We investigated the dependence of luminous efficiency on background chromaticity by measuring 25-Hz heterochromatic flicker photometry (HFP) matches in six genotyped male observers on 21 different 1000-photopic-troland adapting fields: 14 spectral ones ranging from 430 to 670 nm and 7 bichromatic mixtures of 478 and 577 nm that varied in luminance ratio. Each function was analyzed in terms of the best-fitting linear combination of the long- (L) and middle- (M) wavelength sensitive cone fundamentals of A. Stockman and L. T. Sharpe (2000). Taking into account the adapting effects of both the backgrounds and the targets, we found that luminous efficiency between 603 and 535 nm could be predicted by a simple model in which the relative L- and M-cone weights are inversely proportional to the mean cone excitations produced in each cone type multiplied by a single factor, which was roughly independent of background wavelength (and may reflect relative L:M cone numerosity). On backgrounds shorter than 535 nm and longer than 603 nm, the M-cone contribution to luminous efficiency falls short of the proportionality prediction but most likely for different reasons in the two spectral regions.

*V*(

*λ*), for photopic (cone) vision, which has since become the standard for business and industry. In visual science,

*V*(

*λ*) or its variants has been assumed to correspond to the spectral sensitivity of a hypothetical human postreceptoral channel, the so-called “luminance” channel with additive inputs from the long-wavelength sensitive (L-) and middle-wavelength sensitive (M-) cones (e.g., Smith & Pokorny, 1975). The widespread acceptance of these functions, however, downplays the many difficulties inherent in their derivation and application (see, for discussion Sharpe, Stockman, Jagla, & Jägle, 2005).

*V**(

*λ*), based exclusively on HFP measurements made in 40 genotyped observers (Sharpe et al., 2005). The function was obtained under neutral adaptation that corresponded to daylight D

_{65}adaptation and is defined as a linear combination of the Stockman and Sharpe (2000) L- and M- cone fundamentals. The specification of a fixed luminous efficiency function for neutral adaptation, and its definition in terms of standardized L- and M-cone spectral sensitivities, is important for practical photometry and for modeling visual performance under neutral or achromatic conditions. In terms of modeling performance under “real world” conditions, however, it is a gross oversimplification. Photopic luminous efficiency functions are not fixed in spectral sensitivity but change with chromatic adaptation (e.g., De Vries, 1948b; Eisner, 1982; Eisner & MacLeod, 1981; Ikeda & Urakubo, 1968; King-Smith & Webb, 1974; Marks & Bornstein, 1973; Stockman, MacLeod, & Vivien, 1993; Stromeyer, Chaparro, Tolias, & Kronauer, 1997; Stromeyer, Cole, & Kronauer, 1987; Swanson, 1993). Thus, any luminous efficiency function strictly applies only to the limited conditions of chromatic adaptation under which it was measured. Luminous efficiency is therefore quite distinct from color matching functions (CMFs) or cone fundamentals, which do not change in spectral sensitivity (unless the photopigment is bleached at high intensity levels). Consequently, the correspondence between luminous efficiency and the CIE

*λ*) CMF is largely a contrivance: the former

*should*change with chromatic adaptation, but the latter does not.

*approximately*correct for conditions under which Weber's Law applies independently for each cone signal (see Discussion section).

^{180}/ser

^{180}) photopigment polymorphism at amino-acid position 180 in the L-cone photopigment gene: S1, S2, S3, and S4 have L(ser

^{180}), while A1 and A2 have L(ala

^{180}). Their ages ranged between 18 and 53 years.

^{180})] and 44% the alanine variant [identified as L(ala

^{180})] for their L-cone gene (summarized in Table 1 of Stockman & Sharpe, 2000). In contrast, in the M-cone pigment, the ala

^{180}/ser

^{180}polymorphism is much less frequent, 93%–94% of males having the ala

^{180}variant (Neitz & Neitz, 1998; Winderickx, Battisti, Hibiya, Motulsky, & Deeb, 1993). Therefore, we only identified the genotype with respect to the ser

^{180}/ala

^{180}polymorphism in the first (L-cone) photopigment gene in the array of our observers. Given that we used 6 subjects, there is about a 33% chance that one of them will have the ser

^{180}M-cone variant. This variant would cause a modest red shift in the luminous efficiency function and a corresponding increase in the estimated L-cone weight. The genotype was first determined by amplification, using total genomic DNA, of exon 3 followed by digestion with Fnu4H as previously described (Deeb, Hayashi, Winderickx, & Yamaguchi, 2000).

*in situ*at the plane of the observer's pupil with a silicon photodiode (Model SS0-PD50-6-BNC, Gigahertz-Optics, München, Germany), which was calibrated against the German National Standard and a picoammeter (Model 486, Keithley, Germering, Germany). The fixed and variable neutral density filters were calibrated

*in situ*for all test and field wavelengths. Particular care was taken in calibrating the monochromators and interference filters: a spectroradiometer (Compact Array Spectrometer CAS-140, Instrument Systems, München, Germany), with a spectral resolution better than 0.2 nm, was used to measure the center wavelength and the bandpass (full-width at half-maximum, FWHM) at each wavelength. The absolute wavelength accuracy was better than 0.2 nm, whereas the resolution of the wavelength settings was better than 0.15 nm (Sharpe, Stockman, Jägle et al., 1998). The wavelengths of the two CVI monochromators were additionally calibrated against a low-pressure mercury source (Model 6035, L.O.T.-Oriel, Darmstadt, Germany).

