Abstract

An innovative method for the automatic design of optical systems is presented and verified. The proposed method is based on a multi-objective evolutionary memetic optimization algorithm. The multi-objective approach simultaneously, but separately, addresses the image quality, tolerance, and complexity of the system. The memetic technique breaks down the search for optical designs in to three different parts or phases: optical glass selection, exploration, and exploitation. The optical glass selection phase defines the most appropriate set of glasses for the system under design. The glass selection phase limits the available glasses from hundreds to just a few, drastically reducing the design space and significantly increasing the efficiency of the automatic design method. The exploration phase is based on an evolutionary algorithm (EA), more specifically, on a problem-tailored generalized extremal optimization (GEO) algorithm, named Optical GEO (O-GEO). The new EA incorporates many features customized for lens design, such as optical system codification and diversity operators. The trade-off systems found in the exploration phase are refined by a local search, based on the damped least square method in the exploitation phase. As a result, the method returns a set of trade-off solutions, generating a Pareto front. Our method delivers alternative and useful insights for the compromise solutions in a lens design problem. The efficiency of the proposed method is verified through real-world examples, showing excellent results for the tested problems.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
New automatic optical design method based on combination of particle swarm optimization and least squares

Dabo Guo, Liang Yin, and Guang Yuan
Opt. Express 27(12) 17027-17040 (2019)

Automatic Optical Design

Donald P. Feder
Appl. Opt. 2(12) 1209-1226 (1963)

Method of glass selection for color correction in optical system design

Bráulio Fonseca Carneiro de Albuquerque, Jose Sasian, Fabiano Luis de Sousa, and Amauri Silva Montes
Opt. Express 20(13) 13592-13611 (2012)

