Formulae for generating standard and individual human cone spectral sensitivities

Abstract

Normal color perception is complicated. But at its initial stage it is relatively simple, since at photopic levels it depends on the activations of just three photoreceptor types: the long- (L-), middle- (M-) and short- (S-) wavelength-sensitive cones. Knowledge of how each type responds to different wavelengths-the three cone spectral sensitivities-can be used to model human color vision and in practical applications to specify color and predict color matches. The CIE has sanctioned the cone spectral sensitivity estimates of Stockman and Sharpe (Stockman and Sharpe, 2000, Vision Res) and their associated measures of luminous efficiency as "physiologically-relevant" standards for color vision (CIE, 2006; 2015). These LMS cone spectral sensitivities are specified at 5- and 1-nm steps for mean "standard" observers with normal cone photopigments and average ocular transparencies, both of which can vary in the population. Here, we provide formulae for the three cone spectral sensitivities as well as for macular and lens pigment density spectra, all as continuous functions of wavelength from 360 to 850 nm. These functions reproduce the tabulated discrete CIE LMS cone spectral sensitivities for 2-deg and 10-deg with little error in both linear and logarithmic units. Furthermore, these formulae allow the easy computation of non-standard cone spectral sensitivities (and other color matching functions) with individual differences in macular, lens and photopigment optical densities, and with spectrally shifted hybrid or polymorphic L- and M-cone photopigments appropriate for either normal or red-green color vision deficient observers.

Keywords: CIE standards; color matching functions; cone fundamentals; cone spectral sensitivity functions; individual differences.

FIGURE 1

Logarithmic (top panel) and linear (third panel) L‐, M‐, and S‐cone photopigment absorbance spectra on a logarithmic wavelength scale (red, green, and blue solid lines, respectively) defined in the CIE 2006 standard from Stockman and Sharpe. The spectra have been extended to 360 and 850 nm (solid black lines) and the L‐ and M‐cone spectra have been modified slightly between 390 and 400 nm to accommodate a short‐wavelength extension consistent with other photopigment measurements. The 8th order Fourier polynomials that best fit the extended spectra are shown by the yellow dashed lines. See text for details. Errors in the fitted functions compared to the CIE 2006 standards are shown on logarithmic and linear scales in the second and fourth panels, respectively.

FIGURE 2

Standard macular optical density spectrum (upper panel, pink solid line) and lens pigment optical density spectrum (middle panel, orange solid line) extrapolated from the Stockman and Sharpe and CIE color matching standard with best‐fitting 11th order (macular) or 9th order (lens) Fourier polynomials (white dashed lines). See text for details. Errors in the fitted functions for macular (pink solid line) and lens (orange solid line) are shown in the bottom panel.

FIGURE 3

Logarithmic (upper panels) and linear (lower panels) corneal L‐, M‐ and S‐cone spectral sensitivities (yellow dashed lines) calculated from the Fourier polynomials for 2‐deg (left panels) and 10‐deg (right panels) vision. The CIE L‐, M‐ and S‐cone standards are shown by the red, green, and blue solid lines, respectively. The sensitivities are given in relative energy units.

FIGURE 4

The extended logarithmic (top panel) and linear (third panel) L‐, M‐ and S‐cone photopigment absorbance spectra (red, green, and blue solid lines, respectively) fitted by the same 8th order Fourier polynomial (yellow dashed lines) shifted along the log‐wavelength scale. The L‐cone spectrum was assumed to be a linear combination of two underlying common spectra: one for L(ala180) and the other for L(ser180). The black portion of the S‐cone spectrum was not included in the fit. See text for further details.

FIGURE 5

(A) Maximum saturation method of setting a color match. A monochromatic test field of wavelength, λ, can be matched by a mixture of red (645 nm), green (526 nm) and blue (444 nm) primary lights, one of which must be added to the test field to complete the match. In this example, the red primary must be added to match a cyan (c. 490 nm) test light. (B) The amounts of each of the three primaries required to match monochromatic lights spanning the visible spectrum are the $\overline{r}\left(\lambda \right)$, $\overline{g}\left(\lambda \right)$, $\overline{b}\left(\lambda \right)$ CMFs. (C) Color matches produce identical quantum catches in the three cone types. Thus, the cone fundamentals CMFs $\overline{l}\left(\lambda \right)$, $\overline{m}\left(\lambda \right)$, and $\overline{s}\left(\lambda \right)$ (or the cone spectral sensitivities) must be a linear transformation of the $\overline{r}\left(\lambda \right)$, $\overline{g}\left(\lambda \right)$, $\overline{b}\left(\lambda \right)$ CMFs. (D) The l,m cone chromaticity plane with the coordinates of the spectrum locus shown by the solid line and yellow circles. The purple line joins the bottommost points on the locus and is the limit of physically realizable colors in the violet to red region of the chromaticity space (dashed purple line). The red, green, and blue diamonds are the intersections of the L, M and S primary vectors with the l,m cone chromaticity plane. The white dashed lines that join them delimit the imaginary and real colors that can be matched by adding together those primaries. The red, green, and blue squares are the intersections of the vectors representing the R, G and B primaries with this plane. The blacked dashed lines joining them show the real colors that can be matched by adding together those primaries. The figure shows 10‐deg color matching data.

FIGURE 6

Cone fundamentals transformed to other primaries and plotted as chromaticity coordinates calculated from the original tabulated CIE 2006 fundamentals (dashed black lines, and small black circles at 10‐nm steps) and from the formulae presented here (solid white lines, and larger white circles). Upper panel: spectrum locus in l,m chromaticity coordinates. Middle panel: spectrum locus in r,g, chromaticity coordinates for RGB primaries of 444, 526 and 645 nm. Lower panel: spectrum locus in x,y, chromaticity coordinates for CIE XYZ primaries.

FIGURE 7

An expanded view of the short‐wavelength region of the spectrum locus in l,m coordinates. Symbols as in Figure 6.