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A New Method for Image Noise Removal using Chaos-PSO and Nonlinear ICA
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Procedia Engineering
ProcediaProcedia Engineering 00 (2011) Engineering 24 000–000 (2011) 111 – 115 www.elsevier.com/locate/procedia
2011 International Conference on Advances in Engineering
A New Method for Image Noise Removal using Chaos-PSO and Nonlinear ICA Wei Ding a
a*
Nanyang Institute of Technology, Nanyang 470004, China
Abstract The blind source separation (BSS) algorithms, especially the independent component analysis (ICA) algorithms, have been proven to be effective for the image data processing. The noise signal introduced into the image data can be perfectly eliminated using ICA only under the linear mixture condition. However, the images are always mixed nonlinearly with noise perturbations. The traditional linear ICA algorithm is not capable enough to suppress the noise in this situation. Hence, the nonlinear ICA is proposed to deal with the nonlinear mixtures in this paper. The radial basis function (RBF) neural network based post-nonlinear ICA algorithm has been adopted to remove noise from the original image data. To enhance the RBF-ICA operation, the Chaos-Particle Swarm Optimization (PSO) algorithm has been employed to optimize the RBF neural network to obtain satisfactory nonlinear solution of the nonlinear BBS procedure. A series of experiments have been implemented in this work to validate the efficiency of the proposed method. The Chaos-PSO optimized RBF-ICA model has been compared with other ICA models in the image de-noising processing. The comparative results show that the proposed approach is superior to the nonoptimized ICA methods with respect to the image de-noising performance.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of ICAE2011. Keywords: Image processing; nonlinear ICA; Chaos-PSO; noise removal
1. Introduction Noise removal is important for the face recognition of the image data. Heavy noise pollution can make the face identification fail. As a result, the noise removal is essential for the identification of face images. There are many developments to handle the face images in an intelligent manner. These advanced methods include evolution algorithm, artificial neural network (ANN), principal component analysis (PCA) and independent component analysis (ICA) [1]. Though PCA [2] is very powerful for image denoising and benefits the face recognition processing, it has drawbacks when dealing with nonlinear * Corresponding author. Tel.: +86- 377-3121454; fax: +86-377-3121486. E-mail address: [email protected]
1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.11.2611
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Wei /Ding / Procedia Engineering 24 (2011) 111 – 115 Wei Ding Procedia Engineering 00 (2011) 000–000
signals [3-5]. This is because the PCA is proposed on the linear model of the data. The ICA has not such shortcomings and is very effective in the image pre-processing for the nonlinear signals [6]. The ICA is firstly proposed by Giannakis [7] in 1987 to solve the blind source separation (BSS) problem. Following in 1991, Jutten and Herault [8], Sorouchyari [9], Comon [10] have published three important papers about the BSS to make ICA be famous in signal processing. Frankly, the traditional ICA algorithm, such as the Fast ICA, has the ability to tackle the nonlinear image data. However, with the increase of the nonlinearity severity, the linear ICA cannot separate correct signal sources. This is because the nonlinear mixing increases the complexity of nonlinear characteristics. In addition, a unique solution is difficult to attain for the nonlinear BSS problem. For this reason, Tan [11] proposed the RBF neural network based ICA to deal with the nonlinear BSS. The possibility of unique solution emerges for the nonlinearity. Taking the advantage of Tan’s method [11], we have further improved the RBF based on nonlinear ICA algorithm. A new method based on the Chaos-Particle Swarm Optimization (PSO) and RBF-ICA has been proposed to eliminate the image noise. Compared with Tan’s RBF-ICA [11], the proposed approach uses Chaos-PSO to optimize the RBF neural network (RBF NN) and hence better efficiency can be obtained for the nonlinear mixture separation. By the face recognition experiments, the results show that the noise can be removed effectively and the face recognition of the proposed method is superior to the linear ICA and the original RBF-ICA. 2. The proposed noise removal algorithm 2.1. RBF based ICA The RBF based ICA is a typical post-nonlinear model (PNL) for nonlinear BSS problem. The RBF NN is used to map the nonlinear mixture into the linear one such that the reversal of the nonlinear mixing function can be obtained. Then the nonlinear BBS is transformed into linear case and the traditional linear ICA can be employed to find the independent sources (ICs). The RBF based ICA model is defined as [11]
