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Young is famous for stating and experimentally confirming thewave theory of light, based on his observation ofinterference bands created when a single beam of light is passed through two very narrow, closely parallel slits. His ideas were verified and extended by the French civil servant and amateur physicist Augustin Fresnel and the German optics manufacturer Joseph von Fraunhofer in the early 1820's. These wave experiments affirmedNewton's conjecture that each spectral color had a uniquephysical attribute (its wavelength or frequency) that determined theperceptual response of the eye's color receptors.

The logical crux of the trichromatic theory, nicely stated by Young, is that the almost infinite number of perceived colors could not possibly arise from an almost infinite number of different wavelength specific receptors in the eye: they couldn't all fit on the retina. So color perception must originate a small number of primary receptor cells, each tuned to a different part of the spectrum and present in every part of the retina. The blending of these different receptor outputs would create the perceived color variety. Thepainter's trick suggested this small number was three, tuned to colors that Young first suggested were red, yellow and blue (as propoosed by earlier authors, such as Palmer). By 1807, however, Young had changed his primaries tored, green and blue violet.

Nearly a century of research was necessary to confirm the existence of thesecolor receptors in the retina — by microscopic dissection of retinas in 1828, the isolation of rod photopigment ("visual purple") in 1877, and exhaustive color matching experiments in the late 19th and early 20th centuries. These confirmed the essentials of Helmholtz's theory and Young's hypothesis that additive mixtures are best modeled by the additive primary lights red, green and violet (although it turned out that the human light receptors are actually most sensitive toyellow green, green and violet wavelengths).
 

In the original version of Maxwell's color matching experiments, the viewer looked through a small lens or eyepiece to see a round patch of illumination surrounded by a darkened field (right). This patch was divided into matching semicircles. The upper half contained a constanttarget color of "white" light. This was the color to be matched. (Maxwell found that "white" light was a convenient color target because it minimized the errors introduced bychromatic adaptation and theHelmholtz-Kohlrausch effect; and it allowed easy identification of the missing primary incolorblindness.) The lower half of the field contained the additive mixture of two monochromatic "primary" lights (R,G orB) and thetest light (T), the light whose chromaticity he wanted to measure. The viewer was asked to turn a small knob to adjust a neutral density wedge filter, placed across the beam of each "primary" light, until their mixture, with the target light, produced acolor match to the "white" upper half.

Maxwell's method depends on the fact that a "white" light mixture can always be produced by the mixture of any spectral color with two of the three additive primaries. (Which two depends on the hue of the target color:G andB must be used with "yellow" to "red" wavelengths,R andB with "green" wavelengths, andR andG with "blue" wavelengths.) First, the brightnesses of the three primaries are adjusted until they match the white standard: this identifies their relative proportions in a white mixture. Then one of the three primary lights is replaced by the test color, such as an orange light or colored paper disk, and the matching is repeated. By subtracting and renormalizing the contribution of the two primaries in the second white mixture from their contribution in the three primary mixture, Maxwell was able to define the test color in terms of the quantity of three primary values it replaced. 

Maxwell organized his results as a "diagram of colors" — an equilateral triangle now more often called aMaxwell triangle. Any color can be specified in this triangle as the relative proportions of the scarlet, emerald green and blue violet primaries necessary to match it. The red color represented by the pure scarlet primary is located at the red corner of the triangle; the yellow color matched by equal mixtures of the scarlet and green primaries is located midway between the red and green corners; and the white color matched by an equal mixture of all three colors is located at the center — but only if the primaries have been adjusted to have equal luminance or tinting strength.
 

james clerk maxwell's "diagram of colors"
 
. But there was a catch. Maxwell used his color triangle to analyze the primary color composition of many common artists' pigments, only to discover thatsome pigments were more saturated than any mixtures of his three primaries could match. Thus, the artists' pigment natural gamboge (NY24) was a more intense yellow than any additive mixture of vermilion and emerald green on his color top.
 

• light metamers, different spectral emittance curves perceived as the same color

• material metamers, two different surface reflectance curves or filter transmittance curves perceived as the same color when each is viewed with the same light source (right)

• observer metamers, different spectral profiles perceived as the same color due to limitations in the viewer's visual responses (colorblindness or dark adapted vision).

