# Colorimetry/Chapter 2

In 1931, the Commission Internationale de l'Eclairage (CIE) 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 ​Fig. 3.The (x,y) chromaticitiy diagram, showing the spectrum locus and the purple boundary.Wavelength is indicated in millimicrons. The hue names are those proposed by Kelly (77). specify the chromatic aspect of the light. The analogous ratio, $Z/(X+Y+Z)$, is also known as a chromaticity coordinate, $z$, but only two of the three coordinates, $x,y,z$, give independent information since by definition the sum of all three is unity. Table 2a gives the chromaticity coordinates, $x,y,x,$ of the spectrum colors for 2° and table 2b for 10° fields [59]. Figure 3 shows the points representing the spectrum colors in the ($x,y$)-chromaticity diagram. This diagram is also known as a Maxwell triangle because of Maxwell's first use of such a diagram [98]. Furthermore, it has aptly been called a mixture diagram because it indicates in a very simple way the chromaticity of the color resulting from the additive combination of any two lights. The point representing this chromaticity is found on the straight line connecting the points representing two lights. The primary lights are represented by points at the corners of a triangle, and every point within the triangle represents the chromaticity of a mixture of the primary lights whose proportions are indicated by the chromaticity coordinates, $x,y,z$. The spectrum colors are shown by a smooth curve known as the spectrum locus. A few points on this locus are identified by wavelength in nanometers. It will be noted from figure 3 that the spectrum locus is substantially straight from 540 nm to the long-wave extreme. This means that the standard observer would find binary mixtures of, say, 540 nm with 640 nm, closely equivalent to some intermediate portion of the spectrum. But the spectrum locus from 540 nm to the short-wave extreme is curved outward. This means that for the standard observer a binary mixture of 540 nm with, say, 440 nm would differ importantly in chromaticity from the intermediate parts of the spectrum. By drawing straight lines through any central point (such as x = y = ⅓, representing the so-called equal-energy stimulus) and extending them until they cut the spectrum locus, we may find the spectral complementaries relative to a stimulus represented by that point; that is, we may find the two parts of the spectrum that, when combined in proper proportions, will for the standard observer be equivalent to the central stimulus.
The straight line in figure 3 joining the extremes of the spectrum locus represents the chromaticities of the mixtures of the two extremes of the visible spectrum. The area bounded by the closed curve made up of the spectrum locus and this straight line is the locus of all physically realizable chromaticities. Note that the points representing the primaries of the CIE coordinate system, the apices of the triangle ($x=1,y=z=0$; $y=1,x=z=0$; $z=1,x=y=0$), all fall outside this area; that is, the primaries are imaginary. Note also that both the $X$ and $Z$ primaries fall on the line $y=0$, which is unassociated with luminosity and is known as the alychne or lightless line. The short-wave extreme of the spectrum locus comes close to this line; this means that, although it has the power to elicit in the standard observer a considerable $X$ and $Z$ response, resulting in a vivid bluish purple color, radiant flux of wavelength 380 to 420 nm is almost unassociated with luminosity. The areas in figure 3 corresponding to common color designations for lights are those proposed by Kelly [77] and will be discussed later.
At the time of setting up the standard observer and coordinate system, the International Commission on Illumination [135], Commission International de I'Eclairage (CIE), recommended use of three standard sovirces for colorimetry; source A, representative of gas-filled incandescent lamps; source B, representative of noon sunlight; and source C, representative of average daylight such as that from a completely overcast sky. Source A is an incandescent lamp operated at a color temperature of 2854 °K, on the international temperature scale ($c_{2}$ = 14,380). Source B is obtained by using this same lamp in combination with a two-cell Davis-Gibson liquid filter designed to give a color temperature of about 5000 °K. Source C is obtained similarly and results in a source of correlated color temperature about 6800 °K. These sources are recommended for general use, or whenever there is no special reason for using some other source. Table 3 gives the relative spectral irradiance of Sources A, B, C, D$_{5500}$, D$_{6500}$, and D$_{7500}$. Sources D$_{5500}$, D$_{6500}$, D$_{7500}$ represent several phases of daylight, closely represented by the subscripted correlated color temperatures. Tables 4a and 4b give computation forms for evaluation of the colors of non-self-luminous specimens that transmit, scatter, or reflect incident light for the 2° standard ob​server. Table 4a refers to Source A; table 4b, to Source C. Source B is relatively little used except in Great Britain. Table 4c gives corresponding values for the 10° observer.
