Page:Colorimetry104nime.djvu/10

$\Delta \lambda$, to be used in these summations depends upon the irregularity of the curve of spectral irradiance with wavelength; intervals of 20 nm are sometimes sufficient to yield a significant result; intervals of 10 nm often do; and intervals of 5 nm are usually sufficient except for discontinuous spectral distributions such as those characterising gaseous discharge lamps. De Kerf [27] has reported instances where 1-nm intervals have been required.

The summations of eq (2) form the tristimulus values of the color and are customarily given the symbols $X$, $Z$, $Z$, respectively, so that the condition for a color match would ordinarily be written:

${\begin{aligned}X_{1}&=X_{2}\\Y_{1}&=Y_{2}\\Z_{1}&=Z_{2}\end{aligned}}$

(2a)

and would mean that the two colors are identical since their tristimulus values are identical. Thus, to match color one requires $X_{1}$ parts of the $X$ primary, $Y_{1}$ parts of the $Y$ primary, and $Z_{1}$ parts of the $Z$ primary; and because to match color two requires the same amounts of the same primaries as does color one, the two colors are the same.

Any three lights may be used as primaries in a system of tristimulus color specifications, provided only that no one of them is equivalent to a combination of the other two. Tristimulus specifications $X,Y,Z,$ expressed relative to one set of primaries, may be transformed into specifications $R,G,B,$ relative to any other set, by transformation equations of the form:

${\begin{aligned}R&=K_{1}X+K_{2}Y+K_{3}Z\\G&=K_{4}X+K_{5}Y+K_{6}Z\\R&=K_{7}X+K_{8}Y+K_{9}Z\\\end{aligned}}$

(3)

The constants $K_{1}$ to $K_{3}$ may take on any arbitrary values, positive, negative, or zero, provided they are not such as to make one of the new primaries identical to a combination of the other two; that is, provided that:

${\begin{matrix}K_{1}&K_{2}&K_{3}\\K_{4}&K_{5}&K_{6}\\K_{7}&K_{8}&K_{9}\\\end{matrix}}\neq 0$

(3a)

As the exceptions that cause the determinant of the system to vanish are trivial, the choice of coordinate system is very wide. The primaries do not even have to correspond to physically realizable lights. Imaginary lights defined by spectral compositions having negative values for some parts of the spectrum are admissible, and indeed are preferred for routine colorimetry because by their use the computation of tristimulus values from spectrophotometric data is somewhat simplified.

2. The 1931 CIE Standard Colorimetric System and 1964 Supplement

2.1. General Principles

Fig. 2.Spectral tristimulus values according to the 1931 CIE standard observer.

In 1931, the Commission Internationale de l'Eclairage (CIE)^[1] recommended that all subsequent color data be expressed in terms of the same tristimulus system so that the results would be immediately comparable. The standard observer and coordinate system recommended [49, 65, 135, 143] is defined by the tristimulus values of the spectrum colors given in table 1a and plotted in figure 2. The supplementary observer for large fields, adopted by the CIE in 1964 [59] is given in table 1b. It will be noted that the primaries chosen are such that none of these tristimulus values is less than zero. It is further true that the green primary chosen, whose amounts are designated by y, is such as to carry all of the luminosity, the other two primaries (red, blue) whose amounts are designated by $X$ and $Z$, respectively, being unassociated with luminosity. Therefore, the values of ${\overline {y}}$ for the spectrum correspond to the standard luminosity function, and it is convenient and customary to express the $Y$ value of a luminous area as its luminance (photometric brightness) in terms of some recognized unit (such as candles per square meter, milli-lambert, or foot-lambert). The $Y$ value of an opaque specimen may be conveniently expressed as its luminous reflectance (ratio of reflected to incident luminous flux); and the $Y$ value of a transmitting specimen is customarily put in terms of luminous transmittance (ratio of transmitted to incident luminous flux).

If, as is usual, light combinations are not the chief interest, it is convenient to substitute for the tristimulus values, $X,Y,Z,$ the two ratios, $X/(X+Y+Z)$ and $Y/(X+Y+Z)$, combined with the luminous value, $Y$. The two ratios are known as chromaticity coordinates, $x,y$, because they serve to

1. This was formerly called ICI for the initial letters of the English name of this commission.