Fractals are geometric or data structures which do not simplify under magnification. Fractal Image Compression is a technique which associates a fractal to an image. On the one hand, the fractal can be described in terms of a few succinct rules, while on the other, the fractal contains much or all of the image information. Since the rules are described with less bits of data than the image, compression results. Data compression with fractals is an approach to reach high compression ratios for large data streams related to images. The high compression ratios are attained at a cost of large amounts of computation. Both lossless and lossy modes are supported by the technique. The technique is stable in that small errors in codes lead to small errors in image data. Applications to the NASA mission are discussed

Barnsley, Michael F.

Sloan, Alan D.

English

NASA Technical Reports Server

N89-22346 
< 
FRACTAL IMAGE COMPRESSION 
Michael F. Barnsley and Alan D. Sloan 
Mathematics, Georgia Tech 
ABSTRACT 
Fractals are geometric or data structures which do not simplify under 
magnification. Fractal Image Compression is a technique which 
associates a fractal to an image. On the one hand, the fractal can be 
described in terms of a few succinct rules, while on the other, the 
fractal contains much or all of the image information. Since the 
rules are described with less bits of data than the image, compression 
results. 
Data compression with fractals is an approach to reach high 
compression ratios for large data streams related to images. The high 
compression ratios are attained at a cost of large amounts of 
computation. Both lossless and lossy modes are supported by the 
technique. The technique is stable in that small errors in codes lead 
to small errors in image data. Applications to the NASA mission are 
discussed. 
OVERVIEW 
Fractals are geometric or data structures which do not simplify under 
magnification. Fractal Image Compression is a technique which 
associates a fractal to an image. On the one hand, the fractal can be 
described in terms of a few succinct rules, while on the other, the 
fractal contains much or all of the image information. Since the 
rules are described with less bits of data than the image, compression 
results. 
Fractal image compression is a computationally intensive technique. 
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However, the computations required are mainly multiplications and 
accumulations and iterative in nature. The rules consist of low 
dimension matrix transformations. Therefore, high speed hardware 
implementations are possible. A hardware decoder was demonstrated in 
October, 1987. This prototype device can decode 256 x 256 x 
8bits/pixel images at a rate of several frames per second. It 
demonstrates the feasibility of higher performance decoders. 
The Collage Theorem, described in the next section, provides the 
connection between rules and images. It allows for the complete 
control of the fidelity of the encoded image. Compression ratios in 
the lossy mode are typically much higher than in the lossless mode. 
The association between fractals and images is a continuous one in the 
following sense: small changes in the matrices produce small changes 
in images. 
I The observation that a simple set of rules can produce a complex 
image began with abstract fractal pictures known as Julia sets. The 
Georgia Tech mathematics research team set out to explore the limits 
of this observation. The exploration took the form of a search for a 
solution to an llinversell problem (e.g., given an image, find the rules 
which encode it as a fractal). The Collage Theorem is a remarkable 
solution to this inverse problem. High resolution, color images have 
been encoded in several thousand bytes. 
Basic research in this technique at Georgia Tech and other 
universities has been supported by DARPA, AFOSR, NSF and ONR. Georgia 
Tech was specifically funded under the Applied and Computational 
Mathematics Program of DARPA to investigate automation of this 
technique using simulated thermal annealing algorithms. While basic 
research continues, a number of corporations are investigating 
applications of this technique. In particular, Iterated Systems, Inc. 
was formed to commercialize this technology. 
352 
THE TECHNIQUE 
Fractal Image Compr-sion Cod 
The code for an image which has been subjected to fractal image 
compression consists of an iterated function system (ifs). An ifs is 
composed of affine transformations and probabilities. In two 
