Abstract

The grating measurement systems can be used for displacement and angle measurements. They require of zero reference codes to obtain zero reference signals and absolute measures. The zero reference signals are obtained from the autocorrelation of two identical zero reference codes. The design of codes which generate optimum signals is rather complex, especially for larges codes. In this paper we present a global optimization method, a DIRECT algorithm for the design of zero reference codes. This method proves to be a powerful tool for solving this inverse problem.

©2005 Optical Society of America

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References

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  1. Li Yajun, “Autocorrelation function of a bar code system” J. Mod. Opt. 34, 1571–5 (1987).
    [Crossref]
  2. Xiangyang Yang and Chunyong Yin, “A new method for the design of zero reference marks for grating measurement systems” J. Phys. E Sci. Instrum. 19, 34–7 (1986).
    [Crossref]
  3. Li Yajun, “Optical valve using bar codes” Optik 79, 67–74 (1988).
  4. Li Yajun, “Characterization and design of bar code systems for accurate alignment” Appl. Opt. 27, 2612–20 (1988).
    [Crossref]
  5. Li Yajun and F. T. S. Yu, “Design of bar code systems for accurate alignment: a new method” Appl. Opt. 29, 723–5 (1990).
    [Crossref]
  6. D. R. Jones, C. D. Perttunen, and B. E. Stuckman. “Lipschitzian Optimization without the Lipschitz Constant” J. Optim. Theory Appl. 79, 157–181 (1993).
    [Crossref]
  7. Donald R. Jones. DIRECT Global optimization algorithm. Encyclopedia of Optimization. (Kluwer Academic Publishers, Dordrecht, 2001).
  8. Li Yajun, “Bar codes with special correlations like those of the Barker codes” Optics Communications 83, 15–20 (1991).
    [Crossref]
  9. Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).
  10. C. T. Kelley. Iterative Methods for Optimization. (SIAM, Portland1999).
    [Crossref]
  11. Daniel E. Finkel. DIRECT Optimization Algorithm User Guide. (2003).
  12. Daniel E. Finkel and C. T. Kelley. “Convergence analysis of the DIRECT algorithm” Optimization Online (2004).
  13. J. M. Gablonski. DIRECT Version 2.0 User Guide. (CRSC Technical Report, Raleigh, 2001).

1999 (1)

Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).

1993 (1)

D. R. Jones, C. D. Perttunen, and B. E. Stuckman. “Lipschitzian Optimization without the Lipschitz Constant” J. Optim. Theory Appl. 79, 157–181 (1993).
[Crossref]

1991 (1)

Li Yajun, “Bar codes with special correlations like those of the Barker codes” Optics Communications 83, 15–20 (1991).
[Crossref]

1990 (1)

1988 (2)

1987 (1)

Li Yajun, “Autocorrelation function of a bar code system” J. Mod. Opt. 34, 1571–5 (1987).
[Crossref]

1986 (1)

Xiangyang Yang and Chunyong Yin, “A new method for the design of zero reference marks for grating measurement systems” J. Phys. E Sci. Instrum. 19, 34–7 (1986).
[Crossref]

Bjorkman,

Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).

Finkel, Daniel E.

Daniel E. Finkel and C. T. Kelley. “Convergence analysis of the DIRECT algorithm” Optimization Online (2004).

Daniel E. Finkel. DIRECT Optimization Algorithm User Guide. (2003).

Gablonski, J. M.

J. M. Gablonski. DIRECT Version 2.0 User Guide. (CRSC Technical Report, Raleigh, 2001).

Holmstrom,

Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).

Jones, D. R.

D. R. Jones, C. D. Perttunen, and B. E. Stuckman. “Lipschitzian Optimization without the Lipschitz Constant” J. Optim. Theory Appl. 79, 157–181 (1993).
[Crossref]

Jones, Donald R.

Donald R. Jones. DIRECT Global optimization algorithm. Encyclopedia of Optimization. (Kluwer Academic Publishers, Dordrecht, 2001).

Kelley, C. T.

C. T. Kelley. Iterative Methods for Optimization. (SIAM, Portland1999).
[Crossref]

Daniel E. Finkel and C. T. Kelley. “Convergence analysis of the DIRECT algorithm” Optimization Online (2004).

Kenneth,

Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).

Mattias,

Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).

Perttunen, C. D.

D. R. Jones, C. D. Perttunen, and B. E. Stuckman. “Lipschitzian Optimization without the Lipschitz Constant” J. Optim. Theory Appl. 79, 157–181 (1993).
[Crossref]

Stuckman, B. E.

