Abstract

This paper considers the use of autocorrelation properties to design zero reference codes (ZRCs) for optical applications. Based on the properties of the autocorrelation function, the design of an optimum ZRC problem is transformed into a minimization problem with binary variables, and the objective is to minimize the second maximum of the autocorrelation signal σ. However, the considerable computational complexity for an exhaustive search through all combinations of (nn1) different code patterns is a potential problem especially for large codes, where n and n 1 are the length of the ZRC and the number of transparent slits, respectively. To minimize σ while reducing the computational complexity at the same time, we introduce the Cross-Entropy (CE) method, an effective algorithm that solves various combinatorial optimization problems to obtain a good code. The computer simulation results show that compared with the conventional genetic algorithm (GA), the proposed CE obtains the better σ with low computational complexity.

© 2009 Optical Society of America

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  1. J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Eusebio, and A. Bernabeu, "Optimal design of optical reference signals using a genetic algorithm," Opt. Lett. 30, 2734-2736 (2005).
    [CrossRef]
  2. X. Yang and C. Yin, "A new method for the design of zero reference marks for grating measurement systems," J. Phys. E., Sci. Instrum. 19, 34-37 (1986).
    [CrossRef]
  3. Y. Li, "Autocorrelation function of a bar code system," J. Mod. Opt. 34, 1571-1575 (1987).
    [CrossRef]
  4. Y. Li, "Optical valve using bar codes," Optik 79, 67-74 (1988).
  5. Y. Li, "Characterization and design of bar code systems for accurate alignment," Appl. Opt. 27, 2612-2620 (1988).
    [CrossRef] [PubMed]
  6. Y. Li and F. T. S. Yu, "Design of bar code systems for accurate alignment: a new method," Appl. Opt. 29, 723-725 (1990).
    [CrossRef] [PubMed]
  7. J. Saez-Landete, J. Alonso, and E. Bernabeu, "Design of zero reference codes by means of a global optimization method," Opt. Express 13, 195-201 (2005).
    [CrossRef] [PubMed]
  8. R. Y. Rubinstein and D. P. Kroese, The Cross-Entropy Method. (Berlin, Germany, Springer, 2004).
  9. S. Salcedo-Sanz, G. Camps-Valls, F. Perez-Cruz, J. Sepulveda-Sanchis, and C. Bousono-Calzon, "Enhancing genetic feature selection through restricted search and Walsh analysis," IEEE Trans. Syst., Man. Cybern. C, Appl. Rev. 34, 398-406 (2004).
    [CrossRef]

2005 (2)

J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Eusebio, and A. Bernabeu, "Optimal design of optical reference signals using a genetic algorithm," Opt. Lett. 30, 2734-2736 (2005).
[CrossRef]

J. Saez-Landete, J. Alonso, and E. Bernabeu, "Design of zero reference codes by means of a global optimization method," Opt. Express 13, 195-201 (2005).
[CrossRef] [PubMed]

2004 (1)

S. Salcedo-Sanz, G. Camps-Valls, F. Perez-Cruz, J. Sepulveda-Sanchis, and C. Bousono-Calzon, "Enhancing genetic feature selection through restricted search and Walsh analysis," IEEE Trans. Syst., Man. Cybern. C, Appl. Rev. 34, 398-406 (2004).
[CrossRef]

1990 (1)

1988 (2)

1987 (1)

Y. Li, "Autocorrelation function of a bar code system," J. Mod. Opt. 34, 1571-1575 (1987).
[CrossRef]

1986 (1)

X. Yang and C. Yin, "A new method for the design of zero reference marks for grating measurement systems," J. Phys. E., Sci. Instrum. 19, 34-37 (1986).
[CrossRef]

Alonso, J.

Bernabeu, A.

J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Eusebio, and A. Bernabeu, "Optimal design of optical reference signals using a genetic algorithm," Opt. Lett. 30, 2734-2736 (2005).
[CrossRef]

Bernabeu, E.

Bousono-Calzon, C.

S. Salcedo-Sanz, G. Camps-Valls, F. Perez-Cruz, J. Sepulveda-Sanchis, and C. Bousono-Calzon, "Enhancing genetic feature selection through restricted search and Walsh analysis," IEEE Trans. Syst., Man. Cybern. C, Appl. Rev. 34, 398-406 (2004).
[CrossRef]

Camps-Valls, G.

S. Salcedo-Sanz, G. Camps-Valls, F. Perez-Cruz, J. Sepulveda-Sanchis, and C. Bousono-Calzon, "Enhancing genetic feature selection through restricted search and Walsh analysis," IEEE Trans. Syst., Man. Cybern. C, Appl. Rev. 34, 398-406 (2004).
[CrossRef]

Eusebio, J.

J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Eusebio, and A. Bernabeu, "Optimal design of optical reference signals using a genetic algorithm," Opt. Lett. 30, 2734-2736 (2005).
[CrossRef]

Li, Y.

Perez-Cruz, F.

S. Salcedo-Sanz, G. Camps-Valls, F. Perez-Cruz, J. Sepulveda-Sanchis, and C. Bousono-Calzon, "Enhancing genetic feature selection through restricted search and Walsh analysis," IEEE Trans. Syst., Man. Cybern. C, Appl. Rev. 34, 398-406 (2004).
[CrossRef]

Rosa-Zurera, M.

