1、Designation: E 2022 08Standard Practice forCalculation of Weighting Factors for Tristimulus Integration1This standard is issued under the fixed designation E 2022; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last re
2、vision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice describes the method to be used forcalculating tables of weighting factors for tristimulus integra-tion using c
3、ustom spectral power distributions of illuminants orsources, or custom color-matching functions.1.2 This practice provides methods for calculating tables ofvalues for use with spectral reflectance or transmittance data,which are corrected for the influences of finite bandpass. Inaddition, this pract
4、ice provides methods for calculating weight-ing factors from spectral data which has not been bandpasscorrected. In the latter case, a correction for the influence ofbandpass on the resulting tristimulus values is built in to thetristimulus integration through the weighting factors.1.3 The values st
5、ated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.4 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety a
6、nd health practices and determine the applica-bility of regulatory limitations prior to its use.2. Referenced Documents2.1 ASTM Standards:2E 284 Terminology of AppearanceE 308 Practice for Computing the Colors of Objects byUsing the CIE System2.2 CIE Standard:CIE Standard S 002 Colorimetric Observer
7、s33. Terminology3.1 DefinitionsAppearance terms in this practice are inaccordance with Terminology E 284.3.2 Definitions of Terms Specific to This Standard:3.2.1 illuminant, nreal or ideal radiant flux, specified byits spectral distribution over the wavelengths that, in illuminat-ing objects, can af
8、fect their perceived colors.3.2.2 source, nan object that produces light or otherradiant flux, or the spectral power distribution of that light.3.2.2.1 DiscussionA source is an emitter of visible radia-tion. An illuminant is a table of agreed spectral powerdistribution that may represent a source; t
9、hus, Illuminant A is astandard spectral power distribution and Source A is thephysical representation of that distribution. Illuminant D65 is astandard illuminant that represents average north sky daylightbut has no representative source.3.2.3 spectral power distribution, SPD, S(l),nspecification of
10、 an illuminant by the spectral compositionof a radiometric quantity, such as radiance or radiant flux, as afunction of wavelength.4. Summary of Practice4.1 CIE color-matching functions are standardized at 1-nmwavelength intervals. Tristimulus integration by multiplicationof abridged spectral data in
11、to sets of weighting factors occursat larger intervals, typically 10-nm or 20-nm; therefore, inter-mediate 1-nm interval spectral data are missing, but needed.4.2 Lagrange interpolating coefficients are calculated for themissing wavelengths. The Lagrange coefficients, when multi-plied into the appro
12、priate measured spectral data, interpolatethe abridged spectrum to 1-nm interval. The 1-nm intervalspectrum is then multiplied into the CIE 1-nm color-matchingdata, and into the source spectral power distribution. Eachseparate term of this multiplication is collected into a valueassociated with a me
13、asured spectral wavelength, thus formingweighting factors for tristimulus integration.4.3 A correction may be applied to the resulting table ofweighting factors to incorporate a correction for the spectraldatas bandpass dependence.1This practice is under the jurisdiction of ASTM Committee E12 on Col
14、or andAppearance and is the direct responsibility of Subcommittee E12.04 on Color andAppearance Analysis.Current edition approved Dec. 1, 2008. Published January 2009. Originallyapproved in 1999. Last previous edition approved in 2006 as E 2022 - 061.2For referenced ASTM standards, visit the ASTM we
15、bsite, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.3Available from USNC-CIE Publications Office (International Commission onIllumination), C/o Thomas M. Lemons,
16、TLA-Lighting Consultants, Inc., 7 Pond St.,Salem, MA 01970, http:/www.cie-usnc.org.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.5. Significance and Use5.1 This practice is intended to provide a method that willyield uniformity of
17、calculations used in making, matching, orcontrolling colors of objects. This uniformity is accomplishedby providing a method for calculation of weighting factors fortristimulus integration consistent with the methods utilized toobtain the weighting factors for common illuminant-observercombinations
