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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 19 — Sep. 14, 2009
  • pp: 16809–16819
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Optimization of an off-axis three-mirror anastigmatic system with wavefront coding technology based on MTF invariance

Feng Yan and Xuejun Zhang  »View Author Affiliations


Optics Express, Vol. 17, Issue 19, pp. 16809-16819 (2009)
http://dx.doi.org/10.1364/OE.17.016809


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Abstract

In this paper the invariance of modulation transfer function (MTF), which describes the insensitivity to perturbation of MTF, is defined to be the evaluating criterion of the wavefront coding system. The rapid optimization of wavefront coding system based on the MTF invariance is proposed by means of introducing the mathematical program Matlab to normal optical design process. The interface called MZDDE between Matlab and Zemax is applied to realize the fast data exchanging and merit function calculating. The genetic algorithm tool (GA) in Matlab is introduced to the optimizing process, which accelerates the converging efficiency considerably. The MTF invariance of optimized system drops to 0.0119 while that of original system is larger than 0.018. If the all the fields of view is taken into consideration, the MTF invariance of optimized system and original system is less than 0.015 and larger than 0.020 respectively. It is proven that the optimization of the unusual optical system with special property can be executed conveniently and rapidly with the help of external program and dynamic data exchange.

© 2009 OSA

1. Introduction

Different from traditional imaging system, the wavefront coding system is not designed to achieve the best image quality but to preserve the image quality as constant as possible in the extended DOF and reduce the difficulty of digital processing. Therefore the optimizing target of wavefront coding system also becomes different from traditional system. The leading factor lies in that proper quality criterion that exactly describes the coincidence of MTF in the extended DOF should be established and applied in the optimizing process. Several criteria have been proposed to optimize and evaluate the wavefront coding system. Fisher information (FI) is the most widely- used evaluating function as a measure of the sensitivity of point spread function (PSF) to defocus of the wavefront coding system [10

10. Z. Ting-yu, W.-Z. Zhang, Z. Ye, and F.-H. Yu, “Design of wavefront coding system based on evaluation function of fisher information,” Acta Opt. Sin. 27(6), 1096–1101 (2007) (in Chinese).

,11

11. V. Sudhakar Prasad, P. Panca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004). [CrossRef]

]. Quality factor (QF), which is the integral of the defocused MTF with respect to the spatial frequency has been built as a MTF based criterion to describe the correlation of variation of the MTF with the defocus [12

12. S. Mezoouari and A. R. Harvey, “Wavefront coding for aberration compensation in thermal imaging systems,” Proc. SPIE 4442, 34–42 (2001). [CrossRef]

]. Strehl ratio (SR) is also considered as effective evaluation of the wavefront coding system with rotational symmetry [13

13. S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28(10), 771–773 (2003). [CrossRef] [PubMed]

,14

14. S. Mezouari, G. Muyo, and A. R. Harvey, “Circular symmetric phase filters for control of primary third-order aberrations:coma and astigmatism,” J. Opt.Soc.Am.A 23, 1058–1062 (2006).

]. Besides, some designers use the combination of SR and encircled energy (EE, which refers to the percentage of the total energy in the PSF contained within a radius measured from the PSF center) as their criterion in the design of triplet objectives [15

15. L. Guang-zhi, X. Zhang, Z. Jian-pin, Y. Hao-ming, H. Feng-yun, and X. Zhang, “Novel optimization method for wavefront coding system,” Opt.Precision Eng. 16(7), 1171–1176 (2008) (in Chinese).

]. It can be concluded that all the criteria mentioned above are all MTF based directly or indirectly.

Hence, these criteria or merit function can be obtained through simple calculation and enables rapid optimization and convergence.