J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Eusebio, and A. Bernabeu, "Optimal design of optical reference signals using a genetic algorithm," Opt. Lett. 30, 2734-2736 (2005).
[CrossRef]

Saez-Landete, J.

J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Eusebio, and A. Bernabeu, "Optimal design of optical reference signals using a genetic algorithm," Opt. Lett. 30, 2734-2736 (2005).
[CrossRef]

J. Saez-Landete, J. Alonso, and E. Bernabeu, "Design of zero reference codes by means of a global optimization method," Opt. Express 13, 195-201 (2005).
[CrossRef] [PubMed]

Salcedo-Sanz, S.

J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Eusebio, and A. Bernabeu, "Optimal design of optical reference signals using a genetic algorithm," Opt. Lett. 30, 2734-2736 (2005).
[CrossRef]

S. Salcedo-Sanz, G. Camps-Valls, F. Perez-Cruz, J. Sepulveda-Sanchis, and C. Bousono-Calzon, "Enhancing genetic feature selection through restricted search and Walsh analysis," IEEE Trans. Syst., Man. Cybern. C, Appl. Rev. 34, 398-406 (2004).
[CrossRef]

Sepulveda-Sanchis, J.

S. Salcedo-Sanz, G. Camps-Valls, F. Perez-Cruz, J. Sepulveda-Sanchis, and C. Bousono-Calzon, "Enhancing genetic feature selection through restricted search and Walsh analysis," IEEE Trans. Syst., Man. Cybern. C, Appl. Rev. 34, 398-406 (2004).
[CrossRef]

Yang, X.

X. Yang and C. Yin, "A new method for the design of zero reference marks for grating measurement systems," J. Phys. E., Sci. Instrum. 19, 34-37 (1986).
[CrossRef]

Yin, C.

X. Yang and C. Yin, "A new method for the design of zero reference marks for grating measurement systems," J. Phys. E., Sci. Instrum. 19, 34-37 (1986).
[CrossRef]

Yu, F. T. S.

Appl. Opt. (2)

IEEE Trans. Syst., Man, Cybern. C, Appl. Rev. (1)

S. Salcedo-Sanz, G. Camps-Valls, F. Perez-Cruz, J. Sepulveda-Sanchis, and C. Bousono-Calzon, "Enhancing genetic feature selection through restricted search and Walsh analysis," IEEE Trans. Syst., Man. Cybern. C, Appl. Rev. 34, 398-406 (2004).
[CrossRef]

J. Mod. Opt. (1)

Y. Li, "Autocorrelation function of a bar code system," J. Mod. Opt. 34, 1571-1575 (1987).
[CrossRef]

J. Phys. E, Sci. Instrum. (1)

X. Yang and C. Yin, "A new method for the design of zero reference marks for grating measurement systems," J. Phys. E., Sci. Instrum. 19, 34-37 (1986).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Eusebio, and A. Bernabeu, "Optimal design of optical reference signals using a genetic algorithm," Opt. Lett. 30, 2734-2736 (2005).
[CrossRef]

Optik (1)

Y. Li, "Optical valve using bar codes," Optik 79, 67-74 (1988).

Other (1)

R. Y. Rubinstein and D. P. Kroese, The Cross-Entropy Method. (Berlin, Germany, Springer, 2004).

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Figures (5)

Fig. 1.
Fig. 1.

Comparison of the second maximum reached with the GA, the CE method, and the theoretical lower bound. The code length is 200, and n 1 varies from 1 to 199.

Fig. 2.
Fig. 2.

The improved percentage of CE as compared to GA versus the number of slits n 1.

Fig. 3.
Fig. 3.

A ZRC found by the proposed CE method with n=1000 and n 1=100.

Fig. 4.
Fig. 4.

Autocorrelation signal generated with an optimum code. The length of the code is 1000, and n 1=100. The value of the second maximum is 11.

Fig. 5.
Fig. 5.

Average height of the second maximum versus iterations.

Equations (11)

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c = [ c 1 , c 2 , , c n ] , c i { 0 , 1 } ,
S k ( c ) = i = 1 n k c i c i + k ,
ω * = arg min ω q Ω C sel ( ω q ) ,
C sel ( ω q ) = max { S 1 ( ω q ) , , S n 1 ( ω q ) } ,
ω q = { I i } i = 1 n , I i { 0 , 1 } , q = 1 , 2 , , Q
f ( ω q , p ) = i = 1 n p i I i ( ω q ) ( 1 p i ) 1 I i ( ω q ) .
p i ( t ) = m = 1 M I { C sel ( ω q ( m , t ) ) r ( t ) } I i ( ω q ( m , t ) ) m = 1 M I { C sel ( ω q ( m , t ) ) r ( t ) } .
I { C sel ( ω q ( m , t ) ) r ( t ) } = { 1 , if C sel ( ω q ( m , t ) ) r ( t ) 0 , otherwise .
p ( t ) = λ × p ( t ) + ( 1 λ ) × p ( t 1 ) ,
σ ( 2 n + 1 ) ( 2 n + 1 ) 2 4 n 1 ( n 1 1 ) 2 .
improved percentage ( n 1 ) = σ GA ( n 1 ) σ CE ( n 1 ) σ GA ( n 1 ) × 100 % ,

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