Journal of Signal and Information Processing, 2013, 4, 176-181
doi:10.4236/jsip.2013.43B032 Published Online August 2013 (http://www.scirp.org/journal/jsip)
Study of Symmetry Process Behavior in Fractal Gray
Image Compression by Traditional Method
Eman A. Al-Hilo, Kawther H. Al-Khafaji
College of Education for Girls, Physics Departments, Kufa University, Najaf, Iraq.
Received June, 2013
ABSTRACT
This paper studies the effect of symmetry process on the compression parameters of the fractal image compression tech-
nique proposed by Jacquin. Two kinds of tests have been conducted. The first all kind of the symmetry operations [0-7]
were taken; while the second tests were concentrated on studying the effect of the following parameters Block Size,
Step Size, Domain Size on the probability distribution of symmetry operation. The results show that the higher value of
PSNR and the lower value of ET occur at even symmetry operation only, but compression ratio is not affected with
symmetry process. Also the occurrence probability of even symmetry is more than odd symmetry for all compression
parameters. This behaviour can be utilized to reduce the encoding time to 50% with preserving PSNR.
Keywords: Image Compression; Zero-Mean; Fractal Image Compression; Symmetry process
1. Introduction
Image coding through fractal geometry has proven to be
very effective, and fractal image compression is a prom-
ising field that has gained efficiency in both memory
consumption and high speed transmission. Fractal image
coding introduced by Barnsley and Jacquin is the out-
come of the study of the iterated function system devel-
oped in the last decade [1]. By using IFS techniques and
simple deterministic algorithms, images with spatial
complexity can be encoded through certain fractal rela-
tionships that describe mapping of blocks of images
within themselves. This fractal image technique finds
similar patterns that exist on different scales/orientations,
and at different places in an image. Hence, it allows en-
coding of an image file by systematically analyzing it
and saving a much smaller set of instructions that can be
used to iteratively reconstruct the entire image from
those patterns. In fact, this compression technique can be
thought of a method eliminating as much redundancy as
possible. [2-3]
Saup [4] represents selected studies about the influ-
ence of symmetry on the quality of fractal codes. After-
wards an empirically comparison of the efficiency of
domain pools with same size with or without using
symmetry blocks follows. The presented results show
that the using of symmetry doesn't increase the efficiency
of codes.
The aim of our project is studding the behavior of
symmetry process in the fractal gray image compression.
2. Fractal Image Coding
The basic idea of fractal image compression is partition-
ing the image into non overlapping range blocks. For
every range block a similar but larger domain block is
found. There are many ways to partition images. The
fixed size partitioning are used in this research because it
requires less computational time than the other.[5]
For a range block with pixel values (ro,r1,…,rm-1), and
the domain block (do,d1…,dm-1) the contractive affine
approximation is [6]:
osdr ii 


, (1)
where, i
r

is the optimally approximated ith pixel value
in the range block. di is the corresponding pixel value in
the domain block. The symbols s,o represent the scaling
and offset coefficients, respectively.
The scale (s) and offset (o) coefficients are determined
by applying the least mean square difference (χ2) criteria
between (
r

r
) and () values [7].

12
2
11
0
1m
i
rr
m



 


, (2)
22
0, 0,
so
 


 (3)
The straightforward manipulation of the above equa-
tion leads to:

Study of Symmetry Process Behavior in Fractal Gray Image Compression by Traditional Method 177























1
0
2
1
0
2
1
0
1
0
1
0
n
i
n
i
ii
n
i
n
i
i
n
i
iii
ddn
drdrn
s, (4)

























1
0
2
1
0
2
1
0
1
0
1
0
1
0
2
n
i
n
i
ii
n
i
n
i
ii
n
i
i
n
i
ii
ddn
ddrdr
o, (5)
1111
22 2
0000
1
0
22
2
nnnn
iiii
iiii
n
i
i
rssdrd od
ono r






 





 





i 

(6)
3. Symmetry Process
In order to increase the size of domain pool and, conse-
quently, to increase the probability of finding the best
(near optimal) approximations for the range blocks, each
domain block is transformed by using a set of symmetry
transforms (i.e., rotation and flipping), such that eight
versions (blocks) are produced for each domain block [8].
The eight symmetry mappings are (identity, rotation 90,
rotation 180, rotation 270, reflection-x, reflection with
rotation 90, reflection with rotation 180, reflection with
rotation 270), (see Table 1). The number of bits required
to represent the eight symmetry mapping cases is (3) bits.
Table 1. The eight symmetry transformations.
Sym Equations Results
0. Identity




