Abstract

A method is proposed on the basis of a choice of the appropriate solution of simultaneous equations obtained in the course of linearization of the autocorrelation equation of a bar code system that is used for the purpose of accurate alignment. Numerical examples are presented to show the usefulness of this new method.

© 1990 Optical Society of America

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References

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  1. I. F. Blake, Algebraic Coding Theory: History and Development (Dowden, Hutchinson & Ross, Stroudsburg, PA, 1973), pp. 62–67.
  2. R. H. Barker, “Group Synchronizing of Binary Digital Systems,” in Communication Theory, W. Jackson, Ed. (Academic, New York, 1953), pp. 273–287.
  3. Y. Li, “Autocorrelation Function of a Bar Code System,” J. Mod. Opt. 34, 1571–1575 (1987).
    [Crossref]
  4. Y. Li, “Characterization and Design of Bar Code Systems for Accurate Alignment,” Appl. Opt. 27, 2612–2620 (1988).
    [Crossref] [PubMed]
  5. Y. Li, “Optical Valve Using Bar Codes,” Optik 79, 67–74 (1988).
  6. W. J. Dallas, “Digital Computation of Image Complex Amplitude from Image—and Diffraction—Intensity: an Alternative to Holography,” Optik 44, 45–59 (1975).

1988 (2)

1987 (1)

Y. Li, “Autocorrelation Function of a Bar Code System,” J. Mod. Opt. 34, 1571–1575 (1987).
[Crossref]

1975 (1)

W. J. Dallas, “Digital Computation of Image Complex Amplitude from Image—and Diffraction—Intensity: an Alternative to Holography,” Optik 44, 45–59 (1975).

Barker, R. H.

R. H. Barker, “Group Synchronizing of Binary Digital Systems,” in Communication Theory, W. Jackson, Ed. (Academic, New York, 1953), pp. 273–287.

Blake, I. F.

I. F. Blake, Algebraic Coding Theory: History and Development (Dowden, Hutchinson & Ross, Stroudsburg, PA, 1973), pp. 62–67.

Dallas, W. J.

W. J. Dallas, “Digital Computation of Image Complex Amplitude from Image—and Diffraction—Intensity: an Alternative to Holography,” Optik 44, 45–59 (1975).

Li, Y.

Y. Li, “Characterization and Design of Bar Code Systems for Accurate Alignment,” Appl. Opt. 27, 2612–2620 (1988).
[Crossref] [PubMed]

Y. Li, “Optical Valve Using Bar Codes,” Optik 79, 67–74 (1988).

Y. Li, “Autocorrelation Function of a Bar Code System,” J. Mod. Opt. 34, 1571–1575 (1987).
[Crossref]

Appl. Opt. (1)

J. Mod. Opt. (1)

Y. Li, “Autocorrelation Function of a Bar Code System,” J. Mod. Opt. 34, 1571–1575 (1987).
[Crossref]

Optik (2)

Y. Li, “Optical Valve Using Bar Codes,” Optik 79, 67–74 (1988).

W. J. Dallas, “Digital Computation of Image Complex Amplitude from Image—and Diffraction—Intensity: an Alternative to Holography,” Optik 44, 45–59 (1975).

Other (2)

I. F. Blake, Algebraic Coding Theory: History and Development (Dowden, Hutchinson & Ross, Stroudsburg, PA, 1973), pp. 62–67.

R. H. Barker, “Group Synchronizing of Binary Digital Systems,” in Communication Theory, W. Jackson, Ed. (Academic, New York, 1953), pp. 273–287.

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Figures (1)

Fig. 1
Fig. 1

Simple structure of bar code system (a) and the corresponding autocorrelation function (b).

Equations (31)

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[ x ] = [ x 0 , x 1 , , x n ]             ( x i = 0 or 1 ) ,
x 0 = x n = 1.
C k = i = 0 n - k x i x i + k ,
C 0 = N and C n = 1.
C n = x 0 x n = 1 ,
C n - 1 = x 0 x n - 1 + x 1 x n = x n - 1 + x 1 ,
C n - 2 = = x n - 2 + x n - 1 x 1 + x 2 ,
C n - 3 = = x n - 3 + x n - 2 x 1 + x n - 1 x 2 + x 3 ,
C k = = x k + x k + 1 x 1 + x k + 2 x 2 + + x n - 1 x n - k - 1 + x n - k ,
C 0 = N .
2 [ x n - 2 + x 2 ] - [ x n - 1 - x 1 ] 2 = 2 C n - 2 - C n - 1 ,
2 [ x n - 2 + x 2 ] ± [ x n - 1 - x 1 ] = 2 C n - 2 - C n - 1 .
- x 1 + 2 x 2 + 2 x n - 2 + x n - 1 = 2 C n - 2 - C n - 1 .
x 1 + 2 x 2 + 2 x n - 2 - x n - 1 = 2 C n - 2 - C n - 1 ,
4 [ x n - 3 + x 3 ] [ x n - 1 - x 1 ] ± 2 [ x n - 2 - x 1 ] ± 2 [ x n - 1 - x 2 ] = 4 C n - 3 - 2 C n - 2 - C n - 1 .
C 1 = 1 , C 2 = 0 , C 3 = 1.
x 1 + x 3 = 1 4 x 2 ± [ x 3 - x 1 ] = - 1 , 4 [ x 1 + x 3 ] [ x 3 - x 1 ] ± 2 [ x 2 - x 1 ] ± 2 [ x 3 - x 2 ] = 3.
x 1 + x 3 = 1 x 1 - 4 x 2 - x 3 = 1 3 x 1 + 5 x 3 = 3 ,
x 1 + x 3 = 1 x 1 - 4 x 2 - x 3 = 1 7 x 1 - 4 x 2 + 5 x 3 = 3 ,
x 1 + x 3 = 1 x 1 - 4 x 2 - x 3 = 1 3 x 1 + 4 x 2 + x 3 = 3 ,
x 1 + x 3 = 1 x 1 - 4 x 2 - x 3 = 3 7 x 1 + x 3 = 3 ,
x 1 + x 3 = 1 x 1 + 4 x 2 - x 3 = - 1 x 1 + 7 x 3 = 3 ,
x 1 + x 3 = 1 x 1 + 4 x 2 - x 3 = - 1 5 x 1 - 4 x 2 + 7 x 3 = 3 ,
x 1 + x 3 = 1 x 1 + 4 x 2 - x 3 = - 1 x 1 + 4 x 2 + 3 x 3 = 3 ,
x 1 + x 3 = 1 x 1 + 4 x 2 - x 3 = - 1 5 x 1 + 3 x 3 = 3 .
C m 0.5 [ ( 2 n + 1 ) - ( 2 n + 1 ) 2 - 4 N ( N - 1 ) ] ,
C m = max [ C 1 , C 2 , , C n ] .
C 1 + C 2 + + C n = N ( N - 1 ) / 2.
X + Y = 4.
( 0 ) X + ( 1 ) Y = 3 × 2 / 2 = 3.
[ 3 , 0 , 1 , 1 , 1 ] , [ 3 , 1 , 0 , 1 , 1 ] , [ 3 , 1 , 1 , 0 , 1 ] , [ 3 , 1 , 1 , 1 , 0 ] .

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