CS380: Introduction to Computer Graphics Color (2) Chapter ... · 18/05/17 1 Min H. Kim (KAIST) Foundations of 3D Computer Graphics, S. Gortler, MIT Press, 2012 CS380: Introduction

18/05/17

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MinH.Kim(KAIST) Foundationsof3DComputerGraphics,S.Gortler,MITPress,2012

CS380:IntroductiontoComputerGraphicsColor(2)Chapter19

MinH.KimKAISTSchoolofComputing


SUMMARYColor(1)

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ColorBases•  Wecaninsertany(nonsingular)3-by-3matrixManditsinversetoobtain:

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!c(l(λ))= !c(l436)!c(l546)

!c(l700)⎡⎣

⎤⎦M

−1( ) Mk436(λ)l(λ)dλΩ∫k546(λ)l(λ)dλΩ∫k700(λ)l(λ)dλΩ∫

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=!c1!c2!c3

⎡⎣

⎤⎦

k1(λ)l(λ)dλΩ∫k2(λ)l(λ)dλΩ∫k3(λ)l(λ)dλΩ∫

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


ColorBases•  Wherethedescribeanewcolorbasisdefinedas

•  Thek(λ)functionsformthenewassociatedmatchingfunctions,definedby:

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!c1!c2!c3

⎡⎣

⎤⎦=

!c(l436)!c(l546)

!c(l700)⎡⎣

⎤⎦M

−1 ci

k1(λ)k2(λ)k3(λ)

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=Mk436(λ)k546(λ)k700(λ)

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

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Howdoescomputationwork?

•  Illuminationonasurfacecolor(element-by-elementproduct)

•  Reflectedcolor

•  ThreeCMFsforXYZ

•  Trichromaticresponseasscalar(sumofenergy)


COLOR(2)Chapter19

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RememberThisColor


Mapofcolorspace(lassocurve)

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LassocurveinLMScoordinates

NormalizedlassocurveinLMScoordinates

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Mapofcolorspace(lassocurve)•  Asweletλvary,suchvectorswilltraceoutalassocurveinspace.

•  Thelassocurveliescompletelyinthepositiveoctantsinceallresponsesarepositive.

•  Thecurvebothstartsandendsattheoriginsincetheseextremewavelengthsareattheboundariesofthevisibleregion,beyondwhichtheresponsesarezero.

•  ThecurvespendsashorttimeontheSaxis(shownwithbluetintedpoints)

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Mixedinvisible•  Aswelookatallpossiblemixedbeams,the

resultingcoordinatessweepoutsomesetofvectorsin3Dspace.

•  Sincecanbeanypositivefunction,thesweptsetiscomprisedofallpositivelinearcombinationsofvectorsonthelassocurve.

•  Thus,thesweptsetistheconvexconeoverthelassocurve,whichwecallthecolorcone.

•  Vectorsinsidetheconerepresentactualachievablecolorsensations.

•  Vectorsoutsidethecone,suchastheverticalaxisdonotarisethesensationfromanyactuallightbeam,whetherpure(monochromatic)orcomposite

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l(λ)[L,M ,S]t

l(λ)

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CIEXYZcolorspacein3D•  Centralstandardizedspace.•  Specifiedbythethreematchingfunctionscalled

•  Thecoordinatesforsomecolor(aspectrum)withrespecttothisbasisisgivenbyacoordinatevectorthatwecall.

•  Theseparticularmatchingfunctionswerechosensuchthattheyarealwayspositive,andsothattheY-coordinateofacolorpresentsitsoverallperceived“luminance”.ThusYisoftenusedasablackandwhiterepresentationofthecolor.

