Goniocolorimetry: from measurement to representation in the CIELAB color space

Transcription

1 Gonocolomety: fom measuement to epesentaton n the CIELAB colo space Lonel Smonot PHYMAT: Physque des Matéaux, UMR CNRS 6630, Boulevad Mae et Pee Cue, BP 179, 8696 Futuoscope Chasseneul Cedex, Fance Coespondng autho: lonel.smonot@unv-potes.f Matheu Hébet Ecole Polytechnque Fédéale de Lausanne (EPFL), School fo Compute and Communcaton Scences, 1015 Lausanne, Swtzeland Damen Dupaz STIL SA, 595 ue Pee Bethe, Domane de St Hlae, F Ax en Povence Cedex 3, Fance Abstact The classcal defntons of colo ae well adapted to dffusng objects, whose colo s almost ndependent of the vewng angle, and to vey glossy object obseved n the specula decton n espect to the lght souce. Fo glossy o descent objects, the colo s dffcult to chaacteze due to ts dependence on the vewng decton. In ode to cope wth such objects and to epesent the angle-dependent colos n a colometc space, we adapt the CIELAB space to gonocolometc measuements. A cucal pont when defnng ths space s the statement of the vewng sold angle. Fst, we suggest pefomng a BRDF measuement at hgh angula esoluton n ode to chaacteze the gloss of the specmen. Then, snce fo the defnton of colos the CIE ecommends cones of halfangle of o 10, we popose to convet the measued BRDF nto a eflectance facto defned n espect to these sold angles. Ths pocedue s eased by a plana multspectal mage of the BRDF, whee sold angles ae specfed by the pxel sze. At last, the eflectance factos ae conveted nto CIELAB coodnates. By usng ths pocedue, the pefect whte dffuse but also the pefect mo can be epesented n ths colometc space. Keywods Colomety, gloss, geometc aspect, eflectance, spectophotomety, BRDF Pepnt of atcle publshed n Colo Reseach and Applcaton, Vol. 36, Issue 1, Feb 011, pp Wley Peodcals Inc. 1

2 1. Intoducton The vsual appeaance of objects s a majo cteon of acceptablty fo manufactues, especally n the domans of pntng, automotve, cosmetc, textles and food. Appeaance s usually descbed accodng to fve dstnct factos: fom, textue, tanspaency, gloss and colo. These vsual attbutes ae the esponse by the human vsual pecepton system to the lght sgnal ssued fom the object afte possble complex lght-matte nteactons. Colo s the vsual attbute whose mechansm of pecepton by the human vsual system has been the best chaactezed, n patcula thanks to the woks by the Commsson Intenatonale de l Eclaage (CIE) along the 0 th centuy [1]. Colo s elated to the spectal dstbuton of lght n the vsble wavelength doman, whch can be measued usng a spectophotomete. But the stmulated egons n the etna have also an nfluence n the pecepton of colo. The colo scence, called colomety, has been eased by a mathematcal epesentaton whee each colo s epesented by the tstmulus values X, Y and Z defned fom the CIE colo matchng functons of the standad obseve [1,, 3]. They take nto account the dffeences n colo appecaton n tems of the etnal aea eached by the lght. In 1931, the CIE establshed the standad obseve tchomatc functons fo lght contaned wthn a cone of half-angle centeed on the optcal axs. Ths sold angle was selected to stmulate the foveal aea, egon of the etna whee the cone cells ae numeous and the od cells ae almost absent. In ode to obtan a bette matchng between colo defntons and the vewng condtons n the eveyday lfe, the CIE defned n 1964 dffeent standad obseve colo matchng functons by consdeng a lage sold angle,.e. a cone of 10 half-angle. In the case of objects, colo s elated to a spectum modfcaton,.e. to a vaaton of the ncdent spectum due to the eflecton by the object. Befoe beng ntepeted as colo, the sensoal esponse by the etna to the eflected lght (tstmulus values X, Y and Z) s combned wth othe nfomaton, especally the esponse by the etna to the lght eflected by a efeence dffuse (tstmulus values X ef, Y ef and Z ef ). In ode to take nto account the elatve pecepton mechansm wth espect to the llumnaton souce, seveal colo spaces wee defned, e.g. the CIELAB coodnate system [1,, 3]. Colo coodnates ae calculated n espect to a standad llumnant whose spectum s fxed by the CIE. The angula dstbuton of the eflected lght detemnes the type of geomety that should be used fo measuement, especally the sold angle of detecton. When an object s stongly scatteng, e.g. a matte pant o pape, ts spectum s almost ndependent of the ncdent and vewng dectons. Snce the detecto s flled wth the homogenous dffuse lght, the quantty of collected lght s popotonal to the sold angle of the detecto. In contast, when the object s vey glossy, e.g. a metallc plate o a delectc plane suface fo optcal coatng [4], collmated ncdent lght s eflected aound a sngle decton, called the specula decton. If the eflected lght pencl s nfntely thn, the sze of the detecto and ts sold angle have no nfluence on the measuement. In many cases, the object s nethe vey scatteng no vey glossy but n between [5, 6, 7]. A satn-fnsh pantng, fo example, eflects a dffuse coloed component ssued fom the pantng tself as well as an uncoloed component scatteed by the suface. The popoton of these two components vaes wth the obsevaton decton. Snce they ae added, the spectophotomete measues them at the tme and cannot dstngush them. Howeve, they ae peceved by a human obseve as dstnct appeaance attbutes,.e. as colo and gloss. The pecepton of gloss eles on an analyss of angle-

