Abstract

We present a method for obtaining the phase of a noisy simulated interferogram. We find the wave-front aberrations by transforming the problem of fitting a polynomial into an optimization problem, which is then solved using an evolutionary algorithm. Our experimental results show that our method yields a more accurate solution than other methods commonly used to solve this problem.

© 2002 Optical Society of America

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References

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  1. D. Dutton, A. Cornejo, M. Latta, “A semiautomatic method for interpreting shearing interferograms,” Appl. Opt. 7, 125–131 (1968).
    [CrossRef] [PubMed]
  2. G. E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting on a digital computer,” J. Soc. Ind. Appl. Math. 5, 74–80 (1957).
    [CrossRef]
  3. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike Polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  4. A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. I: Computer simulations,” Appl. Opt. 33, 7343–7349 (1994).
    [CrossRef]
  5. A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. II: Analytical study,” Appl. Opt. 33, 7343–7349 (1994).
    [CrossRef]
  6. A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. III: Nonlinear solution,” Appl. Opt. 37, 7983–7987 (1998).
    [CrossRef]
  7. T. Back, Evolutionary Algorithms in Theory and Practice, (Oxford University Press, New York, 1996).
  8. D. Goldberg, Genetic Algorithms in Search Optimization and Machine Learning, (Addison-Wesley, Reading, Mass.1989).
  9. J. Koza, Genetic Programming: on the Programming of Computers by Means of Natural Selection; (MIT Press, Cambridge, Mass., 1992).
  10. O. Fuentes, R. C. Nelson, “Learning dextrous manipulation skills for multifingered robot hands using the evolution strategy,” Machine Learning 31, 223–237 (1998).
    [CrossRef]
  11. J. Born, “Evolutionstrategien zur numerischen Losung von Adaptionsaufgaben” Ph.D. dissertation Humboldt Universitat, Berlin, Germany, 1978).
  12. E. H. L. Aarts, J. Korst, Simulated Annealing and Bolzmann Machines, (Wiley, Chichester, 1989).
  13. A.E. Eiben, E. H. L. Aarts, K. M. Van Hee, “Global convergence of genetic algorithms: an infinite Markov chain analysis,” Proceeding of the First International Conference on Parallel Solving from Nature, (Springer, Berlin, Germany, 1991) pp. 4–17.

1998 (2)

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. III: Nonlinear solution,” Appl. Opt. 37, 7983–7987 (1998).
[CrossRef]

O. Fuentes, R. C. Nelson, “Learning dextrous manipulation skills for multifingered robot hands using the evolution strategy,” Machine Learning 31, 223–237 (1998).
[CrossRef]

1994 (2)

1980 (1)

1968 (1)

1957 (1)

G. E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting on a digital computer,” J. Soc. Ind. Appl. Math. 5, 74–80 (1957).
[CrossRef]

Aarts, E. H. L.

E. H. L. Aarts, J. Korst, Simulated Annealing and Bolzmann Machines, (Wiley, Chichester, 1989).

A.E. Eiben, E. H. L. Aarts, K. M. Van Hee, “Global convergence of genetic algorithms: an infinite Markov chain analysis,” Proceeding of the First International Conference on Parallel Solving from Nature, (Springer, Berlin, Germany, 1991) pp. 4–17.

Back, T.

T. Back, Evolutionary Algorithms in Theory and Practice, (Oxford University Press, New York, 1996).

Born, J.

J. Born, “Evolutionstrategien zur numerischen Losung von Adaptionsaufgaben” Ph.D. dissertation Humboldt Universitat, Berlin, Germany, 1978).

Cardona-Nuñez, O.

Cordero-Dávila, A.

Cornejo, A.

Cornejo-Rodríguez, A.

Dutton, D.

Eiben, A.E.

A.E. Eiben, E. H. L. Aarts, K. M. Van Hee, “Global convergence of genetic algorithms: an infinite Markov chain analysis,” Proceeding of the First International Conference on Parallel Solving from Nature, (Springer, Berlin, Germany, 1991) pp. 4–17.

Forsythe, G. E.

G. E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting on a digital computer,” J. Soc. Ind. Appl. Math. 5, 74–80 (1957).
[CrossRef]

Fuentes, O.