•  Theassociatedbasisismadeupofthreeimaginarycolors;theaxesareoutsideofthecolorcone. 11

kx(λ),ky(λ)andkz(λ)

[X ,Y ,Z ]t

[cx ,cy ,cz ]

t


HowtocomputeCIEXYZ

•  Emittingcolors(radiance)– SincewehavethespectralpowerdistributionsofradianceL(powerperwavelength)

X =Km L(λ)x(λ)Δλλ

∑ ,

Y =Km L(λ)y(λ)Δλ ,λ

∑

Z =Km L(λ)z(λ)Δλ ,λ

∑

whereKm = 683lm/W .– HeretheYvaluecorrespondstoluminance(cd/sqm)

Noticethedifference!

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RECAP:CIEXYZcomputation

•  Illuminationonasurfacecolor(element-by-elementproduct)

•  Reflectedcolor

•  ThreeCMFsforXYZ

•  Trichromaticresponseasscalar(sumofenergy)X Y

Z

CIECMFs

Reflection

ReflectanceIllumination


Reflectionmodeling•  Whenabeamoflightfromanilluminationsourcehitsasurfaceofareflectancefunction

•  Thismultiplicationhappensonaper-wavelengthbasis.Metamerismhappensinourbrain.

•  A3Dcolorrenderingcannothandlethis;instead,weneedtousemultispectralorhyperspectralrendering.

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i(λ)r(λ)

l(λ) = i(λ)r(λ)

!c[i1(λ)ra(λ)]=

!c[i1(λ)rb(λ)]⇔ !c[i2(λ)ra(λ)]=!c[i2(λ)rb(λ)]

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HowtocomputeCIEXYZ

•  Relativereflectivecolors– SincewehavethespectralpowerdistributionsofilluminationIandsurfacereflectanceR

X =k I(λ)R(λ)x(λ)Δλλ

∑ ,

Y =k I(λ)R(λ)y(λ)Δλ ,λ

∑

Z =k I(λ)R(λ)z(λ)Δλ ,λ

∑

wherek = 100I(λ)y(λ)Δλ

λ

∑. NotethereisnoR(λ)inthedenominator!


Chromaticityxyin2D•  2Dplotforchromaticinformation•  What’sleftafterluminanceisfactoredout(thecolorwithoutregardforoverallluminance),thereforecommonlycoupledwithY

x = XX +Y + Z

,

y = YX +Y + Z

,

z = ZX +Y + Z

,

x + y + z = 1

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Chromaticityxyin2D•  Scalesofvectorsintheconecorrespondtobrightnesschangesinourperceivedcolorsensation,soletsnormalizebyscale.

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Chromaticityxyin2D

•  Spectrallocus–  lassoin2D– Plotmonochromaticlightsinthevisiblespectrum(400-700nm)

•  Isthisdiagramperfectforrepresentingcolors?

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UniformChromaticityu’v’

•  Chromaticityxyismathematicallyconvenient,notsuitableforevaluatingcolorinformationduetonon-uniformity

CIE1931xy CIE1976u’v’


UniformChromaticityu’v’

•  CIT1976chromaticitycoordinatesu’v’•  aregivenby

u ' = 4X / (X +15Y + 3Z )= 4x / (−2x +12y + 3)

v ' = 9Y / (X +15Y + 3Z )= 9y / (−2x +12y + 3)

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CorrelatedColorTemperaturein1D

•  Real-worldilluminantcanbeapproximatedasacolortemperatureofPlanckianblackbodyradiation(=thesun)

•  Theclosestcolortemperatureontheblackbodylocusofthereal-worldilluminantiscalledcorrelatedcolortemperature(CCT)

•  Unit:Kelvin


DeviceDependentColorSpaces•  Exampe:RGBvalues•  Pros:–  Simpledescriptionofcolorforthedevice

– Natural,easywaytospecifycolortotheuser

•  Cons:–  Cannotcomparecolorsbetweendevices

– Notperceptuallyuniform

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Perceptualcolorspacein3D•  WeonlydrawcolorsinthegamutoftheRGBmonitor

•  Colorsalongtheboundaryoftheconearevividandareperceivedas“saturated”.

