Bennett, A. Prugel

N., M. H. Tayarani

English

Southampton (e-Prints Soton)

A Novel Initialization for Quantum EvolutionaryAlgorithms Based on Spatial Correlation inImages for Fractal Image CompressionM. H. Tayarani N., A. Prugel Bennett, M. Beheshti and J. SabetAbstract Quantum Evolutionary Algorithm (QEA) is a novel optimization algo-rithm proposed for class of combinatorial optimization problems. While Fractal Im-age Compression problem is considered as a combinatorial problem, QEA is notwidely used in this problem yet. Using the spatial correlation between the neigh-bouring blocks, this paper proposes a novel initialization method for QEA. In theproposed method the information gathered from the previous searches for the neigh-bour blocks is used in the initialization step of search process of range blocks. ThenQEA starts searching the search space to ﬁnd the best matching domain block. Theproposed algorithm is tested on several images for several dimensions and the exper-imental results shows better performance for the proposed algorithm than QEA andGA. In comparison with the full search algorithm, the proposed algorithm reachescomparable results with much less computational complexity.1 IntroductionFractal Image Compression, proposed by Barnsley has, recently become one of themost promising encoding technologies in the generation of image compression [1].The high compression ratio and the quality of the retrieved images attract manyof researchers, but the high computational complexity of the algorithm is its maindrawback. One way of decreasing the time complexity is to move from full searchmethod to some optimization algorithms like Genetic Algorithms. From this pointof view, several works try to improve the performance of fractal image compres-sion algorithms using Genetic algorithm. In [2] a new method for ﬁnding the IFScode of fractal image is developed and the inﬂuence of mutation and the crossoverM. H. Tayarani, N.University of Southampton, e-mail: mhtn1g09@ece.soton.ac.ukAdam Prugel BennettUniversity of Southampton, e-mail: apb@ecs.soton.ac.uk12 M. H. Tayarani N., A. Prugel Bennett, M. Beheshti and J. Sabetis discussed. The low speed of fractal image compression blocks its way to practi-cal application. In [3] a genetic algorithm approach is used to improve the speed ofsearching process in fractal image compression. A new method for genetic fractalimage compression based on an elitist model in proposed in [4]. In the proposedapproach the search space for ﬁnding the best self similarity is greatly decreased.Reference [5] makes an improvement on the fractal image coding algorithm by ap-plying genetic algorithm. Many researches increase the speed of fractal image com-pression but the quality of the image will decrease. In [6] the speed of fractal imagecompression is improved without signiﬁcant loss of image quality. Reference [7]proposes a genetic algorithm approach which increases the speed of the fractal im-age compression without decreasing of the quality of the image. In the proposedapproach a standard Barnsley algorithm, the Y. Fisher based in classiﬁcation andthe genetic compression algorithm with quad-tree partitioning are compared. In GAbased algorithm a population of transformations is evolved for each range block. Inorder to prevent the premature convergence of GA in fractal image compression anew approach is proposed in [8], which controls the parameters of GA adaptively.A spatial correlation genetic algorithm is proposed in [9], which speeds up the frac-tal image compression algorithm. In the proposed algorithm there are two stages,ﬁrst the spatial correlations in image for both the domain pool and the range pool isperformed to exploit local optima. In the second stage if the local optima were notcertiﬁable, the whole of image is searched to ﬁnd the best self similarity. A schemagenetic algorithm for fractal image compression is proposed in [10] to ﬁnd the bestself similarity in fractal image compression.Using spatial correlation between the neighbor range and domain blocks thispaper proposes a novel initialization method for QEA. In the proposed method,based on the information gathered from the search process for the neighbor rangeblocks, the q-individuals are initialized to represent the better parts of the searchspace with higher probability. Performing this new method the q-individuals havemore chance to ﬁnd better solutions in less time. The proposed algorithm is tested onseveral images and experimental results shows better performance for the proposedalgorithm than GA and original form of QEA.The rest of the paper is organized as follows. Section 2 introduces QEA, insection 3 the new algorithm is proposed and in section 4 is experimented on severalimages and ﬁnally section 5 concludes the paper.2 Quantum Evolutionary AlgorithmQEA is inspired from the principles of quantum computation, and its superpositionof