*V*

_{ μ}*(

*λ*). The symbol

*μ*in this context refers to the background chromaticity (which can be defined in the model as the monochromatic wavelength that produces the same ratio of M:L-cone excitations as the actual background). The previously published

*V**(

*λ*) function (Sharpe et al., 2005) is a special case of

*V*

_{μ}*(

*λ*), which we now refer to as

*V*

_{D65}*(

*λ*).

*M*

_{ μ}/

*L*

_{ μ}). Ideally, the mean chromaticity of the combined background, target, and reference lights at the HFP null (

*M*

_{ μ,λ}/

*L*

_{ μ,λ}) should deviate very little from the dashed lines. However, as the symbols show for subject S1, the mean chromaticity at each null (plotted as

*M*

_{ μ,λ}/

*L*

_{ μ,λ}) can diverge substantially from the chromaticity of the background alone. Overall, the target and reference lights shift the mean chromaticity of both short- and long-wavelength fields:

*M*

_{ μ,λ}/

*L*

_{ μ,λ}is increased on short-wavelength fields and decreased on long-wavelength ones. Moreover,

*M*

_{ μ,λ}/

*L*

_{ μ,λ}also changes with target wavelength (

*λ*):

*M*

_{ μ,λ}/

*L*

_{ μ,λ}is decreased by short-wavelength targets and increased by long-wavelength ones. This dependence of adapting field chromaticity on the test wavelength occurs on all backgrounds. Comparable effects can be seen for all six subjects in Figure 4, and Figures A1– A6 in 1, in which the

*M*

_{ μ,λ}/

*L*

_{ μ,λ}ratios produced by the background lights alone are shown by the vertical colored lines, and the combined values at each HFP null are shown by the symbols.

*M*

_{μ,λ}/

*L*

_{μ,λ}in Equations 2–5, below) calculated for

*each*HFP setting.

*M*

_{ μ,λ}/

*L*

_{ μ,λ}value for

*each*HFP match. The matches for the other five subjects are shown in Figures A1– A6 in 1. The matches were made in one run, except for observer S1, who made the bichromatic matches twice, and S2, who made the bichromatic and spectral matches twice, and observer A2, who made no bichromatic matches. The spectral matches and bichromatic mixture matches are shown in the upper and middle panels, respectively, in each figure, with the exception of those for observer S2, whose more extensive data span two figures: his first runs are shown in Appendix Figure A1, and his second in Appendix Figure A2.

*M*

_{ μ,λ}/

*L*

_{ μ,λ}instead of as a conventional function of target wavelength,

*λ*. This unusual scheme reveals the effects of the test and reference lights on the adapting chromaticity. Each spectral sensitivity curve, determined for the different spectral and bichromatic mixture fields, comprises the data for target wavelengths from 420 to 680 nm. Notice that the spectrum is effectively reversed in these plots. Long-wavelength backgrounds and targets produce lower

*M*

_{ μ,λ}/

*L*

_{ μ,λ}values and therefore plot to the left, whereas shorter wavelength backgrounds and targets produce higher values and plot to the right. Consequently, the target wavelengths for each adapting condition plot from right to left.

*μ*and

*λ,*we fit linear combinations of the Stockman and Sharpe (2000) L- and M-cone spectral sensitivities for each adapting condition (

*μ*) not just as a function of target wavelength (

*λ*) but also as a function of the mean adapting chromaticity at each HFP null (

*M*

_{μ,λ}/

*L*

_{μ,λ}). We began with

*a*

_{μ}is the L-cone weight,

*V*

_{μ}*(

*λ*) is the predicted spectral sensitivity (luminous efficiency) function for an adapting field

*μ,*

*λ*) is either the L(ser

^{180}) or the L(ala

^{180}) variant of the Stockman and Sharpe (2000) L-cone fundamentals, and

*λ*) is the Stockman and Sharpe (2000) M-cone spectral sensitivity. The spectral sensitivity functions, which in these formulae are quantum-based, are given in Appendix Table A1. We then set

*a*

_{μ}=

*β*

_{μ}× (

*M*

_{μ,λ}/

*L*

_{μ,λ}), where

*β*

_{μ}is now the “L-cone bias,” and (to reiterate)

*M*

_{μ,λ}/

*L*

_{μ,λ}is the relative M:L-cone excitation produced by the combined test, reference, and background lights at each HFP null. (We ignore the possibility that the M- and L-cone incremental test components might have a greater or lesser influence than expected, because of spatial opponency or spatial integration with respect to the surrounding background.) The excitation ratios were calculated from the radiometric calibrations and null settings using the Stockman & Sharpe cone fundamentals with unity quantal peaks. Next, the sensitivity for each

*μ, λ,*and

*M*

_{μ,λ}/

*L*

_{μ,λ}is normalized relative to the reference wavelength of 560 nm (recall that

*M*

_{μ,λ}/

*L*

_{μ,λ}

*includes*the effects of the fixed 560-nm reference), so that

*λ*and 560 nm for each combination of background, test, and reference lights at an HFP null.

*a*

_{ μ}=

*β*

_{ μ}× (

*M*

_{ μ,λ}/

*L*

_{ μ,λ}) is illustrated by rearranging ( Equation 2) to yield

*β*

_{ μ}, is found to be constant over a range of

*M*

_{ μ,λ}/

*L*

_{ μ,λ}, it suggests that Weber's Law holds over that range. An increase in

*β*

_{ μ}implies a decrease of the M-cone contribution relative to the Weber's Law prediction, whereas a decrease in

*β*

_{ μ}implies a decrease of the L-cone contribution relative to the Weber's Law prediction. Increases or decreases in

*β*

_{ μ}can be potentially due to a variety of causes including relative chromatic suppression or facilitation, or shortfalls from Weber's Law (see above).

