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Section 3.3 Representing Color 78

tion for surface colors as well if we use a standard light for illuminating the surface
(and if the surfaces are equally clean, etc.).
Performing a matching experiment each time we wish to describe a color can
be practical. For example, this is the technique used by paint stores; you take in
a flake of paint, and they mix paint, adjusting the mixture until a color match is
obtained. Paint stores do this because complicated scattering effects within paints
mean that predicting the color of a mixture can be quite difficult. However, Grass-
man’s laws mean that mixtures of colored lights—at least those seen in a simple
display—mix linearly, which means that a much simpler procedure is available.

Color Matching Functions
When colors mix linearly, we can construct a simple algorithm to determine
which weights would be used to match a source of some known spectral energy
density given a fixed set of primaries. The spectral energy density of the source
can be thought of as a weighted sum of single wavelength sources. Because color
matching is linear, the combination of primaries that matches a weighted sum of
single wavelength sources is obtained by matching the primaries to each of the
single wavelength sources and then adding up these match weights.
For any set of primaries, P 1 , P 2 ,and P 3 , we can obtain a set of color matching
functions by experiment. We tune the weight of each primary to match a unit energy
source at every wavelength, and record a table of these weights against wavelengths.
These tables are the color matching functions, which we write as f 1 (λ), f 2 (λ), and
f 3 (λ). Now for some wavelength λ 0 ,we have

U(λ 0 )= f 1 (λ 0 )P 1 + f 2 (λ 0 )P 2 + f 3 (λ 0 )P 3
(i.e., f 1 , f 2 ,and f 3 give the weights required to match a unit energy source at that
wavelength).
We wish to choose the weights to match a source S(λ). This source is a sum
of a vast number of single wavelength sources, each with a different intensity. We
now match the primaries to each of the single wavelength sources and then add up
these match weights, obtaining

S(λ)= w 1 P 1 + w 2 P 2 + w 3 P 3

= f 1 (λ)S(λ)dλ P 1 + f 2 (λ)S(λ)dλ P 2 + f 3 (λ)S(λ)dλ P 3 .
Λ Λ Λ
General Issues for Linear Color Spaces
Linear color naming systems can be obtained by specifying primaries, which
imply color matching functions, or by specifying color matching functions, which
imply primaries. It is an inconvenient fact of life that, if the primaries are real lights,
at least one of the color matching functions is negative for some wavelengths. This
is not a violation of natural law; it just implies that subtractive matching is required
to match some lights, whatever set of primaries is used. It is a nuisance, though.
One way to avoid this problem is to specify color matching functions that are
everywhere positive (which guarantees that the primaries are imaginary because for
some wavelengths their spectral energy density is negative). Although this looks like