Fast Fractal Image Compression using PSO Based Optimization Techniques

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1 Fast Fractal Compression using PSO Base Optimization Techniques A.Krishnamoorthy Visiting faculty Department Of ECE University College of Engineering panruti S.Buvaneswari Visiting faculty Department Of ECE University College of Engineering Panruti ABSTRACT In traitional fractal image compression, the encoing proceure is time-consuming ue to the full search mechanism. In orer to spee up the encoer, we aopt particle swarm optimization metho performe uner classification an Diheral transformation to further ecrease the amount of MSE computations. The classifier partitions all of the blocs in omain pool an range pool into three classes accoring to the thir level wavelet coefficients. Each range bloc searches the most similar bloc only from the blocs of the same class. Furthermore, accoring to the property of Diheral transformation, only four transformations for each omain bloc are consiere so as to reuce the encoing time. Experimental results show that, the encoing time of the propose metho is faster than that of the full search metho. Experimental results show that the propose metho is about 8 times faster with only.56b ecay in image quality. Keywors Fractal image compression, particle swarm optimization, Diheral transformation, Encoing time.. INTRODUCTION The iea of the image reunancies can be efficiently exploite by means of bloc self-affine transformations may call the fractal image compression (FIC), base on the partitione iteration function system (PIFS) which utilize the self-similarity on first practical fractal image compression scheme was introuce in 99 by Jacquin[]. The fractal transform for image compression was introuce in 985 by Barnsley an Demo. The very high encoing time is the main isavantages because of exhaustive search strategy. Therefore, ecreasing the encoing time is an interesting research topic for FIC. One way of ecreasing the encoing time is by using stochastic optimization methos using Genetic Algorithm (GA) this recent topics of GA-base methos are propose to improve the efficiency []. The iea of special correlation of an image is use in these methos while the chromosomes in GA consist of all range blocs which leas to high encoing spee[3]. Other researchers focuse on improvements by tree structure search methos of the search process an parallel search methos [4, 5] or qua tree partitioning of range blocs to mae it faster. transform is use to ecompose the original image to various frequency sub bans in which the attributes can be extracte from the wavelet coefficients belonging to ifferent sub-bans. The istribution of wavelet coefficients can be use in context base multiscale classification of ocument image[6]. The fast an efficient algorithm[7] was applie to triangular mesh to approximate surface ata using wavelet transform coefficients. It irectly etermine local area complexity in an image an ivies square cells epening on complexity. In implemente a hybri image classification metho combining wavelet transform, rough set approach, an artificial neural networ. Zou an Li have propose image classification using wavelet coefficients in low-pass bans [8]. This approach was base on the istribution of histograms of the wavelet coefficients. In this paper, it use particle swarm optimization metho to reuce the search space for FIC[]. If the two blocs are not of the same type no similarity will not be calculate. The classification metho is to partition all of the blocs in omain pool an range pool into three classes accoring to thir level wavelet coefficients. Each range bloc calculates the similarity measure only with the omain bloc from the same class. In the meanwhile, we consier the special property of the Diheral transformation so that only four transformations are require to calculate the similarity. Therefore the encoing time can be further reuce. Experiments are conucte on image using 6 methos incluing the full search metho, iscrete wavelet transform (DWT), PSO, SGA, ANN an the propose metho. Experimental results shows that the propose metho outperform all the other 5 methos. In average, the propose metho is about 8 times faster in comparison to the full search metho with only.56 B ecay in image quality. Comparing to Wu s schema genetic algorithm (SGA) [], the propose metho is better than the performance of SGA an ANN metho[]. Moreover, the encoing spee of the propose metho is faster than that of the full search metho, better retrieve image quality.. FRACTAL IMAGE COMPRESSION In local self-similarity property in a nature images. The funamental iea is coming from the Partitione Iterate Function System (PIFS). Suppose the original gray level image f is of size m m. Let the range pool R be efine as the set of all non-overlapping blocs of size n n of the image f, which maes up (m/n) blocs. For obeying the Contractive Mapping Fixe-Point the omain bloc must 585

