Describing Natural Objects Traditional and other natural effects is not new; it has
role. computer graphics encodes images in been a primary practical application. For
The reason is that digitized images- terms of simple geometrical shapes: instance, through experimentation, you
images converted into bits for processing points, line segments, boxes, circles, and find that a certain fractal generates a pat-
by a computer-demand large amounts of so on. More advanced systems use three- tern similar to tree bark. Later, when you
computer memory. For example, a high- dimensional elements, such as spheres want to render a tree, you put the treebark
detail gray-scale aerial photograph might and cubes, and add color and shading to fractal to work.
be blown up to a 3'h-foot square and then the description. What is new is the ability to start with
resolved to 300 by 300 pixels per square Graphics systems founded on tradi- an actual image and find the fractals that
inch with 8 significant bits per pixel. tional geometry are great for creating will imitate it to any desired degree of ac-
Digitization at this level requires 130 pictures of man-made objects, such as curacy. Since our method includes a com-
megabytes of computer memory-too bricks, wheels, roads, buildings, and pact way of representing these fractals,
much for personal computers to handle. cogs. However, they don't work well at we end up with a highly compressed data
For real-world images such as the all when the problem is to encode a sun- set for reconstructing the original image.
aerial photo, current compression set, a tree, a lump of mud, or the intricate
techniques can achieve ratios of structure of a black spleenwort fern.
between 2 to 1 and 10 to 1. By these Think about using a standard graphics Overview of Fractal Compression We
methods, our photo would still require system to encode a digitized picture of a start with a digitized image. Using image-
between 65 and 13 megabytes. cloud: You'd have to tell the computer processing techniques such as color
In this article, we describe some of the address and color attribute of each separation, edge detection, spectrum
the main ideas behind a new method for point in the cloud. But that's exactly what analysis, and texture-variation analysis,
image compression using fractals. The an uncompressed digitized image is-a we break up the image into segments.
method has yielded compression ratios in long list of addresses and attributes.
excess of 10,000 to 1 (bringing our aerial (Some of the same techniques continued
To escape this difficulty, we need a
photo down to a manageable 13,000 richer library of geometrical shapes.
bytes). The color pictures in figures 1 These shapes need to be flexible and
through 5 were encoded using the new Michael F. Barnsley and Alan D.
controllable so that they can be made to Sloan are professors of mathematics at
technique; actual storage requirements conform to clouds, mosses, feathers,
for these images range from 100 to 2000 the Georgia Institute of Technology
leaves, and faces, not to mention waving (Atlanta, GA 30332) and officers of
bytes. sunflowers and glaring arctic wolves. Iterated Systems Inc. (1266 Holly Lane
A mathematics research team at the Fractal NE, Atlanta, GA 30329).

IMAGE COMPRESSION

form the basis for the automatic coloring you should be able to encode and
of black-and-white motion pictures.) A decode fascinating black-and-white
segment might be a fern, a leaf, a cloud, image segments, such as leaf skeletons,
or a fence post. A segment can also be a tree shadows, spirals, and thunderheads.
more complex collection of pixels: A sea- You should also obtain an overview of
scape, for example, may include spray, how a fully automated fractal
rock, and mist. compression system operates.
We then look up these segments in a Affine transformations can be de-
library of fractals. The library doesn't scribed as combinations of rotations,
contain literal fractals; that would require scalings, and translations of the coordi-
astronomical amounts of storage. In-
stead, our library contains relatively nate axes in n-dimensional space. An ex-
compact sets of numbers, called iterated ample in two dimensions is
Figure 1: IFS-encoded color image of W(x,y) = (y2x+'ky+1, 'lax+'hy+2),
function system (IFS) codes, that will re-
three-dimensional ferns (4 produce the corresponding fractals. Fur- which can also be written in matrix form
transformations, 100 bytes). thermore, the library's cataloging system as
is such that images that look alike are
close together: Nearby codes correspond
to nearby fractals. This makes it feasible R' [xy] = [.5.25 .55] [xy] + [2I]
to set up automated procedures for
searching the library to find fractals that .
approximate a given target image. A This transformation moves the point
mathematical result known as the Collage (0,0) to (1,2) and moves (-1,0.5) to
Theorem (more on that later) guarantees (0.625, 2). To confirm your
that we can always find a suitable IFS understanding of the idea, you should
code-and gives a method for doing so. work out where it moves the point (1,1).
Once we have looked up all the seg- We denote this transformation by W; the
ments in our library and found their IFS notation W(S) denotes the subimage of
codes, we can throw away the original Won a set of points S.
digitized image and keep the codes, Now let's see what W does to a picture
achieving our compression ratio of of a smiling face, F, lying on the x,y plane
10,000 to 1-or even higher. (see figure 6). The result is a new,
squeezed face W(F). The affine transfor-
Iterated Function Systems mation has deformed and moved the
face. Notice that the eyes in the
Figure 2: IFS-encoded color photo of We start by explaining how a set of IFS transformed face W(F) are closer
Black Forest, color set adjusted to give codes can approximate a natural image. together than they are in F. We say that
winter tones (120 transformations, 2000 IFS theory is an extension of classical the transformation W is contractive: It
bytes). geometry. It uses affine transformations, always moves points closer together.
explained below, to express relations be- Another example of a contractive af-
tween parts of an image. Using only fine transformation is shown in figure 7.
these relations, it defines and conveys
intricate pictures. With IFS theory, we This time it acts on a leaf to produce a
can describe a cloud as clearly as an new, smaller leaf.
architect can describe a house. The general form for an affine trans-
By studying the following sections, formation is

