Page:Colorimetry104nime.djvu/49

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.

6.2. Photoelectric Tristimulus Colorimeters

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.

a. Angular Conditions

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].