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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 8, No. 2, 2010

Effective MSE optimization in fractal image
compression
A.Muruganandham Dr.R.S.D.wahida banu
Sona College of Technology, Govt Engineering College
salem-05.,India. Salem-11, India
muruga_salem@rediffmail.com, rsdwb@yahoo.com

Abstract- The Fractal image compression encodes image at low
bitrate with acceptable image quality, but time taken for encoding is II. FRACTAL IMAGE COMPRESSION ALGORITHM
large. In this paper we proposed a fast fractal encoding using The Iteration Function System (IFS) is the fundamental
particle swarm optimization (PSO). Here optimization technique is idea of fractal image compression in which the governing
used to optimize MSE between range block and domain block. PSO theorems are the Collage Theorem and the Contractive
technique speedup the fractal encoder and preserve the image
Mapping Fixed-Point Theorem [7]. The encoding unit of FIC
quality.
for given gray level image of size N x N is (N/L)2 of non-
Keywords- mean square error (MSE) ,particle swarm overlapping range blocks of size L x L which forms the range
optimization (PSO), fractal image compression (FIC), Iteration pool R. For each range block v in R, one search in the (N - 2L +
Function System (IFS) 1)2 overlapping domain blocks of size 2L x 2L which forms the
domain pool D to find the best match. The parameters
describing this fractal affine transformation of domain block
I. INTRODUCTION into range block form the fractal compression code of v.
The idea of the fractal image compression (FIC) is based on
the assumption that the image redundancies can be efficiently The parameters of fractal affine transformation Φ of
exploited by means of block self-affine transformations. The domain block into range block is domain block coordinates-(tx,
fractal transform for image compression was introduced in ty), Dihedral transformation-d, contrast scaling-p, brightness
1985 by Barnsley and Demko [1,2]. The first practical fractal offset-q. Flowchart for this fractal affine transformation is
image compression scheme was introduced in 1992 by Jacquin illustrated in fig. 1 and it is given by
[3,4]. One of the main disadvantages using exhaustive search
strategy is the very high encoding time. Therefore, decreasing
the encoding time is an interesting research topic for FIC [3, 4]. x a11 a12 0 x tx
y a 21 a 22 0 y ty ,
An approach for decreasing encoding time is using the u ( x, y ) 0 0 p u ( x, y ) q
stochastic optimization methods such as genetic algorithm (1)
(GA). Some recent GA-based methods are proposed to improve
the efficiency [5, 6]. The idea of special correlation of an image a11 a12
is used in these methods, which is of great interesting. While
the chromosomes in GA consist of all range blocks which leads a 21 a 22
where the 2 x 2 sub-matrix is one of the
to high encoding speed and particular properties of natural Dihedral transformations in (2)
images have never been used that will results in lose of visual
effect in the retrieved image. 1 0 1 0 1 0
T0 , T1 , T2 ,
Other researchers focused on improvements of the search 0 1 0 1 0 1
process to make it faster by tree structure search methods
[12,13], parallel search methods [14,15] or quad tree
partitioning of range blocks [9,16,].
1 0 0 1 0 1
In this paper, we present a fast fractal encoding using T3 , T4 , T5 ,
particle swarm optimization. The outline of the remaining part 0 1 1 0 1 0
of this paper is as follows: Section II includes fractal image
coding. Section III involves implementation of PSO. Section
IV concerns the proposed fast fractal encoder using PSO, and 0 1 0 1
in Section V experimental results are included. In Section VI, T6 , T7
we present our conclusions. 1 0 1 0
(2)

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Input image of size
NxN
L 1 L 1 L 1 L 1
L2 u k , v i 0 j 0
u k (i, j ) i 0 j 0
v(i, j )
2
Partition input image into (N/L) non overlapping range
pk 2
,
L 1 L 1
blocks of size L x L and (N-2L+1)2 overlapping domain L2 u k , u k i 0 j
u (i, j )
0 k
blocks of size 2L x 2L
(5)

For Range blocks 1 to (N/L) 2
1 L 1 L 1 L 1 L 1
qk v(i, j ) pk u k (i, j ) .
For Domain blocks 1 to (N-2L+1)2 L2 i 0 j 0 i 0 j 0