Theoretically speaking, a perfect wavefront coding system should satisfy the relationship shown in Eq. (1) for given toleranced error budget of defocus aberrationw020:
MTFm,nw020=0,
(1)
Where m denotes the different position of image plane in the extended DOF corresponding to different defocus value w020and n denotes the sampled space frequency which ranges from 0 to fcutoff and is spaced by proper increment [15

15. L. Guang-zhi, X. Zhang, Z. Jian-pin, Y. Hao-ming, H. Feng-yun, and X. Zhang, “Novel optimization method for wavefront coding system,” Opt.Precision Eng. 16(7), 1171–1176 (2008) (in Chinese).

].

Equation (1) demonstrates the most fundamental and important property of wavefront coded system: MTF values or image quality should be uncorrelated to the defocus aberration within the extended DOF. In another words the MTF values will remain perfectly uniform within the extended DOF if the relationship is fulfilled for a wavefront coded system. Thus, Eq. (1) is actually considered to be the most appropriate criterion or merit function of the optimization since it’s the essential character of the wavefront coding system defined by its fundamental theory. However, the practical application of that criterion means that all the MTF/w020 values of each sampled space frequency in each position of image plane in the extended DOF will be calculated within each optimizing cycle. It will be of extremely enormous consumption of time and computation for optical design program, which limits the actual application of the merit function in Eq. (1).

2 Work foundations

The secondary mirror is selected to be the wavefront coding element for it serves as the stop aperture and has relatively small geometrical size. The surface sag of the secondary mirror before optimization is shown in Eq. (2) [16

16. F. Yan, Z. Li-gong, and Z. Xue-jun, “A Design of Off-axis Three Mirror Anastigmatic Optical System with Wavefront Coding Technology,” Opt. Eng. 47(6), 063001 (2008). [CrossRef]

]
z(x,y)=c(x2+y2)1+1(1+k)c2(x2+y2)+β(x3+y3),
(2)
It can be observed that the secondary mirror is an quasi-freeform surface comprised by adding a cubic term to a convex aspherical surface, which yields the pupil function of the new system with redesigned secondary mirror is shown in Eq. (3) [16

16. F. Yan, Z. Li-gong, and Z. Xue-jun, “A Design of Off-axis Three Mirror Anastigmatic Optical System with Wavefront Coding Technology,” Opt. Eng. 47(6), 063001 (2008). [CrossRef]

].
P(x,y)={12exp{j[w020(x2+y2)+α(x3+y3)]}  for (x2+y2)1/21 0otherwiseα=24,
(3)
The MTF curves of the wavefront coding system are shown in Fig. 2
Fig. 2 The comparison of MTF curves between wavefront coding system (left) and traditional system (right) when the defocus aberration varies from 1.25λto 1.25λ with increment of 0.25λ defocus
when the defocus aberration varies from 1.25λto1.25λ with0.25λincrement. It can be observed that these MTF curves are coincided very well with each other and it can be regarded that the DOF of the wavefront coding system is extended as 10 times as the original system.

where x, y is mirror coordinate, z is surface sag, c is the curvature of the vertex, k is the conic constant and β denotes the magnitude of cubic terms.

where x, y is normalized coordinate and α indicates the magnitude of cubic phase term from which β in Eq. (2) is deduced.

The actual cut-off space frequency of the system is about 57 lp/mm, which is determined by the Nyquist frequency of digital image detector (the size of CCD pixel is8.75μm×8.75μm). Considering authentic restoration of mid images, the dynamic MTF must be higher than 0.05 to decrease the difficulty of image restoration, which demand the design MTF should be 0.18 at least because of the push-broom working mode (In a push broom sensor, a line of sensors arranged perpendicular to the flight direction of the spacecraft is used, thus different areas of the surface are imaged as the spacecraft flies forward), residual manufacturing and alignment error and other factor that may decrease the MTF value. Thus, the value of α is set about 24 to satisfy the requirement although the value of α is suggest to be set much larger than 20 to ensure adequate insensitivity to misfocus of the MTF according to the theory of WFC technology. Therefore, the MTF curves differ from each other a little especially in the low-frequency domain, and then further compensation and optimization can be carried out to improve the performance.

