Modelos de Cores

8/9/2019 Modelos de Cores

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modern color models

The first six pages have examined thefundamental aspects of color perception thetrichromatic mechanism, the threecolormaking attributesthat describe colorsensations, the geometry of color(including thehue circle, opponent functions and responsecompression), the many basic forms of color,and the complex effects of context, contrast &

adaptation, and finally the principles of additive& subtractive color mixing. This page describesthe development of modern color models anddiscusses their relationship to these color

fundamentals.

A color modelrepresents the logical orperceptual relationships among colors of lights orsurfaces. From the modern perspective, a colormodel must meet the following four

requirements

(!) a color specificationthat analy"es everylight or surface color into a mixture offundamental attributes (such as "primary"colors, trichromatic cone responses ortristimulus values, or unique hues)#

($) a geometrical frameworkthat locates allthe possible colors in relation to each other and to

the fundamental attributes#(%) a unique color identifieror color notation(now usually the numerical value of the threecolormaking attributes brightness&lightness,hue and hue purity) for every possible color# and

(') a definition of physical exemplars, specificmixtures of lights or paints, that recreate themeasured color perception when viewed within astandard surround under standard lightingconditions.

colorvision

the evolution ofcolor models

wedish atural*olor ystem

+unsell color system

A uniform colorscales

*-/01uniform color space

*-/A2uniform color space

*-*A+color appearance model
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a name to a specific color.

Color aming. The earliest color labels areembedded in common speech. very languagehas words to name colors, though in many

primitive cultures only a handful of words areavailable, and the words ma3e very grossdistinctions between light and dar3, or warm andcool colors. Thus the Black Seais not blac3, butdar3 blue# its color name has been handed downfrom a time when the color ideas green, blue,darkand blackwere denoted by the same word.This doesn6t mean that primitive spea3ers couldnot seethe differences between those colors, 5ustthat the differences were not conceptuali"ed as

abstract color categories.

As human 3nowledge and technology developed inareas such as painting, dyeing, weaving,ceramics, cosmetics, medical diagnosis,mineralogy, botany, horticulture, entomology andchemistry, more precise color comparisons acrossdifferent 3inds of materials became useful. Thisled to the development of color namingsystems. The earliest and most obvious approach

was to anchor color names on a distinctivelycolored flower, fruit, mineral or organic compound(dye) and so we have the colors lemon,primrose, saffron, amber, gold, orange, vermilion,rose, ruby, burgundy, carmine, violet, sapphire,turquoise, emerald, leaf green, ochre, sepia,indigo, ivory, ebonyand so on. The world itselfwas a reference boo3 of color.

9uring the !:th and !;th centuries,naturalists and their rich patrons collectedsamples of sea shells, corals, flowers, gems,minerals, ceramics and insects as a cabinetof curiosities, and these artifacts weresometimes used to identify and preserveunusual color exemplars from the naturalworld. As a result, dyers, printers andnaturalists (especially botanists andentomologists, collectors of plants and insects)

were among the most active color researchersduring the !


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>owever, early color enthusiasts had noassurance that they had sampled the fullrange of possible colors they did notunderstand the physiological limitsof color

vision, and therefore could not be sure that anew insect, coral, flower or gem might notreveal a hue that had never been seen before.

This uncertainty was swept away in !;?'when Isaac ewtonpublished his Opticks,the first scientific analysis of color. Among themany stunning brea3throughs in this boo3 wasewton6s claim to have identified allfundamental colors and by extension all

possible mixed colors. Any color could bereduced either to a pure spectral hue or to ageometrically defined mixture of spectralhues.ewton symboli"ed this intellectualclosure as a closed circle his hue circle. oended doubt about the rangeof possiblecolors.

>owever, the practical problem remained thatprecise spectral mixtures were difficult to

achieve until the mid !=th century, ob5ectcolors were typically duller and more complexthan the colors of mixed spectral lights, and allthose complex colors still had to be identifiedwith a name and captured as a physicalexemplar. -n fact, the last two re8uirementswere the most important and the easiest tofulfill. o naturalists had to derive their coloridentifications and color labels from a coloratlas, a tradition solution that extends from

A.@. 7erner6s Nomenclature of Colors(!;;')up to obert idgeway6s Color standards andcolor nomenclature(!=!$). These provided aseries of hand painted colored patches as avisual standard for each color, with the color6sname and, in earlier versions, a list of flowers,insects or minerals that exemplified the colorin nature. imilar tools are still used today inhorticulture, agriculture and food processing.

Color #ixture. -f namingthe color wasincidental to using it, then colors could bereduced to a recipe or proportional mixture of
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a few standard colorants or 4primary4 colors.The entrepreneurial printer 'akob Christoffele lonfirst applied this approach to theindustrial tas3 of color printing, by showing

that color images could be created from threeprimary or 4primitive4 colored in3s, usuallysupplemented by a fourth blac3 in3, one in3printed over another with separate, carefullyaligned me""otint plates (a techni8uedeveloped earlier in woodbloc3 printing ofboo3 illustrations). +ulticolor printing systemssimilar to /e2lon6s, though not commerciallysuccessful, continued up to the end of the!


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proposed by the @erman astronomer /obias#ayeras a university lecture in !;C

hole system be represented by any othercorrect and complete figure. &nd since eachnuance is given its correct relation to all thepure elements as ell as to all mixtures, thissphere must be considered a universal chart,enabling anybody to orient himself as to theoverall context of all colors.

#ichel2Eug3ne Chevreulfollowed with hisown 4color hemisphere4 in !


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wor3s.

Color #odel !ssumptions. The geometricalframewor3 used in modern color models

consists of the three colormaking attributesarranged to define a three dimensional space.The figure below shows how this is done. Thevertical blac3 to white dimension is thelightness or valueof a color# thecircumference of the hori"ontal dis3perpendicular to the lightness dimension is thehueof a color# and the lateral distance orradius on this dis3, from the center outward, isthe chroma(approximately the saturation) of

a color.

the geometry of colormaking attributesin modern color models

otice that this arrangement allows hues tochange gradually from one to the next, butdefines chroma and lightness as rulers or ratioscalesstarting from a definite "ero point blac3 for lightness, and gray for saturation.

*olor models differ in the specific assumptionsthat are used to define the color geometry andcolor labels and the specific measurementmethods used to lin3 the geometry to colorstimuli and color exemplars. +y selection of
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color models is intended to highlight thesedifferences

E 6eometrical 7niformityD The perceptual

facts of color are tremendously complex, andmany early models emphasi"ed a geometricalor logical framewor3 that summari"ed color ina simplified way. These systems use visualcolor matching to measure colors, appeal tothephenomenologyor sub5ective relationshipsamong colors, and emphasi"e a framewor3 forverbal color communication. The best modernexample is the 8wedish C8color model.

E erceptual 7niformityD >ere the goal is tocreate a color geometry where the units ofmeasurement stand for e8ual apparent colordifferences on all the colorma3ing dimensions.Berceived color differences determine the colorgeometry and exemplification as physical colorsamples. The classic modern example is the#unsell Color 0rder 8ystem# very differentrecent examples include the 08! uniformcolor scalesand the CIE 9:;< 7C8(uniform

chromaticity scales) diagram.E /richromatic ColorimetryD -ncreasingly,modern color models are lin3ed to theelectronic measurement of spectral emissionor reflectance profiles, expressed as a mixtureof three imaginary "primary" lightscalledthe =)> tristimulus values. Thisstandardi"es the 4viewing eye4 used tocompare colors and allows automated colormatching. The modern examples of thisapproach are CIE7?and CIE!.

