Encyclopedia of Color Science and Technology

2016 Edition
| Editors: Ming Ronnier Luo

Cone Fundamentals

  • Andrew Stockman
Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-8071-7_85

Synonyms

Definition

The three cone fundamentals are the spectral sensitivities of the long- (L-), middle- (M-), and short- (S-) wavelength cones measured relative to light entering the cornea. They are also the fundamental color matching functions (CMFs), which in colorimetric notation are referred to as \( \overline{l}\left(\lambda \right) \), \( \overline{m}\left(\lambda \right) \), and \( \overline{s}\left(\lambda \right) \). The simple identity between the cone fundamentals of color matching and the cone spectral sensitivities depends on phototransduction and its property of univariance. The absorption of a photon produces a photoreceptor response that is independent of photon wavelength, so that all information about wavelength is lost. With one cone type, vision is monochromatic. With three cone types, vision is trichromatic.

Trichromacy means that for observers with normal color vision, the color of a test light of any chromaticity can be matched by superimposing three independent primary lights (with the proviso that one of the primaries sometimes must be added to the test light to complete the match). The amounts of the three primary lights required to match test lights as a function of test wavelength are the three CMFs for those primary lights (usually defined for matches to test lights of equal energy). Because the cone spectral sensitivities overlap, it is not possible for any real light to stimulate just one of them. However, the cone fundamentals are the three CMFs for the three imaginary primary lights that would uniquely stimulate individual cone types (i.e., imaginary lights that produce the three “fundamental” sensations that underlie color vision). The cone fundamental CMFs define all other CMFs and must be a linear transformation of them.

The cone fundamentals can be determined directly by measuring cone spectral sensitivities using “color-deficient” observers lacking one or two cone types and/or special conditions to isolate single-cone responses. They can also be derived by the linear transformation of CMFs measured using real primary lights, but for that the coefficients of the linear transformation must be known.

Current estimates of the cone fundamentals [1, 2, 3] use spectral sensitivity measurements to guide the choice of the coefficients of the linear transformation from a set of measured CMFs to the cone fundamental CMFs.

Overview

The spectral properties of the cone fundamentals and that of color matching, in general, are determined principally by the way in which the cone photoreceptors interact with light at the very first stage of vision. They depend, in particular, on photon absorptions by the cone photopigment and on how the probability of photon absorption varies with wavelength. To understand that, we start at the molecular level.

Phototransduction

The cone photopigment molecule is made up of a transmembrane opsin, a G protein-coupled receptor protein, bound to a chromophore, 11-cis retinal. The absorption of a photon provides the energy needed to isomerize the chromophore from its 11-cis form to its all-trans form; this change in shape activates the opsin and triggers the phototransduction cascade and the neural response. The likelihood that a given photon will produce an isomerization depends upon how closely its energy matches the energy required to initiate the isomerization. Crucially, this energy varies with cone type because of differences in key amino acids in those parts of the opsin molecule that surround the chromophore. These amino acids modify the isomerization energy and thus the spectral sensitivity of the photoreceptor [4]. In most observers with normal color vision, there are four photoreceptor classes: three types of cone photoreceptors, L-, M-, and S-cones, and a single type of rod photoreceptor.

Cone fundamentals, since they are measured behaviorally in terms of energies measured at the cornea, also depend on the absorption of photons by the optical media and on the density of photopigment in the photoreceptor outer segment, both of which vary between observers (see below). Thus, the cone fundamentals are not precisely related to the spectral properties of the photopigments (called the absorbance or extinction spectra).

Univariance

The relatively simple relationship between the cone fundamentals and color matching arises because of the way in which cone (and rod) photoreceptors transduce absorbed photons. When a photon is absorbed to initiate the phototransduction cascade, the effect is all or nothing and is consequently independent of photon wavelength. Photoreceptor outputs thus vary univariantly according to the number of photons that they absorb [5], as a result of which wavelength and intensity are confounded, and individual photoreceptors are “color blind.” A change in the rate of photon absorption could be due to a variation in light intensity, but equally, it could be due to a variation in wavelength.

With only one cone type, vision is monochromatic and reduced to a single dimension: two lights of any spectral composition can be made to match simply by equating their intensities (a relationship defined by the cone type’s spectral sensitivity). With only two cone types, vision is dichromatic and reduced to two dimensions: lights of any spectral composition can be matched with a mixture of two other lights. Dichromatic human observers fall into three classes, protanopes, deuteranopes, and tritanopes, depending upon whether they lack L-, M-, or S-cones, respectively. Observers with normal color vision have three classes of cone photoreceptor, and their vision is trichromatic. Color vision depends on comparing the univariant outputs of different cone types.

