Abstract

In this paper, two types of weighting tables are derived by applying the local power expansion method proposed by Oleari [Color Res. Appl. 25, 176 (2000)]. Both tables at two different levels consider the deconvolution of the spectrophotometric data for monochromator triangular transmittance. The first one, named zero-order weighting table, is similar to weighting table 5 of American Society for Testing and Materials (ASTM) used with the measured spectral reflectance factors (SRFs) corrected by the Stearns and Stearns formula. The second one, named second-order weighting table, is similar to weighting table 6 of ASTM and must be used with the undeconvoluted SRFs. It is hoped that the results of this paper will aid the International Commission on Illumination TC 1-71 on tristimulus integration in focusing on ongoing methods, testing, and recommendations.

© 2011 Optical Society of America

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References

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  1. International Commission on Illumination (CIE), “Colorimetry,” 3rd ed., CIE Publ. 15:2004 (CIE Central Bureau, 2004).
  2. International Organization for Standardization (ISO) and International Commission on Illumination (CIE), “CIE standard illuminants for colorimetry,” joint Publ. ISO 11664-2:2008(E)/CIE S 014-2/E:2006 (International Organization for Standardization, 2008; International Commission on Illumination, 2006).
  3. International Commission on Illumination (CIE), “Recommended practice for tabulating spectral data for use in colour computations,” CIE Publ. 167 (CIE Central Bureau, 2005).
  4. International Organization for Standardization (ISO) and International Commission on Illumination (CIE), “CIE standard colorimetric observers,” joint Publ. ISO 11664-1:2008(E)/CIE S 014-1/E:2006 (International Organization for Standardization, 2008; International Commission on Illumination, 2006).
  5. C. J. Li, M. R. Luo, and G. Wang, “Recent progress in computing weighting tables for calculating CIE tristimulus values,” Proc. SPIE 6033, 198–207 (2006).
  6. C. Li, CIE TC 1-71 technical report, draft 1 (personal communication, 2009). Available from cjli.cip@googlemail.com
  7. American Society for Testing and Materials (ASTM), “Standard method for computing the colors of objects by using the CIE system,” ASTM Intl. E308-85 (ASTM, 1985).
  8. American Society for Testing and Materials (ASTM), “Standard practice for computing the colors of objects by using the CIE system,” ASTM E308-95 (ASTM, 1995).
  9. H. S. Fairman, “Results of the ASTM field test of tristimulus weighting functions,” Color Res. Appl. 20, 44–49 (1995).
    [CrossRef]
  10. C. Oleari, “Spectral-reflectance-factor deconvolution and colorimetric calculations by local-power expansion,” Color Res. Appl. 25, 176–185 (2000).
    [CrossRef]
  11. C. J. Li, M. R. Luo, and B. Rigg, “A new method for computing optimum weights for calculating CIE tristimulus values,” Color Res. Appl. 29, 91–103 (2004).
    [CrossRef]
  12. E. I. Stearns and R. E. Stearns, “An example of a method for correcting radiance data for bandpass error,” Color Res. Appl. 13, 257–259 (1988).
    [CrossRef]
  13. W. H. Venable, “Accurate tristimulus values from spectral data,” Color Res. Appl. 14, 260–267 (1989).
    [CrossRef]
  14. J. M. Lerner and A. Thevenon, “The optics of spectroscopy, a tutorial,” http://www.jyhoriba.it/itdivisions/OOS/index.htm.

2009 (1)

C. Li, CIE TC 1-71 technical report, draft 1 (personal communication, 2009). Available from cjli.cip@googlemail.com

2008 (2)

International Organization for Standardization (ISO) and International Commission on Illumination (CIE), “CIE standard illuminants for colorimetry,” joint Publ. ISO 11664-2:2008(E)/CIE S 014-2/E:2006 (International Organization for Standardization, 2008; International Commission on Illumination, 2006).

International Organization for Standardization (ISO) and International Commission on Illumination (CIE), “CIE standard colorimetric observers,” joint Publ. ISO 11664-1:2008(E)/CIE S 014-1/E:2006 (International Organization for Standardization, 2008; International Commission on Illumination, 2006).

2006 (1)

C. J. Li, M. R. Luo, and G. Wang, “Recent progress in computing weighting tables for calculating CIE tristimulus values,” Proc. SPIE 6033, 198–207 (2006).