Table 4a. Computation form
CIE coordinates (2°) Source A (2854 "K.)
| Sample | Source of Trans. Data |
| Submitted by | |
Computed by Checked by
Planck 2854 °K., C2 = 14,380 Planck 2848 °K., C2 = 14,350
Table 4b. Computation form
CIE Coordinates Source C (Davis Gibson, 2854 °K. to 6800°K)
| Sample | Source of Trans. Data |
| Submitted by | |
Planck 2854 "K., $C_{1}$ = 14,380 Planck 2848 °K., $C_{2}$ = 14.350
Computed by Checked by
The fundamental nature of the tristimulus specification of color permits it to be used as a common denominator by means of which colorimeters involving color standards of glass, plastic, or solutions, or systems of material color standards, transparent and opaque, may be inter-compared. In order to demonstrate how the CIE standard observer and coordinate system may be used for this purpose, four printing inks, red purple, greenish yellow, greenish blue, and blue, have been evaluated, and the steps are reproduced here in detail. Figure 4 shows spectral reflectances of these four printing inks obtained on a recording spectrophotometer. Table 5a gives the spectral reflectances read from the originals of the curves of figure 4. These reflectances apply to the specimens measured except for small wave-length-scale and photometric-scale corrections which have not been applied. Table 5b gives for the greenish yellow specimen the products indicated on the form for computation of luminous reflectance, $Y$, and chromaticity coordinates, $x$,$y$,$z$, under standard source C; see table 4b. The sums of these products are the tristimulus values, $X$,$Y$,$Z$. The luminous reflectance is found as $Y/100,000$; and the chromaticity coordinates, $x$,$y$,$z$, are found by dividing $X$,$Y$, and $Z$, respectively, by the corresponding sum, $X+Y+Z$. Table 5c lists these results for all four printing-ink specimens. Figure 5 is the ($x$,$y$)-chromaticity diagram on which have been plotted large dots to represent these chromaticity coordinates, $x$,$y$.
Comparison of figure 5 with figure 3 shows that the chromaticity points of the four printing-ink specimens correspond to the hue designations red purple, greenish yellow, and blue. This accords well with the color designations found by visual inspection of the specimens. Furthermore, it will be noted that one of the blues is greener than the other. The position of the chromaticity point for the greener-ink color is in accord with the greener hue of this ink. Note also that the greenish yellow is much lighter than the red purple or either of the blues; this accords with the luminous reflectance determinations (compare 0.74 with 0.221, 0.242, and 0.246 in table 5c).
The labor of computing $X$,$Y$,$Z$, or $Y$,$x$,$y$, corresponding to pairs of spectrophotometric curves to see how the colors of the corresponding specimens compare is considerably great. Often the degree of metamerism exhibited by the pair is sufficiently small that the comparison can be made directly from the curves themselves, and much product-control work can be handled in this way. There is still frequent need, in the establishment of color standards and tests for conformity to those standards, to compute the tristimulus values, $X$,$Y$,$Z$, by a short-cut method, the selected ordinate method, to reduce spectrophotometric data.
In this method the ordinates of the spectrophotometric curve are read at a series of selected wavelengths different for each source. Instead of multiplying by the tristimulus values of the spectrum of the source, $(E{\overline {x}})_{\lambda }$, $(E{\overline {y}})_{\lambda }$, $(E{\overline {z}})_{\lambda }$, the selected ordinates are spaced proportionately closer in the wavelength regions where the tristimulus values are higher, and the corresponding readings of spectral reflectance are simply added. Tables 6 gives [21, 49] selected ordinates for source A (incandescent lamp light) and source C (average daylight). Table 7 gives the spectral reflectances of the greenish yellow printing-ink specimen read from figure 4b for the selected ordinates for source C together with the sums of these spectral reflectances, both for ten ordinates and for thirty. It will be noted that, after applying the multiplying factors listed in table 6, the tristimulus values, for the greenish yellow print​ing-ink specimen are found again to a close approximation (compare 0.630, 0.704, and 0.145 from table 7 with 0.631, 0.704, and 0.145, respectively, from table 5c.
Ten selected ordinates sometimes give significant information (see table 7); thirty selected ordinates are often sufficient (as above); and one hundred selected ordinates are sufficient for all but a few very irregular spectral distributions (such as produced by gaseous discharge tubes). These wavelengths are available for many sources in other publications [13, 21, 49]. Nickerson [114] and De Kerf [27] have published studies of the reliability of the selected-ordinate method of computation.
Analog and digital techniques have been developed for use with automatic computing devices to abbreviate the labor of computation required by spectrophotometric colorimetry. In the application of the analog technique [25] data are sent directly from the spectrophotometer to the computer without wavelength-scale and photometric-scale corrections. In one application of the digital technique [12, 95] spectral data are punched directly on cards and fed into the computer, while in another [75] the spectral data are corrected and then punched on cards and fed into the computer.