3 dependent colo vaatons by the human vsual system, whch s stll a subject of nvestgaton [8, 9, 10]. Thee exsts one specal class of objects fo whch gloss can be easly dscaded fom colo measuements: the objects composed of a dffusng mateal havng a smooth suface. The coloed dffuse component s spead athe homogeneously ove the hemsphee wheeas the gloss s dected only nto the specula decton. To measue the colo wthout gloss component, one can use a dectonal lght souce and captue lght at an angle dffeent fom the specula decton, e.g. the 45 :0 geomety ecommended by the CIE, whee the ncdent lght comes at 45 and the eflected lght s detected at 0. Fo geometes elyng on an ntegatng sphee, a gloss tap located aound the specula decton enables dscadng the specula component, e.g. the 8 :de geomety also ecommended by the CIE [1]. Many measung nstuments, especally potatve spectophotometes, ely on these geometes and offe the possblty to dscad the gloss. In pactce, one can consde that gloss s effectvely dscaded when the mateal looks suffcently dffusng and the suface looks suffcently smooth, whch aleady concens a wde ange of objects (e.g. glossy pnts, photogaphs, ceamcs ) But the theshold beyond whch the suface s too ough to nsue that no gloss component s captued s dffcult to estmate, and thee s no way to check t. Except fo the specal class of object descbed above, colo and gloss ae caed by mxed lght components. Snce colo vaes as a functon of the vewng decton [5, 6, 7, 11, 1], ntendng to measue "one" colo ndependently of gloss makes no sense. The poblem becomes even moe pegnant n the case of "effect mateals", also called "gonochomatc mateals", such as shot fabcs [13, 14], metalpgmented pants and pealescent coatngs [15, 16, 17, 18]. A specfc colo chaactezaton s needed, able to ende mpotant angula colo vaatons. The soluton that we popose eles on the bdectonal eflectance dstbuton functon (BRDF) of the object. Ths functon specfes fo each couple of ncdent and obsevaton dectons the ato of the emtted adance to the ncdent adance. It s measued usng a gonoeflectomete [19, 0]. Fo each couple of dectons, the measued eflectance spectum s conveted nto CIEXYZ tchomatc values, then nto CIELAB colo coodnates by choosng a whte efeence. We ecommend epesentng two addtonal ponts n the CIELAB colo space, epesentng the colo coodnates of a pefect whte dffuse and the colo of a pefect mo. The two ponts show ntutvely to whch extend the consdeed specmen s athe speculaly o dffusely eflectng. The conveson of eflectance specta nto colo coodnates s well establshed when the measung geomety s fxed (fxed obsevaton decton), but dffcultes appea when obsevaton decton vaes. A vey small sold angle s hghly ecommended f one deses acqung the BRDF of glossy specmen wth a satsfyng accuacy. But snce the colo matchng functons wee establshed by the CIE n espect to cones of and 10 half-angles (CIE1931 and CIE1964 espectvely), t seems pefeable to pefom eflectance measuements wth these sold angles. A soluton would be to pefom two measuements usng fst a vey small sold angle to obtan the BRDF, and then a lage sold angle to calculate the eflectance factos and the colo coodnates. Instead, we show that the second measuement s equvalent to applyng an undesamplng opeaton on the fst one. Thus, a sngle hgh esoluton measuement s needed and the colo coodnates ae calculated by consdeng pecsely the sold angle defned by the CIE fo colomety ( o 10 ). The pesent pape s stuctued as follows: n secton, we ecall some essental adometc defntons fo the chaactezaton of lght eflecton. Secton 3 defnes the colo fo two patcula 3