3. Definition of MTF invariance

The most distinct character of wavefront coding system is the insensitivity of the MTF to various defocus aberration in its given range. The MTF value of arbitrary spatial frequency should maintain nearly the same in the extended DOF. MTF invariance is defined in Eq. (4) to describe the degree of consistency of MTF in the extended DOF.
MTF similarity{PVMTF=max(ΔMTFi)RMSMTF=[(Σn1ΔMTFi2)/n(Σn1ΔMTFi/n)2]1/2,
(4)
whereΔMTFi=MTFtestiMTFrefi, theMTFirefis the MTF of a referenced image plane within the extended DOF, which calls for calculational selection and usually does not locate in the middle of the extended DOF range while the MTFitestis the MTF of other evaluating position in that range as shown in Fig. 3
Fig. 3 The schematic explanation of the definition of MTF invariance
. i=1,2......n, n is the number of sampling point. For the system under research whose cut-off space frequency is 57 lp/mm, if the sampling interval is 1 lp/mm, then n equals 58; while if the sampling interval is 0.2 lp/mm, n equals 286.

4. Definition of optimizing criterion base on MTF invariance

The optimizing criterion is established in Eq. (5):
opt_cri=max(RMSjMTF),
(5)
WhereRMSjMTF=[(Σ1nΔMTFi2)/n(Σ1nΔMTFi/n)2]1/2|i=1n,j=1m ΔMTFi|j=1m=MTFtestiMTFrefi|i=1n,j=1m j=1,2......m, m is the number of sampling positions where the MTFtestis calculated corresponding to different defocus value. The value of m should be assigned properly to balance the tradeoff between calculation speed and sampling intensity. When the value of Eq. (5) is minimized, the optimal design is considered to be achieved. It can be observed in Fig. 4
Fig. 4 The RMSMTFcurve of the WFC system before optimazition with different defocus aberration in the extended DOF with increment of 0.25λ defocus
that the RMSMTFcurve of the original system (β=8×10-4)with different defocus aberration in the extended DOF is not continuous or monotonous but do not oscillate fiercely or vary sharply, it is enough to set m equal 11 to get sufficient sampling intensity with the increment of 0.25λ defocus aberration. It also can be seen from Fig. 4 that the maximum RMSMTFvalue of the original system is about 0.0187. As the assignment of the value of m, the same consideration should be taken into in setting the value of n. Finally, the interval of sampling space frequency is 0.5 lp/mm so that n is set 115.

This criterion is based on the consideration that if the maximum value ofRMSMTFin the extended DOF is minimized, all the RMSMTFvalue will also achieve its minimum. Thus, the optimizing cycles are concentrated on minimizing the maximum value of the 11 samplingRMSMTF.

The surface sag of the secondary mirror is transformed to the optimizing form as shown in Eq. (6):
z(x,y)=c(x2+y2)1+1(1+k)c2(x2+y2)+a1x3+a2x2y+a3xy2+a4y3,
(6)
where x, y is normalized coordinate and ai (i = 1..4) is optimizing variable.

It is presented above that the DOF of the wavefront coding system under research is extended as ten times as traditional system and the image quality will remain the same in the extended DOF. However, the concept of Gaussian image plane of traditional system is not suitable for the wavefront coding system or in another words the extended DOF of wavefront coding system may not be symmetric about the original “Gaussian image plane” for optimum image quality coincidence mainly because of the residual off-axis aberration. A nominal Gaussian image plane can be defined for describe the location of the extended DOF for convenience as shown in Fig. 5
Fig. 5 The schematic explanation of location of the extended DOF
.