18、contained in Practice E 308.5.2 This practice should be utilized by persons desiring tocalculate a set of weighting factors for tristimulus integrationwho have custom source, or illuminant spectral power distri-butions, or custom observer response functions.5.3 This practice assumes that the measure
19、ment interval isequal to the spectral bandwidth integral when applying correc-tion for bandwidth.6. Procedure6.1 Calculation of Lagrange CoeffcientsObtain by calcu-lation, or by table look-up, a set of Lagrange interpolatingcoefficients for each of the missing wavelengths.46.1.1 The coefficients sho
20、uld be quadratic (three-point) inthe first and last missing interval, and cubic (four-point) in allintervals between the first and the last missing interval.6.1.2 Generalized Lagrange CoeffcientsLagrange coeffi-cients may be calculated for any interval and number ofmissing wavelengths by Eq 1:Ljr! 5
21、)i50 ifijnr ri!rj ri!, for j 5 0,1,.n (1)where:n = degree of coefficients beingcalculated,5iand j = indices denoting the locationalong the abscissa,p = repetitive multiplication ofthe terms in the numeratorand the denominator, andindices ofthe interpolant, r= chosen on the same scale asthe values i
22、and j.6.1.2.1 Fig. 1 assist the user in selecting the values of i, j,and r for these calculations.6.1.2.2 Eq 1 is general and is applicable to any measurementinterval or interpolation interval, regular or irregular.6.1.3 10 and 20-nm Lagrange CoeffcientsWhere themeasured spectral data have a regular
23、 or constant interval, theequation reduces to the following:L05r 1!r 2!r 3!6(2)L15r!r 2!r 3!2(3)L25r 1!r!r 3!2(4)L35r 1!r 2!r!6(5)for the cubic case, and toL05r 1!r 2!2(6)4Hildebrand, F. B., Introduction to Numerical Analysis , Second Edition,Dover, New York, 1974, Chapter 3.5Fairman, H. S., “The Ca
24、lculation of Weight Factors for Tristimulus Integra-tion,” Color Research and Application, Vol 10, 1985, pp. 199203.FIG. 1 The Values of i in Eq 1 are Plotted Above the Abscissa and the Values of r are Plotted Below for A) the First MeasurementInterval; B) the Intermediate Measurement Intervals; and
25、, C) the Last Measurement Interval Being InterpolatedE2022082L15r!r 2!1(7)L25r 1!r!2(8)for the quadratic case. In each of the above equations, asmany or as few values of r as required are chosen to generatethe necessary coefficients.6.1.3.1 Eq 2-8 are applicable when the spectral data areabridged at
26、 10-nm or 20-nm intervals, and the interpolatedinterval is regular with respect to the measurement interval,presumably 1-nm.6.1.4 Tables 1-4 provide both quadratic and cubic Lagrangecoefficients for 10-nm and 20-nm intervals.6.2 With the Lagrange coefficients provided, the intermedi-ate missing spec
27、tral data may be predicted as follows:Pl! 5(i50nLimi(9)where:P = the value being interpolated at interval l,L = the Lagrange coefficients, andm = the measured abridged spectral values.Because the measured spectral values are as yet unknown, itmay be best to consider this equation in its expanded for
28、m:Pl! 5 L0m01 L1m11 L2m21 L3m3(10)6.3 Multiply each P(l) by the 1-nm interval relative spectralpower of the source or illuminant being considered.6.3.1 It may be necessary to interpolate missing values ofthe source spectral power distribution S(l), if the source hasbeen measured at other than 1-nm i
29、ntervals.6.3.2 Doing so results in the following equation:Sl!Pl! 5 Sl!L0m01 Sl!L1m11 Sl!L2m21 Sl!L3m3(11)6.4 Multiply the weighted power at each 1-nm wavelengthby the appropriate custom color-matching function value forthat wavelength. Using the CIE color-matching functions as anexample, obtain the
30、CIE 1-nm data from CIE Standard S 002,Colorimetric Observers. Doing so results in the followingequation:x l!Sl!Pl! 5 x l!Sl!L0#m01 x l!Sl!L1#m11 x l!Sl!L2#m21 x l!Sl!L3#m3(12)where:TABLE 1 The Lagrange Quadratic Interpolation CoefficientsApplicable to the First and Last Missing Interval for Calculat
31、ionof 10-nm Weighting Factors for Tristimulus IntegrationIndex of MissingWavelength L0L1L21 0.855 0.190 0.0452 0.720 0.360 0.0803 0.595 0.510 0.1054 0.480 0.640 0.1205 0.375 0.750 0.1256 0.280 0.840 0.1207 0.195 0.910 0.1058 0.120 0.960 0.0809 0.055 0.990 0.045TABLE 2 The Lagrange Cubic Interpolatio
32、n CoefficientsApplicable to the Interior Missing Intervals for Calculation of10-nm Weighting Factors for Tristimulus IntegrationIndex of MissingWavelength L0L1L2L31 0.0285 0.9405 0.1045 0.01652 0.0480 0.8640 0.2160 0.03203 0.0595 0.7735 0.3315 0.04554 0.0640 0.6720 0.4480 0.05605 0.0625 0.5625 0.562
33、5 .06256 0.0560 0.4480 0.6720 0.06407 0.0455 0.3315 0.7735 0.05958 0.0320 0.2160 0.8640 0.04809 0.0165 0.1045 0.9405 0.0285TABLE 3 The Lagrange Quadratic Interpolating CoefficientsApplicable to the First and Last Missing Interval for Calculationof 20-nm Weighting Factors for Tristimulus Integration.