*β*

_{ μ}for each spectral and bichromatic spectral sensitivity condition (

*μ*) by simultaneously fitting all the data for each subject with

*k*

_{ lens}and

*k*

_{ mac}are, respectively, best-fitting lens and macular pigment density multipliers that adjust each subject's HFP curves to be consistent with the mean lens [

*d*

_{ lens}(

*λ*)] and macular [

*d*

_{ mac}(

*λ*)] pigment density spectra implied by the Stockman and Sharpe standard observer. The function

*d*

_{ lens}(

*λ*) is the lens pigment density spectrum of Norren and Vos (1974), slightly modified by Stockman, Sharpe, and Fach (1999) [

*d*

_{lens}(400), for example, is 1.76], while the function

*d*

_{mac}(

*λ*) is the mean macular density spectrum based on measurements by Bone, Landrum, and Cairns (1992) proposed by Stockman et al. (1999) [

*d*

_{mac}(460), for example, is 0.35]. For further details about the fitting procedure, see our earlier papers (Sharpe et al., 2005; Stockman & Sharpe, 2000).

^{180}) and L(ala

^{180}) cone templates are based on the Stockman and Sharpe (2000) L-cone fundamental calculated back to an absorbance spectrum (see their Table 2, Column 9), and then shifted along a logarithmic wavelength scale by −1.51 nm at

*λ*

_{max}for L(ala

^{180}) or by +1.19 nm for L(ser

^{180}) in accordance with the 2.7-nm spectral shift between the L(ala

^{180}) and L(ser

^{180}) spectral sensitivities (Sharpe, Stockman, Jägle et al., 1998). Eisner and MacLeod (1981) also found a 2.7-nm shift between two groups of alleged L-cone isolates but of unknown genotype. The two respective shifted spectra were then corrected back to corneal spectral sensitivities to generate the corneal templates used in the fits. For further details, see Stockman and Sharpe (2000). These two templates, which have not been published before, are provided in 5-nm steps in Table A1 in 1. The use of the appropriate version of the L-cone template is important for avoiding sizable errors that can arise in estimating the relative L-cone weights with the mean L-cone template (e.g., Bieber, Kraft, & Werner, 1998; Carroll, McMahon, Neitz, & Neitz, 2000; Sharpe et al., 2005). The functions in Equation 4 can be downloaded from the website: http://www.cvrl.org.

*β*

_{ μ}for each

*μ*. Second, we assumed that

*β*

_{ μ}varied as some function of

*μ*or was constant (Weber's Law) and determined the optimal fit for all

*μ*.

*β*

_{ μ}for each

*μ*

*β*

_{ μ}were found for

*each*adapting condition,

*μ,*and the best-fitting values of

*k*

_{ lens}and

*k*

_{ mac}were found for each subject for

*all μ*. The fits to the luminous efficiency data are shown in the upper and middle panels of Figure 4 and Appendix Figures A1– A6 as the small symbols in each panel, and the residuals are shown in the lower panels. Overall, the fits are good with

*R*

^{2}values of better than 99% (see Table 1), but some systematic deviations are apparent. This suggests that the model of 25-Hz HFP-determined luminous efficiency embodied in Equation 4, which combines only L- and M-cone signals, is essentially correct.

Serine 180 | Alanine 180 | |||||
---|---|---|---|---|---|---|

S1 | S2 | S3 | S4 | A1 | A2 | |

(i) Separate determinations of β _{ μ} for each μ | ||||||

k _{ lens} | 0.03 ± 0.01 | −0.29 ± 0.02 | 0.07 ± 0.02 | 0.31 ± 0.02 | 0.22 ± 0.02 | −0.06 ± 0.03 |

k _{ mac} | −0.14 ± 0.02 | −0.53 ± 0.02 | −0.29 ± 0.03 | −0.39 ± 0.02 | −0.47 ± 0.02 | −0.73 ± 0.04 |

rms error | 0.034 | 0.038 | 0.049 | 0.041 | 0.043 | 0.049 |

R ^{2} | 99.70 | 99.66 | 99.31 | 99.50 | 99.50 | 99.14 |

(ii) β _{ μ} constant (Weber's Law) | ||||||

β _{ μ} | 1.44 ± 0.06 | 2.12 ± 0.06 | 2.59 ± 0.16 | 2.46 ± 0.11 | 1.82 ± 0.08 | 1.93 ± 0.12 |

k _{ lens} | 0.03 ± 0.01 | −0.29 ± 0.01 | 0.06 ± 0.02 | 0.31 ± 0.02 | 0.21 ± 0.02 | −0.06 ± 0.03 |

k _{ mac} | −0.13 ± 0.02 | −0.54 ± 0.02 | −0.30 ± 0.03 | −0.39 ± 0.02 | −0.48 ± 0.03 | −0.73 ± 0.04 |

rms error | 0.038 | 0.042 | 0.055 | 0.043 | 0.049 | 0.050 |

R ^{2} | 99.58 | 99.58 | 99.12 | 99.44 | 99.36 | 99.38 |

(iii) β _{ μ} threshold power model | ||||||

n | 0.84 ± 0.24 | 1.92 ± 0.08 | 2.37 ± 0.16 | 2.41 ± 0.12 | 1.67 ± 0.08 | 1.90 ± 0.13 |