2 excee the range bloc in length. Let the omain pool D be efine as the set of all possible blocs of size n n of the image f, which maes up (m n + ) blocs. For m is 56 an n is 8, the range pool R is compose of (56/8) (56/8) = 4 blocs of size 8 8 an the omain pool D is compose of ( ) ( ) = 588 blocs of size 6 6. For each range bloc v from the R, in the fractal affine transformation is constructe by searching all of the omain blocs in the D to fin the most similar one an the parameters representing the fractal affine transformation will form the fractal compression coe for v. To execute the similarity measure between range bloc an omain bloc, In the size of the omain bloc must be first sub-sample to 8 8 such that its size is the same as the range bloc v. Let u enote a sub-sample omain bloc. The similarity of two image blocs u an v of size n n is measure by mean square error (MSE) efine S( u, v) n n n n j i ( u( i, j) v( i, j)). The fractal transformation allows the eight Diheral transformations in Table, T : =,, 7, of the omain blocs. If the coorinate origin is assume to locate at the center of the bloc, the transformations T an T correspon to flip the bloc along horizontal an vertical line, respectively. T 3 is the flip along both horizontal an vertical lines. T 4, T 5, T 6 an T 7 are the transformations of T, T, T an T 3 performe by an aitional flip along the main iagonal line, respectively. The fractal coer also allows the ajustment of the contrast scaling p an the brightness offset q on the bloc u. Thus the similarity is to minimize the quantity e = p u + q v, where u = T (u), 7, are the 8 orientations of u. By irect computation, p an q can be compute by () In the ecoing process, In first mae up the 4 affine transformations from the compression coes. We choose one arbitrary image as the initial image an then perform the 4 affine transformations on the image to obtain a new one. The transformation is proceee recursively. Accoring to Contractive Mapping Fixe-Point sequence of image will converge. The stopping criterion of the recursion is esigne accoring to user s application an the converge image is the retrieve image of fractal coing. 3. FAST FRACTAL IMAGE ENCODING 3.. Particle Swarm Optimization (PSO) A population-base algorithm is PSO for searching global optimum. To simulate a simplifie social behavior is the way of original iea of PSO[9]. Similar to the crossover operation of the GA, in PSO the particles are ajuste towar the best iniviual experience (PBEST) an the best social or global experience (GBEST). However, PSO is unlie a GA, why because in that each potential solution, particle is flying through hyperspace with a velocity, the particles an the swarm have memory for process; in the population of the GA memory oes not exist. Let x j, (t) an v j, (t) enote the th imensional value of the vector of position an velocity of j th particle in the swarm, respectively, at time t. The PSO moel can be expresse as v x ( t) v ( t) x ( t ) c..( x ( t ) v ( t), * x ( t )) c..( x # x ( t )), p n u n u, v, u u u v, I p, I u q L, I v, I, I where L is the number of pixels in the bloc of the range pool 8 8 vector with all entries being, an the Eucliean inner prouct of two vectors. Finally, the position (i, j) of the omain bloc (after sub-sample, it is enote by u), the contrast scaling p, the brightness offset q, an the orientation constitute the fractal coe of the given range bloc v. For each range bloc v, it will mae up a fractal coe of i,, p an q. Table.The 8 transformations in the Diheral group T T T T 3 T 4 T 5 T 6 T 7 Where * x (PBEST) enotes the best position of j th particle x # up to time t- an (GBEST) enotes the best position of the whole swarm up to time t-, φ an φ are ranom numbers, an c an c represent the iniviuality an sociality coefficients, respectively. The steps involve here is the population size is first etermine, an the velocity an position of each particle are initialize. Each particle moves accoring to fitness is then calculate. Meanwhile, the best positions of each swarm an particles are recore. Finally, as the stopping criterion is satisfie, the best position of the swarm is the final solution. The bloc iagram of PSO is isplaye in an the main steps are given as follows: Initialize the PSO parameters. For each particle (t x, t y ), fetch the omain bloc at (t x,t y ) in the image. Sub-sam-ple the bloc an enote it by u. Obtain Diheral transforme u, =,, 7. Calculate the contrast scaling p by minimizing the MSEs an brightness offset q by taing the meian of the final resiuals. The corresponing MSEs, which are treate as the cost of the given particle, are obtaine uring the process. 586

3 Upate the pbest an the gbest if require. The corresponing fractal coes are also upate Fig.(a) Original image of Fig.(b) Full Search Fig. (c) Fig.() ANN Fig.(e) SGA Fig.(f) PSO accoringly. If stopping criterion is met, then stop. Fig.(g) ) Propose metho 3.. Diheral Transformation To each range bloc, we must calculate the similarity measure with all eight transforme blocs of omain bloc accoring to the Diheral transformations fin the best match. The relations of F an F after applying Diheral transformations, It is clear that all items are ± F or ± F. This fact can be utilize to further reuce the encoing time. For example, the transformation T is to flip the bloc along the horizontal line. relations between the coefficients of the T transforme bloc to the original bloc f can be easily calculate as it separate the unit circle in the first quarant into two regions Ω an Ω accoring to the line θ = 45. For a given range bloc v, we pic a omain bloc u. If u an v are locate in the same region, say Ω, then we nee only four Diheral transformations, T : = ~3 performe on u because the transformations T : = 4~7 will move u another region i.e., Ω. The 4 transforms T : = ~3 or T : = 4~7 have the same ege irections since their F an F are the same. Therefore, there are only four MSE computations require. On the other han, if u an v are locate in the ifferent region, then we nee only four Diheral transformations, i.e., T : = 4~7. From the argument above, the amount of MSE computations will be reuce two times. 4. PROPOSED METHOD In the propose fast fractal encoing using PSO, reuce the encoing time by reucing the searching time to fin a best match omain bloc for the given range bloc from all omain blocs. Set the swarm size an particle s parameters. If the type of omain bloc is the same as that of range bloc, go to step 3, otherwise, go to step 5. 3 If u an v belong to the same region, only T: = ~3 are performe. Otherwise, only T: = 4~7 are performe. 4 Calculate the MSE. 5 Upate particle s position an velocity. 587