Figure 3: IFS-encoded color photo of a Figure 4: IFS-encoded color photo of Figure 5: IFS-encoded color image

Bolivian girl (120 transformations, 2000 the Monterey coast (60 transformations, from A Cloud Study (30
photographs © 1987 Georgia Tech Research Corp. Figure 1: Barnsley; Figures 2 and 3: Barnsley,
lacquin, Malassenet. Rarer, Sloan; Figures 4 and 5: Barnsley, Jarquin, Reuter, Sloan.

IMAGE COMPRESSION

W [Yx] _ Ca d ] [ x Y ] + al a+a2b+e=
0,a+a2b+e=
-y,a+_y2b+e =1 ',,
_ ax+by+e - I
cx+dy+f I ' and find c , d , and f in similar fashion
from these equations:
where the coefficients a, b, c, d , e, and f a,c+a2d + f = a2, (4)
are real numbers. 0,c+02d + f = N2, (5)
If we know in advance the y,c+ti2d + f = y 2. (6)
translations, rotations, and scalings that
combine to produce W, we can generate We recommend the use of an equation
coefficient values as follows: solver such as TK Solver Plus
(Universal Technical Systems,
a = r cos 6, b = -s sin ~, Rockford, Illinois) or Eureka (Borland Figure 6: An aff ne transformation W
c=rsin0,d=scos0, International, Scoffs Valley, California) moves the smiling face F to a new
for finding the coefficient values. Doing face W(F). The transformation is
where r is the scaling factor on x, s is the it manually can be tedious. called contractive because it moves
scaling factor on y, 9 is the angle of Now that we know what a contractive points closer together.
rotation on x, 0 is the angle of rotation continued
on y, e is the translation on x, and f i s
the translation on y.
How can you find an affine transfor-
mation that produces a desired effect?
Let's show how to find the affine trans- ((31,82)
formation that takes the big leaf to the
little leaf in figure 7. We wish to find the
numbers a, b, c, d, e, and f for which ~ - ( « 1 , C - 0
the transformation W has the property fY1,l`z) _ i- (a~,az)

W(big leaf) = little leaf.
Begin by introducing x and y coordinate
axes, as already shown in the figure.
Mark three points on the big leaf (we've
chosen the leaf tip, a side spike, and the
point where the stem joins the leaf) and
determine their coordinates (a,,a2),
(N1,N2), and (y,,72). Mark the corre-
sponding points on the little leaf and
determine their coordinates (ix,,a2),
(N2), and (j,,y2), respectively.
Determine values for the
coefficients a, b, and e by solving the Figure 7: Two leaves fix an affine transformation
three linear equations ivy W.