dimensions, an affine transformation, T, takes the form 
T(X,Y) = (X'rY') 
where x' = Ax + By + C 
and y' = Dx + Ey + F 
The six coefficients A,B,C,D,E,F define T and must be specified in 
log(reso1ution) bits. For example, 100 affine transformations on a 
1024 x 1024 screen require 100*6*10 bits. Affine transformations 
consists of scalings, rotations and translations and so have a 
geometrical interpretation. The affine transformations which appear 
in ifs codes should be contractive in the following sense. If IP-QI 
denotes the Euclidean distance between two points, P and Q, and if T 
is a contractive transformation, then 
where the contractivity factor s < 1. 
The probabilities form a linkage matrix used in the decoding process. 
In the most common usage to date, only one probability is specified 
for each transformation and these may be specified in log(number of 
colors) bits. A color look-up table must also be specified for use in 
the decoder. Typically, several color values are given and linear 
interpolation is used to generate intermediate colors. In typical 
examples this can be done in less than log(number of colors) bits/map. 
Thus, encoding a 1024 x 1024 x 9 bits/pixel image in N maps results in 
3 5 3  
a code of length 72* N bits. As an example, choosing N = 100 results 
in a length of 7200 bits. The original image is 1024 * 1024 * 9 bits 
so the compression ratio exceeds 1000 : 1. 
Some ifs codes are given on Tables 1-4. 
, Hausdorff Image Distance 
Precise statements concerning fractal image encoding and decoding 
refer to the Hausdorff distance between images. Fix a screen, S, 
consisting of R rows with P pixels/row. A monochrome image (1 
bit/pixel) is simply a collection, I, of pixel sites on the screen 
which are illuminated. The distance from a screen location a to an 
image B is defined as the closest Euclidean distance. That is, 
I d(a,b) = minimum ( la-bl : for b in B). 
The distance from an image A to the image B is given by 
d(A,B) = maximum { d(a,B) : for a in A}. 
This max-min type distance function is not symmetric. 
may not coincide with d(B,A). See Figure 2. 
That is, d(A,B) 
The Hausdorff distance between A and B is 
H(A,B) = maximum ( d(A,B), d(B,A)). 
If the Hausdorff distance between two images is zero, then the images 
are identical. If the Hausdorff distance is less than the resolution 
of the screen, the two images are indistinguishable. 
The Hausdorff distance definition may be generalized to color and 
grey-scale images by viewing the color information as a third 
coordinate. From this point of view, images are surfaces and the 
354 
Hausdorff metric is used to measure distances between surfaces. 
The Decoder 
An ifs code can be thought of as an image processing operation. Let C 
be an ifs code consisting of the affine transformations T1, T2, ..., 
Tn. If R is any image, then Ti(R) means the image under Ti of all 
the points in R. C(R) is then defined by 
C(R) = Tl(R) U T2(R) U ... U Tn(R). 
As an example, let the image R consist of one point and suppose the 
code consists of two transformations. Then typically, C(R) will 
consist of two points. Starting now with C(R), C(C(R)) will typically 
consist of four points. Since, however, the affine transformations in 
an ifs code are contractions, a transformation may coalesce several 
points into a single point. 
Associated to every ifs code is a unique set called the attractor of 
the code. The attractor A = attr(C) of the ifs code C is defined as 
the only set with the property that 
A = Tl(A) U T2(A) U ... U Tn(A). 
The maximum of all the contractivity factors of the affine 
transformations in an ifs code is the called the contractivity factor 
of the code. 
Let A(0) be any non-zero subset. Inductively define A(i) = 
C(A(i-1)). Then the sequence A(O), A(l), A(2) converges to the 
attractor A of the ifs code C, in the Hausdorff metric. That is, as i 
gets large H(A(i), A) becomes small. For typical examples, the 
Hausdorff distance becomes less than screen resolution when i is 
between 10 and 50. 
355 
The attractor attr(C) associated with the code C is, in this way, the 
image encoded by the code C. 
Figure 3 illustrates the decoding process for a four transformation 
code corresponding to a fern. The initial rectangle which initiates 
the decoder does not affect the attractor. It could just as well be a 
sin curve or a random screen. Thirty iterations separate the initial 
and final images. 
The Encoder 
The basis for fractal image encoding is the COLLAGE THEOREM: Let B be 
a target image and let C be an ifs code with contractivity factor 