D. R. Jones, C. D. Perttunen, and B. E. Stuckman. “Lipschitzian Optimization without the Lipschitz Constant” J. Optim. Theory Appl. 79, 157–181 (1993).
[Crossref]

Yajun, Li

Li Yajun, “Bar codes with special correlations like those of the Barker codes” Optics Communications 83, 15–20 (1991).
[Crossref]

Li Yajun and F. T. S. Yu, “Design of bar code systems for accurate alignment: a new method” Appl. Opt. 29, 723–5 (1990).
[Crossref]

Li Yajun, “Optical valve using bar codes” Optik 79, 67–74 (1988).

Li Yajun, “Characterization and design of bar code systems for accurate alignment” Appl. Opt. 27, 2612–20 (1988).
[Crossref]

Li Yajun, “Autocorrelation function of a bar code system” J. Mod. Opt. 34, 1571–5 (1987).
[Crossref]

Yang, Xiangyang

Xiangyang Yang and Chunyong Yin, “A new method for the design of zero reference marks for grating measurement systems” J. Phys. E Sci. Instrum. 19, 34–7 (1986).
[Crossref]

Yin, Chunyong

Xiangyang Yang and Chunyong Yin, “A new method for the design of zero reference marks for grating measurement systems” J. Phys. E Sci. Instrum. 19, 34–7 (1986).
[Crossref]

Yu, F. T. S.

Advanced Modeling and Optimization (1)

Bjorkman, Mattias, Holmstrom, and Kenneth. “Global Optimization Using the DIRECT Algorithm in Matlab” Advanced Modeling and Optimization,  1, 17–37 (1999).

Appl. Opt. (2)

J. Mod. Opt. (1)

Li Yajun, “Autocorrelation function of a bar code system” J. Mod. Opt. 34, 1571–5 (1987).
[Crossref]

J. Optim. Theory Appl. (1)

D. R. Jones, C. D. Perttunen, and B. E. Stuckman. “Lipschitzian Optimization without the Lipschitz Constant” J. Optim. Theory Appl. 79, 157–181 (1993).
[Crossref]

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

Xiangyang Yang and Chunyong Yin, “A new method for the design of zero reference marks for grating measurement systems” J. Phys. E Sci. Instrum. 19, 34–7 (1986).
[Crossref]

Optics Communications (1)

Li Yajun, “Bar codes with special correlations like those of the Barker codes” Optics Communications 83, 15–20 (1991).
[Crossref]

Optik (1)

Li Yajun, “Optical valve using bar codes” Optik 79, 67–74 (1988).

Other (5)

Donald R. Jones. DIRECT Global optimization algorithm. Encyclopedia of Optimization. (Kluwer Academic Publishers, Dordrecht, 2001).

C. T. Kelley. Iterative Methods for Optimization. (SIAM, Portland1999).
[Crossref]

Daniel E. Finkel. DIRECT Optimization Algorithm User Guide. (2003).

Daniel E. Finkel and C. T. Kelley. “Convergence analysis of the DIRECT algorithm” Optimization Online (2004).

J. M. Gablonski. DIRECT Version 2.0 User Guide. (CRSC Technical Report, Raleigh, 2001).

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Figures (3)

Fig. 1.
Fig. 1. Height of the second maximum of the autocorrelation signal respect to the number of slits in the ZRC. The ZRC has 50 elements. The continuous graph is the reached with the optimization and the dotted one is a lower bound calculated theoretically.

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Fig. 2.
Fig. 2. Six of the 136 autocorrelation signals found by the algorithm with n=50 and s=25. The value of the second maximum is S1 =11.

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Fig. 3.
Fig. 3. Six of the 136 ZRC’s found by the algorithm with n=50 and s=25.

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Equations (17)

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c = [ c 0 , c 1 , c 2 , … , c n ]
t ( x ) = ∑ j = 0 n c j rect ( x − j b )
rect ( x ) = { 1 if ∣ x ∣ < 1 2 , 0 if ∣ x ∣ ≥ 1 2 .
S ( τ ) = ∫ − ∞ + ∞ t ( x ) t ( x − τ ) dx = ∑ j = − n n a j · Λ ( τ − j b )
Λ ( x ) = { 1 − ∣ x ∣ if ∣ x ∣ ≤ 1 0 if ∣ x ∣ > 1
a k = a − k = ∑ j = 0 n − k c j c j + k , k = 0 , 1 , … , n .
S ( kb ) = S k = a k , k = 0 , 1 , … , n .
K = σ S 0 ,
σ ≥ [ ( 2 n + 1 ) − ( 2 n + 1 ) 2 − 4 n 1 ( n 1 − 1 ) ] 2 .
min x f ( x )
L x ≤ x ≤ U x
LB ≤ A · X ′ ≤ UB
x i ∈ I integer .
f ( c ) = max { S 1 , … , S n } , S k = ∑ j = 0 n − k c j c j − k
[ 0 , … , 0 ] ≤ c ≤ [ 1 , … , 1 ]
n 1 L ≤ [ 1 , … , 1 ] · [ c 1 ⋮ c n ] ≤ n 1 U
c ∈ ℤ n + 1

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