2005 (1)

International Commission on Illumination (CIE), “Recommended practice for tabulating spectral data for use in colour computations,” CIE Publ. 167 (CIE Central Bureau, 2005).

2004 (2)

International Commission on Illumination (CIE), “Colorimetry,” 3rd ed., CIE Publ. 15:2004 (CIE Central Bureau, 2004).

C. J. Li, M. R. Luo, and B. Rigg, “A new method for computing optimum weights for calculating CIE tristimulus values,” Color Res. Appl. 29, 91–103 (2004).
[CrossRef]

2000 (1)

C. Oleari, “Spectral-reflectance-factor deconvolution and colorimetric calculations by local-power expansion,” Color Res. Appl. 25, 176–185 (2000).
[CrossRef]

1995 (2)

American Society for Testing and Materials (ASTM), “Standard practice for computing the colors of objects by using the CIE system,” ASTM E308-95 (ASTM, 1995).

H. S. Fairman, “Results of the ASTM field test of tristimulus weighting functions,” Color Res. Appl. 20, 44–49 (1995).
[CrossRef]

1989 (1)

W. H. Venable, “Accurate tristimulus values from spectral data,” Color Res. Appl. 14, 260–267 (1989).
[CrossRef]

1988 (1)

E. I. Stearns and R. E. Stearns, “An example of a method for correcting radiance data for bandpass error,” Color Res. Appl. 13, 257–259 (1988).
[CrossRef]

1985 (1)

American Society for Testing and Materials (ASTM), “Standard method for computing the colors of objects by using the CIE system,” ASTM Intl. E308-85 (ASTM, 1985).

Fairman, H. S.

H. S. Fairman, “Results of the ASTM field test of tristimulus weighting functions,” Color Res. Appl. 20, 44–49 (1995).
[CrossRef]

Lerner, J. M.

J. M. Lerner and A. Thevenon, “The optics of spectroscopy, a tutorial,” http://www.jyhoriba.it/itdivisions/OOS/index.htm.

Li, C.

C. Li, CIE TC 1-71 technical report, draft 1 (personal communication, 2009). Available from cjli.cip@googlemail.com

Li, C. J.

C. J. Li, M. R. Luo, and G. Wang, “Recent progress in computing weighting tables for calculating CIE tristimulus values,” Proc. SPIE 6033, 198–207 (2006).

C. J. Li, M. R. Luo, and B. Rigg, “A new method for computing optimum weights for calculating CIE tristimulus values,” Color Res. Appl. 29, 91–103 (2004).
[CrossRef]

Luo, M. R.

C. J. Li, M. R. Luo, and G. Wang, “Recent progress in computing weighting tables for calculating CIE tristimulus values,” Proc. SPIE 6033, 198–207 (2006).

C. J. Li, M. R. Luo, and B. Rigg, “A new method for computing optimum weights for calculating CIE tristimulus values,” Color Res. Appl. 29, 91–103 (2004).
[CrossRef]

Oleari, C.

C. Oleari, “Spectral-reflectance-factor deconvolution and colorimetric calculations by local-power expansion,” Color Res. Appl. 25, 176–185 (2000).
[CrossRef]

Rigg, B.

C. J. Li, M. R. Luo, and B. Rigg, “A new method for computing optimum weights for calculating CIE tristimulus values,” Color Res. Appl. 29, 91–103 (2004).
[CrossRef]

Stearns, E. I.

E. I. Stearns and R. E. Stearns, “An example of a method for correcting radiance data for bandpass error,” Color Res. Appl. 13, 257–259 (1988).
[CrossRef]

Stearns, R. E.

E. I. Stearns and R. E. Stearns, “An example of a method for correcting radiance data for bandpass error,” Color Res. Appl. 13, 257–259 (1988).
[CrossRef]

Thevenon, A.

J. M. Lerner and A. Thevenon, “The optics of spectroscopy, a tutorial,” http://www.jyhoriba.it/itdivisions/OOS/index.htm.

Venable, W. H.

W. H. Venable, “Accurate tristimulus values from spectral data,” Color Res. Appl. 14, 260–267 (1989).
[CrossRef]

Wang, G.