Table 1: IFS codes for a Sierpinski triangle. f P Table 3 : I F S codes
for a fern. d e f P
W a b c d e W abc
1 0.5 0 0 0.5 0 0 0.33 1 000 0.16 0 0 0.01
2 0.5 0 0 0.5 1 0 0.33 2 0.2 -0.26 0.23 0.22 0 1.6 0.07
3 0.5 0 0 0.5 0.5 0.5 0.34 3 -0.15 0.28 0.26 0.24 0 0.44 0.07
4 0.85 0.04 -0.04 0.85 0 1.6 0.85

Table 2 : I F S codes for a square. e f P Table 4 : I F S
codes for fractal tree. e t P
W a b c d W a bcd
1 0.5 0 0 0.5 0 0 0.25 1 0 0 0 0.5 0 0 0.05
2 0.5 0 0 0.5 0.5 0 0.25 2 0.1 0 0 0.1 0 0.2 0.15
3 0.5 0 0 0.5 0 0.5 0.25 3 0.42 -0.42 0.42 0.42 0 0.2 0.4
4 0.5 0 0 0.5 0.5 0.5 0.25 4 0.42 0.42 -0.42 0.42 0 0.2 0.4

IMAGE COMPRESSION

affine transformation is and how to find formations. In the present case we might (i) Initialize: x=0, y=0.
one that maps a source image onto a de- have p,, ps , and p,. Notice that the
sired target image, we can describe an probabilities must add up to 1. That is, (ii) For n=1 to 2500, do steps (iii)-(vii).
iterated function system. An IFS is a col- p, +
p,+p,=1. (iii) Choose k to be one of the numbers
lection of contractive affine transforma- 1, 2 , . . ., m, with probability P,
tions. Here's an example of an IFS of Of course, the above notation for an (iv) Apply the transformation W, to the
three transformations: IFS is cumbersome. Table 1 expresses point (x,y) to obtain
the same information in tabular form. y=y.
Other examples of IFS codes are given
W' 1yI _ [0.0 0.5] Cy] + [00] ' in tables 2 through 4. Notice that an IFS (vi) If n > 10, plot (x,y).
can contain any number of affine (vii) Loop.
transformations. Applying this procedure to the trans-
The Random Iteration Algorithm Now formation in table 1 produces the figure
WZ Cy] = [o:o oa] Cy] + Co] ' let's see how to decode an arbitrary IFS shown in figure 8-a fractal known as the
code using the random iteration method. Sierpinski triangle. Increasing the number
Remember that in general an IFS can of iterations n adds points to the image.
W, [y] _ 00 0.5 Cy] + 1.55] contain any number, say m, of affine Figure 9 shows the result of the random
transformations, W,, W2, W3,.. ., W,., iteration algorithm applied to the data in
Each transformation must also have an each with an associated probability. The table 3, at several stages during the
associated probability, pi, determining its following code summarizes the method: process. By increasing the scale factor
"importance" relative to the other trans used in plotting, you can zoom in on the
image (see figure 10). The text box on
page 221 contains a BASIC implementa-
tion of the method with additional com-
ments on programming.
You may wonder why the first 10
points are not plotted (step (vi)). This is
to give the randomly dancing point time
to settle down on the image. It is like a
soccer ball thrown onto a field of expert
players: Until someone gains control of
the ball, its motion is unpredictable, or at
least is independent of the players' ac-
tions. But eventually a player gets the
ball, and its motion then becomes a
direct result of the skill of the players.
The fact that our transformation is
contractive guarantees that the "ball" will
eventually get to one of the "players,"
and that it will stay under control after
Figure 8: The result of applying the random iteration algorithm to the IFS code that.
in table 1. It is called the Sierpinski triangle. How do we know that the random iter-
co nt inue d

Figure 9: A fern appears when the random iteration algorithm is applied to the IFS code in table
3.