4 cases: the Lambetan and the specula samples. Then, Secton 4 pesents the gonocolometc space and ecaps the dffeent steps fo convetng measued specta nto colo coodnates. Secton 5 llustates the methodology wth the example of a glossy and dffuse blue sample. Fnally, Secton 6 daws the conclusons.. Radometc defntons The angle-dependent eflecton popetes of an object ae chaactezed by angula functons defned as atos of adometc quanttes, e.g. the bdectonal eflectance dstbuton functon (BRDF). Ths functon can be measued usng an nstument called gonoeflectomete whch contans a movable collmated lght souce and a movable detecto. Let us consde a plana sample llumnated by a pefectly collmated lght comng fom a gven decton ( θ, ). The sample eflects n decton ( θ, ) a adance L( θ,, θ, ) defned as [1]: d Φ( θ, ) L( θ,, θ, ) = (1) dscosθdω whee d Φ( θ, ) s the flux element eflected n the decton ( θ, ), ds s an elemental aea on the sample and dω = snθ dθ d s an nfntesmal sold angle n the consdeed decton. Radance facto and BRDF eflected by the sample to the adance L ef eflected by the pefect dffuse n the same llumnaton and obsevaton condtons: L( θ,, θ, ) βθ (,, θ, ) = () L The adance facto, β, s defned as the ato of the adance L( θ,, θ, ) The adance coeffcent q (n s -1 ) s now wdely pefeed to the adance facto. It s defned as the ato of the adance L( θ,, θ, ) to the ncdent adance E : L( θ,, θ, ) q( θ,, θ, ) = (3) E Howeve, n most setups, the ncdent adance s not measued dectly but evaluated fom a measuement on a pefect dffuse,.e., a whte dffusng sample obeyng Lambet s law (appeang equally bght fom all dectons). The pefect whte dffuse eflects a adance L ef popotonal to the ncdent adance E [1]: L = E / π (4) ef Ths yelds the followng elaton between adance coeffcent and adance facto: β( θ,, θ, ) q ( θ,, θ, ) = (5) π The BRDF s the functon that descbes the evoluton of the adance coeffcent accodng to the dectons of llumnaton and eflecton. ef 4

5 Reflectance facto In pactce a detecto does not measue a adance but a flux n a fnte sold angle. Snce adance facto and BRDF ae defned n espect to an nfntesmal sold angle, they can not be measued dectly. Hence, a eflectance facto R s defned as the ato of the flux ΔΦ ( θ, ) sample n the decton ( θ, ) and n a gven sold angle to the flux (, ) eflected by the ΔΦef θ eflected by the pefect dffuse exactly n the same llumnaton and obsevaton condtons: ΔΦ( θ, ) R ( θ,, θ, ) = ΔΦ θ, (6) ( ) ef Let us calculate the flux ΔΦ by assumng that the detecto suface s pependcula to the vewng decton and has a fxed aea Δ Sd. The vewng sold angle ΔΩ d, also fxed, s a cone centeed on the detecto suface nomal. Though ths cone, the detecto obseves an ellptcal egon of the sample, whose aea vaes wth the cosne of the vewng angle θ. The obseved aea s geneally dffeent fom the llumnated aea,.e. t s often wholly ncluded wthn the llumnated aea at small vewng angles and ovepasses t at gazng angles. When calculatng the measued flux, one should consde only the facton of obseved aea that s llumnated, called the emttng aea S θ,. Δ. Ths latte s a functon of the obsevaton decton ( ) Fgue 1: Geomety of a gonometc measuement. The detecto eceves a collecton of lght ays, contaned wthn the sold angle ΔΩ based on the emttng aea and subtended by ts suface Δ Sd (Fgue 1). Each lght ay, popagatng nto a decton ( θ, ), coesponds to a adance L ( θ,, θ, ) defned n the same manne as n Eq.(1), and theefoe to a flux element gven by d Φ θ, = L θ,, θ, ΔScosθ snθ dθ dφ (7) ( ) ( ) The sum of the collected flux elements gves the measued flux. It may be expessed as a functon of the adance coeffcent accodng to Eq. (3): ΔΦ θ, = EΔS q θ,, θ, cosθ snθ dθ d (8) ( ) ( ) ( θ, ) ΔΩ The flux eflected by the pefect dffuse s obtaned wth q = 1/ π EΔS ΔΦef ( θ, ) = cosθ sn (, θ dθ d θ π (9) ) ΔΩ 5

6 Fom Eqs. (6), (8) and (9), one obtans the followng expesson fo the eflectance facto (,,, ) R θ θ =π ( θ, ) ΔΩ ( ) q θ,, θ, cosθ snθ dθ d ( θ, ) ΔΩ cosθ snθ dθ d The ntegal of the denomnato of Eq. (10) can be gven a geometcal ntepetaton, descbed by Fgue, leadng to the followng esult: (10) ΔS d cosθ snθ θ = cosθ ( θ, ) ΔΩ d d whee s the dstance between the detecto and the emttng suface. (11) Fgue : Intecepton of the sold angle ΔΩ wth the unt sphee and pojecton on the hozontal plane. In the specal case whee the detecto s located at θ = 0, ΔΩ s then called ΔΩ 0 and Eq. (11) can be wtten as: ΔSd cosθ sn (, θ dθ d θ = (1) ) ΔΩ0 and snce the detecton was assumed to be n a cone of half-angle α, we also have: α cosθ sn cos sn sn ( ) (, θ dθ d = π ) 0 d θ ΔΩ θ θ θ θ = =π α. (13) 0 Fnally, accodng to Eqs. (10) to (13), we obtan Reflectance (,,, ) ( ) sn ( α) q θ,, θ, cos θ sn θ θ ( θ, d d ) ΔΩ R θ θ = (14) cosθ The eflectance ρ s the ato of the total flux eflected by the sample Φ to the ncdent flux Φ : 6