C. J. Li, M. R. Luo, and G. Wang, “Recent progress in computing weighting tables for calculating CIE tristimulus values,” Proc. SPIE 6033, 198–207 (2006).

Color Res. Appl. (5)

H. S. Fairman, “Results of the ASTM field test of tristimulus weighting functions,” Color Res. Appl. 20, 44–49 (1995).
[CrossRef]

C. Oleari, “Spectral-reflectance-factor deconvolution and colorimetric calculations by local-power expansion,” Color Res. Appl. 25, 176–185 (2000).
[CrossRef]

C. J. Li, M. R. Luo, and B. Rigg, “A new method for computing optimum weights for calculating CIE tristimulus values,” Color Res. Appl. 29, 91–103 (2004).
[CrossRef]

E. I. Stearns and R. E. Stearns, “An example of a method for correcting radiance data for bandpass error,” Color Res. Appl. 13, 257–259 (1988).
[CrossRef]

W. H. Venable, “Accurate tristimulus values from spectral data,” Color Res. Appl. 14, 260–267 (1989).
[CrossRef]

Proc. SPIE (1)

C. J. Li, M. R. Luo, and G. Wang, “Recent progress in computing weighting tables for calculating CIE tristimulus values,” Proc. SPIE 6033, 198–207 (2006).

Other (8)

C. Li, CIE TC 1-71 technical report, draft 1 (personal communication, 2009). Available from cjli.cip@googlemail.com

American Society for Testing and Materials (ASTM), “Standard method for computing the colors of objects by using the CIE system,” ASTM Intl. E308-85 (ASTM, 1985).

American Society for Testing and Materials (ASTM), “Standard practice for computing the colors of objects by using the CIE system,” ASTM E308-95 (ASTM, 1995).

International Commission on Illumination (CIE), “Colorimetry,” 3rd ed., CIE Publ. 15:2004 (CIE Central Bureau, 2004).

International Organization for Standardization (ISO) and International Commission on Illumination (CIE), “CIE standard illuminants for colorimetry,” joint Publ. ISO 11664-2:2008(E)/CIE S 014-2/E:2006 (International Organization for Standardization, 2008; International Commission on Illumination, 2006).

International Commission on Illumination (CIE), “Recommended practice for tabulating spectral data for use in colour computations,” CIE Publ. 167 (CIE Central Bureau, 2005).

International Organization for Standardization (ISO) and International Commission on Illumination (CIE), “CIE standard colorimetric observers,” joint Publ. ISO 11664-1:2008(E)/CIE S 014-1/E:2006 (International Organization for Standardization, 2008; International Commission on Illumination, 2006).

J. M. Lerner and A. Thevenon, “The optics of spectroscopy, a tutorial,” http://www.jyhoriba.it/itdivisions/OOS/index.htm.

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Tables (10)

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Table 1 Algorithm 1: Computing coefficients for the second-order local expansions of S λ x ¯ ( λ ) , S λ y ¯ ( λ ) , and S λ z ¯ ( λ ) .

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Table 2 Algorithm 2: Computing zero-order weighting table WT 0 , Δ λ , X , i , WT 0 , Δ λ , Y , i , WT 0 , Δ λ , Z , i

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Table 3 Algorithm 3: Computing second-order weighting table WT 2 , Δ λ , X , i , WT 2 , Δ λ , Y , i , WT 2 , Δ λ , Z , i .

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Table 1 Formulas for Computing Tristimulus Values for Oleari’s Method

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Table 2 1 nm Weighting Table (Partial Results) for the Illuminant D50 and the CIE 1964 Standard Colorimetric Observer

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Table 3 Coefficients x S , i , 0 , x S , i , 1 , x S , i , 2 Using Algorithm 1 with 1 nm Weighting Table

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Table 4 Coefficients y S , i , 0 , y S , i , 1 , y S , i , 2 Using Algorithm 1 with 1 nm Weighting Table

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Table 5 Coefficients z S , i , 0 , z S , i , 1 , z S , i , 2 Using Algorithm 1 with 1 nm Weighting Table

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Table 6 10 nm Zero-Order Weighting Table for Illuminant D50 and the CIE 1964 Standard Colorimetric Observer

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Table 7 10 nm Second-Order Weighting Table for Illuminant D50 and the CIE 1964 Standard Colorimetric Observer

Equations (38)