Light metamers are extremely common, for two reasons: (1) lights may comprise any combination of wavelengths, from monochromatic to full spectrum; and (2) all lights are summarized by the eye as just three cone outputs. There is great flexibility in mixing lights, and great limitation in perceiving lights, so that the samecolor sensation can be produced in many very different ways.

The basis for color matching through the additive mixture ofthree "primary" lights is that light metamers can be generated by a specific combination ofa small number of monochromatic lights. This is summarized as theadditive metameric generalization: provided they arearranged correctly,three "primary" lights canalways produce a metameric color match toany color of light — provided that none of the primary lights can be matched by a mixture of the other two, and the luminance (brightness) of the lights can be freely adjusted.

The primary lights provide a common currency in whichany spectral profile can be summarized as the relative power or luminance of just three monochromatic lights. This means, in turn, that two spectral profiles can be said to produce avisual color match if their trichromatic metamers are exactly the same. In this way, metameric color matching provides a procedure by which any color can be objectively compared with any other color — the foundation of the colorimetric framework.

the colorimetric framework
 
. The usefulness of these "primary" color matches might be trivial — if they only applied in situations that were "arranged correctly". But a few principles known asGrassmann's Laws greatly increase the importance of additive metameric matches by greatly expanding therange of circumstances in which the test light and its equivalent "primary" mixture will appear to be the same color. In modern colorimetry, Grassmann's original statements are summarized as three algebraic principles:

• Additivity: if a third light is mixed in equal amounts with both the test light and the metameric mixture of three "primary" lights, the color match (metamerism) remains unchanged. (In algebra: if x = y [the colors match], then x+z = y+z [the match is unchanged].)

• Proportionality: if theluminance of both the test light and the three "primary" lights in the matching mixture are increased or decreased by an equal proportional amount (such as 10%), the color match remains unchanged. (If x = y then x*z = y*z or x/z = y/z.);

• Transitivity: If either the test light or the "primary" mixture is metameric with any third light or mixture, then either (a) the test light or (b) the "primary" mixture can be replaced by this third light, and both the additivity and proportionality laws will still govern the new color match. (If x = y and x = a, then a = y; and if also y = b, then x = b and a = b.)

Grassmann's Laws mean thata color match persists despite a change in color appearance. The test light and its matching primary mixture can be made dimmer or brighter, or mixed equally with another color of light, or replaced by a third matching light, and the two lights still appear to match, even though the color or brightness of the lights has visibly changed.

The last point means that all the color matches produced by mixing monochromatic (single wavelength) primary lights can be duplicated by passing "white" light through narrowpass (multiple wavelength) color filters, or even by changing the color of the "primary" lights themselves:one choice of "primary" lights can be replaced by another. Colorimetry builds on the fact that additive color mixing is both promiscuous and precise.

As an aside: it turns out that Grassmann's Laws are not true for many additive mixtures, most famously in theHelmholtz-Kohlrausch effect, and do not hold across changes in luminance adaptation, but they are true enough often enough so that color matches by themselves can unlock the basic perceptual structure of color.
 

First, it was discovered that Maxwell'scolor matching method, in which a trichromatic mixture is adjusted to match a "white" standard, tends to produce errors in the trichromatic specification of highly saturated colors such as monochromatic lights. So an alternative method of matching colors with a mixture of three "primaries," known as themaximum saturation method (right), was used instead.

The target color in the match is no longer white, as in Maxwell's method, but any single wavelength or complex (broadband) light mixture we choose. The viewer attempts to match this target color by a mixture of threereal (visible)RGB "primaries" of monochromatic light, typically at wavelengths around 645 nm (R), 526 nm (G) and 444 nm (B). For all moderately saturated and near white colors, this arrangement leads to a direct color match; for all wavelengths above "yellow" (~570 nm), only theR andG monochromatic primaries are needed to match a spectral hue. 

However, asMaxwell discovered, there are many colors, including many monochromatic lights, that are more saturated than can be matched by any mixture of three standard primary lights. In the conventional choice ofRGB primaries, these hues include "green", "cyan", "blue" and "violet" lights on or near the spectrum locus, and highly saturated extraspectral "magenta" and "purple" mixtures. In these cases, the third primary is mixed with the out of gamuttarget color (shown in the diagram asR added to a "blue green" monochromatic light in the upper semicircular field), to desaturate the target color — move it toward "white" — until it can be matched by a mixture of the remaining two primaries (shown in the diagram asG andB mixed in the lower semicircular field).
 