b | 1.21 ± 0.55 | 4.71 ± 1.44 | 9.90 ± 4.38 | 11.13 ± 7.88 | 1.24 ± 3.67 | 1.13 ± 25.90 |

c | 1.08 ± 0.52 | 1.64 ± 0.09 | 1.46 ± 0.07 | 1.64 ± 0.11 | 1.36 ± 0.04 | 1.81 ± 0.56 |

k _{ lens} | 0.03 ± 0.01 | −0.29 ± 0.01 | 0.06 ± 0.02 | 0.31 ± 0.02 | 0.21 ± 0.02 | −0.06 ± 0.03 |

k _{ mac} | −0.09 ± 0.02 | −0.52 ± 0.02 | −0.28 ± 0.03 | −0.37 ± 0.02 | −0.44 ± 0.02 | −0.72 ± 0.04 |

rms error | 0.035 | 0.041 | 0.053 | 0.043 | 0.045 | 0.050 |

R ^{2} | 99.66 | 99.61 | 99.20 | 99.46 | 99.46 | 99.39 |

*β*

_{ μ}, for each subject, plotted as a function of the mean

*M*

_{ μ,λ}/

*L*

_{ μ,λ}for each condition (dotted, yellow circles). The best-fitting values, ± their standard error, and the percentage coefficient of determination (

*R*

^{2}) are given in section (i) of Table 1.

*β*

_{ μ}is roughly constant at low

*M*

_{ μ,λ}/

*L*

_{ μ,λ}(i.e., Weber's Law holds), but for four out of six subjects (S1, S2, S3, and A1) it increases at higher mean

*M*

_{ μ,λ}/

*L*

_{ μ,λ}(i.e., the M-cone contribution falls below the Weber's Law prediction). The fall in the M-cone contribution on shorter wavelength fields (lower mean

*M*

_{ μ,λ}/

*L*

_{ μ,λ}) is expected from previous work (e.g., Eisner & MacLeod, 1981). Note that although the L-cone bias,

*β*

_{μ}, is constant at low

*M*

_{μ,λ}/

*L*

_{μ,λ}, the weight,

*a*

_{μ}, decreases as

*M*

_{μ,λ}/

*L*

_{μ,λ}decreases. In general, on shorter wavelength fields (i.e., higher values of

*M*

_{μ,λ}/

*L*

_{μ,λμ}),

*a*

_{μ}increases so that the spectral luminous efficiency becomes more L-cone-like, whereas on longer wavelength fields (i.e., lower values of

*M*

_{μ,λ}/

*L*

_{μ,λμ}), it decreases so that the efficiency becomes more M-cone-like (compare the red and black continuous lines in Figure 6, below).

*β*

_{ μ}increases, so too does the standard error of its fit. This is a general property of these fits that arises because as

*β*

_{ μ}gets larger, its effect on spectral sensitivity gets smaller (for further discussion, see Sharpe et al., 2005). Thus, apparently large discrepancies between large values of

*β*

_{μ}, such as that between the two

*β*

_{μ}values for the repeated 462-nm adapting field measurements for S2 (i.e., the two rightmost yellow circles in the middle left panel of Figure 5), correspond to only relatively small differences in the underlying spectral luminous efficiency functions.

*k*

_{ lens}were 0.03, −0.29, 0.07, 0.31, 0.22, and −0.05 for subjects S1–S4, A1, and A2, respectively, and the mean values of

*k*

_{ mac}were −0.14, −0.53, −0.29, −0.39, −0.47, and −0.73, respectively. These values correspond to the factor by which the pigment density spectrum template in question must be adjusted to bring each subject's luminous efficiency data into best agreement with the linear combination of

*λ*) and

*λ*). Note that a negative value means that a particular subject has a higher pigment density than the Stockman and Sharpe (2000) mean observer, and a positive value, a lower pigment density. Thus, our observers have on average 0.05 times less lens density as the Stockman and Sharpe (2000) mean observer (so that their mean density at 400 nm is 1.67 compared with 1.76) and, on average, 0.43 times more macular density (so that their mean peak density is 0.50 compared with 0.35). Our observers, therefore, are more heavily macular pigmented, on average, than the Stockman and Sharpe (2000) mean observer, but their density values all lie within the normal range. We are confident that these densities are not overinflated by the fitting procedure. The macular pigment density of three of the six subjects has been estimated before. Two of them (S2 and A2) participated in a study (Sharpe, Stockman, Knau, & Jägle, 1998) in which the actual macular density spectra were determined. S2 was found to have a peak density of 0.50 (compared with 0.54 here) and A2 was found to have a peak density of approximately 0.60 (compared with 0.61 here). S1 has carried out more limited determinations but is known to be typical in having a peak density of ca. 0.35 (compared with 0.40 here).

*β*

_{ μ}becomes increasingly large, suggesting the contribution of the M-cones falls below the Weber's Law predictions. However, this is one of the complexities discussed in The effects of background luminance section; the lowered M-cone contribution occurs probably because the M-cones are relatively unadapted.

