curve of the cone is broader than that of the dilute photopigment.
As the optical density of the photopigment increases, so the sensitivity
curve of the cone becomes broader (Fig. 1). Two cones
expressing the same photopigment at different optical densities
will, therefore, have different spectral sensitivities, and comparison
of their output will yield a color signal. A cone with a higher optical
density will also be more sensitive across the spectrum (more
photopigment means more photoisomerisations). There are reports
that observers exist who gain color discrimination by such
a comparison: Neitz et al. (1999) describe protanomalous observers
who, according to genetic analysis, possess only two spectrally
distinct photopigments (M and S), but achieve trichromatic vision.
The authors suggest that the M photopigment is expressed at two
different optical densities, thus supporting a limited discrimination
in the red–green range.
In the literature, there is substantial variation in the values reported
for photopigment optical density (OD). It also remains a
matter of debate whether the optical density is usually equal for
the different cone classes of a given observer. Miller (1972) reports
ODs of 0.4–0.5 for the M cone in protanopes and 0.5–0.6 for the L
cone in deuteranopes. Smith and Pokorny (1973) place these values
at 0.3 and 0.4 respectively, while Burns and Elsner (1993) find
a greater disparity, with values of 0.27 and 0.48. Shevell and He
(1997) suggest that the OD of L may be higher than that of L0 in
the deuteranomalous observer. Berendschot, van de Kraats, and
van Norren (1996) report values of 0.39 for M and 0.42 for L in
dichromats. Renner et al. (2004) find no significant difference between
the OD of M and L: 0.66 and 0.65 respectively. The literature
does not, then, provide us with any consensus on the amount and
nature of OD variation among normal or anomalous trichromats. It
does, however, support the notion that such variation may exist,
and that the differences in OD between cone classes may be as
large as 0.2 or more. Multiple factors will underlie this variation
in OD, including the length of the cone outer segment, the concentration
at which photopigment is expressed, and the quantal
efficiency of the individual photopigment molecules (Penn &
Williams, 1986). The stability of the photopigment will also affect
the OD, and could be of particular importance in anomalous
trichromats whose ‘‘hybrid’’ photopigments may have reduced stability
(Williams et al., 1992). Several of these factors may vary over
time within an individual. For example, the density of photopigment
expression and the length of the rod outer segment are, in
part, determined by the ambient light levels in rats through a phenomenon
known as photostasis (Penn & Williams, 1986), and such
a mechanism may exist in humans (Beaulieu et al., 2009).
In this work we estimate the contribution that optical density
variation could make to the real-world color vision of anomalous
trichromats. In order to model this, we need to know the values
of two factors: first, the cone sensitivities of the theoretical anomalous
trichromat; second, the spectral composition of incident light
from each point in real-world scenes.
1.1. Cone spectral sensitivities
There is no exhaustive database of human cone sensitivities
that we can use to generate observers with photopigments of
any peak wavelength and expressed at any optical density. Fortunately,
it was noted by Dartnall (1953) that although photopigments
vary in their wavelength of peak sensitivity (kmax), they
retain the same fundamental shape. He described this shape with
a nomogram of sensitivity plotted against 1/k 1/kmax. Ebrey and
Honig (1977) noted that the bandwidth of the sensitivity curves
varied with kmax, and introduced three separate nomograms to
cover different parts of the spectrum. Mansfield (1985) found that
description of a template on a normalized frequency axis allowed
the return to a single template to cover the entire spectrum. Following
this realization, a number of generalized templates have
been developed (Baylor, Nunn, & Schnapf, 1987; Govardovskii
et al., 2000; Lamb, 1995).
In this work, we use the Lamb (1995) template to define sensitivity
spectra for photopigments of any given kmax (in earlier work,
we used the Baylor, Nunn, and Schnapf (1987) template, which
gave very similar results). Lamb (1995) validated his template
against data from eight psychophysical and electrophysiological
studies on human, bovine, monkey, and squirrel subjects. Having
generated the photopigment spectrum, we correct it for a given
optical density. Thus, we can produce cone sensitivity triplets for
all theoretical deuteranomalous and protanomalous observers
(i.e. all combinations of wavelength of peak sensitivity and photopigment
optical density).
1.2. The spectral composition of real world scenes
We use the hyperspectral images of Foster, Nascimento, and
Amano (2004) and Nascimento, Ferreira, and Foster (2002). To construct
these images, multiple photographs were taken of the same
scene through narrowband filters centered on different wavelengths.
In this way, the spectral flux from each point could be
determined. The technique amounts to spectroradiometry with
preservation of spatial information.
With knowledge of the cone spectral sensitivities and of the
spectral reflectances of real world scenes, we can calculate the
cone excitations produced in any observer by any of our scenes under
any illuminant. We then use simple metrics to estimate the impact
of small changes in peak sensitivity and optical density on the
gamut of colors potentially available to the observer, and in doing
so we assess the relative importance of peak separation and optical
density to the color vision of the anomalous trichromat.