1.2 The three kinds of cone

The retina, the light-sensitive layer at the back of the eye, does not record the reflectance curve. It has three kinds of cone cell, and each kind responds to a broad range of wavelengths that overlaps with the ranges of the other two. A cone simply counts the light it absorbs. It cannot tell which wavelength the light had, so a lot of light at a wavelength it is not very sensitive to gives the same signal as a little light at its favorite one. All the eye ever learns about a patch of color is therefore three numbers: how strongly the patch excites each kind of cone. However complicated the reflectance curve, the eye reduces it to those three numbers, and every color we can see is a particular combination of them.

The three kinds are named for where their sensitivity peaks: long-wavelength or L cones in the yellow-green, middle-wavelength or M cones in the green, and short-wavelength or S cones in the blue-violet. They are often called red, green and blue cones, which is misleading, since the L cones peak in the yellow-green and all three respond over wide ranges. Figure 2(a) shows how sensitive each kind is at each wavelength, from the measurements of Stockman and Sharpe that the CIE has adopted as its standard [70, 12]. The curves are broad and the L and M curves overlap almost completely, so no wavelength excites only one kind of cone.

Figure 2. (a) How sensitive each of the three kinds of cone is at each wavelength, from Stockman and Sharpe [70], each scaled to its own peak; the peaks are marked. (b) The signal each kind of cone receives from four Golden paints in daylight, as a percentage of what it receives from a perfect white.

Panel (b) gives the three signals for four paints in daylight, each as a share of what a perfect white would produce. Cadmium yellow medium excites the L and M cones strongly and the S cones hardly at all, and that combination is what the eye sees as a light, vivid yellow. Phthalo blue straight from the tube gives all three very little, with the S cones slightly ahead: a very dark blue. Chromium oxide gives three similar, low signals with the M cones a little ahead, which the eye reads as a dull, dark green.

The eye also has rods, which handle vision in dim light and play no part in the colors a painter sees. The three kinds of cone are far from equal in number. Only about one cone in twenty is an S cone [17, 18], and the ratio of L to M cones varies enormously from one person to another, from about one to one to more than sixteen to one in a study of people with normal color vision [33]. People with very different mixes of cones still see much the same colors, because the later stages of vision calibrate themselves to whatever cones they are given.

Because the eye works with three numbers, color can be measured with three numbers. In 1931 the CIE defined a standard way to compute three such numbers from a reflectance curve and an illuminant, and they carry very nearly the same information as the three cone signals. One of them, called Y, is the share of the light the surface sends back, counted the way the eye counts it, and it sets how light the color looks. Every color calculation in these notes starts from these three standard numbers, and the math box at the end of this lesson gives the formulas.

Working from three signals also means that two different spectra giving the same three look identical, the subject of Section 1.4. And no single wavelength of light is “pure” in any sense the cones recognize, since every wavelength excites at least two kinds of cone.

TipThe math behind this lesson

Cone signals and tristimulus values. The human retina has about 4.6 million cones and about 92 million rods [17]. The S cones make up about 5 percent of the cones and are absent from the very center of the fovea [17, 18], and the ratio of L to M cones ranges from about 1 to 1 to more than 16 to 1 in one study of observers with normal color vision [33]. The L, M and S sensitivities peak near 570, 545 and \(445\,\mathrm{nm}\); the L and M peaks are only about \(25\,\mathrm{nm}\) apart [70]. Each cone contains a photopigment that absorbs light with a probability depending on wavelength, and the cone’s output depends only on the total number of photons absorbed. This is the principle of univariance: a cone cannot tell a few photons at its best wavelength from many photons at a poor one. The signal from a cone of type \(i\) exposed to a light \(L(\lambda)\) is therefore

\[ c_i = \int \bar{c}_i(\lambda)\,L(\lambda)\,d\lambda, \qquad i = L, M, S, \tag{2} \]

where \(\bar{c}_i(\lambda)\) is the cone’s spectral sensitivity, and two spectra that give the same three numbers are indistinguishable.

The cone signals are rarely used directly in color work. The CIE’s 1931 standard observer, built from the matching experiments of Wright and Guild [77, 27], defines three color matching functions \(\bar{x}(\lambda)\), \(\bar{y}(\lambda)\), \(\bar{z}(\lambda)\) that are linear combinations of the cone sensitivities, and the tristimulus values

\[ X = k\!\int \bar{x}(\lambda) E(\lambda) R(\lambda)\, d\lambda, \quad Y = k\!\int \bar{y}(\lambda) E(\lambda) R(\lambda)\, d\lambda, \quad Z = k\!\int \bar{z}(\lambda) E(\lambda) R(\lambda)\, d\lambda, \tag{3} \]

with \(k\) chosen so that a perfect white has \(Y = 1\) (or 100). \(\bar{y}\) is the eye’s sensitivity to brightness, the luminous efficiency function, so \(Y\) is the luminance factor, the fraction of the light the surface sends back as the eye weights it. Because \(X\), \(Y\) and \(Z\) are, to a good approximation, linear combinations of the cone signals, they carry the same information as \(L\), \(M\) and \(S\): two lights match for the standard observer if and only if they have the same \(XYZ\). Every color calculation in these notes starts from equation (3), evaluated as a sum over the wavelengths at which the data are tabulated [11]. In Figure 2(b), cadmium yellow medium gives the L, M and S cones 71, 59 and 6 percent of what a white gives them, phthalo blue in masstone 4, 4 and 9 percent, and chromium oxide 14, 15 and 9.

Sources

[11] CIE, Colorimetry, 4th ed., CIE 015:2018, Commission Internationale de l’Éclairage, Vienna, 2018.

[12] CIE 170-1:2006, Fundamental Chromaticity Diagram with Physiological Axes, Part 1; CIE 170-2:2015, Part 2: Spectral Luminous Efficiency Functions and Chromaticity Diagrams.

[17] C. A. Curcio, K. R. Sloan, R. E. Kalina and A. E. Hendrickson, Human photoreceptor topography, Journal of Comparative Neurology 292 (1990), 497–523.

[18] C. A. Curcio, K. A. Allen, K. R. Sloan, C. L. Lerea, J. B. Hurley, I. B. Klock and A. H. Milam, Distribution and morphology of human cone photoreceptors stained with anti-blue opsin, Journal of Comparative Neurology 312 (1991), 610–624.

[27] J. Guild, The colorimetric properties of the spectrum, Philosophical Transactions of the Royal Society of London, Series A 230 (1932), 149–187.

[33] H. Hofer, J. Carroll, J. Neitz, M. Neitz and D. R. Williams, Organization of the human trichromatic cone mosaic, Journal of Neuroscience 25 (2005), 9669–9679.

[70] A. Stockman and L. T. Sharpe, The spectral sensitivities of the middle- and long-wavelength-sensitive cones derived from measurements in observers of known genotype, Vision Research 40 (2000), 1711–1737.

[77] W. D. Wright, A re-determination of the trichromatic coefficients of the spectral colours, Transactions of the Optical Society 30 (1929), 141–164.