References

  • View by:
  • |
  • |
  • |

  1. J. L. Rayces and M. Rosete-Aguilar, “Critical view of three lens design methods: damped least squares, Spencers and Glatzels,” Proc. SPIE 4927, 77–89 (2002).
    [Crossref]
  2. M. J. Colaço and G. S. Dulikravich, “A survey of basic deterministic, heuristic and hybrid methods for single objective optimization and response surface generation,” in Thermal Measurements and Inverse Techniques, H. R. B. Orlande, O. Fudym, D. Maillet, and R. M. Cotta, eds. (CRC, 2011), pp. 355–405.
  3. M.-H. Lin, J.-F. Tsai, and C.-S. Yu, “A review of deterministic optimization methods in engineering and management,” Math. Probl. Eng. 2012, 756023 (2012).
    [Crossref]
  4. M. Cavazzuti, Optimization Methods, from Theory to Design Scientific and Technological Aspects in Mechanics (Springer, 2013).
  5. C. Blum and A. Roli, “Metaheuristics in combinatorial optimization: overview and conceptual comparison,” ACM Comput. Surv. 35(3), 268–308 (2003).
    [Crossref]
  6. T. F. Gonzalez, Handbook of Approximation Algorithms and Metaheuristics, Computer & Information Science Series (Chapman & Hall/CRC, 2007).
  7. L. Bianchi, M. Dorigo, L. Gambardella, and W. Gutjahr, “A survey on metaheuristics for stochastic combinatorial optimization,” Nat. Comput. 8(2), 239–287 (2009).
    [Crossref]
  8. E.-G. Talbi, Metaheuristics: From Design to Implementation (J. Wiley & Sons, 2009).
  9. V. K. Viswanathan, I. O. Bohachevsky, and T. P. Cotter, “An attempt to develop an “intelligent” lens design program,” Proc. SPIE 0554, 10–17 (1986).
    [Crossref]
  10. M. van Turnhout, P. van Grol, F. Bociort, and H. P. Urbach, “Obtaining new local minima in lens design by constructing saddle points,” Opt. Express 23(5), 6679–6691 (2015).
    [Crossref] [PubMed]
  11. D. Shafer, “Global optimization in optical design,” Comput. Phys. 8(2), 188–195 (1994).
    [Crossref]
  12. H. Qin, “Particle swarm optimization applied to automatic lens design,” Opt. Commun. 284(12), 2763–2766 (2011).
    [Crossref]
  13. Z. Li and B. Yang, “Optical design and optimization of a miniature projector with liquid lenses via modified ant colony algorithm,” Opt. Eng. 51(7), 073001 (2012).
    [Crossref]
  14. E. G. M. Lacerda and A. C. P. L. F. Carvalho, “Introdução oas algoritmos genéticos,” in Sistemas Inteligentes: Aplicações a Recursos Hídricos e Ciências Ambientais, Coleção ABRH de Recursos Hidricos; 7, C. O. Galvão, and M. Valença, eds. (UFRGS, 1999), pp. 99–150.
  15. A. E. Eiben and J. E. Smith, Introduction to Evolutionary Computing, Natural Computing Series 1st ed. (Springer, 2003).
  16. D. Vasiljevic, Classical and Evolutionary Algorithms in The Optimization of Optical Systems, Genetic Algorithms and Evolutionary Computation Series, 1st ed. (Kluwer Academic Publishers, 2002).
  17. 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]
  18. I. Ono, S. Kobayashi, and K. Yoshida, “Global and multi-objective optimization for lens design by real-coded genetic algorithms,” Proc. SPIE 3482, 110–121 (1998).
    [Crossref]
  19. K. E. Moore, “Algorithm for global optimization of optical systems based on genetic competition,” Proc. SPIE 3780, 40–47 (1999).
    [Crossref]
  20. X. Chen and K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44(9), 1693–1702 (1997).
    [Crossref]
  21. J. Beaulieu, C. Gagné, and M. Parizeau, “Lens system design and re-engineering with evolutionary algorithms,” in Proceedings of Genetic and Evolutionary Computation Conference (GECCO, 2002), pp. 155–162.
  22. J. Sakuma and S. Kobayashi, “Latent variable crossover for k-tablet structures and its application to lens design problems,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2005), pp. 1347–1354.
    [Crossref]
  23. Y.-C. Fang, C.-M. Tsai, J. Macdonald, and Y.-C. Pai, “Eliminating chromatic aberration in Gauss-type lens design using a novel genetic algorithm,” Appl. Opt. 46(13), 2401–2410 (2007).
    [Crossref] [PubMed]
  24. S. Thibault, C. Gagné, J. Beaulieu, and M. Parizeau, “Evolutionary algorithms applied to lens design,” Proc. SPIE 5962, 66–76 (2005).
    [Crossref]
  25. Y. Nagata, “The lens design using the CMA-ES algorithm,” in Genetic and Evolutionary Computation-GECCO 2004, Vol. 3103 of Lecture Notes in Computer Science, K. Deb, ed. (Springer, 2004), pp. 1189–1200.
  26. J. R. Koza, S. H. Al-Sakran, and L. W. Jones, “Automated re-invention of six patented optical lens systems using genetic programming,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2005), pp. 1953–1960.
    [Crossref]
  27. L. W. Jones, S. H. Al-Sakran, and J. R. Koza, “Automated synthesis of a human-competitive solution to the challenge problem of the 2002 international optical design conference by means of genetic programming and a multi-dimensional mutation operation,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2006), pp. 823–830.
    [Crossref]
  28. L. N. Hazra and S. Chatterjee, “A Prophylactic strategy for global synthesis in lens design,” Opt. Rev. 12(3), 247–254 (2005).
    [Crossref]
  29. J. Branke, K. Deb, K. Miettinen, and R. Slowinski, Multiobjective Optimization: Interactive and Evolutionary Approaches (Springer-Verlag, 2008).
  30. S. Joseph, H. W. Kang, and U. K. Chakraborty, “Optical design with epsilon-dominated multi-objective evolutionary algorithm,” in Adaptive and Natural Computing Algorithms, Vol. 4431 of Lecture Notes in Computer Science, B. Beliczynski, A. Dzielinski, M. Iwanowski, and B. Ribeiro, eds. (Springer, 2007), pp. 77–84.
  31. C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
    [Crossref]
  32. L. Li, Q.-H. Wang, X.-Q. Xu, and D.-H. Li, “Two-step method for lens system design,” Opt. Express 18(12), 13285–13300 (2010).
    [Crossref] [PubMed]
  33. I. Ono, Y. Tatsuzawa, S. Kobayashi, and K. Yoshida, “Designing lens systems taking account of glass selection by real-coded genetic algorithms,” in Proceedings of IEEE International Conference on Systems, Man, and Cybernetics (SMS),3, (IEEE, 1999), pp. 592–597.
    [Crossref]
  34. J. C. Tesar, “Mozart, dice, and glass selection,” SPIE 4092, 1–6 (2000).
  35. F. L. De Sousa, F. M. Ramos, P. Paglione, and R. M. Girardi, “New stochastic algorithm for design optimization,” AIAA J. 41(9), 1808–1818 (2003).
    [Crossref]
  36. P. Moscato, “On evolution, search, optimization, genetic algorithms and martial arts,” Tech. Rep., CALTECH Report 826 (1989).
  37. P. Moscato and C. Cotta, “A gentle introduction to memetic algorithms,” in Handbook of Metaheuristics, Vol. 57 of International Series in Operations Research & Management Science, F. Glover and G. A. Kochenberger, eds. (Springer, 2003), pp. 105–144.
  38. B. F. C. de Albuquerque, J. Sasian, F. L. de Sousa, and A. S. Montes, “Method of glass selection for color correction in optical system design,” Opt. Express 20(13), 13592–13611 (2012).
    [Crossref] [PubMed]
  39. R. L. Galski, F. L. De Sousa, and F. M. Ramos, “Application of a new evolutionary algorithm to the optimum design of a remote sensing satellite constellation,” in Proceedings of 5th International Conference on Inverse Problems in Engineering: Theory and Practice,2, (Leeds University, 2005), paper G01.
  40. I. Mainenti-Lopes, F. L. De Sousa, and L. C. G. de Souza, “The generalized extremal optimization with real codification,” in Proceedings of International Conference on Engineering Optimization (EngOpt) (EngOpt, 2008).
  41. M. Isshiki, L. Gardner, and G. G. Gregory, “Automated control of manufacturing sensitivity during optimization,” Proc. SPIE 5249, 343–352 (2004).
    [Crossref]
  42. M. Jeffs, “Reduced manufacturing sensitivity in multi-element lens systems,” in International Optical Design Conference, 2002 OSA Technical Digest Series (Optical Society of America, 2002), paper IMC4.
    [Crossref]
  43. K. Fuse, “Method for designing a refractive or reflective optical system and method for designing a diffraction optical element,” United States Patent 6,567,226 (2003).
  44. J. P. McGuire, Jr., “Designing easily manufactured lenses using a global method,” in International Optical Design Conference, 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper TuA6.
    [Crossref]
  45. M. Isshiki, D. Sinclair, and S. Kaneko, “Lens design: Global optimization of both performance and tolerance sensitivity in International Optical Design Conference, 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper TuA5.
  46. A. Epple and H. Wang, “Design to manufacture- from the perspective of optical design and fabrication,” in Optical Fabrication and Testing Conference, 2008 OSA Technical Digest Series (Optical Society of America, 2008), paper OFB1.
    [Crossref]
  47. B. F. C. de Albuquerque, L.-Y. Liao, A. S. Montes, F. L. de Sousa, and J. Sasián, “A multi-objective approach in the optimization of optical systems taking into account tolerancing,” Proc. SPIE 8131, 813105 (2011).
    [Crossref]
  48. W. Smith, Modern Lens Design, 2nd ed. (McGraw-Hill, 2004).
  49. A. Girard, “Calcul automatique en optique géométrique,” Rev. Opt. Theor. Instrum. 37, 225–241 (1958).
  50. J. Sasián, “Theory of sixth-order wave aberrations,” Appl. Opt. 49(16), D69–D95 (2010).
    [Crossref] [PubMed]
  51. V. N. Mahajan, Optical Imaging and Aberration (SPIE Press, 1998).
  52. G. W. Forbes, “Optical system assessment for design: numerical ray tracing in the Gaussian pupil,” J. Opt. Soc. Am. A 5(11), 1943–1956 (1988).
    [Crossref]
  53. J. L. Rayces, “Classical methods of optimization in lens design,” 5890 N. Placita Alberca, Tucson-AZ, 85718 (personal communication, 2009).
  54. J. Sasian and M. R. Descour, “Power distribution and symmetry in lens systems,” Opt. Eng. 37(3), 1001–1004 (1998).
    [Crossref]
  55. L. Wang and J. Sasian, “Merit figures for fast estimating tolerance sensitivity in lens systems,” Proc. SPIE 7652, 76521P (2010).
    [Crossref]
  56. J. Sasian, “Tolerance II: Opt 517 lens design”, Tucson, AZ: College of Optical Sciences, The University of Arizona, Fall 2011. (Lecture notes, 2011).
  57. J. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33(6), 2045–2061 (1994).
    [Crossref]
  58. SCHOTT N. America, Inc., “Optical glass catalogue- ZEMAX format, status as of 13th September 2011, http://www.us.schott.com/advanced_optics/english/tools_downloads/download/index.html?PHPSESSID=utt2cbk96nlk3gf7gjpb7ggt54#Optical%20Glass