Furthermore, in the unoptimizable system the location of the extended DOF is fixed and just symmetric about the original Gaussian image plane. Nevertheless for the system to be optimized the location of the extended DOF is made another variable a5to gain the most appropriate position, which means the range of the extended DOF can be moved as a whole to obtain the global minimum value of MTF similarit. In the unoptimizable system the position of MTFrefis just set in the middle of the extended DOF, which is namely the Gaussian image. But for the system to be optimized, the position of Gaussian image may not the optimal choice to obtain the minimum ofRMSMTF. Hence the position where theMTFrefis calculated is also made variable a6to obtain the most suitable MTFreffor minimizing the RMSMTF, which means that the position ofMTFrefcan be select within the whole extended DOF in optimization. Thus, the optimizing variable vector is established asA=[ai]i=6. Several restrictions should be imposed on the variable to avoid invalid results. Firstly, a1anda4should be limited a little smaller than the initial value of β to maintain the fundamental property of the system. Secondly, the maximum value ofa2and a3should be limited less than one tenth of β for the same purpose. Thirdly, the absolute values of a5 should be restricted less than 3 to limit the deviation of the referenced image plane from original Gaussian image plane not too far away. Finally, a6should be limited properly so as to maintain the reference position where the MTFrefis calculated within the bound of the extended DOF. All the restrictions are appended to prevent the nonsensical result which achieves the small RMSMTF at the cost of excessively low MTF value.

5. Optimization of the wavefront coding system with MZDDE and GA tool

  • 1 Create an optimizing vector A=[ai]i=6 in Matlab.
  • 2 Push these data to Zemax and make them take effect through MZDDE.
  • 3 Calculate the MTF of the 12 particular positions with the given sampled space frequency in Zemax.
  • 4 Extract the MTF value from Zemax through MZDDE and create the variable vector of Matlab according to the MTF value.
  • 5 Calculate the value of opt_cri for comparison and filtration.

It can be seen from Fig. 7
Fig. 7 The MTF curves of optimized wavefront coding system with different defocus aberration varying from 1.25λto 1.25λwith increment of 0.25λ defocus
that these MTF curves seem to coincide with each other not so closer as the original system shown in Fig. (2) left. But in fact t the PVMTFof optimized system drops to 0.046 from 0.057 of original system since the selection ofMTFrefis also optimized. The MTF value has nearly no reduction compared with the original system. The most distinguished advantage of this optimizing method is much more time-saving than the optimization directly operated in Zemax. The total execution time of the optimizing process is about 30 minutes. Nevertheless the direct optimization with the same criterion merit function with Zemax is very difficult to operate, which always terminate abnormally without any significative result mainly because the system resource is exhausted by the massive computation. And it can be seen from the Fig. 8
Fig. 8 The comparison of RMSMTFcurve of different FOV between optimized system (left) and original system (right) when the defocus aberration varies from 1.25λto 1.25λwith increment of 0.25λ defocus
that the optimization result is also valid for all the fields of view (FOV).

TheRMSMTFof the optimized system of all the other FOV is less than 0.015 while the correspondingRMSMTFof the original system is larger than 0.020. It can be observed that the RMSMTFof the other FOV is not as fine as the one of the central FOV used in optimization, which is obvious because the RMSMTFof the other FOV is not considered adequately in optimization but just supposed to be minimized so long as the RMSMTFof central FOV approaches its optimum instead. The assumption is applied practically although it is not absolutely exact. After all the optimizing result can be accepted without insufferable deviation and the dominating excuse is that the cost time will be extremely prolonged based on the existing criterion if the FOV ingredient is involved. The improved optimizing criterion including the FOV and requiring less computational time and resource is under research

6. Conclusions

In this paper the MTF invariance is defined as a new criterion for the optimization of wavefront coding system. The new optimizing method based on MZDDE and GA tool is proposed to deal with the complex computation of MTF invariance. The optimizing result and the cost time are satisfied. In the future work the factor of FOV is planned to be introduced to the optimization and the more suitable result for all FOV may be obtained. Another important work calls for great attention and effort lies in the manufacturing and testing of the optimized secondary mirror with adequate accuracy, which is actually a free-form surface.

Acknowledgement

This research is supported by a grant from the National High Technology Research and Development Program of China (863 Program) (No. 2009AA12Z105).

Reference and Links:

1.

E. R. Dowski Jr and T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

2.