•  Aswecirclearoundtheboundary,wemovethroughthedifferent“hues”ofcolor.

•  StartingfromtheLaxis,wemovealongtherainbowcolorsfromredtogreentoviolet.– Achievablebymonochromaticbeams.

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Perceptualcolorspacesin3D•  Colorcone’sboundaryhasaplanarwedge(alinesegmentinthe2Dfigure).– Thecolorsonthiswedgearethepinksandpurples.– Theydonotappearintherainbowandcanonlybeachievedbyappropriatelycombiningbeamsofredandviolet.

•  Aswemoveinfromtheboundarytowardsthecentralregionofthecone,thecolors,whilemaintainingtheirhue,de-saturate,becomingpastelandeventuallygrayishorwhitish.

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DeviceIndependentColorSpaces

•  Pros:– Basedonhumanvisualperception– Colorspecificationindependentofdevice–  Interchangeablecoloramongdevices– Comparison,computationofsmallcolordifferences

•  Cons:– CIEXYZ:notuniform– CIELAB,CIELUV,CIEXYZ,Munsell:alldependentontheilluminant


Perceptualcolorspacesin3D•  Uniformperceptualdistanceofdifferentcolors

•  Opponentprimaries•  Threedimensions:lightness,colorfulness,andhue(L,C,H)

•  Relatedtoprocessesofhumanvisualperception

•  Meaningfulwayofdescribingcolor

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HSVColorSpace•  notperceptuallydriven!•  Value:

•  Saturation:•  Hue:

V = M = max(R,G,B).

m = min(R,G,B),C = M −m,S = C /V ,

H =360 + 60(G − B) /C if M = R120 + 60(B − R) /C240 + 60(R −G) /C

if M = Gif M = B

⎧

⎨⎪

⎩⎪

JacobRu

s


MunsellSystem(1915)•  Fiveprimaryhues:•  Valuerange:•  Chromarange:

YellowRed Green Blue Purple

…0 5 … ∞

10RP4/10=aspecificreddishpurplehueof10RP,avalueof4,andachromaof10

… 105…0

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CIEUniformColorSpaces(1976)•  OriginatedfromHunterLab1948•  Perceptuallyuniformcolordefinition

•  DrivenfromCIEXYZ

L*=43.31a*=47.63b*=14.12


CIELABMath•  Simplifiedconeresponse(XYZandacubicrootfunc.)•  Coloropponenttheoryforcomputingchromaandhue•  Lightness:•  Coloropponents:

•  Chroma:•  Hue:

L* =116 f (Y /Yn)−16,a* = 500[ f (X / Xn )− f (Y /Yn )],

b* = 200[ f (Y /Yn )− f (Z / Zn )],

Cab* = (a*)2 + (b*)2 ,

hab = tan−1(b* / a*).

f (x) = x1/3, x > 0.0088567.787x +16 /116, x ≤ 0.008856

⎧⎨⎪

⎩⎪

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RememberThisColor


ColorDifferences

•  ConventionalEuclideanmetricinaperceptuallyuniformcolorspace(CIELAB)

ΔEab* = ΔL*( )2 + Δa*( )2 + Δb*( )2

CIE ΔEab*

L*

b*

a*

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sRGBcolorspace•  ThereareavarietyofRGBstandards.•  CurrentoneiscalledRec.709RGBspace(so-calledsRGB).

•  Basisismadeupofthreeactualcolorsintendedtomatchthecolorsofthethreephosphorsofanidealmonitor/TVCRT(cathoderaytube)display.

•  Colorswithnon-negativeRGBcoordinatescanbeproducedonamonitorandaresaidtolieinsidethegamutofthecolorspace.Thesecolorsareinthefirstoctantofthefigure.

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[cr ,cg ,cb ]

t


sRGBvs.Pointer’sgamut•  SomeactualcolorslieoutsidethesRGBgamut.