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J ( λ i ) = 0 R ( λ ) κ ( λ , λ i ) d λ ,
κ ( λ , λ i ) = S λ T ( λ λ i ) E λ
R s ( λ i ) = J ( λ i ) J WS ( λ i ) R WS ( λ i ) ,
J WS ( λ i ) = 0 R WS ( λ ) κ ( λ , λ i ) d λ
R ( λ ) = n = 0 R i , n ( λ λ i ) n = R i , 0 + R i , 1 ( λ λ i ) + R i , 2 ( λ λ i ) 2 + R i , 0 + R i , 1 ( λ λ i ) + R i , 2 ( λ λ i ) 2 .
T ( λ λ i ) = { 0 for     λ < λ i Δ λ Δ λ + λ λ i Δ λ for     λ i Δ λ λ < λ i Δ λ λ + λ i Δ λ for     λ i λ λ i + Δ λ 0 for     λ i + Δ λ < λ .
R i , 0 = 1 12 [ R s ( λ i 1 ) + 14 R s ( λ i ) R s ( λ i + 1 ) ] ; R i , 1 = 1 2 Δ λ [ R s ( λ i 1 ) + R s ( λ i + 1 ) ] ; R i , 2 = 1 2 ( Δ λ ) 2 [ R s ( λ i 1 ) 2 R s ( λ i ) + R s ( λ i + 1 ) ] .
X = K a b S λ x ¯ ( λ ) R ( λ ) d λ , Y = K a b S λ y ¯ ( λ ) R ( λ ) d λ , Z = K a b S λ z ¯ ( λ ) R ( λ ) d λ ,
K = 100 / a b S λ y ¯ ( λ ) d λ
a = λ 0 , b = λ N , and Δ λ = λ i + 1 λ i , i = 0 , 1 , , N 1
i = 0 N T ( λ λ i ) = 1 for     λ [ a , b ] .
X = K a b S λ x ¯ ( λ ) R ( λ ) d λ = K i = 0 N a b S λ x ¯ ( λ ) R ( λ ) T ( λ λ i ) d λ = K i = 0 N λ i 1 λ i + 1 S λ x ¯ ( λ ) R ( λ ) T ( λ λ i ) d λ = K i = 0 N λ i 1 λ i + 1 W X ( λ ) T ( λ λ i ) d λ ,
W X ( λ ) = S λ x ¯ ( λ ) R ( λ ) .
S λ x ¯ ( λ ) = n = 0 x S , i , n ( λ λ i ) n = x S , i , 0 + x S , i , 1 ( λ λ i ) + x S , i , 2 ( λ λ i ) 2 +
W X ( λ ) = n = 0 W X , i , n ( λ λ i ) n = W X , i , 0 + W X , i , 1 ( λ λ i ) + W X , i , 2 ( λ λ i ) 2 + ,
W X , i , 0 = R i , 0 x S , i , 0 , W X , i , 1 = R i , 0 x S , i , 1 + R i , 1 x S , i , 0 , W X , i , 2 = R i , 0 x S , i , 2 + R i , 1 x S , i , 1 + R i , 2 x S , i , 0 .
x S , i , 0 = 1 A [ B x , i , 0 j = Δ λ Δ λ j 4 B x , i , 2 j = Δ λ Δ λ j 2 ] , x S , i , 1 = B x , i , 1 / j = Δ λ Δ λ j 2 , x S , i , 2 = 1 A [ B x , i , 0 j = Δ λ Δ λ j 2 + B x , i , 2 ( 2 Δ λ + 1 ) ] ,
A = ( 2 Δ λ + 1 ) j = Δ λ Δ λ j 4 [ j = Δ λ Δ λ j 2 ] 2 ,
B x , i , 0 = j = Δ λ Δ λ [ S λ i + j x ¯ ( λ i + j ) ] , B x , i , 1 = j = Δ λ Δ λ { [ S λ i + j x ¯ ( λ i + j ) ] j } , B x , i , 2 = j = Δ λ Δ λ { [ S λ i + j x ¯ ( λ i + j ) ] j 2 } .
