A proprietary fractal image compression algorithm and the Joint Photographic Experts Group (JPEG) industry standard algorithm for image compression are compared. In every case, the JPEG algorithm was superior to the fractal method at a given compression ratio according to a root mean square criterion and a peak signal to noise criterion

Cheung, K.-M.

Shahshahani, M.

English

NASA Technical Reports Server

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TDA Progress Report42-107 November 15, 1991
Y-C,
N92-14240
A Comparison of the Fractal and JPEG Algorithms
K.-M. Cheung and M. Shahshahani
Communications Systems Research Section
A proprietary fractal image-compression algorithm and the Joint Photographic
Experts Group (JPEG) industry standard algorithm for image compression are
compared. In every case, the JPEG algorithm was superior to the fractal method
at a given compression ratio according to a root-mean-square criterion and a peak
signal-to-noise criterion.
I. Introduction
Fractal image compression has attracted much public-
ity in recent years. It has been suggested that one can
achieve compression ratios of the order of thousands to
one by the application of fractal algorithms. Some re-
searchers successfully generated certain images, with very
small databases, by using fractal algorithms. These im-
ages generally consisted of natural objects, and the mem-
ory requirement for, e.g., a realistic looking tree, was about
120 bytes. However, the applicability of these methods to
general image compression and the achievement of the phe-
nomenal compression ratios of thousands to one have been
viewed with general skepticism. In order to make a com-
parative study of the fractal versus Joint Photographic Ex-
perts Group (JPEG) standard algorithms, the authors sent
ten images to a vendor, Iterated Systems Inc., Norcross,
Georgia. These images were compressed by their propri-
etary fracta] algorithms, and reconstructed using their de-
compression package. The compression ratios were from
about five to one to twenty to one. The mean square er-
rors and peak signal-to-noise ratio (SNR) were compared
" to the corresponding ones for the JPEG algorithms at the
same compression ratios. The latter approach proved to
be superior in every case according to these criteria.
The theoretical foundation and the practical implemen-
tation of tile fractal method for image generation is de-
scribed in Section II. The relevant portion of the JPEG
algorithm is briefly described in Section III, and finally in
Section IV the results of the comparative study are given.
II. Fractals
There are various ways of defining fractals. Tile frame-
work proposed by J. IIutchinson [1] has been the most
successful approach for tile study of fractals. To describe
this method let S = {S1,'",S,_} be a finite set of affine
transformations of R q. This means that if x E Rq, then
the effect of Si = (Ai, vi) on z is given by
Si(x) = Ai(x) + vi
where Ai is a linear transformation of R q and vi E R q. IL
will be assumed that Ai's are nonsingular and Si's are con-
tracting, i.e., HSi(x) - Si(y)]] < [Ix -- y]] for all x,y E Rq.
Then the affine transformation Si_ ." Sir is also contract-
ing and, therefore, has a unique fixed point that will be
21
https://ntrs.nasa.gov/search.jsp?R=19920005022 2020-03-17T14:38:09+00:00Z
denotedbyF/_ ...,i k, The fractal set .7"(8) associated to S
is, by definition,
.r(s) = closure({Fi,,. ,kJ all 05 ix,..., ik < n
and k = 1,2,3,...})
Y'(S) is a compaft subset of R q. The self-similarity prop-
erty of fractals is expressed by the fundamental equation
r_
7(s) = [_J
i=l
In fact,
Theorem 1. From [1], .T'(S) is the unique compact set
K C R q with the property
K -" CJ Si(K) (1)
i=l
Notice that each S/(K) is a replica of K, so that Eq. (1)
does indeed express the self-similarity property of fractals.
This characterization of fractals is also important in prac-
tical applications.
Example 1. Let K be a convex polygon in R 2 with ver-
tices vl,...,vn, and let SI,...,Sn be the affine transfor-
mations Si(x) = vi + cq(x - vi) , where 0 < cri < 1. If
the o_i's are not too small, then [.JSi(K) = K and, con-
sequently, F(S) = K by Theorem 1 of fractal sets. Thus,
every convex polygon can be realized as an .T'(S) for some
S. On the other hand, it is not hard to see that the bound-
ary 0[( of the convex polygon K cannot be realized as a
fractal set.
To generate a fractal image, one starts with a set
S = {S1,...,S,} of aftlne transformations. For every
product Si, ""Si=Sil = S = (A,v) of length l < k, for
some pre-assigned value k, one computes its unique fixed
point Fi,,.'.,Fi, = F = (I- A)-X(v). Note that since S
is contracting, I - A is invertible. The coordinates F1 and
F2 of F are multiplied by a normalizing factor N, and then
quantized to the nearest integer to obtain a point p with
integral coordinates (Pl,P2) E Z 2. A point q = (qx,q2)
on the screen is white (black) as it is (or is not) of form
