3.2 Perceptual color and trichromatic vision⧉
A display can match the yellow of a flower without emitting the flower's spectrum. Different physical lights can produce the same visual response. That makes color reproduction possible, but also means that a color match tells us little about the underlying spectrum.
We distinguish the light's spectrum, the cone responses it produces, and its appearance in a particular context. Color matching is a measurable relation between stimuli for an observer under specified conditions. A standard observer summarizes such measurements; it does not eliminate variation between people or predict every aspect of appearance.
3.2.1 Cone photoreceptors and perceptual color⧉
Even though cones are known to be the color vision receptors, a single cone does not measure wavelength; it produces one number, and stimuli of different wavelengths can yield the same response.
A given cone type $k$ has a spectral sensitivity $c_k(\lambda)$: a curve saying how strongly it responds to light of each wavelength (Figure 3.2.1).
For a fixed state of adaptation, we model the initial cone excitation from an energy spectrum $E(\lambda)$ as a weighted integral, the spectrum weighted by its sensitivity and summed over all wavelengths (Figure 3.2.2):
This linear excitation model precedes the nonlinear adaptation and neural response. Use energy-based sensitivities with an energy spectrum, or quantum-based sensitivities with photon counts; mixing the two conventions gives a wavelength-dependent error. Mathematically the integral is a continuous dot product. In the case of monochromatic light (a single wavelength $\lambda$) it reduces to the scalar product of the cone's sensitivity at $\lambda$ and the light's intensity for $\lambda$.
As Figure 3.2.3 shows, lights at different wavelengths can yield the same response if their intensity ratio is the inverse of the cone's spectral response ratio for those two wavelengths. A single cone conflates wavelength with intensity, a property called univariance.
While a single cone cannot distinguish wavelength, the combination of three cone responses provides much more information (click on "show other cones" in Figure 3.2.3 ). This is the trichromatic theory of color vision, pioneered by Thomas Young and put on a quantitative footing by Hermann von Helmholtz.
For photopic color matching, a fixed observer is modeled by three cone excitations $(r_L,r_M,r_S)$. Equal excitations predict a match when spatial, temporal, and adaptation conditions are held fixed. Rod contributions, receptor differences between observers, and contextual appearance require additional considerations.
Hermann von Helmholtz (1821–1894) was the physicist and physiologist who turned Thomas Young's guess into a working theory. Young had argued decades earlier that the eye cannot possibly carry a separate detector for every wavelength and must instead make do with a few receptor types; Helmholtz developed this quantitatively, and the three-receptor account of color vision is still called the Young–Helmholtz theory. His Handbook of Physiological Optics founded the field, and he invented the ophthalmoscope that let physicians look into the living eye. Two of his ideas run straight through this chapter. The first is trichromacy itself, the L/M/S projection that everything here is built on. The second is unconscious inference, his claim that perception is not a direct readout of the retina but an inference the brain makes about the most likely cause of its input. That is exactly the stance we take later on color constancy, white balance, and "the dress," where the visual system guesses the illuminant and quietly discounts it. Beyond vision he is everywhere in physics as well, from the conservation of energy to the Helmholtz free energy and the acoustic resonators that carry his name. Portrait: 19th-century photograph, public domain (via Wikimedia Commons).
To compute the integral, sample an energy spectrum at wavelengths $\lambda_i$ with bin widths $\Delta\lambda_i$. Put the spectral-density values into $\mathbf E$, and define the matrix entries as $C_{ki}=c_k(\lambda_i)\Delta\lambda_i$. Then
Each term is the excitation contributed by one wavelength bin. The bin width matters: doubling the number of samples should improve the approximation, not double the response. With nonuniform samples, use the appropriate quadrature weights. Figure 3.2.5 illustrates the matrix computation.
response = [0, 0, 0] for each wavelength bin i: for each cone type k: response[k] += spectrum[i] * sensitivity[k, i] * bin_width[i]
Keep spectrum and sensitivity on the same wavelength grid and use one wavelength unit throughout.