Trichromacy

A consequence of trichromacy is that the color of any light can be matched with three specially selected or “independent” primary lights of variable intensity (chosen so that no two will match the third). These primary lights are frequently red (R), green (G), and blue (B), but many other triplets are possible. The upper panel of Fig. 1 shows a typical color matching experiment, in which an observer is presented with a half-field illuminated by a “test” light of variable wavelength λ and a second half-field illuminated by a mixture of red, green, and blue primary lights. At each λ, the observer adjusts the intensities of the three primary lights, so that the test field is perfectly matched by the mixture of primary lights. The lower left-hand panel of Fig. 1 shows the mean \( \overline{r}\left(\lambda \right) \), \( \overline{g}\left(\lambda \right) \), and \( \overline{b}\left(\lambda \right) \) CMFs obtained by Stiles and Burch [6] for primary lights of 645, 526, and 444 nm. Notice that except at the primary wavelengths one of the CMFs is negative. There is no “negative light”; rather, these negative values indicate that the primary in question must be added to the spectral test light to make a match (as illustrated in the panel for the red primary). Real primaries give rise to negative values because real lights do not uniquely stimulate single-cone photoreceptors (see Figs. 2 and 3). The cone fundamental CMFs are always positive.
Cone Fundamentals, Fig. 1

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 (upper panel). The amounts of each of the three primaries required to match monochromatic lights spanning the visible spectrum are the \( \overline{r}\left(\lambda \right) \), \( \overline{g}\left(\lambda \right) \), and \( \overline{b}\left(\lambda \right) \) CMFs (red, green, and blue lines, respectively) shown in the lower left-hand panel. These CMFs were measured using 10-deg diameter targets by Stiles and Burch [6]. A negative sign means that the primary must be added to the target to complete the match. CMFs can be linearly transformed from one set of primaries to another and to the fundamental primaries. Shown in the lower right-hand panel are the are 10-deg \( \overline{r}\left(\lambda \right) \), \( \overline{g}\left(\lambda \right) \), and \( \overline{b}\left(\lambda \right) \) CMFs linearly transformed to give the \( \overline{l}\left(\lambda \right) \), \( \overline{m}\left(\lambda \right) \), and \( \overline{s}\left(\lambda \right) \) 10-deg cone fundamental primary CMFs (red, green, and blue lines, respectively)

Cone Fundamentals, Fig. 2

Mean cone spectral sensitivity data [10, 11]. L-cone data from deuteranopes with either the L(S180) (redsquares, n  =  17) or L(A180) (yellow circles, n  =  3) polymorphism, M-cone data from protanopes (green diamonds, n  =  9), and S-cone data (blue triangles) from S-cone monochromats (n  =  3) and below 540 nm (measured under intense long-wavelength adaptation) from normal observers (n  =  5)

Cone Fundamentals, Fig. 3

Comparisons between estimates of the 2-deg L-, M-, and S-cone fundamentals by Stockman and Sharpe [2] (solid-colored lines), by Smith and Pokorny [1] (dashed lines), and by König and Dieterici [17] (symbols)

CMFs, such as the ones shown in the lower-left panel, can be linearly transformed to any other set of real primary lights and to the fundamental primaries that are illustrated in the lower right-hand panel of Fig. 1. The three fundamental primaries (or “Grundempfindungen” – fundamental sensations) are the three imaginary primary lights that would uniquely stimulate each of the three cones to yield the L-, M-, and S-cone spectral sensitivity functions (such lights are not physically realizable because of the overlapping spectral sensitivities of the cone photopigments). All other sets of CMFs depend on the fundamental CMFs and should be a linear transformation of them. Note that the fundamental CMFs shown here are comparable to those shown in Fig. 3 but are plotted as linear sensitivities rather than as the more usual logarithmic sensitivities.