X = K i = 0 N λ i 1 λ i + 1 W X ( λ ) T ( λ λ i ) d λ = K i = 0 N [ W X , i , 0 Δ λ Δ λ T ( ξ ) d ξ + W X , i , 1 Δ λ Δ λ ξ T ( ξ ) d ξ + W X , i , 2 Δ λ Δ λ ξ 2 T ( ξ ) d ξ + ] .
Δ λ Δ λ ξ 2 j T ( ξ ) d ξ = ( Δ λ ) 2 j + 1 ( j + 1 ) ( 2 j + 1 ) and Δ λ Δ λ ξ 2 j + 1 T ( ξ ) d ξ = 0 ,
X = K i = 0 N j = 0 [ W X , i , 2 j ( Δ λ ) 2 j + 1 ( j + 1 ) ( 2 j + 1 ) ] = K i = 0 N [ W X , i , 0 + W X , i , 2 ( Δ λ ) 2 6 + ] Δ λ .
X K Δ λ i = 0 N W X , i , 0 = K Δ λ i = 0 N x s , i , 0 R i , 0 , Y K Δ λ i = 0 N W Y , i , 0 = K Δ λ i = 0 N y s , i , 0 R i , 0 , Z K Δ λ i = 0 N W Z , i , 0 = K Δ λ i = 0 N z s , i , 0 R i , 0 , K = 100 / [ Δ λ i = 0 N y s , i , 0 ] .
X K Δ λ i = 0 N [ W X , i , 0 + W X , i , 2 ( Δ λ ) 2 6 ] , Y K Δ λ i = 0 N [ W Y , i , 0 + W Y , i , 2 ( Δ λ ) 2 6 ] , Z K Δ λ i = 0 N [ W Z , i , 0 + W Z , i , 2 ( Δ λ ) 2 6 ]
W X , i , 0 + W X , i , 2 ( Δ λ ) 2 6 = R i , 0 x S , i , 0 + ( Δ λ ) 2 6 [ R i , 0 x S , i , 2 + R i , 1 x S , i , 1 + R i , 2 x S , i , 0 ] = ( x S , i , 0 + ( Δ λ ) 2 6 x S , i , 2 ) 1 12 [ R S ( λ i 1 ) + 14 R S ( λ i ) R S ( λ i + 1 ) ] + ( Δ λ ) 2 6 x S , i , 1 1 2 Δ λ [ R S ( λ i 1 ) + R S ( λ i + 1 ) ] + ( Δ λ ) 2 6 x S , i , 0 1 2 ( Δ λ ) 2 [ R S ( λ i 1 ) 2 R S ( λ i ) + R S ( λ i + 1 ) ] = X i , i 1 R S ( λ i 1 ) + X i , i R S ( λ i ) + X i , i + 1 R S ( λ i + 1 )
X i , i 1 = ( Δ λ ) 2 72 x S , i , 2 Δ λ 12 x S , i , 1 , X i , i = 14 ( Δ λ ) 2 72 x S , i , 2 + x S , i , 0 , X i , i + 1 = ( Δ λ ) 2 72 x S , i , 2 + Δ λ 12 x S , i , 1 .
R S ( λ 1 ) = R S ( λ 0 ) and R S ( λ N + 1 ) = R S ( λ N ) .
X K Δ λ i = 0 N [ W X , i , 0 + W X , i , 2 ( Δ λ ) 2 6 ] = K Δ λ [ i = 0 N X i , i 1 R S ( λ i 1 ) + i = 0 N X i , i R S ( λ i ) + i = 0 N X i , i + 1 R S ( λ i + 1 ) ] = K Δ λ [ i = 1 N 1 X i + 1 , i R S ( λ i ) + i = 0 N X i , i R S ( λ i ) + i = 1 N + 1 X i 1 , i R S ( λ i ) ] = K Δ λ [ X 0 , 1 R S ( λ 1 ) + ( X 1 , 0 + X 0 , 0 ) R S ( λ 0 ) + i = 1 N 1 ( X i + 1 , i + X i , i + X i , i 1 ) R S ( λ i ) + ( X N , N + X N 1 , N ) R S ( λ N ) + X N , N + 1 R S ( λ N + 1 ) ] = K Δ λ i = 0 N V X , i R S ( λ i ) .