IMAGE COMPRESSION

at on algorithm will produce the same plane, with its vertices at (0,0), (1,0), (1, tions 1 through 3 and 4 through 6. The
image over and over again, independent 1), and (0,1) (see figure 11). The ob- values one finds in the present case are
of the particular sequence of random jective is to choose a set of contractive
choices that are made? This remarkable affine transformations, in this case W , given in table 2. When the random itera-
result was first suggested by computer- W2, W,, W4, so that S is approximated as tion algorithm is applied to this IFS
graphical mathematics experiments and well as possible by the union of the four code, the square is regenerated.
later given a rigorous theoretical subimages WI(S) U W2(S) U W3(S) U The preceding example typifies the
foundation by Georgia Tech W4(S). Figure 11 shows, on the left, S general situation: You need to find a set
mathematician John Elton. together with four noncovering affine of affine transformations that shrink dis-
The Collage Theorem transformations of it; on the right, the tances and that cause the target image to
affine transformations have been be approximated by the union of the af-
adjusted to make the union of the fine transformations of the image. The
Our next goal is to show a systematic transformed images cover up the square. Collage Theorem says that the more ac-
method for finding the affine transforma- curately the image is described in this
tions that will produce an IFS encoding To find the coefficients of these trans-
formations, we use the method described way, the more accurately the transforma-
of a desired image. This is achieved with tions provide an IFS encoding of it.
the help of the Collage Theorem. earlier in the section on iterated function Figure 12 provides another
To illustrate the method, we start systems, leading to simultaneous equa illustration of the Collage Theorem. At
from a picture of a filled-in square S in the bottom left is shown a polygonalized
leaf boundary, together with four affine
transformations of that boundary. The
transformed leaves taken together do
not form a very good approximation of
the leaf, in consequence, the
corresponding IFS image (bottom right),
computed using the random iteration
algorithm, does not look much like the
original leaf image. However, as the
collage is made more accurate (upper
left), the decoded image (upper right)
becomes more accurate.
So, there's a fundamental stability
here. You don't have to get the IFS code
exactly right in order to capture a good
likeness of your original image. More-
over, the IFS code is robust: Small per-
turbations in the code will not result in
unacceptable damage to the image. In
each of the above examples, we have
used four transformations to encode the
image. However, any number can be
used.
For example, the spiral image in
figure 13 can be encoded with just two
contractive affine transformations. See if
you can find them. Then determine the
Figure 10: Successive zooms on pieces of an IFS-encoded fern. IFS transformation coefficients and
input them to the random iteration
algorithm to get the spiral back again.

Figure 11: The collage theorem is used to encode a classical square S. The correct IFS code is obtained when the four affine
transformations of S cover S, as shown on the right.

IMAGE COMPRESSION

Assigning Probabilities filled in. Let the affine transformations W, amount of time that the randomly dancii.
corresponding to an image I be g point should spend in the subimage W,
Once you have defined your transforma- is approximately equal to
tions, you need to assign probabilities to W, [xY] - [c, d,] [xy] +
them. Different choices of probabilities do
not in general lead to different images, but
they do affect the rate at which various where i = 1, 2, 3,.. ., n. Then the continued
regions or attributes of the image are

IFS Decoding Listing A: A BASIC program demonstrating the use of the random
iteration algorithm to reconstruct an IFS-compressed image.
in BASIC 10 'Allow for a maximum of 4 transformations in the IFS
20 DIM a(4), b(4), c(4), d(4), e(4), f(4), p(4)
isting A is a BASIC 30
implementation of the random iteration 40 'Transformation data, Sierpinski triangle 50
'First comes the number of transformations 60 'then
algorithm. It includes the data for the the coefficients a through f and probability 70 'The pk
Sierpinski triangle, but you can use it to values for pk should be in descending order. 80 DATA
process any IFS tables. In particular, you 3
90 DATA .5,0,0,.5,0,0,.34 100 DATA .5,0,0,.5,1,0,.33
will want to try the data in tables 2, 3, 110 DATA .5,0,0,.5,.5,.5,.33
and 4. Be sure to set the variable m
correctly; it tells the program how many
120
transformations are in the IFS. 130 'Read in the
It is also essential that the probabilities 140 data READ m
in p ( ) add up to 1. For speed, the 150 pt = 0 'Cumulative probability
160 FOR j = 1 TO m
transformations should be listed in de- 170 READ a(j), b(j), c(j), d(j), e(j), f(j), pk
scending order of probability: the highest 180 pt = pt + pk
probability transformation first, and the 190 P(j) =
lowest probability last. 200 pt NEXT j
210
The program includes variables for 220
resealing and translating the origin to 230 'Set up for Graphics
accommodate the range of the points be- 240
258 SCREEN 3 'Select graphics screen
ing plotted to the limits of your screen. If 260
the image is too wide, decrease xscale; if
xscale = 350 'Map [0,11 onto [0,3501
the points are too close horizontally, yscale = 325 'Map [8,11 onto [0,3251
increase xscale. Adjust yscale similarly to 270 xoffset = 0
280 'Initialize x and y
get a good vertical point spread. To move 290 x = 8
the image, adjust xoffset and yoffset. 300 y = 0
You can do these adjustments by trial 310
and error: Run the program; interrupt it 320
330 'Do 2500
and change the offsets and scale factors; 340 iterations FOR n =
and run it again. Or, you can replace the 350 1 TO 2580 pk = RND
plot command pset with a command to 360 'The next line works for m<=4. It must be
print the values of x and y and run the 370 modified 'for values of m > 4.
380 IF pk <= p(1) THEN k = 1 ELSE IF pk <= p(2) THEN k
program to get an exact idea of the range = ELSE IF pk <= p(3) THEN k = 3 ELSE k = 4
2
of points being plotted, so you can adjust
the scale and offsets more precisely. 390 newx = a (k) * x + b (k) * y + e
(k)
Another way to arrange the program 400 newy = c (k) * x + d (k)
is to have it read all the data-m, a () , ( ), * y + f (k) 410 x =
c( ), d( ), e( ), f( ), p( ) > xscale, yscale, newx
xoffset, and yoffset-from a disk file 420 y = newy
430 Use PRINT x,y instead of the PSET line
specified by the user. Instead of reading 440 'to see the range of coordinates. Then fix yscal
in the coefficients a, b, c, and d, you may 450 'xscale, yscale, xoffset, and yoffset e
want to read in angles 0 and 0 and scale 460 IF n > 10 THEN PSET (x *
470 NEXT n
factors r and s, and then calculate the 480
coefficients. 490 LOCATE 24, 35
The random iteration method is com- 500
510 PRINT "Press any key to end.";
putation-intensive, so we recommend use 520 WHILE INKEY$
of a compiler such as Microsoft's 530 WEND
QuickBASIC or Borland's Turbo BASIC. 540
550 'Return to text screen SCREEN 0
If your computer has a floatingpoint 560 END
coprocessor and your compiler supports
one, so much the better.