(pl,p2), as described above.
The fl'actal set .7"(S), thus constructed, corresponds to
a black-and-white image. In order to introduce grey levels
into 9r(S), one would like to use the density of the points
{Fi_ ,..,, Fik} as a measure of the brightness of the pixels.
To do so, it is convenient to regard the procedure for the
generation of a fractal set as a random process. In fact, let
Pl,"' ,pn be positive real numbers such that Epi = 1, and
z E R q. Consider the random process X where x ---* Si(x)
with probability pi- This process has a unique stationary
distribution/t. The stationarity condition is expressed by
the equation
u = pis;,(F,) (2)
where Si.(It) is the transform of the measure/_ under the
affine transformation Si. Note that Eq. (2) is a more
precise version of the fundamental self-similarity prop-
erty expressed by Theorem 1. The measure p is the
mathematical representation of a grey-scale image on the
screen. The support of the measure /t is the fractal set
-¢'(S), and is independent of the choice of positive num-
bers Pi. Furthermore, if _Tk denotes the discrete probabil-
ity measure naturally assigned to the set {Fi_ _-.., Fi_ [ l <
k and all ix,... ,it}, then
C_ ---*/t weakly
if all Pi = 1In.
By introducing an alternative method for generating
the fractal set .7"(S), one can take advantage of the prob-
abilities {Pi) in actual image generation. Consider a
realization of the random process X. This can be in-
terpreted as follows: Using a random-number generator,
one generates a sequence of integers {i1,i2,'"}, where
1 < ik < n and the integer j is chosen with probability
pj. A realization of the process 2" is then the sequence of
points Si, (z), Si_Sq (z),.... Let ._4k be the discrete prob-
ability measure assigned to the set {Sit(z),Si_Six(z),...,
Sik'" Si_Si_ (z)}. Then, one can show by standard argu-
ments that
Theorem 2, With probability 1, Mk ---* # weakly.
After possibly multiplying by a factor N and quantiz-
ing, one can regard a point Si_ ... Si_Si, (z) in a realization
of the process X as the point qi,..,i, = q = (qa,qz) _ Z 2,
One can make a histogram of the number of times the
points of the integer lattice Z 2 are generated in this fash-
ion. Grey levels are accordingly assigned to the points so
that the points generated more often have higher intensity
(are whiter) than those generated fewer times. The addi-
tional input consisting of the positive numbers {Px, "" ", P,}
22
allowsoneto havebettercontroloverthedensityof the
points,i.e.,thegreylevels.
In orderfor anyprocedureto havepracticalapplica-
tions,it mustbestablerelativeto the variationof the
parameters.In thecaseoffractals,onewouldlikethefun-
damentalpropertyofself-similarity,asexpressedbyThe-
orcm1, to have the necessary stability. This means that
if the transformations Si are such that Uinl Si(K) is ap-
proximately identical with K, then the fractal set _'(S) is
also approximately identical with K. It is not difficult to
establish, in a quantitative sense, the validity of this sta-
bility, which yields a slight generalization of Theorem 1.
It is this mild generalization of Hutchinson's theorem that
has been widely publicized by M. Barnsley as the Collage
Theorem [2].
Just as in Example 1, where Theorem 1 was used to
generate convex polygons as fractal sets, other natural-
looking objects were generated by appealing to the ap-
proximate version and the experimental insight into the
effects of the variation of the parameters on p. In actual
applications, it is also convenient to decompose the matrix
Ai as a product
A_ = 0 0 b_a sin0 cos0 )
where [bbq = 1 and 0 < c_ < 1. The advantage of using
this decomposition is that the effects of the parameters
can be more readily understood. The matrix
cos 0 - sin 0'_
sinO cosO)
is a rotation through angle 0. The matrix
b% )
represents scaling by factors ba and-b%, and
;)
can be regarded as a twist. The numbers b, b', and a should
be chosen such that [ba[, ]b'a[ < 1. By specializing the
parameters b, b', and a to l, 1, and 0, respectively, one can
already generate a number of very complex and natural-
looking images. The interested reader should view these
points as hints or general guidelines for experimentation
with fractal image generation.
Using the fractal method, a number of images were gen-
erated by one of the authors and others (see, e.g., [2] and
[31). FracLM methods have been used recently at JPL for
computer simulation of certain images. Since these im-
ages were generated by storing only the parameters of a
few afflne transformations, one would like to believe that
the fractal method can be used for efficient data, or more
specifically, image compression. The systematization of
the fractal method so that it becomes applicable to gen-
eral image compression has been attempted by a num-
ber of researchers. One approach is to try to find a set
S = {S1,"',Sn} of affine transformations and positive
