direct brightness matching, a photopic brightness-matching function is likely to be more appropriate than a luminance function. Brightness matching, however, does not obey Abney’s law (see the section on psychophysical measures of luminous efficiency).

In an attempt to overcome the additivity failures that are inherent in the use of the direct brightness matching method, He and coworkers used a binocular synchronicity method (which they referred to as a reaction time difference method) to measure mesopic visual performance [90, 94]. Although promising, binocular synchronicity measures must be influenced by the complex changes in rod-cone delay that accompany changes in adaptation level (see [50]), as well as by the changes in delay caused by changes in the relative rod and cone contributions to the detection of the two flashes, neither of which are likely to be simple. An obvious complication, given that the adapting field wavelength in one eye is varied from long to short wavelengths, is that the luminous efficiency will be distorted by the additional suppression of the rods by the cones, which are excited more by long-wavelength background fields.

International Standard

Given the inherent complexity of defining mesopic luminous efficiency, no international standard is currently available. Any practical model of mesopic luminous efficiency will have to incorporate the effects of adaptation, spectral composition, spatial frequency, temporal frequency, retinal location, and retinal area. Moreover, it is likely to be nonadditive. If accuracy is essential, then the only consistently reliable way of estimating mesopic luminous efficiency is to measure it for each new application and stimulus condition.

INDIVIDUAL DIFFERENCES INFLUENCING LUMINOUS EFFICIENCY

All luminous efficiency functions vary between observers because of individual differences in the spectral filtering by the ocular media (predominantly by the crystalline lens), the macular pigment, and possibly other as yet unidentified prereceptoral, intraretinal pigments. Other significant individual differences that affect photopic and mesopic luminous efficiency functions include shifts in the spectral positions of the L- and M-cone photopigments owing to genetically encoded polymorphic variants (for reviews, see [95, 96]); regional variations in the optical densities of the cone photopigments; large variations in the relative numbers of L and M cones in the retina; and variations in the contribution of chromatic channels to luminosity (see, e.g., [97]).

Attenuation of Spectral Light by the Lens and Other Ocular Media

Light to all parts of the retina has to pass through the same anterior eye medium, which absorbs greatest at very short wavelengths. Stockman, Sharpe, and Fach [98] proposed a slightly adjusted version of the mean lens density spectrum of van Norren and Vos [99], which is shown in Fig. 5. The main attenuation is caused by the yellowish pigmentation of the crystalline lens. The average density at 400 nm for an average standard observer of about 30 years of age is 1.76 log unit. However, the value varies between individuals by a factor of at least ±25% of the mean density in young observers (<30 years old) [22, 98–100]. Because lens density increases (becomes progressively more yellow) with the

age of the observer (e.g., [31, 101, 102]), the variability in the general population is even larger. As a consequence, photopic luminous efficiency functions, measured by different techniques (including HFP and HBM), reveal gradual decreases in average sensitivity at short wavelengths, between about 420 and 560 nm, with increasing age, consistent with age-related increases in the density of the ocular media [103, 104]. The short-wave- length decline in sensitivity is lower in magnitude for functions based on HBM than for those based on HFP [103]. In contrast, infants, because they tend to have very clear optic media, tend to show an elevation in efficiency at short wavelengths [105].

A two-factor model has been proposed to estimate the change in the density of the

optic media dlens(λ) with age [102, 106]. The density can be separated into two components: dlens1(λ), which represents the portion affected by aging after age 20, and dlens2(λ), which represents the portion stable after age 20.

Thus, the optical density of the lens of an average observer between the ages of 20 and 60 can be estimated as

where A is the observer’s age. Tables for determining the values of dlens1(λ) and dlens2(λ) as a function of wavelength are provided by the CIE [107].

Likewise, that of an average observer over the age of 60, can be estimated as

The influence of the change in the optical density of the lens pigment dlens(λ) with aging on luminous efficiency can then be approximately compensated for by appropri-

ately adjusting the lens density multiplier klens in Eq. 6 after calculating the age-specific value for 400 nm from the appropriate Eq. 4 or 5:

the optical density of the macular pigment, kmac is the macular density multiplier, and c is simply a unity-normalizing constant (for details, see [47]). The lens pigment density multiplier klens is adjusted to increase or decrease the mean standard observer values—dlens (λ) = 1.48 at 400 nm—to coincide with the actual (age-relevant) optical density of the individual observer.

Attenuation of Spectral Light by the Macular Pigment

The macular pigment absorbs light mainly of short wavelengths with a peak optical density around 460 nm (see Fig. 5, which is a mean macular density spectrum proposed by [23]). The absorbance spectrum of the macular pigment, like that of the optic media, does not vary between observers. However, individual differences in its density dmac(λ) can be very large, with a range of peak density from 0.0 to about 1.2 at 460 nm [30, 108, 109]. In addition, the optical density of the macular pigment diminishes with retinal location, tending to become more transparent with eccentricity and being wholly or largely absent

by a retinal eccentricity of 10° (e.g., [110]). Thus, the effective screening of the macular pigment is much less for large centrally viewed fields than for small ones. Representative averages of the maximum optical density at 460 nm are 0.35 for a 2° diameter centrally fixated visual view and 0.095 for a 10° diameter one (for a review, [23]).