In the late 1920's W.D. Wright realized that most of these individual differences or "errors" could be eliminated mathematically. The saving resource had been described by Newton: passing a monochromatic light through a colored filterdoes not change the color of the light, it onlymakes it dimmer. So the logical fix to the "yellow" filtering of light was to eliminate differences in theperceived brightness and tinting strength of the monochromaticRGB primary lights.
 

wright's WDW chromaticity coefficients

 
. In order to revert these WDW corrected color matching proportions back to the relative proportions of monochromatic light required in the original metameric mixtures, or equate the colormatching results from different colored "primary" lights, the proportionalradiance (radiant power) orluminance of the three lights must also be specified.

Either choice determines a fourth primary color, thepure white that results when the three monochromatic lights are mixed. The location of this mixture in a chromaticity diagram is called thewhite point — the only color with exactly zero chroma. Thus,all trichromatic systems are defined by four primary lights — referred to as thecardinal stimuli.
 

RGB color matching functions

• The earliest color matching curves (circa 1890 to 1930) were based on a2° or foveal presentation of color stimuli. (A 2° visual field is about the size of 1 inch viewed from a distance of 29 inches, or 10 cm viewed from 2.9 meters.) This was done because it was not then technically feasible to produce bright and homogeneous color stimuli across large visual areas, and because it minimizedrod intrusion, or the mixture of rod and cone responses, in color matching of dim lights. As a result, the 2° mixtures are more robust across moderate ("reading light") to bright (noon daylight) levels of illumination. These curves are denoted by the date they were adopted (1931) or the field size (2°). Unfortunately, these early studies combined data using different primary lights (including filtered "white" lights), and estimated the relative luminances of the primary lights from the 1924luminous efficiency function, which was later found to underestimate the luminance of short wavelengths. Judgmental revisions were imposed in 1951 to correct for this, but the uncorrected curves are still occasionally used.

• Later (circa 1960), color matching curves were measured in a10° or wide field presentation of color areas that extended well outside the fovea and filtering by themacular pigment. (A 10° visual field is about the size of 1 inch viewed from 5 3/4 inches, or 10 cm viewed from 57 cm.) These curves are based on two large, carefully screened samples of subjects who viewed monochromatic "primary" light mixtures at radiances well above rod saturation (except for extreme "red" mixtures, which were corrected for rod intrusion). These curves are also denoted by the publication date (1964) or the field size (10°). I use the wide field data throughout this site because human wide field color discrimination is about 2 to 3 times more accurate than foveal color discrimination, and because the "primary" light radiances were directly measured rather than inferred from a flawed luminous efficiency function; the 10° curves are also preferred for industrial colorimetry. The drawback is that the curves are most accurate at daylight levels of illumination and can succumb to additivity failures undermesopic light levels. 

. But we're still not done. The tristimulus values contain negative (subtracted)r(λ) andg(λ) quantities for the out of gamut wavelengths in the original color matching mixtures. As these negative values conceptually amount to saying that the photoreceptor pigments sometimesemit rather than absorb light, and as they are both an artifact of the maximum saturation method with real primary lights and of the overlap in theL,M andS cone fundamentals, a mathematical manipulation is applied to create three newimaginary XYZ primaries (as shown in the diagramabove right).

The new mixture proportionsx(λ),y(λ) andz(λ) are found by multiplying theRGB tristimulus values at each wavelength by atransformation matrix to get theXYZ color matching functions:

x₁₀(λ) = 0.341r₁₀(λ) + 0.189g₁₀(λ) + 0.388b₁₀(λ)

y₁₀(λ) = 0.139r₁₀(λ) + 0.837g₁₀(λ) + 0.073b₁₀(λ)

z₁₀(λ) = 0.000r₁₀(λ) + 0.040g₁₀(λ) + 2.026b₁₀(λ)

where as beforeλ denotes a specific wavelength.