Each cone excitation is an integral of spectral density times the appropriate sensitivity. A sampled implementation must include wavelength-bin widths (or absorb them into the matrix). Three excitations summarize photopic color matching for a fixed observer and viewing condition; they do not recover the spectrum.
We established above that a single cone cannot distinguish wavelengths. Only by comparing the three cones' outputs can the brain begin to separate the two. However, this ability is limited. This is unsurprising because trichromatic vision summarizes a whole spectrum with only three numbers.
The three-bin examples below are a model of the spectrum, not three isolated cone channels. A bin containing long-wavelength light generally excites both L and M cones. Changing its power changes their ratio as well as their absolute responses. Equal bin powers need not produce equal cone excitations or a neutral white: those depend on the bin boundaries, sensitivity normalization, and reference white. The displayed swatches show an RGB approximation of the resulting color, limited by the screen's gamut.
A few cases make the three-number code concrete. Put the power in the long-wavelength bin and L responds most, M somewhat less, S hardly at all: red. Put it in the short bin and S leads: blue. In the middle bin, M leads with L close behind: green. Fill the long and middle bins together, with little short, and L and M both respond strongly, with L further ahead than for the middle bin alone: yellow, a color with no bin of its own, which is the sign that yellow sits next to green in cone space. All three bins at comparable power give a color near white, depending on the normalization. Orange (mostly long, some middle), cyan (middle plus short) and purple (long plus short, skipping the middle) fill in the hue circle. Every named color is a ratio of three cone catches, and very different spectra can land on the same ratio.
3.2.2 Metamers⧉
Different nonnegative spectra can produce the same cone-response triple. Such spectra are metamers for that observer and viewing condition. In the sampled model, a spectral perturbation in the null space of $\mathbf C$ leaves the responses unchanged, provided the modified spectrum remains nonnegative (Figure 3.2.8). Displays exploit this: they reproduce an in-gamut color by emitting a metamer rather than the original spectrum. Three fixed physical primaries cannot reproduce every visible color (Chapter 4.1, Linear color reproduction and CIE XYZ).
Different spectra can yield the same cone excitations for a specified observer. This enables color reproduction without reproducing the spectrum, but a fixed display can match only colors within its gamut. Matches between reflecting materials can fail when the illuminant changes.
Overlapping cone sensitivities limit the response combinations obtainable by changing a spatially uniform spectrum. Fong, Doyle, Wang and colleagues used adaptive optics, cone classification, and eye tracking to direct light preferentially to selected cone types Fong et al. 2025 (Oz / "olo") Roorda & Williams 1999. Observers reported a highly saturated blue-green, named “olo,” outside the gamut of conventional spatially uniform spectral stimulation. The system changes where light lands on the receptor mosaic; it still uses light. A conventional display cannot reproduce that experience. See Chapter 37.6, Retinal projection.
Dolby 3D uses separate spectral bands to deliver different images to the two eyes. Each eye receives its own red, green, and blue primary bands through matching filters Jorke & Fritz 2003 (Infitec / Dolby 3D). Color correction seeks to match the two eyes' rendered colors within the gamuts the primary sets share. The individual red bands need not be metamers by themselves: the match uses mixtures of each set of primaries. Filter leakage, viewing angle, and observer differences limit the separation and color match.
Two fabrics can match in a shop and differ outdoors. For a nonfluorescent surface in the simple reflectance model, the spectrum reaching the eye is proportional to illuminant times reflectance. A change of illuminant weights the reflectance differences differently, so equality of the three integrals need not survive (Figure 3.2.13). Narrow spectral peaks can make matches sensitive to particular reflectance differences, but smooth illuminants can also break a match. Color-critical work specifies the illuminant and observer. Figure 3.2.11 shows a printed metameric pair under daylight and halogen illumination.
A metameric match is a match between two products of reflectance and illuminant (L2.6). Swap the illuminant and you multiply by a different spectral shape, so two surfaces that looked identical under one light can diverge under another: illuminant metamerism. It is the headache behind prints that shift between the proofing booth and the living room, and fabrics that match in the shop but clash in daylight; color-critical work must always name the light a match is judged under.