IMAGE COMPRESSION

So long as ad - cd is not 0, it is a stan-
dard calculus result that our ratio equals
the determinant of the transformation
matrix for W,.. So a good choice for the
probability p; is
a; d; - b;
c;
I akdk-b,Ckl

provided none of these numbers p;
comes out to be 0. A 0 value should be
replaced by a very small positive value,
such as 0.001, and the other probabilities
correspondingly adjusted to keep the
sum of all the probabilities equal to 1.
We now summarize the compression
and decompression process: An input
image is broken up into segments
through image-processing techniques.
These image components are looked up
in the IFS library using the Collage
Theorem, and their IFS codes are
recorded. When the image is to be
reconstructed, the IFS codes are input to
Figure 12: The Collage Theorem is applied to a leaf. The collage at lower left isn't the random iteration algorithm. The
much good, so the corresponding IFS image, shown at lower right, is a poor accuracy of the reconstructed image
approximation. But as the collage improves, upper left, so does the IFS image. depends only on the tolerance setting
Applications

For graphics applications, we use a more
sophisticated procedure that allows full-
color images to be encoded. Combina-
torial searching algorithms can be used to
automate the collage mapping stage. Fig-
ures 2, 3, and 4 were obtained using IFS
theory at compression ratios in excess of
10,000 to 1. These images were based on
photographs in recent issues of National
Geographic. A full-sequence video ani-
mation, A Cloud Study, was shown at
SIGGRAPH '87. This was encoded at a
ratio exceeding 1,000,000 to 1 and can be
transmitted in encoded form at video
rates over ISDN lines (ISDN stands for
integrated services digital network, a
concept for integrated voice and data
communications). A frame from the ani-
mation is shown in figure 5.
The IFS compression technique is
computation-intensive in both the
encoding and decoding phases.
Computations for the color images were
all carried out on Masscomp 5600
workstations (dual 68020-based systems)
with Aurora graphics. Complex color
images require about 100 hours each to
encode and 30 minutes to decode on the
Masscomp.
For practical applications, you need
custom hardware that can speed the en-
coding and decoding process. An experi-
mental prototype, the IFSIS (iterated
Figure 13: Can you find the IFS codes for this spiral image? Only two function system-image synthesizer), de-
transformations are needed. codes at the rate of several frames per
second. The IFSIS device was produced
from a cooperative effort between