0sin0cos yxx 

  0cos0sin yxy  xx


yy


1. Rot.(+90)
  90sin90cos yxx 





90cos90sin yxy 

yx


xy



2. Rot.(+180)




180sin180cos yxx

  180cos180sin yxy 

xx



yy



3. Rot.(+270)
  270sin270cos yxx 

  270cos270sin yxy 

yx



xy


4. Ref. at
x-axis




0sin0cos yxx 

  0cos0sin yxy 

xx



yy


5. Ref.& Rot.
(90)
  90sin90cos yxx 





90cos90sin yxy 

yx



xy



6. Ref.& Rot.
(180)
  180sin180cos yxx 


  180cos180sin yxy 

xx 

yy



7. Ref & Rot.
(270)
  270sin270cos yxx

  270cos270sin yxy 

yx


xy


The main disadvantage of using the symmetry map-
pings in the encoding process is that more computational
time will be required to perform the extra matching
processes.
4. Encoding Process
The encoding method could be summarized by the fol-
lowing steps:
1) Load BMP image and put it in (2D arrays).
2) Establish the range image (array).
3) Down sample the range image to produce the do-
main array.
4) Great range and domain pool by partitioning:
a) The range array must be partitioned into non-over-
lapping fixed blocks, to generate the range blocks
(r1,….,rn).
b) The domain must be partitioned into overlapping
blocks, using specific step size, to generate the domain
blocks (d1,…,dn). They should have the same size of
range blocks.
5) Searching: For each range block do the following:
a) Pick up a domain block from the domain pool.
b) Perform one of the symmetry mappings that men-
tioned in table (1).
c) Calculate the scale (s) and offset (o) coefficient us-
ing equations (4) and (5).
d) Apply the following condition to bound the value of
(s) and offset (o) coefficient:
If s< s
min then s=smin
Else if s >smax then s=smax
If o< o
min then o=omin
Else if o >omax then o=omax
e) Quantize the value (s) and offset (o) using equations
that referred in [9].
f) Compute the approximation error (χ2) using equa-
tion (6).
g) After the computation of IFS code and the sum of
error (χ2) of the matching between the range and the
tested domain block, the (χ2) is compared with registered
minimum error (χ2
min); such that:
If χ2< (χ2
min) then
sopt=is; oopt=io, χ2
min= χ2
PosI=domain block index
Sym=symmetry index
End if
h) If χ2
min <

then the search across the domain
blocks is stopped, and the registered domain block is
considered as the best matched block.
i) Repeat steps (4) to (10) for all symmetry states of
the tested domain block.
j) Repeat steps (3) to (11) for all the domain blocks
listed in the domain pool.
k) The output is the set of IFS parameters

Study of Symmetry Process Behavior in Fractal Gray Image Compression by Traditional Method
178
 SymposIiiei os ,,,.,. 
which should be registered as
a set of fractal coding parameters for the tested range
block.
l) Repeat steps (1) to (12) for all range blocks listed in
the range pool
m) Store all IFS mapping parameters as an array of
record. The length of this array is equal to the number of
range blocks in the range pool.
5. Decoding Process
The decoding process can be summarized by the follow-
ing steps:
1) Generate arbitrary the domain pool, the domain
pool could be initialized as a blank image or a piece of
image extracted from any available image.
2) The values of the indices of (is) and (io) for each
range block should be mapped to reconstruct the quan-
tized values of the scale (sq) and offset (oq) coefficients.
3) Choose the number of possible iterations, and the
threshold value of the mean square error (TMSE). At
each iteration, do the following steps:
a) For each range block determine the coordinates
(xd,yd), of the best matched domain, from the IFS pa-
rameters (posI), in order to extract the domain block (d)
from the arbitrary domain image.
b) For each range block, its approximation i
r

is ob-
tained by multiplying the corresponding best matched
domain block (d) by the scale value (sq) and adding to the
result the offset value (oq), according to equation (1).
c) The generated i block is transformed (rotated,
reflected, or both) according to its corresponding IFS
symmetry parameter value (Iso).
r
d) Put the generated iblock in its position in the de-
coded image array (i.e., range image).
r
e) Check whether there is another range block, if yes
then repeat steps (b,c,d)
f) Down sample the reconstructed image (range pool)
in order to produce the domain pool using the averaging
sampling.
g) Calculate the mean square error MSE between the
reconstructed range and the previous reconstructed range
image. If the MSE is greater than TMSE value then the
iteration continues and the above steps (a-f) should re-
peated; this iteration is continued till reaching the attrac-
tor state (i.e., the newly reconstructed range image is
very similar to the previous reconstructed image). Oth-
erwise the iteration continues till reaching the predefined
maximum number of iterations.
6. Tests Results
The proposed system was established using Visual Basic
(Ver.6.0) and tested on Aser laptop with (2.20GHZ,
RMA 956MB)
The proposed system had been tested on Lena image
(256x256 pixels, 8 bits). The value of the parameters
MaxOffset and MinOffset were fixed in all these tests at
(255) and (-256) respectively but the other coding pa-
rameters were taken as: BlockSize=(4x4), StepSize=2,
DomSize= (128x128), ScaleBits=6, OffsetBits=6, Min-
Scale=-1.5, MaxScale=3,