This transformation does not undo thedesaturating physiological obstacle to the measurement of color vision — it just turns it on its head as asupersaturated definition of the primary lights! That is, theXYZ primaries are outside the gamut of all real colors and are thereforeinvisible. No color of light or surface can reproduce them. However, their imaginary mixing triangle completely contains the chromaticity space of all real, visible colors, so all colors can be described as thepositive mixture of theXYZ imaginary lights. 

So here, at last, are the 10° (wide field)XYZ color matching functions:
 

1964 XYZ color matching functions

 
. Each of the tristimulus values combines information about chromaticity (radiance in its part of the spectrum) and luminance (its part of the radiance across the whole spectrum). To obtain coordinates for a two dimensionalchromaticity diagram, the effect of luminance on the tristimulus values ismade constant, which is done bynormalizing the values. That is,X andY (thex₁₀(λ) andy₁₀(λ) values summed across all wavelengths) are divided by the sum of all three tristimulus values:

The normalizedz value is redundant, since the normalized weights sum to 1.0, soz can be recovered by subtraction:

z =1.0 –x –y.
 

 
. The WDW correctedXYZ values, in the form of thex,y chromaticity diagram andY luminous efficiency function, represent theCIE Yxy standard observer — first published from 2° color matching data in 1931 and supplemented by 10° data in 1964. The standard observer is anidealized human retina that does very well at the limited task of predicting additive color mixtures and color matches that are viewed in isolation and atmid mesopic to photopic luminance levels.

Color matching data, repeated and varied across hundreds of different color samples and viewing situations for both normal and color deficient viewers, and archived in the color vision literature withlight sensitivity andhue cancellation data, are the foundation texts of color research. They provide a quantitative basis for evaluating theories of color vision, and they stand for all the practical situations in which people say two differentcolor stimuli do or do not create the samecolor sensation. 

In addition, differenttransformation matrices can be used to convert theXYZ color matching curves intocone sensitivity curves, or to define theL*,u* andv* uniform perceptual dimensions of theCIELUV color model, or to define theL*,a* andb* opponent dimensions of theCIELAB orCIECAM02 color appearance models. Here, for example, are the CIE transformation equations that define theequal area cone fundamentals:

L(λ) =  0.390X(λ)+0.690Y(λ)–0.079Z(λ)

M(λ) =–0.230X(λ)+1.183Y(λ)+0.046Z(λ)

S(λ) =  1.000Z(λ).

Smith & Pokorny devised population weighted curves from the 2° standard observer so that theL andM functions sum to the photopic sensitivity functionV(λ):

L(λ) =  0.15516X(λ)+0.54308Y(λ)–0.03287Z(λ)

M(λ) =–0.15516X(λ)+0.45692Y(λ)+0.03287Z(λ)

S(λ) =  1.000Z(λ).

And here are Stockman, MacLeod & Johnson (1993) 10° cone fundamentals, calculated from the 1964 10° standard observer:

L(λ) =  0.23616X(λ)+0.82643Y(λ)–0.04571Z(λ)

M(λ) =–0.43112X(λ)+1.20692Y(λ)+0.09002Z(λ)

S(λ) =  0.04056X(λ)–0.01968Y(λ)+0.48619Z(λ)

We might pity an observer who, like Lieutenant Kije, exists only in official documents, but he has had an unusually productive career. TheXYZ color matching functions enable electronic color measurement in thousands of practical applications. Light intensities are measured through three colored filters or photometric diodes with transmission profiles that match each function, or light intensities are measured at regular (usually 1 nm to 5 nm) intervals across the spectrum, then multiplied by thex₁₀,y₁₀ andz₁₀ weights at each point and summed to get the totalXYZ tristimulus values. These estimate the color's brightness and chromaticity as it appears to a "typical" viewer with "normal" color vision. 

The chromaticity diagram, and the trichromatic or trilinear system it is based on, have several cool and useful properties — as backround, you may want to refer to the discussion of thetrilinear mixing triangle:

• Three Number Color Specification. Every visible color must lie within the chromaticity diagram, which meansall possible colors can be defined as a proportional mixture of theXYZ primary lights.

• Illuminant Adaptation. The definition of thewhite point in the chromaticity diagram can be adjusted in relation to a second set ofXYZ values, which define the brightness and chromaticity of the light source. In this way the standard observer can "adapt" to a wide range of viewing situations and predictilluminant based metamers.