IMAGE COMPRESSION

DARPA, Atlantic Aerospace Electronics Corporation, and Iterated Systems, and it was
demonstrated on October 5, 1987, at thethirdannualmeetingoftheAppliedand Computational
Mathematics Program of DARPA. It can be connected to a personal computer through a serial
port; the personal computer sends the IFS codes to the device, whichrespondsbyproducingcom-
plex color images on a monitor.
The IFSIS is a proof of concept for faster devices with higher resolution. Once the higher-
performance IFSIS devices are combined with ISDN telecommunication, full-color animation at
video rates over phone lines will be a reality.
Another area for future application of IFS encoding is automatic image analysis. What's in a
picture? Does it show a spotted sandpiper or a robin? The more complex the image or the more
subtle the question, the harder it becomes for an algorithmic answer to be formulated. But here's
the point: Whatever the answer, it will proceed faster if stable, compressed images are used. The
reason for this is that image-recognition problems involve combinatorial searching, and searching
times increase factorially with the size of the image file.
During the spring of 1987, Iterated Systems was incorporated to develop commercial
applications of IFS image compression. It is exciting to see how an abstract field of mathematics
research is leading to new technology with implications ranging from commercial and industrial
work to personal computing. E

ACKNOWLEDGMENTS

Figures 2 through S were encoded by graduate students Francois Malassenet, Laurie Reuter, and
Arnaud Jacquin. All color images were produced in the Computergraphical Mathematics
Laboratory at Georgia Institute of Technology and are copyright 1987, GTRC.

BIBLIOGRAPHY

Barnsley, M. F. and S. Demko. "Iterated
Function Systems and the Global Con-
struction of Fractals." The Proceedings of
the Royal Society of London, A399, 1985,
pp. 243-275.
Barnsley, M. F., V. Ervin, D. Hardin, and
J. Lancaster. "Solution of an Inverse
Problem for Fractals and Other Sets."
Proceedings of the National Academy of
Science, vol. 83, April 1985.
Barnsley, M. F. Fractals Everywhere. Ac-
ademic Press, 1988. Forthcoming. Elton, J.
"An Ergodic Theorem for Iterated Maps."

BYTE
Journal of Ergodic Theory and Dynamical
Systems. Forthcoming. Mandelbrot, B. The
Fractal Geometry of Nature. San Francisco,
CA: W. H. Freeman and Co., 1982.
THE SMALL SYSTEMS JOURNAL

Reprinted with permission from the January 1988 issue of BYTE magazine. Copyright c 1988 by
McGraw-Hill, Inc., New York, 10020. All rights reserved.

COMPUTER
GRAPHICS
WORLD GRAPHICS

TECHNOLOGY

Chaotic Compression
A new twist on fractal theory speeds
complex image transmission to video rates
pounding the problems of cost takes three hours at a total trans-
By Michael F. Barnsley & and delay, extended transmission mission cost of $42 (25 cents per
Alan D. Sloan times expose data to degradation. minute) to send our sample image
A one-megabyte-per-second 1000 miles on a 2400-bps toll line.
Ethernet line transmits the same The same image can be sent to the

T he massive volumes of data
associated with computer graphics
image in one second, but two
users who are each exchanging one
minute of animation of such
images consume an hour of time.
same address in 16 seconds at a
cost of 51 cents on a 56,000-bps
T1 line.
Confounding Performance
severely check the use of low- Given these burdens, it's easy to Paradoxically, continuing improve-
bandwidth communications see"why there is no multiple-user ments in computer graphics hard-
network for animating high- ware and software only compound
devices for image transfer. resolution images at video rates.
Multiple users can easily overload Transmission times are propor- the problem. Faster hardware in-
tional to transmission bandwidths. creases the rate of graphics
even the markedly higher production and thus increases
bandwidth of local area networks. At current rates, for example, it demand on transmission. The
Potential applications go quest for realism
unrealized as the cost of Ten thousand-to-one compression ratios can be realized with
transmitting graphics increases a new method based on chaos theory, fractal geometry, and affine
directly with improved per- transformations. While transmission can approach or exceed video
formance and greater complexity. rates, the challenge now is to reconstruct images as rapidly.
Below: "Sunflower Field" and "Arctic Wolf," images compressed
But algorithms based on fractal at 10,000:1 at Georgia Tech.
geometry can encode image data at
high compression ratios-in excess
of 10,000 to one. It is possible to
transmit such compressed image
files at video rates (30 frames per
second) over normal telephone
lines. A new graphics device, the
Iterated Function System Image
Synthesizer, makes Transmission
A Perspective on it possible to
A 1000-by-1000-resolution com-
puter graphics image with eight
bits of color information per pixel
needs one megabyte of data for
specification. The transmission of
this image over a 2400-bit-per-
second line takes over 55 minutes.
Com
Michael F. Barnsley and Alan D. Sloan are
professors of mathematics at the Georgia
Institute of Technology (Atlanta). Barnsley
and Sloan are also officers of Iterated Sys-
tems Inc. (Atlanta).

