# Colorimetry/Chapter 6

Spectrophotometric colorimetry, the most fundamental color-measurement technique, suffers from two major sources of difficulty. First, the 1931 standard observer color-matching data were obtained for only some 17 observers [61] under 2° angular subtense and quite low illumination levels. Extrapolation of the use of this system for all observers and conditions may require justification. Secondly, there is usually some lack of precision and accuracy of spectral measurement of exitance, transmittance, and reflectance. Spectrophotometrs are subject to errors of wavelength and photometric scales, stray-energy and slit-width effects, multiple reflections, and errors due to samples that are temperature dependent, wedge-shaped, and translucent. A compilation of these errors and recommendation for their accounting has been published by Gibson [35].
Although the CIE standard observer system for colorimetry has been generally satisfactory since its recommendation in 1931, there have been some difficulties with the system. Most notable is its; inability to resolve the chromaticity difference of anafase and rutile titanium dioxide [62, 71] usually viewed with angular subtense much greater than 2°. These difficulties have led the CIE to seek a new system (the 1964 CIE supplementary observer) through the Stiles and Burch [146] and Speranskaya [144] color-matching data of some 75 observers for a 10° photometric field. A complete analysis of these data permits the establishment of estimates of within and between-observer variability of the system. Estimation of the variability in the color-matching functions, r(red), g(green), and b(blue) requires information about the between-observer deviations, $\sigma _{r}$, $\sigma _{g}$, $\sigma _{b}$, the within- observer deviations, $_{w}\sigma _{r}$, $_{w}\sigma _{g}$, $_{w}\sigma _{b}$, and the covariances, $\sigma _{r}g$, $\sigma _{r}b$, $\sigma _{b}g$ The covariances, $\sigma _{i}j$, are the products of the correlation coefficients,$\sigma _{r}g$, and the deviations, $\sigma _{i}$, $\sigma _{j}$; thus $\rho _{i}j=\sigma _{i}\sigma _{j}$ where $i$ and $j$ refer to the color-matching functions. Figure 29 shows the between-observer deviations and the correlation coefficients determined by Nimeroff [123] ​Fig. 29.The standard between-observer deviations (upper graph) for 10°-field color-matching functions by Stiles and Burch [146] and Speranskarya [144] and the correlation-coefficients (lower graph) derived by Nimeroff [123]. from the Stiles-Burch and Speranskaya data. The laverage ratio of the between-observer deviations, $\sigma _{i}$, to within-observer deviations,$_{u}\sigma _{i}$, for the three color-matching functions was estimated as 5.7.
Judd and Kelly [74] have transformed the average color-matching data, r,g,b, to the tristimulus ' values, x,y,z by the transformation equations:
${\begin{aligned}{\overline {x}}&=k_{1}{\overline {r}}+k_{2}{\overline {g}}+k_{3}{\overline {b}}\\{\overline {y}}&=k_{4}{\overline {r}}+k_{5}{\overline {g}}+k_{6}{\overline {b}}\\{\overline {z}}&=k_{7}{\overline {r}}+k_{8}{\overline {g}}+k_{9}{\overline {b}}\\\end{aligned}}$
(9)
(The variances $V(u)$ and the covariances $C(uv)$ in the transformed system are computed by applying the propagation of error theory, thus:
${\begin{aligned}V(u)&=\sum {\left({\frac {\delta u}{\delta i}}\right)^{2}\sigma _{i}^{2}}+2\sum {\left({\frac {\delta u}{\delta i}}\right)\left({\frac {\delta u}{\delta j}}\right)\sigma _{ij}}\\C(u)&=\sum {\left({\frac {\delta u}{\delta i}}\right)\left({\frac {\delta v}{\delta i}}\right)\sigma _{i}^{2}}+\sum {\left[\left({\frac {\delta u}{\delta i}}\right)\left({\frac {\delta v}{\delta j}}\right)+\left({\frac {\delta u}{\delta j}}\right)\left({\frac {\delta v}{\delta i}}\right)\right]\sigma _{ij}}\\\end{aligned}}$
(10)
In table 20 are listed the tristimulus values derived by Judd and Kelly. Also listed are the between-observer variances and covariances in these values derived by Nimeroff [123] on the assumption that the variance and covariance terms involving the constants $k_{i}$ of the transformation are zero. These data permit estimation of the ranges of color within which matches set by all observers for any one color will fall. Use of the factor 0.175 may be useful in estimating ranges by one observer from the ranges by many observers.