states is based on qubits, the smallest unit of information stored in a two-statequantum computer. A qubit could be either in state ”0” or ”1”, or in any superposi-tion of the two as described below:jyi = aj0i+b j1i (1)Title Suppressed Due to Excessive Length 3Where a and b are complex numbers, which denote the corresponding stateappearance probability, following below constraint:jaj2+jbj2 = 1 (2)This probabilistic representation implies that if there is a system of m qubits, thesystem can represent 2m states simultaneously. At each observation, a qubits quan-tum state collapses to a single state as determined by its corresponding probabilities.Consider i th individual in t  th generation deﬁned as an m-qubit as below:ati1 ati2 ::: atij ::: atimbti1 bti2 ::: btij ::: btim(3)Where atij 2+ btij 2= 1 , j = 1;2;;m;m is the number of qubits, i.e., the stringlength of the qubit individual, i = 1;2;;n;n is the number of possible solution inpopulation and t is generation number of the evolution. If there is, for instance, athree-qubits (m = 3) individual such as 4:qti =" 1 p21 p3121 p2p2 p3p32#(4)Or alternatively, the possible states of the individual can be represented as:qti =12p6j000i+12p2j001i+12p3j010i+12j011i+12p6j100i+12p2j101i+12p3j110i+12j111i (5)In QEA, only one qubit individual such as 4 is enough to represent eight states,whereas in classical representation eight individuals are needed. Additionally, alongwith the convergence of the quantum individuals, the diversity will gradually fadeaway and the algorithm converges.2.1 QEA StructureIn the initialization step of QEA, [atij btij]T of all q0i are initialized with 1 p2. Thisimplies that each qubit individual q0i represents the linear superposition of all pos-sible states with equal probability. The next step makes a set of binary instants; xtiby observing Q(t) = fqt1;qt2;:::;qtng states, where X(t) = fxt1;xt2;:::;xtng at genera-tion t is a random instant of qubit population. Each binary instant, xti of length m, isformed by selecting each bit using the probability of qubit, either jatijj or jbtijj of qti.Each instant xti is evaluated to give some measure of its ﬁtness. The initial best solu-tion b = maxni=1ff(xti)g is then selected and stored from among the binary instantsof X(t). Then, in ’update’ Q(t), quantum gatesU update this set of qubit individuals4 M. H. Tayarani N., A. Prugel Bennett, M. Beheshti and J. SabetQ(t) as discussed below. This process is repeated in a while loop until convergenceis achieved. The appropriate quantum gate is usually designed in accordance withproblems under consideration.2.2 Quantum Gates AssignmentThe common mutation is a random disturbance of each individual, promoting ex-ploration while also slowing convergence. Here, the quantum bit representation canbe simply interpreted as a biased mutation operator. Therefore, the current best indi-vidual can be used to steer the direction of this mutation operator, which will speedup the convergence. The evolutionary process of quantum individual is completedthrough the step of ”update Q(t)”. A crossover operator, quantum rotation gate, isdescribed below. Speciﬁcally, a qubit individual qti is updated by using the rotationgate U(q) in this algorithm. The j th qubit value of i th quantum individual ingeneration t, [atij btij]T is updated as:atijbtij=cos(Dq)  sin(Dq)sin(Dq) cos(Dq)"at 1ijbt 1ij#(6)Where Dq is rotation angle and controls the speed of convergence and deter-mined from Table 1. Reference [11] shows that these values for Dq have betterperformance.Table 1 Lookup Table of Dq, the rotation angle.xi bi f(x)  f(b) Dq0 0 false 00 0 true 00 1 false 0:01p0 1 true 01 0 false 0:01p1 0 true 01 1 false 01 1 true 03 Proposed MethodStatistical studies on fractal image compression problems show for the range blocks,the potential quasi-afﬁne matched domain centralized around its vicinity [1]. Itmeans that the neighbor range blocks have similar fractal codes. Using this idea, [1]Title Suppressed Due to Excessive Length 5proposes a two stage fractal image coding method. The ﬁrst stage searches aroundthe best matched domain blocks of neighbor range blocks. If the result is not satis-fying, the search process starts to ﬁnd a good domain block. Inspiring this idea, thispaper proposes a novel initialization method for Quantum Evolutionary Algorithmin solving fractal image compression problem. In solving fractal image compres-sion problem using genetic algorithm, for each range block, GA searches amongthe domain pool to ﬁnd the best matched domain block [10]. Basically the searchprocess for each range block is performed independently and the results of the pre-vious searches are not used in future searches for other range blocks. This paperproposes a novel method which uses the information from previous searches to helpthe evolutionary algorithm ﬁnding better solutions. In the proposed method, basedon the information