Tristimulus values, may be obtained by direct comparison of the unknown light with an optical mixture of three primary lights in a divided photometric field. Since the primaries of the CIE standard colorimetric coordinate system are imaginery, such a tristimulus colorimeter cannot be made to read directly. It must be calibrated by measurements of four known stimuli, and then may yield tristimulus values, $X$,$Y$,$Z$, by a transformation the reverse of that indicated in eq. (3). Since the color matches set up in a tristimulus colorimeter designed to cover any substantial part of all possible colors with a single set of primaries exhibit serious metamerism, the field has to be relatively small, subtrending about 2 deg at the observer's eye. This restriction to a small ​angular size of field severely limits the precision of setting compared to what is possible by direct comparison of large specimens in daylight. Furthermore, the metamerism also prevents one normal observer from getting the same reading as another except by accident. If a reasonable approximation to the standard values of $X,Y,$ and $Z$ is to be assured, either the readings of a group of five or ten observers must be averaged, or a color standard yielding a spectral composition similar to that of the unknown specimen must be used. Because of industrial interest in large-field color matching Stiles and Burch [146] and Speranskaya [144] determined the color-matching functions for 10°-field viewing. In this determination either the observers were instructed to ignore the Maxwell spot [73, 98, 103, 104, 156] or it was masked. The color-matching functions thus found are significantly different from the 2°-field functions of the 1931 CIE Standard Observer. The difference is chiefly that expected from the removal of an intervening yellow filter, the macular pigment, from the field of view.
It will be seen that tristimulus colorimeters give only poor information regarding the unknown specimen. Their application to product-control problems is negligible, but because of the ease of calibration and simplicity of the theory they are very useful research tools. Tristimulus colorime​ters Fig. 5.Points representing the colors of four printing ink specimens whose spectral reflectances are shown in figure 9.4.The colors of the ideal closed-cavity radiator are also shown, the temperatures of the radiators being indicated in degrees Kelvin. The smooth curve connecting these points is often called the Planckian locus. have been built and described by Allen [3], Donaldson [28], Guild [43], McAdam [96], Newhall [110], Stiles [145], Verbeek [154], and Wright [157]. The Wright instrument has spectrum primaries; the other four have primaries formed by combining a light source with glass filters. To the Guild and Wright instruments we owe our accurate information regarding the properties of the normal visual system which have been expressed in terms of the standard observer.
The foregoing instruments make up the comparison-field mixture by optical combination of light beams from different sources so that a sum of the separate effects is obtained. A similar optical effect is obtained if the beams are caused to fall upon the same portion of the retina in such rapid succession that a nonflickering spot of color is seen. The effect is that of a time-weighted average of the separate beams. A very simple and widely used tristimulus colorimeter is obtained by taking four disks that have been cut along a radius, interlocking them so as to expose a sector of each, and causing them to rotate on the spindle of a motor so rapidly that neither the separate sectors nor even flicker is perceived. Such an arrangement for combining colors by rotary mixture is called a Maxwell disk. The four disks provide the necessary three degrees of freedom in the adjustment for a match, and if the tristimulus values of the component disks be known ($X_{1},Y_{1},Z_{1};X_{2},Y_{2},Z_{2};X_{3},Y_{3},Z_{3};X_{4},Y_{4},Z_{4})$, the tristimulus values of the mixture can be computed from the fractions of the total area occupied by the respective sectors, $f_{1},f_{2},f_{3},f_{4}$:
${\begin{alignedat}{7}X&=f_{1}X_{1}&+&f_{2}X_{2}&+&f_{3}X_{3}&+&f_{4}X_{4}\\Y&=f_{1}Y_{1}&+&f_{2}Y_{2}&+&f_{3}Y_{3}&+&f_{4}Y_{4}\\Z&=f_{1}Z_{1}&+&f_{2}Z_{2}&+&f_{3}Z_{3}&+&f_{4}Z_{4}.\end{alignedat}}$
(4)
If the disks are chosen anew for each kind of unknown color to be measured so as to be all fairly similar in color to the unknown, the spectral composition of the mixture color is usually sufficiently nonmetameric that no restriction to the central 2 deg of the retina is required. Furthermore any two normal observers with some experience at making the adjustment can check each other closely. The chief drawback of this simple arrangement for product-control work in color is the time lost in adjustment of the sector disk areas. The motor must be turned off, brought to a stop, the disks loosened and readjusted, the motor turned on and allowed to resume speed several times to obtain a final setting of reasonably good, precision.