7 Φ ρ= Φ The total flux element Φ eflected by the aea element ΔS s gven by the sum of the flux elements (, ) ( ΔΩ = π) d Φ θ ove the whole hemsphee. By extenson of Eq. (8) to a hemsphecal sold angle, we obtan ( ) π/ π,,, cos sn θ = 0 = 0 (15) Φ = EΔS q θ θ θ θ dθ d (16) As the ncdent flux Φ s equal to EΔ S, whee Δ S s the llumnated aea, Eq. (15) becomes ΔS π/ π ρ = ( θ,, θ, ) cosθ snθ θ Δ q d d (17) S θ = 0 = 0 Reflectance can be measued wth an ntegatng sphee, whose geomety usually ensues that Δ S =Δ S. The eflectance facto tends towads the adance facto when the sold angle becomes nfntesmal ( ΔΩ 0) and tends towads the eflectance when all the uppe hemsphee s taken nto account ( ΔΩ = π). Note that adance facto and eflectance facto may each values hghe than 1, wheeas eflectance s always lowe than Colo of Lambetan and specula samples The colometc coodnates can be calculated fom the eflectance facto and epesented n 3- dmensonal spaces defned by the CIE [1,, 3]. Ths calculaton can be pefomed fo any object but two types of samples ae of patcula mpotance when studyng the angula colometc vaatons: the Lambetan and the specula samples CIEXYZ and CIELAB colometc spaces The colo of an object s epesented by a set of thee values deduced fom the object eflectance facto. The CIEXYZ tstmulus values ae obtaned by multplyng the standadzed llumnant adance spectum S(λ), the object eflectance spectum R(λ) and each of the matchng colo functons x( λ ), y( λ ) and ( ) z λ, then summng the esultng spectum ove the vsble wavelength doman: X = k R( λ) S( λ) x( λ) λ Y = k R( λ) S( λ) y( λ) λ Z = k R( λ) S( λ) z( λ) λ whee facto k s a nomalzaton facto gvng Y = 100 fo the pefect whte dffuse whose eflectance spectum s R ( λ ) = 1,.e. k = λ 100 S( λ) y( λ) The colo matchng functons x( λ ), y( λ ) and ( ) z λ have been detemned fo a fxed obsevaton sold angle wth half-angle α = n the standad of CIE1931 and α = 10 n the standad of CIE1964. The CIELAB colo space, defned by the CIE n 1976, expesses thee coodnates L *, a * and b * deved fom the CIEXYZ tstmulus values accodng to the followng fomulas: (18) 7

8 * L = f Y Y ( ef ) ( ef ) ( ef ) ( ef ) ( ef ) 116 / 16 * a = 500 f X/ X f Y/ Y * b = 00 f Y/ Y f Z/ Z (19) whee 3 1/3 4 f ( A) = A, A> f ( A) = A+, A (Paul's coecton) (0) and whee X ef, Y ef and Z ef ae the CIEXYZ tstmulus values fo a efeence usually chosen as the pefect whte dffuse, calculated fo a constant spectum R ( λ ) = Lambetan samples A Lambetan sample s defned by a adance facto β ndependent of the ncdent and obsevaton dectons. Its BRDF s constant, equal to β/ π. By angula ntegaton accodng to Eqs. (10) and (17), the eflectance facto R and the eflectance ρ ae equal to β. The Lambetan sample s sad to be a pefect whte when t absobs no lght,.e. when R ( λ ) = 1. When t s tself chosen as the whte efeence, ts CIELAB coodnates ae * * * ( w, w, w) ( 100,0,0) L a b = Specula samples A sample s sad to be pefectly specula when t eflects a collmated ncdent lght nto a sngle decton. Its eflectance ρ( θ ) s a functon of the llumnaton angle θ. Accodng to Snell s laws, lght s eflected at the angle θ and the azmuth angle +π. In the othe dectons, no lght s eflected. The BRDF s gven by the followng b-dmensonal Dac Delta functon δ ( ) ( ) ( θ θ,,, ) δ ( +π θ θ =ρ θ ) q (1) cosθsnθ Fom Eq. (14), we obtan the followng expesson fo the eflectance facto ( ) ( ) ρ θ, ( ) ( θ, +π) ΔΩ R θ, = sn αcosθ () 0 othewse ρ θ can be obtaned by angula ntegaton accodng to Eq. (17) wth Δ S =Δ S and q gven by Eq. (1), o by settng α=π / and θ = 0 nto Eq. (). When ρ( θ ) = 1, the ncdent lght s eflected wthout attenuaton and the sample s called a pefect mo. Its eflectance facto, called γ, depends only on the geomety of the measung system and s gven by Eq. () wth R( θ ) = 1 and θ =θ n the specula decton: γ= sn ( α) 1 cosθ (3) 8