x = As a = f ( x) , b = g (a ) = y Wb ≈ s
(1)
where s is the source signals, A is the linear mixing matrix, x is the linear mixtures, a is the nonlinear mixtures, f(·) is the nonlinear mixing function, b is the nonlinear de-mixtures, g(·) is the nonlinear demixing function, y is the estimated sources, and W is the linear de-mixing matrix. The system diagram of the RBF based ICA is shown in Fig. 1. Sources s
Linear mixing by A
Nonlinear mixing using f(·)
Nonlinear demixing via RBF
De-mixing by W
Estimated sources y
Fig.1. The system diagram of the RBF based ICA
The RBF NN herein acts as the nonlinear de-mixing solver. The detail of the RBF based ICA for the nonlinear BSS can be found in [11]. To further develop Tan’s theory, the Chaos-PSO has been applied to the parameter optimization for the RBF NN. 2.2. Chaos-PSO
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Wei Ding Ding // Procedia Procedia Engineering Engineering 00 24 (2011) (2011) 000–000 111 – 115 Wei
Particle Swarm Optimization (PSO) algorithm is an effective evolutionary algorithm inspired by the social behavior of bird flocks and fish schools [12]. The extremum searching of PSO is very simple and effective and hence PSO has been widely used in the optimization of ANNs. Nevertheless, PSO has problems of suffering from premature convergence [13]. Research for addressing the shortcomings of PSO is ongoing and includes such changes as dynamic or exotic sociometries [13], enriched particle diversity [14], and tunable parameters in the updating [15]. Many methods have been proposed to overcome this problem, but the door for improving PSO still opens. To avoid premature convergence, the Chaos search processing has been employed to improve PSO optimization in this paper. Through the increase of particle diversity by Chaos search, the premature convergence can be depressed significantly. PSO is similar to genetic algorithm, but it updates offspring by the adjustment of the flight speed and position [16]. The optimal solution of the PSO search is the best position that the particle experiences. The PSO algorithm for updating flight speed υ and flight position x can be expressed by [12]
υi ( k = + 1) ωiυi ( k ) + c1r1[ xd − xi ( k )] + c2 r2 [ x g − xi ( k )]
(2)
xi ( k + 1) = xi ( k ) + υi ( k + 1)
(3)
where, ωi is the inertial weights, c1 and c2 are the learning factors, and r1 and r2 denote the random coefficients among [0,1]. xd is the individual extremum and x g is the global extremum. The Chaos search is used for the secondary optimization to diverse PSO’s new generation. The Logistic equation has been adopted for Chaos. The Logistic equation is defined by [16]
xi ( k= + 1) µ xi ( k )(1 − xi ( k ))
(4)
where µ is the control coefficient. The secondary optimization using Chaos can be interpreted as follows. (1) Mapping the flying coordinate x into the Chaos variables in [0 1] by the below carrier
d= ( xi ( k ) − α i ) / β i i (k )
(5)
where α i and β i need to be specified. Then the Chaos behavior can be calculated by
d i ( k= + 1) µ ci ( k )(1 − d i ( k ))
(6)
(2) Iterating the Chaos behavior until find the best d i , and then Chaos variables will be mapped back to the original space by the second carrier
xi ( k= ) α i + βi di (k )
(7)
Through this secondary optimization, the chaos algorithm can help PSO find proper global extremum. 3. Experiment results The face recognition experiments have been implemented in this paper to validate the performance of the proposed algorithm. In the Chaos-PSO optimization, set c= c= 1.3 , ω = 0.55 , the size of particle 1 1 swarm selects 50 for PSO, and set µ = 3.5 , α = 1.7 and β = 0.6 for Logistic chaos. Fig. 2 shows the performance of the image noise removal. One can note from Fig. 2 that the original noisy image can be recovered precisely by the proposed method.
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Wei /Ding / Procedia Engineering 24 (2011) 111 – 115 Wei Ding Procedia Engineering 00 (2011) 000–000
Fig. 2. The original noisy image (left), the skin with noise removal (middle) and the recovered face image (right).