2015 (1)

2012 (3)

M.-H. Lin, J.-F. Tsai, and C.-S. Yu, “A review of deterministic optimization methods in engineering and management,” Math. Probl. Eng. 2012, 756023 (2012).
[Crossref]

Z. Li and B. Yang, “Optical design and optimization of a miniature projector with liquid lenses via modified ant colony algorithm,” Opt. Eng. 51(7), 073001 (2012).
[Crossref]

B. F. C. de Albuquerque, J. Sasian, F. L. de Sousa, and A. S. Montes, “Method of glass selection for color correction in optical system design,” Opt. Express 20(13), 13592–13611 (2012).
[Crossref] [PubMed]

2011 (2)

B. F. C. de Albuquerque, L.-Y. Liao, A. S. Montes, F. L. de Sousa, and J. Sasián, “A multi-objective approach in the optimization of optical systems taking into account tolerancing,” Proc. SPIE 8131, 813105 (2011).
[Crossref]

H. Qin, “Particle swarm optimization applied to automatic lens design,” Opt. Commun. 284(12), 2763–2766 (2011).
[Crossref]

2010 (3)

2009 (1)

L. Bianchi, M. Dorigo, L. Gambardella, and W. Gutjahr, “A survey on metaheuristics for stochastic combinatorial optimization,” Nat. Comput. 8(2), 239–287 (2009).
[Crossref]

2008 (1)

C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
[Crossref]

2007 (1)

2005 (2)

S. Thibault, C. Gagné, J. Beaulieu, and M. Parizeau, “Evolutionary algorithms applied to lens design,” Proc. SPIE 5962, 66–76 (2005).
[Crossref]

L. N. Hazra and S. Chatterjee, “A Prophylactic strategy for global synthesis in lens design,” Opt. Rev. 12(3), 247–254 (2005).
[Crossref]

2004 (1)

M. Isshiki, L. Gardner, and G. G. Gregory, “Automated control of manufacturing sensitivity during optimization,” Proc. SPIE 5249, 343–352 (2004).
[Crossref]

2003 (2)

F. L. De Sousa, F. M. Ramos, P. Paglione, and R. M. Girardi, “New stochastic algorithm for design optimization,” AIAA J. 41(9), 1808–1818 (2003).
[Crossref]

C. Blum and A. Roli, “Metaheuristics in combinatorial optimization: overview and conceptual comparison,” ACM Comput. Surv. 35(3), 268–308 (2003).
[Crossref]

2002 (1)

J. L. Rayces and M. Rosete-Aguilar, “Critical view of three lens design methods: damped least squares, Spencers and Glatzels,” Proc. SPIE 4927, 77–89 (2002).
[Crossref]

2000 (1)

J. C. Tesar, “Mozart, dice, and glass selection,” SPIE 4092, 1–6 (2000).

1999 (1)

K. E. Moore, “Algorithm for global optimization of optical systems based on genetic competition,” Proc. SPIE 3780, 40–47 (1999).
[Crossref]

1998 (2)

I. Ono, S. Kobayashi, and K. Yoshida, “Global and multi-objective optimization for lens design by real-coded genetic algorithms,” Proc. SPIE 3482, 110–121 (1998).
[Crossref]

J. Sasian and M. R. Descour, “Power distribution and symmetry in lens systems,” Opt. Eng. 37(3), 1001–1004 (1998).
[Crossref]

1997 (1)

X. Chen and K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44(9), 1693–1702 (1997).
[Crossref]

1996 (1)

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]

1994 (2)

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8(2), 188–195 (1994).
[Crossref]

J. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33(6), 2045–2061 (1994).
[Crossref]

1988 (1)

1986 (1)

V. K. Viswanathan, I. O. Bohachevsky, and T. P. Cotter, “An attempt to develop an “intelligent” lens design program,” Proc. SPIE 0554, 10–17 (1986).
[Crossref]

1958 (1)

A. Girard, “Calcul automatique en optique géométrique,” Rev. Opt. Theor. Instrum. 37, 225–241 (1958).

Al-Sakran, S. H.

J. R. Koza, S. H. Al-Sakran, and L. W. Jones, “Automated re-invention of six patented optical lens systems using genetic programming,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2005), pp. 1953–1960.
[Crossref]

L. W. Jones, S. H. Al-Sakran, and J. R. Koza, “Automated synthesis of a human-competitive solution to the challenge problem of the 2002 international optical design conference by means of genetic programming and a multi-dimensional mutation operation,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2006), pp. 823–830.
[Crossref]

Beaulieu, J.

C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
[Crossref]

S. Thibault, C. Gagné, J. Beaulieu, and M. Parizeau, “Evolutionary algorithms applied to lens design,” Proc. SPIE 5962, 66–76 (2005).
[Crossref]

J. Beaulieu, C. Gagné, and M. Parizeau, “Lens system design and re-engineering with evolutionary algorithms,” in Proceedings of Genetic and Evolutionary Computation Conference (GECCO, 2002), pp. 155–162.

