207

Colorimetry

Andrew StockmanInstitute of Ophthalmology, 11–43 Bath Street, London EC1V 9EL, UKPhone: +44 (0)20 7608 6914; Fax: +44 (0)20 7608 6850; e-mail: a.stockman@ucl.ac.uk

AbstractColorimetry is concerned with the measurement and numerical specification of the colorof visual stimuli. Color depends on the underlying biology of the human visual system,and in the first instance on the spectral properties of the three types of light-sensitivecone photoreceptors in the eye. Color and color matches are trichromatic, meaning theycan be specified by three variables that are linearly related to the outputs of the threecone types. The primary aim of colorimetry is to be able to specify those variables forlights of any spectral composition.

Keywordscolor matching; color matching functions; cone fundamentals; colorimetry;photometry; trichromacy; univariance; cone spectral sensitivities; chromaticitydiagrams; color, CIE.

1 Introduction 2081.1 Trichromacy and Univariance 2091.2 Color Matching 2091.2.1 Maximum Saturation Method 2101.2.2 Maxwell’s Matching Method 2111.2.3 Linearity and Additivity of Color Matches 2121.2.4 Complex Stimuli 2121.2.5 Transformability of CMFs 2121.2.6 Specificity of CMFs 2131.2.7 Existing CMFs 214

208 Colorimetry

2 Cone Spectral Sensitivities 2143 Factors that Influence Color Matches 2153.1 Individual Differences 2153.1.1 Lens Pigment 2153.1.2 Macular Pigment 2153.1.3 Photopigment Optical Density 2153.1.4 Photopigment Variability 2163.1.5 Color Deficiency 2173.2 Changes with Eccentricity 2173.3 Bleaching 2173.4 Rod Influence 2174 Color Spaces 2184.1 Two- and Three-dimensional Color Spaces 2184.2 Other Cone Spaces 2194.2.1 Equal-luminance Cone Excitation Space 2194.2.2 Cone Contrast Space 2204.3 Uniform Color Spaces 2205 Existing Color-matching Functions 2215.1 CIE (1931) 2-deg CMFs 2215.2 Judd, Vos-modified CIE 2-deg CMFs 2215.3 Stiles & Burch (1955) 2-deg CMFs 2215.4 Stiles & Burch (1959) 10-deg CMFs 2215.5 CIE (1964) 10-deg CMFs 2225.6 Luminance and the Luminosity Function, V(λ) 2226 Conclusion 222

Acknowledgments 223Glossary 223References 224Further Reading 225

1Introduction

Colorimetry is a branch of color sciencepertaining to the measurement and nu-merical specification of the color of visualstimuli. Since perceived color is a propertyof the human eye and brain, and not aproperty of physics [1], colorimetry is inex-tricably linked to the underlying biology ofvision (see also PHYSIOLOGICAL OPTICS).

Color and color perception are limitedat the first stage of vision by the spectral

properties of the layer of light-sensitivephotoreceptors that carpet the rear sur-face of the eye (upon which an invertedimage of the world is projected by theeye’s optics). These photoreceptors trans-duce arriving photons to produce thepatterns of electrical signals that eventu-ally lead to perception. Daytime (photopic)color vision depends mainly upon thethree classes of cone photoreceptors, eachwith different spectral sensitivity, whichare referred to as long-, middle- and

Colorimetry 209

short-wavelength-sensitive (L, M, and S),according to the part of the visible spec-trum to which they are most sensitive(see Fig. 5). Nighttime (scotopic) vision,by contrast, depends on a single class ofphotoreceptor, the rod.

1.1Trichromacy and Univariance

The range of colors that can be producedby the additive combination of just threelights is simulated in Fig. 1. Overlappingred, green, and blue lights produce regionsthat appear cyan, purple, yellow, and white.Other intermediate colors can be producedby varying the relative intensities of thethree lights.

A fundamental property of normalhuman vision is that it is trichromatic:observers can match a test light of anyspectral composition to a suitably adjustedmixture of just three other lights (with theproviso that one of the three primaries

Additive color mixing

Fig. 1 Additive color mixing. Simulated overlapof projected red, green and blue lights. Theadditive combination of red and green is seen asyellow, red and blue as purple, green and blue ascyan, and red, green and blue as white

invariably has to be added to the firstlight in order to complete the match – seeSect. 1.2.1, and Eq. (1)). As a result, colorscan be defined by just three variables: theintensities of the three primary lights withwhich they match (the three ‘‘Tristimulus’’values).

Trichromacy arises because there areonly three classes of cone photoreceptorin the human eye, each of which re-sponds univariantly to light in a way thatdepends on the rate of photon absorp-tion [2, 3]. Although the probability that aphoton is absorbed by a given photore-ceptor varies by many orders of mag-nitude with wavelength, its effect, onceabsorbed, is independent of wavelength.An absorbed long-wavelength photon hasthe same effect as a short-wavelength one.Consequently, a photoreceptor is like a so-phisticated photon counter, the output ofwhich varies according to the rate of ab-sorbed photons. Since a change in ratecould result from a change in the wave-lengths of the photons, from a changein the number of photons, or from both,individual photoreceptors are effectivelycolor-blind. Color is encoded by the relativeoutputs of the three, individually color-blind cone types. It is therefore trivariantor trichromatic.