9 The colo of the pefect mo n the specula decton, n espect to the pefect whte dffuse s gven by the CIELAB coodnates ( * * * ) ( 1/3 m, m, m = γ ,0,0) L a b. Both colos of the pefect whte dffuse and the pefect mo ae located on the achomatc axs ( a, b ) ( 0,0) =. Snce γ s lage than 1, the pefect mo has a bghte colo than the whte dffuse,.e. Lm > Lw. Fo example, f the pefect mo s measued at θ =θ = 0 and f α=, we have γ 81 and the lghtness of the pefect mo s L * m When α = 10, we have γ 33 and * L 357. m 4. Gonocolomety We pesent n ths secton a methodology to defne colo vaatons wth the vewng decton n the CIELAB colometc space as a functon of vewng decton Choosng the vewng sold angle Some objects have shap angula eflecton popetes. Fo example, gloss s often focused on a pecse decton. Measung the BRDF of such objects eques a hgh angula pecson, theefoe a detecto wth small sold angle (cf. [8] fo pactcal examples). The ntepetaton of gloss dstngushes the specula gloss, elated to the eflecton n the specula decton, the dstnctness of mage elated to the specula peak wdth, and the gloss contast elated to the ato between the specula and the dffuse fluxes []. To measue gloss accuately, especally the dstnctness of mage, the vewng sold angle has to be smalle than the ocula acuty, whch s estmated to 1 mnute of ac. Nevetheless, Eq. (3) ndcates that a small value of α nduces a vey hgh value fo the geomety facto γ and theefoe an mpotant souce of mpecson n the colometc computatons. Fo the colo chaactezaton, t s pefeable to use a lage vewng sold angle and t looks judcous to select the one consdeed by the CIE when establshng the colo matchng functons,.e. cones wth half-angle α = o 10. At fst sght, the chaactezatons of gloss and of colo look ncompatble snce they eque dffeent vewng sold angles, but n pactce, t s suffcent to pefom a sngle measuement at hgh angula esoluton. Ths latte s subsequently undesampled fo the colo chaactezaton, wth a samplng angle of o 10 to epoduce CIE s standad sold angles. Ths undesamplng opeaton s at the base of so-called abdged gonophotomety. 4.. Hgh angula esoluton measuement Let us measue the fluxes eflected by the object and by the pefect whte dffuse usng a vey thn vewng sold angle, whch may be wtten as δω = snθdθd whee angles θ and denote the vewng decton. Snce lght ays flowng though t popagate nealy along the same decton ( θ, ), the ntegals n Eq. (10) ae educed to a smple fom: R θ,, θ, =πq θ,, θ, (4) ( ) ( ) Reflectance facto s theefoe popotonal to adance coeffcent n ths case. Sève [8] ecommends usng a detecto wth vewng sold angle smalle than the ones usually found n glossmetes, e.g. wth half-angle sold angle,.e. δω =.10 s. 9

10 4.3. Undesamplng We want to convet the measued spectal eflectance facto nto colo coodnates,.e. nto X, Y, Z tstmulus values, then nto CIELAB coodnates. The colo matchng functons used fo the computaton of the tstmulus values have been establshed fo a vewng sold angle ΔΩ of ethe o 10 half-angle. It looks theefoe sutable to calculate colo coodnates fom specta defned n espect to ths sold angle. Obtanng these specta s pemtted by the followng method. R θ,, θ, s a contnuous functon of angles Let us assume that spectal eflectance facto ( ) ( θ, ). If t was measued at dscete postons wth hgh angula esoluton, ntepolaton may be used. The adance coeffcent q can theefoe be deduced fom Eq. (4). Then, Eq. (10) yelds a new eflectance facto ρ ( θ,, θ, ) that would be measued wth the new vewng sold angle ΔΩ. Ths eflectance facto s consstent wth colo calculatons. In ode to smplfy the computatons, n patcula to avod calculatng the ntegals of Eq. (10), we ecommend a dscete appoach based on a plana mappng of the BRDF. Ths mappng, aleady used by Elas [3] fo the epesentaton of BRDFs, s a Lambet azmuthal equal-aea pojecton [4]. Evey pont P on the hemsphee, specfed by ts sphecal coodnates ( θ, ), s mapped to a pont P' of pola coodnates (, ) contaned wthn a hozontal dsk of adus tangent to the hemsphee at the Noth pole N (Fg. 3). The azmuth coodnate s the same n the two coodnate systems. Coodnate, whch coesponds the dstance NP, s gven by = sn ( θ /) Pont P' s also be specfed by the followng Catesan coodnates u = sn ( θ/) cos v = sn θ/ sn ( ) (5) Fgue 3: Mappng of the hemsphee onto a dsk of adus accodng to Lambet azmuthal equal-aea pojecton appled at the Noth pole N. 10