Bianchi, L.

L. Bianchi, M. Dorigo, L. Gambardella, and W. Gutjahr, “A survey on metaheuristics for stochastic combinatorial optimization,” Nat. Comput. 8(2), 239–287 (2009).
[Crossref]

Blum, C.

C. Blum and A. Roli, “Metaheuristics in combinatorial optimization: overview and conceptual comparison,” ACM Comput. Surv. 35(3), 268–308 (2003).
[Crossref]

Bociort, F.

Bohachevsky, I. O.

V. K. Viswanathan, I. O. Bohachevsky, and T. P. Cotter, “An attempt to develop an “intelligent” lens design program,” Proc. SPIE 0554, 10–17 (1986).
[Crossref]

Chatterjee, S.

L. N. Hazra and S. Chatterjee, “A Prophylactic strategy for global synthesis in lens design,” Opt. Rev. 12(3), 247–254 (2005).
[Crossref]

Chen, X.

X. Chen and K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44(9), 1693–1702 (1997).
[Crossref]

Cotter, T. P.

V. K. Viswanathan, I. O. Bohachevsky, and T. P. Cotter, “An attempt to develop an “intelligent” lens design program,” Proc. SPIE 0554, 10–17 (1986).
[Crossref]

de Albuquerque, B. F. C.

B. F. C. de Albuquerque, J. Sasian, F. L. de Sousa, and A. S. Montes, “Method of glass selection for color correction in optical system design,” Opt. Express 20(13), 13592–13611 (2012).
[Crossref] [PubMed]

B. F. C. de Albuquerque, L.-Y. Liao, A. S. Montes, F. L. de Sousa, and J. Sasián, “A multi-objective approach in the optimization of optical systems taking into account tolerancing,” Proc. SPIE 8131, 813105 (2011).
[Crossref]

de Sousa, F. L.

B. F. C. de Albuquerque, J. Sasian, F. L. de Sousa, and A. S. Montes, “Method of glass selection for color correction in optical system design,” Opt. Express 20(13), 13592–13611 (2012).
[Crossref] [PubMed]

B. F. C. de Albuquerque, L.-Y. Liao, A. S. Montes, F. L. de Sousa, and J. Sasián, “A multi-objective approach in the optimization of optical systems taking into account tolerancing,” Proc. SPIE 8131, 813105 (2011).
[Crossref]

F. L. De Sousa, F. M. Ramos, P. Paglione, and R. M. Girardi, “New stochastic algorithm for design optimization,” AIAA J. 41(9), 1808–1818 (2003).
[Crossref]

Descour, M. R.

J. Sasian and M. R. Descour, “Power distribution and symmetry in lens systems,” Opt. Eng. 37(3), 1001–1004 (1998).
[Crossref]

Dorigo, M.

L. Bianchi, M. Dorigo, L. Gambardella, and W. Gutjahr, “A survey on metaheuristics for stochastic combinatorial optimization,” Nat. Comput. 8(2), 239–287 (2009).
[Crossref]

Fang, Y.-C.

Forbes, G. W.

Gagné, C.

C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
[Crossref]

S. Thibault, C. Gagné, J. Beaulieu, and M. Parizeau, “Evolutionary algorithms applied to lens design,” Proc. SPIE 5962, 66–76 (2005).
[Crossref]

J. Beaulieu, C. Gagné, and M. Parizeau, “Lens system design and re-engineering with evolutionary algorithms,” in Proceedings of Genetic and Evolutionary Computation Conference (GECCO, 2002), pp. 155–162.

Gambardella, L.

L. Bianchi, M. Dorigo, L. Gambardella, and W. Gutjahr, “A survey on metaheuristics for stochastic combinatorial optimization,” Nat. Comput. 8(2), 239–287 (2009).
[Crossref]

Gardner, L.

M. Isshiki, L. Gardner, and G. G. Gregory, “Automated control of manufacturing sensitivity during optimization,” Proc. SPIE 5249, 343–352 (2004).
[Crossref]

Girard, A.

A. Girard, “Calcul automatique en optique géométrique,” Rev. Opt. Theor. Instrum. 37, 225–241 (1958).

Girardi, R. M.

F. L. De Sousa, F. M. Ramos, P. Paglione, and R. M. Girardi, “New stochastic algorithm for design optimization,” AIAA J. 41(9), 1808–1818 (2003).
[Crossref]

Gregory, G. G.

M. Isshiki, L. Gardner, and G. G. Gregory, “Automated control of manufacturing sensitivity during optimization,” Proc. SPIE 5249, 343–352 (2004).
[Crossref]

Gutjahr, W.

L. Bianchi, M. Dorigo, L. Gambardella, and W. Gutjahr, “A survey on metaheuristics for stochastic combinatorial optimization,” Nat. Comput. 8(2), 239–287 (2009).
[Crossref]

Hazra, L. N.

L. N. Hazra and S. Chatterjee, “A Prophylactic strategy for global synthesis in lens design,” Opt. Rev. 12(3), 247–254 (2005).
[Crossref]

Isshiki, M.

M. Isshiki, L. Gardner, and G. G. Gregory, “Automated control of manufacturing sensitivity during optimization,” Proc. SPIE 5249, 343–352 (2004).
[Crossref]

Jones, L. W.

J. R. Koza, S. H. Al-Sakran, and L. W. Jones, “Automated re-invention of six patented optical lens systems using genetic programming,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2005), pp. 1953–1960.
[Crossref]

L. W. Jones, S. H. Al-Sakran, and J. R. Koza, “Automated synthesis of a human-competitive solution to the challenge problem of the 2002 international optical design conference by means of genetic programming and a multi-dimensional mutation operation,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2006), pp. 823–830.
[Crossref]

Kobayashi, S.