*β*

_{ μ}as a function of

*μ*

*β*

_{ μ}, for each

*μ,*we have effectively assumed that local variations of

*M*

_{ μ,λ}/

*L*

_{ μ,λ}around the mean for each adapting condition induce changes in the L-cone weight that are consistent with Weber's Law (i.e., we assume that, locally,

*β*

_{ μ}is constant). The goodness of the fits shown in Figure 4 and Appendix Figures A1– A6 suggests that this is a reasonable approximation. However, for the majority of subjects the

*β*

_{ μ}values, as shown in Figure 5, increase as

*M*

_{ μ,λ}/

*L*

_{ μ,λ}increases. In this section, we try to capture this increase by assuming that

*β*

_{ μ}increases as some function of

*M*

_{ μ,λ}/

*L*

_{ μ,λ}. In principle, this should yield a better local fit in those regions in which

*M*

_{ μ,λ}/

*L*

_{ μ,λ}increases.

*β*

_{ μ}is therefore fixed across all

*M*

_{ μ,λ}/

*L*

_{ μ,λ}. Equation 4 was fitted simultaneously to all the data for each subject to find the best-fitting values of

*β*

_{ μ},

*k*

_{ lens}, and

*k*

_{ mac}for all

*μ*. This reduced the number of fitted parameters from 20 (or more for S1 and S2, less for A1) to just 3. Figure 5 shows the best-fitting values of the fixed L-cone bias,

*β*

_{ μ}, for each subject, plotted as a horizontal gray line. The best-fitting values, ± their standard error, and the percentage coefficient of determination (

*R*

^{2}) are given in section (ii) of Table 1. Despite the large reduction in the number of parameters, the fits are only slightly worse than the individual fits. However, the fixed value clearly underestimates

*β*

_{ μ}at low

*M*

_{ μ,λ}/

*L*

_{ μ,λ}values and overestimates it at high values, which is undesirable in any predictive model. To overcome this problem, we sought a simple continuous function that could be used to describe the change in

*β*

_{ μ}for all 6 subjects just by changing its parameters. We finally settled on the following version of a power function:

*b*is the power to which

*M*

_{ μ,λ}/

*L*

_{ μ,λ}is raised,

*c*determines the “threshold” level of

*M*

_{ μ,λ}/

*L*

_{ μ,λ}after which the power term becomes important, and

*n*is a multiplier that scales the whole function.

*β*

_{ μ}estimates for each subject (dotted yellow circles). This fit was weighted according to the reciprocal of the standard errors of

*β*

_{ μ}. Weighting the fits in this way is important because, as

*β*

_{ μ}(or

*a*

_{ μ}) increases, so too does its standard error (see Figure 7 of Sharpe et al., 2005). As can be seen, the same function does a reasonable job of characterizing the change in

*β*

_{μ}with

*M*

_{μ,λ}/

*L*

_{μ,λ}for all six subjects.

*β*

_{ μ}shown in Figure 5, we next determined the best-fitting parameters

*directly*from the HFP data. To achieve this, we inserted Equation 5 into Equation 4, and found for each subject the best-fitting values of

*n, c, b, k*

_{ lens}and

*k*

_{ mac}for all

*μ*. The best-fitting values, ± their standard error, and the percentage coefficient of determination (

*R*

^{2}) are given in section (iii) of Table 1. Figure 5 shows the best-fitting form of Equation 5 as the red continuous lines. The goodness of this fit is intermediate between the fixed

*β*

_{ μ}fits and the fits in which

*β*

_{ μ}values were determined for each

*μ*. Indeed, the combination of Equations 4 and 5 provides a reasonably simple description of the 25 HFP data for each subject using just five parameters.

*β*

_{ μ}for each effective background wavelength

*μ,*or by assuming that

*β*

_{ μ}varies as some simple function of

*M*

_{ μ,λ}/

*L*

_{ μ,λ}. We can extend this model to predict luminous efficiency of a “typical” observer, by linking it to the mean luminous efficiency data of Sharpe et al. (2005).

*V*

_{ μ}*(

*λ*) for a “typical” observer

*V*

_{ μ}*(

*λ*) for a typical observer according to

*c*

_{ μ}is simply a unity normalizing constant that varies with

*μ*and can be calculated once the other parameters are known (it is, in fact, the value of the first part of the equation when

*λ*equals

*λ*

_{max}, the wavelength of peak efficiency). Note that in defining these formulae, the luminous efficiency and the state of adaptation are assumed to be

*independent*of

*λ*. As for our six individual observers, we use Equation 5 to define how

*β*

_{ μ}in Equation 6 changes with

*M*

_{ μ}/

*L*

_{ μ}. But, which values of

*n, c,*and

*b*are appropriate for the typical observer represented by the

*V**(

*λ*) function (Sharpe et al., 2005)? Note that an L-cone weight (

*a*

_{μ}) of 1.55 was initially determined for the

*V**(

*λ*) function. However, the analyses carried out for this paper made clear that the mean adapting chromaticity had also varied as a function of target wavelength in the earlier

*V**(

*λ*) measurements. Accordingly, we have reanalyzed the original

*V**(

*λ*) data making appropriate corrections. In relative quantal units, with

*λ*) and

*λ*) both normalized to unity quantal peak,

*a*= 1.89, while in relative energy terms, with

*λ*) and

*λ*) normalized to unity energy peak,

*a*= 1.98. These values supersede the values of 1.55 and 1.62 for quantal and energy units, respectively, given in the original paper (Sharpe et al., 2005). Given this reassessment, the luminous efficiency measurements obtained in 40 observers on a daylight D

_{65}field (Sharpe et al., 2005) suggest that

*β*

_{μ}

*xM*

_{μ}/

*L*

_{μ}(or

*a*

_{μ}) should equal 1.89, and thus

*β*

_{μ}should equal 2.29, for the standard observer.