Reprinted from the November, 1987 edition of Computer
Graphics World Copyright 1987 by PennWell Publishing
Company

drives graphics software to on a line between P(n) and Vk
produce images of increasing where Vl = A, V2 = B, and Va = C.
complexity. Greater image
complexity means more Throw away the first 25 points and
information that must be trans- have been working on image com- plot the rest. The result is indepen-
mitted for each image. pression standards. But the com- dent of the initial point P(0) as well
Sophisticated printers also increase pression ratios being discussed are as the sequence of random
transmission requirements. A new basically pretty low.
For example, one exact numbers which were chosen in its
color printer from Tektronix is a generation.
case in point. Producing an A-size, compression method-a type of
method calling for exact When suitably generalized, this
24-bit-deep color printout at 300
dpi can chew up over 9M of data. reconstruction of an image-calls for process leads to full-color images
Even at the Integrated Services a ratio of just two to one. There are like the sunflowers and Arctic wolf
several exact methods, but they on the preceding page. Inspired by
Digital Network bandwidth of National Geographic photographs,
64,000 bps, it will take over 15 aren't particularly useful. In one
minutes to supply the necessary experiment, a photo montage was Georgia Institute of Technology
run-length encoded, producing a graduate students Laurie Reuter and
data to the printer. Data
compression and reconstruction 60-percent increase in data length. Arnaud Jacquin encoded each of
clearly offer the greatest potential Unix provides a Huffman encoder these images at a ratio exceeding
for relief from the costs associated which yields a compression ratio of 10,000 to one. Both images are
decoded by a random iteration al-
with even the fastest communica- An experimental device reconstructs
tions technologies.
Digital data compression compressed video images at the rate
consists of encoding and decoding of about four frames per second.
phases. In the encoding phase, an
input string of integer-based (I) bits Such ratios are common to exact An experimental prototype device
is transformed into a coded string compression methods. introduced at the Meeting of the
of complex number-based (C) bits. General Electric has announced the Applied and Computational Mathe-
The ratio of I to C bits is the Digital Video Interactive com- matics Program of the Defense Ad-
compression ratio. In the decoding pression system developed at the vanced Research Projects Agency
phase, the compressed string David Sarnoff Research Center (DARPA) in October, the Iterated
regenerates the original data. If the ("DVI Video/Graphics," July, Function System Image Synthesizer
compression ratio is two to one, p.125). As the DVI compression (IFSIS) decoded video images at the
then in the time it would take to ratio increases above 10 to one, the rate of about four frames per
transmit one uncompressed image, loss of high-frequency information second. Because of the extreme
two compressed images can be becomes more apparent. But fine compression ratios, this device can
transmitted. detail isn't as critical for motion be coupled loosely to any host,
High-ratio image compression video as for still images, so the DVI which will treat it as if it were a
slashes transmission time and can produce a credible video at a printer-only it produces complex
costs. For example, an 100-to-one ratio. color images instead of text.
uncompressed minute of 30 The new method described here
has been used to encode complex Networking and high-resolution,
frames-per-second animation for a real-time animated graphics can be
24-bit-deep, 1000-by-1000 color graphics at exact compression
ratios of 10,000 to one. It can be realized simultaneously by means of
image takes seven months to used with classical compression IFSIS-type devices. The demon-
transmit over a 24-bps line at a techniques to increase yields even
further-up to one million to one in stration at the DARPA meeting
cost of $75,000. Compressed at a served as proof-of-concept for
10,000-toone ratio, the same the case of "A Cloud Study,"
shown at SIGGRAPH '87. The faster transmission devices with
minute of animation takes 30 method is based on chaotic higher resolution. Once the higher-
minutes to transmit at a cost of dynamic systems theory. A simple
example illustrates the procedure. performance IFSIS devices are
about $7.50. On a 1.544 million- combined with 64,000-bps
bps line, uncompressed Given a triangle with vertices A, B,
and C, choose an initial point P(0) telecommunications, fullcolor
transmission takes eight hours at a animation transmission and
cost of $339; compressed at 10,000 at random in the triangle. An itera-
tive procedure (affine reconstruction at video rates over
to one, it takes just three seconds phone lines will become a commer-