I. Ono, S. Kobayashi, and K. Yoshida, “Global and multi-objective optimization for lens design by real-coded genetic algorithms,” Proc. SPIE 3482, 110–121 (1998).
[Crossref]

J. Sakuma and S. Kobayashi, “Latent variable crossover for k-tablet structures and its application to lens design problems,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2005), pp. 1347–1354.
[Crossref]

I. Ono, Y. Tatsuzawa, S. Kobayashi, and K. Yoshida, “Designing lens systems taking account of glass selection by real-coded genetic algorithms,” in Proceedings of IEEE International Conference on Systems, Man, and Cybernetics (SMS),3, (IEEE, 1999), pp. 592–597.
[Crossref]

Koza, J. R.

J. R. Koza, S. H. Al-Sakran, and L. W. Jones, “Automated re-invention of six patented optical lens systems using genetic programming,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2005), pp. 1953–1960.
[Crossref]

L. W. Jones, S. H. Al-Sakran, and J. R. Koza, “Automated synthesis of a human-competitive solution to the challenge problem of the 2002 international optical design conference by means of genetic programming and a multi-dimensional mutation operation,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2006), pp. 823–830.
[Crossref]

Li, D.-H.

Li, L.

Li, Z.

Z. Li and B. Yang, “Optical design and optimization of a miniature projector with liquid lenses via modified ant colony algorithm,” Opt. Eng. 51(7), 073001 (2012).
[Crossref]

Liao, L.-Y.

B. F. C. de Albuquerque, L.-Y. Liao, A. S. Montes, F. L. de Sousa, and J. Sasián, “A multi-objective approach in the optimization of optical systems taking into account tolerancing,” Proc. SPIE 8131, 813105 (2011).
[Crossref]

Lin, M.-H.

M.-H. Lin, J.-F. Tsai, and C.-S. Yu, “A review of deterministic optimization methods in engineering and management,” Math. Probl. Eng. 2012, 756023 (2012).
[Crossref]

Lucasius, C. B.

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]

Macdonald, J.

Montes, A. S.

B. F. C. de Albuquerque, J. Sasian, F. L. de Sousa, and A. S. Montes, “Method of glass selection for color correction in optical system design,” Opt. Express 20(13), 13592–13611 (2012).
[Crossref] [PubMed]

B. F. C. de Albuquerque, L.-Y. Liao, A. S. Montes, F. L. de Sousa, and J. Sasián, “A multi-objective approach in the optimization of optical systems taking into account tolerancing,” Proc. SPIE 8131, 813105 (2011).
[Crossref]

Moore, K. E.

K. E. Moore, “Algorithm for global optimization of optical systems based on genetic competition,” Proc. SPIE 3780, 40–47 (1999).
[Crossref]

Ono, I.

I. Ono, S. Kobayashi, and K. Yoshida, “Global and multi-objective optimization for lens design by real-coded genetic algorithms,” Proc. SPIE 3482, 110–121 (1998).
[Crossref]

I. Ono, Y. Tatsuzawa, S. Kobayashi, and K. Yoshida, “Designing lens systems taking account of glass selection by real-coded genetic algorithms,” in Proceedings of IEEE International Conference on Systems, Man, and Cybernetics (SMS),3, (IEEE, 1999), pp. 592–597.
[Crossref]

Paglione, P.

F. L. De Sousa, F. M. Ramos, P. Paglione, and R. M. Girardi, “New stochastic algorithm for design optimization,” AIAA J. 41(9), 1808–1818 (2003).
[Crossref]

Pai, Y.-C.

Parizeau, M.

C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
[Crossref]

S. Thibault, C. Gagné, J. Beaulieu, and M. Parizeau, “Evolutionary algorithms applied to lens design,” Proc. SPIE 5962, 66–76 (2005).
[Crossref]

J. Beaulieu, C. Gagné, and M. Parizeau, “Lens system design and re-engineering with evolutionary algorithms,” in Proceedings of Genetic and Evolutionary Computation Conference (GECCO, 2002), pp. 155–162.

Qin, H.

H. Qin, “Particle swarm optimization applied to automatic lens design,” Opt. Commun. 284(12), 2763–2766 (2011).
[Crossref]

Ramos, F. M.

F. L. De Sousa, F. M. Ramos, P. Paglione, and R. M. Girardi, “New stochastic algorithm for design optimization,” AIAA J. 41(9), 1808–1818 (2003).
[Crossref]

Rayces, J. L.

J. L. Rayces and M. Rosete-Aguilar, “Critical view of three lens design methods: damped least squares, Spencers and Glatzels,” Proc. SPIE 4927, 77–89 (2002).
[Crossref]

Roli, A.

C. Blum and A. Roli, “Metaheuristics in combinatorial optimization: overview and conceptual comparison,” ACM Comput. Surv. 35(3), 268–308 (2003).
[Crossref]

Rosete-Aguilar, M.

J. L. Rayces and M. Rosete-Aguilar, “Critical view of three lens design methods: damped least squares, Spencers and Glatzels,” Proc. SPIE 4927, 77–89 (2002).
[Crossref]

Sakuma, J.

J. Sakuma and S. Kobayashi, “Latent variable crossover for k-tablet structures and its application to lens design problems,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2005), pp. 1347–1354.
[Crossref]

Sasian, J.

B. F. C. de Albuquerque, J. Sasian, F. L. de Sousa, and A. S. Montes, “Method of glass selection for color correction in optical system design,” Opt. Express 20(13), 13592–13611 (2012).
[Crossref] [PubMed]

L. Wang and J. Sasian, “Merit figures for fast estimating tolerance sensitivity in lens systems,” Proc. SPIE 7652, 76521P (2010).
[Crossref]

J. Sasian and M. R. Descour, “Power distribution and symmetry in lens systems,” Opt. Eng. 37(3), 1001–1004 (1998).
[Crossref]

J. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33(6), 2045–2061 (1994).
[Crossref]

Sasián, J.

B. F. C. de Albuquerque, L.-Y. Liao, A. S. Montes, F. L. de Sousa, and J. Sasián, “A multi-objective approach in the optimization of optical systems taking into account tolerancing,” Proc. SPIE 8131, 813105 (2011).
[Crossref]

J. Sasián, “Theory of sixth-order wave aberrations,” Appl. Opt. 49(16), D69–D95 (2010).
[Crossref] [PubMed]

Shafer, D.

