��

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

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

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.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

380 420 460 500 540 580 620 660 700nm

p1_

p2_

p3_

λ

Conesensitivities[3]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

380 420 460 500 540 580 620 660 700nm

p1_

p2_

p3_

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

0.0

0.1

0.2

0.3

0.4

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).

300435.8546.�700.0800

R,G,B +4.5907

+�.0000

+0.060�

CIEprimariesR,G,B

77

6. XYZ Color-Matching Functions

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

380 420 460 500 540 580 620 660 700nm

x_

y_

z_

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

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

2000

2105

2222

2353

2500

2677

2857

3077

3333

3636

4000

4444

5000

5714

6667

8000

1000

0

D50

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.

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

0.05

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

�6�6

% 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

�7�7

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

�9�9

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

42.373944.582748.5532

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.

2626

sRGB

Worthey

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

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

2828

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

Settings in Acrobat Pro

Edit / Preferences / General / Page Display

Use Custom Resolution 72 dpiDo not use 2D GPU acceleration

The PDF document is pixel synchronized for zoom 100% / 200% / 72dpi

2929

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.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.

�

2

3

4

6 78

9

�

9

8

7

6

5

4

3

2

3030

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

3232

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 22 / 2015

Med.White: D65Ref.White: D50Input: RGB’

Primaries: Rec.709Trc: sRGBBradford: No

Intent: AbsCol

Set: 9

XYZ

xyz

L*a*b*

RGB

R’G’B’

CCTRGB

0.9504561.0000001.089058

0.3127000.3290000.358300

100.0000-2.40361- 9.3869

255.0000255.0000255.0000

255.0000255.0000255.0000

6493 Kin-gam

0.4123910.2126390.019331

0.6400000.3300000.030000

53.237178.270562.1461

255.00000.00000.0000

255.00000.00000.0000

nonein-gam

0.3575840.7151690.119195

0.3000000.6000000.100000

87.73558- 7.916873.9132

0.0000255.0000

0.0000

0.0000255.0000

0.0000

nonein-gam

0.1804810.0721920.950532

0.1500000.0600000.790000

32.300977.8137

1- 26.3828

0.00000.0000

255.0000

0.00000.0000

255.0000

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.

3333

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 22 / 2015

Med.White: D65Ref.White: D50Input: RGB’

Primaries: AdobeRGBTrc: 2.2Bradford: No

Intent: AbsCol

Set: 10

XYZ

xyz

L*a*b*

RGB

R’G’B’

CCTRGB

0.9504561.0000001.089058

0.3127000.3290000.358300

100.0000-2.40361- 9.3869

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

255.00000.00000.0000

nonein-gam

0.1855580.6273640.070689

0.2100000.7100000.080000

83.30351- 39.3677

83.0449

0.0000255.0000

0.0000

0.0000255.0000

0.0000

nonein-gam

0.1882290.0752910.991337

0.1500000.0600000.790000

32.982478.9117

1- 28.1660

0.00000.0000

255.0000

0.00000.0000

255.0000

nonein-gam

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.

3434

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: No

Intent: AbsCol

Set: 10

XYZ

xyz

L*a*b*

RGB

R’G’B’

CCTRGB

0.9504561.0000001.089058

0.3127000.3290000.358300

100.0000-2.40351- 9.3869

255.0000254.9999255.0001

255.0000255.0000255.0000

6493 Kin-gam

0.5766690.2973450.027031

0.6400000.3300000.030000

61.424587.526069.4952

255.00000.0000

-0.0001

255.00000.09160.0000

nonein-gam

0.1855580.6273640.070689

0.2100000.7100000.080000

83.30351- 39.3679

83.0449

-0.0002255.0002

0.0000

0.0000255.0000

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

nonein-gam

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

nonein-gam

0.1804810.0721920.950532

0.1500000.0600000.790000

32.300877.8142

1- 26.3829

0.0002-0.0002

244.5037

0.37770.0000

250.1742

nonein-gam

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

3535

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 22 / 2015

Med.White: D65Ref.White: D50Input: RGB’

Primaries: Rec.709Trc: sRGBBradford: Yes

Intent: RelCol

Set: 5

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

0.4360660.2224930.013924

0.6484410.3308530.020705

54.290580.804969.8910

255.00000.00000.0000

255.00000.00000.0000

nonein-gam

0.3851510.7168870.097081

0.3211950.5978440.080960

87.81857- 9.271180.9946

0.0000255.0000

0.0000

0.0000255.0000

0.0000

nonein-gam

0.1430780.0606200.714099

0.1558930.0660490.778058

29.568368.2874

1- 12.0297

0.00000.0000

255.0000

0.00000.0000

255.0000

nonein-gam

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.

3636

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 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.