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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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                                                                                                                ISSN 1947-5500
                                                                         (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                            Vol. 8, No. 2, 2010
                          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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                                                                                                                           ISSN 1947-5500
                                                                             (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                                Vol. 8, No. 2, 2010
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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                                                                                                                          ISSN 1947-5500
                                                                         (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                            Vol. 8, No. 2, 2010
    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



                  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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                                                                                                                          ISSN 1947-5500
                                                                         (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                            Vol. 8, No. 2, 2010
                                                                                                          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

                                                                                       [1]    M.F. BARNSLEY, S. DEMKO, ITERATED FUNCTION SYSTEMS AND
                                                                                              THE GLOBAL CONSTRUCTION OF FRACTALS, PROC . ROY. SOC.
                                                                                              LOND A399 (1985) 243–275.
                                                                                       [2]    A.E. Jacquin, Fractal image coding: a review, Proc. IEEE 10
                                                                                              (1993)1451–1465.
                                                                                       [3]    A.E. JACQUIN, IMAGE CODING BASED ON A FRACTAL THEORY OF
                                                                                              ITERATED CONTRACTIVE IMAGE TRANSFORMATIONS, IEEE
                                                                                              TRANSACTIONS ON SIGNAL PROCESSING 1 (1992) 18–30.
                                                                                       [4]    M.F. Barnsley, A.D. Sloan, A better way to compress images,
                                                                                              BYTE Magazine (1988) 215–233
                                                                                       [5]    M. POLVERE, M. N APPI, SPEED-UP IN FRACTAL IMAGE CODING:
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                                                                                              ENCODING ALGORITHM FOR FRACTAL IMAGE COMPRESSION USING
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                                                                                       [9]    Y. FISHER, FRACTAL IMAGE COMPRESSION—THEORY AND
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                    (a)                                (b)                                    NEURAL NETWORKS, PERTH, AUSTRALIA, VOL. 4, 1995, PP. 1942–
                                                                                              1948.
                                                                                       [11]   R.C. EBERHART, J. KENNEDY, A NEW OPTIMIZER USING PARTICLE
                                                                                              SWARM THEORY, IN: PROCEEDINGS OF IEEE INTERNATIONAL
  Fig. 5. Decoded Lena image at 0.5 bits per pixel for (a) Full search FIC                    SYMPOSIUM ON MICRO MACHINE AND HUMAN SCIENCE, NAGOYA,
              (b) proposed fast fractal encodind using PSO                                    JAPAN, 1995, PP. 39–43.
                                                                                       [12]   B. BANI-EQBAL, SPEEDING UP FRACTAL IMAGE COMPRESSION,
                               TABLE I
                                                                                              PROC . SPIE: STILL-IMAGE COMPRESS. 2418 (1995) 67–74.
                            NUMERIC RESULTS
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                 Bitrate                                            PSNR
Input image                     Method              Time                               [14]   C. HUFNAGL, A. UHL, ALGORITHMS FOR FRACTAL IMAGE
                 (bpp)                                               (dB)
                                                 (hh:mm:ss)                                   COMPRESSION ON MASSIVELY PARALLEL SIMD ARRAYS, REAL-
                              Full search                                                     TIME IMAGING 6 (2000) 267–281.
                                                   09:07:20          35.80
                                 FIC                                                   [15]   D. VIDYA, R. PARTHASARATHY, T.C. B INA, N.G. SWAROOPA,
   Lena             0.5                                                                       ARCHITECTURE FOR FRACTAL IMAGE COMPRESSION, J. SYST.
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                                                   00:15:34          35.03                    ARCHIT. 46 (2000) 1275–1291.
                                method                                                 [16]   Y. FISHER, FRACTAL IMAGE COMPRESSION, SIGGRAPH’92
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                                                   09:02:12          33.64
                                 FIC
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                                                   00:17:37          32.77
                                method
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                                                   09:02:49          35.11
 Camera                          FIC
                    0.5
  man                          Proposed
                                                   00:15:24          34.23
                                method




                                                                             242                                  http://sites.google.com/site/ijcsis/
                                                                                                                  ISSN 1947-5500
                                                                          (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                             Vol. 8, No. 2, 2010

                                                                                                            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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