•  Additionally,onamonitor,eachphosphormaxesoutat“1”,whichalsolimitstheachievableoutputs.

•  Imageswithcolorsoutsidethegamutneedsomekindofmapping/clippingtokeepinthegamut,so-calledgamutmapping.

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Rec.709/sRGBvs.Pointer’sgamut(69.4%ofPointer’sgamut)

http://www.tftcentral.co.uk/articles/pointers_gamut.htm

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XYZvs.sRGB

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CIEXYZ sRGB


Gammacorrection

•  InheritedfromthetonereproductioncurveoftheCRTphosphors

•  Tocompensatenon-linearresponse(^2.2)ofthedisplay,apply(^1/2.2)tothedisplaysignals(sRGB)

•  Computationalredundancy(replacedwithLUT)

•  RemovedfromHDTVsignals

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Gammacorrection•  Eachpixelonadisplayisdrivenbythreevoltages,say.

•  Lettingtheoutgoinglightfromthispixelhaveacolorwithcoordinates

•  Wewanttoobtainsomespecificoutputfromapixel,thenweneedtodriveitwithvoltages:

•  valuesarecalledthegammacorrectedRGBcoordiantes.

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( ′R , ′G , ′B )

[R,G,B]t

R=( ʹR )2.2 ,G =( ʹG )2.2 ,B =( ʹB )2.2

[R,G,B]t

ʹR =(R)0.45 , ʹG =(G)0.45 , ʹB =(B)0.45

[ ′R , ′G , ′B ]t


Gammacorrection•  Linearvs.gamma-corrected

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Linear

Gamma-corrected

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NonlinearSRGBtolinearXYZ?

•  (Step1)NormalizeRGBvalues•  (Step2)Inversegammacorrection(γ=2.2)

•  (Step3)TransformationfromsRGBtoCIEXYZ•  sRGBàXYZ

•  (cf)Inv.Trans:XYZàsRGB

XYZ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

0.4124 0.3576 0.18050.2126 0.7152 0.07220.0193 0.1192 0.9505

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

RGB

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

RGB

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

3.2406 −1.5372 −0.4986−0.9689 1.8758 0.04150.0557 −0.2040 1.0570

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

XYZ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

R=( ʹR )2.2 ,G =( ʹG )2.2 ,B =( ʹB )2.2


Quantization•  sRGBcoordinatesareintherealrange[0…1]•  Afixedpointrepresentationisusedwithvalues[0…255](8-bitcolor)àunsignedcharinC

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(realàbyte)byteR=round(realR*255);(byteàreal)realR=byteR/255.0;

(realàbyte)byteR=round(realR>=1.0?255:(realR*256)–0.5);(byteàreal)realR=(byteR+0.5)/256.0;e.g.:(realàbyte)0=round(0.7–0.5);1=round(1.0–0.5)

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ColorsinGLSL•  Imagesaretypicallystoredingammacorrectedcoordinates,andthemonitorscreenisexpectingcolorsingammacorrectedcoordinates.

•  Computergraphicssimulatesprocessesthatarelinearlyrelatedtolightbeams.Assuch,mostcomputergraphicscomputationsshouldbedoneinalinearcolorrepresentation.

•  Inprofessionalcomputergraphics,weuselinearHDRradianceintheformatofOpenEXR

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[R,G,B]t


ColorsinGLSL•  ByusingthecallglEnable(GL_FRAMEBUFF_SRGB),•  Wecanpasslinearvaluesoutfromthefragmentshader,andtheywillbegammacorrectedintothesRGBformatbeforebeingsenttothescreen.

•  glTexImage2D(GL_TEXTURE_2D,0,GL_SRGB,twidth,theight,0,GL_RGB,GL_UNSIGNED_BYTE,pixdata)

•  Then,wheneverthistextureisaccessedinafragmentshader,thedataisfirstconvertedtolinearcoordinatesbeforegiventotheshader.

