# Colorimetry/Chapter 3

In the fundamental colorimetry of lights and objects a single standard is used for each class of specimen. Opaque surfaces are referred to the ideal perfect diffuser, or to physically realizable near-perfect diffusers, such as a sufficiently thick layer of magnesium oxide deposited from the smoke of magnesium turnings or ribbon [108] or layers of barium sulfate [141, 6]. Transparent objects, such as gelatin films and crystal or glass plates, are referred to the equivalent thickness of air; transparent solutions, to the equivalent thickness of distilled water or solvent. Self-luminous objects, such as fluorescent lamps, cathode-ray tubes, and incandescent lamps, are measured relative to one of the standard sources, usually source A [19]. The colors of specimens closely resembling the respective standards can be evaluated quite precisely and accurately; those differing radically in spectral composition, only with relative uncertainty. That is, near-white specimens, nearly clear glass plates, and incandescent lamps nearly equivalent to source A present the simplest colorimetric problem; highly selective absorbers and emitters, like the rare-earth glasses and gaseous discharge tubes, present difficult measurement problems. In general, the greater the deviation in spectral composition between the unknown specimen and the standard, the greater the uncertainty of the result obtained by a visual or a photoelectric colorimeter.
Automatic spectrophotometry has greatly extended the application of both visual and photo-electric colorimetry. It has supplied a practical way to calibrate working standards of color. If a fairly large group of specimens is at hand to be measured, say twenty or more, all of similar spectral composition, the most satisfactory way to measure them in the present state of colorimetric science is to evaluate one or two of them carefully by means of the spectrophotometer to serve as working standards, then obtain the color specifications of the remainder by visual or photoelectric determination of the difference between specimen and standard.
In the interpretation of the importance of chromaticity differences based upon separation of the points representing the two chromaticities in the ($x,y$)-diagram a warning is necessary. This diagram is considerably expanded in the green portion relative to the other portions, much as the Mercator projection of the earth's surface is expanded near the poles. Thus, two points separated by a given distance in the green portion of this diagram correspond to chromaticities that are Fig. 7.Chromaticity spacing in the (x,y)-diagram.Between any center and any point on the corresponding ellipse there are approximately 100 just noticeable chromaticity steps [67]. much harder to distinguish under ordinary viewing conditions than two chromaticities separated by the same amount in other portions of the diagram. Furthermore, the bluish purple portion of the diagram is correspondingly compressed. The system of ellipses shown on figure 7 serves to indicate approximately the metric properties of the ($x,y$) -diagram. Under moderately good observing conditions, the distances from the central point of each ellipse to any point on its boundary correspond approximately to one hundred times the chromaticity difference just perceptible with certainty. These ellipses were drawn from a review of the literature in 1936 [66, 67], and subsequent extensive work published by Wright [160, 161] and by MacAdam [92, 94] has corroborated the essential correctness of the indicated chromaticity spacing. Figure 7 not only indicates the extent to which the green portion of the diagram is expanded, and the bluish purple compressed, but also indicates that, in general, the chromatic importance of a distance on the ($x,y$)-diagram is a function both of the position of the central point and the direction of the deviation from it.
When sets of primaries other than those of the CIE standard observer system are expressed by transformations of the form of eq (3) , the chromaticity spacings in the resulting Maxwell triangle may be made to vary widely. There have been several attempts to select primary sets that yield uniform chromaticness scales in which the chromaticity spacing corresponds to perceptibility [14, 55, 66, 90, 144] . The transformation equations for the chromaticity coordinates, r,g, of the uniform-​chromaticnss-scale Fig. 8.Uniform-chromaticity-scale triangle according to Judd [66].The length of the straight line connecting the points representing any two chromaticities on this triangle is approximately proportional to the perceptibility of the chromaticity difference. (UCS) System [66], from the CIE chromaticity coordinates, x,y, are
${\begin{aligned}r&={\frac {2.7760x;+2.1543y-0.1129}{-1.0000x;+6.3553y+1.5405}}\\\\g&={\frac {-2.9446x;+5.0323y+0.8283}{-1.0000x+6.3553y+1.5405}}\\\end{aligned}}$
(5)
The triangle resulting from this transformation is shown in figure 8.
In 1960 the CIE adopted a provisional recommendation that the transformation suggested by MacAdam in 1937 [89] be used whenever a spacing perceptually more uniform than that of the ($x'',y$)- diagram is desired. This transformation is:
${\begin{aligned}u&={\frac {4x}{12y-2x+3}}\\\\v&={\frac {6y}{12y-2x+3}}\\\end{aligned}}$
(6)
Although the transformation coefficients are sim- ple, the ($u,v$)-diagram (fig. 9) has a spacing that closely resembles the UCS diagram.
One of the most useful visual devices for determining relative luminance is the Martens photometer. Figure 10 shows the Martens photometer combined with a diffuse illuminator to form the Priest-Lange reflectometer [134]. This reflectometer is intended for the measurement of luminous reflectance of opaque specimens relative to reflecting standards of similar spectral reflectance. The Priest-Lange instrument is also adaptable to the measurement of luminous transmittance of transparent plates relative to transmitting standards similar in spectral transmittance to the unknown. Finally, the Martens photometer, removed from the mounting, may be used for the determination of the luminance of an unknown self-luminous surface relative to a spectrally similar standard of known luminance. The superior usefulness of the Martens photometer arises from the convenience of the adjustment for equality of brightness between the two halves of the photometer field and from the fact that the dividing line between the half-fields is exceptionally narrow so that it is often invisible when a brightness match has been set.