For background reading on color vision,see, for example, Kaiser & Boynton (1996).See also the COLOR VISION article.

1.2Color Matching

Trichromacy means that the color-match-ing behavior of an individual can bestraightforwardly assessed by determin-ing the intensities of three independentfixed-wavelength ‘‘primary’’ lights (inde-pendent in the sense that no two primarieswill match the third) required to match

210 Colorimetry

a series of monochromatic spectral lightsspanning the visible spectrum. Two prin-cipal methods have been used to de-termine color matches: the maximumsaturation method and Maxwell’s method.Most colorimetry is based on the for-mer, though it is arguably the inferiormethod.

1.2.1 Maximum Saturation MethodThe most commonly used matchingmethod is the maximum saturationmethod, which was used by Wright [4]

and Guild [5] to obtain matches that weresubsequently used to generate the CIE1931 Color Matching Functions (CMFs)(see Sect. 5.1). As illustrated in Fig. 2(a),the observer is presented with a halffield illuminated by a ‘‘test’’ light of vari-able wavelength, λ, and a second halffield illuminated by a mixture of red (R),green (G) and blue (B) primary lights (notethat bold capitals denote primary lights).At each λ, the observer adjusts the in-tensities of the three primary lights, sothat the test field is perfectly matched by

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Fig. 2 Maximum saturation method of color matching. Amonochromatic test field of wavelength, λ, can be matched by amixture of red (645 nm), green (526 nm) and blue (444 nm)primary lights, one of which must be added to the test field tocomplete the match. The amounts of each of the three primariesrequired to match monochromatic lights spanning the visiblespectrum are known as the red, r(λ), green, g(λ), and blue, b(λ)

CMFs (red, green, and blue lines respectively) shown in the lowerpanel. The data are from Stiles & Burch [6]. A negative sign meansthat that primary must be added to the target to complete the match

Colorimetry 211

the mixture of primary lights. Figure 2(b)shows the mean r(λ), g(λ), and b(λ) CMFsobtained by Stiles & Burch [6] for pri-mary lights of 645, 526, and 444 nm.Notice that except at the primary wave-lengths one of the CMFs is invariablynegative. There is no ‘‘negative light’’;rather, these negative values indicate thatthe primary in question has been addedto the test light in order to make amatch. Real primaries always give riseto negative values because lights cannotuniquely stimulate single cone photore-ceptors, since their sensitivities overlapthroughout the visible spectrum (seeFig. 5).

The general form of the maximumsaturation match between Eλ, a monochro-matic constituent of the equal unit en-ergy stimulus, E, of wavelength λ, andthe three primaries lights (A1, A2, andA3) is

Eλ + a1(λ)A1 = a2(λ)A2 + a3(λ)A3, (1)

where a1(λ), a2(λ), and a3(λ) are the threeCMFs set by the observer. For the specialcase of the unit energy stimulus, thetristimulus values, A1, A2, and A3 (note theuppercase, italic convention), are simply

the three CMF values. Otherwise, theyare the CMF values scaled by the radiantpower of the monochromatic test stimulus.Although CMFs are usually defined for astimulus, E, that has equal unit energythroughout the spectrum, in practice, thespectral power of the test light usedin most matching experiments is variedwith wavelength. CMFs and the spectralpower distributions of lights are nearlyalways discrete functions of wavelength,typically defined in steps of 1, 5, or10 nm.

1.2.2 Maxwell’s Matching MethodThe first careful, quantitative measure-ments of color matching and trichromacywere made by Maxwell [7]. In Maxwell’smethod, which is illustrated in Fig. 3, thematched fields always appear white, so thatat the match point, the eye is always in thesame state of adaptation whatever the testwavelength (in contrast to the maximumsaturation method in which the chromatic-ity of the match varies with wavelength).In a matching experiment, the subjectis first presented with a white standardhalf field, and is asked to match it withthe three primary lights. The test light

Mixture half-fieldWhite half-field

Test (l)Green (526 nm)

“Blue” (444 nm)

Standardwhite light

Two primary lightsplus test light

White

Fig. 3 Maxwell’s method of color matching. A monochromatic testfield of wavelength, λ, replaces the primary light to which it is mostsimilar, and a match is made to the white standard by adjusting theintensities of the two remaining primaries and the test field

212 Colorimetry

then replaces the primary light to whichit is most similar and the match is re-peated.

1.2.3 Linearity and Additivity of ColorMatchesColor-matching data, obtained for spectrallights, are useful in general only if they canbe used to predict matches for nonspectrallights of any arbitrary spectral powerdistribution, and by extension for othertriplets of primary lights. Color matches,in other words, must be linear and additive.If not, a new match would have tobe established for each new matchingcondition. Grassmann’s laws embody testsof linearity and additivity. They are asfollows [8, 9]:

1. Symmetry: If light A matches light B,then B matches A.2. Transitivity: If light A matches light Band B matches light C, then A matches C.

3. Proportionality: If light A matches lightB, then nA matches nB (where n is aconstant of proportionality).

4. Additivity: If A matches B and Cmatches D, then the combination of Aand C matches the combination of B andD (and similarly the combination of B andC matches A and D).