0 < s < 1. If the Hausdorff distance between B and C(B) is less than 
E then the Hausdorff distance between B and attr(C) is less than E / ( l -  
s )  
This theorem says that to find the ifs compression code for an image 
or image segment, one can solve the following puzzle. Small (affine) 
deformed copies of the target must be arranged so that they cover up 
the target as exactly as possible. This ltcollagetl of deformed copies 
determines an ifs code since each deformation is an affine 
transformation of the target. The better the collage, as measured by 
the Hausdorff distance, the closer will be the attractor of the ifs to 
the target. 
Application of the collage theorem so that E < (1-s)*resolution 
assures a lossless compression. One can search for transformations 
which have contractivity factors < .7, for example. Then lossless 
compression requires E to be less than .3*resolution. More generally, 
upper bounds on contractivity produces a priori bounds on the errors 
in the encoded image during the encoding process. 
I 
Another consequence of the collage theorem is that if the matrix 
entries in two codes are close then the attractors of the codes are 
356 
also close. This has an important interpretation related to error 
propagation. Small errors in codes lead to small error in images. 
Figures 4 and 5 demonstrate the collage theorem. 
NASA APPLICATIONS 
Data compression can provide service to the NASA mission in both 
space based and ground based operations. 
Data Quality 
One issue which arises in space and ground use of lossy compression is 
that of data quality. Common measures of error in reconstructed data 
are based on mean-square or root-mean square computations. This type 
of error calculation is often chosen for convenience, rather than for 
scientific merit. Scientific analyses attempt to compensate for 
spatial errors through a registration procedure. Fractal image 
compression has focused on a different error metric, the Hausdorff 
distance. It integrates both spatial and spectral data distortion 
into a single measure of error. Evaluation of the Hausdorff distance 
as a relevant discriminant of data quality can begin immediately using 
existing experiment data. 
Compression procedures are generally sensitive to transmission bit 
errors. Sensitivity generally increases with increasing compression 
ratios. Most high compression ratio techniques are therefore risky in 
a noisy environment. Fractal image compression contains error 
propagation independently of the source of the errors. As discussed 
above the collage theorem provides bounds on the error in the 
reconstructed data from bounds on errors in the codes. This error 
containment need not suffer with increased compression ratios. 
357 
Scientific Utility 
The capture of data from space sensors or experiments and its 
transmission to earth is a major NASA activity. Often this data is 
destined to be analyzed by scientists in the form of images. Space 
sensors and experiments can easily produce enough image data to 
overload all available data communication bands to earth. Twice as 
much two-to-one compressed data as uncompressed data can be 
transmitted in the presence of a bandwidth bottleneck. Fractal 
compressed images can provide orders of magnitude more images through 
a bandwidth bottleneck. To achieve 1000-to-1 compression on 1024 x 
1024 images, subsampling, for example, would produce 32 x 32 images. 
Such coarse images may not be of any use to a scientist. Fractal 
image compression can be used by a scientist to achieve such 
compression and yet be structured so as to retain certain recognizable 
features on a fine scale. While such high compression ratio encoding 
may not be of universal interest, scientists should be given the 
choice. 
High data rate sensors may be operational for only a small percent of 
their lifetime due to bandwidth bottleneck. Fractal image compression 
can use additional sensor data during the compression process to 
increase the quality of the transmitted data. 
While cost of storage media decreases and read/write speeds and 
bandwidth increases, these trends are not able to match the increase 
in available data. As a consequence, data compression should form an 
important part of any data management system. Moreover, even given 
unlimited and inexpensive memory and bandwidth, image analysis would 
remain as an outstanding problem of overriding importance. Fractal 
image compression does not simply produce an unintelligible code from 
