Cone spectral sensitivities and colour matching
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 of wavelength λ are referred to as the red, green and blue colour 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
.
Colour 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:
,
,
and
,
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 of
wavelength, λ,
which is also known, is set to be equal in energy or quantal units throughout
the spectrum, and so is discounted from the above equations.)
There are, therefore, only eight unknowns required for the linear transformation:
.
Because we are usually unconcerned about
the absolute sizes of ,
and
,
the eight unknowns collapse to just five:
,
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 one example (Smith
& Pokorny, 1975),
sum to
,
the luminosity function.
The above equations 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 λ-1).
References
Helmholtz, 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.
Young, T. (1807). Lectures on Natural Philosophy. London: Johnson, Vol. II.