gathered from the previous searches for neighbor range blocks,the q-individuals are initialized to represent the better parts of the search space withhigher probability. The new algorithm is proposed as follows:The Proposed Algorithmbegint = 01. Initialize Q(0) based on Hb, Hm and Hw1. Make X0 by observing the states of Q(0)3. Evaluate X(0)4. Store X(0) into B(0). Store the best solution among X(0) into b5. While not termination condition dobegint=t+16. Make Xt by observing the states of Q(t  1)7. Evaluate X(t)8. Update Q(t) using Q-Gates9. Store the best solutions among B(t  1) and X(t) into B(t)10. Store the best solution among B(t) into bendendQEA has a population of quantum individuals Q(t) = fqt1;qt2;:::;qtng, where t isgeneration step and n is the size of population.A comprehensive description of QEA can be found in [11]. The QEA procedureis described as:1. In QEA the possible solutions are initialized with the values of 1 p2 to representthe whole search space with the same probability. Giving these values to q-gatesmeans a complete random initialization. Random initialization works well when wedo not have previous information about the search space. Knowing where to search,helps QEA searching better parts of the search space and ﬁnding better solutionswith less searching time. Here we propose a novel initialization approach using theinformation from previous searches. In the proposed method the best, worst andmedian solutions of each iteration in the search process for each range block are6 M. H. Tayarani N., A. Prugel Bennett, M. Beheshti and J. Sabetstored in Hb, Hm and Hw respectively. Where H is the history of X, containing thebest, median and worst possible solutions in previous searches. The initializationstep in the proposed algorithm is performed as follows:Bl;i =1TTåj=1Hbl;ij; Ml;i =1TTåj=1Hml;ij; Wl;i =1TTåj=1Hwl;ij (7)Where Hb, Hm and Hw are the history of the best, median and worst possiblesolutions respectively, Hbl;ij is the i th bit of j th possible solution in Hb, and theindex l shows the l th neighbor range block, Bl;i is the average of i th bit amongall the possible solutions in Hb of l  th neighbor range block. Here l = 1;2;:::;8shows the l th neighbor of the range block. Each range block has 8 neighbor rangeblocks. Here Bl;i, Ml;i and Wl;i contain the percentage of ones in the best, medianand worst possible solutions during the previous searches respectively. The valueof Bi shows the importance of the i th bit in possible solutions, if Bi is near 1,it means that in most of better possible solutions, this bit has the value of 1, andit is better to give a value to this q-bit which represents 1 with more probability.Analogously, if Wi is near 1, it means that in most of worse possible solutions,this bit is 1, therefore it is better to give a value of to this q-bit which represents 0with more probability. Accordingly, this paper proposes the following method forreinitialization step:qtij =p4+[2Bi+Mi 3Wi]p16(8)Where j = 1;2;:::;l , l is the number of neighbor blocks, i = 1;2;;m, m is thenumber of q-bits in the q-individuals, i. e. the dimension of the problem. In orderto prevent the q-individuals getting stuck in local optima of previous searches theother q-individuals are initialized randomly:qtij =p4(9)For j = l+1;l+2;:::;n where n is the size of the population.Theproposedinitializationoperatorgathersinformationfromtheprevioussearchesand initializes the q-individuals with the values representing better parts of searchspace.2. In this step the binary solutions X(0) = fx01;x02;:::;x0ng at generation t = 0 arecreated by observing Q(0). Observing xtij from qubit [atij btij]T is performed asbelow:xtij =(0 if U(0;1) < jatijj21 otherwise(10)Where U(:;:), is a uniform random number generator.3. All solutions in X(t) are evaluated with ﬁtness function.4. Store X(0) into B(0). Select best solution among X(0) and store it to b.Title Suppressed Due to Excessive Length 75. The while loop is running until termination condition is satisﬁed. Termina-tion condition can be considered as maximum generation condition or convergencecondition.6. Observing X(t) from Q(t  1).7. Evaluate X(t) by ﬁtness function.8. Update Q(t).9, 10. Store the best solutions among B(t  1) and X(t) to B(t). If the ﬁttestsolution among B(t) is ﬁtter than b then store the best solution into b.3.1 CodingIn the proposed algorithm for each range block, QEA searches among all the domainpool to ﬁnd the best match domain block. The domain blocks are coded by theirhorizontal and vertical address in the image. Therefore a solution is a binary stringhaving3parts, px;py;pT,representingthehorizontalandverticallocationofdomainblock in the image and the transformation respectively. The length of the possiblesolution for a MN image is calculated as follows:m = dlog2(M)e+dlog2(N)e+3 (11)Where m is the size of the possible solutions. Here 8 ordinary transformationare considered: rotate 0;90;180;270, ﬂip vertically, horizontally, ﬂip relative to45, and relative to 135.4 Experimental ResultsThis section experiments the proposed algorithm and compares the proposed algo-rithm with the performance of GA and original version of QEA in fractal imagecompression. The proposed algorithm is examined on images Lena, Pepper and Ba-boonwiththesizeof256256andgrayscale.Thesizeofrangeblocksisconsideredas 88 and the size of domain blocks is considered as 1616. In order to comparethe quality of results, the PSNR test is performed:PSNR = 10log 25521MN åNi=1åMj=1(f(i; j) g(i; j))2!