Abstract

In colorimetric work, the calculation of tristimulus values is typically accomplished by one of the standard numerical-integration schemes, such as Simpson’s rule. However, great computational savings can be achieved by use of a more-sophisticated approach. By computing the orthogonal polynomials associated with the CIE color-matching functions, the method of gaussian quadratures can be applied. For certain types of color-matching problems, the gaussian technique is at least twice as fast, for the same accuracy, as conventional methods.

© 1975 Optical Society of America

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References

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  1. G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967).
  2. A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965).
  3. N. Ohta, Appl. Opt. 10, 2183 (1971).
    [Crossref] [PubMed]
  4. P. Beckmann, Orthogonal Polynomials for Engineers and Physicists (Golem, Boulder, Colo., 1973).
  5. Technical Notes, J. Opt. Soc. Am. 64, 896 (1974).

1974 (1)

1971 (1)

Beckmann, P.

P. Beckmann, Orthogonal Polynomials for Engineers and Physicists (Golem, Boulder, Colo., 1973).

Ohta, N.

Ralston, A.

A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965).

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967).

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Other (3)

P. Beckmann, Orthogonal Polynomials for Engineers and Physicists (Golem, Boulder, Colo., 1973).

G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967).

A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965).

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Figures (1)

FIG. 1
FIG. 1

Spectral distributions of test colors used in the example.

Tables (4)

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TABLE I Weights (Hi) and abscissas (λi) for gaussian-quadrature method, with no illumination bias. Wavelengths are ex pressed in nanometers.

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TABLE II Weights (Hi) and abscissas (λi) for gaussian-quadrature method, biased with illuminant A. Wavelengths are expressed in nanometers.

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TABLE III Weights (Hi) and abscissas (λi) for gaussian-quadrature method, biased with illuminant C. Wavelengths are expressed in nanometers.

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TABLE IV Estimates of tristimulus values for the spectral distributions of Fig. 1.

Equations (20)

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X = λ x ¯ ( λ ) C ( λ ) d λ ,
Y = λ y ¯ ( λ ) C ( λ ) d λ ,
Z = λ z ¯ ( λ ) C ( λ ) d λ ,
X = λ x ¯ ( λ ) I ( λ ) R ( d 1 , d 2 , d 3 , λ ) d λ ,
Y = λ y ¯ ( λ ) I ( λ ) R ( d 1 , d 2 , d 3 , λ ) d λ ,
Z = λ z ¯ ( λ ) I ( λ ) R ( d 1 , d 2 , d 3 , λ ) d λ ,
α β w ( x ) f ( x ) d x i = 1 n H i f ( x i ) ,
α β w ( x ) Q m ( x ) Q n ( x ) d x = { h n 2 if m = n 0 if m n ,
H i = 1 Q n ( x i ) α β w ( x ) Q n ( x ) x x i d x .
Q n + 1 ( x ) = ( x B n ) Q n ( x ) ( h n h n 1 ) 2 Q n 1 ( x ) ,
h n 2 = α β w ( x ) Q n 2 ( x ) d x ,
B n = 1 h n 2 α β x w ( x ) Q n 2 ( x ) d x ,
Q 0 ( x ) = 1 ,
Q 1 ( x ) = x α β x w ( x ) d x α β ω ( x ) d x .
Q n ( x ) = x n + b n x n 1 + c n x n 2 + + r n = i = 1 n ( x x i ) .
Y = λ y ¯ ( λ ) C ( λ ) d λ i = 1 n H i C ( λ i ) .
Y = λ y ¯ ( λ ) C ( λ ) d λ 0.1582 C ( 487.3 ) + 0.6616 C ( 558.4 ) + 0.1802 C ( 630.6 ) .
Y = λ y ¯ ( λ ) R ( λ ) I ( λ ) d λ 0.1601 R ( 484.3 ) + 0.6679 R ( 556.4 ) + 0.1719 R ( 628.9 ) .
E 1 = ( Δ X 2 + Δ Y 2 + Δ Z 2 ) .
E 2 = [ ( Δ L * ) 2 + ( Δ u * ) 2 + ( Δ υ * ) 2 ] 1 / 2 .