By the Chaos-PSO optimized nonlinear ICA pre-processing, the face recognition rate can be enhanced greatly. The RBF NN has been adopted as the classifiers to identify the face images. In this paper, 50 pieces face have been prepared for the face recognition experiment. The face recognition performance is shown in Table 1. The recognition accuracy of the proposed method has been compared with some existing approaches. The proposed method has been compared with no de-noising processing, PCA removal, ICA removal and RBF-ICA removal in the face recognition. It can be seen form Table 1 that the nonlinear ICAs increase the face recognition rate apparently, and the Chaos-PSO optimized one obtains the best identification among the five methods. This is because the Chaos-PSO can prevent the RBF from local optimization. Table 1. The face recognition results Method
Recognition accuracy
RBF NN PCA-RBF NN Linear ICA-RBF NN RBF based ICA-RBF
87.3% 88.7% 88.3%
The proposed method
93.3%
NN
90.6%
4. Conclusions Pre-processing on the original noisy images can increase the image identification efficiency. The linear ICA has been proved to be useful for the image noise removal. However, for images which are mixed with nonlinear noise, the traditional ICA may fail to work. Hence, this paper proposed a new face image pre-processing method based on Chaos-PSO based RBF-ICA. The advantage of the approach is that it adopts Chaos-PSO to optimize the RBF parameter in the nonlinear ICA algorithm, and thus the nonlinear ICA can achieve higher performance than the original one. The face image experiments have been carried out to evaluate and validate the newly proposed method. The analysis results demonstrate that the proposed method is superior to the linear ICA based noise removal. Thus, the Chaos-PSO based RBFICA can be used in industrial applications.
Wei Ding Ding // Procedia Procedia Engineering Engineering 00 24 (2011) (2011) 000–000 111 – 115 Wei
References [1] Zhao X, Jing R, Gu M. Adaptive intrusion detection algorithm based on rough sets. Qinghua Daxue Xuebao 2008; 48: 1165– 1168. [2] Widodo A, Yang B. Wavelet support vector machine for induction machine fault diagnosis based on transient current signal. Expert Sys Appl 2008; 35: 307–316. [3] Li Z, Yan X, Yuan C, Zhao J, Peng Z. Fault detection and diagnosis of the gearbox in marine propulsion system based on bispectrum analysis and artificial neural networks. J Marine Sci Appl 2011; 10: 17–24. [4] Belkin M, Niyogi P. Laplacian Eigenmaps for Dimensionality Reduction and Data Representation. Neural Comput 2003; 15: 1373–1396 [5] Li Z, Yan X, Yuan C, Peng Z, Li L. Virtual prototype and experimental research on gear multi-fault diagnosis using wavelet-autoregressive model and principal component analysis method. Mech Syst Signal Process 2011; 25: 2589–2607. [6] Li Z, Yan X. Independent component analysis and manifold learning with applications to fault diagnosis of VSC-HVDC systems. Hsi-An Chiao Tung Ta Hsueh 2011; 45: 44–48. [7] Giannakis G, Swami A. New results on state-space and input-output identification of non-Gaussian processes using cumulants. In: Advanced Signal Processing Alg., Arch. and Implem., SPIE'87, San Diego, CA, SPIE; 1987; 826: 199–205. [8] Jutten C, Herault J. Blind separation of sources, part I. An adaptive algorithm based on neuromimetic architecture. Signal Process 1991; 24: 1–10. [9] Sorouchyari E. Blind separation of sources, part III. Stability analysis. Signal Process 1991; 24: 21–29. [10] Comon P, Jutten C, Herault J. Blind separation of sources, part II. Problems statement. Signal Process 1991; 24: 11–20. [11] Tan Y, Wang J, Zuradam M. Nonlinear blind source separation using a radial basis function network. IEEE Trans Neural Networks 2001; 12: 124–134. [12] Kennedy J, Eberhart R. Particle swarm optimization. In: Proc. of IEEE international conference on neural networks, Perth Australia, IEEE Service Centre, Piscataway, NJ USA; 1995 p. 1942–1948. [13] Mendes R, Kennedy J, Neves J. Watch thy neighbor or how the swarm can learn from its environment. In: Proc. of the IEEE Swarm Intelligence Symposium 2003, Indianapolis, Indiana, IEEE Service Centre, Piscataway, NJ USA; 2003, p. 88–94. [14] Riget J, Vesterstroem J. A diversity-guided particle swarm optimizer–the ARPSO. Technical Report 2002-02, Department of Computer Science, University of Aarhus, 2002. [15] Clerc M, Kennedy J. The particle swarm: Explosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evol Comput 2002; 6: 58–73,. [16] S. Wang and B. Meng: Chaos Particle Swarm Optimization for Resource Allocation Problem. Proc. of the IEEE International Conference on Automation and Logistics, Jinan, China, pp. 464-467, 2007.