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[R,G,B]t

[R,G,B]t

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Colorconstancy•  Althoughthespectralpowerdistributionofsceneilluminationchanges,wecanperceivecolors(reflective)consistently,so-calledcolorconstancy

•  Thisvisualphenomenonisimplementedaswhitebalancingindigitalcameras.

•  ThisisoftenimplementedasavonKriestransformintheLMSorXYZspacefromagivenilluminationtoatargetillumination.

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i1(λ)i2 (λ)

M =

L2 /L1 0 00 M2 /M1 00 0 S2 /S1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

whereL1 ,M1 ,S1 aretheconeresponsesundergiveni1(λ),L2 ,M2 ,S2 aretheconeresponsesundertargeti2(λ).


Colorimetriccalculation

•  AllcolorimetricvaluesarecomputedfromCIEXYZ

Radiance Reflectance CIEXYZ CIELuv

CIELAB

xy

sRGB

Drgb

Energyperunitareapersolidangle

Energyatagivenangle,relativetoenergyreflectedbyperfectdiffuser

Relativeamountsofthreeimaginaryprimariesrequiredtomatchthecolorappearance

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ColorReproduction•  Imaginewehaveaspectrums;wanttomatchonRGBdisplay

•  Practically,wecannotachieveaphysicallyidenticalspectrumbecausetheyaredifferentmedia

•  Butcouldfindaspectrumsathatthedisplaycanproduce,whichisametamerofs

ssa


ColorReproductionasLinearAlgebra

•  WewanttocomputesathecombinationofR,G,B

•  whichwillprojecttothesamevisualresponseass

•  sawillbeametamerofs

RGB

XYZ

Spanofeye’sspectralresponsefunctions

Spanofdisplay’sprimaries

Adap

tedfrom

SteveM

arschn

er

Visualresponsetosandsa

Spectrums

Spectrumsa C

V

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•  Theprojectionontothethreeresponsefunctionscanbewritteninamatrixform:

•  or,

XYZ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= rX rY

rZ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

s

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

SpectralresponsivityofXYZ

V = MXYZs.



•  ThespectrumthatisproducedbythedisplayforthecolorsignalsR,G,Bis:

•  Againthediscreteformcanbewrittenasamatrix:

•  or,

Sa (λ) = Rsr (λ)+Gsg (λ)+ Bsb (λ).

sa

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

sR sG sB

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

RGB

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

sa = MRGBC.SpectraofRGBphosphors

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•  Whatcolordoweseewhenwelookatthedisplay?

•  FeedC(R,G,B)todisplay•  Displayproducessa•  EyeslookatsaandproduceV

V = MXYZMRGBC.

XYZ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

rX ⋅ sR rX ⋅ sG rX ⋅ sBrY ⋅ sR rY ⋅ sG rY ⋅ sBrZ ⋅ sR rZ ⋅ sG rZ ⋅ sB

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

RGB

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

RGB

XYZ

saC

V



•  Goalofreproduction:visualresponsetosandsaisthesame:

•  Substitutingintheexpressionforsa,

MXYZ s = MXYZ sa .

MXYZ s = MXYZMRGBC.

C = (MXYZMRGB )−1MXYZ s.

Colorreproductionmodelfordisplay

RGB

XYZ sa≈s

s

saC

V

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RGB

XYZ

Spanofeye’sspectralresponsefunctions

Spanofdisplay’sprimaries

Visualresponsetosandsa

Spectrums

Spectrumsa C

V


Wherearethecolortransforms?

•  Nowadays,ineverydigitalimagingdevices:– TV,digitalcameras,camcorders,inkjetprinters,laserprinters,LCDdisplays,etc…

•  Otherwise…

≠

Documents

CS380: Introduction to Computer Graphics Color (2) Chapter ... · 18/05/17 1 Min H. Kim (KAIST) Foundations of 3D Computer Graphics, S. Gortler, MIT Press, 2012 CS380: Introduction