11 Fgue 4: Sampled mage of the BRDF pojecton dsk whee the pxel aea coesponds to a vewng angle of half-angle. Ths mappng conseves aeas. A poton of hemsphee wth aea A s mapped nto a poton of dsk wth same aea A. Convesely, the aea of a dsk element equals the aea of the coespondng sphee element, theeby the value n steadan of the coespondng sold angle. By pattonng the dsk accodng to a unfom gd (Fg. 4), one obtans a patton of the hemsphee whee all the sold angles ae equal. Applyng ths mappng and ths pattonng to the BRDF yelds a multspectal mage,.e. a dscete mage contanng as many channels as wavelengths. Each pxel of the mage coesponds to a same sold angle. Pxels wth sde d epesent sold angles of d steadans. When one deses consdeng a patton wth lage sold angles, one can undesample the mage n such manne that the aea of the new pxels coesponds pecsely to the desed sold angle. Classcal esamplng techncs n mage pocessng, e.g. blnea esamplng, may be used. In ou case, we ae nteested n concal sold angle wth half-angle α= o 10, whose aea n steadan s 4πsn ( α /). The pxel sde should theefoe be d = πsn ( α /) (6) Note that the pxelzaton wthout ovelappng n the (u,v) plane nduces to choose a squae base cone nstead of the classcal ccula base cone. It s also possble to use a patton gd whose cells have a moe ccula shape, e.g. a hexagonal gd Colo coodnates Now, the undesampled multspectal mage can be conveted nto CIELAB coodnates usng Eqs. (18) and (19). Consdeng a gven α value ( o 10 ), one obtans a 3-channel mage whee each pxel contans the thee colo coodnates, e.g. the L *, a * and b * coodnates of the CIELAB colo system. One may select a subset of pxels, coespondng to a subset of dectons, and epesent the assocated colo n the CIELAB space. The geomety facto γ, gven by Eq.(3), enables epesentng both the pefect whte dffuse * * * ( L w, a w, b w) = ( 100,0,0) and the pefect mo ( L *, *, * ) ( 1/3 m a m b m = γ ,0,0) n the CIELAB space. Thanks to the undesamplng pocedue, the sold angle selected fo colo vewng s ndependent of 11

12 the one used fo spectal measuements and the geomety facto γ, theeby the lghtnesses of the pefect whte dffuse and the pefect mo, become ndependent of the expemental devce Gonocolomety n 3 steps The complete pocedue to follow fom flux measuement to CIELAB coodnates may be summazed by the thee followng steps: 1. Measuement: usng a fxed collmated lght and a gonoeflectomete wth small sold angle, measue at hgh angle esoluton the flux eflected by the sample to study and by the pefect whte dffuse. Dvde the measued flux of the sample by the measued flux of the pefect whte dffuse. One obtans the eflectance facto of the sample.. Undesamplng: choose a vewng sold angle fo the colo calculaton ( o 10 ). Undesample the eflectance facto calculated n step 1, wth a samplng ate coespondng to the selected vewng sold angle. 3. Colomety: calculate the colo coodnates. Ths deal pocedue must be modfed n pactce because, nowadays, the devces seldom acheve to obtan both a hgh angula esoluton (e.g. ΔΩ=10-4 s) and a suffcent spectal esoluton fo colometc calculatons (e.g. Δλ=5 nm). Moe sophstcated potocol should be poposed wth shap measuements fo lage BRDF vaatons, n patcula aound the specula decton, and moe blu measuements elsewhee. In ths last case, the second step should athe be called esamplng. The opeaton conssts n changng the ntal non-unfom samplng to a unfom one. Howeve, the same classcal esamplng techncs adapted to an egula gd can be appled. 5. Example We popose to llustate the methodology ntoduced above wth the example of a glossy sample llumnated at nomal ncdence ( θ = = 0). Its eflectance facto s supposed to be wtten as the supeposton of a Lambetan dffuse component and a specula peak [5, 6] 1 tan ξ R( θ, ) = Rd( λ ) +ρs exp 3 (7) m cos ξ m whee R d s the volume dffuse eflectance facto, ρ s s the suface specula eflectance, m s the.m.s. slope of the ough nteface and ξ s the local ncdence angle,.e. half the angle between the llumnaton decton ( θ, ) and the obsevaton decton ( θ, ) [7]. At nomal ncdence, we have ξ=θ /. The dffuse component R d s a functon of wavelength and coesponds to a blue colo n the example wheeas the eflectance ρ s s assumed to be wavelength-ndependent (Fgue 5). 1