V X , 0 = x S , 0 , 0 ( x S , 0 , 1 12 + x S , 1 , 1 12 ) Δ λ + ( 13 x S , 0 , 2 72 x S , 1 , 2 72 ) ( Δ λ ) 2 , V X , i = x S , i , 0 + ( x S , i 1 , 1 12 x S , i + 1 , 1 12 ) Δ λ ( x S , i 1 , 2 72 7 x S , i , 2 36 + x S , i + 1 , 2 72 ) ( Δ λ ) 2 , V X , N = x S , N , 0 + ( x S , N 1 , 1 12 + x S , N , 1 12 ) Δ λ ( x S , N 1 , 2 72 13 x S , N , 2 72 ) ( Δ λ ) 2 .
Y K Δ λ i = 0 N V Y , i R S ( λ i ) and Z K Δ λ i = 0 N V Z , i R S ( λ i )
V Y , 0 = y S , 0 , 0 ( y S , 0 , 1 12 + y S , 1 , 1 12 ) Δ λ + ( 13 y S , 0 , 2 72 y S , 1 , 2 72 ) ( Δ λ ) 2 , V Y , i = y S , i , 0 + ( y S , i 1 , 1 12 y S , i + 1 , 1 12 ) Δ λ ( y S , i 1 , 2 72 7 y S , i , 2 36 + y S , i + 1 , 2 72 ) ( Δ λ ) 2 , V Y , N = y S , N , 0 + ( y S , N 1 , 1 12 + y S , N , 1 12 ) Δ λ ( y S , N 1 , 2 72 13 y S , N , 2 72 ) ( Δ λ ) 2 ,
V Z , 0 = z S , 0 , 0 ( z S , 0 , 1 12 + z S , 1 , 1 12 ) Δ λ + ( 13 z S , 0 , 2 72 z S , 1 , 2 72 ) ( Δ λ ) 2 , V Z , i = z S , i , 0 + ( z S , i 1 , 1 12 z S , i + 1 , 1 12 ) Δ λ ( z S , i 1 , 2 72 7 z S , i , 2 36 + z S , i + 1 , 2 72 ) ( Δ λ ) 2 , V Z , N = z S , N , 0 + ( z S , N 1 , 1 12 + z S , N , 1 12 ) Δ λ ( z S , N 1 , 2 72 13 z S , N , 2 72 ) ( Δ λ ) 2 .
S λ x ¯ ( λ ) = x S , i , 0 + x S , i , 1 ( λ λ i ) + x S , i , 2 ( λ λ i ) 2 for     λ [ λ i Δ λ , λ i + Δ λ ] ,
φ ( x i , 0 , x i , 1 , x i , 2 ) = j = Δ λ Δ λ [ S λ i + j x ¯ ( λ i + j ) ( x S , i , 0 + x S , i , 1 j + x S , i , 2 j 2 ) ] 2 ,
φ ( x S , i , 0 , x S , i , 1 , x S , i , 2 ) x S , i , 0 = 2 j = Δ λ Δ λ [ S ( λ i + j ) x ¯ ( λ i + j ) ( x S , i , 0 + x S , i , 1 j + x S , i , 2 j 2 ) ] = 0 ,
φ ( x S , i , 0 , x S , i , 1 , x S , i , 2 ) x S , i , 1 = 2 j = Δ λ Δ λ { [ S ( λ i + j ) x ¯ ( λ i + j ) ( x S , i , 0 + x S , i , 1 j + x S , i , 2 j 2 ) ] j } = 0 ,
φ ( x S , i , 0 , x S , i , 1 , x S , i , 2 ) x S , i , 2 = 2 j = Δ λ Δ λ { [ S ( λ i + j ) x ¯ ( λ i + j ) ( x S , i , 0 + x S , i , 1 j + x S , i , 2 j 2 ) ] j 2 } = 0 ,
{ ( 2 Δ λ + 1 ) x S , i , 0 + ( j = Δ λ Δ λ j 2 ) x S , i , 2 = j = Δ λ Δ λ [ S ( λ i + j ) x ¯ ( λ i + j ) ] ( j = Δ λ Δ λ j 2 ) x S , i , 1 = j = Δ λ Δ λ [ S ( λ i + j ) x ¯ ( λ i + j ) j ] ( j = Δ λ Δ λ j 2 ) x S , i , 0 + ( j = Δ λ Δ λ j 4 ) x S , i , 2 = j = Δ λ Δ λ [ S ( λ i + j ) x ¯ ( λ i + j ) j 2 ] .

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