To a first approximation, color matchingis linear and additive [9, 10].

1.2.4 Complex StimuliGiven additivity, the tristimulus values,A1, A2, and A3 for an arbitrarily complexspectral radiant power distribution, P(λ)

can be obtained from the a1(λ), a2(λ), anda3(λ) CMFs, thus

A1 =∫

P(λ)a1(λ) dλ,

A2 =∫

P(λ)a2(λ) dλ,

and A3 =∫

P(λ)a3(λ) dλ. (2)

The chromaticity coordinates (see Sect. 4)are then a1 = A1/(A1 + A2 + A3) anda2 = A2/(A1 + A2 + A3).

Since spectral power distributions andCMFs are discrete functions, the inte-gration in Eq. (2) is usually replaced bya sum.

1.2.5 Transformability of CMFsThe r(λ), g(λ), and b(λ) CMFs shown inFig. 2 are for the RGB (red–green–blue)primaries of 645, 526, and 444 nm. TheseCMFs can be linearly transformed to anyother set of real primary lights, and, asillustrated in Fig. 4, to imaginary primarylights (i.e., physically unrealizable lights),such as the X, Y, and Z primaries fa-vored by the CIE, or to the L-, M-, andS-cone fundamental primaries. Each trans-formation is accomplished by multiplyingthe CMFs by a 3 × 3 matrix [Eq. (4)]. SeeSect. 3.2.5 of Wyszecki and Stiles [8] formore details about transformations be-tween primaries.

The X, Y, and Z primaries were se-lected by the CIE because x(λ), y(λ),and z(λ) are always positive, thus sim-plifying computations using logarithms(which was important before comput-ers); and, because y(λ) is the luminosityfunction (so incorporating luminosity in-formation into the CMFs, see Sects. 5.1and 5.6).

The three fundamental primaries (or‘‘Grundempfindungen’’ – fundamentalsensations) are the three imaginary pri-mary lights that would uniquely stimulateeach of the three cones to yield l(λ), m(λ),and s(λ), or the L-, M-, and S-cone spectral

Colorimetry 213

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Fig. 4 CMFs can be linearly transformed from one set of primaries to another. Illustrated hereare CMFs (a) for R, G, and B primaries; (b) for the imaginary X, Y, and Z primaries; and (c) forthe cone fundamental L, M, and S primaries (bottom). The CMFs are Judd–Vos modified CIE1931 functions (see Sect. 5.2) and the Smith–Pokorny cone fundamentals (see Sect. 2)

sensitivity functions that underlie and de-termine all color matches. Although l(λ)

and m(λ) cannot be obtained directly fromcolor matches, they should, however, bea linear transformation of standard CMFs(in contrast, s(λ) can be determined fromcolor matches; see Sect. 2).

1.2.6 Specificity of CMFsColor-matching data are specific to the con-ditions under which they were measured,and strictly to the individual observers inwhom they were measured. By applyingthe data to other conditions and usingthem to predict other observer’s matches,some errors will inevitably be introduced.

An important consideration is the area ofthe retina within which the color matcheswere made. Standard color-matching data(see Sect. 1.2.7) have been obtained forcentrally viewed fields with diameters of

either 2 deg or 10 deg of visual angle.These visual angles refer to the anglessubtended by objects in the external fieldof view at the effective optical center ofthe eye. The size of a circular matchingfield used in colorimetry is defined asthe angular difference subtended at theeye between diametrically opposite pointson the circumference of the field. Thus,matches are defined according to theretinal size of the matching field, not itsphysical size. A 2-deg diameter field isknown as a ‘‘small’’ field, whereas a 10-deg one is known as a ‘‘large’’ field. (1 degof visual angle is roughly equivalent to thewidth of the fingernail of the index fingerheld at arm’s length.) Color matches varywith retinal size and with retinal positionbecause of changes in macular pigmentdensity and photopigment optical densitywith visual angle (see Sect. 3).

214 Colorimetry

The CMFs are mean data that are alsoknown as ‘‘standard observer’’ data in thesense that they are assumed to representthe color-matching behavior of a typical,standard observer. The color matches ofindividual observers, however, can varysubstantially from these mean matches.Individual differences in lens pigmentdensity, macular pigment density, pho-topigment optical density, and in thephotopigments themselves can all influ-ence color matches (see Sect. 3).

1.2.7 Existing CMFsThere are several existing sets of CMFs.For the central 2 deg of vision (the small-field-matching conditions), they includethe CIE 1931 CMFs [11], the Judd–Vosmodified 1931 CMFs [12, 13], and theStiles & Burch [6] CMFs. For the central10 deg of vision (the large-field-matchingconditions), they include the 10-deg CMFsof Stiles & Burch [14], and the related 10-deg CIE 1964 CMFs. The CIE functionsare available as r(λ), g(λ) and b(λ) or x(λ),y(λ) and z(λ) CMFs. The advantages anddisadvantages of these CMFs are discussedin Sect. 5. The most widely used functions,the CIE 1931 CMFs, for example, areflawed at shorter wavelengths.