13 Fgue 5: Spectal vaatons of the specula component ρ s and of the dffuse component R d fo the eflectance facto smulaton (Eq.(7)). Fst, the flux eflected by the sample s acqued at a hgh angula esoluton,.e. wth a small vewng sold angle δω. Then we poceed to the undesamplng opeaton, choosng a samplng sold angle wth o 10 half-angle. Snce θ = 0, the geomety facto s accodngly γ = sn ( ) o γ 10 = sn ( 10 ). Fgues 6 and 7 epesent the eflectance facto befoe and afte the two types of undesamplng at the wavelength λ= 700 nm whee R d = 0.5, fo a suface.m.s. slope m = 0.1 and a suface eflectance ρ s = The eflectance factos ae epesented n Fgue 6 n (u,v)-coodnates, deved fom the ( θ, ) -coodnates accodng to tansfomaton (5). In Fgue 7, they ae epesented n the ncdent plane as a functon of the vewng angle. Fgue 6: Reflectance facto at a nomal ncdence, smulated wth Eq. (7) wth m = 0.1, R d = 0.5 and ρ s = 0.04 and pojected onto the (u,v)-plane accodng to the tansfomaton defned by Eq. (5); (a) befoe undesamplng, (b) afte undesamplng wth α = (gd spacng Δu = 0.049) and (c) afte undesamplng wth α = 10 (gd spacng Δu = 0.47). 13

14 Fgue 7: Pofle of the eflectance facto plotted n Fgue 6 along the u-axs, befoe and afte each type of undesamplng. In the last step, the eflectance specta calculated fo each gd cell ae conveted nto CIELAB coodnates, as well as the eflectance specta of the pefect whte dffuse and of the pefect mo. All the coespondng ponts ae placed n the gonocolometc CIELAB space. Note that lghtness ovepasses 100 when, n a gven decton, moe lght s eflected by the sample than by the pefect whte dffuse. Fo vey glossy sufaces, lghtness may vay ove decades. Hence, t s pefeable to epesent lghtness usng a logathm scale L* pefect mo ( ) pefect mo (10 ) a* achomatc axe specula colo pefect whte dffuse 0 specula colo θ >30 dffuse colo θ >30 dffuse colo C* b* -0 Fgue 8: Colo coodnates of a sample obseved at dffeent oentatons θ epesented n the (C*,logL*)-dagam (left) and the (a*,b*)-dagam (ght) of the CIELAB space. Coespondng eflectance factos ae smulated accodng to Eq. (7) at nomal ncdence, wth R d (λ) and ρ s (λ) shown n Fgue 5 and m = 0.1. The vewng sold angles ae defned by α = ( ) and α = 10 ( ). Fgue 8 shows the colo vaatons obtaned fo vewng sold angles wth and 10 half-angle. We see that nea the specula decton (θ tendng to 0 ), the colo becomes pale (L * nceases and C * deceases). Ths s the usual gonocolometc behavou of glossy mateals. It should be notced that the two undesamplng opeatons ( α = and α = 10 ) gve dffeent colo coodnates fo a 14

15 same decton due to dffeent vewng sold angle ove whch the eflectance s aveaged. At θ = 0, 94.6,.1, 8.3. Howeve, these the CIELAB coodnates ae espectvely ( 98.3, 0.3, 8.8) and ( ) vaatons ae small compaed to the hgh lghtness vaaton of the pefect mo, detaled n Secton 3.3 ( L * m = fo α = and fo α = 10 ). The small coodnate dffeences (especally a hue 63.7, 0.7, 16.8 and angle dffeence) obseved n the dffuse pat when θ >30, ( ) ( 64.5, 4.0, 14.8), ae due to dffeences between the standad obseve colo matchng functons defned by the CIE n 1931 and n 1964, and moe pecsely to the dffeent foveal aea stmulated by the vewng sold angle wth α = and α = Concluson The gonocolometc space ntoduced n ths pape s a CIELAB colo space whee the colos of a same object vewed at dffeent angles ae epesented. Snce the amount of lght eflected n the specula decton may be much hghe than the amount of lght eflected by the pefect whte dffuse used as efeence, lghtness may ovepass 100. The hghest lghtness value s obtaned fom a pefect mo vewed n the specula decton. Ths value s systematcally epesented n the gonocolometc space n such manne to have a scale of gloss,.e. a scale fo eadng hgh lghtness values. As an advantage of the poposed methodology, colo coodnates ae calculated by consdeng the vewng sold angles consdeed by the CIE when establshng the colo matchng functons,.e. sold angles wth o 10 half-angle. Ths vewng sold angle s ntoduced whle convetng the BRDF, measued wth a thnne sold angle, nto a multspectal plana mage whose pxel sze s dectly elated to the consdeed sold angle. Ths mathematcal pocedue allows the ndependence of the colo calculaton fom the measuement setup. The method poposed hee needs to be tested on a wde ange of samples n ode to evaluate ts lmtatons. At fst sght, t s dffcult to evaluate the petnence of the colo coodnates computed n dectons whee the spectal BRDF vaes vey stongly wthn the consdeed sold angle. Ths queston mght necesstate a wde expement based on vsual testng. On the othe hand, the connecton between the BRDF (cone ssued fom the suface) and the vsual feld (cone ssued fom the detecto) s not as smple as supposed n ou pape. When the sold angle s vey small, the adance nvaance pncple ensues that the adance emtted by the suface and the one eceved by the detecto ae equal. But when a lage sold angle s consdeed, the textue o the cuvatue of the suface may have an nfluence on the pecepton. It appeas that ou wok s lmted to samples whose suface s athe homogenous and plane. The case of objects pesentng a stongly textued aspect o a non neglgble adus of cuvatue s theefoe beyond the scope of the pesent study and epesents a challenge fo futhe nvestgaton. The noton of gonocolometc space was pesented fo eflectance, but t can be extended n a staghtfowad manne to tansmttance. The only dffeence concens the efeence samples. Fo dect tansmsson (the equvalent fo specula n eflecton), the efeence s the dect measuement of the ncdent lght wthout sample (n pactce, ths dect measuement s also often used as efeence fo specula eflecton measuements nstead of pefect mo). A Lambetan efeence fo dffuse tansmsson measuements s not usual. In ths case, the pefect whte dffuse, efeence fo dffuse eflecton measuements, can also be used as efeence fo dffuse tansmsson measuements. 15