Nickerson has described a disk colorimeter [105] that avoids the difficulties of the elementary Maxwell disk. Light reflected from the unknown specimen fills one-half of a photometric field, and that from a stationary sector disk fills the other. By having the observer look at the sector disk through a rapidly rotating glass wedge, each sector is presented to view in sufficiently quick succession that no flicker is produced; and at the same time the sectors, since they are stationary, may be continuously adjusted until a color match is obtained. A further advantage is obtained by extending the rotary scanning to the unknown specimen. In this way the average color of a notably nonuniform specimen such as that made up of coarse salt crystals may be obtained. The disk, colorimeter has been extensively used by the United States Department of Agriculture for the color-grading of food products and is well adapted to product-control colorimetry of many kinds.
Another way to identify a combination of lights to specify a color, alternate to the tristimulus method, is to determine the luminance (photometric brightness) of one spot of light of fixed spectral composition (such as average daylight) and the luminance of a spot of light of continuously variable spectral composition separately identified (as by wavelength in the spectrum). In this way the requisite three degrees of freedom in adjustment to a color match are supplied. This form of identification leads naturally into a specification in which the luminance [20] of the unknown is given and the chromaticity is specified by two variables in polar coordinates. One of these variables is an angle, the other a radius, and both ​can be computed from the chromaticity coordinates of the fixed spot of light, the variable spot of light, and the unknown, these coordinates serving to locate the respective positions in a chromaticity diagram. If the fixed light is nearly achromatic, the angle often correlates well with the hue of the color perception, and the radius fairly well with its saturation. The most fundamental way to specify the direction on a chromaticity diagram from the point representing the fixed light to the point representing the unknown light is by wavelength of the part of the spectrum required to make the match. If the unknown color can be matched by adding some part of the spectrum to the fixed light, it is said to have a spectral color, and the required wavelength is called dominant wavelength. But if a color match is produced for the fixed light by adding some part of the spectrum to the unknown color, the unknown is said to be nonspectral, and the required wavelength is called the complementary wavelength. Either dominant wavelength or complementary wavelength may be obtained for the standard observer by drawing on a chromaticity diagram a straight line through the point representing the fixed light and that representing the unknown color, and then by reading the wavelength corresponding to the point at which this line extended intersects the locus of spectrum colors. If the unknown color is plotted between the fixed light and the spectrum, the intersection gives the dominant wavelength; but if the fixed light is represented by a point intermediate to the unknown and the intersection of the straight line with the spectrum locus, the intersection indicates the complementary wave-length.
The degree of approach of the unknown color to the spectrum color is commonly indicated by the ratio of the amount of the spectrum color to the total amount of the two-part combination; this ratio is called purity, and if the amounts are specified in luminance units, the ratio is called luminance (formerly colorimetric) purity. By far the most common convention, however, is to express the amounts in units of the excitation sum $X+Y+Z$; the resulting ratio is called excitation purity and corresponds simply to distance ratios on the chromaticity diagram of a colorimetric coordinate system [49, 64, 133]. Formulas have been derived by Hardy [49] and MacAdam [91] to ​convert from luminance purity to excitation purity, and the reverse.
*r1Values for calculation with 10 selected ordinates.
Figure 6 indicates how dominant wavelength and excitation purity of the four printing-ink specimens would be found from their chromaticity coordinates $x,y$, relative to source C taken as the fixed light. Table 8 gives the dominant and complementary (C) wavelengths found as in figure 10 by the intersections of the straight lines with the spectrum locus. Table 8 also gives the excitation purities found by dividing the distance from the fixed point (source C) to the specimen point by the total distance from the fixed point to the boundary (spectrum locus plus straight line connecting its extremes). Large-scale charts for reading dominant wavelength and purity relative to source C are provided in the Hardy Handbook of Colorimetry [49].
Apparatus for the direct measurement of dominant wavelength and luminance purity has been designated by Nutting [125] and by Priest [132]. The degree of metamerism ordinarily obtained with such apparatus leaves it open to the same objections as have prevented tristimulus colorimeters with single sets of primaries from being useful for production control. There is a further disadvantage in the direct measurement of luminance purity in that the luminance of the spectrum component has to be determined relative to the luminance of the mixture by separate photometry. Since there is usually a large chromatic difference between these two fields, simple equality-of-brightness settings are not reliable, and an auxiliary flicker photometers, as in Priest's apparatus [132], must be used. This method has been found to exaggerate individual-observer differences; often-times observers will differ only slightly in the mixtures of spectrum light and fixed light that they find to be equivalent to an unknown color, but they will disagree importantly in their photomerty of the components.
* ↑ Extrapolated.
* ↑ Values for calculation with 10 selected ordinates.
* ↑ C denotes complementary wavelength.