2Cone Spectral Sensitivities

With the establishment of trichromaticcolor theory [15–17], a central goal in colorscience has been the definition of the lineartransformation between r(λ), g(λ), andb(λ) and the three fundamental CMFs, thethree cone spectral sensitivities, l(λ), m(λ),and s(λ), the first plausible estimates ofwhich were obtained as early as 1886 [18].

When an observer matches the testand mixture fields in a color-matching

experiment, the two fields are matchedfor each of his or her three cone types. Thematch, in other words, is a match at thelevel of the cones. Put more formally, it canbe defined for each cone type thus:

lRr(λ) + lGg(λ) + lBb(λ) = l(λ);mRr(λ) + mGg(λ) + mBb(λ) = m(λ);

and sRr(λ) + sGg(λ) + sBb(λ) = s(λ),

(3)

where lR, lG, and lB are respectively theL-cone sensitivities to the R, G, and Bprimary lights, and similarly mR, mG,and mB are the M-cone sensitivities tothe primary lights, and sR, sG, and sB

are the S-cone sensitivities. Since the S-cones are insensitive in the red, it canbe assumed that sR is effectively zero fora long-wavelength R primary. There aretherefore eight unknowns required for thelinear transformation: lR lG lB

mR mG mB0 sG sB

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Moreover, since we are often more con-cerned about the relative shapes of thel(λ), m(λ), and s(λ), rather than their ab-solute values, the eight unknowns collapseto just five

lR/lB lG/lB 1

mR/mB mG/mB 10 sG/sB 1

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g(λ)

b(λ)

= kll(λ)

kmm(λ)

kss(λ)

, (5)

where the absolute values of kl(1/lB),km(1/mB), and ks(1/sB) remain unknown,but are typically chosen to scale threefunctions in some way so that, forexample, kll(λ), kmm(λ), and kss(λ) peakat unity. In the Smith–Pokorny solution

Colorimetry 215

(below), kll(λ) + kmm(λ) sum to V(λ), theluminosity function.

The unknowns in Eq. (5) can be es-timated by making spectral sensitivitymeasurements in dichromatic observers(who lack one of the three cone types;see Sect. 3.1.5) and normal observers un-der special conditions that isolate theresponses of single cones, or by compar-ing color matches made by normal anddichromatic observers. Derivations basedon dichromats depend on the ‘‘loss,’’‘‘reduction,’’ or ‘‘Konig’’ hypothesis thatdichromatic observers lack one of the threecone types, but retain two that are iden-tical to their counterparts in normals [7,19] (so that they accept all normal colormatches). This assumption is now moresecure, since it is possible to sequence andidentify the photopigment opsin genes ofnormal, dichromat, and monochromat ob-servers [20, 21], and so distinguish thoseindividuals who conform, genetically, tothe ‘‘reduction’’ hypothesis. The unknownvalue, sG/sB, can also be derived directlyfrom normal color-matching data [22, 23].

Several estimates of the normal conespectral sensitivities have been basedon the loss hypothesis [23–31]. Figure 5shows the current estimates by Smith &Pokorny [29] and Stockman & Sharpe [32].

The Smith–Pokorny estimates are atransformation of the Judd–Vos correctedCIE 1931 functions, while the Stock-man–Sharpe estimates are a transforma-tion of the Stiles & Burch 10-deg CMFs(adjusted to 2 deg). See Stockman & Sharpe(1999) for further information.

3Factors that Influence Color Matches

Several factors can influence colormatches. Many of them vary between

observers (differences which are referredto as individual differences), and so shouldbe taken into account when trying topredict the color matches of an individualfrom standard CMFs. Some of themvary with retinal position, and so shouldbe taken into account when trying topredict the color matches for retinal areasor retinal positions that differ from thecentrally viewed 2-deg or 10-deg areas usedto obtain the standard functions. See alsoSect. 9 of the article on COLOR VISION.

3.1Individual Differences

3.1.1 Lens PigmentLight is brought into focus on the retina bythe cornea and the pigmented crystallinelens. The pigment in the lens absorbslight mainly of short wavelengths (seeFig. 5b). Individual differences in lenspigment density can be large with arange of approximately ±25% of the meandensity in young observers (<30 yearsold) [33]. Since lens density increaseswith the age of the observer [34, 35], thevariability in the general population is evenlarger.

3.1.2 Macular PigmentBefore reaching the photoreceptor, lightmust pass through the ocular media,including at the fovea, the macula lutea,which contains the macular pigment. Thispigment also absorbs light mainly ofshort wavelengths (see Fig. 5b). Individualdifferences in its density can also be large,with a range of peak density from 0.0 toca. 1.2 at 460 nm [36–38].

3.1.3 Photopigment Optical DensityThe axial optical density of the pho-topigment in the receptor outer seg-ment varies between individuals. De-creases in photopigment optical density

216 Colorimetry

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Fig. 5 (a) S-, M-, and L-cone spectral sensitivity estimates of Stockman &Sharpe [32] (colored lines) compared with the estimates of Smith &Pokorny [29] (dashed lines). (b) shows the lens pigment optical densityspectrum (black line) and the macular pigment optical density spectrum(yellow line) from Stockman and Sharpe [32]. Note the logarithmic verticalscale – commonly used in such plots to emphasize small sensitivities

result in a narrowing of cone spectralsensitivity curves, which cause corre-sponding changes to their linear trans-formations, the CMFs. Any correctionsare most easily applied to the cone fun-damentals rather than the CMFs. Esti-mates of photopigment optical densityvary considerably depending to a largeextent on the method used to estimate

them, but all estimates show sizableindividual differences [39–46]. Photopig-ment bleaching has the effect of reduc-ing photopigment optical density (seeSect. 3.3).

