,
and 
. If the CMF is negative, the
primary light in question must be added to the test field to complete the match. The trichromacy of individuals with normal color
vision is evident in their ability to match any test light to a mixture of three
"primary" lights. The relative intensities of the primary lights required to
match equal energy test lights, l, are referred to as
the red, green and blue color matching functions (CMFs),
respectively, and written ,
and . If the CMF is negative, the
primary light in question must be added to the test field to complete the match.
CMFs can be linearly transformed to other sets of
real and imaginary primary lights, such as the X, Y and Z primaries favored by the CIE, or
the L, M and S cone fundamental primaries that underlie all trichromatic color matches.
Each transformation is accomplished by multiplying the CMFs by a 3x3 matrix. The goal is
to determine the unknown 3x3 matrix that will transform the CMFs, 
, 
and 
, to the three cone spectral sensitivities, 
, 
and 
.
Color matches are determined at the cone level. When matched, the test and mixture fields appear identical to all three cone classes. Thus, for matched fields, the following relationships apply:
[1] 
where 
, 
and 
are, respectively, the L-cone sensitivities to the R, G and
B primary lights; and similarly 
, 
and 
are the M-cone sensitivities
to the primary lights; and 
, 
and 
are the S-cone sensitivities.
We know 
, 
and 
, and
we assume, for the red R primary, that 
is effectively zero, since the S-cones are insensitive to
long-wavelength lights. (The intensity of the spectral light l,
which is also known, is equal in energy units throughout the spectrum, and so is
discounted from the above equations.)
There are, therefore, only eight unknowns required for the linear transformation:
[2] 
.
However, because we are usually unconcerned about
the absolute sizes of 
, 
and 
, the eight unknowns collapse
to just five:
[3] 
,
where the absolute values of 
(or 
), 
(or 
), and 
(or 
) remain unknown, but are
typically chosen to scale three functions in some way: for example, so that 
, 
and 
peak at unity. In the
well-known solution of Eqn [3] by Smith & Pokorny (1975), 
sum to 
, the luminosity function.
Equations [1] to [3] (and [4] and [5], below) could be for an
equal-energy or an equal-quanta spectrum. Since the CMFs are invariably tabulated for test
lights of equal energy, most workers, use an equal-energy spectrum to define the unknowns
in the equations and to calculate the cone spectral sensitivities from the CMFs. They then
convert the relative cone spectral sensitivities from energy to quantal sensitivities (by
multiplying by 
).
ReferencesHelmholtz, H. (1866). Handbuch der Physiologischen Optik, 1st ed. Leipzig: Voss.
König, A. & Dieterici, C. (1886). Die Grundempfindungen und ihre Intensitäts-Vertheilung im Spektrum. Siz. Akad. Wiss. Berlin, 1886, 805-829.
König, A. & Dieterici, C. (1893). Die Grundempfindungen in normalen und anomalen Farbensystemen und ihre Intensitätsverteilung im Spektrum. Z. Psychol. Physiol. Sinnesorg. 4, 241-347.
Smith, V. C. & Pokorny, J. (1975). Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm. Vision Research, 15, 161-171.
Stockman, A., & Sharpe, L. T. (1999). Cone spectral sensitivities and color matching. In K. Gegenfurtner & L. T. Sharpe (Eds.), Color vision: from genes to perception (pp. 53-87) Cambridge: Cambridge University Press.
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