(6)
For Orientation 1 to 8
4. As u runs over all of the domain blocks in D to find
Calculate Scaling, Offset and the best match, the terms tx and ty can be obtained
Mean Square Error together with d and the specific p and q corresponding
this d, the affine transformation (1) is found for the
given range block v.
In practice, tx, ty, d, p, and q can be encoded using log2(N),
Store quantized scaling and offset, log2(N), 3, 5, and 7 bits, respectively, which are regarded as the
orientation and position of the domain
block of minimum MSE compression code of v. Finally, the encoding process is
completed as v runs over all of the (N/L)2 range blocks in R.
Fig. 2 show the MSE vs. quantization parameter for an
randomly selected range block of size 8 x 8 from 256 x 256
Fig. 1 fractal affine transformation of domain block into Lena image. From fig. 2, choosing 5 bits and 7 bits as
range block quantization parameter for scale and offset value is justified.
The above parameters are found using the following 16
procedure scale bits = 3
4
1. the domain block is first down-sampled to L x L and 14
5
denoted by u 6
12 7
2. The down-sampled block is transformed subject to the 8
eight transformations Tk:k = 0,. . . ,7 in the Dihedral 9
MSE

on the pixel positions and are denoted by uk, k = 0,1, . 10 10

. . ,7, where u0 = u. The transformations T1 and T2 11

correspond to the flips of u along the horizontal and 8
vertical lines, respectively. T3 is the flip along both the
horizontal and vertical lines. T4, T5, T6, and T7 are the 6
transformations of T0, T1, T2, and T3 performed by an
additional flip along the main diagonal line, 4
respectively. 3 4 5 6 7 8 9 10 11
no. of bits for Offset quantization
3. For each domain block, there are eight separate MSE
computations required to find the index d such that Fig. 2 MSE Vs. quantization parameter
To decode, chooses any image as the initial one and makes
d arg min{ MSE (( p k u k q k ), v) : k 0,1,...,7} up the (N/L)2 affine transformations from the compression
(3) codes to obtain a new image, and proceeds recursively.
According to Partitioned Iteration Function Theorem (PIFS),
where the sequence of images will converge. According to user’s
1 L 1 application the stopping criterion of the recursion is designed.
MSE (u , v) (u (i, j 0) v(i, j )) 2 The final image is the retrieved image of fractal coding.
L2 i, j 0
(4) III. PARTICLE SWARM OPTIMIZATION (PSO)
Here, pk and qk can be computed directly as PSO is a population-based algorithm for searching global
optimum. It ties to artificial life, like fish schooling or bird
flocking, and has some common features of evolutionary

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(IJCSIS) International Journal of Computer Science and Information Security,
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computation such as fitness evaluation. The original idea of (Start)
PSO is to simulate a simplified social behavior [8, 9]. Similar
to the crossover operation of the GA, in PSO the particles are
adjusted toward the best individual experience (PBEST) and
Initialization
the best social experience (GBEST). However, PSO is unlike a
GA in that each potential solution, particle is ‘‘flying” through
hyperspace with a velocity. Moreover, the particles and the Fitness evaluation
swarm have memory; in the population of the GA memory
does not exist. Experience
Let xj,d(t) and vj,d(t) denote the dth dimensional value of the updating
vector of position and velocity of jth particle in the swarm,
respectively, at time t. The PSO model can be expressed as

v j ,d (t ) v j ,d (t 1) c1. 1.(x*,d
j x j ,d (t 1)) No Stopping
Moving criterion
#
c2 . 2 .(x d x j ,d (t 1)), reached?