3.1.4 Photopigment VariabilityThere is now clear molecular genetic andother evidence that the M- and L-cone

Colorimetry 217

photopigments can vary in spectral posi-tion between observers [47], thus confirm-ing earlier evidence for such shifts [48–52].

Large shifts away from the normal spec-tral position result in anomalous trichro-macy (see Sect. 3.1.5). Smaller shifts occurwithin the normal population, becauseof different polymorphisms (commonlyoccurring allelic differences) of the M-and L-cone photopigment opsin genes.The two common polymorphic variantsof the L-cone photopigment (which haveeither alanine or serine at position 180of the L photopigment opsin gene) dif-fer in spectral position by 2.7 nm ormore [47].

3.1.5 Color DeficiencySome color-deficient observers makedichromatic matches, since they requirejust two primaries to match all spectrallights. Their deficiency is therefore con-sistent with the lack of one of the threecone types. If their two surviving conetypes are normal (see Sect. 2), they will ac-cept the matches of normal observers. Thethree types of dichromacy are referred to asdeuteranopia (lack of M-cones), protanopia(lack of L-cones), and tritanopia (lack ofS-cones).

Other color-deficient observers makeanomalous trichromatic color matches.These individuals typically have an anoma-lous L-cone or M-cone pigment that isshifted away from its normal spectral posi-tion. In protanomaly, the spectral positionof the L-cone photopigment is shifted to-ward that of M, while in deuteranomalythat of the M-cone photopigment is shiftedtoward that of L. The size of the shift affectsthe severity of the deficiency.

The causes of most deficiencies nowhave a sound molecular genetic explana-tion [47]. See also Sect. 10 of the articleon COLOR VISION.

3.2Changes with Eccentricity

Macular pigment and photopigment opti-cal density both decline with eccentricity.Consequently, the available CMFs andcone fundamentals, which are for cen-trally viewed 2- or 10-deg diameter fields,must be adjusted in order to predict ac-curately color matches for other viewingconditions – either for different field sizesor for different viewing angles.

One additional complication is that theS-cones are absent in approximately thecentral 25-min diameter of vision, sothat in that region color matches becometritanopic [53].

3.3Bleaching

Color matches are disturbed at highbleaching levels, because bleaching lowersthe optical density of the photopigmentwithin the photoreceptor. As noted above,this has the effect of narrowing the shapesof the cone spectral sensitivities. Correc-tions for the effect of bleaching can beapplied directly to the cone fundamen-tals [54]. Corrections to other CMFs canbe most easily accomplished by transform-ing them to cone fundamentals, correctingthem, and transforming them back. Ap-proximately half of the photopigment isbleached by a steady white light of 4.3 logphotopic trolands [55].

3.4Rod Influence

At mesopic levels (levels at which bothrods and cones operate), trichromatic colormatches can be disturbed by the influenceof rods, as a result of which predicted colormatches may fail [see Sect. 5.6.2 of [8]].

218 Colorimetry

4Color Spaces

4.1Two- and Three-dimensional Color Spaces

For simplicity, color-matching data areoften represented not as CMFs, but inrelative units called chromaticity coordi-nates. The chromaticity coordinates (r(λ),g(λ), and b(λ)) of the spectrum locus (the

locus of monochromatic spectral lights)are related to the CMFs (r(λ), g(λ), andb(λ)) as follows:

r(λ) = r(λ)

r(λ) + g(λ) + b(λ),

g(λ) = g(λ)

r(λ) + g(λ) + b(λ),

and b(λ) = b(λ)

r(λ) + g(λ) + b(λ). (6)

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Fig. 6 CMFs, cone fundamentals, and chromaticity coordinates. (a) Spectrum locus(continuous line) and selected wavelengths (yellow circles) plotted in the Stiles & Burch [6]2-deg r, g chromaticity space, and the projection of the 2-deg L- (red diamond), M- (greencircle) and S- (blue square) cone fundamentals of Stockman & Sharpe [32]. (b) Spectrumlocus (continuous line) and selected wavelengths (filled circles) plotted in the Stiles &Burch [6] 2-deg RGB space, and the L- (red line), M- (green line) and S- (blue line) conevectors. (c) Version of CMF space with the plane R + G + B = 1 shown in gray. Theprojections of selected points on the spectrum locus onto this plane are shown by the yellowcircles, and the projections of the L-, M- and S-cone vectors are shown by the red diamond,green circle, and blue square, respectively

Colorimetry 219

Given that r(λ) + g(λ) + b(λ) = 1, onlyr(λ) and g(λ) are typically plotted, sinceb(λ) is simply 1 − (r(λ) + g(λ)).