4 6. If the pre-specifie number of iteration is reache, then stop. 7 Go bac to step. 5. EXPERIMENTAL RESULTS The results have been compare to the full search FIC mentione in the previous sections in terms of encoing time an PSNR of fast fractal encoing using PSO. The istortion or error between the original image f an the ecoe image g cause by lossy compression process is measure in pea signal to noise ratio (PSNR) efine by PSNR log 55 MSE( f, g) where MSE is efine () in The teste image on in which each is a gray scale image of size selecte from the CVG-UGR image atabase[] The relate PSO parameters swarm size, number of clusters, an inertial weight, are set as 4, 4 an.9, respectively. The velocity in is limite in an the maximal number of iterations is set as 3. Table. Simulation results for PSNR an comparison. Full search ANN 5.73 PSO SGA Propose Metho 9. 5 Table 3. Simulation results for CPU time(s) an comparison. Full search ANN 45.3 PSO SGA 4. 6 Propose Metho 8. 5 Table 4. Simulation results for MSE Computation an comparison. Full search 475,799,54 58,664,88 ANN,657,67 PSO,66,6 SGA,835,477 Propose Metho,4,44 See Figure, (a) show the original image for ecoing, see Figures, (b)- (c) () (e) (f) (g) show retrieve images of Full Search,, ANN, PSO,SGA an propose methos have PSNR 6.9 B, 6.8 B, B, B, 9.5 B respectively. Executing times, The results of the propose metho is liste in the tabular column. Compare to the Full search metho, the speeup ratio is about three times faster. The etaile results of PSNR, executing time, an the amount of MSE computations an CPU time of the image liste in Tables (-3-4) respectively. The retrieve image qualities are very close. The CPU time of the propose metho is 8.5 secons, which is the least. The speeup ratio with respect to the full search metho shown in that is low. Uner the conition of similar quality of ecoe images, the encoing time of the propose metho reuces about 8 times which is better than that of the SGA metho. As an complexity analysis, the amount of MSE computations in of the propose metho for lena image is,4,44 which is 5 times of the amount of the full search metho, which is shown in the last column. As emonstrate, the propose metho has better performance that that of other methos. 6. CONCLUSION In this paper, particle swarm optimization metho is aopte with classification an Diheral transformation in orer to speeup the fractal image encoer. By using particle swarm optimization (PSO) base propose metho for fractal coing can reuce CPU time, PSNR, MSE Values an prouces better compression ratio at acceptable quality, when comparing with existing full search, wavelet classification ANN, PSO an SGA methos. 7. REFERENCES [] E. Jacquin, coing base on a fractal theory of iterate contractive image transformations, IEEE Transactions on Processing, Vol., 99, pp [] M. S. Wu, J. H. Jeng, an J. G. Hsieh, Schema genetic algorithm for fractal image compression, Engineering Applications of Artificial Intelligence, Vol., 7, pp [3] T. K. Truong, C. M. Kung, J. H. Jeng, an M. L. Hsieh, Fast fractal image compression using spatial correlation, Chaos, Solitons an Fractals, Vol., 4, pp [4] C. Hufnagl, A. Uhl, Algorithms for fractal image compression on massively parallel SIMD arrays, Real- Time Imaging 6 () [5] D. Viya, R. Parthasarathy, T.C. Bina, N.G. Swaroopa, Architecture for fractal image compression, J. Syst. Archit. 46 () [6] J. Li an R. M. Gray, Context-base multiscale classification of ocument images using wavelet coefficient istributions, IEEE Transactions on Processing, Vol. 9,, pp [7] H. J. Yu an J. B. Ra, Fast triangular mesh approximation of surface ata using wavelet coefficients, The Visual Computer, Vol. 5, 999, pp

5 [8] W. Zou an Y. Li, classification using wavelet coefficients in low-pass bans, in Proceeings of International Joint Conference on Neural Networs, 7, pp [9] M.S. Wu, W.C. Teng, J.H. Jeng, J.G. Hsieh, Spatial correlation genetic algorithm for fractal image compression, Chaos Solitons & Fractals 8 () (6) [] CVG- UG Database,7, genes/inex.php. [] A. Murugananham an Dr. R.S.D. Wahia Banu, Aaptive Fractal Compression using PSO, Proceia Computer Science () , Publishe by Elsevier Lt [] A. K. M. Ashiur Rahman an Chowhury Mofizur Rahman, A New Approach for Compressing Color s using Neural Networ, Proceeings/ISBN ,-4 February 3 A.Krishnamoorthy receive the B.E egree from the Department of Electronics an Communication Engineering, IFET college of Engineering, Villupurm Tamilnau in 9 an ME egrees from the Department of Process Control An Instrumentation Engineering in Annamalai University, Chiambaram in, Currently, he is woring as a Visiting Faculty(Teaching Fellow) in university college of Engineering pantruti(anna university)tamilnau Inia. In the Department of Electronics an Communication Engineering. Current research interests are an Optimization Techniques. 589