*β*

_{ μ}of 2.29 ties Equation 5 at one value of

*M*

_{ μ}/

*L*

_{ μ}, but what about the other parameter values? Our solution is to take advantage of the common feature of the individual

*β*

_{ μ}versus

*M*

_{ μ,λ}/

*L*

_{ μ,λ}functions across subjects (see Figure 6); namely, that the functions are roughly constant at low

*M*

_{ μ,λ}/

*L*

_{ μ,λ}(i.e., they follow Weber's Law). Figure 6 shows these individual functions unaligned in the upper panel and vertically aligned at

*M*

_{ μ,λ}/

*L*

_{ μ,λ}< 1 in the lower panel. The alignment was achieved by scaling the functions and minimizing the squared differences between them and the mean. The fit of Equation 5 to the aligned data was weighted by the reciprocal of the standard errors shown in the figure. The best-fitting version of Equation 5 to account for the aligned data was then found. It is shown by the red line in Figure 6 and is given by

*n*= 2.1699 (

*n*is left undefined in the equation, because it can be allowed to vary for individual subjects). The vertical position of the aligned functions in the lower panel of Figure 6 was chosen so that Equation 7 with

*n*= 2.1699 is equal to 2.29 for a D

_{65}background (as indicated by the vertical and upper dashed horizontal lines in Figure 6). The L-cone weights,

*a*

_{ μ}, corresponding to Equation 7 with

*n*= 2.1699 are shown by the continuous black line. As required by

*V*

_{ D65}*(

*λ*),

*a*

_{ D65}is equal to 1.89 (as indicated by the vertical and lower dashed horizontal lines).

*V*

_{ μ}*(

*λ*) for a typical observer. As in Equation 6,

*M*

_{ μ}/

*L*

_{ μ}in Equation 7 is assumed for the general formulae to be independent of

*λ*. To implement this combined equation,

*M*

_{ μ}/

*L*

_{ μ}, the ratio of the M:L-cone excitations, must be computed for each adapting condition. For monochromatic or nearly monochromatic lights, this value can simply be read off Table A1 for the L(mean)-cone template or for the appropriate L(ser

^{180})- or L(ala

^{180})-cone template when the relevant polymorphic variant is known. For spectrally complex lights, the ratio must be calculated by cross-multiplying the spectral power distribution of the field in question with the Stockman and Sharpe (2000) L- and M-cone spectral sensitivities (after choosing the appropriate L-cone polymorphic variant), separately summing the L- and M-cone cross-multiplications and obtaining the ratio between them. For the estimate of

*V*

_{μ}*(

*λ*) to be more accurate for individual observers,

*n*in Equation 7 can be individually determined.

nm | log M | log Lmean | M/Lmean | log L(ser ^{180}) | M/L(ser ^{180}) | log L(ala ^{180}) | M/L(ala ^{180}) |
---|---|---|---|---|---|---|---|