Figure 6(b) shows a three-dimensionalplot of the spectrum locus (continuousblack line, with some wavelengths plottedas circles) in the Stiles & Burch [6] 2-deg RGB tristimulus space. Figure 6(a)shows the same data plotted as r, gchromaticity coordinates. Figure 7 showsthe spectrum locus in the 1931 CIEx, y chromaticity space with a veryapproximate representation of the colorsassociated with each coordinate. Whilechromaticity coordinates are a convenientway of plotting spectral distributions, andpredicting color mixtures, they inevitablythrow away all information about intensity.

Figure 6(c) (bottom right) shows theprojections of the cone vectors onto a two-dimensional unit plane R + G + B = 1(gray plane). This plane corresponds to that

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Fig. 7 CIE (1931) x, y chromaticity spaceshowing the spectrum locus (continuous line)and spectral wavelengths at every 10 nm (filledcircles). A representation of the color of eachcoordinate is shown, which is necessarily veryapproximate, since it is impossible to reproducespectral lights in print

of the chromaticity diagram. Estimates ofthe cone primaries in the Stiles & Burch2-deg r, g space by Stockman & Sharpe [32]are plotted as the red (L), green (M) andblue (S) symbols in Fig. 6(a) and in theRGB tristimulus space as the red (L), green(M) and blue (S) vectors in Fig. 6(b). Theprojections of the fundamental primariesin the chromaticity space lie entirelyoutside the spectrum locus.

The counterintuitive way in which thecone primaries plot in r, g (and x, y)chromaticity diagrams provides a strongargument for using spaces, the axes ofwhich represent cone excitations. Colorspaces are more straightforward and intu-itive when they are defined by the conefundamentals and represent cone excita-tion. Figure 8 shows the spectrum locusplotted as l, m chromaticity coordinates(a) and as LMS cone excitations (b).

4.2Other Cone Spaces

Other transformations of the cone spacesare favored in vision research because theymake clear the operation of mechanismsthat work after the photoreceptors, such ascolor opponent mechanisms, which com-pute the difference between cone signals(L − M and S-[L + M]), or the luminancemechanism, which sums them (L + M).

4.2.1 Equal-luminance Cone ExcitationSpaceA common projection of the LMScone space used in vision research isthe MacLeod–Boynton equal-luminanceplane [56, 57]. Its popularity rests, in part,on current models about postreceptoral or-ganization, and, in particular, on the theorythat only L- and M-cones contribute addi-tively to the luminance channel so thatV(λ) = l(λ) + m(λ).

220 Colorimetry

0.0

0.5

1.0

0.5

1.0 0.0

0.5

1.0

S

LM

440

460

480

500520

560580600

620

420

540

640

SS

M

L

l

0.00 0.25 0.50 0.75 1.00

0.00

0.25

0.50

0.75

1.00

500

640

520

480

560540

460

M

L

580

600

620

S

m

(a)

(b)

Fig. 8 (a) Spectrum locus (continuous line) andselected wavelengths (yellow circles) plotted inthe Stockman & Sharpe [32] 2-deg l-, m-conechromaticity space. The L- (red diamond), M-(green circle), and S- (blue square) conefundamentals plot at (1,0), (0,1), and (0,0),respectively. (b) Spectrum locus (continuousline) and selected wavelengths (filled circles)plotted in the Stockman & Sharpe [32] 2-degLMS cone excitation space, and the L- (red line),M- (green line), and S- (blue line) cone vectors

The MacLeod–Boynton chromaticity co-ordinates are defined as

rMB(λ) = l(λ)

V(λ),

gMB(λ) = m(λ)

V(λ)

and bMB(λ) = s(λ)

V(λ),

where V(λ) = l(λ) + m(λ). (7)

Other variations of this space have beenproposed [58, 59]. See also the Appendixby Brainard in Kaiser & Boynton [60]. SeeFig. 4 of COLOR VISION article.

4.2.2 Cone Contrast SpaceThe cone contrast space is a scaled, nonlin-ear version of the cone fundamental space,in which the cone excitations produced bya stimulus are scaled separately for eachcone type, according to Weber’s law (�S/S,�M/M and �L/L, where �S, �M and �Lare the differential cone excitations pro-duced by the stimulus, and S, M and L arethe unchanging cone excitations producedby, for example, a background). Accordingto Weber’s Law, the incremental thresholdintensity �I is proportional to the overallintensity, I, so that �I/I = constant. Thisspace is useful for understanding postre-ceptoral visual mechanisms [61].

4.3Uniform Color Spaces

The sizes and orientations of isodiscrimi-nation ellipes plotted in the CIE 1931 XYZcolor space vary substantially with position(such ellipses define the distances in colorspace that must be moved for a change inchromaticity to be discriminable). The CIEhas proposed two nonlinear transforma-tions of the CIE 1931 color coordinates totwo new sets of coordinates within whichthe sizes of the isodiscrimination ellipsesare approximately constant. The transfor-mations are to either the CIE 1976 L*u*v(CIELUV) color coordinates or to the CIE1976 L*a*b (CIELAB) color coordinates.Further details of these transformationscan be found elsewhere [see Sect. 3.3.9of [8]].