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8(2), 188–195 (1994).
[Crossref]

Tatsuzawa, Y.

I. Ono, Y. Tatsuzawa, S. Kobayashi, and K. Yoshida, “Designing lens systems taking account of glass selection by real-coded genetic algorithms,” in Proceedings of IEEE International Conference on Systems, Man, and Cybernetics (SMS),3, (IEEE, 1999), pp. 592–597.
[Crossref]

Tesar, J. C.

J. C. Tesar, “Mozart, dice, and glass selection,” SPIE 4092, 1–6 (2000).

Thibault, S.

C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
[Crossref]

S. Thibault, C. Gagné, J. Beaulieu, and M. Parizeau, “Evolutionary algorithms applied to lens design,” Proc. SPIE 5962, 66–76 (2005).
[Crossref]

Thijssen, J. M.

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]

Tsai, C.-M.

Tsai, J.-F.

M.-H. Lin, J.-F. Tsai, and C.-S. Yu, “A review of deterministic optimization methods in engineering and management,” Math. Probl. Eng. 2012, 756023 (2012).
[Crossref]

Urbach, H. P.

van Grol, P.

van Leijenhorst, D. C.

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]

van Turnhout, M.

Viswanathan, V. K.

V. K. Viswanathan, I. O. Bohachevsky, and T. P. Cotter, “An attempt to develop an “intelligent” lens design program,” Proc. SPIE 0554, 10–17 (1986).
[Crossref]

Wang, L.

L. Wang and J. Sasian, “Merit figures for fast estimating tolerance sensitivity in lens systems,” Proc. SPIE 7652, 76521P (2010).
[Crossref]

Wang, Q.-H.

Xu, X.-Q.

Yamamoto, K.

X. Chen and K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44(9), 1693–1702 (1997).
[Crossref]

Yang, B.

Z. Li and B. Yang, “Optical design and optimization of a miniature projector with liquid lenses via modified ant colony algorithm,” Opt. Eng. 51(7), 073001 (2012).
[Crossref]

Yoshida, K.

I. Ono, S. Kobayashi, and K. Yoshida, “Global and multi-objective optimization for lens design by real-coded genetic algorithms,” Proc. SPIE 3482, 110–121 (1998).
[Crossref]

I. Ono, Y. Tatsuzawa, S. Kobayashi, and K. Yoshida, “Designing lens systems taking account of glass selection by real-coded genetic algorithms,” in Proceedings of IEEE International Conference on Systems, Man, and Cybernetics (SMS),3, (IEEE, 1999), pp. 592–597.
[Crossref]

Yu, C.-S.

M.-H. Lin, J.-F. Tsai, and C.-S. Yu, “A review of deterministic optimization methods in engineering and management,” Math. Probl. Eng. 2012, 756023 (2012).
[Crossref]

ACM Comput. Surv. (1)

C. Blum and A. Roli, “Metaheuristics in combinatorial optimization: overview and conceptual comparison,” ACM Comput. Surv. 35(3), 268–308 (2003).
[Crossref]

AIAA J. (1)

F. L. De Sousa, F. M. Ramos, P. Paglione, and R. M. Girardi, “New stochastic algorithm for design optimization,” AIAA J. 41(9), 1808–1818 (2003).
[Crossref]

Appl. Opt. (2)

Appl. Soft Comput. (1)

C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
[Crossref]

Biosystems (1)

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]

Comput. Phys. (1)

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8(2), 188–195 (1994).
[Crossref]

J. Mod. Opt. (1)

X. Chen and K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44(9), 1693–1702 (1997).
[Crossref]

J. Opt. Soc. Am. A (1)

Math. Probl. Eng. (1)

M.-H. Lin, J.-F. Tsai, and C.-S. Yu, “A review of deterministic optimization methods in engineering and management,” Math. Probl. Eng. 2012, 756023 (2012).
[Crossref]

Nat. Comput. (1)

L. Bianchi, M. Dorigo, L. Gambardella, and W. Gutjahr, “A survey on metaheuristics for stochastic combinatorial optimization,” Nat. Comput. 8(2), 239–287 (2009).
[Crossref]

Opt. Commun. (1)

H. Qin, “Particle swarm optimization applied to automatic lens design,” Opt. Commun. 284(12), 2763–2766 (2011).
[Crossref]

Opt. Eng. (3)

Z. Li and B. Yang, “Optical design and optimization of a miniature projector with liquid lenses via modified ant colony algorithm,” Opt. Eng. 51(7), 073001 (2012).
[Crossref]

J. Sasian and M. R. Descour, “Power distribution and symmetry in lens systems,” Opt. Eng. 37(3), 1001–1004 (1998).
[Crossref]

J. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33(6), 2045–2061 (1994).
[Crossref]

Opt. Express (3)

Opt. Rev. (1)

L. N. Hazra and S. Chatterjee, “A Prophylactic strategy for global synthesis in lens design,” Opt. Rev. 12(3), 247–254 (2005).
[Crossref]

Proc. SPIE (8)

S. Thibault, C. Gagné, J. Beaulieu, and M. Parizeau, “Evolutionary algorithms applied to lens design,” Proc. SPIE 5962, 66–76 (2005).
[Crossref]

M. Isshiki, L. Gardner, and G. G. Gregory, “Automated control of manufacturing sensitivity during optimization,” Proc. SPIE 5249, 343–352 (2004).
[Crossref]

V. K. Viswanathan, I. O. Bohachevsky, and T. P. Cotter, “An attempt to develop an “intelligent” lens design program,” Proc. SPIE 0554, 10–17 (1986).
[Crossref]

I. Ono, S. Kobayashi, and K. Yoshida, “Global and multi-objective optimization for lens design by real-coded genetic algorithms,” Proc. SPIE 3482, 110–121 (1998).
[Crossref]

K. E. Moore, “Algorithm for global optimization of optical systems based on genetic competition,” Proc. SPIE 3780, 40–47 (1999).
[Crossref]