J. Ojeda-Castañeda, J. E. Landgrave, and C. M. Gómez-Sarabia, “Conjugate phase plate use in analysis of the frequency response of imaging systems designed for extended depth of field,” Appl. Opt. 47(22), E99–E105 (2008). [CrossRef] [PubMed]

3.

S. Bagheri, P. E. X. Silveira, and D. Pucci de Farias, “Analytical optimal solution of the extension of the depth of field using cubic phase Wavefront Coding,” J. Opt. Soc. Am. 25(5), 1051–1063 (2008). [CrossRef]

4.

W. Chi, and N. George, “Smart Camera with Extended Depth of Field,” Proc. SPIE 6024, 602424–1—602424–6(2005).

5.

M. Somayaji and M. P. Christensen, “Frequency analysis of the wavefront-coding odd-symmetric quadratic phase mask,” Appl. Opt. 46(2), 216–226 (2007). [CrossRef] [PubMed]

6.

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983). [CrossRef]

7.

S. Bradburn, W. T. Cathey, and E. R. Dowski, “Realizations of focus invariance in optical-digital systems with wave-front coding,” Appl. Opt. 36(35), 9157–9166 (1997). [CrossRef]

8.

J. van der Gracht, J. G. Nagy, V. Pauca, and R. J. Plemmons, “Iterative restoration of wavefront coded imagery for focus invariance,” in Integrated Computational Imaging Systems, OSA Technical Digest Series (Optical Society of America, 2001), paper ITuA1.

9.

F. Yan, Z. Li-gong, and Z. Xue-jun, “Image Restoration of an Off-axis Three mirror Anastigmatic Optical System with Wavefront Coding Technology,” Opt. Eng. 47(1), 0170081–0170088 (2008).

10.

Z. Ting-yu, W.-Z. Zhang, Z. Ye, and F.-H. Yu, “Design of wavefront coding system based on evaluation function of fisher information,” Acta Opt. Sin. 27(6), 1096–1101 (2007) (in Chinese).

11.

V. Sudhakar Prasad, P. Panca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004). [CrossRef]

12.

S. Mezoouari and A. R. Harvey, “Wavefront coding for aberration compensation in thermal imaging systems,” Proc. SPIE 4442, 34–42 (2001). [CrossRef]

13.

S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28(10), 771–773 (2003). [CrossRef] [PubMed]

14.

S. Mezouari, G. Muyo, and A. R. Harvey, “Circular symmetric phase filters for control of primary third-order aberrations:coma and astigmatism,” J. Opt.Soc.Am.A 23, 1058–1062 (2006).

15.

L. Guang-zhi, X. Zhang, Z. Jian-pin, Y. Hao-ming, H. Feng-yun, and X. Zhang, “Novel optimization method for wavefront coding system,” Opt.Precision Eng. 16(7), 1171–1176 (2008) (in Chinese).

16.

F. Yan, Z. Li-gong, and Z. Xue-jun, “A Design of Off-axis Three Mirror Anastigmatic Optical System with Wavefront Coding Technology,” Opt. Eng. 47(6), 063001 (2008). [CrossRef]

17.

http://www.mathworks.com/matlabcentral/fileexchange/7507.

18.

D. C. van Leijenhorst, C. B. Lucasius, and J. M. Thijssen, “Optical design with the aid of a genetic algorithm,” Biosystems 37(3), 177–187 (1996). [CrossRef] [PubMed]

19.