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Procedia Engineering
ProcediaProcedia Engineering 00 (2011) Engineering 24 000–000 (2011) 111 – 115 www.elsevier.com/locate/procedia
2011 International Conference on Advances in Engineering
A New Method for Image Noise Removal using Chaos-PSO and Nonlinear ICA Wei Ding a
a*
Nanyang Institute of Technology, Nanyang 470004, China
Abstract The blind source separation (BSS) algorithms, especially the independent component analysis (ICA) algorithms, have been proven to be effective for the image data processing. The noise signal introduced into the image data can be perfectly eliminated using ICA only under the linear mixture condition. However, the images are always mixed nonlinearly with noise perturbations. The traditional linear ICA algorithm is not capable enough to suppress the noise in this situation. Hence, the nonlinear ICA is proposed to deal with the nonlinear mixtures in this paper. The radial basis function (RBF) neural network based post-nonlinear ICA algorithm has been adopted to remove noise from the original image data. To enhance the RBF-ICA operation, the Chaos-Particle Swarm Optimization (PSO) algorithm has been employed to optimize the RBF neural network to obtain satisfactory nonlinear solution of the nonlinear BBS procedure. A series of experiments have been implemented in this work to validate the efficiency of the proposed method. The Chaos-PSO optimized RBF-ICA model has been compared with other ICA models in the image de-noising processing. The comparative results show that the proposed approach is superior to the nonoptimized ICA methods with respect to the image de-noising performance.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of ICAE2011. Keywords: Image processing; nonlinear ICA; Chaos-PSO; noise removal
1. Introduction Noise removal is important for the face recognition of the image data. Heavy noise pollution can make the face identification fail. As a result, the noise removal is essential for the identification of face images. There are many developments to handle the face images in an intelligent manner. These advanced methods include evolution algorithm, artificial neural network (ANN), principal component analysis (PCA) and independent component analysis (ICA) [1]. Though PCA [2] is very powerful for image denoising and benefits the face recognition processing, it has drawbacks when dealing with nonlinear * Corresponding author. Tel.: +86- 377-3121454; fax: +86-377-3121486. E-mail address: [email protected]
1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.11.2611
2112
Wei /Ding / Procedia Engineering 24 (2011) 111 – 115 Wei Ding Procedia Engineering 00 (2011) 000–000
signals [3-5]. This is because the PCA is proposed on the linear model of the data. The ICA has not such shortcomings and is very effective in the image pre-processing for the nonlinear signals [6]. The ICA is firstly proposed by Giannakis [7] in 1987 to solve the blind source separation (BSS) problem. Following in 1991, Jutten and Herault [8], Sorouchyari [9], Comon [10] have published three important papers about the BSS to make ICA be famous in signal processing. Frankly, the traditional ICA algorithm, such as the Fast ICA, has the ability to tackle the nonlinear image data. However, with the increase of the nonlinearity severity, the linear ICA cannot separate correct signal sources. This is because the nonlinear mixing increases the complexity of nonlinear characteristics. In addition, a unique solution is difficult to attain for the nonlinear BSS problem. For this reason, Tan [11] proposed the RBF neural network based ICA to deal with the nonlinear BSS. The possibility of unique solution emerges for the nonlinearity. Taking the advantage of Tan’s method [11], we have further improved the RBF based on nonlinear ICA algorithm. A new method based on the Chaos-Particle Swarm Optimization (PSO) and RBF-ICA has been proposed to eliminate the image noise. Compared with Tan’s RBF-ICA [11], the proposed approach uses Chaos-PSO to optimize the RBF neural network (RBF NN) and hence better efficiency can be obtained for the nonlinear mixture separation. By the face recognition experiments, the results show that the noise can be removed effectively and the face recognition of the proposed method is superior to the linear ICA and the original RBF-ICA. 2. The proposed noise removal algorithm 2.1. RBF based ICA The RBF based ICA is a typical post-nonlinear model (PNL) for nonlinear BSS problem. The RBF NN is used to map the nonlinear mixture into the linear one such that the reversal of the nonlinear mixing function can be obtained. Then the nonlinear BBS is transformed into linear case and the traditional linear ICA can be employed to find the independent sources (ICs). The RBF based ICA model is defined as [11]