Colorimetry 221

5Existing Color-matching Functions

The available color-matching data andfunctions (and as a result, the cone spectralsensitivities derived from them) vary con-siderably in quality. In this section, five setsof data are discussed. The most commonlyused are not necessarily the best at predict-ing normal color matches. The luminosityfunction, V(λ), which has been artificiallylinked to the CIE CMFs, is also discussed.

5.1CIE (1931) 2-deg CMFs

The most widely used CMFs are theCIE (1931) 2-deg CMFs [11], which arebased on color-matching data obtainedby Wright [4] and Guild [5]. Those data,however, are relative data, which give onlythe ratios of the three primaries requiredto match spectral test lights. Althoughsufficient for chromaticity coordinates (seeSect. 4.1), a knowledge of the absoluteradiances of the matching primaries isrequired to generate CMFs. The CIEattempted to reconstruct this informationby assuming that a linear combinationof the three unknown CMFs must equalthe 1924 CIE V(λ) function [11, 62]. Inaddition to uncertainties about the validityof this assumption [63], the V(λ) curve thatwas used is far too insensitive at shortwavelengths [see Fig. 13 of [54]].

5.2Judd, Vos-modified CIE 2-deg CMFs

In an attempt to correct the CIE 1924 V(λ)function, Judd [12] increased the sensitiv-ity of V(λ) below 460 nm, and derived anew set of CMFs also based on the Wrightand Guild data (see Wyszecki & Stiles [8]Table 1 (5.5.2)), which were later slightly

modified by Vos [13, Table 1]. These CMFsare in common use in vision research astheir transformation, the Smith–Pokornycone fundamentals (see Sect. 2).

The substantial modifications to the V(λ)function introduced by Judd had the un-wanted effect of producing CMFs that arerelatively insensitive near 460 nm (wherethey were unchanged). Although this in-sensitivity can be roughly characterizedas being consistent with a high macu-lar pigment density [6, 32, 64] (see alsoSect. 3.1.2), the CMFs are somewhat ar-tificial and thus removed from real colormatches.

5.3Stiles & Burch (1955) 2-deg CMFs

The assumption that V(λ) is a linear combi-nation of the CMFs is entirely unnecessary,since CMFs can be measured without anyrecourse to photometric data. The Stiles &Burch [6] 2-deg CMFs are an example ofdirectly measured functions. Though re-ferred to by Stiles as ‘‘pilot’’ data, theseCMFs are the most extensive set of di-rectly measured color-matching data for2-deg vision available, being averaged frommatches made by ten observers. Even com-pared in relative terms, there are real differ-ences between the CIE 1931 and the Stiles& Burch [6] 2-deg color-matching data inthe range between 430 and 490 nm [seeFig. 1 of [6]]. These CMFs are seldom used.

5.4Stiles & Burch (1959) 10-deg CMFs

The most comprehensive set of color-matching data are the large-field, centrallyviewed 10-deg CMFs of Stiles & Burch [14].Measured in 49 subjects from approxi-mately 390 to 730 nm (and in 9 subjects

222 Colorimetry

from 730 to 830 nm), these data are prob-ably the most secure set of existing CMFs.Like the Stiles & Burch [6] 2-deg functions,the 10-deg functions represent directlymeasured CMFs, and so do not dependon measures of V(λ). These CMFs arethe basis of the Stockman and Sharpe [32]cone fundamentals (see Sect. 2).

5.5CIE (1964) 10-deg CMFs

The large field CIE 1964 CMFs are basedmainly on the 10-deg CMFs of Stiles& Burch [14], and to a lesser extent onthe arguably inferior and possibly rod-contaminated 10-deg CMFs of Speran-skaya [65]. While the CIE 1964 CMFs aresimilar to the 10-deg CMFs of Stiles &Burch (1959), they differ in several waysthat compromise their use as the basis forcone fundamentals [32].

5.6Luminance and the Luminosity Function,V(λ)

The CIE combined luminosity functionsfor 2-deg [V(λ) or y(λ)] or 10-deg [(V10 (λ)or y10(λ)] vision with color-matching databy making one of the CMFs equal to theluminosity function (see above). It shouldbe recognized, however, that luminosityfunctions are quite distinct from CMFsand cone fundamentals. In particular, theshapes of luminosity functions change withchromatic adaptation [66, 67], whereas theCMFs and cone spectral sensitivities donot (until bleaching levels). Biologically,the cone spectral sensitivities are recep-toral, whereas the luminosity function ispostreceptoral.

V(λ) is a photometric measure of lumi-nous efficiency or spectral sensitivity that is

defined as the effectiveness of lights of dif-ferent wavelength in specific photometricmatching tasks, which today are usu-ally heterochromatic flicker photometry(HFP), in which rapidly alternating lightsare matched in intensity to eliminate theperception of flicker, or a version of side-by-side matching, in which the relative in-tensities of the two half fields are set so thatthe border between them appears ‘‘mini-mally distinct’’ (MDB). These tasks arefavored because they produce, in contrastwith some of the methods used to obtainthe 1924 V(λ), additive results [63, 68]. Itshould be noted that luminance does notdefine how bright things actually appear.Fields that are equal in luminance oftendiffer substantially in apparent brightness.

