Wavelength (nm)

Fig. 3. Photopic luminous efficiency functions for 2° and 10° central viewing conditions. The CIE (Commission Internationale de l’ Éclairage, International Lighting Commission) 1924

International Standards

Additive Functions for 2° Viewing Fields

In 1924, the CIE [42] established a luminous efficiency function V(λ) for 2° photopic (cone) vision (see Fig. 3; the values can be downloaded from the Web site http://www.cvrl.org), which has since become the standard (mean) photopic observer for business and industry [42, 43]. In visual science, V(λ) or its variants have been assumed to correspond to the spectral sensitivity of a hypothetical human postreceptoral “luminance” channel with additive inputs from the L and M cones (e.g., [8]). Nevertheless, such wide acceptance overlooks serious flaws and complications in V(λ)’s derivation. First, the derivation of V(λ) was not based exclusively on psychophysical techniques that obey additivity. Rather, it is a speculative hybrid function, artificially smoothed and assembled from divergent data measured using both “additive” and “nonadditive” techniques at several laboratories (see [11, 12, 43, 44]). Its most conspicuous flaw is that it seriously underestimates luminous efficiency at short wavelengths (see, for discussion, [22]). Although two attempts were later made to correct this problem [45, 46], the modifications generated functions that are far removed from actual data.

The CIE did not adequately specify V(λ) for adaptation level. In fact, the specific adaptive or desensitization states of the L and M cones were not held constant in the various

336 Sharpe and Stockman

experiments (see, for a comment, [44]). The CIE was also unable to take into account variations in genotype, in particular the L-cone polymorphic variation that primarily affects long-wavelength sensitivity (see Chapter 14 on human cone spectral sensitivities and color vision deficiencies).

To correct these problems, Sharpe et al. [44] determined a new luminous efficiency function for 2° photopic viewing conditions, V*(λ) (see Fig. 3; the values can be downloaded from the Web site http://www.cvrl.org or from the online article), based exclusively on HFP measurements, which obey additivity, in 40 genotyped observers whose L-cone polymorphic variant was determined. The V*(λ) and the CIE 1924 V(λ) functions can be directly compared in Fig. 3. The two functions differ most conspicuously at short wavelengths below 500 nm.

The new function was obtained under neutral adaptation that corresponds to a specific and reproducible phase of natural daylight (CIE standard illuminant D65) adaptation. Moreover, it is defined as a linear combination of the Stockman and Sharpe [23] L- and

where a is the relative L-cone weight, and c is a constant that scales the function so

¯ l − l

that it peaks at unity. In relative quantal units, with l( ) and m ( ) themselves both normalized to unity quantal peak, a = 1.890000 and c = 2.80361; in relative energy units,

¯ l − l

with l( ) and m( ) both normalized to unity energy peak, a = 1.98065 and c = 2.87091 [44]. The S-cone contribution is so small, and anyhow is so complexly dependent on temporal frequency and adaptation, that it can be safely ignored in defining photopic luminous efficiency under conditions that obey additivity (for a full discussion of this point, see [44, 47]).

Note that these constants define photopic luminous efficiency when the eye is adapted to a specific state of “daylight” (D65) adaptation. Necessarily, different states of adaptation will lead to different constants and different luminous efficiency functions, so that the applicability of V*(λ) is unavoidably limited. However, once photopic luminous efficiency is defined in terms of the L- and M-cone sensitivities, it becomes possible to define it for other states of chromatic adaptation merely by investigating and modeling the changes that occur in the relative contributions of the L and M cones as a function of the effective wavelength of the adapting field.

Accordingly, Stockman, Jägle, Jagla, and Sharpe [47] have investigated the dependence of V*(λ) on adapting field chromaticity. As expected, through selective cone adaptation or desensitization, short-wavelength fields increase a by decreasing the relative contribution of the M cones, and long-wavelength fields decrease a by decreasing the relative contribution of the L cones (see Fig. 4).

A generalized formula is suggested in [47].

Additive Functions for 10° Viewing Fields

Until recently, the photopic luminous efficiency function that was considered most representative for large centrally viewed fields was the 1964 CIE V10 (λ) function for 10° fields, which was derived from large-field color-matching data and luminous

Fig. 4. The 25-Hz heterochromatic flicker photometric (HFP) luminous efficiency functions for the same subject measured on spectral backgrounds ranging from 430 to 670 nm. The curve drawn through each set of data is the best-fitting version of Eq. 1. The background wavelength (in nanometers) and the derived L-cone weighting factor a (from Eq. 1) are noted to the right of each curve. (From [47].) Generally, the L-cone weight or relative contribution of the L cones increases for short-wavelength adaptation and decreases for long-wavelength adaptation. However, the change is not linear with adapting wavelength and is complicated by changes in chromatic adaptation caused by the targets (see [47]).

efficiency measurements made in a subset of subjects at four wavelengths [12].

− l

It is by design identical to the y10( ) color-matching function of the CIE 1964 supplementary standard colorimetric observer. This function is shown in Fig. 3 together with the 2° V(λ) and V*(λ) functions (the values can be downloaded from the Web site http://www.cvrl.org). However, a technical committee of the CIE has recommended that a new photopic luminosity function V10*(λ), derived from the V*(λ) function by making corrections for the differences in the mean optical density of the

Wavelength (nm)

Fig. 5. Spectral filtering by the optic media. The standard or average optical density of the lens and other ocular media as a function of wavelength is shown (solid black line; [23]). Also shown is the standard or average optical density of the macular pigment as a function of wavelength for small (2°; solid gray line) and large (10°; dashed gray line) centrally viewed fields. (After [23].)

macular pigment for 2° and 10° viewing fields (see the section on attenuation of spectral light by the macular pigment and Fig. 5), should replace the 1964 CIE V10(λ) function in physiologically relevant colorimetric and photometric systems. The new V10*(λ) function is shown in Fig. 3, in which it can be compared directly with the 1964 CIE V10 (λ) function. There are only slight discrepancies, mostly at short wavelengths.

Other Photopic (Nonadditive) Luminous Efficiency Functions

The 1924 CIE V(λ) function and the V*(λ) function are not representative of luminous efficiency functions based on HBM. These tend to be broader and to exhibit more than one peak. The CIE Technical Committee 1.4 Vision (TC 1.4) recommended a 2° brightness-matching luminous efficiency function based on a number of studies to supplement V(λ) [12]. Revised brightness-matching standard luminous efficiency functions for 2° and 10° viewing fields [48] and for point sources [49] have since been derived. The 10° field function differs from the 2° function only at short wavelengths, from 410 through 520 nm, when the functions are normalized at 570 nm. The difference is attributed to absorption by the macular pigment. The point source function can be approximated by Judd’s [45] modification of the CIE 1924 V(λ) function. It is pertinent to point out that these functions, based on HBM, will not obey Abney’s law, so their application is very limited.