x = As a = f ( x) , b = g (a ) = y Wb ≈ s
(1)
where s is the source signals, A is the linear mixing matrix, x is the linear mixtures, a is the nonlinear mixtures, f(·) is the nonlinear mixing function, b is the nonlinear de-mixtures, g(·) is the nonlinear demixing function, y is the estimated sources, and W is the linear de-mixing matrix. The system diagram of the RBF based ICA is shown in Fig. 1. Sources s
Linear mixing by A
Nonlinear mixing using f(·)
Nonlinear demixing via RBF
De-mixing by W
Estimated sources y
Fig.1. The system diagram of the RBF based ICA
The RBF NN herein acts as the nonlinear de-mixing solver. The detail of the RBF based ICA for the nonlinear BSS can be found in [11]. To further develop Tan’s theory, the Chaos-PSO has been applied to the parameter optimization for the RBF NN. 2.2. Chaos-PSO
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Wei Ding Ding // Procedia Procedia Engineering Engineering 00 24 (2011) (2011) 000–000 111 – 115 Wei
Particle Swarm Optimization (PSO) algorithm is an effective evolutionary algorithm inspired by the social behavior of bird flocks and fish schools [12]. The extremum searching of PSO is very simple and effective and hence PSO has been widely used in the optimization of ANNs. Nevertheless, PSO has problems of suffering from premature convergence [13]. Research for addressing the shortcomings of PSO is ongoing and includes such changes as dynamic or exotic sociometries [13], enriched particle diversity [14], and tunable parameters in the updating [15]. Many methods have been proposed to overcome this problem, but the door for improving PSO still opens. To avoid premature convergence, the Chaos search processing has been employed to improve PSO optimization in this paper. Through the increase of particle diversity by Chaos search, the premature convergence can be depressed significantly. PSO is similar to genetic algorithm, but it updates offspring by the adjustment of the flight speed and position [16]. The optimal solution of the PSO search is the best position that the particle experiences. The PSO algorithm for updating flight speed υ and flight position x can be expressed by [12]
υi ( k = + 1) ωiυi ( k ) + c1r1[ xd − xi ( k )] + c2 r2 [ x g − xi ( k )]
(2)
xi ( k + 1) = xi ( k ) + υi ( k + 1)
(3)
where, ωi is the inertial weights, c1 and c2 are the learning factors, and r1 and r2 denote the random coefficients among [0,1]. xd is the individual extremum and x g is the global extremum. The Chaos search is used for the secondary optimization to diverse PSO’s new generation. The Logistic equation has been adopted for Chaos. The Logistic equation is defined by [16]
xi ( k= + 1) µ xi ( k )(1 − xi ( k ))
(4)
where µ is the control coefficient. The secondary optimization using Chaos can be interpreted as follows. (1) Mapping the flying coordinate x into the Chaos variables in [0 1] by the below carrier
d= ( xi ( k ) − α i ) / β i i (k )
(5)
where α i and β i need to be specified. Then the Chaos behavior can be calculated by
d i ( k= + 1) µ ci ( k )(1 − d i ( k ))
(6)
(2) Iterating the Chaos behavior until find the best d i , and then Chaos variables will be mapped back to the original space by the second carrier
xi ( k= ) α i + βi di (k )
(7)
Through this secondary optimization, the chaos algorithm can help PSO find proper global extremum. 3. Experiment results The face recognition experiments have been implemented in this paper to validate the performance of the proposed algorithm. In the Chaos-PSO optimization, set c= c= 1.3 , ω = 0.55 , the size of particle 1 1 swarm selects 50 for PSO, and set µ = 3.5 , α = 1.7 and β = 0.6 for Logistic chaos. Fig. 2 shows the performance of the image noise removal. One can note from Fig. 2 that the original noisy image can be recovered precisely by the proposed method.
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Wei /Ding / Procedia Engineering 24 (2011) 111 – 115 Wei Ding Procedia Engineering 00 (2011) 000–000
Fig. 2. The original noisy image (left), the skin with noise removal (middle) and the recovered face image (right).