numbers {Pi} such that the corresponding stationary dis-
tribution is a good approximation to a given image. While
in theory this is possible, the number of the afflne transfor-
mations may be so large that the result will have no practi-
cal value. Furthermore, since most objects do not exhibit,
even remotely, the self-similarity property that is an es-
sential feature of fractals, this direct approach is probably
a futile one. By segmentation of an image, one will have
better control over the choice of the afflne transformations.
However, the startling compression ratios of thousands to
one will not be achievable. The most successful attempt of
the application of fractals to image compression has been
by Iterated Systems Inc. While their methodology is a
well-guarded trade secret, the authors have tested the re-
sults of their fractal compression scheme against the JPEG
baseline algorithm. This will be discussed in more detail
in Section IV.
III. JPEG Algorithms
With the advent of multi-media services offered by the
64-Kbit/sec Integrated Services Digital Networks (ISDN),
there is a strong urge to define a standard for applica-
tions as diverse as photo-videotex, desktop publishing,
graphic arts, color facsimile, photojournalism, medical sys-
tems, and many others. The JPEG was formed under
the joint auspices of the International Standards Organi-
zation (ISO) and the Comit6 Consultatif International de
T616phone et T616communication (CCITT) at the end of
1986 for the purpose of developing an international stan-
dard for the compression and decompression of continuous-
tone, still-frame, monochrome, and color images.
The JPEG-proposed algorithm has three major com-
ponents. The first is a baseline system that provides a
23
simpleandefficientalgorithmthat is adequatefor most
image-codingapplications.Thesecondisasetofextended
systemfeaturesthatallowsthebaselinesystemto satisfya
broaderrangeof applications.Amongtheseoptionalfea-
turesare12-bit/pixelinput,progressives quentialndhi-
erarchicalbuild-up,andarithmeticoding.Thethirdisan
independent,differential,pulse-codemodulation(DPCM)
schemefor applicationsthat requirelosslesscompression.
Thefollowingis abriefdescriptionof theJPEGbaseline
systemonly.Thereaderisreferredto theJPEGproposal
[4]for acompletedescriptionof thecomponentsandthe
algorithms.
TheJPEGbaselinesystemis a transform-basedalgo-
rithmconsistingof threestages.Thefirst stageis adis-
cretecosinetransform(DCT).Theoutputof theDCTis
thenquantizedandin thefinalstagethequantizedoutput
isencodedbyvariablelengthcodes.Tileoriginalimageis
partitionedinto8x 8pixelblocksandeachblockis inde-
pendentlytransformedbytheDCT.Thetransformcoeffi-
cientsarethenquantizedusingauser-definedquantization
templatethat is fixedfor all blocks.Eachcomponentof
thequantizationtemplateisan8-bitintegerandispassed
tothereceiveraspartoftheheaderinformationthatis re-
quiredfor everyimage.Up to fourdifferentquantization
templatescanbespecified;for example,differentquan-
tizationtemplatesmaybeusedfor thedifferentcompo-
nentsofacolorimage.TheJPEGbaselinesystemsupplies
twodefaultquantizationtemplates:onefortheluminance
component( heY-component) and the other for the two
chrominance components (the I and Q components). The
top-left coefficient in the two-dimensional DCT block [i.e.,
the (0, 0) coefficient] is the DC coefficient and is propor-
tional to the average brightness of the spatial block. The
remaining coefficients are called the AC coefficients. After
quantization, the DC coefficient is encoded with a lossless
DPCM scheme using the quantized DC coefficient from
the previous block as a one-dimensional predictor. For the
baseline system, up to two separate Huffman tables for
encoding the resulting differential signal can be specified
in the header information. A default Huffman table for
DC encoding is given in the JPEG proposal. The encod-
ing of the quantized AC coefficients uses a combination
of runlength and tIuffman coding techniques. There are
many zeros in the quantized AC coefficients, especially in
the high frequencies. The AC coefficients that are close
(respectively, far) in location to (0, 0) are the low (respec-
tively, high) frequencies. Typically, high frequencies have
low energies. The two-dimensional block of quantized coef-
ficients istransformed into a one-dimensional vector using
a zigzag reordering so that the coefficients are arranged
in approximately decreasing order of their average energy.
This creates a combination of nonzero values at the begin-
ning of the vector and long runs of zeros thereafter. To
encode the AC coefficients, each nonzero coefficient is first
described by a composite 8-bit value, denoted by I, of the
form (in binary notation)
I = NNNNSSSS
The four least significant, bits, SSSS, define a category
for the coefficient amplitude. The values in category k
are in the range (2_-1,2 k - l) or (-2 k + 1,-2k-1), where
1 < k _< 10. Given a category, it is necessary to send