E Color !ppearanceD The earliest colormodels only described colors that were viewedor compared in a standardi@ed viewingsituation. -n the past few decades, moreambitious color models have been developedthat ad5ust the color space to represent theeffects of the visual context (the illuminant,surround and bac3ground colors) and extremeviewing conditions (small or large colorsamples, dim or bright illumination). This
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raises them to the status of colorappearance models color models thatpredict a wide range of real world colorexperiences. The current example of this

approach is CIEC!#$%.

An authoritative summary of the development andempirical foundations of modern color models is olf @.uehni6s Color 8pace and Its Aivisions4 Color 0rderfrom !ntiquity to the resent(7iley -nterscience,$??%). For an interactive tutorial on historical colormodels, see G. Frans @erritsen6s Evolution in Color(chiffer, !=


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sensations, organi"ed as verbal or geometricalmodels of color relationships. The wedishatural Color 8ystem(C8), proposed byTryggve Gohansson in the !=%?6s and

eventually reali"ed by ven >esselgren andAnders >Hrd at the candinavian *olour-nstitute in the !=:?6s, is a modern colormodel that exemplifies this approach. The *is currently a color standard in severalcandinavian and 0 countries.

The experimental pedigree of the * goesbac3 to wald >ering6s -as natrliche!arbsystem(!=?C), which describes colors as

the result of opponent color processingandthe mixture of four unique huesplus blac3and white. >ering6s was aphenomenologicalor sub5ective approach to color he tried todescribe the universal patterns or laws of ourcolor experience without specifying what 3indof external stimulus or internal color receptorsmight produce it. This phenomenologicalapproach can be traced from >ering bac3 tothe omantic @erman philosophers Arthur

chopenhauer, @eorg 7.F. >egel andultimately to 'BB von 6oethe althoughthose authors and their philosophical concernswere of little practical use to the artists oftheir time.

Dour Components of C8. Thisphenomenological or 4in the head4 approachto color modeling is based on fourcomponents.

The first is a single, universal colorspecification formula

color C(u9Fu%+ F 8 F

where Crepresents the apparent proportion ofpure hue (chromaticness) in the color,8(@erman Schart) is the apparentproportion of blac3 in the color, and (@erman /eiss) is the apparent proportion of

white. The chromatic ingredient consists eitherof a single unique hue,or more commonly of
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the mixture of two ad5acent uni8ue hues,shown as (u!Iu$). >ering called any colormixed with either white or blac3 (or both) aveiled color, and by these mixtures

accounted for all dull, diluted and near neutralcolors.

The basic formula has a long pedigree. The7estern painting tradition used the techni8ueof broken colorto ad5ust pure pigment (highchroma) colors by mixture with white and&orblac3 paints, a method described by *ennini,Alberti and /eonardo. The same mixturegeometry is used in +ichelDugJne *hevreul6s

color hemisphere. >ering6s colorspecification was also used by the @ermanchemist and obel laureate ilhelm 0stwald(!ering uni8ue hues arearranged to create four cardinal pointsaround the hue circle, as shown in the figure.Following >ering6s original concept, the y&band r&g opponent functionsdefine twodimensions at right angles within the huecircle. Antagonistic uni8ue hues, which are not

allowed to mix in >ering6s theory, are onopposite ends of the two dimensions. The thirdopponent dimension defined by white andblac3 is placed perpendicular to the first two.
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three color dimensions defined by thehering colors

The third component consists of two colormixing rules

(!) nly uni8ue hues on different opponentdimensions can be mixed# uni8ue hues atopposite ends of the same dimension cannotbe mixed. (That is, we are not allowed toma3e 4yellowish blue4 or 4reddish green4mixtures.)

($) 7hite and blac3 can mix with each other,and with any hue mixture, in any proportions#

this provides the full range of shades, tonesand tintsfor any hue.

The fourth and last component is a fixed sumfor the color specification

C(u9Fu%+ F 8 F 9$$

in the same way that mixtures in Tobias+ayer6s color hexahedronmust sum to !.

This forces the color geometry into atriangular (conical, hexahedral) form.

/he C8 Color #odel. These four designcomponents interconnect the six >ering colorsas !% elementary scales a single grayscale (white to blac3), and three chromaticscales for each hue (e.g., red to white, red toblac3, and red to gray) that measure offmixtures among the six cardinal colors.

The basic mixing formula color C F 8 F
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9$$dictates that the model is mostconveniently represented as vertical sections,called hue triangles, that include one purehue or hue mixture, pure white and pure blac3

as the three corners of an e8uilateral triangle.This is e8uivalent the geometry of two cones5oined at their base that was used in stwald6ssystem, as shown below for the !% elementaryscales and a purple (mixed red and blue) huesection.

geometrical framework of the swedishC8 color model

The * is exemplified as the C8 coloratlas, a large boo3 of standard colorswatches. The atlas contains '? of these huetriangles at e8ual intervals around the huecircle nine steps (increments of !?Lmixture) between each pair of ad5acent uni8uehues. The atlas also includes a representationof the complete hue circle at maximumchroma, a lightness scale from white to blac3,and several 4off white4 color samples as a helpto specifying pastel colors.

ach hue triangle displays a wide selectioncolor samples made from mixtures of the purehue with blac3 (8) and&or white (). /ines ofe8ual blac3 mixture are shown as diagonal

rows parallel to the top edge of the triangle#blac3ness increases from the top edge to the


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bottom corner. /ines of e8ual whiteness areshown as diagonal rows parallel to the bottomedge# whiteness increases from the bottomedge to the top corner. As a result, gradations

in chroma (C) are represented by verticalcolumns through the triangle, from the leftcorner (the maximum chroma), which containsno white or blac3, to the gray scale along thevertical edge. *olor samples for the pure hues,white and blac3 are omitted due to thelimitations of available pigments, and verydar3 or near neutral samples are usuallyomitted to reduce production costs and visualclutter.

a page from the C8 color atlas :ering colors thatcreates it, finds the matching color in the *color atlas, and determines the color6s *color notation. A textile mill or constructioncompany then uses the * notation toidentify the color sample they should match infabric or wall paint. The ma5or drawbac3 tothis method is that most people find it verydifficult to identify dull or dar3 colors (browns,tans or maroons in particular) in terms of

saturated uni8ue hue mixtures.

The geometrical simplicity of the *introduces some significant distortions orcomplications into theperceptualcolor space.2ecause the four >ering colors are notequally spacedaround a perceptual colorcircle (as shown for example on the CIEC!#acbcplane), the hue differences around the

* color circle are not perceptually e8ual,either. 9ifferences between the yellow greenhue triangles are perceptually very small,while hue differences between the blue greenand purple hue triangles are 8uite large. Thismeans fine color discriminations are possiblefor yellow green colors, while the blue greenand purple colors re8uire some guesswor3. -naddition, the chroma spacing is une8ual fromone hue to the next, and from neutral to

intense colors within any hue (right).-n addition, for most parts of the color circle,opposite colors are not 4mix to gray4 visualcomplements, because the uni8ue huesthemselves are not visual complements. Thevisual complement of uni8ue red is not uni8uegreen, but a blue green located at roughly6H$# the visual complement of uni8ue blueis not uni8ue yellow, but a deep yellow atroughly )$1. Finally, the artist6s mixingcomplements are also different from the *complements. And while the * establishes

lines of constant C8 ha

chroma plotted onCIE! aJbJ pl

after 3uehni #4)
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when e8uating the apparent whiteness ofvarious hues in pastel color schemes.

-n general, however, the phenomenological

and geometrical approach did not generatemuch interest among American and 2ritishresearch psychologists, who turned instead topsychophysical methods of color scaling.