=0.4,TMSE=0.05. These
tests were taking two aspects:
6.1. Symmetry Tests
These tests were conducted to investigate the effect of
each symmetry operation, subset [0-3], subset [4-7], and
full symmetry [0-7], on compression performance pa-
rameters. Table 1 shows these effects on the compres-
sion parameters MSE, RMSE, PSNR, CR and ET.
Table 1. Effect of each, subsets, and full Symmetry on the
compression performance parameters.
Sym MSE RMSE PSNR CR ET(sec)
0 31.73 5.63 33.12 4.741 11.97
1 40.60 6.37 32.05 4.741 12.38
2 31.71 5.63 33.12 4.741 12.14
3 39.21 6.26 32.19 4.741 12.37
4 33.09 5.75 32.93 4.741 12.03
5 38.60 6.21 32.26 4.741 12.38
6 32.73 5.72 32.98 4.741 12.01
7 38.39 6.19 32.29 4.741 12.38
Subsets and Full Symmetry Operation
(0-3) 25.16 5.02 34.12 4.741 45.42
(4-7) 24.39 4.94 34.26 4.741 45.49
(0-7) 21.79 4.67 34.75 4.741 89.67
The results show that best PSNR occur when all sym-
metry operations are used. The symmetry (0) appears
higher PSNR and lower ET among (8) symmetry opera-
tion. CR is not affected by symmetry operation.
Figure 1 shows the effects of each symmetry opera-
tion alone on the compression performance parameters
MSE, RMSE, PSNR, and ET respectively.
The results show that the value of MSE, RMSE and
ET appear lower at symmetry operations (0,2,4,6) with
respect to their values at symmetry (1,3,5,7), But PSNR
appears higher at symmetry operations (0,2,4,6) with
respect to their values at symmetry operations (1,3,5,7).
This indicate that PSNR is behaves inversely with ET.
Figure 2 shows a set of the reconstructed images when
no symmetry (identical) and full symmetry operations
were applied.

Study of Symmetry Process Behavior in Fractal Gray Image Compression by Traditional Method 179
Figure 1. Effect of each Symmetry operation on MSE,
RMSE, PSNR and ET respectively.
Original No Symmetry Full (0-7)
PSNR 33.12 34.75
ET 12.36 89.67
Figure 2. Effects of different symmetry on the compression
performance parameters.
6.2. Symmetry Distribution Tests
In these tests, the numbers of blocks that have the same
Symmetry status from [0-7] were counted. The results of
these tests are useful to know the most redundant sym-
metry case, which is useful to reduce the consumed en-
coding time.
These tests were concentrated on studying the effect of
the following parameters:
1) Block Size: In this test set different BlockSize val-
ues (i.e., 2x2, 4x4, 8x8, 16x16) were taken into consid-
eration, while the values of the StepSize = 2, DomSize =
(128x128) were kept fixed. Table 2 shows the effect of
BlockSize parameter on symmetry distribution.
The results show that the occurrence probability of
even symmetry increases with the increase of BlockSize
value. The highest probability is (76%) at BlockSize
(16x16)
Figure 3 shows the effect of BlockSize on the sym-
metry distribution. The results show that the even states
(0,2,4,6) have the highest populations of symmetry.
2) Step Size: This subset of tests investigates the effect
of different StepSize on symmetry distribution. In this
test, different values of StepSize (1,2,3,4) were taken and
the values of other coding parameters were fixed at
BlockSize = (4x4), DomSize = (128x128). Table 3 illus-
trates the effect of StepSize on symmetry distribution.
Figure 3. BlockSize effect on the symmetry distribution.
Table 2. Effect of BlockSize on the Sym distribution.
(BlockSize)
Sym
2x2 4x4 8x8 16x16
0 3369 689 236 74
1 2209 366 65 15
2 2792 685 183 45
3 2125 371 66 11
4 1735 648 154 45
5 1210 404 75 17
6 1698 568 156 32
7 1246 365 89 17
Tot 16384 4096 1024 256
Even% 58.56% 63.23% 71.19% 76.56%
Odd% 41.44% 36.77% 28.8% 23.44%
Table 3. Effect of StepSize on the symmetry distri bution.
(StepSize)
Sym
1 2 3 4
0 735 689 725 686
1 365 366 368 380
2 665 685 674 674
3 369 371 371 366
4 620 648 578 598
5 401 404 405 399
6 589 568 602 623
7 352 365 373 370
Tot 4096 4096 4096 4096
Even% 63.69% 63.23% 62.96% 63.01%
Odd% 36.3% 36.76% 37.04% 36.99%