By the Chaos-PSO optimized nonlinear ICA pre-processing, the face recognition rate can be enhanced greatly. The RBF NN has been adopted as the classifiers to identify the face images. In this paper, 50 pieces face have been prepared for the face recognition experiment. The face recognition performance is shown in Table 1. The recognition accuracy of the proposed method has been compared with some existing approaches. The proposed method has been compared with no de-noising processing, PCA removal, ICA removal and RBF-ICA removal in the face recognition. It can be seen form Table 1 that the nonlinear ICAs increase the face recognition rate apparently, and the Chaos-PSO optimized one obtains the best identification among the five methods. This is because the Chaos-PSO can prevent the RBF from local optimization. Table 1. The face recognition results Method
Recognition accuracy
RBF NN PCA-RBF NN Linear ICA-RBF NN RBF based ICA-RBF
87.3% 88.7% 88.3%
The proposed method
93.3%
NN
90.6%
4. Conclusions Pre-processing on the original noisy images can increase the image identification efficiency. The linear ICA has been proved to be useful for the image noise removal. However, for images which are mixed with nonlinear noise, the traditional ICA may fail to work. Hence, this paper proposed a new face image pre-processing method based on Chaos-PSO based RBF-ICA. The advantage of the approach is that it adopts Chaos-PSO to optimize the RBF parameter in the nonlinear ICA algorithm, and thus the nonlinear ICA can achieve higher performance than the original one. The face image experiments have been carried out to evaluate and validate the newly proposed method. The analysis results demonstrate that the proposed method is superior to the linear ICA based noise removal. Thus, the Chaos-PSO based RBFICA can be used in industrial applications.
Wei Ding Ding // Procedia Procedia Engineering Engineering 00 24 (2011) (2011) 000–000 111 – 115 Wei
References [1] Zhao X, Jing R, Gu M. Adaptive intrusion detection algorithm based on rough sets. Qinghua Daxue Xuebao 2008; 48: 1165– 1168. [2] Widodo A, Yang B. Wavelet support vector machine for induction machine fault diagnosis based on transient current signal. Expert Sys Appl 2008; 35: 307–316. [3] Li Z, Yan X, Yuan C, Zhao J, Peng Z. Fault detection and diagnosis of the gearbox in marine propulsion system based on bispectrum analysis and artificial neural networks. J Marine Sci Appl 2011; 10: 17–24. [4] Belkin M, Niyogi P. Laplacian Eigenmaps for Dimensionality Reduction and Data Representation. Neural Comput 2003; 15: 1373–1396 [5] Li Z, Yan X, Yuan C, Peng Z, Li L. Virtual prototype and experimental research on gear multi-fault diagnosis using wavelet-autoregressive model and principal component analysis method. Mech Syst Signal Process 2011; 25: 2589–2607. [6] Li Z, Yan X. Independent component analysis and manifold learning with applications to fault diagnosis of VSC-HVDC systems. Hsi-An Chiao Tung Ta Hsueh 2011; 45: 44–48. [7] Giannakis G, Swami A. New results on state-space and input-output identification of non-Gaussian processes using cumulants. In: Advanced Signal Processing Alg., Arch. and Implem., SPIE'87, San Diego, CA, SPIE; 1987; 826: 199–205. [8] Jutten C, Herault J. Blind separation of sources, part I. An adaptive algorithm based on neuromimetic architecture. Signal Process 1991; 24: 1–10. [9] Sorouchyari E. Blind separation of sources, part III. Stability analysis. Signal Process 1991; 24: 21–29. [10] Comon P, Jutten C, Herault J. Blind separation of sources, part II. Problems statement. Signal Process 1991; 24: 11–20. [11] Tan Y, Wang J, Zuradam M. Nonlinear blind source separation using a radial basis function network. IEEE Trans Neural Networks 2001; 12: 124–134. [12] Kennedy J, Eberhart R. Particle swarm optimization. In: Proc. of IEEE international conference on neural networks, Perth Australia, IEEE Service Centre, Piscataway, NJ USA; 1995 p. 1942–1948. [13] Mendes R, Kennedy J, Neves J. Watch thy neighbor or how the swarm can learn from its environment. In: Proc. of the IEEE Swarm Intelligence Symposium 2003, Indianapolis, Indiana, IEEE Service Centre, Piscataway, NJ USA; 2003, p. 88–94. [14] Riget J, Vesterstroem J. A diversity-guided particle swarm optimizer–the ARPSO. Technical Report 2002-02, Department of Computer Science, University of Aarhus, 2002. [15] Clerc M, Kennedy J. The particle swarm: Explosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evol Comput 2002; 6: 58–73,. [16] S. Wang and B. Meng: Chaos Particle Swarm Optimization for Resource Allocation Problem. Proc. of the IEEE International Conference on Automation and Logistics, Jinan, China, pp. 464-467, 2007.
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