drives graphics software to on a line between P(n) and Vk
produce images of increasing where Vl = A, VZ = B, and V$ = C.
complexity. Greater image
complexity means more Throw away the first 25 points and
information that must be trans- have been working on image com- plot the rest. The result is indepen-
mitted for each image. pression standards. But the com- dent of the initial point P(0) as well
Sophisticated printers also increase pression ratios being discussed are as the sequence of random
transmission requirements. A new basically pretty low.
For example, one exact numbers which were chosen in its
color printer from Tektronix is a generation.
case in point. Producing an A-size, compression method-a type of
method calling for exact When suitably generalized, this
24-bit-deep color printout at 300
dpi can chew up over 9M of data. reconstruction of an image-calls for process leads to full-color images
Even at the Integrated Services a ratio of just two to one. There are like the sunflowers and Arctic wolf
several exact methods, but they on the preceding page. Inspired by
Digital Network bandwidth of National Geographic photographs,
64,000 bps, it will take over 15 aren't particularly useful. In one
minutes to supply the necessary experiment, a photo montage was Georgia Institute of Technology
run-length encoded, producing a graduate students Laurie Reuter and
data to the printer. Data
compression and reconstruction 60-percent increase in data length. Arnaud Jacquin encoded each of
clearly offer the greatest potential Unix provides a Huffman encoder these images at a ratio exceeding
for relief from the costs associated which yields a compression ratio of 10,000 to one. Both images are
decoded by a random iteration al-
with even the fastest communica- An experimental device reconstructs
tions technologies.
Digital data compression consists compressed video images at the rate
of encoding and decoding phases. of about four frames per second.
In the encoding phase, an input
string of integer-based (I) bits is Such ratios are common to exact An experimental prototype device
transformed into a coded string of compression methods. introduced at the Meeting of the
complex number-based (C) bits. General Electric has announced the Applied and Computational Mathe-
The ratio of I to C bits is the Digital Video Interactive com- matics Program of the Defense Ad-
compression ratio. In the decoding pression system developed at the vanced Research Projects Agency
phase, the compressed string David Sarnoff Research Center (DARPA) in October, the Iterated
regenerates the original data. If the ("DVI Video/Graphics," July, Function System Image Synthesizer
compression ratio is two to one, p.125). As the DVI compression (IFSIS) decoded video images at the
then in the time it would take to ratio increases above 10 to one, the rate of about four frames per
transmit one uncompressed image, loss of high-frequency information second. Because of the extreme
two compressed images can be becomes more apparent. But fine compression ratios, this device can
transmitted. detail isn't as critical for motion be coupled loosely to any host,
High-ratio image compression video as for still images, so the DVI which will treat it as if it were a
slashes transmission time and can produce a credible video at a printer-only it produces complex
costs. For example, an 100-to-one ratio. color images instead of text.
uncompressed minute of 30 The new method described here
has been used to encode complex Networking and high-resolution,
frames-per-second animation for a
graphics at exact compression real-time animated graphics can be
24-bit-deep, 1000-by-1000 color ratios of 10,000 to one. It can be realized simultaneously by means of
image takes seven months to used with classical compression IFSIS-type devices. The demon-
transmit over a 24-bps line at a techniques to increase yields even
further-up to one million to one in stration at the DARPA meeting
cost of $75,000. Compressed at a
the case of "A Cloud Study," served as proof-of-concept for
10,000-toone ratio, the same shown at SIGGRAPH '87. The faster transmission devices with
minute of animation takes 30 method is based on chaotic higher resolution. Once the higher-
minutes to transmit at a cost of dynamic systems theory. A simple
example illustrates the procedure. performance IFSIS devices are
about $7.50. On a 1.544 million-
Given a triangle with vertices A, B, combined with 64,000-bps
bps line, uncompressed and C, choose an initial point P(0) telecommunications, fullcolor
transmission takes eight hours at a at random in the triangle. An itera- animation transmission and
cost of $339; compressed at 10,000 tive procedure (affine
transformation) transforms a point reconstruction at video rates over
to one, it takes just three seconds
phone lines will become a commer-

Image compression

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