Study of Symmetry Process Behavior in Fractal Gray Image Compression by Traditional Method
180
The results show that the probability of occurrence of
even and odd symmetry is not affected by the variation
of StepSize value. The highest probability at even sym-
metry is (63.69%) at StepSize (1) and at odd symmetry is
(37.04) at StepSize (3).
Figure 4 shows the effect of StepSize on symmetry
distribution.
Figure 4. StepSize effect on the Symme try distribution.
3) Domain Size: This subset of tests was conducted to
investigate the effect of DomSize on the symmetry dis-
tribution. The value of DomSize was varied to (128x128,
64x64, 32x32, 16x16), while the values of other coding
parameters were taken at BlockSize = (4x4), StepSize =
(2). Table 4 shows the effect of DomSize on symmetry
distribution.
Table 4. Effect of DomSize on the Symmetry distribution.
(DomSize)
Sym
128 64 32 16
0 689 676 697 855
1 366 353 333 312
2 685 693 728 720
3 371 355 337 300
4 648 624 639 617
5 404 359 346 315
6 568 679 639 622
7 365 357 377 355
Tot 4096 4096 4096 4096
Even% 63.23% 65.23% 65.99% 68.70%
Odd% 36.77% 34.77% 34.01% 31.29%
The results show that the probability of occurrence of
even symmetry increases with the increase of DomSize
value, but at odd symmetry decrease with the increase of
DomSize value. The highest probability is (68%) at
DomSize (16x16) at even symmetry and (36.77%) at
DomSize (128x128) at odd symmetry. Figure 5 shows
the effect of DomSize on symmetry distribution.
Figure 5. DomSize effect on the symmetry distribution.
7. Conclusions
From the above listed results could be concluding the
following:
1) The higher value of PSNR at even symmetry and
the lower value of ET at even symmetry also, but CR is
not affected with symmetry process.
2) The Most probable symmetry state is (0: identical
symmetry) for all compression parameters (BlockSize,
StepSize and DomSize).
3) The occurrence probability of even symmetry (0,2,4,
6) is more than odd symmetry
The high probability of occurrence of even symmetry
could be utilized as an efficient way to reduce the en-
coding time in matching process. So, instead of using (8)
symmetry operations one can use only the even symme-
try operations (2,4,6,8). In such a case, the expected re-
duction in encoding time will be around 50%.
REFERENCES
[1] A. Jacquin, “Fractal Image Coding a Review,” Process
ding of the IEEE, Vol. 81, 1993, pp. 1451-1465.
[2] M. F. Barnsley, “Fractals Everywhere,” New York Aca-
demic, second edition.
[3] A. Kapoor, K. Arora, A. Jain and G. P. Kapoor, “Stochas-
tic Image Compression Using Fractals,” International
conference on Information Technology: Coding and
Computing (ITCC 2003) Las Vegas, Nevada, Pg. 574-579,
28-30 April 2003.
[4] D. Saupe, “The Futility of Square Isometries in Fractal
Image Compression,” IEEE Int. Conf. On Image Proc-
essing (ICIP96), Lausanne, Sept. 1996.
[5] M. Schebe, “Square Isometries as Integer Part of Fractal
Transformation- An Analysis ,” FREQENZ,12 De,1996.
[6] Y. Fisher, “Fractal Image Compression Theory and Ap-
plication,” University of California, Institute for Nonlin-
ear Science, Springer-Verlay, New York, Inc, 1995.
[7] Y. Fisher, “Fractal Image Compression,” SIGARAPH 92
Course Notes, the San Diego Super Computer Center,

Study of Symmetry Process Behavior in Fractal Gray Image Compression by Traditional Method
181
University of California, an Diego, 1992
[8] C. Frigaard, “Fast Fractal 2D/3D Image Compression,
Report,” Institute of Electronic Systems, Alborg Univer-
sity, Laboratory of Image Analysis, 1995.
[9] E. A. Al-Hilo, “Loay E. George, "Fractal Color Image
Compression,” 4th International Conference on Nov.
2008 (ICIMU’ 2008), Malaysia.