uncompressed data. Rather, fractal codes themselves contain geometric 
and measure-theoretic information about the data sets. Analysis of an 
image can, in part, be done on the compressed code. In partic. .ar, 
experiments with texture and object identification and classification 
358 
based on compressed codes can begin with existing experimental data. 
1 efficient format for animation. The collage theorem guarantees that 
animation can be accomplished through small changes in codes that are 
already highly compressed. Fast decoders will be required to view 
the animation at video rates. A prototype decoder was displayed in 
demonstrated the feasibility of higher performance video rate 
decoders. 
I 
October, 1987 at DARPA's ACMP conference in Washington, D.C. It 
Scientific Justification 
The nature of the data collected by space sensors suggest a vast 
potential for compression, well beyond that indicated by standard 
entropy calculations. Much of the data is collected from repeated 
observations over similar areas. Multispectral data is expected to be 
correlated over multiple channels. The data itself, generated by 
natural laws, though complex is far from random. As a concluding 
example, consider the data in Figures 6-9. These graphs of voltage as 
a function of time measure laser scattering and voltage across a wire 
in a turbulent jet experiment. Both data streams, which come from the 
359 
Data Management 
Interactive access to scientific data increases its usefulness. 
Users often do not have a specific data address to examine but rather 
wish to browse through data samples of a generic type. In the 
browsing. mode, only accuracy to some level of detail is required. 
Fractal image compression can provide interactive browsing on existing 
networks by simulating a virtual bandwidth orders of magnitude higher 
than the actual bandwidth. 
I probes of a single system yield the same fractal dimension, 1.5 as 
indicated in Figure 9. Such correlations are not exploited in 
classical compression schemes. Fractal image compression can find the 
hidden redundancy suggested by such data. 
360 
Then Table 1 t a tidiu way of conveying the oame iterated function system. 
1 0.6 0 0 0.5 1 1 0.33 
2 0.5 0 0 0.5 1 so 0.33 
3 0.5 0 0 0.5 25 so 0.34 
TABLE 1 IFS code for a Siupinski triangle. 
W a b C d e t P 
1 0.5 0 0 0.5 1 1 O S  
2 0.5 0 0 0.5 50 1 0-25 
3 0.5 0 0 0.5 1 50 0% 
4 0.6 0 0 0.6 50 60 0.25 
& 
TABLE 2 IFS d e  for a Square. 
W 8 b c d t f P 
0 0 0 0.16 0 0 0.01 
0.85 0.04 -0.04 0.85 0 1.6 0.85 
0.2 -0.26 0.23 0.22 0 1.6 0.07 
4 -0.15 0.28 0.26 0.24 0 0.44 0.07 
TABLE 3 IFS code for a Fern. 
W 8 b C d e I P 
1 0 0 0 0.5 0 0 0.05 
2 0.42 -0.42 0.42 0.42 0 0.2 0.4 
3 0.42 0.42 -0.42 0.42 0 0.2 0.4 
4 0.1 0 0 0.1 0 0.2 0.15 
TABLE 4 ITS code for a Fractal Tree. 
361 
FIGURE 1 
0 
FIRST MAKE A LINEAR 
A F F I N E  
TRANSFORMATIONS 
EXAMPLE: 
F I G U R E  2 
HAUSDORFF DISTANCE is 
H(A,B) = maximum (d(A,B), d(B,A)) 
362  
3.1 
3.6 
4. 
3.2 
- 
3.7 
I .  
Figure 3 
ILLUSTRATION OF DECODING 
U 3.4 3.5 
3.10 
3.1 is an initial computer screen. This initial state can be chosen at random but in this example it 
is a small square, in the upper left hand corner of the screen. Four affine transformations are 
applied to each point in the square and give the four parallelograms in 3.2. A0 is the initial square. 
The image AI  is the four parallelograms in 3.2. The same four transformations are applied to each 
of the parallelograms in 3.2 and produceA2 in 3.3 which consists of sixteen parallelograms. Some 
intermediate screens are not shown. After about thirty iterations the fern appears in Figure 3.10. 
= A  d 
Fig1 
(a) Collage 
( b )  A t t r ac to r  
Two app l i ca t ions  of the co l lage  
theorem. ( a )  and ( c )  a re  co l l ages  
of a l ea f  under a f f i n e  transforma- 
t i o n s .  Four t ransformations a re  
used i n  each case.  The Hausdorff 
d i s t a n c e  between the  co l lage  and 
the t a r g e t  l e a f  i s  smaller i n  ( a )  
than i n  ( c ) .  ( b )  i s  the  l ea f  
reconstructed from ( a )  while ( d )  
i s  reconstructed from ( c ) .  The 
recons t ruc t ion  from ( a )  t o  ( b )  i s  
super ior  than t h a t  from ( c )  t o  ( d )  
a s  suggested by t h e  co l l age  theorem. 
r e  4 
Collage of four s imi l i t udes .  
Target leaf  i s  o u t l i n e  i n  s o l i d  
s t roke .  Aff inely deformed copies  
have broken ou t l ines .  
Decoded l ea f  from co l l age  above. 
Figure 5 
3 64 
W 
36 5 


Fractal image compression

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