(7) Yes

(Stop)
x j ,d (t ) x j ,d (t 1) v j ,d (t ),
(8)
Fig. 3 Block diagram of PSO
*
x
where j ,d
(PBEST) denotes the best position of jth IV. PROPOSED METHOD
#
xd
particle up to time t-1 and (GBEST) denotes the best In the proposed fast fractal encoding using PSO, we reduce
position of the whole swarm up to time t-1, φ1 and φ2 are the encoding time by reducing the searching time to find a best
random numbers, and c1 and c2 represent the individuality and match domain block for the given range block from all domain
sociality coefficients, respectively. blocks.
The population size is first determined, and the Flowchart of the fractal encoding of the proposed method is
velocity and position of each particle are initialized. Each shown in fig. 4.
particle moves according to (7) and (8), and the fitness is then
calculated. Meanwhile, the best positions of each swarm and
particles are recorded. Finally, as the stopping criterion is Input image of size
NxN
satisfied, the best position of the swarm is the final solution.
The block diagram of PSO is displayed in Fig. 3 and the main
steps are given as follows: Partition input image into (N/L)2 non overlapping range
1. Set the swarm size. Initialize the velocity and the blocks of size L x L and (N-2L+1)2 overlapping domain
blocks of size 2L x 2L
position of each particle randomly.
2. For each j, evaluate the fitness value of xj and update
For Range blocks 1 to (N/L) 2
x *,d
j
the individual best position if better fitness is
found.
3. Find the new best position of the whole swarm. Calculate scale, offset, orientation and position of the
Update the swarm best position x# if the fitness of the domain block of minimum MSE from all domain
new best position is better than that of the previous blocks using PSO
swarm.
4. If the stopping criterion is satisfied, then stop.
Store quantized scaling and offset,
5. For each particle, update the position and the velocity orientation and position of the domain block
according (8) and (7). Go to step 2. of minimum MSE

Fig.4 Fractal encoding of proposed method
Domain block of minimum MSE is found by PSO
using the steps given below:

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1. Set the swarm size proportional to (N- Fig. 7 shows the variation in PSNR by varying the
2L+1)2/(maximum no. of iterations for PSO) and stopping criterion in fast fractal encoding using PSO by
initialize the velocity and the position of each particle changing the percentage of maximum iteration of PSO. From
randomly fig.7 variation of PSNR with variation of stopping criterion is
2. Fitness value includes finding MSE between domain very less. Hence a 10% of maximum iteration for PSO is
block specified by the particles position and given choose as stopping criterion.
range block using eqn. (3)
34.34
3. Update swarm best position if the fitness of the new
34.338
best position is better than that of the previous swarm.
4. If swarm best position is not changed for some 34.336

percentage of maximum iteration for PSO, then stop
34.334
PSNR (dB)
5. The best position of the particles is updated using eqn.
(7) and (8) and goto step 2. 34.332

34.33
V. XPERIMENTAL RESULTS
34.328
The fast fractal encoding using PSO results have been
compared to the full search FIC mentioned in the previous 34.326
sections in terms of encoding time and PSNR.
Fig. 5 shows the original image Lena 256 x 256 at 8bpp. 34.324
10 20 30 40 50 60 70 80
Fig. 6 shows the decoded Lena image using full search FIC Stopping criterion (percentage of maximum iteration for PSO)
and fast fractal encoding using PSO. Fig. 7 variation in PSNR by varying the stopping criterion
The numeric results containing bitrate, encoding time and
PSNR of decoded image for various images are tabulated in
table I.

Fig. 4. Original 256 x 256 Lena image

(a) (b) (a) (b)
Fig. 5. Decoded Lena image at 0.5 bits per pixel for (a) Full search FIC
(b) proposed fast fractal encodind using PSO

Fig. 5. Decoded Lena image at 0.5 bits per pixel for (a) Full search FIC
(b) proposed fast fractal encodind using PSO

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VI. CONCLUSION
Fractal image compression can produce better compression
ratio at acceptable quality. By using PSO for fractal coding we
can reduce the encoding time with 1.2dB loss in image quality.

VII. REFERENCES

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Dr.R.S.D.Wahidabanu obtained her B.E. degree in
A.Muruganandham obtained her B.E. degree in 1981 and M.E. degree in 1984 from Madras
1993 in from Madras University and M.E. degree University and Ph.D in 1998 from Anna University,
in 2000 from Bharathithasan University and doing Chennai. Her areas of interest are Pattern
Ph.D in Anna University, Coimbatorei. His areas Recognition, Application of ANN for Image
of interest are Image processing, He is a life Processing, Network Security, Knowledge
member of ISTE and member of IEEE. He is Management and Grid Computing. She is a life member of IE, ISTE, SSI,
currently working as a Asst. Professor and Electronics and Communication CSI India and ISOC and IAENG. She is currently working as a Professor
Engineering Sona College of Technology, Salem-05, Tamilnadu, India and and Head of Electronics and Communication Engineering, Government
has a teaching experience of 15 years. Authored national and international College of Engineering, Salem, Tamilnadu, India and has a teaching
conferences and journals. experience of 27 years. Authored and coauthored 180 research journals in
national and international conferences and journals.

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