Page 41 - Computational Colour Science Using MATLAB

28 COMPUTING CIE TRISTIMULUS VALUES
where E(l) is the relative spectral power distribution of an illuminant, x(l), y(l)
and z(l) are the colour-matching functions for the CIE 1931 or 1964 standard
observers, P(l) is the spectral reflectance of a surface and k is a normalizing
R
factor given by 100/ E(l)y(l)dl. The integration was originally specified to be
performed over the visible range of the electromagnetic spectrum between the
wavelengths 360 nm and 830 nm. Unfortunately, analytical expressions for the
colour-matching functions do not exist and so it is not possible in practice to
calculate the tristimulus values according to Equation (4.1). Furthermore, the
reflectance spectrum P(l) usually is measured at discrete intervals using
commercially available reflectance spectrophotometers and is therefore also
not available as an analytic expression. In 1986 the CIE adopted an alternative
practice for calculating tristimulus values based upon numerical integration
using wavelength intervals of 1 nm (CIE, 1986a). This leads to Equation (4.2)
where the summation is carried out over the visible range of wavelengths as
before:

830
X
X ¼ k EðlÞxðlÞPðlÞ,
360
830
X
Y ¼ k EðlÞyðlÞPðlÞ, ð4:2Þ
360
830
X
Z ¼ k EðlÞzðlÞPðlÞ.
360

The colour-matching functions (for both the 1931 and 1964 standard observers)
are provided with seven significant figures by the CIE in tabular form at 1-nm
intervals between the wavelengths 360 nm and 830 nm in CIE Publication
number S2 (CIE, 1986a). These are the official sets of colour-matching functions
recommended by the CIE. However, for most practical applications it is
suggested that an abridged set of colour-matching functions may be used at 5-nm
intervals over the range 380–780 nm and these are provided in CIE Publication
number 15.2 (CIE, 1986b).
The 1931 colour-matching functions are recommended whenever correlation
with visual colour matching of fields of angular subtense between approximately
18 and 48 at the eye of the observer is desired. For larger angular subtenses the
1964 colour-matching functions should be used.
The use of the 5-nm colour-matching functions requires that the spectral
reflectance data (for surfaces) be known at 5-nm intervals. For practical
applications, the required data are often not available in an appropriate format
because of abridgement (measurement at intervals greater than 5 nm) or
truncation (omission of the data at the spectral extremes). Many modern
reflectance spectrophotometers, for example, provide data at 10-nm intervals in
the range 400–700 nm. For situations where the spectral data are abridged or