If the field size is known, the effective optical density dmac(λ), at its maximum value of 460 nm, can be calculated by an exponential formula [111]:

in which s is the field size in degrees.

The influence of the change in the effective optical density of the macular pigment with field size on luminous efficiency can be approximately compensated for in the V*(λ) photopic luminosity function by appropriately adjusting the macular density dmac(λ) by the macular density multiplier kmac in Eq. 6, after calculating the relevant field-size density value at 460 nm from Eq. 7.

Only a few studies have investigated how the absorption by the macular pigment depends on age [110, 112–116]. None has shown a strong or significant relationship (see [116]). For more information, see [107].

Optical Densities of the Photopigments

The axial optical density of the photopigment in the receptor outer segment, where the photopigment molecules are stacked, varies considerably between individuals (e.g., [117–124]). In addition, for the cones, but not for the rods, it decreases significantly with retinal eccentricity owing to morphological changes in the cone photoreceptors: The peripheral cones are squatter than the foveal ones, with correspondingly shorter outer segments. Changes in axial optical density do not affect the pigment peak sensitivity, but they do change its spectral absorbance (and hence spectral sensitivity), with consequences for luminous efficiency. For instance, peripheral cones, which have low optical densities relative to those of central foveal cones, have narrower spectral sensitivity curves. Thus, photopic luminous efficiencies measured in the peripheral retina or with participation of the peripheral cones will be relatively less sensitive at the spectral extremes than those measured foveally, all other factors being equal.

At present, there is no truly reliable way of estimating or correcting for photopigment optical density differences between individuals. But, the dependence on field size of the

cone pigment optical density dpigment(max)—it tends to decrease on average as centrally viewed fields increase in size—can be roughly described by an exponential function

with an asymptotic value [125]. Some evidence suggests different maximal optical density values for the S cone than for the L and M cones (see [22, 98]):

in which s is the field size in degrees. These formulas lead to values of 0.50 and 0.38 at, respectively, 2° and 10° for the L and M cones and of 0.40 and 0.30 at, respectively, 2° and 10° for the S cones [107].

Changes in the optical densities of the cone photopigments on photopic luminous efficiency can only be compensated for by adjusting the optical densities of the L- and M-cone spectral sensitivities themselves (see Chapter 14 on human cone spectral sensitivities and color vision deficiencies) and then recalculating the V*(λ) from Eq. 2. Additional information on how to do this is provided by a CIE technical report [107].

There is some indication that the peak optical densities of the visual pigments in the central visual field decrease gradually as a function of age [116, 126].

Relative Numbers of L and M Cones

Photopic luminous efficiency may be affected by the relative numbers of the L and M cones (L:M cone ratio) in the normal human retina. The ratio varies greatly between individuals, with estimates based on the various techniques ranging at least from 1:3 to 19:1 [127–135]. The normal or typical mean L:M cone ratio is believed to be close 2:1 [44, 128, 129, 131–133].

Several research groups have used luminous efficiency functions as a way of estimating the relative number of L and M cones in the retinal area within which it is measured (e.g., [8, 127, 128, 131, 136–143]). The assumption underlying such estimates is that the L-cone weight (i.e., a in Eq. 2) directly reflects the relative numbers of the L and M cones contributing to luminous efficiency. This assumption is, however, questionable because the outputs of each cone type are modified by receptoral adaptation and by postreceptoral adaptation before their signals are combined postreceptorally. Thus, a could, in principle, have little or nothing to do with relative L- and M-cone numbers but instead reflect the relative L- and M-cone contrast gains. This extreme view is unlikely given that L:M cone ratio estimates derived from luminous efficiency functions correlate with estimates derived in the same subjects using other methods (e.g., [131, 133, 137, 140, 141, 144–146]). It does, therefore, seem likely that some of the differences in luminous efficiency functions between individuals is caused by the large individual differences in L:M cone ratio. Nevertheless, to whatever extent a actually corresponds to relative cone numbers, it is also strongly affected by chromatic adaptation (see the section on additive functions for 2° viewing fields and [47].

Cone Pigment Polymorphisms

The derivation of photopic luminous efficiency functions is complicated by polymorphisms in the normal population, the most common of which is the frequent replacement of serine by alanine at codon 180 in exon 3 of the X-chromosome-linked opsin gene. Approximately 56% of a large sample of 304 Caucasian males with normal and deutan color vision had the serine variant [identified as L(ser180)] and 44% the alanine variant [identified as L(ala180)] for their L-cone gene (summarized in Table 1 of [23]). In contrast, in the M-cone pigment, the ala180/ser180 polymorphism is much less frequent, 93–94% of males having the ala180 variant [147, 148]. The substitution of alanine for serine causes a shift of about 2.7 nm (see [149]) to shorter wavelengths of the spectral sensitivity of the L cones. This will cause those individuals with the L(ser180) genotype to have a slightly higher luminous efficiency to long wavelengths than those with the L(ala180) genotype.

If the individual observer’s genotype is known, the influence of the L-cone polymorphism on photopic luminous efficiency for the V*(λ) (see Eq. 2) can be compensated for by replacing the L-cone [l-(λ)] spectral sensitivity values, which are population-corrected