• Luminosity Specification. TheYxy system defines thebrightness of a color as its totalY value; its lightness is the ratio between theY values of the color and a white surface under the same illumination.

• Hue Specification. Thedominant wavelength of a color is defined as the point where the spectrum locus intercepts a line drawn from thewhite point through the color'sx,y location in a chromaticity diagram.

• Chroma Specification. Thehue purity of a color is approximately defined by its chromaticity distance from the white point. (In the originalx,y chromaticity diagram, the relationship between chromaticity distance and chroma varied across hue, a problem largely fixed inCIELAB.)

• Straight Mixing Lines. The mixtures between any twocolors of light, defined as points within the chromaticity diagram, are described by a straight line between them. (Beware! Mixtures ofcolors of paint, because they aresurface colors, do not makestraight lines in a chromaticity diagram or color space!)

• Visual Complements. The visual complement of any hue is defined by a straight line from that hue through the white point to the opposite side of the diagram. Thus, the visual complement of a "deep yellow" (580 nm) is a "greenish blue" (480 nm).

• Perceived Color Differences. The distance between any two colors on the chromaticity diagram approximates theperceived difference between the colors. Unfortunately, this approximation is very poor in the originalx,y chromaticity diagram andY brightness measure, but was substantially improved in the CIELUVu,v chromaticity diagram (for emitting colors) and inCIELAB andCIECAM (for reflecting colors).

The intellectual elegance of this colorimetric edifice, built over a century of continuous work, is that all the essential information about anunrelated color — any color seen in isolation, from a single wavelength light to the most complex surfacereflectance curve — can be captured, compared and contrasted with other colors through the mechanism of three numbers.

 
. The primary paradoxes were already known in the 18th century, at least in the recognition that paints and dyes could only"imitate the hues (though not always the splendor) of those almost numberless differing colors" of nature, and that differentcolors (different pigments or dyes) could equally serve as a "primary" red, yellow or blue.

Unfortunately, the recognition came long before scientific explanations of overlapping photoreceptor sensitivities, additive mixtures and gamut mapping were available. As a result,four misconceptions developed in the 18th century to explain the problems with primary color mixtures ... and many artists repeat them even today.

The first misconception is that"primary" colors must bevisible colors, in the sense that an artist can pick a color of sticky paint or a wavelength of visible light and say, "there,that is the primary yellow". But the first primary paradox shows this belief is false. The "primary" colors that describe color vision are imaginary colors — and the mind never has direct experience of the "primary" signals from our three photoreceptors. The choice is not between one shade of color and another but between avisible color and animaginary color. It is no more possible to find a paint that matches "primary" yellow than it is to find a horse that matches Pegasus.

The second misconception is that"primary" colors must bespecific colors, in the sense that an artist can pick one color of primary yellow paint as the "nearest match" to the "true" primary yellow color, or that one primary yellow paint is the "best" primary yellow. (See for example the "color theory" book byJim Ames.) But, as we have seen, the selection of real colorants is always arbitrary: it has much more to do with perceived image quality than with "objective" color characteristics. Almostany three colors can serve as primary colors, depending on how you want to use them. The only relevant issues are (1) theactual range of mixtures (gamut) you are able make with the colorants, and (2) whether this gamut produces the desired visual effect in the images you want to represent.

The third misconception is that none of the various colors of primary paints are the "true" primary color becauseall paint colors are "impure" or polluted by added light from the other two primaries. (See for example the justification offered for thesplit "primary" palette.) This is an especially quaint anachronism from the 18th century, and it is wrong from several points of view. If a paint really were "pure" and only reflected a single wavelength of light (which is the "purest" possible color stimulus), the paint would have a luminance factor near zero and would appear blacker than the "purest" black paint! And that monochromatic "yellow" light is no more saturated than a mixture of a monochromatic "orange" and "yellow green" light, solight purity isnot the cause ofhue purity. Finally, even if our primaries were three "pure" colors of light (regardless of their hue), we stillcouldn't mix all other colors. "Purity" or pollution has absolutely nothing to do with the limitations of primary colors of paints or dyes.