The relationship between the fundamental CMFs and a set of real CMFs (obtained using, e.g., red, green, and blue primaries) can be stated formally: When an observer matches the test and mixture fields in a color matching experiment, the two fields cause identical absorptions in each of his or her three cone types. The match, in other words, is a match at the cones. The matched test and mixture fields appear identical to S-cones, to M-cones, and to L-cones. For matched fields, the following relationships apply:
$$ {\overline{l}}_R\overline{r}\left(\lambda \right)+{\overline{l}}_G\overline{g}\left(\lambda \right)+{\overline{l}}_B\overline{b}\left(\lambda \right)=\overline{l}\left(\lambda \right) $$
$$ {\overline{m}}_R\overline{r}\left(\lambda \right)+{\overline{m}}_G\overline{g}\left(\lambda \right)+{\overline{m}}_B\overline{b}\left(\lambda \right)=\overline{m}\left(\lambda \right) $$
(1)
$$ {\overline{s}}_R\overline{r}\left(\lambda \right)+{\overline{s}}_G\overline{g}\left(\lambda \right)+{\overline{s}}_B\overline{b}\left(\lambda \right)=\overline{s}\left(\lambda \right) $$
where \( {\overline{l}}_R \), \( {\overline{l}}_G \), and \( {\overline{l}}_B \) are, respectively; the L-cone sensitivities to the R, G, and B primary lights; and similarly, \( {\overline{m}}_R \), \( {\overline{m}}_G \), and \( {\overline{m}}_B \) and \( {\overline{s}}_R \), \( {\overline{s}}_G \), and \( {\overline{s}}_B \) are the analogous M- and S-cone sensitivities. Since the S-cones are insensitive in the long-wavelength (red) part of the spectrum, \( {\overline{s}}_R \) can be assumed to be zero. There are therefore eight unknowns required for the linear transformation:
$$ \left(\begin{array}{ccc}{\overline{l}}_R& {\overline{l}}_G& {\overline{l}}_B\\ {}{\overline{m}}_R& {\overline{m}}_G& {\overline{m}}_B\\ {}0& {\overline{s}}_G& {\overline{s}}_B\end{array}\right)\left(\begin{array}{c}\overline{r}\left(\lambda \right)\\ {}\overline{g}\left(\lambda \right)\\ {}\overline{b}\left(\lambda \right)\end{array}\right)=\left(\begin{array}{c}\overline{l}\left(\lambda \right)\\ {}\overline{m}\left(\lambda \right)\\ {}\overline{s}\left(\lambda \right)\end{array}\right). $$
(2)
Since only relative cone spectral sensitivities are required, the eight unknowns reduce to five:
$$ \left(\begin{array}{ccc}{\overline{l}}_R/{\overline{l}}_B& {\overline{l}}_G/{\overline{l}}_B& 1\\ {}{\overline{m}}_R/{\overline{m}}_B& {\overline{m}}_G/{\overline{m}}_B& 1\\ {}0& {\overline{s}}_G/{\overline{s}}_B& 1\end{array}\right)\left(\begin{array}{c}\overline{r}\left(\lambda \right)\\ {}\overline{g}\left(\lambda \right)\\ {}\overline{b}\left(\lambda \right)\end{array}\right)=\left(\begin{array}{c}\overline{l}\left(\lambda \right)\\ {}\overline{m}\left(\lambda \right)\\ {}\overline{s}\left(\lambda \right)\end{array}\right) $$
(3)
A definition of the cone fundamental CMFs in terms of real CMFs requires a knowledge of the coefficients of the transformation.

Spectral Sensitivity Measurements

The transformation in Eq. 3 can be estimated by comparing dichromatic and normal color matches [7]. Dichromats confuse pairs of colors that trichromats do not. When these confusions are plotted in a normal chromaticity diagram, they yield characteristic lines of confusion that converge to a different confusion point for each type of dichromat. The three confusion points correspond to the chromaticities of the missing imaginary fundamental primaries, from which the transformation matrix can be derived.

An alternative, straightforward way of estimating the transformation matrix is to measure the three cone spectral sensitivities directly. This can be achieved using steady or transient chromatic backgrounds to selectively adapt one or two of the cone types to isolate the third [8, 9]. However, cone isolation can more easily be achieved by using chromatic adaptation in observers in dichromats lacking one (or two) of the three cone types. With the S-cones selectively adapted, L- and M-cone spectral sensitivities can be directly measured in deuteranopes without M-cone function and in protanopes without L-cone function. Figure 2 shows the mean spectral sensitivity data obtained from nine protanopes (green diamonds), from seventeen single-gene L(S180) deuteranopes with serine at position 180 of their L-cone photopigment opsin gene (red squares), and from five single-gene L(A180) deuteranopes with alanine at position 180 (orange circles) [2, 10]. L(A180) and L(S180) are two commonly occurring L-cone photopigment polymorphisms in the normal human population that differ in λmax by about 2.5 nm [10]. Figure 2 also shows the mean S-cone spectral sensitivities obtained from three S-cone monochromats, who lack L- and M-cones, and under intense long-wavelength adaptation at wavelengths shorter than 540 nm obtained from five normal subjects [11]. Importantly, thanks to molecular genetics, we can now choose dichromats or monochromats for these experiments whose remaining cone photopigments are normal.