Consider a printer matching a fabric. The inks and the fabric can have quite different reflectance spectra while sending the same three cone responses to an observer under the proofing light. Moving both under another light changes the wavelength-by-wavelength weights before those responses are summed. The differences that canceled under the first light need not cancel under the second. A narrow spectral peak can emphasize a region where the two reflectances differ; a smooth change of illumination can also break the match. This is why specifying only three color coordinates does not promise a match under every illuminant. A proofing agreement needs a viewing illuminant and observer as well as a numerical tolerance.
3.2.3 Color blindness⧉
In dichromatic vision, two functionally distinct cone classes supply the photopic color signal. More spectra then share the same response than for a trichromatic observer. In anomalous trichromacy, three classes remain but their spectral sensitivities differ from the usual model. This can reduce discrimination between particular colors. “Color blindness” covers several conditions; it does not usually mean seeing only gray.
Protan and deutan deficiencies affect the L/M system; tritan deficiencies affect the S-cone system. Complete achromatopsia involves severely impaired cone function and affects acuity and light sensitivity as well as color. Oliver Sacks's The Island of the Colorblind describes its unusually high prevalence on Pingelap Sacks, The Island of the Colorblind. These conditions cannot all be modeled by simply deleting an RGB channel.
Common inherited red–green deficiencies are associated with L/M opsin genes on the X chromosome and occur more frequently in males. Prevalence varies by population. Other deficiencies have different genetic or acquired causes (National Eye Institute).
For imaging interfaces, do not make hue the only way to distinguish information. Use labels, shapes, or adequate lightness contrast as well. Simulations can reveal likely confusions, but they do not reproduce every observer's experience. Screen versions of test plates depend on the display and viewing conditions and should not be treated as a diagnosis. Recoloring and spectral filters can change which differences are visible; they do not add a missing receptor class.
Anyone can check their own color vision in a few minutes. Ishihara plates screen for red-green deficiency; arrangement tests such as the Farnsworth–Munsell 100 hue test, in which the observer orders colored caps around the hue circle, measure discrimination along that circle, and the pattern of errors shows which axis is affected. Online versions are useful screening, with the display caveats above. Help comes in two kinds, neither a cure. Software daltonization remaps an image to push confusable colors apart, and phones and design tools can simulate a deficiency so a designer can check a chart. Notch-filter glasses remove a narrow band where the L and M sensitivities overlap most, which can enlarge the difference between their responses for some anomalous trichromats; they add no pigment and do little for a dichromat.
Both tools start from the same model. To simulate a dichromat, convert each pixel to cone responses and replace them with the closest triple the dichromat's two remaining cone classes cannot tell from it: the response of the missing class is filled in from the other two, so the pixel moves onto a surface in cone space where every color it could be confused with lands on the same point. Brettel, Viénot and Mollon built that surface from two half-planes anchored on colors that dichromats and trichromats describe alike Brettel, Viénot & Mollon 1997; Machado, Oliveira and Fernandes extended the model to anomalous trichromacy by shifting a pigment's sensitivity by a chosen amount rather than removing it Machado, Oliveira & Fernandes 2009. Daltonization then uses the difference between the original and the simulated image, which is exactly the information the viewer loses, and adds it back along a direction the viewer can see, typically into lightness or the blue-yellow axis. Early web tools such as Vischeck made simulation widely available; the same computation now ships in accessibility checkers and in browser developer tools. A simulation predicts which colors collapse, not what the viewer experiences, and daltonization trades the natural look of the colors for their separability.