For more information about the * and related colorproducts, see the * web site.

Munsell Color System

The #unsell Color 8ystemis arguably thefirst modern color order system based onthe three colorma3ing attributes, andimplemented through careful colormeasurement. *onceived in the !. +unsell(!


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changes in color stimulus composition, in afreely branching geometry called a color tree.

+unsell was inspired by and heavily relied on

the expertise of the American physicist andcolor researcher 0gden 1ood, who showed+unsell how to use a color topto analy"ecomplex colors into additive color mixtures of4primary4 colors, and to develop perceptuallye8ual scales of gray or color chroma based onproportional mixtures of blac3 and white. 7iththis method +unsell was able to simulateincremental steps in chroma(proportionalmixtures with gray) or hue (by proportional

mixtures of 4primary4 colors, or analogouscolors such as red and yellow), and to identifyvisual complementary colors(saturatedhues that additively mix to a pure gray). vera decade of patient, incremental wor3, +unsellanaly"ed reflective or surface colors into theircomponent lightness, hue and chroma,published in his atlas as e8ual apparentdifferences on each of these colorma3ingattributes.

6eometry of the #unsell 8ystem. Thebac3bone of the +unsell system, the 4trun34 ofthe color tree, is a vertical dimension oflightness or value, which was developedthrough human perceptual 5udgments of e8ualdifferences in lightness across visual (colortop) mixtures of white and blac3 paints. The+unsell value scaleranges from pure blac3(?) to pure white (!?)# each step is divided

into decimal increments, resulting in a !??step lightness scale.
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At the middle value level is a hue circledefined by five equally spaced huedimensions red (1), yellow ()), green (6),blue () and purple () in cloc3wise order(yellow at the top, red on the left). These

principal colorsare separated by fivemixture hues between them yellow red()1), green yellow (6)), blue green (6),purple blue (), and red purple (1) eachthe visual complementary colorof theprincipal hue directly opposite it. Theseprincipal and mixture hues divide the huecircle into ten e8ual hue segments, and eachhue segment is again divided by ten (countedin the cloc3wise direction), resulting in a !??

step hue circle. The 4standard4 or centralexample of each color segment is located atstep C (diagram, right).

Finally, each hue was calibrated in e8ualchromasteps from "ero (pure gray) to themaximum color intensity in the paints or in3sused for each hue# in the ideali"ed (aim)colors, chroma intervals are calculated out tothe optimal colorlimits.

The approximately cylindrical framewor3 of the+unsell system (below), which includes avalue scale limited by the luminance factor ofsurfaces and chroma steps bounded by theoptimal color limits for each hue. Thedimensions are not otherwise connected,which corresponds to the separate perceptualmeasurement procedures used to define them.

the H$ standard munhue circle divisi
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conceptual framework of the munsellcolor order system (9:$G+

*olors are 4named4 through the standardnotation se8uence hue value6chroma.Thus,the pigment vermilionwould be notated asKBG1 GBGL9% a hue of


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color sample is as large as the atlas samples,and all colors are viewed on a graybac3ground under the same daylight orincandescent illumination (usually, illuminant

*).

>owever, conversion formulas and loo3uptables published by the AT+ are nowavailable to produce a +unsell e8uivalent forany spectrophotometric (*- NOP or Oxy) colorspecification. These allow translation to orfrom +unsell and any other modern colorsystem also on the *- standards.

/he #unsell 1enotations. -n the !=$?6ssome of the !=!C +unsell samples were firstmeasured with a spectrophotometer, andthese were mapped into the *- %M )xy colorspaceshortly after it was developed in !=%!.

The early measurements were made by the0A ational 2ureau of tandards, which too3an interest in developing the +unsell systemas a color standard. -n the !=%?6s the ptical

ociety of America also began a technicalstudy of the system. The entire !=$= 2oo3 of*olor was measured spectrophotometrically in!=%C and published as )xyvalues in !='?.These revealed ma5or discrepancies betweenthe !=!C and !=$= color samples, and ma5orirregularities in the spacing of color sampleson the )xychromaticity diagram, particularlyin the chroma intervals but also in the lines ofe8ual hue.

The A committee primarily focused ondefining the +unsell value scale as amathematical formula, based on visual5udgments of gray samples by a large sampleof viewers, and on 5udgmentally ad5usting or4smoothing4 the contours of +unsell hue andchroma on !=%! xy chromaticity diagrams.These calculations were used to revise thevalue scale, to e8uali"e the hue and chromaintervals, to extend chroma intervals out tothe optimal color limits, and to redefinematerial color specifications. This renotated
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systemwas published in !='% as specific )xyvalues for $;?? uni8ue color locations, as amathematical formula to calculate +unsellvalue from =)>tristimulus values, and as

several !=%! xy chromaticity charts used forthe visual estimation of +unsell hue andchroma values from xy values. ubse8uenteditions of the 2oo3 of *olor are blended tomatch these !='% targets. The renotatedsystem is the standard specificationof+unsell colors used today.

ubse8uently, the A committee wor3ing onuniform color scales (discussed below)

revisited the renotation data, generally toexamine the effects of imposing a genuinelythreedimensional definition of colordifferences, but particularly to examine theeffects of response compressionon thechroma of different hues. The data for this re2renotation systemwere published in !=:;for more than $=?? color samples, andrepresent a ma5or revision to the +unsellstandards. -t is not in general use.

Today all the +unsell colors are available asideali"ed tristimulus valuesor aim colors,and )xycolor locations or colorspectrophotometric data can be translated intothe +unsell notation entirely by computersoftware. Thus the color system designed forteaching color to children has ascended to therealm of digital exemplification and industrialcolor standards.

/he !symmetry of Color 8pace. +unselland his successors wor3ed hard to producecolor samples that were perceptuallyequidistantor e8ually different fromneighboring colors on the individualdimensions of value, chroma and hue. Thus,the difference between a chroma of $ and achroma of ' is intended to appear to theviewer as large as the chroma differencebetween !$ and !'. >owever, thesemeasurement units are not comparableacross dimensions# the principle of e8ual
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perceived distance applies ithineachdimension of the model separately, notbetween any combination of color attributes.This is obvious from the fact that !? units

encompass the entire range of values (fromblac3 to white), but only !?L of the totalrange of hues (for example, from yellow toorange) and, for most hues, only half the totalrange of chroma.

/i3e ood, +unsell originally conceived of hiscolor model as a sphere. 2ut the empiricalmeasurements that have been at the heart ofthe +unsell system from the beginning

compelled +unsell to an important reali"ationthe natural color space is highly irregularwhen it is represented without geometricalpreconceptions. As a result +unsell allowed forune8ual dimensions of chroma at differentlevels of lightness and across different hues.Two 4pages4 from the +unsell system at thecomplementary hues G)and Gshow thisclearly.

two pages from the munsell book of color(9:%:+

These hues are directly opposite each other on


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The range of colors represented in a +unsellatlas is limited by the gamut of paints or in3sused to create the color samples. -n fact, twodifferent versions of +unsell color samples, in4glossy4 and 4matte4 preparations, are

available. The glossy colors have a slightlylarger gamut or range of chroma.

2ut even the glossy colors are incomplete. Thefigure at right shows the range of surfacecolors represented by the +unsell color solidas irregular colored shapes, enclosed within awhite envelope that represents the optimalcolor stimulior maximum chroma for surfacecolors of each hue. (*olors with a higher color

intensity than the white boundary will appearfluorescent or self luminous.) The +unsellsystem represents the range of colors that canbe achieved by high 8uality acrylic paints, orin3s printed on highly reflective white paper#the range of colors possible with watercolorpigments is significantly smaller.

*omparison of the +unsell color range withthe optimal color envelope shows that the

yellow, orange, red and magenta (CB)pigments do a fairly good 5ob of filling out thepotential color space, while modern lightfastblue (C2), blue green (C2@)and green (C@)pigments reach less than half the chromapotentialfor their hues. Thus, the +unsellcolor atlas, and in fact any color atlas in anycolor system, exemplifies an incomplete ands3ewed subset of the total asymmetrical colorspace. ote, for example, that the 6

dimension of the +unsell atlas is smaller thanthe 1dimension, although in the perceptualsurface color space 5ust the reverse is true.

OSA uniform color scales

As we6ve seen, the +unsell system containstwo significant problems (!) a variety ofdiscrepancies were found in the perceptual

spacing of colors, depending on their locationin the color space, and ($) the quantitative

maximum chromamunsell color samp

compared to the limitssurface color chroma% a

3uehni #4))5$7erales, 0ora et al. #4)
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difference between colors could only bedefined on a single color attribute (lightness,chroma or hue) at a time. These problemswere not resolved in the +unsell renotation,and the +unsell system remained an

inconvenient and uncertain basis for8uantifying perceived color differencesthathad to be defined on all three colorma3ing

attributes.

The subse8uent development of thecolorimetric systemdid not resolve matters,as the standard chromaticity diagramalsobadly distorted perceived color differences the chromaticity differences between greens

are much too large, and differences betweenblues are much too small.

+ethods to 8uantify perceived colordifferences were very important in theimplementation of color measurement, forpurposes as diverse as photographic dyepatents and governmental standards, and formanufacturing color control in industries asdissimilar as textiles, food products and

automotives. These re8uirements stimulatedan effort to develop a truly uniform colorspace, from the ground up, as a set of uniformcolor scalesacross a limited value range ofmoderately saturated colors. This tas3 wasta3en up by a committee under the auspicesof the ptical ociety of America (A), incollaboration with the 0A ational 2ureau oftandards, and led by 9eane Gudd and 9avid+acAdam. The final 0* aim color tristimulus

values, color scales, recommendations fortheir use and conclusions regarding the humancolor space, were published in the !=;?6s asthe ptical ociety of America6s 7niformColor 8cales (7C8+.

1hombohedral attice. The first issue wasthe specification of the color geometry. Thedecision was made to shift from the 4wheeland spo3e4 arrangement of +unsell to arigorously cartesian or three dimensional setof dimensions, as in the *- )xy color space.
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-n addition, this color geometry should supporte8uivalent measurement of perceived colordifferences in all directions (includingdirections in which all three colorma3ing

attribute change), which meant that colordifferences along the diagonal dimensions inthe cartesian space should be e8uivalent tocolor differences in any direction parallel toone of the three defining dimensions.

The A committee determined that themetric solution to both problems was to use arhombohedral lattice, based on thecuboctohedron, as the organi"ing geometry for

locating aim colors within the system. This isdifficult to represent as a single image, but thelogic behind it is straightforward and builds onthe idea of a regular stacking.

rhombohedral lattice structure of 08!7C8

(left) a triangle of three spheres fitted into a hexagon ofseven spheres, with three more spheres fitted from

below# (right) the resulting cuboctahedron with edgesof e8ual length (blac3), lin3ed by six scales passing

through a central color# (bottom) the six sets of parallelcolor scales identified by color

The densest possible stac3ing of three spheres


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in two dimensions is with the center of thespheres forming an e8uilateral triangle (asbilliard balls are stac3ed in the rac3). Thedensest possible stac3ing in three dimensions

is pyramidal a single sphere is fitted into thegap between three spheres below, as orangesare stac3ed in a grocery bin or carbon atomsare stac3ed in a diamond crystal. Thecuboctahedron is the geometric solid thatdefines this stac3ing. tarting from a singlesphere that mar3s a neutral, mid valued gray,we continue this stac3ing in all directions outto the perceptual limits of the color space,creating a lattice of color 4atoms4 of color in

this stac3ing arrangement.

cuboctahedral geometry of the osa

uniform color scaleslightness is measured on the FLNdimension,yellow&blue on theOFLONdimension and red&green onthe gFLgNdimension# rhombohedral stac3ing of color4atoms4 defines a total of seven uni8ue cleavage planesthrough the color space (labeled), with additional twocleavage planes (not shown) defined perpendicular to

theOand gdimensions

As in a diamond crystal, the stac3ing alignsthe 0* aim colors (the 4atoms4 in the lattice)into nine planar color groupings or cleavage


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planes(diagram, above), and six perceptualscalesdefined by the intersections betweenany two cleavage planes. The three cleavageplanes perpendicular to the three

measurement dimensions stac3 the targetcolors in a s8uare lattice# all other planes arestac3ed in a hexagonal lattice. This create arigid measurement framewor3 within a threedimensional (euclidean) color space, intendedto reveal for color scientists any irregularitiesin the color space as discrepancies in colordistances calculated across the different scalesor cleavage planes. -n addition, the variouscolor scales and color planes were suggested

to artists as dimensions of harmonious colorvariation.

The lattice naturally suggests fourmeasurement benchmar3s (!) the intervalbetween any two ad5acent color samplesrepresents aOust noticeable differencebetween the colors# ($) the smallest intervalsor steps between colors must be perceptuallye8ual in all directions and all parts of the color

space# (%) the perceptual uncertainty ordisagreement between observers in identifyinga specific color must approximately form aspherearound the average colorspecification# and (') the spheres around allcolor samples will be the same si"e not theellipses of different si"es and shapes thatappear in the *- chromaticity diagram. -nfact, both the color stimuli originally used tomeasure the perceptual color space, and the

A 0* aim colors (the standard tabularexemplification of the system), were spacedby about $? 5nd6s, but in theory this shouldonly change the perceived distance betweencolor samples, not the geometrical regularityof the color scales. 0nfortunately, as the Acommittee discovered, this assumption turnsout to be false.
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/he 08! 7niform Color 8pace. The 0*exemplification pro5ects the rhombohedrallattice into the color space so that a cleavageplane of s8uare stac3ing defines thehue&chroma planes at constant lightness# each

s8uare defines a measurement interval of twounits. These planes are staggered or shifted inrelation to each other, so that chromaticitycoordinates are even or odd numbered onalternating lightness levels. The lightnessdimension is perpendicular to these planes.

The three perpendicular dimensions of theuniform color spaceare the lightnessdimension and two chromatic dimensions,O

(for2aune,approximately a y&b opponentdimension) and g(for green,approximately ag&r opponent dimension). The mid valued graythat is the anchor of the system has a value of? on all dimensions# colors with negativevalues are dar3er, bluer or redder.

The 0* dimensions are computed from thestimulus *- NOP tristimulus values for the9$M standard observer, as viewed under the

A


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the 08! 7C8 color solidconceptual side view of the A 0* whole step aim

colors, with the vertical scale expanded for visualclarity# adapted from )xytristimulus values in Table

-(:.:.') of 7ys"ec3i Q tiles (!=


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value of :. - reproduce below the '< A aimcolors for /R! as approximate e8uivalents tothese color stimuli. As the example shows, therange of colors studied is relatively muted the

most intense color sample (the orange at topright) has a *-/A2 chroma of ;'.

08! 7C8 constant lightness planeA 0* aim colors for R ! (*-/A2 JR :% to :


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results to different values and to ad5ust for the.elmholt@2*ohlrausch effect. The finalrhombohedral model of C?? color samples andC< pastel samples was built on these

foundations.

The diagram (right) shows the C?? A 0*4full step4 aim colors pro5ected into theCIE!color space. The average distancebetween aim colors on the *-/A2 , aJor bJdimensions is about PEabR !$ that is,about !$ 5nd6s. The image shows the samplingis fairly complete (when compared for example

to a similar representation of watercolorpigments) and is generally regular, with atendency for *-/A2 chroma spacing to growlarger in yellow and green colors. (The slightdownward bowing visible in the color rows is acompensation for the >elmholt"Dohlrauscheffect.)

The various diagonal scales or cleavage planesthrough the color space were intended to

identify palettes of complex color change thatmight be useful to graphic artists andcolor designers. They turned out to be wildlyunpopular for that purpose. The legacy of theA 0* is as a research landmar3 and auniform tiling of the color space that is usefulto assess color models or imaging systems.

Color 8pace Is oneuclidean. 9espite theextensive and patient research program andthe elegant color model it produced, the Acommittee had already in !=:; come to adispiriting conclusion

/e ill affirm to the orld that no regularrhombohedral lattice sampling of color space,ith a fixed background, can exist% e illproduce the best approximation to such alattice for a neutral value : background thate can design.

-n plain language it is impossible torepresent uniform color differences as a

08! 7C8 aim colors inCIE! color sp

under illuminant -:;% la

OSA L = 1shon in magenthegdimension is reverse

all C"9 color mod
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three dimensional color model. The humancolor space is noneuclidean, which means athree dimensional framewor3 cannot describeit. Any color difference formula or color model

that represents color in only three dimensions(such as lightness, hue and hue purity) mustproduce a distorted or inconsistent picture ofcolor differences and color relationships.

The first and perhaps most important reasonwhy this is so was identified by 9eane Gudd in!=:< as hue superimportance# the effectmight better be described as chromacompression.-t appears as a fundamental

discrepancy between measures of 5ustnoticeable difference in chroma and in hue.The diameter of any hue circle at constantchroma is always perceptually shorter than thecircumference across all hues as calculatedfrom that chroma 4radius4. This discrepancy is8uite large if the ratio of the circumference tothe diameter of a euclidean circle is roughly%.!' to !, the same ratio in a hue circle isgreater than : to !.

Gudd represented this fact as a 4pleated dis34(diagram, top right) chroma is measuredoutward along the folds in the pleats, whilehue is measured "ig"ag along the edge of thefan, which has roughly double the length of aflat circumference. Alternately, if perceivedhue and chroma differences are representedon a plane of e8ual lightness, then chroma

differences within the same hue must beelongated relative to hue differences withinthe same chroma, producing ellipticaldiscrimination ellipses. -n addition, theseellipses must become larger (chromadiscrimination becomes wea3er) at higherchroma values (diagram, bottom right). TheA committee concluded that both lightnessand chroma had to be ad5usted by a cube rootor exponential compressionto correct forthis discrepancy.

The second problem involves chromatic

the hue circle as a plead

the circumference is mt

six times the length ofdiame

after


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contrast, in particular crispening, whichma3es small color differences, or differences5udged against a similar bac3ground, appearrelatively larger. +ost of us are familiar with

the 4reading magnifiers4 that enlarge wordswhen rested on a printed page. *rispeningacts li3e a magnifying bubble inside the colorspace, enlarging the apparent contrastbetween similar colors on lightness, chromaand hue. As a result, all small colordifferences are magnified in comparisonto large color differences, and 4smalldistance4 color difference formulas tend tooverestimate the perceived color difference

between colors that are very different.

Gust as a reading magnifier can be shiftedaround a page, the crispening locus canmigrate around the color space, depending onthe color and lightness of the bac3ground(surround) used to 5udge colors, thechromaticity of the illuminant. To control this,color vision experiments and color atlasviewing standards use a mid valued gray

surroundin color presentations, so thatcrispening occurs as an increased chromasensitivity around near neutral colors.

The third problem is that hue superimportanceis not e8ual in all directions chromacompression is strongest in yellow andgreen huesand wea3est in red, purple andblue hues. -t also changes across differentvalues, becoming more extreme in lighter

valued (more luminous) colors.The final problem, related to the previous one,is the unequal spacing of hues, which isexplored on a previous page. ome huesproduce a greater number of perceptible colorgradations than others# in particular, colordiscriminations are generally more acute forwarm hues than for green hues.

hueLchroma discriminatiellipses in CIE

from 3uehni #4)

The two fundamental problems in color modelsuncovered by the A 0* research strangely
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parallel the two primary color paradoxes

E The first is that all color models e can seecannot represent colors relationshipsaccurately,and color models that represent

color relationships accurately cannot be seen(physically constructed or represented as animage).

E The second is that all color differenceformulas are arbitrary,in the sense that theymust include many 5udgmental corrections forspecific discrepancies, can only providedifference estimates that are approximatelycorrect for small (or large) color differences,

and provide difference estimates whoseaccuracy depends on where in the color spacethe color difference is calculated.

The primary color paradoxes can be explainedas resulting from the overlap in three conesensitivity curves. The color model paradoxescan be generally explained by the fact that theperceptual color space changescontinuallyunder the influence of many

contextual factors. -t6s not that a changingmetric cannot be expressed as a threedimensional color model# it6s that the changeswithin a perceptual space vary depending onthe color stimuli, color difference, contrastbac3grounds, illuminance levels and lightsources that are being used to assess the colordifference.

A detailed and very useful explanation of the AD0*scales and their application in color design issues, by

9anny Bascale, is available from abelColorBcom.

CIELUV uniform color space

The previous section described the efforts bythe ptical ociety of America to build a trulyuniform color space in three dimensions basedon rigorous psychometric and geometricalfoundations, and the discovery that such a

goal is fundamentally impossible. ow weturn to a second strand of color research,
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which had a more limited aim in view todiscover a uniform color differenceformulathat could be used in specific

practical applications.

Color #atching and Color Aifferences.ince the early !=$?6s, a ma5or goal ofcolorimetrywas the prediction of anunrelated colorresponse through thephotometric measurement of a physical colorstimulus. The approach that seemed mostefficient was to measure the spectral powerdistributionof the light emitted or reflectedby the color stimulus, translate this intophotometric units by means of a luminous

efficiency functionor a standard set ofcolor matching functions, then use these8uantities to estimate the sub5ectivebrightness or chromaticity of the stimuluswhen it is viewed as an isolated patch of color.

>owever, the =)>tristimulus values that cameout of color matching tas3s literally onlypredict color identity, that is, whether twocolors will appear the same.The whole

problem of how much two colors will appeardifferentwas left undetermined. The CIE xychromaticity diagramdoes represent therelativedifference between two colors thatone color sensation is redder, or brighter, orduller than another, if both colors aremeasured in the same way. 2ut the desirednext step was to subtract one set of =)>tristimulus values from another, and use thenumerical difference to predict the

perceived color differencebetween them.This estimation procedure did not wor3 well. Ama5or problem was that the *- xychromaticity diagram is not a uniformchromaticity scalesdiagram or 7C8. -n a true0* an e8ual chromaticity distance is an e8ualperceptual difference, no matter which colorattributes are measured. 2ut two colorsseparated by an e8ual )xy(or =)>)difference can appear nearly identical (if thecolors are green or light valued) or verydifferent (if the colors are violet or dar3
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valued). They cannot consistently predictwhether a color difference will be unnoticeableor obviousto an average viewer. This was ama5or obstacle to using colorimetry in the

industrial control of dyes (textiles andphotography), in3s (printing and pac3aging)and paints (automotive and architecturalcolors) especially after inexpensive andreliable spectrophotometers became availablein the !=C?6s.

As the measurement standards were alreadyin place, the easiest solution was a hac3. The

original chromaticity diagram could bemathematically stretched or compressed untilthe numericaldifference between thetristimulus values of two colors approximatedtheperceiveddifference between the colors.(These differences would be applied only tosimilar colors large color differences were notimportant to 8uality control problems.) Thesemathematical distortions are called proOectivetransformations, and they are identical to

the warping or 43eystoning4 that occurs whena slide is pro5ected onto a curved surface oronto a wall at an obli8ue angle (right). Thesetransforms don6t change any of the foundationprocedures of colorimetry or the 4straight line4color mixing relationships of the chromaticitydiagram, 5ust the color differences calculatedfrom them.

Bro5ective transformations were proposed

independently by 9eane Gudd and 9avid+acAdam in the !=%?6s# Gudd6s version isshown below. -t radically compresses the4green4 part of the chromaticity space whileexpanding the 4blue4 and 4red4. The whitepoint is located close to the S@ mixing line,because the and #cones dominate theperception of whiteness (compare with mycone chromaticity diagrams).

"keystoning" asexample of a proOect

transformati
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an early proOective transformationthe *- !=%! chromaticity diagram within a +axwell

triangle defined by three imaginary primary lights thatproduce an approximately uniform color space (after

Gudd, !=%C)

Bro5ective transformations were only one ofseveral solutions to the color differenceproblem. 2y the !=:?6s anywhere from !% to

$? different formulas were being used fordifferent applications in ngland and the 0A.To corral this diversity, the CommissionInternationale de l5Qclairage(*-), aninternational clearinghouse for color researchat universities and research laboratories,adopted in !=;: two color difference formulasthat seemed most promising for further wor3CIE7?, described here, and CIE!.

/he CIE7? Aimensions. *-/01 is a muchrevised and expanded version of the pro5ectivetransformation proposed by 9avid +acAdam in!=%; and adapted as a color difference spaceby @nter 7ys"ec3i in !=:'. -t builds on twouniform chromaticity scales(0*),denoted u5and v5, derived from the standard=)>tristimulus values.

ecall that a basic problem in colorimetry isdevising a measure of chromaticity (hue and

hue purity) that is not affected by changes inthe luminance (brightness or lightness) of the
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color. The *- xy chromaticity diagramaccomplishes this by normaliingthe =)>values, or dividing each score by the sum of=)> values

x = L (= F ) F >+y ) L (= F ) F >+

The *- 0* adopts the same normali"ingstrategy for the color6s tristimulus values, butwith a specific reweighting of the =c, )cand >cscores that changes the shape and orientationof the chromaticity diagram, and the locationof the white pointinside it

u5c H=cL (=cF 9G)cF >c+v5c :)cL (=cF 9G)cF >c+

The u5and v5scales alter the dimensions ofthe chromaticity space but preserve thetrichromatic color geometry. That is, in the *-0* and *-/01, 6rassmann5s awsof colormixture are still valid all colors are defined byadditive mixtures of three imaginary 4primary4

lights, and therefore color mixtures aredescribed by straight mixing lines.

The next problem is representing theluminance or intensity of the illumination andchromatic adaptationto its color. Toincorporate these effects, the luminance factor)cis used to expand the dimensions of theu5cv5cdiagram to approximate the increase incolorfulness that occurs with increasedilluminance, and to shift the location of thewhite point to center it on the chromaticity ofthe light source, defined by a second set oftristimulus values (=w)w>w). This yields therelativebrightness dimension Jand therelativechromaticity dimensions uJand vJ(u5wand v5ware computed from =w)w>wasshown above)

J 99


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These ad5ustments center the uJand vJdimensions on the achromatic point at theorigin of a uJvJcartesian plot, insert a cuberoot compressioninto the brightness

contrast, and (by the factor 9) mimic the.unt effectwhich causes colors to increase incolorfulness as their luminance increases.These calculations apply to colors perceived aslights, so extremely dar3 colors withluminance ratios ()c&)w) less than ?.?! areout of bounds.

At last we get a definition of chromaacrossdifferent levels of brightness as the euclidean

chromaticity distance between the color andthe white point

Cuv RuJ%F vJ%S9L%

... we get a simple numerical definition of hueas the hue anglemeasured from the whitepoint in relation to the uJdimension

huv arctanRvJLuJS

... and we end up with a version of the colordifference score that was the origin of themodel, PEuv, as the euclidean distancebetween the J, uJand vJvalues of twosimilarcolors

PEuv R(J9NJ%+%F(uJ9NuJ%+%F(vJ9NvJ%+%S9L%

where 4similar4 means colors that areseparated by a PEuvvalue of !? or less.

/he 7niform Color 8pace. The result is the!=;: C"9 >?u?v? uniform color spaceorCIE7?, in which the brightness andchromaticity difference between two colors isproportional to theirperceived differencewhenviewed side by side in a gray field at moderatebrightness.

*-/01 provides the same chromaticitydimensions as the u5v5 uniform chromaticity
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The diagram below shows the effect of theu5v5scaling on e8ual chromaticity differencesmeasured in different directions around thechromaticity surface. The e8uali"ing effect is

especially noticeable by comparing the colordifferences around the spectrum locus, andthe color distances from 4blue4 toward 4green4and 4blue4 toward 4red4.

comparison of equal perceptual distancese8ual perceptual distances, drawn at !? times actualsi"e, in *- xy and u6v6 chromaticity diagrams# after

7right (!=:=) and >unt ($??%)

Throughout the $?th century, the +unsell

system figured prominently in efforts toidentify color scaling problems or confirm thata color difference formula accurately describedcolor differences. The diagram below showsthe relative improvement in the spacing of+unsell lines of e8ual hue and chroma forsurface colors of value :.


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comparison of munsell hue and chromalines of e8ual +unsell chroma (steps of $) and hue

(steps of C) at value : in *- xy and u6v6 chromaticitydiagrams# after 7ys"ec3i Q tiles (!=owever, significant discrepancies(chroma compression or expansion) appear inthe yellow green to red parts of the diagram,with the greatest compression approximatelyalong the yellow (CO) hue dimension.

Current 7se of CIE7?. The *- !=;: u5v5diagram is the 4least wrong4 chromaticitydiagram in general use, and is a hugeimprovement over the distorted !=%! or !=:'diagrams which still lurch through the colorliterature. -t gets the relative si"es of thegreen, blue and red areas of color space aboutright, as well as the perceived hue purityof

colors, including spectral hues, as the distancefrom the white point. -t should always be used
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to display basic chromaticity data.

*-/01 has been a popular color space forlighting, video and photographic applications

which re8uire measurement of lights, lightfilters or optical systems, because *-/01represents the additive mixture of two lightsas a straight lineacross the color space.(.7.@ >unt6s authoritative $??' text, @heAeproduction of Color, :th edition,consistentlyrelies on it.) *-/01 is 8uite useful to predictthe appearance or mixture of colors producedby luminous displays such as computermonitors, television screens, or transparencies

pro5ected onto a uniformly reflecting surface where the brightness of colors is a constantproportion of the luminance of the sourceillumination (there is no 4external4 lightsource), although *-/01 is also used tospecify the color of gems or optical glassesilluminated by a standard 4white4 light source,and the color of photographic dyes and filters.

*onsidered in terms of the four

requirementsof a modern color model,*-/01 (and the )xystandard observersystem on which it is based) represent ama5or departure. This isn6t that the colorspecificationis exclusively in terms ofcontinuous numerical values, because that wasinnovated centuries ago by Isaac ewtonand /obias #ayer. The innovation is thatthere is no color exemplification. All colormeasurements are made electronically, using

a spectrophotometerthat measures theseparate =)>values directly and thencalculates the coordinate values for anystandard color system. The exemplification isimplicit in all physical color stimuli, whetherlights or surfaces, that have identical or similarcolor coordinates.

>owever, *-/01 suffers from a ma5ordrawbac3 the PEuVvVcolor difference formulayields a poor prediction of colordifferences, especially for surface colors. -n
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fact, when a variety of color models are testedfor their ability to predict surface colordifferences, *-/01 is almost always at thebottom of the heap.

*-/01 does even worse at predictingcorresponding colors, or colors that matchunder different colors of illumination(illuminants). The problem is that *-/01represents relative color differences in relationto the white point as a differenceor simplesubtraction (e.g., u5cN u5w), rather than as aratioor proportion. This means that if thewhite point (the illuminant) is changed, or isstrongly chromatic, all colors are shifted in a

parallel direction by a constant amount (as ifprinted on a moveable transparency),regardless of where they are located in thechromaticity space.

-f colors are shifted from yellow toward blue,the compressed chroma intervals in the yellowgreen to orange part of the uJvJchromaticitydiagram do not change, so *-/01 grosslyoverestimates chroma differences wherever

these boundaries are relocated. -f colors areshifted from blue to yellow (for example, thecolor of a gem viewed under tungsten ratherthan halogen light), *-/01 thrusts thepredicted color locations completely outsidethe spectrum locus, which is perceptuallyimpossible.

As corresponding color predictions are aminimal re8uirement for modern color

appearance models, *-/01 has beendiscarded as the basis for further wor3.

CIELA uniform color space

The second color difference formula adoptedby the *- in !=;: (along with CIE7?) wasthe C"9 >?a?b? uniform color spaceorCIE!. *-/A2 is based on a color difference

formula developed by .K. Adams and 9.ic3erson in the !=%?6s to measure color
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fading in textiles. /i3e *-/01, it is an attemptto hac3 the =)>tristimulus values to producea reliable method for calculating colordifference scores. >owever, *-/A2 is based

on a very different transformation

strategy.

/he CIE! Aimensions. The *-/A2formula comprises four transformation steps,which are intuitive when examined separately.

ecall that the color geometrycommon to allmodern color models consists of abrightness&lightness dimension that isperpendicular to (separate from) the

chromaticity plane or hue circle. Thisgeometry is implicit in the =)> tristimulusvalues the )value represents the photopicluminosity functionof the specified color.7e start with the color6s )tristimulus value()c) as the measure of its luminance orluminance factor (labeled , as in CIE7?),and use the =cand >cvalues as measures ofchromaticity (hue and hue purity).

The =value increases primarily as the coneresponse to a color becomes relatively large,so it is the 4red4 anchor for an rLgdimension(labeled a)# the >value increases with 8coneresponse, so it is given a negative sign torepresent the complementary 4yellow4 of ayLb(b) dimension

)ca =c

b N>c.

The first transformation step is an ad5ustmentof the tristimulus values to represent achromatic adaptationto the illuminant(color of illumination). The correct method (avon *ries transform) divides the coneexcitationsproduced by the color area by thecone excitations produced by the illuminant.*-/A2 instead uses the 4wrong4 ratiobetween the tristimulus values=w)w>wof theillumination. These are a standard both for theluminance upper limit represented by a
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perfectly reflecting 4white4 surface ()wisalways e8ual to !??) and for the chromaticityinherent in the light itself (in the values of =wand >wrelative to !??).

)cL)wa =cL=w

b N>cL>w.

The tristimulus values of a perfectly white orequal energy illuminantare defined as =w)w >w 9$$. An illuminant with aperceptible 4warm4 (yellow or red)chromaticity has values of =wgreater than !??

and values of >wless than !??# 4cool4 or blueilluminants have low =wand high >w. Theadaptation ratios reduce or increase the valuesof =cand >cin proportion.

-n the second step, *-/A2 applies anexponential compressionto all threeadaptation ratios. This inflates the perceivedchromaticity and lightness differences amongdar3 colors, and inflates the chromaticity of

yellows relative to blues. At the time *-/A2was published, different compression fractionshad been tried in color difference formulas,and the cube rootseemed to perform about aswell as any

()cL)w+9L

a (=cL=w+9L

b ()cL)w+9L.

Although the )tristimulus value representsthe luminance of the color area, all threetristimulus values contain similar luminanceinformation, because the =)>values all getlarger as the surface reflectance increasesfrom dar3 to light. As a result, the =)>valuesare highly correlated in actual colormeasurements, from ?.C$ W)and >X or ?.C:W=and >X to ?.curves all pea3 at different parts of thespectrum.
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o the third transformation step separates thechromaticity information from the luminance

information contained in all three values.*-/01 does this by normaliingtheluminance information (dividing all valuesby the =)>sum). *-/A2 does it bysubtractingone value from another

()cL)w+9L

a (=cL=w+9LN ()cL)w+9L

b ()cL)w+9LN (>cL>w+9L.

This creates two undulating curves for thedifference values of spectral hues (diagram,right).

An explanation when applied to 8uantities ofthe same thing, such as dollars or gallons,subtraction 5ust identifies the 8uantity of thedifference. 7hen applied to 8uantities ofdissimilarthings measured within a similarrange of values, subtraction purges any

information the two measures have incommon(in this case, information about theluminance) and leaves as a single number allthe unique informationin both (the combinedinformation about hue and hue purity). As the=c)c>cvalues are already standardi"ed on thewhite point (by the adaptation transform), thesubtraction also centers the values of aand bat "ero for a surface color that exactlymatches the chromaticity of the illuminant apale yellow surface under a pale yellow lightwill appear to be a 4pure4 white.

Finally, the , aand bdimensions areexpanded by different factors so that a unitscale value on any combination of *-/A2dimensions represents an approximately e8ualand 5ust noticeable perceptual difference. Asthe )N>curve (bdimension) has a largeramplitude than the =N)curve (adimension,diagram above right), it is multiplied by a

smaller amount. The slope and intercept of thefunction is also ad5usted to match the units

opponent functions definby tristimulus differen

=2) and
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of +unsell value

J 99cL >w+9LS.

These formulas ta3e an alternate form, notshown here, for extremely dar3 colors (whereany adaptation ratio is less than ?.??=). Fordetails see the references at the end of thissection.

*-/A2 yields a reliable correlate (relativemeasurement) of chromaas the euclidean

distance between the chromaticity of the colorand the achromatic point

Cab RaJ%F bJ%S9L%

... and a numerical definition of hue as thehue angle

hab arctanRbJLaJS

... and its own color difference score PEab,obtained as the euclidean distance betweenthe J, aJand bJvalues of two similarcolors

PEab R(J9NJ%+%F(aJ9NaJ%+%F(bJ9NbJ%+%S9L%

where 4similar4 means colors that areseparated by a PEabvalue of around !? orless. As a rule of thumb, !? units of *-/A2

lightness exactly matches ! unit of #unsellvalue, and !? units of *-/A2 chromaapproximately matches $ units of +unsellchroma. The average PEabfor the first chromastep in +unsell aim colors across all hues andvalues is about C.


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perpendicular dimensions that are oftenrepresented as a cylindrical geometry. Thatis, the Jdimension is the axis of the cylinder,through the achromatic point# habrepresents

the hue angle or circumferential direction ofthe color, and Cabthe distance of the colorfrom the cylinder axis.

geometry of the CIE! color model

The diagram is somewhat misleading. -npractice, the color difference (euclideandistance) score is based on the cartesian(aJbJ) coordinates and not cylindrical radiansor degrees# and the distribution of surface

colors in *-/A2 tapers toward the achromaticcenter at very high or low lightness.