O. Fuentes, R. C. Nelson, “Learning dextrous manipulation skills for multifingered robot hands using the evolution strategy,” Machine Learning 31, 223–237 (1998).
[CrossRef]

Goldberg, D.

D. Goldberg, Genetic Algorithms in Search Optimization and Machine Learning, (Addison-Wesley, Reading, Mass.1989).

Korst, J.

E. H. L. Aarts, J. Korst, Simulated Annealing and Bolzmann Machines, (Wiley, Chichester, 1989).

Koza, J.

J. Koza, Genetic Programming: on the Programming of Computers by Means of Natural Selection; (MIT Press, Cambridge, Mass., 1992).

Latta, M.

Nelson, R. C.

O. Fuentes, R. C. Nelson, “Learning dextrous manipulation skills for multifingered robot hands using the evolution strategy,” Machine Learning 31, 223–237 (1998).
[CrossRef]

Silva, D. E.

Van Hee, K. M.

A.E. Eiben, E. H. L. Aarts, K. M. Van Hee, “Global convergence of genetic algorithms: an infinite Markov chain analysis,” Proceeding of the First International Conference on Parallel Solving from Nature, (Springer, Berlin, Germany, 1991) pp. 4–17.

Wang, J. Y.

Appl. Opt. (5)

J. Soc. Ind. Appl. Math. (1)

G. E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting on a digital computer,” J. Soc. Ind. Appl. Math. 5, 74–80 (1957).
[CrossRef]

Machine Learning (1)

O. Fuentes, R. C. Nelson, “Learning dextrous manipulation skills for multifingered robot hands using the evolution strategy,” Machine Learning 31, 223–237 (1998).
[CrossRef]

Other (6)

J. Born, “Evolutionstrategien zur numerischen Losung von Adaptionsaufgaben” Ph.D. dissertation Humboldt Universitat, Berlin, Germany, 1978).

E. H. L. Aarts, J. Korst, Simulated Annealing and Bolzmann Machines, (Wiley, Chichester, 1989).

A.E. Eiben, E. H. L. Aarts, K. M. Van Hee, “Global convergence of genetic algorithms: an infinite Markov chain analysis,” Proceeding of the First International Conference on Parallel Solving from Nature, (Springer, Berlin, Germany, 1991) pp. 4–17.

T. Back, Evolutionary Algorithms in Theory and Practice, (Oxford University Press, New York, 1996).

D. Goldberg, Genetic Algorithms in Search Optimization and Machine Learning, (Addison-Wesley, Reading, Mass.1989).

J. Koza, Genetic Programming: on the Programming of Computers by Means of Natural Selection; (MIT Press, Cambridge, Mass., 1992).

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Figures (3)

Fig. 1
Fig. 1

Noisy simulated interferograms.

Fig. 2
Fig. 2

Interferograms obtained in different generations within the optimization process.

Fig. 3
Fig. 3

Behavior of the objective function. The evolution strategy converges to the solution in generation 430.

Tables (1)

Tables Icon

Table 1 Comparison between the Initial Aberrations and the Results of the Algorithm

Equations (10)

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I x, y= A+B cos2πλ W x, y+ Gaussian error.
Wx, y=A x2+y22+By x2+y2+C x2+3y2+D x2+y2+Ey+Fx,
t := 0;Initial Population Pparents0= a10,, aμ0, where ak= xk,1, , xk,i, σk,1, , σk,i, k 1, ., μ.Evaluate fitness of Pparents0: f a10, , f aμ0.While Termination condition  true doPoffspringt= Generate μ *ra offspring using recombination,b1t, , bμ*rat.Generate μ * 1-ra offspring using mutation,bμ*ra+1 t, , bμt.
Evaluate fitness of Poffspringt: f b1t, , f bμt.Let Pparentst+1 be the μ individuals from PoffspringtYPparentst with the highest fitness t=t+1;end.
DistanceGK, I= i=1mj=1nGKi,j - Ii,j2.
brec1k=rk aρ1,K+ 1 - rkaρ2,k,
brec2k=aρk,k,
bmut k= xk,1+M 0,σk,1,xk,2 + M 0, σk,2, , xk,i + M0, σk,i.
If Pf bmut k>f bk>1/5 σk,i=σk,i * 1+c, i=1, , n,
σk,i=σk,i/c, i=1, , n,

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