The genetics explains much of the difference in prevalence. The L- and M-opsin genes are close together on the X chromosome. Recombination between these similar sequences can remove a gene or produce a hybrid pigment gene, changing the set of spectral sensitivities. With only one X chromosome, an affected copy is sufficient for an X-linked deficiency; with two, another functioning copy can protect against it. MedlinePlus Genetics gives about one in twelve males and one in two hundred females for inherited red-green deficiency in populations of Northern European ancestry, with lower frequencies in many other populations. Blue-yellow deficiencies have different inheritance and are much rarer.
| Type | What changes | Males | Females |
|---|---|---|---|
| Protanopia | L cones absent | about 1% | about 0.02% |
| Protanomaly | L pigment shifted | about 1% | about 0.03% |
| Deuteranopia | M cones absent | about 1% | about 0.01% |
| Deuteranomaly | M pigment shifted | about 5% | about 0.4% |
| Tritanopia | S cones absent | about 0.002% | about 0.002% |
| Achromatopsia | little or no cone function | about 0.003% | about 0.003% |
Approximate prevalence in populations of European ancestry, after Sharpe and colleagues Sharpe et al. 1999; red-green rates are lower in many other populations, and published estimates vary with the test used.
This is a useful distinction for the projection model. Anomalous trichromacy changes the measurement curves but retains three functionally distinct classes. Dichromacy reduces their number to two. Monochromacy is a further loss of chromatic discrimination, although different clinical conditions can produce it. A software simulation should model the relevant change in sensitivities, not merely turn off the screen's red or green primary.
Trichromacy itself is also a limitation: three measurements cannot identify an arbitrary spectrum. A trichromat can distinguish some pairs that a dichromat cannot, while both discard most of the spectral information. Calling one observer color deficient should not obscure the same information loss in the standard observer used by imaging systems.
3.2.4 Opponent process and the multistage model⧉
A yellow light need not stimulate a “yellow cone.” Comparisons between cone signals contribute to chromatic processing. Physiological opponent responses and psychophysical hue-cancellation measurements support multistage accounts of color vision De Valois et al. 1966 Hurvich & Jameson 1957. They do not establish that one fixed matrix explains unique hues, afterimages, and color names.
A useful teaching model forms one sum and two differences of the cone signals. For cone contrasts relative to positive background excitations $r_{k,0}$, define
Here $q$ averages the L and M contrasts, $a$ compares them, and $b$ compares S contrast with that average. Equal fractional increases in all three cone excitations change $q$ but leave $a$ and $b$ zero. This is a simplified opponent representation, not a calibrated model of an individual observer's luminance or hue perception.
Derrington, Krauskopf, and Lennie studied chromatic mechanisms in macaque lateral geniculate neurons Derrington, Krauskopf & Lennie 1984 (DKL cardinal directions). The resulting DKL framework describes modulations around an adapting background along luminance and chromatic cardinal directions. Its numerical transform depends on cone fundamentals, background, luminance weights, and axis normalization. Our equal-weight matrix illustrates sum-and-difference processing; it should not be substituted for a calibrated DKL transform in an experiment. Cardinal directions also need not coincide with an observer's unique red, green, blue, and yellow.
Cone excitations are followed by neural comparisons and adaptation. Sum-and-difference models illustrate this processing, but their axes are not exact perceptual hue axes. Engineering luma/chroma transforms exploit related sensitivity differences without reproducing the neural pathway.
Afterimages involve adaptation across the visual system, including cone and postreceptoral mechanisms (Chapter 3.3, Adaptation and constancy). Compression exploits lower sensitivity to fine chromatic detail, but that is an empirical property of spatial vision, not a consequence of taking differences alone. The engineering channel $Y\prime$ is luma computed from nonlinear RGB, distinct from physical luminance. Thin colored text and saturated edges can reveal chroma-subsampling artifacts.
3.2.5 So what are the primary colors?⧉
A screen adds light from its primaries, usually red, green, and blue. Inks and filters instead modify the spectrum of incident light; cyan, magenta, and yellow are useful subtractive choices. Neither list is a property of light itself. Different primary spectra give different gamuts.
Psychological unique hues describe appearances such as a yellow that seems neither reddish nor greenish. They are not display primaries or cone classes. Traditional red–yellow–blue painting instruction is another use of “primary”; actual pigment mixtures depend on absorption and scattering, and their gamut depends on the pigments chosen. Chapter 4.7, Synthesizing color develops these physical mixing processes.