Propagation of error theory has also been applied [120] to solve the problem of finding the limitations of spectral measurements used in evaluating chromaticity coordinates. On the assumption that the standard observer is free from error, uncertainty ellipses for specimens illuminated by CIE source C may be derived from the equation
$T\epsilon ={\frac {\left(x-x_{n}\right)^{2}\sigma _{y}^{2}+(x-x_{n})(y-y_{n})\sigma _{xy}+(y-y_{n})^{2}\sigma _{x}^{2}}{\sigma _{x}^{2}\sigma _{y}^{2}-\sigma _{xy}^{2}}}$
(11)
where $T\epsilon$ depends on the level of significance, x and y are coordinates of the ellipse, $x_{n}$ and $y_{n}$ are the chromaticity coordinates computed from the mean spectral data of a specimen, $\sigma _{x}^{2}$ and $\sigma _{y}^{2}$, are the variances, and $\sigma _{xy}$ is the covariance of the chromaticity coordinates. The results of this investigation are shown in figure 30 compared with the perceptibility ellipses of MacAdam [91]. This error theory has been applied also to the problem of estimating the uncertainty ellipses resulting from imprecision in measuring spectral exitance of fluorescent lamps [122].
Errors introduced by using summation as in eq (2) instead of integration to evaluate tristimulus values X.Y,Z, have been treated by Nickerson [114] and by De Kerf [27]. Integration is not possible with the present CIE system because the spectral tristimulus values are specified for small but finite intervals. De Kerf compares colorimetric results from summation with intervals of 1, 5, 10, and 20 nm. The conclusion reached by both investigators is that the size of the interval is determined by the type of problem. The most spectrally selective specimens require the smallest summation intervals.
Errors in measurement by photoelectric tristimulus colorimeters result mainly from three failings [58, 153]: (1) to duplicate the geometric conditions of the real situation, (2) to illuminate the specimen with light of applicable spectral character, and (3) to duplicate the required filter-detector spectral sensitivity for the color-matching functions.
Many photoelectric colorimeters use the standard 45°0° conditions, or the equivalent 0°45° conditions. Errors expected for nonglossy specimens by substitution of 0°-hemispherical condition for the standard 45°0° conditions have been analyzed by Budde and Wyszecki [18].
​Fig. 30.Uncertainty ellipses compared with MacAdam perceptibility ellipses.Correlated uncertainty, solid line ellipses; MacAdam perceptibility, dotted line ellipses. (All ellipses are plotted on a times-ten scale.)
Most photoelectric colorimeters are designed for one standard source, usually CIE source C. Such colorimeters do not indicate how closely specimen and standard match for other sources, which depend on the degree of metamerism. Several color imeters now have provision for inserting an auxiliary filter to approximate a source other than the one supplied.
Most photoelectric tristimulus colorimeters use three filter-detector combinations and approximate the short-wave lobe of the x-f unction by a suitable fraction of the z-function. Errors encountered in this type of colorimeter have been treated by Van den Akker [152J.
There is some degree of failure to duplicate the CIE color-matching functions even in colorimeters with four filters, one for y, one for s, and two for the two-lobed x. This failure restricts utility of these colorimeters to measurement of the color of specimens of spectral characteristics similar to that of the standard.
If the limitations are appreciated, methods using photoelectric tristimulus colorimeters are useful in product-control colorimetry of nonfluores cent specimens. The National Bureau of Standards has developed three sets of color standards, Non selective standards, KB-Chromatic standards, and S-Chromatic standards, for use with these methods. These standards are now available, either in dividually or in sets, from the Gardner Laboratory. Inc., Bethesda, Md. 20014, and Hunter, Associates Laboratory, Inc., Fairfax, Va. 22030.
Limitations of visual comparisons result from inability to make precise interpolation between two colors. Use of the Munsell color system requires interpolation of hue, value, and chroma. Experience shows that the best a trained observer does in such interpolations is correct within 0.5 hue step for chromas 4 or more, within 0.2 value step, and 0.4 chroma step.
Visual comparisons are limited also because of the difficulty of choosing permanent materials that have suitable spectral characteristics. The most permanent of these are vitreous enamel and glass, which, however, are so limited in variety of spectral characteristics that metamerism cannot be avoided. This limitation requires control of the spectral character of the light source and disqualifies observers with abnormal vision.
Material standards of glass used in one-dimensional color scales cause difficulty because the luminous-transmittance match with the specimen may not match in chromaticity; see, for example, table 17. Hence, one-dimensional scales as represented by material standards are not always an adequate solution to a problem.
Gross errors in visual judgment may result from marked dissimilarity of surrounds. For example, the orchid standard of the KB-chromatic set issued by the NBS is perceived to have a moderately saturated light reddish-purple color when viewed against a middle gray surround. When it is viewed against a brilliant magenta surround it is seen as greenish gray. Spectrophotometric colorimetry does not account for the influence of surround.