Page 47 - Computational Colour Science Using MATLAB

34 COMPUTING CIE TRISTIMULUS VALUES
recommended by the CIE is to extend the reflectance data by using the most
extreme value as an estimate of all values beyond that extreme (CIE, 1986b). So,
for example, if the calculation is being carried out at 10-nm intervals and the
reflectance data are in the range 400–700 nm, then values of reflectance at 710,
720, . . ., 780 nm are set equal to the value at 700 nm. A similar procedure applies
to the shorter wavelength. Although it could be suggested that more accurate
extrapolation methods could be employed it should be remembered that
extrapolation is far more dangerous than interpolation. Also, the fact that the
colour-matching functions have very small values below 400 nm and above
700 nm means that the errors that result from the CIE method generally are very
small and the risk of using sophisticated extrapolation techniques is not justified.

4.5 Tables of weights

Some practitioners prefer to use weighting tables where the terms E(l)x(l),
E(l)y(l) and E(l)z(l), as used in Equation (4.2), are pre-computed at each
wavelength interval. These weighting tables can be computed from the CIE
colour-matching functions and illuminants. The benefit to the user in using these
tables is that Equation (4.2) can be replaced by Equation (4.6),

P
W x ðlÞPðlÞ,
X ¼
W y ðlÞPðlÞ,
P
Y ¼ ð4:6Þ
P
W z ðlÞPðlÞ,
Z ¼
where the weight vectors W , W and W also include the normalizing constant k
x y z
from Equation (4.2). The CIE recommends that such tables of weighting factors
should be provided for the full range of wavelengths, 360–830 nm, so that they
may be used for any degree of truncation by adding the weights at the
unmeasured wavelengths to those at the extreme measured wavelengths.
A set of useful weights is provided by the American Society for Testing and
Materials in E308-01 (ASTM, 2001). The E308-01 tables are provided only for
the range of wavelengths 360–780 nm but are suitable for most practical
applications. They are provided at 10- and 20-nm intervals. The fact that the
E308-01 tables are abridged to intervals of 10 and 20 nm has resulted in them
probably being the most widely used method for computing tristimulus values
since the 10-nm data, in particular, are suitable for direct use with reflectance
data obtained from most reflectance spectrophotometers without interpolation.
The ASTM publication provides the data in two main tables: ASTM Table 5
should be used with reflectance data that have been corrected for the spectral
bandpass of the instrument whereas ASTM Table 6 has the spectral-bandpass
correction built in and should be used with reflectance data that have not been