P. Siarry, A. Petrowski, and M. Bessaou, “A multipopulation genetic algorithm aimed at multimodal optimization,” Adv. Eng. Software 33(4), 207–213 (2002). [CrossRef]

OCIS Codes
(110.0110) Imaging systems : Imaging systems
(110.4100) Imaging systems : Modulation transfer function
(220.0220) Optical design and fabrication : Optical design and fabrication
(220.1250) Optical design and fabrication : Aspherics

ToC Category:
Imaging Systems

History
Original Manuscript: July 17, 2009
Revised Manuscript: August 24, 2009
Manuscript Accepted: August 25, 2009
Published: September 4, 2009

Citation
Feng Yan and Xuejun Zhang, "Optimization of an off-axis three-mirror anastigmatic system with wavefront coding technology based on MTF invariance," Opt. Express 17, 16809-16819 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16809


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References

  1. E. R. Dowski and T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]
  2. J. Ojeda-Castañeda, J. E. Landgrave, and C. M. Gómez-Sarabia, “Conjugate phase plate use in analysis of the frequency response of imaging systems designed for extended depth of field,” Appl. Opt. 47(22), E99–E105 (2008). [CrossRef] [PubMed]
  3. S. Bagheri, P. E. X. Silveira, and D. Pucci de Farias, “Analytical optimal solution of the extension of the depth of field using cubic phase Wavefront Coding,” J. Opt. Soc. Am. 25(5), 1051–1063 (2008). [CrossRef]
  4. W. Chi, and N. George, “Smart Camera with Extended Depth of Field,” Proc. SPIE 6024, 602424–1—602424–6(2005).
  5. M. Somayaji and M. P. Christensen, “Frequency analysis of the wavefront-coding odd-symmetric quadratic phase mask,” Appl. Opt. 46(2), 216–226 (2007). [CrossRef] [PubMed]
  6. K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983). [CrossRef]
  7. S. Bradburn, W. T. Cathey, and E. R. Dowski, “Realizations of focus invariance in optical-digital systems with wave-front coding,” Appl. Opt. 36(35), 9157–9166 (1997). [CrossRef]
  8. J. van der Gracht, J. G. Nagy, V. Pauca, and R. J. Plemmons, “Iterative restoration of wavefront coded imagery for focus invariance,” in Integrated Computational Imaging Systems, OSA Technical Digest Series (Optical Society of America, 2001), paper ITuA1.
  9. F. Yan, Z. Li-gong, and Z. Xue-jun, “Image Restoration of an Off-axis Three mirror Anastigmatic Optical System with Wavefront Coding Technology,” Opt. Eng. 47(1), 0170081–0170088 (2008).
  10. Z. Ting-yu, W.-Z. Zhang, Z. Ye, and F.-H. Yu, “Design of wavefront coding system based on evaluation function of fisher information,” Acta Opt. Sin. 27(6), 1096–1101 (2007) (in Chinese).
  11. V. Sudhakar Prasad, P. Panca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004). [CrossRef]
  12. S. Mezoouari and A. R. Harvey, “Wavefront coding for aberration compensation in thermal imaging systems,” Proc. SPIE 4442, 34–42 (2001). [CrossRef]
  13. S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28(10), 771–773 (2003). [CrossRef] [PubMed]
  14. S. Mezouari, G. Muyo, and A. R. Harvey, “Circular symmetric phase filters for control of primary third-order aberrations:coma and astigmatism,” J. Opt. Soc. Am. A 23, 1058–1062 (2006).
  15. L. Guang-zhi, X. Zhang, Z. Jian-pin, Y. Hao-ming, H. Feng-yun, and X. Zhang, “Novel optimization method for wavefront coding system,” Opt. Precision Eng. 16(7), 1171–1176 (2008) (in Chinese).
  16. F. Yan, Z. Li-gong, and Z. Xue-jun, “A Design of Off-axis Three Mirror Anastigmatic Optical System with Wavefront Coding Technology,” Opt. Eng. 47(6), 063001 (2008). [CrossRef]
  17. http://www.mathworks.com/matlabcentral/fileexchange/7507 .
  18. D. C. van Leijenhorst, C. B. Lucasius, and J. M. Thijssen, “Optical design with the aid of a genetic algorithm,” Biosystems 37(3), 177–187 (1996). [CrossRef] [PubMed]
  19. P. Siarry, A. Petrowski, and M. Bessaou, “A multipopulation genetic algorithm aimed at multimodal optimization,” Adv. Eng. Software 33(4), 207–213 (2002). [CrossRef]

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