Cone Fundamentals

The spectral sensitivities shown in Fig. 2 were used by Stockman and Sharpe [2] to find the linear combinations of the \( \overline{r}\left(\lambda \right) \), \( \overline{g}\left(\lambda \right) \), and \( \overline{b}\left(\lambda \right) \) CMFs that best fit each measured cone spectral sensitivity, allowing adjustments in the densities of pre-receptoral filtering and photopigment optical density in order to account for differences in the mean densities between different populations and different target sizes. The transformation matrix for the Stockman and Sharpe 10-deg cone fundamentals is
$$ \left(\begin{array}{ccc}2.846201& 11.092490& 1\\ {}0.168926& 8.265895\;& 1\\ {}0& 0.010600\;& 1\end{array}\right)\left(\begin{array}{c}\overline{r}\left(\lambda \right)\\ {}\overline{g}\left(\lambda \right)\\ {}\overline{b}\left(\lambda \right)\end{array}\right) $$
(4)
where \( \overline{r}\left(\lambda \right) \), \( \overline{g}\left(\lambda \right) \), and \( \overline{b}\left(\lambda \right) \) CMFs are the Stiles and Burch CMFs measured with a 10-deg diameter test field [6]. The Stockman and Sharpe 2-deg estimates are based on the same transformation but the cone fundamentals have been adjusted to macular and photopigment optical densities appropriate for a 2-deg target field. The 2-deg functions are shown in Fig. 3 as the solid-colored lines. The Stockman and Sharpe 2-deg and 10-deg functions have been adopted by the Commission Internationale de l′ Éclairage (CIE) as the 2006 physiologically relevant cone fundamental CMFs [3].

The quality of cone fundamentals depends not only on the correct transformation matrix but also in the CMFs from which they are transformed. The Stiles and Burch 10-deg CMFs [6], which were measured in 49 subjects from approximately 390–730 nm (and in nine subjects from 730 to 830 nm), are probably the most secure and accurate set of existing color matching data. Other CMFs are less secure and typically flawed [12].

Most other cone fundamentals are also given in the form of Eq. 4 but for different underlying CMFs [1, 2, 13, 14, 15, 16]. The most widely used have been those by Smith and Pokorny [1]. Their transformation matrix is
$$ \left(\begin{array}{ccc}0.15514& 0.54312& -0.03286\\ {}-0.15514& 0.45684\;& 0.03286\\ {}0& 0.00801\;& 1\end{array}\right)\left(\begin{array}{c}\overline{x}\left(\lambda \right)\\ {}\overline{y}\left(\lambda \right)\\ {}\overline{z}\left(\lambda \right)\end{array}\right) $$
(5)
where \( \overline{x}\left(\lambda \right) \), \( \overline{y}\left(\lambda \right) \), and \( \overline{z}\left(\lambda \right) \) are the Judd-Vos-modified 2-deg CMFs [15].

Figure 3 shows the Smith and Pokorny estimates as dashed lines and for historical context the much earlier estimates obtained 125 years ago by König and Dieterici [17] as symbols. For the L- and M-cone fundamentals, the discrepancies between the more modern fundamentals are mainly at shorter wavelengths; the discrepancies between the S-cone fundamentals are more extensive.

The functions mentioned here can be found at http://www.cvrl.org

Other Factors that Influence Cone Fundamentals

Factors other than the properties of the photopigment also affect the cone spectral sensitivities. They include the density of the pigment in the lens that absorbs light mainly of short wavelengths, the density of macular pigment at the fovea, and the axial optical density of the photopigment in the photoreceptor. All three factors exhibit individual differences between observers, and the last two vary with retinal eccentricity. These factors should all be taken into account when trying to predict the spectral sensitivities of an individual from standard functions such as those defined by Eqs. 4 and 5 for a given target size and eccentricity. See, for example, Brainard and Stockman [18] for further details.

Cross-References

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Visual NeuroscienceUCL Institute of OphthalmologyLondonUK