J. L. Rayces and M. Rosete-Aguilar, “Critical view of three lens design methods: damped least squares, Spencers and Glatzels,” Proc. SPIE 4927, 77–89 (2002).
[Crossref]

B. F. C. de Albuquerque, L.-Y. Liao, A. S. Montes, F. L. de Sousa, and J. Sasián, “A multi-objective approach in the optimization of optical systems taking into account tolerancing,” Proc. SPIE 8131, 813105 (2011).
[Crossref]

L. Wang and J. Sasian, “Merit figures for fast estimating tolerance sensitivity in lens systems,” Proc. SPIE 7652, 76521P (2010).
[Crossref]

Rev. Opt. Theor. Instrum. (1)

A. Girard, “Calcul automatique en optique géométrique,” Rev. Opt. Theor. Instrum. 37, 225–241 (1958).

SPIE (1)

J. C. Tesar, “Mozart, dice, and glass selection,” SPIE 4092, 1–6 (2000).

Other (29)

V. N. Mahajan, Optical Imaging and Aberration (SPIE Press, 1998).

W. Smith, Modern Lens Design, 2nd ed. (McGraw-Hill, 2004).

R. L. Galski, F. L. De Sousa, and F. M. Ramos, “Application of a new evolutionary algorithm to the optimum design of a remote sensing satellite constellation,” in Proceedings of 5th International Conference on Inverse Problems in Engineering: Theory and Practice,2, (Leeds University, 2005), paper G01.

I. Mainenti-Lopes, F. L. De Sousa, and L. C. G. de Souza, “The generalized extremal optimization with real codification,” in Proceedings of International Conference on Engineering Optimization (EngOpt) (EngOpt, 2008).

J. Sasian, “Tolerance II: Opt 517 lens design”, Tucson, AZ: College of Optical Sciences, The University of Arizona, Fall 2011. (Lecture notes, 2011).

J. L. Rayces, “Classical methods of optimization in lens design,” 5890 N. Placita Alberca, Tucson-AZ, 85718 (personal communication, 2009).

SCHOTT N. America, Inc., “Optical glass catalogue- ZEMAX format, status as of 13th September 2011, http://www.us.schott.com/advanced_optics/english/tools_downloads/download/index.html?PHPSESSID=utt2cbk96nlk3gf7gjpb7ggt54#Optical%20Glass

M. J. Colaço and G. S. Dulikravich, “A survey of basic deterministic, heuristic and hybrid methods for single objective optimization and response surface generation,” in Thermal Measurements and Inverse Techniques, H. R. B. Orlande, O. Fudym, D. Maillet, and R. M. Cotta, eds. (CRC, 2011), pp. 355–405.

M. Cavazzuti, Optimization Methods, from Theory to Design Scientific and Technological Aspects in Mechanics (Springer, 2013).

E.-G. Talbi, Metaheuristics: From Design to Implementation (J. Wiley & Sons, 2009).

T. F. Gonzalez, Handbook of Approximation Algorithms and Metaheuristics, Computer & Information Science Series (Chapman & Hall/CRC, 2007).

E. G. M. Lacerda and A. C. P. L. F. Carvalho, “Introdução oas algoritmos genéticos,” in Sistemas Inteligentes: Aplicações a Recursos Hídricos e Ciências Ambientais, Coleção ABRH de Recursos Hidricos; 7, C. O. Galvão, and M. Valença, eds. (UFRGS, 1999), pp. 99–150.

A. E. Eiben and J. E. Smith, Introduction to Evolutionary Computing, Natural Computing Series 1st ed. (Springer, 2003).

D. Vasiljevic, Classical and Evolutionary Algorithms in The Optimization of Optical Systems, Genetic Algorithms and Evolutionary Computation Series, 1st ed. (Kluwer Academic Publishers, 2002).

M. Jeffs, “Reduced manufacturing sensitivity in multi-element lens systems,” in International Optical Design Conference, 2002 OSA Technical Digest Series (Optical Society of America, 2002), paper IMC4.
[Crossref]

K. Fuse, “Method for designing a refractive or reflective optical system and method for designing a diffraction optical element,” United States Patent 6,567,226 (2003).

J. P. McGuire, Jr., “Designing easily manufactured lenses using a global method,” in International Optical Design Conference, 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper TuA6.
[Crossref]

M. Isshiki, D. Sinclair, and S. Kaneko, “Lens design: Global optimization of both performance and tolerance sensitivity in International Optical Design Conference, 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper TuA5.

A. Epple and H. Wang, “Design to manufacture- from the perspective of optical design and fabrication,” in Optical Fabrication and Testing Conference, 2008 OSA Technical Digest Series (Optical Society of America, 2008), paper OFB1.
[Crossref]

I. Ono, Y. Tatsuzawa, S. Kobayashi, and K. Yoshida, “Designing lens systems taking account of glass selection by real-coded genetic algorithms,” in Proceedings of IEEE International Conference on Systems, Man, and Cybernetics (SMS),3, (IEEE, 1999), pp. 592–597.
[Crossref]

P. Moscato, “On evolution, search, optimization, genetic algorithms and martial arts,” Tech. Rep., CALTECH Report 826 (1989).

P. Moscato and C. Cotta, “A gentle introduction to memetic algorithms,” in Handbook of Metaheuristics, Vol. 57 of International Series in Operations Research & Management Science, F. Glover and G. A. Kochenberger, eds. (Springer, 2003), pp. 105–144.

Y. Nagata, “The lens design using the CMA-ES algorithm,” in Genetic and Evolutionary Computation-GECCO 2004, Vol. 3103 of Lecture Notes in Computer Science, K. Deb, ed. (Springer, 2004), pp. 1189–1200.

J. R. Koza, S. H. Al-Sakran, and L. W. Jones, “Automated re-invention of six patented optical lens systems using genetic programming,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2005), pp. 1953–1960.
[Crossref]

L. W. Jones, S. H. Al-Sakran, and J. R. Koza, “Automated synthesis of a human-competitive solution to the challenge problem of the 2002 international optical design conference by means of genetic programming and a multi-dimensional mutation operation,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2006), pp. 823–830.
[Crossref]

J. Branke, K. Deb, K. Miettinen, and R. Slowinski, Multiobjective Optimization: Interactive and Evolutionary Approaches (Springer-Verlag, 2008).