16 Refeences 1. Colomety, 3d edton, Techncal Repot CIE 15:004.. Wyszeck G, Stles WS. Colo scence: Concepts and methods, quanttatve Data and Fomulae. nd ed. New Yok: Wley Intescence Publcaton; Shama G. Colo fundamentals fo dgtal magng. In: Dgtal Colo Imagng Handbook, Ed. G. Shama, CRC Pess; 003. p Olea C. Colomety n Optcal Coatng. Poc of SPIE 005;5963: Baba G, Suzuk K. Gono-spectophotometc analyss of whte and chomatc efeence mateals. Analytca Chmca Acta 1999;380: Mkula M, Ceppan M, Vasko K. Gloss and Gonocolomety of Pnted Mateals. Col. Res. and appl. 003;8: Westlund HB, Meye GW. Applyng appeaance standads to lght eflecton models. ACM SIGGRAPH Confeence on compute gaphcs 001; Seve R. Poblems connected wth the concept of gloss, Col. Res. and Appl. 1993;18: Oben G, Knoblauch K, Vénot F. Dffeence scalng of gloss: Nonlneaty, bnoculaty, and constancy. Jounal of Vson 004;4: J W, Ponte MR, Luo RM, Dakn J. Gloss as an aspect of the measuement of appeaance. JOSA A 11. Smonot L, Elas M. Colo change due to suface state modfcaton. Col. Res. and appl. 003;8: Smonot L, Elas M. Colo change due to a vansh laye. Col. Res. and appl. 004;9: ;3; Kuehn RG. Cangante : a fabc and a colostc devce n the at of the enassance. Col. Res. and appl. 1996;1: Lu R, Koendenk JJ, Kappes AML. Optcal popetes (bdectonal eflectance dstbuton functon) of shot fabc. Appled Optcs. 000;39: McCamy CS. Obsevaton and measuement of the appeaance of metallc mateals. Pat I. Maco appeaance. Col. Res. and appl. 1996;1: McCamy CS. Obsevaton and measuement of the appeaance of metallc mateals. Pat II. Mco appeaance. Col. Res. and appl. 1998;3: Nadal ME, Ealy EA. Colo Measuements fo Pealescent Coatngs. Col. Res. and appl. 004;9: Eshov S, Kolchn K, Myszkowsk K. Rendeng appeaance based on pant-composton modelng. Compute Gaphcs Foum 001;0: Oben G, Bousquet R, Nadal ME. New NIST efeence gonospectomete. Poc. SPIE 005;5880: Hünehoff D, Gusemann U, Höpe A. New obot-based gonoeflectomete fo measung spectal dffuse eflecton. Metologa 006;43:S11 S McCluney WR. Intoducton to adomety and photomety. Boston: Atech House; p

17 . Hunte RS. The measuement of appeaance. New Yok: John Wley & sons; Elas M, Elas G. Radatve tansfe n nhomogeneous statfed scatteng meda wth use of the auxlay functon method. JOSA A 004;1: Snyde J P. Map Pojectons A Wokng Manual. U. S. Geologcal Suvey Pofessonal Pape Washngton, DC: U. S. Govenment Pntng Offce; p Toance KM, Spaow EM. Theoy fo Off-Specula Reflecton fom Roughened Sufaces. JOSA A 1967;9: Hébet M, Hesch RD. Extendng the Clappe-Yule model to ough pntng suppot. JOSA A 005;: Smonot L, Oben G. Geometcal consdeatons n analyzng sotopc o ansotopc suface eflectons. Appled optcs 007;46: