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Gernot Hoffmann
�. CIEChromaticityDiagram(�93�) 2 2. ColorPerceptionbyEyeandBrain 3 3. RGBColor-MatchingFunctions 4 4. XYZCoordinates 5 5. XYZPrimaries 6 6. XYZColor-MatchingFunctions 7 7. ChromaticityValues 8 8. ColorSpaceVisualization 9 9. ColorTemperatureandWhitePoints �0�0. CIERGBGamutinxyY ����. ColorSpaceCalculations �2�2. Matrices �7�3. sRGB 23�4. BarycentricCoordinates 24�5. OptimalPrimaries 25�6. References 27AppendixA ColorMatching 29AppendixB FurtherExplanationsforChapter5 30AppendixCComparePrimariessRGBandAdobeRGB 3�
CIE Color Space
Contents
22
CIENTSC sRGB
AdobeRGB
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1. CIE Chromaticity Diagram (1931)
ThethreedimensionalcolorspaceCIEXYZisthebasisforallcolormanagementsystems.Thiscolorspacecontainsallperceivablecolors-thehumangamut.Manyofthemcannotbeshownonmonitorsorprinted.
ThetwodimensionalCIEchromaticitydiagramxyY(below)showsaspecialprojectionofthethreedimensionalCIEcolorspaceXYZ.SomeinterpretationsarepossibleinxyY,othersrequirethethreedimensionalspaceXYZortherelatedthreedimensionalspaceCIELab.
Purpleline
Wavelengthsinnm
sRGBusesITU-RBT.709primaries RedGreen Blue Whitex 0.64 0.30 0.�5 0.3�27y 0.33 0.60 0.06 0.3290
AdobeRGB(98)usesRedandBluelikesRGBandGreenlikeNTSC
CIE-RGBaretheprimariesforcolormatchingtests:700/546.�/435.8nm
33
2. Color Perception by Eye and Brain
Theretinacontainstwogroupsofsensors,therodsandthecones.Ineacheyeareabout�00millionsofrodsresponsiblefortheluminance.About6millionsofconesmeasurecolor.Thesensorsarealready’wired’intheretina–only�millionnervefibrescarrytheinformationtothebrain.Theperceptionofcolorsbyconesrequiresanabsoluteluminanceofatleastsomecd/m2(candelapersquaremeter).Amonitordeliversabout�00cd/m2forwhiteand�cd/m2forblack.
Threetypesofcones(togetherwiththerods)formatristimulusmeasuringsystem.Spectralinformationislostandonlythreecolorinformationsareleft.Wemaycallthesecolorsblue,greenandredbuttheredsensorisinfactanorangesensor.Theopticalsystemisnotcolorcorrected.Itwouldbeimpossibletofocussimultaneouslyforthreedifferentwavelengths.Theoverlappingsensitivitiesofthegreenandtheredsensormayindicatethatthefocussinghappensmainlyintheoverlappingrangewhereasblueisgenerallyoutoffocus.Thissoundsstrange,butthegapforimagepartsontheblindspotiscorrectedaswell–anotherexampleforthesurprisingfeaturesofeyeandbrain.
Thesediagramsshowtwoofseveralmodelsfortheconesensitivities.Theseandsimilarfunc-tionscannotbemeasureddirectly–theyaremathematicalinterpretationsofcolormatchingexperiments.Thesensitivitybetween700nmand800nmisverylow,thereforeallthediagramsaredrawnfortherange380nmto700nm.
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p1_
p2_
p3_
λ
Conesensitivities[3]
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p2_
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λ
Conesensitivities[�]
44
3. RGB Color-Matching
The color matching experiment was invented byHermannGraßmann(�809–�877)about�853.
Three lamps with spectral distributions R,G,B andweight factors R,G,B = 0..�00 generate the colorimpressionC =RR+GG+BB.
The three lamps must have linearly independentspectra,withoutanyotherspecialspecification.AfourthlampgeneratesthecolorimpressionD.
Can we match the color impressions C and D byadjustingR,G,B?Inmanycaseswecan:
BlueGreen=7R+33G+39BInothercaseswehavetomoveoneofthethreelampstotheleftsideandmatchindirectly:
Vibrant BlueGreen+38R =42G+9�BVibrant BlueGreen =-38R+42G+9�BThisistheintroductionof’negative’colors.Theequalsignmeans’matchedby’.Itisgenerallypossibletomatchacolorbythreeweightfactors,butoneoreventwocanbenegative(onlyoneforCIE-RGB).DatafortheexampleareshowninAppendixA.
View
ColorDColorC
ThenormalizedweightfactorsarecalledCIEColor-MatchingFunctions r ( )λ ,g( )λ ,b( )λ .
Thediagramshowsforexamplethethreevaluesformatchingaspectralpurecolor(monochromat)withwavelength λ=540nm.Thisrequiresanegativevalueforred.
Colormatchingexperiment
300435.8546.�700.0800
R,G,B +4.5907
+�.0000
+0.060�
-0.1
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380 420 460 500 540 580 620 660 700nm
r_
g_
b_
λ
CIEStandardPrimaries
RGBColor-matchingfunctions
TheCIEStandardPrimaries(�93�)arenarrowbandlight sources,monochromats, linespectraordeltafunctionsR,700nm,G,546.�nmandB,435.8nm.Theyreplacethered,greenandbluelampsinthedrawingabove.Infactthesesourceswereactuallynotused–allresultswerecalculatedfortheseprimariesaftertestswithothersources.
R k P r d
G k P g d
B k P b d
=
=
=
∫∫∫
( ) ( )
( ) ( )
( ) ( )
λ λ λ
λ λ λ
λ λ λ
RGBcolors foraspectrumP(λ)arecalculatedbytheseintegrals intherangefrom380nmto700nmor800nm:
55
4. XYZ Coordinates
InordertoavoidnegativeRGBnumbers, the CIE consortiumhad introduced a new coordi-natesystemXYZ.TheRGBsystemisessentiallydefinedbythreenon-orthogonalbasevectorsinXYZ.
Thebottomimageexplainsthesitutionfor2DcoordinatesR,GandX,Yalittlesimplified.The shaded area shows thehumangamut.Aplanedividesthespaceintwohalfspaces.The new coordinates X,Y arechosen so, that the gamut isentirely accessible for positivevalues.Thiscanbegeneralizedforthe3Dspace.
In the upper image the axesXYZ are drawn orthogonally,in the lower image the axesRGB. X
Z
X
R
Y
G
Plane
RGBbasevectorsandcolorcubeinXYZ
2DvisualizationforRGandXY
R0 490000 176970 00000
.
.
.
G0 310000 812400 01000
.
.
.
B0 200000 010630 99000
.
.
.
ThecoordinatesofthebasevectorsinXYZ(coordinatesoftheprimariesasshownabove)foranyRGBsystemarefoundascolumnsofthematrixCxrinchapter��.
66
5. XYZ Primaries (see App. B for further Explanations)
ThecoordinatesystemsXYZandRGBarerelatedtoeachotherbylinearequations.
X
-2.36499
+2.3646�
+0.0003�
Z
0.40747
-0.46807
0.06065
Y
+6.54822
-0.89654
-0.00087
Anotherviewispossiblebyintroducingsyntheticalor’imaginary’primariesX,Y,Z.
TheStandardPrimariesR,G,Baremonochromaticstimuli.Mathematicallytheyaresingledeltafunc-tionswithwelldefinedareas.Inthediagramtheheightrepresentsthecontribu-tiontotheluminance.Theratiosare�.0 :4.5907:0.060�.
Thespectra X,Y,Z arecalculatedbytheapplicationofthematrixoperation(2)andthescalefactors.
Anexample:
X=�,Y=0,Z=0:
TheprimariesX,Y,Zaresumsofdeltafunctions.XandZdonotcontributetotheluminance.ThisisaspecialtrickintheCIEsystem.Theintegralsarezero,hererepresentedbythesumoftheheights.TheluminanceisdefinedbyYonly.
SyntheticalprimariesX,Y, Z
X C R== + + += + + +
xr
X R G B
Y R G
0 49000 0 31000 0 20000
0 17697 0 81240 0
. . .
. . .001063 1
0 00000 0 01000 0 99000
2 36461 0
B
Z R G B
R Xrx
( )
. . .
.
= + + +== + −
R C X
.. .
. . . ( )
.
89654 0 46807
0 51517 1 42641 0 08876 2
0 0052
Y Z
G X Y Z
B
−= − + += + 00 0 01441 1 00920X Y Z− +. .
X R
G
B
X
= + ⋅− ⋅+ ⋅
= +
2 36461 1 0000
0 51517 4 5907
0 00520 0 0601
2 364
. .
. .
. .
. 661 2 36499 0 00031R G B− +. .
IncolormatchingexperimentsnegativevaluesorweightfactorsR,G,Bareallowed.Some matchable colors cannot be generated bythe Standard Primaries. Other light sources arenecessary,especiallyspectralpuresources(mono-chromats).
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+�.0000
+0.060�
CIEprimariesR,G,B
77
6. XYZ Color-Matching Functions
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x_
y_
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λ
The functions x( )λ ,y( )λ ,z( )λ can be understoodasweightfactors.ForaspectralpurecolorCwithafixedwavelengthλreadinthediagramthethreevalues.ThenthecolorcanbemixedbythethreeStandardPrimaries:
C=x( )λ X+y( )λ Y+z( )λ Z
Generallywewrite
C=XX+YY+ZZ
andagivenspectralcolordistributionP(λ)deliversthethreecoordinatesXYZbytheseintegralsintherangefrom380nmto700nmor800nm:
XYZColor-matchingfunctions
Thenewcolor-matchingfunctionsx( )λ ,y( )λ ,z( )λ havenon-negativevalues,asexpected.Theyarecalculatedfrom r ( )λ ,g( )λ ,b( )λ byusingthematrixCxrinchapter5.
X Y
ThisdiagramshowsalreadythehumangamutinXYZ.Itisanirregularlyshapedcone.Thein-tersectionwiththeblue-ishcoloredplaneinthecornerwilldeliverthechromaticitydiagram.
HumangamutinXYZ
Mostly,thearbitraryfactorkischosenforanormalizedvalueY=�orY=�00.MatrixoperationsarealwaysnormalizedforR,G,B,Y=0to�.
X k P x d
Y k P y d
Z k P z d
=
=
=
∫∫∫
( ) ( )
( ) ( )
( ) ( )
λ λ λ
λ λ λ
λ λ λ
88
Thechromaticityvaluesx,y,zdependonlyon thehue or dominant wavelength and the saturation.Theyareindependendoftheluminance:
7. Chromaticity Values
x
y
z
�
�
�
View
Projectionandchromaticityplane
ArbitrarycolorXYZ
Theverticalprojectionontothexy-planeisthechromaticitydiagramxyY(viewdirection).To reconstruct a color triple XYZ from the chromaticity values xy, we need an additionalinformation,theluminanceY .
All visible (matchable)colorswhichdifferonlybyluminancemaptothesamepointinthechromati-citydiagram.Thisissometimescalled’horseshoediagram’(page2).Therightimageshowsa3Dviewofthecolor-match-ingfunctions,connectedbyrayswiththeorigin.Thecontourisherecalled‘locusofunitmono-chromats’[�8].ForspectralcolorsthisisthesameasXYZ.Then the contour is mapped onto the plane asabove.Thespectrallociforblueandforredendnearlyintheorigin:colorswithshortandlongwavelengthsappearratherdark,theyarealmostinvisibleforareasonablylimitedpower.Thechromaticitydiagramconcealsthis importantfact.Thepurplelinecanbeconsideredasafake.Realpurplesareinsidethehorseshoecontour.
Rendering primaries445535606Halfaxis length 1.0
X
Y
Z
z x y
Xxy
Y
Zzy
Y
= − −
=
=
1
Rendering primaries445535606Halfaxis length 1.0
X
Y
Z
xX
X Y Z
yY
X Y Z
zZ
X Y Z
=+ +
=+ +
=+ +
Obviouslywehavex+y+z=�.Allthevaluesareonthetriangleplane,projectedbyalinethroughthearbitrarycolorXYZandtheorigin,ifwedrawXYZandxyzinonediagram.Thisisaplanarprojection.Thecenterofprojectionisintheorigin.
99
These imagesarecomputergraphics.Accurate transformationsanda fewapplicationsofimageprocessing.ThecontourofthehorseshoeismappedtoXYZforluminancesY=0..�.Thepurpleplaneisshowntransparent.Allcolorswereselectedforreadabilty.Thecolorsarenotcorrect,thisisanywayimpossible.Moreimportantisherethegeometry.Thegamutvolumeisconfinedbythecolorsurface(purespectralcolors),thepurpleplaneandtheplaneY=�.TheregionswithsmallvaluesYappearextremelydistorted-neartoasingularity.ForblueveryhighvaluesZarenecessarytomatchacolorwithspecifiedluminanceY=�.
8. Color Space Visualization
YX
� �
2 2
X Y
Z
�0�0
ThegraphicshowsthecolortemperatureforthePlanckradiatorfrom2000Kto�0000K,thedirectionsofcorrelatedcolortemperaturesandthewhitepointsfordaylightD50andD65.Uncalibratedmonitorshaveabout9300KwhichisheresimplycalledD93.Databy[3].TheEPSgraphicisavailablehere[�5].
9. Color Temperature and White Points
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0
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D65
D93
2000 0.52669 0.41331 1.33101 2105 0.51541 0.41465 1.39021 2222 0.50338 0.41525 1.45962 2353 0.49059 0.41498 1.54240 2500 0.47701 0.41368 1.64291 2677 0.463 0.41121 1.76811 % error in table [3], estimated values 2857 0.446 0.40742 1.92863 3077 0.43156 0.40216 2.14300 3333 0.41502 0.39535 2.44455 3636 0.39792 0.38690 2.90309 4000 0.38045 0.37676 3.68730 4444 0.36276 0.36496 5.34398 5000 0.34510 0.35162 11.17883 5714 0.32775 0.33690 -39.34888 6667 0.31101 0.32116 -6.18336 8000 0.29518 0.30477 -3.08425 10000 0.28063 0.28828 -1.93507
T/K x y Dir y/x
����
10. CIE RGB Gamut in xyY
ThegamutofanyRGBsystemismostlyvisualizedbyatriangleinxyY.Fordifferentlumi-nancesY=const.wegettheintersectionofaverticalplaneandtheRGBcube(chapter4).Theintersectiondeliversatriangle,aquadrilateral,apentagonorahexagon.Thesepolygonsareprojectedontothexy-planeThechromaticitydiagrambelowshows theactual gamut fordifferent luminancesY. Lowluminancesseemtoproducealargegamut.Butthatisafake–aresultoftheperspectiveprojectionfromXYZtoxyY.ThegamutappearssimilarlyinallRGBsystems.Acoloroutsidethetriangle(whichisdefinedbytheprimaries)isalwaysout-of-gamut.Acolorinsidethetriangleisnotnecessarilyin-gamut.
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x
y
0.05
0.15
0.25
0.55
0.75 0.95
0.35
0.65
0.45
Y = 0.05 .. 0.950.85
�2�2
11.1 Color Space Calculations / General
InthischapterwederivetherelationsbetweenCIExyY,CIEXYZandanyarbitraryRGBspace.ItisessentialtounderstandtheprincipleofRGBbasisvectorsintheXYZcoordi-natesystem.Thiswasshownonpreviouspages.
Given are the coordinates for the primaries in CIE xyY and for the white point:xr,yr, xg,yg,xb,yb,xw,yw .CIExyYisthehorseshoediagram.Furtheronweneedthelumi-nanceV.
Wewanttoderivetherelationbetweenanycolorsetr,g,bandthecoordinatesX,Y,Z.
( ) /
/
8 X V x y
Y V
Z V z y
=
=
=
( ) ( , , )
( ) ( , , )
1
2
r
X
=
=
r g b
X Y Z
T
T
Color values in RGB
Color values iin XYZ
Color values in xyY
Scaling v
( ) ( , , )
( )
3
4
x =
= + +
x y z
L X Y Z
T
aalue
( ) /
/
/
( )
( )
5
6 1
7
x X L
y Y L
z Z L
z x y
L
=
=
=
= − −
=X x
( ) ( , , )
( , , )
( , , )
(
9 R x
G x
B x
= =
= =
= =
L L x y z
L L x y z
L L x y z
r r r rT
g g g gT
b b b bT
110) ( , , )W w= =L L x y zw w wT
( ) ( , , )11 u = u v w T
Vistheluminanceofthestimulus,accordingtotheluminousefficiencyfunctionV(λ)in[3 ].WeshouldnotcallthisimmediatelyYbecauseYismostlynormalizedfor�or�00.
BasisvectorsfortheprimariesandwhitepointinXYZ:
Setofscalefactorsforthewhitepointcorrection:
�3�3
11.2 Color Space Calculations / General
Forthewhitepointcorrection,thebasisvectorsR,G,Barescaledbyu,v,w.ThisdoesnotchangetheircoordinatesinxyY.ThemappingfromXYZtoxyYisacentralplanarprojection.
TheselinearequationsaresolvedbyCramer’srule.
( ) ( , , )12 X R G B= = + +L x y z ru gv bwT
( ) ( , , ) ( , , ) ( , , ) ( , ,13 W = = + +L x y z Lu x y z L v x y z L w x y zw w wT
r r rT
g g gT
b b b))T
( )14xyz
x x xy y yz z z
uvw
w
w
w
r g b
r g b
r g b
=
= Puuvw
( )
( )
15 1
161
w u v
xy
x x xy y y
uvu v
w
w
r g b
r g b
= − −
=
− −
= − + − +
= − + − +
( ) ( ) ( )
( ) ( )
17 x x x u x x v x
y y y u y y v yw r b g b b
w r b g b b
( ) ( ) ( ) ( ) ( )
( ) ( ) (
18 D x x y y y y x x
U x x y y y yr b g b r b g b
w b g b w b
= − − − − −
= − − − − )) ( )
( ) ( ) ( ) ( )
( ) /
/
x x
V x x y y y y x x
u U D
v V D
w
g b
r b w b r b w b
−
= − − − − −=== −
19
1 uu v−
Inthenextstepweassumethatu,v,warealreadycalculatedandweusethegeneralcolortransformationEq.(�2)andfurtheronEq.(8).WegetthematricesCxrandCrx.
( )/ / // / //
20XYZ
Vux y v x y w x yuy y v y y w y yuz y
r w g w b w
r w g w b w
r
=ww g w b w
xr
xr
v z y w z y
rgb
V
V
/ /
(
( ) ( / )
=
= −
21
22 1
X C r
r C 11 1X C X= ( / )V rx
Itisnotnecessarytoinvertthewholematrixnumerically.WecansimplifythecalculationbyaddingthefirsttworowstothethirdrowandfindsoimmediatelyEq.(�5),whichisanywayclear:
Thiscanbere-arranged,Lcancelsonbothsides.:
Forthewhitepointwehaver=g=b=�.
�4�4
11.3 Color Space Calculations / General
Forbetterreadabilityweshowthelasttwoequationsagain,butnowwithV=�,asinmostpublications.
AnexampleshowstheconversionofRec.709/D65toD50andD93.TheresultingmatrixC2�isdiagonal,becausethesourceanddestinationprimariesarethesame.TheexplanationasaboveisvalidfortherepresentationofthesamephysicalcolorintwodifferentRGBsystems.ForthesimulationofD50orD93effectsinthesameD65RGBsystemonehastoapplytheinversematrix.
Rec.709 xr= 0.6400 yr= 0.3300 zr= 0.0300 xg= 0.3000 yg= 0.6000 zg= 0.1000 xb= 0.1500 yb= 0.0600 zb= 0.7900
D65 xw= 0.3127 yw= 0.3290 zw= 0.3583
D93 xw= 0.2857 yw= 0.2941 zw= 0.4202
Matrix Cxr: X=Cxr*R65 0.4124 0.3576 0.1805 0.2126 0.7152 0.0722 0.0193 0.1192 0.9505
Matrix Crx: R65=Crx*X 3.2410 -1.5374 -0.4986 -0.9692 1.8760 0.0416 0.0556 -0.2040 1.0570
Matrix Dxr: X=Dxr*R93 0.3706 0.3554 0.2455 0.1911 0.7107 0.0982 0.0174 0.1185 1.2929
Matrix Drx: R93=Drx*X 3.6066 -1.7108 -0.5549 -0.9753 1.8877 0.0418 0.0409 -0.1500 0.7771
Matrix Err: R93=Err*R65=Drx*Cxr*R65 1.1128 0.0000 0.0000 0.0000 1.0063 0.0000 0.0000 0.0000 0.7352
Matrix Frr: R65=Frr*R93=Crx*Dxr*R93 0.8986 0.0000 0.0000 0.0000 0.9938 0.0000 0.0000 0.0000 1.3602
Rec.709 xr= 0.6400 yr= 0.3300 zr= 0.0300 xg= 0.3000 yg= 0.6000 zg= 0.1000 xb= 0.1500 yb= 0.0600 zb= 0.7900
D65 xw= 0.3127 yw= 0.3290 zw= 0.3583
D50 xw= 0.3457 yw= 0.3585 zw= 0.2958
Matrix Cxr: X=Cxr*R65 0.4124 0.3576 0.1805 0.2126 0.7152 0.0722 0.0193 0.1192 0.9505
Matrix Crx: R65=Crx*X 3.2410 -1.5374 -0.4986 -0.9692 1.8760 0.0416 0.0556 -0.2040 1.0570
Matrix Dxr: X=Dxr*R50 0.4852 0.3489 0.1303 0.2502 0.6977 0.0521 0.0227 0.1163 0.6861
Matrix Drx: R50=Drx*X 2.7548 -1.3068 -0.4238 -0.9935 1.9229 0.0426 0.0771 -0.2826 1.4644
Matrix Err: R50=Err*R65=Drx*Cxr*R65 0.8500 0.0000 0.0000 0.0000 1.0250 0.0000 0.0000 0.0000 1.3855
Matrix Frr: R65=Frr*R50=Crx*Dxr*R50 1.1765 0.0000 0.0000 0.0000 0.9756 0.0000 0.0000 0.0000 0.7218
( )
( )
23
24 1
X C r
r C X C X
=
= =−xr
xr rx
NowwecaneasilyderivetherelationbetweentwodifferentRGBspaces,e.g.workingspacesandimagesourcespaces.
( )
( )
( )
( )
25
26
27
28
1 1
2 2
2 21
1 1
2 21 1
X C r
X C r
r C C r
r C r
==
==
−
xr
xr
xr xr
�5�5
11.4 Color Space Calculations / Simplified
Nowwecleanupthemathematics.Eq.(�4)delivers:
( )29 1u P w= −
( )
( )
30
31
X PDr
X C r
== xr
Eq.(�2)andEq.(20)canbewrittenusingthediagonalmatrixDwithelementsu/ywetc.:
TogetherwithEq.(29)wefindthissimpleformulaforthematrixCxr:
( )/
//
320 0
0 00 0
C Pxr
w
w
w
u yv y
w y=
The examples in chapter �2 were written by Pascal. Here is a new example in MatLab.CalculationofthematricesforsRGB:
% January 14 / 2005% Matrix Cxr and Crx for sRGB
xr=0.6400; yr=0.3300; zr=1-xr-yr;xg=0.3000; yg=0.6000; zg=1-xg-yg;xb=0.1500; yb=0.0600; zb=1-xb-yb;xw=0.3127; yw=0.3290; zw=1-xw-yw;
W=[xw; yw; zw];P=[xr xg xb; yr yg yb; zr zg zb]; u=inv(P)*W
% D=[u(1) 0 0;% 0 u(2) 0;% 0 0 u(3)]/yw
D=diag(u/yw)
Cxr=P*D Crx=inv(Cxr)
% Result:
% Cxr 0.4124 0.3576 0.1805% 0.2126 0.7152 0.0722% 0.0193 0.1192 0.9505
% Crx 3.2410 -1.5374 -0.4986% -0.9692 1.8760 0.0416% 0.0556 -0.2040 1.0570
% G.Hoffmann
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% G.Hoffmann% January 19 / 2005% Calculations for CIE primaries% x-bar,y-bar,z-bar interpolated% 700.0 546.1 435.8 nmxbr=0.011359; xbg=0.375540; xbb=0.333181; ybr=0.004102; ybg=0.984430; ybb=0.017769;zbr=0.000000; zbg=0.012207; zbb=1.649716;% Equal Energy WPXw=1; Yw=1; Zw=1;
%Chromaticity coordinatesD=xbr+ybr+zbr; xr=xbr/D; yr=ybr/D; zr=zbr/D;D=xbg+ybg+zbg; xg=xbg/D; yg=ybg/D; zg=zbg/D;D=xbb+ybb+zbb; xb=xbb/D; yb=ybb/D; zb=zbb/D;D=Xw +Yw+ Zw; xw=Xw/D; yw=Yw/D; zw=Zw/D;
w=[xw; yw; zw];P=[xr xg xb; yr yg yb; zr zg zb];
u=inv(P)*wD=diag(u/yw)
Cxr=P*D% 0.4902 0.3099 0.1999% 0.1770 0.8123 0.0107% 0.0000 0.0101 0.9899Crx=inv(Cxr)% 2.3635 -0.8958 -0.4677% -0.5151 1.4265 0.0887% 0.0052 -0.0145 1.0093
% Radiant power ratiosXbar=[xbr xbg xbb; ybr ybg ybb; zbr zbg zbb];W=[Xw; Yw; Zw];
R=inv(Xbar)*W
R=R/R(3)% 71.9166 1.3751 1.0000% 72.0962 1.3791 1.0000 Wyszecki & Stiles
% Luminous efficiency ratiosL=[R(1)*ybr; R(2)*ybg; R(3)*ybb]L=L/L(1)% 1.0000 4.5889 0.0602% 1.0000 4.5907 0.0601 Wyszecki & Stiles
% G.Hoffmann% January 19 / 2005% Calculations for Laser primaries% x-bar,y-bar,z-bar interpolated% 671 532 473 nmxbr=0.0819; xbg=0.1891; xbb=0.1627; ybr=0.0300; ybg=0.8850; ybb=0.1034;zbr=0.0000; zbg=0.0369; zbb=1.1388;% D65Xw=0.9504; Yw=1.0000; Zw=1.0890;
%Chromaticity coordinatesD=xbr+ybr+zbr; xr=xbr/D; yr=ybr/D; zr=zbr/D;D=xbg+ybg+zbg; xg=xbg/D; yg=ybg/D; zg=zbg/D;D=xbb+ybb+zbb; xb=xbb/D; yb=ybb/D; zb=zbb/D;D=Xw +Yw+ Zw; xw=Xw/D; yw=Yw/D; zw=Zw/D;
w=[xw; yw; zw];P=[xr xg xb; yr yg yb; zr zg zb];
u=inv(P)*wD=diag(u/yw)
Cxr=P*D% 0.6571 0.1416 0.1516% 0.2407 0.6629 0.0964% 0 0.0276 1.0614Crx=inv(Cxr)% 1.6476 -0.3435 -0.2042% -0.6005 1.6394 -0.0631% 0.0156 -0.0427 0.9438
% Radiant power ratiosXbar=[xbr xbg xbb; ybr ybg ybb; zbr zbg zbb];W=[Xw; Yw; Zw];
R=inv(Xbar)*W
R=R/R(1)% 1.0000 0.0934 0.1162R=R/R(2)% 10.7111 1.0000 1.2442R=R/R(3)% 8.6088 0.8037 1.0000
% Luminous efficiency ratiosL=[R(1)*ybr; R(2)*ybg; R(3)*ybb];L=L/L(1)% 1.0000 2.7542 0.4004L=L/L(2)% 0.3631 1.0000 0.1454L=L/L(3)% 2.4977 6.8791 1.0000
11.5 Color Space Calculations / Application
Thetask:red,greenandbluelasersgeneratemonochromaticlightatwavelengths67�nm,532nmand473nm.ThepowersaretobeadjustedsothatthethreelaserstogetherdeliverwhitelightD65.Calculatethematrices,theradiantpowerratiosandthephotometricratios.
InordertotestthealgorithmswearedoingthesameforCIEprimariesandEqualEnergyWhite,justasifthelasershadtheseprimaries.Theresultsareknowninadvance,basedonstandardtextbooks.ThankstoGerhardFuernkranzforimportantclarifications.
LaserprimariesandwhitepointD65CIEprimariesandwhitepointE
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12.1 Matrices / CIE + E
CIEPrimariesandwhitepointE[3].Page5showsthesameresults.DataareinthePascalsourcecode.
Program CiCalcCi;{ Calculations RGB—CIE }{ G.Hoffmann February 01, 2002 }Uses Crt,Dos,Zgraph00;Var r,g,b,x,y,z,u,v,w,d : Extended; i,j,k,flag : Integer; xr,yr,zr,xg,yg,zg,xb,yb,zb,xw,yw,zw : Extended; prn,cie : Text;Var Cxr,Crx: ANN;BeginClrScr;{ CIE Primaries }xr:=0.73467;yr:=0.26533;zr:=1-xr-yr;xg:=0.27376;yg:=0.71741;zg:=1-xg-yg;xb:=0.16658;yb:=0.00886;zb:=1-xb-yb;{ CIE White Point }xw:=1/3;yw:=1/3;zw:=1-xw-yw;{ White Point Correction }D:=(xr-xb)*(yg-yb)-(yr-yb)*(xg-xb);U:=(xw-xb)*(yg-yb)-(yw-yb)*(xg-xb);V:=(xr-xb)*(yw-yb)-(yr-yb)*(xw-xb);u:=U/D;v:=V/D;w:=1-u-v;{ Matrix Cxr }Cxr[1,1]:=u*xr/yw; Cxr[1,2]:=v*xg/yw; Cxr[1,3]:=w*xb/yw;Cxr[2,1]:=u*yr/yw; Cxr[2,2]:=v*yg/yw; Cxr[2,3]:=w*yb/yw;Cxr[3,1]:=u*zr/yw; Cxr[3,2]:=v*zg/yw; Cxr[3,3]:=w*zb/yw;{ Matrix Crx }HoInvers (3,Cxr,Crx,D,flag);
Assign (prn,’C:\CiMalcCi.txt’); ReWrite(prn);
Writeln (prn,’ Matrix Cxr’);Writeln (prn,Cxr[1,1]:12:4, Cxr[1,2]:12:4, Cxr[1,3]:12:4);Writeln (prn,Cxr[2,1]:12:4, Cxr[2,2]:12:4, Cxr[2,3]:12:4);Writeln (prn,Cxr[3,1]:12:4, Cxr[3,2]:12:4, Cxr[3,3]:12:4);
Writeln (prn,’ Matrix Crx’);Writeln (prn,Crx[1,1]:12:4, Crx[1,2]:12:4, Crx[1,3]:12:4);Writeln (prn,Crx[2,1]:12:4, Crx[2,2]:12:4, Crx[2,3]:12:4);Writeln (prn,Crx[3,1]:12:4, Crx[3,2]:12:4, Crx[3,3]:12:4);Close(prn);Readln;End.
Matrix Cxr X 0.4900 0.3100 0.2000 Y 0.1770 0.8124 0.0106 Z -0.0000 0.0100 0.9900
Matrix Crx R 2.3647 -0.8966 -0.4681 G -0.5152 1.4264 0.0887 B 0.0052 -0.0144 1.0092
X=CxrR
R=CrxX
�8�8
12.2 Matrices / 709 + D65 / sRGB
ITU-RBT.709PrimariesandwhitepointD65[9].ValidforsRGB.DataareinthePascalsourcecode.
Program CiCalc65;{ Calculations RGB—CIE }{ G.Hoffmann February 01, 2002 }Uses Crt,Dos,Zgraph00;Var r,g,b,x,y,z,u,v,w,d : Extended; i,j,k,flag : Integer; xr,yr,zr,xg,yg,zg,xb,yb,zb,xw,yw,zw : Extended; prn,cie : Text;Var Cxr,Crx: ANN;BeginClrScr;{ Rec 709 Primaries }xr:=0.6400;yr:=0.3300;zr:=1-xr-yr;xg:=0.3000;yg:=0.6000;zg:=1-xg-yg;xb:=0.1500;yb:=0.0600;zb:=1-xb-yb;{ D65 White Point }xw:=0.3127;yw:=0.3290;zw:=1-xw-yw;{ White Point Correction }D:=(xr-xb)*(yg-yb)-(yr-yb)*(xg-xb);U:=(xw-xb)*(yg-yb)-(yw-yb)*(xg-xb);V:=(xr-xb)*(yw-yb)-(yr-yb)*(xw-xb);u:=U/D;v:=V/D;w:=1-u-v;{ Matrix Cxr }Cxr[1,1]:=u*xr/yw; Cxr[1,2]:=v*xg/yw; Cxr[1,3]:=w*xb/yw;Cxr[2,1]:=u*yr/yw; Cxr[2,2]:=v*yg/yw; Cxr[2,3]:=w*yb/yw;Cxr[3,1]:=u*zr/yw; Cxr[3,2]:=v*zg/yw; Cxr[3,3]:=w*zb/yw;{ Matrix Crx }HoInvers (3,Cxr,Crx,D,flag);
Assign (prn,’C:\CiMalc65.txt’); ReWrite(prn);
Writeln (prn,’ Matrix Cxr’);Writeln (prn,Cxr[1,1]:12:4, Cxr[1,2]:12:4, Cxr[1,3]:12:4);Writeln (prn,Cxr[2,1]:12:4, Cxr[2,2]:12:4, Cxr[2,3]:12:4);Writeln (prn,Cxr[3,1]:12:4, Cxr[3,2]:12:4, Cxr[3,3]:12:4);
Writeln (prn,’ Matrix Crx’);Writeln (prn,Crx[1,1]:12:4, Crx[1,2]:12:4, Crx[1,3]:12:4);Writeln (prn,Crx[2,1]:12:4, Crx[2,2]:12:4, Crx[2,3]:12:4);Writeln (prn,Crx[3,1]:12:4, Crx[3,2]:12:4, Crx[3,3]:12:4);Close(prn);Readln;End.
Matrix Cxr X 0.4124 0.3576 0.1805 Y 0.2126 0.7152 0.0722 Z 0.0193 0.1192 0.9505
Matrix Crx R 3.2410 -1.5374 -0.4986 G -0.9692 1.8760 0.0416 B 0.0556 -0.2040 1.0570
X=CxrR
R=CrxX
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12.3 Matrices / AdobeRGB + D65
AdobeRGB(98),D65.DataareinthePascalsourcecode.
Program CiCalc98;{ Calculations RGB—AdobeRGB98 }{ G.Hoffmann März 28, 2004 }Uses Crt,Dos,Zgraph00;Var r,g,b,x,y,z,u,v,w,d : Double; i,j,k,flag : Integer; xr,yr,zr,xg,yg,zg,xb,yb,zb,xw,yw,zw : Double; prn,cie : Text;Var Cxr,Crx: ANN;BeginClrScr;{ AdobeRGB(98) }xr:=0.6400;yr:=0.3300;zr:=1-xr-yr;xg:=0.2100;yg:=0.7100;zg:=1-xg-yg;xb:=0.1500;yb:=0.0600;zb:=1-xb-yb;{ D65 White Point }xw:=0.3127;yw:=0.3290;zw:=1-xw-yw;{ White Point Correction }D:=(xr-xb)*(yg-yb)-(yr-yb)*(xg-xb);U:=(xw-xb)*(yg-yb)-(yw-yb)*(xg-xb);V:=(xr-xb)*(yw-yb)-(yr-yb)*(xw-xb);u:=U/D;v:=V/D;w:=1-u-v;{ Matrix Cxr }Cxr[1,1]:=u*xr/yw; Cxr[1,2]:=v*xg/yw; Cxr[1,3]:=w*xb/yw;Cxr[2,1]:=u*yr/yw; Cxr[2,2]:=v*yg/yw; Cxr[2,3]:=w*yb/yw;Cxr[3,1]:=u*zr/yw; Cxr[3,2]:=v*zg/yw; Cxr[3,3]:=w*zb/yw;{ Matrix Crx }HoInvers (3,Cxr,Crx,D,flag);
Assign (prn,’C:\CiMalc98.txt’); ReWrite(prn);
Writeln (prn,’ Matrix Cxr’);Writeln (prn,Cxr[1,1]:12:4, Cxr[1,2]:12:4, Cxr[1,3]:12:4);Writeln (prn,Cxr[2,1]:12:4, Cxr[2,2]:12:4, Cxr[2,3]:12:4);Writeln (prn,Cxr[3,1]:12:4, Cxr[3,2]:12:4, Cxr[3,3]:12:4);Writeln (prn,’’);Writeln (prn,’ Matrix Crx’);Writeln (prn,Crx[1,1]:12:4, Crx[1,2]:12:4, Crx[1,3]:12:4);Writeln (prn,Crx[2,1]:12:4, Crx[2,2]:12:4, Crx[2,3]:12:4);Writeln (prn,Crx[3,1]:12:4, Crx[3,2]:12:4, Crx[3,3]:12:4);Writeln (prn,’dummy’);Readln;End.
Matrix Cxr X 0.5767 0.1856 0.1882 Y 0.2973 0.6274 0.0753 Z 0.0270 0.0707 0.9913
Matrix Crx R 2.0416 -0.5650 -0.3447 G -0.9692 1.8760 0.0416 B 0.0134 -0.1184 1.0152
X=CxrR
R=CrxX
2020
12.4 Matrices / NTSC + C
NTSCPrimariesandwhitepointC[4],alsousedasYIQModel.DataareinthePascalsourcecode.
Program CiCalcNT;{ Calculations RGB—NTSC }{ G.Hoffmann April 01, 2002 }Uses Crt,Dos,Zgraph00;Var r,g,b,x,y,z,u,v,w,d : Extended; i,j,k,flag : Integer; xr,yr,zr,xg,yg,zg,xb,yb,zb,xw,yw,zw : Extended; prn,cie : Text;Var Cxr,Crx: ANN;BeginClrScr;{ NTSC Primaries }xr:=0.6700;yr:=0.3300;zr:=1-xr-yr;xg:=0.2100;yg:=0.7100;zg:=1-xg-yg;xb:=0.1400;yb:=0.0800;zb:=1-xb-yb;{ NTSC White Point }xw:=0.3100;yw:=0.3160;zw:=1-xw-yw;{ White Point Correction }D:=(xr-xb)*(yg-yb)-(yr-yb)*(xg-xb);U:=(xw-xb)*(yg-yb)-(yw-yb)*(xg-xb);V:=(xr-xb)*(yw-yb)-(yr-yb)*(xw-xb);u:=U/D;v:=V/D;w:=1-u-v;{ Matrix Cxr }Cxr[1,1]:=u*xr/yw; Cxr[1,2]:=v*xg/yw; Cxr[1,3]:=w*xb/yw;Cxr[2,1]:=u*yr/yw; Cxr[2,2]:=v*yg/yw; Cxr[2,3]:=w*yb/yw;Cxr[3,1]:=u*zr/yw; Cxr[3,2]:=v*zg/yw; Cxr[3,3]:=w*zb/yw;{ Matrix Crx }HoInvers (3,Cxr,Crx,D,flag);
Assign (prn,’C:\CiMalcNT.txt’); ReWrite(prn);
Writeln (prn,’ Matrix Cxr’);Writeln (prn,Cxr[1,1]:12:4, Cxr[1,2]:12:4, Cxr[1,3]:12:4);Writeln (prn,Cxr[2,1]:12:4, Cxr[2,2]:12:4, Cxr[2,3]:12:4);Writeln (prn,Cxr[3,1]:12:4, Cxr[3,2]:12:4, Cxr[3,3]:12:4);Writeln (prn,’’);Writeln (prn,’ Matrix Crx’);Writeln (prn,Crx[1,1]:12:4, Crx[1,2]:12:4, Crx[1,3]:12:4);Writeln (prn,Crx[2,1]:12:4, Crx[2,2]:12:4, Crx[2,3]:12:4);Writeln (prn,Crx[3,1]:12:4, Crx[3,2]:12:4, Crx[3,3]:12:4);Close(prn);Readln;End.
Matrix Cxr X 0.6070 0.1734 0.2006 Y 0.2990 0.5864 0.1146 Z -0.0000 0.0661 1.1175
Matrix Crx R 1.9097 -0.5324 -0.2882 G -0.9850 1.9998 -0.0283 B 0.0582 -0.1182 0.8966
X=CxrR
R=CrxX
2�2�
12.5 Matrices / NTSC + C + YIQ
NTSCPrimariesandwhitepointC[4],YIQConversion.DataareinthePascalsourcecode.
Program CiCalcYI;{ Calculations RGB—NTSC YIQ }{ G.Hoffmann April 01, 2002 }Uses Crt,Dos,Zgraph00;Var r,g,b,x,y,z,u,v,w,d : Extended; i,j,k,flag : Integer; xr,yr,zr,xg,yg,zg,xb,yb,zb,xw,yw,zw : Extended; prn,cie : Text;Var Cyr,Cry: ANN;BeginClrScr;{ NTSC Primaries }xr:=0.6700;yr:=0.3300;zr:=1-xr-yr;xg:=0.2100;yg:=0.7100;zg:=1-xg-yg;xb:=0.1400;yb:=0.0800;zb:=1-xb-yb;{ NTSC White Point }xw:=0.3100;yw:=0.3160;zw:=1-xw-yw;{ Matrix Cyr, Sequence Y I Q }Cyr[1,1]:= 0.299; Cyr[1,2]:= 0.587; Cyr[1,3]:= 0.114;Cyr[2,1]:= 0.596; Cyr[2,2]:=-0.275; Cyr[2,3]:=-0.321;Cyr[3,1]:= 0.212; Cyr[3,2]:=-0.528; Cyr[3,3]:= 0.311;{ Matrix Cry }HoInvers (3,Cyr,Cry,D,flag);
Assign (prn,’C:\CiMalcYI.txt’); ReWrite(prn);
Writeln (prn,’ Matrix Cyr’);Writeln (prn,Cyr[1,1]:12:4, Cyr[1,2]:12:4, Cyr[1,3]:12:4);Writeln (prn,Cyr[2,1]:12:4, Cyr[2,2]:12:4, Cyr[2,3]:12:4);Writeln (prn,Cyr[3,1]:12:4, Cyr[3,2]:12:4, Cyr[3,3]:12:4);Writeln (prn,’’);Writeln (prn,’ Matrix Cry’);Writeln (prn,Cry[1,1]:12:4, Cry[1,2]:12:4, Cry[1,3]:12:4);Writeln (prn,Cry[2,1]:12:4, Cry[2,2]:12:4, Cry[2,3]:12:4);Writeln (prn,Cry[3,1]:12:4, Cry[3,2]:12:4, Cry[3,3]:12:4);Close(prn);Readln;End.
Matrix Cyr Y 0.2990 0.5870 0.1140 I 0.5960 -0.2750 -0.3210 Q 0.2120 -0.5280 0.3110
Matrix Cry R 1.0031 0.9548 0.6179 G 0.9968 -0.2707 -0.6448 B 1.0085 -1.1105 1.6996
Y=CyrR
R=CryY
2222
12.6 Matrices / NTSC + C + YCbCr
NTSCPrimariesandwhitepointC[4],YCbCrConversion.DataareinthePascalsourcecode.
Program CiCalcYC;{ Calculations RGB—NTSC YCbCr }{ G.Hoffmann April 03, 2002 }Uses Crt,Dos,Zgraph00;Var r,g,b,x,y,z,u,v,w,d : Extended; i,j,k,flag : Integer; xr,yr,zr,xg,yg,zg,xb,yb,zb,xw,yw,zw : Extended; prn,cie : Text;Var Cyr,Cry: ANN;BeginClrScr;{ NTSC Primaries }xr:=0.6700;yr:=0.3300;zr:=1-xr-yr;xg:=0.2100;yg:=0.7100;zg:=1-xg-yg;xb:=0.1400;yb:=0.0800;zb:=1-xb-yb;{ NTSC White Point }xw:=0.3100;yw:=0.3160;zw:=1-xw-yw;{ Matrix Cxr, Sequence Y Cb Cr }Cyr[1,1]:= 0.2990; Cyr[1,2]:= 0.5870; Cyr[1,3]:= 0.1140;Cyr[2,1]:=-0.1687; Cyr[2,2]:=-0.3313; Cyr[2,3]:=+0.5000;Cyr[3,1]:= 0.5000; Cyr[3,2]:=-0.4187; Cyr[3,3]:=-0.0813;{ Matrix Cry }HoInvers (3,Cyr,Cry,D,flag);
Assign (prn,’C:\CiMalcYC.txt’); ReWrite(prn);
Writeln (prn,’ Matrix Cyr’);Writeln (prn,Cyr[1,1]:12:4, Cyr[1,2]:12:4, Cyr[1,3]:12:4);Writeln (prn,Cyr[2,1]:12:4, Cyr[2,2]:12:4, Cyr[2,3]:12:4);Writeln (prn,Cyr[3,1]:12:4, Cyr[3,2]:12:4, Cyr[3,3]:12:4);Writeln (prn,’’);Writeln (prn,’ Matrix Cry’);Writeln (prn,Cry[1,1]:12:4, Cry[1,2]:12:4, Cry[1,3]:12:4);Writeln (prn,Cry[2,1]:12:4, Cry[2,2]:12:4, Cry[2,3]:12:4);Writeln (prn,Cry[3,1]:12:4, Cry[3,2]:12:4, Cry[3,3]:12:4);Close(prn);Readln;End.
Matrix Cyr Y 0.2990 0.5870 0.1140 Note Cb -0.1687 -0.3313 0.5000 This is a linear conversion, as used for JPEG Cr 0.5000 -0.4187 -0.0813 In TV systems the conversion is different
Matrix Cry R 1.0000 0.0000 1.4020 Note G 1.0000 -0.3441 -0.7141 Rounded for structural zeros B 1.0000 1.7722 0.0000
2323
13. sRGB
TheconversionforD65RGBtoD65XYZusesthematrixonpage�4,ITU-RBT.709Primaries.D65XYZmeansXYZwithoutchangingtheilluminant.
TheconversionforD65RGBtoD50XYZappliesadditionally(bymultiplication)theBradfordcorrection,whichtakestheadaptationoftheeyesintoaccount.ThiscorrectionisanimprovedalternativetotheVonKriescorrrection[�].
MonitorsareassumedD65,butforprintedpaperthestandardilluminantisD50.Thereforethistransformationisrecommendedifthedataareusedforprinting:
sRGBisastandardcolorspace,definedbycompanies,mainlyHewlett-PackardandMicrosoft[9],[�2 ].ThetransformationofRGBimagedatatoCIEXYZrequiresprimarilyaGammacorrection,whichcompensatesanexpectedinverseGammacorrection,comparedtolinearlightdata,herefornormalizedvaluesC=R,G,B=0...�:
IfC≤0.03928 Then C=C/�2.92 Else C=( (0.055+C)/�.055)2.4
Theformulainthedocument[�2 ]ismisleadingbecauseabracketwasforgotten.
Black C=C2.2
Red sRGB,asabove
Green tentimesthedifference
0�
�
0
XYZ
D
=65
0 4124 0 3576 0 18050 2126 0 7152 0 07220 0193 0
. . .
. . .
. .. .1192 0 950565
RGB
D
XYZ
D
=50
0 4361 0 3851 0 14310 2225 0 7169 0 06060 0139 0
. . .. . .. .. .0971 0 7141
65
RGB
D
2424
ThecornersR,G,Bofatriangulargamut,e.g.foramonitor,aredescribedinCIExyYbythree vectors r,g,bwhichhavetwocomponentsx,yeach.AcolorC isdescribedeitherbycwithtwovaluescx,cyorbythreevaluesR,G,B.ThesearethebarycentriccoordinatesofC.Allpointsinsideandonthetrianglearereachableby0≤ R,G,B≤�.Pointsoutsidehaveatleastonenegativecoordinate.ThecornersR,G,Bhavebarycentriccoordinates(�,0,0),(0,�,0)and(0,0,�).
14.1 Barycentric Coordinates / Concept
B R
G
Cr+g
bB’
r+g =(R+G)B B’
b = BBB’
br
bg
LineB = 0LineR= 0
LineG= 0
rg
rb
gb gr
rg = R RG
rb = R RB
gr = GGR
gb = GGB
bg = B BG
br = B BR
Underlinemeanslengthof..B R
G
C
( )
( )
( )
1
2 1
3 1
c r g b= + += + +
= − −
R G B
R G B
R G
Substitute R in(1) by (2):
BB
G B( ) ( ) ( )4 g r b r c r− + − = −(4) consists of two linear equations for G,B, which can be solved by rule.
R is calcu
Cramer’s
llated by (3).
are the edge vectors from to ( ) ( )g r b r− −and R GG R B and to .
The edge vectors are used in (4) as a vectorr base.
Any point inside the triangle is reached by G + < 1,B which leads to R + G 1.+ =B
2525
380460
470475
480
485
490
495
500
505
510
515520 525
530535
540545
550
555
560
565
570
575
580
585590
595600
605610
620635
700
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x
y
0.20700.36040.4326
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14.2 Barycentric Coordinates / Wrong
RGBrGB
rGb
RGb
RgB
rgB
Rgb
G
R
B
rg
gr
D65
Sofarthebarycentriccoordinatesremindmuchtotheexplanationsin[3],chapter3.2.2.ItshouldbepossibletofindtherelativevaluesR,G,Bforagivenpointc=(cx,cy)bymeasuringtheproportionsR=rg/RG,G=gr/GRwithRG=GR,thenB=�-R-G.
Unfortunatelythisinterpretationiswrong.ThedrawingshowstheD65whitepointandthemeasurable values R=0.2�9, G=0.385 and B=0.396 instead of the correct values R=�/3,G=�/3,B=�/3.
ThebasevectorsR,G,B inCIEXYZ(chapter4 forCIEprimaries)donothave thesamelengths.In[3]themathematicswereexplainedforunitvectors.Sofaritisnotclear,howthegeometricallyinterpretationforbarycentriccoordinatescouldbeappliedtotheactualtask.
Thediagrambelowshowsadditionallysevensectors.’RGB’means,allvaluesarepositive(insidethetriangle).’rGB’meansR<0,G>0,B>0andsoon.Negativevaluesarenotprohibitedbythedefinitionofcoordinates.TheyjustdonotappearintechnicalRGBsystem.Ofcoursetheyareessentialforthecolormatchingtheory.
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15. Optimal Primaries
JamesA.Wortheyhadshowninrecentpublications[�8]howtofindoptimalprimaries.Thisapproachisbasedon’Amplitudenotleftout’.Whichprimariesshouldbeusedifthepowerislimitedforeachlightsource?Theresultingwavelengthsareshownbythecornersofthetrianglebelow:445,536,604nm.Atleast,thewavelengthsshouldbeneartothesevalues.Forarealsystem(besidestestsinalaboratory)purespectralcolorscannotbeused.Thecornershavetobeshiftedonaradiustowardsthewhitepoint(whichishereindicatedbythecircleforD65).Theoptimalredat604nmishardlyagoodcandidatefortechnicalsystems–itismoreakindoforangeinsteadofvibrantred.
AdditionalillustrationsforJ.Worthey’sconceptsareshownin[�9].EverythingPostScriptvectorgraphics.
Purpleline
Wavelengthsinnm
2727
16.1 References
[�] R.W.G.Hunt MeasuringColour FountainPress,England,�998
[2] E.J.Giorgianni+Th.E.Madden DigitalColorManagement Addison-Wesley,ReadingMassachusetts,...,�998
[3] G.Wyszecki+W.S.Stiles ColorScience JohnWiley&Sons,NewYork,...,�982
[4] J.D.Foley+A.vanDam+St.K.Feiner+J.F.Hughes ComputerGraphics Addison-Wesley,ReadingMassachusetts,...,�993
[5] C.H.Chen+L.F.Pau+P.S.P.Wang HandbookofPtternrecognitionandComputerVision WorldScientific,Singapore,...,�995
[6] J.J.Marchesi HandbuchderFotografieVol.�-3 VerlagFotografie,Schaffhausen,�993
[7] T.Autiokari AccurateImageProcessing http://www.aim-dtp.net 200�
[8] Ch.Poynton FrequentlyaskedquestionsaboutGamma http://www.inforamp.net/~poynton/ �997[9] M.Stokes+M.Anderson+S.Chandrasekar+R.Motta AStandardDefaultColorSpacefortheInternet-sRGB http://www.w3.org/graphics/color/srgb.html �996
[�0] G.Hoffmann CorrectionsforPerceptuallyOptimizedGrayscales http://docs-hoffmann.de/optigray06�0200�.pdf 200�
[��] G.Hoffmann HardwareMonitorCalibration http://docs-hoffmann.de/caltutor270900.pdf 200�
[�2] M.Nielsen+M.Stokes TheCreationofthesRGBICCProfile http://www.srgb.com/c55.pdf Yearunknown,after�998
[�3] G.Hoffmann CieLabColorSpace http://docs-hoffmann.decielab03022003.pdf
[�4] EverythingaboutColorandComputers http://www.efg2.com
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16.2 References
[�5] CIEChromaticityDiagram,EPSGraphic http://docs-hoffmann.de/ciesuper.txt
[�6] Color-MatchingFunctionsRGB,EPSGraphic http://docs-hoffmann.de/matchrgb.txt
[�7] Color-MatchingFunctionsXYZ,EPSGraphic http://docs-hoffmann.de/matchxyz.txt
[�8] JamesA.Worthey ColorMatchingwithAmplitudeNotLeftOut http://users.starpower.net/jworthey/FinalScotts2004Aug25.pdf
[�9] G.Hoffmann LocusofUnitMonochromats http://docs-hoffmann.de/jimcolor�2062004.pdf
[20] HughS.Fairman,MichaelH.Brill,HenryHemmendinger HowtheCIE�93�Color-MatchingFunctionsWereDerivedfromWright–GuildData �996.Found2/20�4.UseGooglesearchbytitle„How…Data“.
[2�] G.Hoffmann ColorCalc:ColorMathematicsbyPostScript http://docs-hoffmann.de/colcalc03022006.txt (renameas*.eps)
GernotHoffmannAugust29/2000+March23/20�5
WebsiteLoadBrowser/Clickhere
Thisdocument http://docs-hoffmann.de/ciexyz29082000.pdf
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ColorCalcG.HoffmannDec.06 / 2006
Med.White: Eq.Energy Ref.White: D50Input: Lab
Primaries: CIETrc: 1.0Bradford: No
Intent: AbsCol
Set: 13
XYZ
xyz
L*a*b*
RGB
R’G’B’
CCTRGB
0.0909360.2812331.271889
0.0553120.1710600.773627
60.00001- 00.00001- 00.0000
6- 3.255346.7170
127.9629
0.000046.7170
100.0000
noneout-gam
0.1243170.2812330.902067
0.0950710.2150730.689856
60.00007- 5.00007- 5.0000
3- 8.047741.715590.6690
0.000041.715590.6690
noneout-gam
0.1650040.2812330.611932
0.1559330.2657740.578293
60.00005- 0.00005- 0.0000
1- 4.842637.044861.4185
0.000037.044861.4185
noneout-gam
0.2137210.2812330.391815
0.2410110.3171440.441845
60.00002- 5.00002- 5.0000
6.983832.581839.2362
6.983832.581839.2362
12527 Kin-gam
0.2711920.2812330.232047
0.3457000.3585000.295800
60.00000.00000.0000
28.055228.203423.1472
28.055228.203423.1472
5001 Kin-gam
0.3381400.2812330.122959
0.4555100.3788510.165639
60.000025.000025.0000
48.995223.786512.1761
48.995223.786512.1761
2504 Kin-gam
0.4152870.2812330.054882
0.5526830.3742780.073039
60.000050.000050.0000
70.427619.2082
5.3480
70.427619.2082
5.3480
nonein-gam
0.5033580.2812330.018146
0.6270520.3503430.022605
60.000075.000075.0000
92.976114.3453
1.6875
92.976114.3453
1.6875
nonein-gam
0.6030750.2812330.001827
0.6805680.3173710.002062
60.0000100.0000100.0000
117.32319.06360.0930
100.00009.06360.0930
noneout-gam
Matrix Crx
2.364998 -0.896709 -0.468149-0.515142 1.426371 0.0887440.005202 -0.014403 1.008898
Matrix Cxr
0.489921 0.310016 0.2000630.176937 0.812422 0.0106410.000000 0.009999 0.990301
Trc
Appendix A Color Matching
ThecalculationbyprogramColCalc[2�]showscolorsasdefinedbyequaldistancesinCIELab.Thecorrespondingvaluesaredrawninthechromaticitydiagram.BlueGreen(4)canbematchedbypositiveweightsRGBforCIEprimaries.VibrantBlueGreen(2)requiresnegativeR.BlueGreen(�)isoutofhumangamut.RGBvaluesareherenormalizedfor0...�00.
�
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Appendix B Further Explanations for Chapter 5
Chapter5hasalwaysbeenenigmatic-sincethebeginningabouttenyearsago.NowIamverygratefultoMonsieurJean-YvesChasleforgivingfurtherexplanations,herepostedun-changed.
Photometric luminance of color (page 4)
The CIE Photopic Luminous Efficiency function V is related to r_bar, g_bar and b_bar:V(λ) = 1.0000*r_bar(λ) + 4.5907*g_bar(λ) + 0.0601*b_bar(λ) (1)
The theorical light efficacy k equals 683 lm/W, based on the luminous flux measured at around 555 nm where V(λ) equals 1. As on page 4, considering a light with a spectral power diffusion P in W/sr.m², the photometric luminance (in cd/m²) can be calculated as:L = k*integral{P(λ)*V(λ)*dλ} (2)where k is the efficacy of the source light.
Substituting (1) in (2):L = k*integral{P(λ)*(1.0000*r_bar(λ) + 4.5907*g_bar(λ) + 0.0601*b_bar(λ))*dλ} = 1.0000*k*integral{P(λ)*r_bar(λ)*dλ} + 4.5907*k*integral{P(λ)*g_bar(λ)*dλ} + 0.0601*k*integral{P(λ)*b_bar(λ)*dλ}
Using notations from page 4 (in cd/m²):L = 1.0000*R + 4.5907*G + 0.0601*B (3)in cd/m².
The photometric luminance (in cd/m²) can be separated in terms of tristimulus values Lr, Lg and Lb:Lr = 1.0000*R (4)Lg = 4.5907*G (5)Lb = 0.0601*B (6)Lr, Lg and Lb represent the photometric luminance (in cd/m²) at each wavelength (700, 546.1 and 435.8 respectively). These luminances are reported on the graph named „R,G,B“ on page 4 and 6 for a matched white light of coordinates (1,1,1) in the CIE RGB space.
In practice, the light efficacy k is less than 683 lm/W. In [1], Hunt publishes samples of this value depending on the light type (page 75-79, and table 4.2 page 97).
Application (page 6)
These results can be applied on page 6, where X = 1, Y = 0 and Z = 0 representing X in the CIE XYZ space is converted to the CIE RGB space in order to evaluate its photometric luminance at each wavelength (700, 546.1 and 435.8 respectively):R = +2.36461*X - 0.89654*Y - 0.46807*Z = +2.36461G = -0.51517*X + 1.42641*Y + 0.08876*Z = -0.51517B = +0.00520*X - 0.01441*Y + 1.00920*Z = +0.00520in colorimetric luminance of red, green and blue.
From (4), (5) and (6):Lr = 1.0000*R = 1.0000 * +2.36461 = +2.36461Lg = 4.5907*G = 4.5907 * -0.51517 = -2.36499Lb = 0.0601*B = 0.0601 * +0.00520 = +0.00031(in cd/m²)
These luminances are reported on the graph named „X“ on page 6. Using (3), they sum to 0 as expected.
Same calculations for Y and Z, leading to the graphs named „Y“ and „Z“ on page 6.
3�3�
Appendix C Compare Primaries sRGB and Adobe RGB
OnawidegamutmonitoronecanchoosealternativelysRGBandaRGB(abbreviationforAdobeRGB),moreexactly:emulationsfortheseworkingspacesasmonitorprofiles.
SwitchingfromsRGBtoaRGB,onecanseethatforthewholescreencontentnotonlythegreensbutalsotheredsandbluesbecomemorevibrant.Accordingtop.2itseemsthattheredprimariesforbothspacesareidentical,andtheblueprimariesaswell.Butthatisnottrue–onlythechromaticitycoordinatesareidentical,notthebasevectorsthemselves.
R,GandB aretheprimariesinCIEXYZ.R,GandBareweightfactorsintherange0to�.AcolorC isgivenbyC =RR+GG+BB.ForR=G=�wegetwhite,hereD65whiteforsRGBandaRGB.Thesituationisvisualizedbelowin2D,Zisomitted.ComingfromsRGB,GsisshiftedtoGa.ThenewprimaryRahasthesamechromaticitycoordinatesasRs,butthevectorhastobelongerlonger.ThusaRGBredismorevibrantthansRGBredforthesameR.
Rs
Ra
Ga
Gs
Ws = Wa = WD65
X
Y
(xr,yr)s,a
(xg,yg)a
(xg,yg)s
chromaticity diagram
ThecomponentsX,YandZforany primary are found in thematricesCxr,ascolumnvectors,seep.�8andp.�9.
Cxr=(R G B )
Two-dimensionalvisualizationofthethree-dimensionalsituation
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Med.White: D65Ref.White: D50Input: RGB’
Primaries: Rec.709Trc: sRGBBradford: No
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Set: 9
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0.9504561.0000001.089058
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100.0000-2.40361- 9.3869
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6493 Kin-gam
0.4123910.2126390.019331
0.6400000.3300000.030000
53.237178.270562.1461
255.00000.00000.0000
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nonein-gam
0.3575840.7151690.119195
0.3000000.6000000.100000
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0.1804810.0721920.950532
0.1500000.0600000.790000
32.300977.8137
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0.00000.0000
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nonein-gam
Matrix Crx
3.240970 -1.537383 -0.498611-0.969243 1.875967 0.0415550.055630 -0.203977 1.056971
Matrix Cxr
0.412391 0.357584 0.1804810.212639 0.715169 0.0721920.019331 0.119195 0.950532
Trc
Appendix C Compare Primaries…cont. sRGB-noCAT
WhereastheprimariesredandbluehavethesamechromaticitycoordinatesinsRGBandaRGB,theyareinfactdifferentinCIEXYZandinCIELab.HerewehaveallcolorparametersforsRGB.sRGBwhiteisD65(firstcolumn).ReferencewhiteinCIELabisD50.Nochromaticadaptationtransformisapplied(Bradford).Thetonereproductioncurve(TRC)isthatofsrRGB:alinearslopeforsmallvaluesandapowerfunctionwithexponent2.4forlargevalues,whichleadstoaneffectiveTRCwithexponent2.2.InputforallcalculationsisRGB.