34、Index of MissingWavelength L0L1L21 0.92625 0.0975 0.023752 0.85500 0.1900 0.045003 0.78625 0.2775 0.063754 0.72000 0.3600 0.080005 0.65625 0.4375 0.093756 0.59500 0.5100 0.105007 0.53625 0.5775 0.113758 0.48000 0.6400 0.120009 0.42625 0.6975 0.1237510 0.37500 0.7500 0.1250011 0.32625 0.7975 0.123751
35、2 0.28000 0.8400 0.1200013 0.23625 0.8775 0.1137514 0.19500 0.9100 0.1050015 0.15625 0.9375 0.0937516 0.12000 0.9600 0.0800017 0.08625 0.9775 0.0637518 0.05500 0.9900 0.0450019 0.02625 0.9975 0.02375TABLE 4 The Lagrange Cubic Interpolating CoefficientsApplicable to the Interior Missing Intervals for
36、 Calculation of20-nm Weighting Factors for Tristimulus IntegrationIndex of MissingWavelength L0L1L2L31 0.0154375 0.9725625 0.0511875 0.00831252 0.028500 0.940500 0.104500 0.0165003 0.0393125 0.9041875 0.1595625 0.02443754 0.048000 0.864000 0.216000 0.0320005 0.0546875 0.8203125 0.2734375 0.03906256
37、0.059500 0.773500 0.331500 0.0455007 0.0625625 0.7239375 0.3898125 0.05118758 0.064000 0.672000 0.448000 0.0560009 0.0639375 0.6180625 0.5056875 0.059812510 0.062500 0.562500 0.562500 0.06250011 0.0598125 0.5056875 0.6180625 0.063937512 0.056000 0.448000 0.672000 0.06400013 0.0511875 0.3898125 0.723
38、9375 0.062562514 0.045500 0.331500 0.773500 0.05950015 0.0390625 0.2734375 0.8203125 0.054687516 0.032000 0.216000 0.864000 0.04800017 0.0244375 0.1595625 0.9041875 0.039312518 0.016500 0.104500 0.940500 0.02850019 0.0083125 0.0511875 0.9725625 0.0154375E2022083x(l) = the value of the CIE X color-ma
39、tching function atwavelength l, and the calculations are carried outfor each of the three CIE color-matching functions,x(l), y(l), and z(l).6.5 In the four terms on the right-hand side of this equation,the numerical values of the three factors in the brackets areknown and should be multiplied into a
40、 single coefficient. Thefourth factor, mi, in each of the four additive terms is associatedwith a different measured wavelength.6.6 Add all multiplicative coefficients dependent upon eachdifferent measured wavelength into a single coefficient appli-cable to that wavelength. This results in a single
41、set ofweighting factors that then will contain one value for eachmeasured wavelength in each of three color-matching func-tions. The partial contribution to the tristimulus value atwavelength m0is:x l0!Sl0!L0! 1 x l1!Sl1!L0!1 . m05 wt0m0(13)6.7 Normalize the weighting factors by calculating thefollo
42、wing normalizing coefficient:k 5100(Sl!y l!(14)where:k = the normalizing coefficient,S(l) = the power in the 1-nm spectrum, andy(l) = the CIE Y color-matching function.6.8 Multiply the weighting factors by k to normalize the setto Y = 100 for the perfect reflecting diffuser.6.9 Correction for Bandpa
43、ss DependenceIf it is desired tocorrect the resulting weighting factors for the bandpass depen-dence of the measured spectral data, apply the followingcorrection to the interior passbands.6Wci! 5 0.083 WMi 1! 1 1.166 WMi! 0.083 WMi 1 1!(15)where:W = the indexed weight,c = a corrected weight, andm =
44、a weight calculated without bandpass correction.The index i varies from the second measured passband to thenext to last measured passband. The following correctionapplies to the first and last measured passband:Wci! 5 1.083 WMi! 0.083 WMi 6 1! (16)where the symbols are the same as those of Eq 15 and
45、 theindex i and 6 refer to the first and last measured passbands,respectively.7. Precision7.1 The precision of the practice is limited only by theprecision of the data provided for the source spectral powerdistribution. The CIE color-matching functions are precise tosix digits by definition. The Lag
46、range coefficients are precise toseven digits.8. Keywords8.1 color-matching functions; illuminant; illuminant-observer weights; source; tristimulus weighting factorsAPPENDIX(Nonmandatory Information)X1. EXAMPLE OF THE CALCULATIONSX1.1 Table X1.1 gives the spectral power distribution(SPD) of a typica
47、l 3-band fluorescent lamp with a correlatedcolor temperature of about 3000K. The first step is to multiplyeach value of the SPD by the appropriate CIE color matchingfunction ( y in this case), wavelength by wavelength, which isshown in Table X1.2 for three spectral regions: near 360 nm,560 nm, and 8
48、30 nm. Table X1.3 shows a typical interpolationof a measured reflectance curve from a 10-nm reported intervalto the 1-nm interval that matches the SPD- y product in thesame three spectral regions. Tables X1.4-X1.6 illustrate howthe same measured data, used to interpolate the missingreflectance data
49、in several different intervals, can be combinedwith the illuminant-color matching function product to form asingle weight at a single measurement point. Finally, TableX1.7 shows the resulting weight set for this 3000K source andthe 1964 10 color matching functions. Table X1.7 is compat-ible with Tables 5 in Practice E 308. The weights in Table X1.7then can be adjusted by the Stearns6ban