The determination of chromaticity coordinates, $x,y,$ by comparison of the unknown specimen with a working standard of similar spectral reflectance can be carried out visually with high precision by means of a colorimeter described by Judd [64]. The adjustment of the chromaticity of the comparison field to match the standard field is by two double wedges, one of greenish and the other of yellowish glass. Since the light from the comparison field must pass through both the yellow and the green wedge, some of the radiant energy being subtracted by each, it is sometimes called a subtractive colorimeter; see figure 11 which gives a schematic diagram. The standard and comparison fields are brought into juxtaposition by means of a Lummer-Brodhun cube having a double-trapezoid pattern subtending 9 × 13° at the observer's eye. The adjustment to near equality of brightness to facilitate detection of chromaticity differences isby movement of the projection lamp that illuminates both standard and comparison surfaces.
A substitution method is usually employed with this colorimeter, a match first being set up between the standard and comparison surfaces by adjustment of the wedges. Then the unknown specimen is substituted for the standard, and the wedges readjusted to restore the match. The differences in wedge settings can be calibrated in terms of differences in the chromaticity coordinates, $x,y,$ of the CIE standard coordinate system, provided the spectral reflectances of the comparison surface are known approximately, from the known spectral transmittances of the wedges. This calibration has been carried out for about 100 widely differing: comparison surfaces. It has been found that the calibration is chiefly a function of the chromaticity coordinates, $x,y,$ of the comparison surface; so calibrations for comparison surfaces intermediate in chromaticity to those already calibrated may usually be found satisfactorily by interpolation.
ever, of requiring a skilled operator. Furthermore the calibrations, carried out in accord with table 4b, are time-consuming; and the instrument is not applicable to some highly selective samples because nearly homogeneous energy is too little changed in spectral composition by passage through the wedges. This colorimeter has been used in setting up a color standard for ruby mica [70]; for inspection of working standards, transparent and opaque, for conformity to a master standard; and for general colorimetry by difference, for both fluorescent and nonfluorescent specimens [137] .
If three photocells could be adjusted, as by glass filters, so that their responses were proportional throughout the visible spectrum to some linear combination [as in eq. (3) of the standard CIE distribution curves (see fig. 2)], then they could be used to test whether any two light beams have the same color according to eq. (2) and could be made to yield direct measurements of tristimulus values, $X$,$Y$,$Z$ [34, 45]. The $X$-function filter is the most difficult to design because it has two lobes, one short-wave lobe and one long-wave lobe. Two separate filters to cover a portion of one photocell have been designed for this bilobal function by Barnes [10] and by NimerofT and Wilson [121]. Figure 12 shows the $X$-function match achieved in the Nimeroff-Wilson colorimeter. This function has also been fitted by a filter-photocell combination to approximate the long-wave lobe to which a portion of the $Z$-function is added, either electrically, as in the Color Difference Meter of Hunter [56, 57] and arithmetically, as in the Multipurpose refiectometer of Hunter [55], the Colormaster of Glasser and Troy [38] and the Color Eye of Bentley [11]. In these instruments the transformation equations to obtain $X$,$Y$,$Z$ for source C may be represented as:
${\begin{aligned}X&=0.80R+0.18B\\Y&=1.006G\\Z&=1.18B\\\end{aligned}}$
(7)
$X$
where R,G, and B are settings with red, green, and blue filters, respectively. In some of these instruments the filter for the long-wave lobe of the $X$-function may be amber, hence settings with this filter may be designated A. Van den Akker [152] has discussed the incompleteness of success of these colorimeters to duplicate the CIE standard observer system. Recent versions of Color Difference meters and the Color Eye are shown in figure 13.
Figure 14 shows the degree of success achieved by the filters designed by Hunter [55] to duplicate the CIE standard observer and simultaneously to adjust the spectral distribution of a projection lamp to that for CIE source C. Figure 15 shows the discrepancies that the filters of his multipurpose reflectometer introduce. These discrepancies are roughly proportional to the distance from the point representing the magnesium-oxide standard, and are frequently larger than 0.02 in $x$ or $y$; that is, more than 10 times a reasonable chromaticity tolerance for most colorimetric work. However, for the comparison of near-white surfaecs this degree of duplication is sufficient. Figure 16 refers to the small rectangle near the center of figure 15 and indicates that the discrepancies are less than 0.001 in $x$ or $y$ for comparison of near-white surfaces with magnesium oxide. In general, a similar agreement can be expected in using this photoelectric tristimulus colorimeter for the determination of small chromaticity differences between nonmetameric pairs. And even for measurement of fairly sizable nonmetameric chromaticity differences, such as analyzed spectrophotometrically in the upper portion of figure 17 (BPB 8/2 versus MgO, BG 7 /4 versus BG 6/4) , and small chromaticity differences with a moderate metameric component, such as shown in the lower left portion of figure l7 ($Y_{1}$ versus $Y_{2}$), the discrepancy is in the neighborhood of 0.002 in $x$ or $y$, which is negligible for many purposes. However, for highly metameric pairs, such as shown in the lower right portion of figure 17, the discrepancy may be expected to be in the neighborhood of 0.02, just as it is for large chromaticity differences.
If the limitations of photoelectric tristimulus colorimetry are appreciated, the method is most useful in product-control colorimetry of non-fluorescent specimens by difference from a working standard. The precision of the method is comparable, though perhaps not quite equal, to the best that can be done by eye. No unusual qualifications or extended special training is required by the operator; and, compared to visual colorimetry or to indirect colorimetry by the spectrophotometer, the results are obtained very rapidly.