S. Joseph, H. W. Kang, and U. K. Chakraborty, “Optical design with epsilon-dominated multi-objective evolutionary algorithm,” in Adaptive and Natural Computing Algorithms, Vol. 4431 of Lecture Notes in Computer Science, B. Beliczynski, A. Dzielinski, M. Iwanowski, and B. Ribeiro, eds. (Springer, 2007), pp. 77–84.

J. Beaulieu, C. Gagné, and M. Parizeau, “Lens system design and re-engineering with evolutionary algorithms,” in Proceedings of Genetic and Evolutionary Computation Conference (GECCO, 2002), pp. 155–162.

J. Sakuma and S. Kobayashi, “Latent variable crossover for k-tablet structures and its application to lens design problems,” in Proceedings of Conference on Genetic and Evolutionary Computation (GECCO) (ACM, 2005), pp. 1347–1354.
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1 A flowchart overview of the global search algorithm proposed for the automatic design of optical systems.
Fig. 2
Fig. 2 O-GEO flowchart.
Fig. 3
Fig. 3 O-GEO solution representation.
Fig. 4
Fig. 4 Pareto front for the star tracker lens experiment.
Fig. 5
Fig. 5 Layout and MTF plot for one of the non-dominated designs from thestar tracker lens experiment. This 7-lens system has an image quality metric of 1.19 waves and a sensitivity metric of 3.55 waves.
Fig. 6
Fig. 6 Pareto front for the multi-spectral remote-sensing camera lens experiment.
Fig. 7
Fig. 7 Layout and MTF plots for one of the non-dominated systems from the multi-spectral, remote-sensing camera lens experiment. This 8-lens system has an image quality metric of 0.098 waves and a sensitivity metric of 0.68 waves.

Tables (5)

Tables Icon

Table 1 Basic requirements and constraints for the star tracker lens.

Tables Icon

Table 2 Output table from the glass selection method for 3 glasses sorted by | g ¯ i | .

Tables Icon

Table 3 Basic requirements and constraints for the star tracker lens.

Tables Icon

Table 4 Prescription data for the system shown in Fig. 5.

Tables Icon

Table 5 Prescription data for the system shown in Fig. 7.

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

R
δ j =σVrang e j .
W 020 (λ, λ 0 )= n ΔS(λ, λ 0 ) 2 R 2 r 2 ,
W 111 (λ, λ 0 )= n Δ H (λ, λ 0 ) R r.
σ 2 ( H ,λ)= n=1 m=0 n Z nm 2 ( H ,λ) ,
W RMS 2 = H = H 1 H l λ= λ 1 λ k n=1 m=0 n Z nm 2 ( H ,λ) .
δ W i ( h i , r i , θ i ,λ)=( E i a ci (λ)+4 ε i a si (λ) ) r i 3 cos θ i 2( E i a ai (λ)+ ε i a ci (λ) ) h i r i 2 cos 2 θ i ( 2 E i a di (λ)+ ε i a ci (λ) ) h i r i 2 [ 3 E i a ti (λ)+2 ε i ( a ai (λ)+ a di (λ) ) ] h i 2 r i cos θ i ,
E i = M i Δ i ,
ε i = m i Δ i ,
E i = M i S i β i ,
ε i = m i s i β i .
δ W i ( h i , r i , θ i ,λ)= a cci (λ) r i 3 cos θ i + a lai (λ) h i r i 2 cos 2 θ i + a fdi (λ) h i r i 2 + a qti (λ) h i 2 r i cos θ i ,
a cci (λ)=( E i a ci (λ)+4 ε i a si (λ) ),
a lai (λ)=2( E i a ai (λ)+ ε i a ci (λ) ),
a fdi (λ)=( 2 E i a di (λ)+ ε i a ci (λ) ),
a qti (λ)=[ 3 E i a ti (λ)+2 ε i ( a ai (λ)+ a di (λ) ) ].
δ W i ( H ,ρ,θ,λ)= W 031i (λ) ρ 3 cosθ+ W 122i (λ) H ρ 2 cos 2 θ+ W 120i (λ) H ρ 2 + W 211i (λ) H 2 ρcosθ,
W 031i (λ)= a cci (λ) a 3 k=i+1 j m k 3 ,
W 122i (λ)= a lai (λ) h max a 2 k=i+1 j M k m k 2 ,
W 120i (λ)= a fdi (λ) h max a 2 k=i+1 j M k m k 2 ,
W 211i (λ)= a qti (λ) h 2 max a k=i+1 j M k 2 m k .
δ W RSS = i=1 j δ W rm s i 2 ,
δ W rm s i 2 = 0 1 ς i 2 ( H )d H 0 1 H d H ,
ς i 2 ( H )= 1 π 0 2π 0 1 δ W i 2 ( H ,ρ,θ, λ 0 )ρdρdθ 1 π 2 [ 0 2π 0 1 δ W i ( H ,ρ,θ, λ 0 )ρdρdθ ] 2 .
W 040i (λ)= a si (λ)( a 4 k=i+1 j m k 4 ),
W 131i (λ)= a ci (λ)( h max a 3 k=i+1 j M k m k 3 ),
W 222i (λ)= a ai (λ)( h 2 max a 2 k=i+1 j M k 2 m k 2 ),
W 220i (λ)= a di (λ)( h 2 max a 2 k=i+1 j M k 2 m k 2 ),
W 311i (λ)= a ti (λ)( h 3 max a k=i+1 j M k 3 m k ).
W 031i (λ)=( W 131i k=1 j E k a+4 W 040i k=1 j ε k h max h max a ),
W 122i (λ)=2( W 222i k=1 j E k a+ W 131i k=1 j ε k h max h max a ),
W 120i (λ)=( 2 W 220i k=1 j E k a+ W 131i k=1 j ε k h max h max a ),

Metrics