390 | −3.2908 | −3.2186 | 0.8470 | −3.2459 | 0.9018 | −3.2024 | 0.8159 |

395 | −2.8809 | −2.8202 | 0.8696 | −2.8197 | 0.8686 | −2.8119 | 0.8531 |

400 | −2.5120 | −2.4660 | 0.8994 | −2.4737 | 0.9155 | −2.4738 | 0.9158 |

405 | −2.2013 | −2.1688 | 0.9279 | −2.1722 | 0.9352 | −2.1746 | 0.9403 |

410 | −1.9346 | −1.9178 | 0.9622 | −1.9245 | 0.9771 | −1.9270 | 0.9828 |

415 | −1.7218 | −1.7371 | 1.0358 | −1.7353 | 1.0316 | −1.7368 | 1.0351 |

420 | −1.5535 | −1.6029 | 1.1206 | −1.5999 | 1.1129 | −1.5995 | 1.1119 |

425 | −1.4235 | −1.5136 | 1.2305 | −1.5149 | 1.2342 | −1.5120 | 1.2262 |

430 | −1.3033 | −1.4290 | 1.3357 | −1.4345 | 1.3527 | −1.4288 | 1.3350 |

435 | −1.1900 | −1.3513 | 1.4499 | −1.3637 | 1.4921 | −1.3548 | 1.4617 |

440 | −1.0980 | −1.2842 | 1.5355 | −1.2938 | 1.5698 | −1.2815 | 1.5259 |

445 | −1.0342 | −1.2414 | 1.6113 | −1.2500 | 1.6436 | −1.2343 | 1.5852 |

450 | −0.9794 | −1.2010 | 1.6659 | −1.2085 | 1.6946 | −1.1895 | 1.6222 |

455 | −0.9319 | −1.1606 | 1.6931 | −1.1683 | 1.7234 | −1.1463 | 1.6385 |

460 | −0.8632 | −1.0974 | 1.7144 | −1.1113 | 1.7702 | −1.0868 | 1.6734 |

465 | −0.7734 | −1.0062 | 1.7093 | −1.0260 | 1.7889 | −0.9996 | 1.6835 |

470 | −0.6928 | −0.9200 | 1.6873 | −0.9395 | 1.7647 | −0.9118 | 1.6557 |

475 | −0.6301 | −0.8475 | 1.6498 | −0.8597 | 1.6968 | −0.8313 | 1.5895 |

480 | −0.5747 | −0.7803 | 1.6052 | −0.7913 | 1.6464 | −0.7628 | 1.5421 |

485 | −0.5235 | −0.7166 | 1.5602 | −0.7289 | 1.6050 | −0.7010 | 1.5051 |

490 | −0.4738 | −0.6535 | 1.5125 | −0.6626 | 1.5446 | −0.6358 | 1.4520 |

495 | −0.4078 | −0.5730 | 1.4628 | −0.5874 | 1.5120 | −0.5620 | 1.4262 |

500 | −0.3337 | −0.4837 | 1.4126 | −0.4980 | 1.4597 | −0.4744 | 1.3825 |

505 | −0.2569 | −0.3929 | 1.3677 | −0.4068 | 1.4122 | −0.3852 | 1.3436 |

510 | −0.1843 | −0.3061 | 1.3238 | −0.3191 | 1.3640 | −0.2995 | 1.3039 |

515 | −0.1209 | −0.2279 | 1.2791 | −0.2401 | 1.3157 | −0.2226 | 1.2638 |

520 | −0.0699 | −0.1633 | 1.2397 | −0.1789 | 1.2851 | −0.1635 | 1.2403 |

525 | −0.0389 | −0.1178 | 1.1991 | −0.1310 | 1.2363 | −0.1176 | 1.1987 |

530 | −0.0191 | −0.0830 | 1.1586 | −0.0914 | 1.1811 | −0.0799 | 1.1501 |

535 | −0.0081 | −0.0571 | 1.1197 | −0.0638 | 1.1369 | −0.0540 | 1.1116 |

540 | −0.0004 | −0.0330 | 1.0779 | −0.0421 | 1.1007 | −0.0340 | 1.0803 |

545 | −0.0036 | −0.0187 | 1.0353 | −0.0254 | 1.0516 | −0.0189 | 1.0359 |

550 | −0.0163 | −0.0128 | 0.9918 | −0.0131 | 0.9925 | −0.0082 | 0.9813 |

555 | −0.0295 | −0.0050 | 0.9452 | −0.0054 | 0.9460 | −0.0021 | 0.9387 |

560 | −0.0514 | −0.0019 | 0.8923 | −0.0017 | 0.8919 | 0.0000 | 0.8884 |

565 | −0.0769 | −0.0001 | 0.8379 | 0.0000 | 0.8377 | 0.0000 | 0.8377 |

570 | −0.1115 | −0.0015 | 0.7763 | −0.0014 | 0.7761 | −0.0033 | 0.7795 |

575 | −0.1562 | −0.0086 | 0.7119 | −0.0062 | 0.7079 | −0.0101 | 0.7143 |

580 | −0.2143 | −0.0225 | 0.6430 | −0.0146 | 0.6315 | −0.0209 | 0.6406 |

585 | −0.2753 | −0.0325 | 0.5718 | −0.0282 | 0.5662 | −0.0370 | 0.5778 |

590 | −0.3443 | −0.0491 | 0.5067 | −0.0462 | 0.5034 | −0.0579 | 0.5171 |

595 | −0.4264 | −0.0727 | 0.4429 | −0.0693 | 0.4395 | −0.0843 | 0.4549 |

600 | −0.5198 | −0.1026 | 0.3826 | −0.1000 | 0.3803 | −0.1186 | 0.3970 |

605 | −0.6247 | −0.1380 | 0.3261 | −0.1357 | 0.3243 | −0.1583 | 0.3416 |

610 | −0.7390 | −0.1823 | 0.2776 | −0.1790 | 0.2755 | −0.2060 | 0.2931 |

615 | −0.8610 | −0.2346 | 0.2364 | −0.2295 | 0.2336 | −0.2611 | 0.2512 |

620 | −0.9915 | −0.2943 | 0.2008 | −0.2885 | 0.1982 | −0.3252 | 0.2156 |

625 | −1.1294 | −0.3603 | 0.1702 | −0.3555 | 0.1683 | −0.3975 | 0.1854 |

630 | −1.2721 | −0.4421 | 0.1479 | −0.4296 | 0.1437 | −0.4771 | 0.1603 |

635 | −1.4205 | −0.5327 | 0.1295 | −0.5121 | 0.1235 | −0.5652 | 0.1395 |

640 | −1.5748 | −0.6273 | 0.1128 | −0.6031 | 0.1067 | −0.6618 | 0.1222 |

645 | −1.7370 | −0.7262 | 0.0976 | −0.7046 | 0.0928 | −0.7690 | 0.1076 |

650 | −1.8900 | −0.8407 | 0.0893 | −0.8143 | 0.0840 | −0.8844 | 0.0987 |

655 | −2.0523 | −0.9658 | 0.0819 | −0.9311 | 0.0756 | −1.0066 | 0.0900 |

660 | −2.2220 | −1.0966 | 0.0749 | −1.0566 | 0.0683 | −1.1375 | 0.0823 |

665 | −2.3923 | −1.2327 | 0.0692 | −1.1904 | 0.0628 | −1.2764 | 0.0766 |

670 | −2.5559 | −1.3739 | 0.0658 | −1.3311 | 0.0596 | −1.4219 | 0.0734 |

675 | −2.7194 | −1.5208 | 0.0633 | −1.4779 | 0.0573 | −1.5731 | 0.0714 |

680 | −2.8843 | −1.6736 | 0.0616 | −1.6303 | 0.0557 | −1.7294 | 0.0700 |

685 | −3.0519 | −1.8328 | 0.0604 | −1.7874 | 0.0544 | −1.8900 | 0.0689 |

690 | −3.2234 | −1.9992 | 0.0597 | −1.9484 | 0.0531 | −2.0539 | 0.0677 |

695 | −3.3874 | −2.1596 | 0.0592 | −2.1124 | 0.0531 | −2.2201 | 0.0680 |

700 | −3.5484 | −2.3200 | 0.0591 | −2.2783 | 0.0537 | −2.3876 | 0.0691 |

705 | −3.7103 | −2.4819 | 0.0591 | −2.4452 | 0.0543 | −2.5552 | 0.0700 |

