Abstract
As explained in Chap. 1, the objective method of evaluating the appearance of a color consists of determining a stimulus that is equivalent under the viewing conditions. The stimulus ordinarily employed is an additive mixture of three basic or primary stimuli. In the calculation of the equivalent stimulus from spectrophotometric data, it is necessary to use the tristimulus values for the spectrum colors. Table 1.1 lists the values of x̄, ȳ, z̄ that were adopted by the CIE in 1931. Those values indicate the amounts of each of the chosen primaries that are required by a normal observer to match the colors of equal amounts of power at the indicated wavelengths.
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References
Tables of the products of x̄S, z̄S, and z̄S at 10 nm intervals from 380 to 770 nm for both the 1931 and 1964 CIE observers and for illuminants A, C, D65 and three other less frequently used illuminants are included in D. B. Judd, G. Wyszecki: Color in Business, Science, and Industry, 3rd ed. (Wiley, New York 1975). The values in each of those tables have been multiplied by a common factor such that the sum of ȳS values is 100. Therefore, when the values in those tables are multiplied by decimal values of transmittance (or reflectance) the totals of the products are tristimulus values expressed as percent. No division is necessary, provided that values of transmittance (or reflectance) at 10 nm intervals over the entire range from 380 to 770 nm were used. However, if only values at 20 nm intervals are used, the sums of the products should be divided by the sum of ȳS at only those intervals. That sum is, in general, somewhat different than 50. The result of the divisions must be multiplied by 100 to get the tristimulus values in percent. To guard against forgetting division by the total of ȳS for the utilized wavelength range and intervals, the products x̄S, ȳS, and z̄S are given without any adjustments in Tables 5.2–4. For convenience, the totals for all values given are printed at the bottoms of the corresponding columns in Tables 5.2–4, 8–10.
The symbol Δ indicates a small change or difference of the quantity represented by the symbol that follows Δ. In the present case, Δλ indicates the difference between successive values of λ.
The symbol Σ indicates that all quantities of the kind represented by the succeeding expression are to be summed (added together). Numerals (or symbols) placed under and over Σ identify the first and last (respectively) quantities to be included in the sum. The symbol ∫ is the integral sign. The process indicated by it, called integration, consists of summation but with intervals (indicated here by dλ) so small that the result of the integration would not be changed if they were made smaller.
The numbers of ordinates are multiples of 9. For X and Z they are different from 99 in order to keep the multiplying factors as nearly as possible equal to those for Y. However, for illuminante A the factors for Z, for the 1931 and 1964 observer data, are only about 1/3 as much as for Y when 99 ordinates are used for both Z and Y, but 99 wavelengths are listed in the tables. If factors for Z nearly as great as those for X and Y are desired, the abridged set of 33 wavelengths marked by asterisks can be used, with the corresponding factors. Greater accuracy is obtained, however, by using the full list of 99 wavelengths with the smaller factors.
A. C. Hardy: History of the design of the recording spectrophotometer. J. Opt. Soc. Am. 28, 360–364 (1938);
D. L. MacAdam, W. E. White: Universal, digital tristimulus integrator. J. Opt. Soc. Am. 47, 605–611 (1957).
The symbol δ indicates the change of the quantity specified by the letter that follows δ in the numerator, caused by a small change of only the quantity specified by the letter that follows δ in the denominator. All other quantities are kept unchanged. The term “partial derivative” is the name for the ratio of the change indicated by the numerator divided by the amount of the change that causes it, that is, the quantity indicated by the denominator.
The wavelengths and weights by which the ordinates for gaussian quadrature should be multiplied can be calculated by the method published by Robert Wallis: Fast computation of tristimulus values by use of gaussian quadrature. J. Opt. Soc. Am. 65, 91–94 (1975). Wallis included wavelengths and weights for orders 3 to 6 for illuminants A and C for the CIE 1931 observer. 5 nm intervals from 380 to 770 nm were used for the results given in Tables 5.20–25.
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© 1981 Springer-Verlag Berlin Heidelberg
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MacAdam, D.L. (1981). Determination of Tristimulus Values. In: Color Measurement. Springer Series in Optical Sciences, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-13508-2_5
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