There is no a priori reason why colori-metric data should depend on photometricdata, as was the case for the CIE 1931functions. Given modern calibration meth-ods, there is no justification for usingphotometric data to alter or adjust col-orimetric data.

6Conclusion

Colorimetry, the numerical specificationof the color of visual stimuli, is relatedto the spectral sensitivities of the threecone photoreceptors. Colorimetry is moreintuitive when defined in terms of coneexcitations than when defined in termsof imaginary primaries, such as the CIEXYZ primaries. In the past, cone spectralsensitivities were too uncertain to form thebasis of a colorimetric system, but that isno longer the case.

Most of the CMFs, chromaticity co-ordinates of the spectrum locus, conefundamentals, and other functions de-scribed here can be downloaded from:http://www.cvrl.org

Colorimetry 223

Color-matching data and CMFs tell uswhich spectral distributions will match un-der a given set of viewing conditions fora given observer. However, they tell us lit-tle about the actual color appearance of thematch, which can vary enormously withthe viewing conditions.

Acknowledgments

Thanks to Rhea Eskew for comments. Sup-ported by The Wellcome Trust. The colorsin the diagram of Fig. 7 are based in part onsoftware freely distributed by Earl F. GlynnII at http://www.efg2.com/Lab/Graphics/Colors/Chromaticity.htm

Glossary

2 deg: Shorthand for the ‘‘small-field’’color matches made with centrally viewed,circular fields subtending 2-deg diameterof visual angle.

10 deg: Shorthand for the ‘‘large-field’’color matches made with centrally viewed,circular fields subtending 10-deg diameterof visual angle.

Chromaticity Coordinates: x, y, which interms of the tristimulus values are X/(X +Y + Z) and Y/(X + Y + Z), respectively(or r, g for RGB space, or l, m for LMSspace).

CIE: Commission Internationale de l’E-clairage or International Commission onIllumination. An organization, a goalof which is to generate internationalstandards in color and lighting.

Color Match: A subjective match betweentwo lights of different spectral power

distributions (which are therefore met-amers).

Color-matching Functions: x(λ), y(λ), andz(λ). Tristimulus values of the equal-energy spectrum locus.

Colorimetry: The measurement and nu-merical specification of color.

Cone Fundamentals: Cone spectral sen-sitivities. l(λ), m(λ), and s(λ) in CMFnotation. These are the color-matchingfunctions that would result if primariesthat uniquely stimulated the three conescould be used.

Large-field Matches: Color matches forcentrally viewed, circular fields subtending10-deg diameter of visual angle.

Metamers: Two lights that match, but arephysically different. An example is thematch between a spectral yellow and amixture of spectral red and green (seeFig. 1).

Photometry: The measurement and nu-merical specification of the luminousefficiency of lights. Intended to be inde-pendent of the chromaticities of the lights.

Photopic Luminosity Function: Photomet-ric measure of luminance efficiency asa function of wavelength under photopic(i.e., rod-free) conditions: V(λ) or y(λ).

Photoreceptors: The light-sensitive recep-tors that lie on the rear surface of the eyethat transduce photons into electrical sig-nals. They can be sensitive, nighttime rodsor relatively insensitive, daytime cones.

Primary lights: e.g., R, G, B: The three in-dependent primaries (real or imaginary) towhich the test light is matched (actually orhypothetically). They must be independent

224 Colorimetry

in the sense that no combination of twocan match the third.

Small-field Matches: Color matches forcentrally viewed, circular fields subtending2-deg diameter of visual angle.

Standard Observer: The standard observeris the hypothetical person whose color-matching behavior is represented by aparticular set of mean CMFs.

Trichromacy: The ability of normal ob-servers to match test lights with a mixtureof three independent primary lights, oneof which invariably has to be added to thetest light to complete the match.

Tristimulus Values: R, G, B, the amountsof the three primaries required to match agiven stimulus.

Univariance: The output of a photorecep-tor varies unidimensionally only accordingto the rate of photon absorption.

Visual Angle: The angle subtended by anobject in the external field at the effectiveoptical center of the eye.

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Further Reading

Berns, R. S. (2000), Billmeyer and Saltzman’sPrinciples of Color Technology, (3rd ed.), NewYork: Wiley.

Brainard, D. H. (1995), in M. Bass (Ed.), OSAHandbook of Optics. New York: McGraw-Hill,pp. 26.1–26.54.

226 Colorimetry

Kaiser, P. K., Boynton, R. M. (1996), HumanColor Vision, (2nd ed.), Washington, DC:Optical Society of America.

Sharpe, L. T., Stockman, A., Jagle, H., Nathans, J.(1999), in K. Gegenfurtner, L. T. Sharpe (Eds.),Color Vision: From Genes to Perception.Cambridge: Cambridge University Press,pp. 3–51.

Stockman, A., Sharpe, L. T. (1999), in K. Gegen-furtner, L. T. Sharpe (Eds.), Color Vision: FromGenes to Perception. Cambridge: CambridgeUniversity Press, pp. 53–87.

Wyszecki, G., Stiles, W. S. (1982), Color Science:Concepts and Methods, Quantitative Data andFormulae, (2nd ed.), New York: Wiley.