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538 21. Color
ent amounts of each. The amount of each required to match a given wavelength λ
is encoded in color matching functions, given by ¯r(λ), ¯g(λ),and
¯
b(λ) and plotted
in Figure 21.4. Tristimulus values associated with these color matching functions
are termed R, G,andB.
Given that we are adding light, and light cannot be negative, you may have
noticed an anomaly in Figure 21.4: to create a match for some wavelengths, it
is necessary to subtract light. Although there is no such thing as negative light,
we can use Grassmann’s laws once more, and instead of subtracting light from
the mixture of primaries, we can add the same amount of light to the color that is
being matched.
The CIE ¯r(λ), ¯g(λ),and
¯
b(λ) color matching functions allow us to determine
if a spectral distribution Φ
1
matches a second spectral distribution Φ
2
by simply
comparing the resulting tristimulus values obtained by integrating with these color
matching functions:
λ
Φ
1
(λ)¯r(λ)=
λ
Φ
2
(λ)¯r(λ),
λ
Φ
1
(λ)¯g(λ)=
λ
Φ
2
(λ)¯g(λ),
λ
Φ
1
(λ)
¯
b(λ)=
λ
Φ
2
(λ)
¯
b(λ).
Of course, a color match is only guaranteed if all three tristimulus values match.
The importance of these color matching functions lies in the fact that we are
now able to communicate and describe colors compactly by means of tristimulus
values. For a given spectral function, the CIE color matching functions provide a
precise way in which to calculate tristimulus values. As long as everybody uses
the same color matching functions, it should always be possible to generate a
match.
If the same color matching functions are not available, then it is possible to
transform one set of tristimulus values into a different set of tristimulus values
appropriate for a corresponding set of primaries. The CIE has defined one such
a transform for two specific reasons. First, in the 1930s numerical integrations
were difficult to perform, and even more so for functions that can be both posi-
tive and negative. Second, the CIE had already developed the photopic luminance
response function, CIE V (λ). It became desirable to have three integrating func-
tions, of which V (λ) is one and all three being positive over the visible range.
To create a set of positive color matching functions, it is necessary to define
imaginary primaries. In other words, to reproduce any color in the visible spec-
trum, we need light sources that cannot be physically realized. The color match-
ing functions that were settled upon by the CIE are named ¯x(λ), ¯y(λ),and¯z(λ)