an additional k bits to completely specify the sign and
magnitude of a coefficient within that category. The four
most significant bits in the composite value, i.e., NNNN,
give the position of the current coefficient relative to the
previous nonzero coefficients. The runlengths specified by
NNNN can range from 0 to 15, and a separate symbol
I = 11110000 = 240 is defined to represent a runlength
of 16 zeros. In addition, a special symbol, I = 0, is used
to code the e_nd of a block (EOB), which signals that all
the remaining coefficients in the block are zero. Therefore,
the total symbol set contains 162 members (10 categories
× 16 runlength values + 2 additional symbols). The out-
put symbols for each block are then Huffman coded and
are followed by the additional bits required to specify the
sign and exact magnitude of the coefficient in each of the
categories. Up to two separate Huffman tables for the AC
coefficients can be specified in the baseline system. A de-
fault runlength/IIuffman table for AC encoding is given in
the JPEG proposal.
IV. The Comparison
For the comparison of the fractal and the JPEG algo-
rithms, a set of ten 320 x 200 8-bit grey-scale images in
Sun raster format was sent to Iterated Systems Inc. Some
of these images are often used for image-compression ex-
periments, and others are planetary images. The complete
list is (1) an air scene, (2) baboons, (3) a couple in a room,
(4) Lena (portrait of a young woman often used in image
compression tests), (5) peppers, (6) light effects pattern,
(7) the moon, (8) Miranda, and (9) and (10) two images
of the planet Saturn. Each of the images was compressed
to eleven files of sizes ranging from 3 to 13.5 Kbytes.
The reconstruction was done by the decompression soft-
ware P.OEM TM Developers' Kit--Gra_lscale Still devel-
oped by Iterated Systems Inc. for this purpose and the
format conversion developed by one of the authors. The
recopstructed images were then compared to the original
to obtain the mean-square-error (MSE) and the peak-SNR
24
values versus the number of bits per pixel (i.e., compres-
sion ratio). Peak SNR is defined as lOloglo(255)2/MSE,
and is measured in dB's. Similar curves were obtained for
the JPEG baseline algorithm as implemented at JPL [5].
The results are given in Figs. 1 and 2 for images (5) and
(8). It is clear from the figures that the JPEG algorithm is
superior to the fractal algorithm by as much as 6 dB at low
compression ratios and 2 dB at high compression ratios.
The curves presented in this article contain the range of
compression ratios obtained by Iterated Systems Inc. for
the ten images referred to earlier. It should be pointed out
that no conclusion regarding the effectiveness of the fractal
method at higher compression ratios is warranted at this
time. Qualitatively, at high compression ratios of about
16:1 to 20:1, the fractal scheme exhibits solarftare, while
the JPEG baseline algorithm suffers from the tiling effect.
No post-processing was done with the application of the
JPEG algorithm. The authors are unaware of any post-
processing in the fraetal algorithm. The tiling effect can
be somewhat alleviated by the application of certain filters
which involve weighted averages of nearby neighbors. The
authors know of no method for dealing with the solar flare
effect.
V. Conclusion
A comparative study of the JPEG baseline algorithm
and the fractal method was conducted by the authors. In
terms of the mean square error and peak SNR, the JPEG
algorithm was superior in every ease.
References
[1] J. Hutchinson, "Fractals and Self-Similarity," Indiana University Journal of
Mathematics, vol. 30, no. 5, pp. 713-747, 1981.
[2] M. Barnsley, Fractals Everywhere, New York: Academic Press, 1988.
[3] P. Diaconis and M. Shahshahani, "Products of Random Matrices and Computer
Image Generation," Contemporary Mathematics, vol. 50, pp. 163-173, American
Mathematical Society, Providence, Rhode Island, 1986.
[4] W. B. Pennebaker to ISO Group X3L2.8, JPEG Draft Technical Specification
Revision 8, IBM, Yorktown Heights, New York, August 17, 1990.
[5] F. Pollara and S. Arnold, "Emerging Standards for Still Image Compression: A
Software Implementation and Simulation Study," TDA Progress Report 42-104,
vol. October-December 1990, Jet Propulsion Laboratory, Pasadena, California,
pp. 98-101, February 15, 1991.
25
za.
52-
4B"
44-
40.
36-
32
. FRACTAL COMPRESSION
o JPEG COMPRESSION
T'+++gt II II _i+I+)I+I_i+i+14 pill[ In I111+I+ II ,I+4+AII It vl+l+ll+llllpltl+ll
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
BITS PER PIXEL, bpp
Fig. 1. Fractal performance on peppers,
42 --
40
rn
"o 36
If
Z
co
u_ 34 ¸
n
32
30 ¸
28
, FRACTAL COMPRESSION
o JPEG COMPRESSION
......... I ......... L ......... t ........ 210 ........ I0 0.5 1.0 1.5 2.5
BITS PER PIXEL, bpp
Fig. 2. Fractal performance on Miranda.
26


A comparison of the fractal and JPEG algorithms

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