Specify the purpose when naming primaries: sources whose light is added, inks or filters that absorb light, or perceptual hue judgments. RGB and CMY describe common engineering choices, not unique sets required by nature. The chosen spectra and mixing process determine what can be reproduced.
The ambiguity is why an argument about “the primary colors” so often has no useful answer. A physiologist may mean receptor classes, a painter may mean a set of pigments for mixing, and a display engineer may mean the emitters whose powers can be controlled. A perceptual experiment may instead ask which hues appear unmixed. State which experiment or technology the word refers to before choosing a list. Red-yellow-blue painting instruction is a mixing convention, not a competing account of the cone sensitivities.
3.2.6 Color as linear algebra⧉
Cone sensing is a linear measurement from many spectral samples to three excitations. It is not an expansion of the spectrum in three basis functions. The sensitivity curves are measurement weights; three of them cannot span an arbitrary spectrum.
Reproduction asks a different question. Put the cone-response triples of three chosen primary lights in the columns of a $3\times3$ matrix $\mathbf A$. To match target responses $\mathbf r$, solve
The entries of $\mathbf p$ are primary powers. Even if $\mathbf A$ is invertible, a negative entry means the target cannot be emitted by adding those primaries. A finite display also imposes upper limits. This separates two issues: solving a linear system and producing its solution with physical light. Chapter 4.1, Linear color reproduction and CIE XYZ develops the distinction.
A two-dimensional basis-and-dual diagram (Figure 3.2.20) illustrates why dotting with nonorthogonal synthesis vectors does not directly recover their coefficients. It is a linear-algebra analogy, not a model in which cone curves form a basis for all spectra.
Cone sensitivity curves overlap, and physical spectra and emitted primary powers are nonnegative. Cone responses are measurements, not spectrum coefficients. Matching a response triple with three primary lights requires solving for their powers; a negative solution identifies a color outside those primaries' additive gamut.
The overlap of the sensitivity curves and the nonnegativity of light are separate constraints. Because L and M sensitivities overlap strongly, an ordinary spectral change usually affects both responses. We cannot freely prescribe one response while holding the others at zero simply because three coordinates are available on paper. A mathematical solution can call for subtracting light at some wavelengths; a physical source can only supply a nonnegative spectrum. This is the connection between the cone-sensitivity plot and the gamut restrictions on the reproduction solve.
Recap: big lessons of this chapter
Each cone excitation is an integral of spectral density times the appropriate sensitivity. A sampled implementation must include wavelength-bin widths (or absorb them into the matrix). Three excitations summarize photopic color matching for a fixed observer and viewing condition; they do not recover the spectrum.
Different spectra can yield the same cone excitations for a specified observer. This enables color reproduction without reproducing the spectrum, but a fixed display can match only colors within its gamut. Matches between reflecting materials can fail when the illuminant changes.
A metameric match is a match between two products of reflectance and illuminant (L2.6). Swap the illuminant and you multiply by a different spectral shape, so two surfaces that looked identical under one light can diverge under another: illuminant metamerism. It is the headache behind prints that shift between the proofing booth and the living room, and fabrics that match in the shop but clash in daylight; color-critical work must always name the light a match is judged under.
Cone excitations are followed by neural comparisons and adaptation. Sum-and-difference models illustrate this processing, but their axes are not exact perceptual hue axes. Engineering luma/chroma transforms exploit related sensitivity differences without reproducing the neural pathway.
Specify the purpose when naming primaries: sources whose light is added, inks or filters that absorb light, or perceptual hue judgments. RGB and CMY describe common engineering choices, not unique sets required by nature. The chosen spectra and mixing process determine what can be reproduced.
Cone sensitivity curves overlap, and physical spectra and emitted primary powers are nonnegative. Cone responses are measurements, not spectrum coefficients. Matching a response triple with three primary lights requires solving for their powers; a negative solution identifies a color outside those primaries' additive gamut.