710 | −3.8757 | −2.6490 | 0.0593 | −2.6118 | 0.0545 | −2.7217 | 0.0702 |

715 | −4.0389 | −2.8165 | 0.0599 | −2.7769 | 0.0547 | −2.8859 | 0.0703 |

720 | −4.1981 | −2.9801 | 0.0605 | −2.9394 | 0.0551 | −3.0466 | 0.0705 |

725 | −4.3559 | −3.1432 | 0.0613 | −3.0979 | 0.0552 | −3.2025 | 0.0702 |

730 | −4.5101 | −3.3032 | 0.0621 | −3.2514 | 0.0551 | −3.3525 | 0.0696 |

D _{65}* | 0.8170 | 0.8200 | 0.8280 | ||||

A* | 0.6590 | 0.6560 | 0.6750 |

*relative*suppression on long-wavelength fields is, however, of the L-cones (Eisner & MacLeod, 1981).

*a*

_{μ}(

*β*

_{μ}

*xM*

_{μ,λ}/

*L*

_{μ,λ}, in our model) directly reflects the relative numbers of the L- and M-cones contributing to luminous efficiency. This assumption is, however, highly questionable, because the outputs of each cone type are modified not only by receptoral adaptation, but also, as our results suggest on short-wavelength fields that show M-cone suppression, by postreceptoral adaptation before the signals are combined. Indeed, the strong dependence of

*a*

_{μ}on chromatic adaptation begs the question of which condition of chromatic adaptation should be considered truly “neutral”—in the sense of not altering the relative contributions of the M- and L-cones to luminous efficiency away from those due to relative numerosity. One way of potentially simplifying the problem is, as we have done, to consider the effects of selective proportional chromatic adaptation separately from other factors by considering the L-cone weight

*a*

_{μ}as

*β*

_{μ}

*xM*

_{μ,λ}/

*L*

_{μ,λ}, where

*β*

_{μ}is the L-cone bias. Indeed, it is tempting to conclude that the roughly constant value of

*β*

_{μ}found for low

*M*

_{μ,λ}/

*L*

_{μ,λ}(where Weber's Law approximately holds) actually reflects relative L- and M-cone numerosity.

*M*

_{ μ,λ}/

*L*

_{ μ,λ},

*β*

_{ μ}is still likely to be influenced by factors other than numerosity, such as neural weighting differences. In principle,

*β*

_{ μ}could 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 view is doubtful, however, 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., Albrecht, Jägle, Hood, & Sharpe, 2002; Brainard et al., 2000; Kremers et al., 2000; Lutze et al., 1990; Rushton & Baker, 1964; Sharpe, de Luca, Hansen, Jägle, & Gegenfurtner, 2006; Vimal et al., 1989; Wesner, Pokorny, Shevell, & Smith, 1991). Nevertheless, any claims that luminous efficiency can be used to derive cone numerosity directly should be treated with extreme caution. Other workers have pointed out the problems of linking luminous efficiency with cone numerosity (e.g., Chaparro, Stromeyer, Kronauer, & Eskew, 1994; Eskew, McLellan, & Giulianini, 1999).

^{−1}·deg

^{−2}. Given this assumption, we can use the

*V*

_{μ}*(

*λ*) function to estimate the ERG flicker matches and then calculate from those matches the changes in adapting chromaticity with target wavelength. The estimated chromaticities for the ERG measurements are shown as the continuous white line in the upper panel of Figure 3. As can be seen, the changes in adapting chromaticity are substantial—much larger than the changes found when a background is used. Over the typical range of their ERG measurements (460 to 680 nm), the M/L cone excitation ratio changes from 1.25 (equivalent to a background of about 520 nm) to 0.40 (equivalent to a background of about 600 nm). These considerable changes in adapting chromaticity with target wavelength will distort the ERG spectral sensitivities (see Figure 2, above, for the expected changes between comparable spectral fields). The 32.5-Hz ERG measurements, therefore, are unlikely to reflect accurately relative L- and M-cone numerosity, as the authors claim. Although the curves may still be describable as a linear combination of the M- and L-cone spectral sensitivities (as we also found in a preliminary analysis of the HFP data shown here), the weights will be systematically offset from values that would be obtained if there had been no change in chromatic adaptation with target wavelength. Errors of this type are expected from the work of De Vries (1948b), who showed HFP additivity failures for combined test and reference targets that exceeded 1.7 log trolands.

*M*

_{ μ,λ}/

*L*

_{ μ,λ}value for

*each*HFP match. Figures A1 and A2 show the repeated matches for subject S2, who made the bichromatic and spectral matches twice. Figures A3– A6 show the matches for S3, S4, A1, and A2, respectively. The spectral matches and bichromatic mixture matches are shown in the upper and middle panels, respectively, in each figure, with the exception of the figure for A2, who made no bichromatic matches. Data for S1 are shown in Figure 4 in the main text.