(12)Where MN is the size of image.The crossover rate in GA is 0.8 and the probability of mutation is 0.003 foreach allele. Table 2 shows the experimental results using proposed algorithm andGA. The number of iterations for GA, QEA and the proposed algorithm for all theexperiments is 200. According to Table 2 the proposed algorithm improves theperformance of fractal image compression for all the experimental results.8 M. H. Tayarani N., A. Prugel Bennett, M. Beheshti and J. SabetTable 2 Experimental results on Lena, Pepper, and BaboonPicture Method Pop Size MSE Computations PSNRLena Full Search - 59,474,944 28.85QEA 30 6,144,000 28.4925 5,120,000 28.2820 4,096,000 28.9515 3,072,000 27.43Proposed Method 30 6,144,000 28.5825 5,120,000 28.3920 4,096,000 29.0215 3,072,000 27.56GA 30 6,144,000 28.1125 5,120,000 28.0420 4,096,000 27.5515 3,072,000 27.27Pepper Full Search - 59,474,944 29.85QEA 30 6,144,000 29.5525 5,120,000 29.0920 4,096,000 28.8715 3,072,000 28.12Proposed Method 30 6,144,000 29.6225 5,120,000 29.2820 4,096,000 28.9315 3,072,000 28.51GA 30 6,144,000 29.1425 5,120,000 28.9220 4,096,000 28.6415 3,072,000 28.11Baboon Full Search - 59,474,944 20.04QEA 30 6,144,000 19.2825 5,120,000 19.1820 4,096,000 18.9515 3,072,000 18.62Proposed Method 30 6,144,000 19.6325 5,120,000 19.2520 4,096,000 19.0915 3,072,000 18.77GA 30 6,144,000 19.1725 5,120,000 19.0220 4,096,000 18.6515 3,072,000 18.41Title Suppressed Due to Excessive Length 95 ConclusionUsing spatial correlation between the neighbor blocks in images, this paper pro-poses a novel initialization method for QEA in fractal image compression. In theproposed method, during the search process for each range block, some informationabout the best, worst and median possible solutions is gathered. Using this infor-mation in the initialization step in the search process of neighbor range blocks, thispaper proposes a novel method in solving fractal image compression problem. Sev-eral experiments on Lena, Pepper, and Baboon pictures show an improvement onevolutionary algorithms solving fractal image compression.References1. W. Xing-yuan, L. Fan-ping, W. Shu-guo Fractal image compression based on spatial corre-lation and hybrid genetic algorithm, Elsevier, Journal of vis. commun. image R. pp. 505-510.2009.2. Yang Xuan, Liang Dequn, An improved genetic algorithm of solving IFS code of fractal im-age, IEEE 3rd international conference on signal processing 1996.3. X. Chen, G. Zhu, and Y. Zhu, Fractal image coding method based on genetic algorithms,International Symposium on Multispectral Image Processing 1998.4. S.K. Mitra, C.A. Murthy, M.K Kundu, Technique for fractal image compression using geneticalgorithm, IEEE Trans on Image Processing Vol 7 no 4 pp 586-593 1998.5. L. Xun and Y. Zhongqiu The application of GA in fractal image compression, 3rd IEEE WorldCongress on Intelligent Control and Automation, 2000.6. Gafour, A. Faraoun, K. Lehireche, A Genetic fractal image compression, ACS/IEEE Interna-tional Conference on Computer Systems and Applications, 2003.7. F. K. Mohamed, B. Aoued, Speeding Up Fractal Image Compression by Genetic Algorithms,Springer Journal of Multidimention Systems and Signal processing. Vol 16, No 2, 2005.8. L. Xi, L. Zhang,A Study of Fractal Image Compression Based on an Improved Genetic Algo-rithm, International Journal of Nonlinear Science Vol 3, No 2, pp 116-124m 2007.9. M. Wu, W. Teng, J. Jeng and J. Hsieh, Spatial correlation genetic algorithm for fractal imagecompression, Elsevier Journal of Chaos, Solitons and Fractals. Volume 28, Issue 2, pp 497-510, 2006.10. M. Wu, J. Jeng, and J. Hsieh,Schema genetic algorithm for fractal image compression, Else-vier Journal of Engineering Applications of Artiﬁcial Intelligence Vol 20, Issue 4, pp 531-5382007.11. K. Han and J. Kim, Quantum-inspired evolutionary algorithm for a class of combinatorialoptimization, IEEE Transactions on. Evolutionary Computing, Vol. 6. No 6, 2002.

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A Novel Initialization for Quantum Evolutionary Algorithms Based on Spatial Correlation in Images for Fractal Image Compression

A Novel Initialization for Quantum Evolutionary Algorithms Based on Spatial Correlation in Images for Fractal Image Compression

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