The point is that *-/A2 color relationshipsare not distorted by forcing them into aconceptual geometry,such as the ipsative huetriangles of *. *olor measurements arelimited vertically by the luminance factor ofsurfaces, and the chroma extent is limited bythe domain of real surface colors or lights, but

these are boundaries imposed by colorperception, not by arbitrary geometry or logic.
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surface of optimal color stimuli in CIE!

composite from a wireframe animationY $??; 2ruce/indbloom

The illustration below shows the location of;?? commercial watercolor paints in the*-/A2 space, with color icons grouped ontoseparate lightness planes for clarity. Thesedefine a roughly ellipsoidal or 4footballshaped4 color domain inside the *-/A2 colorspace.
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;$$ watercolor paints in the CIE! color

spacebased on spectrophotometric measurements reported inthe guide to watercolor pigments# red (aF) placed

on left to correspond to artist6s color wheels

ote the similarity in the side view of thepigment distribution (at left) and the chromalimits of the yellow&blue color samples in the#unsell color pagesdiscussed above.*ompare also to the outline of 08! 7C8 aim

colorspro5ected into the *-/A2 space. (For a4top down4 view of pigment locations, see theCIE! aJbJ plane.)

>ow closely does *-/A2 approximate auniform color space-f we set aside the smallproblem that a 0* in three dimensions isimpossible, then *-/A2 does remar3ablywell, given its origins as a hac3. A standardbasis for comparison the distribution of

+unsell aim colors at +unsell values ', : and


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munsell aim colors on the CIE! aJbJplane

under illuminant 9:C# aim colors at $.C hue intervalsand &$ chroma intervals for values ', : and < out to theoptimal color limits# constant *-/A2 chroma indicatedby gray circles. ote that all *- systems reverse the

ordering of +unsell hues.

-deally, this diagram would loo3 li3e radialspo3es within concentric circles. (The 4wedge4shaped outline results from the optimal colorlimits see, for example, this diagramand is not a fault per se.) The exaggeratedspacing of +unsell chroma into the yellow andyellow green hues, the displacement in the

lines of constant hue as lightness increases(especially in the blue greens), the curvinglines of constant violet and green hue, and thewide gaps in the hue spacing of green colors,are all primarily or partly due to irregularitiesin *-/A2 (although some are actuallyproblems in +unsell). The distribution of 08!aim colorsma3es a much more favorableimpression of uniformity. *-/A2 shouldalways be used to represent surface colors,and not CIE7?.

2ecause *-/A2 includes an ad5ustment for
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chromatic adaptation, the location of colorswill shift, depending on the chromaticity andluminance specified for the achromatic point,as shown below.

corresponding colors in CIE!+unsell aim colors at value ?


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aJ red (aF+ vsB green (aN+bJ yellow (bF+ vsB blue (bN+

This interpretation re8uires three

8ualifications. First, the "ero end of thelightness dimension Jonly represents theabsence of light (surface reflectance), and notblac3 as a distinct color sensation, asconceived by >ering and observed inlightness induction.

econd, the connection between the aJbJdimensions and hue depends on how theconnection is defined the opponent contrasts

do not consistently identify four hues as thecontrast anchors

E Thephysiological contrastdefined by the aor bdimensions is found in the dominantwavelength or pea3 of the positive andnegative sections of the =N)and )N>curves(above), interpreted as cone sensitivitycurves or transmittance profiles. These identifythe contrast hues as magenta (aF, a mixture

of orange and violet), middle green (aN),middle yellow (bF) and violet (bN).

E Across the total sample of +unsell surfacecolor exemplars, the aand bdimensions arecorrelated around S?.%C. 9escriptively, thismeans that the most intense yellow colors(bF) tend to be greenish (aN), and the mostintense violet colors (bN) tend to be reddish(aF). Berceptually, it implies that the yLbandrLgopponent functionsare not independentof each other that is, a change on onedimension will have no perceptual effect onthe other although this is how they arenormally presented.

E The hue of surface colors located at the endsof perpendicular aJand bJdimensions, asshown here, are magenta (aF), blue green(aN), light yellow (bF) and cobalt blue (bN).

E The >ering uni8ue hues are displaced fromthe ends of the perpendicular aJbJ
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dimensions by a large amount, as shownhere. 0ni8ue red and uni8ue green are bothshifted about $?U toward yellow, and theseparation between uni8ue red and uni8ue

blue is more than one third (!%?U) of the huecircumference.

Third, the cube root compression applied tothe tristimulus values creates a nasty 3in3 inthe aJchromatic dimension (diagram right,top), which significantly departs from theundulating profile of empirically measuredopponent functions. This especially stretches

out the distance between blue hues with4green4 content, and causes the aFred tocontain too much blue, compressing thedistance between yellow and red colors.

2ecause they are based on difference scores,*-/A2 chromaticity dimensions do notredistribute or normali"e the luminanceinformation across the chromaticitycoordinates they expunge it. As a result, the

*-/A2 color metrics cannot be expresseddirectly as the additive mixture of three4primary4 colors of real or imaginary lights.7hen narrowband optimal colors arerepresented in this cone opponent space, theyproduce an erratic, double lobed circuit, withthe spectrum ends diving toward theachromatic origin (diagram right, bottom).This is recogni"ably an upside down image ofthe cone excitationspace, pinched and

s3ewed by the cube root transformation. otealso the exaggerated spacing of the blue hues.

*-/A2 measures chroma, the chromaticcontent of a color independent of itsluminance# the aVbV plane does not representsaturation. 0nli3e the straight line, additivemixture between any two colors of light in the*- 0* (or any chromaticity diagram), colormixtures can ta3e curving or unpredictablepathson the *-/A2 aVbV plane. This is not aresult of the subtractive mixture of the twophysical colorants (paints or dyes) but is basic

opponent geometrCIE

#top$ opponent functionand b% #bottom$ trace

optimal color stimuli ona?b? pla

both displays under eqenergy illumin
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to the *-/A2 space.

Current 7se of CIE!. /i3e *-/01, colorspecification in *-/A2 is tied entirely to

spectrophotometric measurementsofcolor# it can6t be used routinely without a colormeasurement device, and there is no standard*-/A2 color atlas available. >owever, thewide use and practicality of the *-/A2 systemhave spawned a number of proprietary colororder systems or color atlases based in wholeor part on the *-/A2 geometry, including theAmerican Colorcurve 8ystem, the @erman1! Aesign 8ystemand the nglish

Eurocolor !tlas.

Fairchild ($??C) reports validation studies thatpit several color appearance models againsteach other, using datasets that measured colormatching under illuminant adaptation(corresponding colors) or preferred contrastand chroma in color image reproductionsviewed at different luminance levels. verall,these studies showed that different models

performed well in predicting some types ofcolor 5udgments but not others. -n every case,however, *-/A2 typically performed wellagainst more complex models and occasionallywas as good as the best# *-/01 was oftenthe worst.

A ma5or problem is that *-/A2 uses what iscalled a wrong von *ries transformtomodel chromatic adaptation. This computesthe contrast ratio between a color and thewhite standard or illuminantdirectly on the=)>tristimulus values, rather than on conefundamentals. -n addition, the cube rootcompression is not optimal when applied tothe tristimulus values, and results in chromaintervals that can be misplaced in some partsof the hue plane in particular, yellow chromais not compressed enough, and blue chroma iscompressed too much.

These failings aside, *-/A2 has become astandard colorimetric space and is one of the
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current (%$$


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