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Med.White: D65Ref.White: D50Input: RGB’
Primaries: AdobeRGBTrc: 2.2Bradford: No
Intent: AbsCol
Set: 10
XYZ
xyz
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0.9504561.0000001.089058
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255.0000255.0000255.0000
255.0000255.0000255.0000
6493 Kin-gam
0.5766690.2973450.027031
0.6400000.3300000.030000
61.424587.526069.4949
255.00000.00000.0000
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nonein-gam
0.1855580.6273640.070689
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83.30351- 39.3677
83.0449
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0.0000
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0.1882290.0752910.991337
0.1500000.0600000.790000
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Matrix Crx
2.041588 -0.565007 -0.344731-0.969244 1.875967 0.0415550.013444 -0.118362 1.015175
Matrix Cxr
0.576669 0.185558 0.1882290.297345 0.627364 0.0752910.027031 0.070689 0.991337
Trc
Appendix C Compare Primaries…cont. aRGB-noCAT
HerewehaveallcolorparametersforaRGB.aRGBwhiteisD65(firstcolumn).ReferencewhiteinCIELabisD50.Nochromaticadaptationtransformisapplied(Bradford).Thetonereproductioncurve(TRC)isthatofaRGB:apowerfunctionwithexponent2.2.InputforallcalculationsisRGB.
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ColorCalcG.HoffmannMarch 23 / 2015
Med.White: D65Ref.White: D50Input: XYZ
Primaries: AdobeRGBTrc: 2.2Bradford: No
Intent: AbsCol
Set: 10
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xyz
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0.9504561.0000001.089058
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6493 Kin-gam
0.5766690.2973450.027031
0.6400000.3300000.030000
61.424587.526069.4952
255.00000.0000
-0.0001
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nonein-gam
0.1855580.6273640.070689
0.2100000.7100000.080000
83.30351- 39.3679
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-0.0002255.0002
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0.1164
nonein-gam
0.1882290.0752910.991337
0.1500000.0600000.790000
32.982378.9123
1- 28.1662
0.0003-0.0003
254.9999
0.49950.0000
254.9999
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0.4123910.2126390.019331
0.6400000.3300000.030000
53.237178.270662.1459
182.3571-0.00010.0000
218.95280.00000.2246
nonein-gam
0.3575840.7151690.119195
0.3000000.6000000.100000
87.73558- 7.916973.9131
72.6427255.0002
10.4963
144.0969255.0000
59.8090
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0.1804810.0721920.950532
0.1500000.0600000.790000
32.300877.8142
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244.5037
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250.1742
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Matrix Crx
2.041588 -0.565007 -0.344731-0.969244 1.875967 0.0415550.013444 -0.118362 1.015175
Matrix Cxr
0.576669 0.185558 0.1882290.297345 0.627364 0.0752910.027031 0.070689 0.991337
Trc
Appendix C Compare Primaries… aRGB+sRGB-noCATHerewehaveallcolorparametersforaRGBandsRGB.Nochromaticadaptationtransformisapplied(Bradford).Thetonereproductioncurve(TRC)isthatofaRGB:apowerfunctionwithexponent2.2.InputforallcalculationsisXYZ(valuestakenfrompreviouscalculations).NowwecancompareaRGBdirectlywithsRGB.GreenandredaremorevibrantinaRGB,comparedtosRGB.Blueispracticallynotaffected.a*foraRGB-greenexceedsthecoderange–�28fornegativenumbers.R‘G‘B‘aretheinversegammaencodedvalues(likeasquarerootfunction).
aRGB sRGB
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ColorCalcG.HoffmannMarch 22 / 2015
Med.White: D65Ref.White: D50Input: RGB’
Primaries: Rec.709Trc: sRGBBradford: Yes
Intent: RelCol
Set: 5
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0.9642951.0000000.825105
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5001 Kin-gam
0.4360660.2224930.013924
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54.290580.804969.8910
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0.3851510.7168870.097081
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0.1430780.0606200.714099
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Matrix Mrx
3.134137 -1.617386 -0.490662-0.978796 1.916254 0.0334430.071955 -0.228977 1.405386
Matrix Mxr
0.436066 0.385151 0.1430780.222493 0.716887 0.0606200.013924 0.097081 0.714099
Trc
Appendix C Compare Primaries…cont. sRGB-CAT
Thechromaticadaptationtransform(CAT)typeBradfordisapplied.TheInputisRGB.Thegraphicshowsso-calledadaptedprimariesforsRGB.ThismodeisusedbyPhotoshop.
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515
520 525530
535540
545
550
555
560
565
570
575
580585
590595
600605
610620
635700
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x
y
2000
2105
2222
2353
2500
2677
2857
3077
3333
3636
4000
4444
5000
5714
6667
8000
1000
0
2500
0
D50
D65
D93
100 a*
100b*
ColorCalcG.HoffmannMarch 22 / 2015
Med.White: D65Ref.White: D50Input: RGB’
Primaries: AdobeRGBTrc: 2.2Bradford: Yes
Intent: RelCol
Set: 6
XYZ
xyz
L*a*b*
RGB
R’G’B’
CCTRGB
0.9642951.0000000.825105
0.3457000.3585000.295800
100.0000-0.00000.0000
255.0000255.0000255.0000
255.0000255.0000255.0000
5001 Kin-gam
0.6097750.3111250.019471
0.6484410.3308530.020705
62.602590.360178.1556
255.00000.00000.0000
255.00000.00000.0000
nonein-gam
0.2053000.6256530.060879
0.2302000.7015370.068263
83.21311- 29.0843
87.1724
0.0000255.0000
0.0000
0.0000255.0000
0.0000
nonein-gam
0.1492210.0632220.744755
0.1558930.0660490.778058
30.211269.2509
1- 13.6104
0.00000.0000
255.0000
0.00000.0000
255.0000
nonein-gam
Matrix Mrx
1.962467 -0.610742 -0.341358-0.978796 1.916255 0.0334430.028705 -0.140675 1.348914
Matrix Mxr
0.609775 0.205300 0.1492210.311125 0.625653 0.0632220.019471 0.060879 0.744755
Trc
Appendix C Compare Primaries…cont. aRGB-CAT
Thechromaticadaptationtransform(CAT)typeBradfordisapplied.TheInputisRGB.Thegraphicshowsso-calledadaptedprimariesforaRGB.ThismodeisusedbyPhotoshop.
3737
380460
470475
480
485
490
495
500
505
510
515
520 525530
535540
545
550
555
560
565
570
575
580585
590595
600605
610620
635700
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x
y
2000
2105
2222
2353
2500
2677
2857
3077
3333
3636
4000
4444
5000
5714
6667
8000
1000
0
2500
0
D50
D65
D93
100 a*
100b*
ColorCalcG.HoffmannMarch 23 / 2015
Med.White: D65Ref.White: D50Input: XYZ
Primaries: AdobeRGBTrc: 2.2Bradford: Yes
Intent: RelCol
Set: 6
XYZ
xyz
L*a*b*
RGB
R’G’B’
CCTRGB
0.9642951.0000000.825105
0.3457000.3585000.295800
100.0000-0.0001-0.0000
254.9998255.0001255.0001
254.9999255.0000255.0000
5001 Kin-gam
0.6097750.3111250.019471
0.6484410.3308540.020706
62.602590.360078.1553
254.99990.00020.0001
255.00000.39080.3490
nonein-gam
0.2053000.6256530.060879
0.2302000.7015370.068263
83.21311- 29.0842
87.1724
0.0001254.9999
-0.0000
0.2499254.9999
0.0000
nonein-gam
0.1492210.0632220.744755
0.1558940.0660490.778057
30.211269.2513
1- 13.6105
0.0002-0.0002
255.0001
0.42550.0000
255.0000
nonein-gam
0.4360660.2224930.013924
0.6484420.3308530.020705
54.290580.805169.8908
182.3572-0.00020.0000
218.95280.00000.2011
nonein-gam
0.3851510.7168870.097081
0.3211950.5978450.080960
87.81857- 9.271280.9947
72.6428255.0001
10.4962
144.0970255.0000
59.8086
nonein-gam
0.1430780.0606200.714099
0.1558930.0660490.778058
29.568368.2869
1- 12.0296
-0.00020.0002
244.5036
0.00000.4246
250.1742
nonein-gam
Matrix Mrx
1.962467 -0.610742 -0.341358-0.978796 1.916255 0.0334430.028705 -0.140675 1.348914
Matrix Mxr
0.609775 0.205300 0.1492210.311125 0.625653 0.0632220.019471 0.060879 0.744755
Trc
Appendix C Compare Primaries… aRGB+sRGB-CAT
aRGB sRGB
Thechromaticadaptationtransform(CAT)typeBradfordisapplied.TheInputisnowXYZ.Thevalueshavetobetakenfromthetwopreviouspages.Thegraphicshowsso-calledadaptedprimariesforaRGBandsRGB.ThismodeisusedbyPhotoshop.Forinstance:inaRGBwecangetsRGB-redby2�9/0/0,-greenby�44/255/60and-blueby0/0/250.InaRGBweseeaRGB-greenwitha*=–�29justoutsidethecoderange–�28.