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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) 238 http://sites.google.com/site/ijcsis/ 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 239 http://sites.google.com/site/ijcsis/ 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: 240 http://sites.google.com/site/ijcsis/ 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 241 http://sites.google.com/site/ijcsis/ 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: COMPARISON OF METHODS, IEEE T RANSACTIONS ON IMAGE Fig. 4. Original 256 x 256 Lena image PROCESSING 9 (2000) 1002–1009. [6] T.K. TRUONG, J.H. JENG, I.S. REED, P.C. LEE, A.Q. LI, A FAST ENCODING ALGORITHM FOR FRACTAL IMAGE COMPRESSION USING THE DCT INNER PRODUCT, IEEE T RANSACTIONS ON IMAGE PROCESSING 9 (4) (2000) 529–535. [7] M.S. WU, J.H. JENG, J.G. HSIEH, SCHEMA GENETIC ALGORITHM FOR FRACTAL IMAGE COMPRESSION, E NGINEERING APPLICATIONS OF ARTIFICIAL I NTELLIGENCE 20 (2007) 531–538. [8] M.S. WU, W.C. TENG, J.H. JENG, J.G. HSIEH, SPATIAL CORRELATION GENETIC ALGORITHM FOR FRACTAL IMAGE COMPRESSION, CHAOS SOLITONS & FRACTALS 28 (2) (2006) 497– 510. [9] Y. FISHER, FRACTAL IMAGE COMPRESSION—THEORY AND APPLICATION. NEW YORK: SPRINGER -VERLAG, 1994. [10] J. KENNEDY, R.C. EBERHART, PARTICLE SWARM OPTIMIZATION, IN: PROCEEDINGS OF IEEE INTERNATIONAL C ONFERENCE ON (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 [13] B. HURTGEN, C. STILLER, FAST HIERARCHICAL CODEBOOK SEARCH FOR FRACTAL CODING STILL IMAGES, SPIE VISUAL C OMMUN. Encoding PACS MED. APPL. (1993) 397–408. 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. Proposed 00:15:34 35.03 ARCHIT. 46 (2000) 1275–1291. method [16] Y. FISHER, FRACTAL IMAGE COMPRESSION, SIGGRAPH’92 Full search COURSE NOTES 12 (1992) 7.1–7.19. 09:02:12 33.64 FIC Goldhill 0.5 Proposed 00:17:37 32.77 method Full search 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. 243 http://sites.google.com/site/ijcsis/ ISSN 1947-5500