1、Designation: E 2022 06Standard 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 (e) 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
3、custom 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 prac
4、tice 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 This standar
5、d 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 and health practices and determine the applica-bility of regulatory limitations prior to its use.2. Referenced Documents2.1
6、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 Observers33. Terminology3.1 DefinitionsAppearance terms in this practice are inaccordance with Terminology E 284.3.2 Definitions of
7、 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 affect their perceived colors.3.2.2 source, nan object that produces light or otherradiant flux, or the spectral power distri
8、bution 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; thus, Illuminant A is astandard spectral power distribution and Source A is thephysical representation of that distribution.
9、 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 an illuminant by the spectral compositionof a radiometric quantity, such as radiance or radiant flux, as afunction of wave
10、length.4. Summary of Practice4.1 CIE color-matching functions are standardized at 1-nmwavelength intervals. Tristimulus integration by multiplicationof abridged spectral data into sets of weighting factors occursat larger intervals, typically 10-nm or 20-nm; therefore, inter-mediate 1-nm interval sp
11、ectral data are missing, but needed.4.2 Lagrange interpolating coefficients are calculated for themissing wavelengths. The Lagrange coefficients, when multi-plied into the appropriate measured spectral data, interpolatethe abridged spectrum to 1-nm interval. The 1-nm intervalspectrum is then multipl
12、ied 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 measured spectral wavelength, thus formingweighting factors for tristimulus integration.4.3 A correction may be applied to th
13、e resulting table ofweighting factors to incorporate a correction for the spectraldatas bandpass dependence.5. Significance and Use5.1 This practice is intended to provide a method that willyield uniformity of calculations used in making, matching, orcontrolling colors of objects. This uniformity is
14、 accomplishedby providing a method for calculation of weighting factors fortristimulus integration consistent with the methods utilized to1This practice is under the jurisdiction of ASTM Committee E12 on Color andAppearance and is the direct responsibility of Subcommittee E12.04 on Color andAppearan
15、ce Analysis.Current edition approved July 1, 2006. Published July 2006. Originally approvedin 1999. Last previous edition approved in 2001 as E 2022 - 01.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMS
16、tandards volume information, refer to the standards Document Summary page onthe ASTM website.3Available from USNC-CIE Publications Office, TLA Lighting Consultants, 7Pond Street, Salem, MA 01970.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, Unit
17、ed States.obtain the weighting factors for common illuminant-observercombinations 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
18、, or custom observer response functions.5.3 This practice assumes that the measurement 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 interpo
19、latingcoefficients for each of the missing wavelengths.46.1.1 The coefficients should 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
20、may be calculated for any interval and number ofmissing wavelengths by Eq 1:Ljr! 5)i50 ifijnr ri!rj ri!, for j 5 0,1,.n (1)where:n = degree of coefficients beingcalculated,5i and j = indices denoting the locationalong the abscissa,p = repetitive multiplication ofthe terms in the numeratorand the den
21、ominator, andindices ofthe interpolant, r= chosen on the same scale asthe values i 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.
22、1.3 10 and 20-nm Lagrange CoeffcientsWhere themeasured spectral data have a regular 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)L15r!r 2!1(7)L25r 1!r!2(8)for the quadratic ca
23、se. 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 10-nm or 20-nm intervals, and the interpolatedinterval is regular with respect to the measurement interval
24、,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 spectral data may be predicted as follows:4Hildebrand, F. B., Introduction to Numerical Analysis, Second Editio
25、n, Dover,New York, 1974, Chapter 3.5Fairman, H. S., “The Calculation 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 Measurem
26、entInterval; B) the Intermediate Measurement Intervals; and, C) the Last Measurement Interval Being InterpolatedE2022062Pl! 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 valu
27、es are as yet unknown, itmay be best to consider this equation in its expanded form: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 spe
28、ctral power distribution S(l), if the source hasbeen measured at other than 1-nm intervals.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 fo
29、rthat wavelength. Using the CIE color-matching functions as anexample, obtain the 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:x(l) = the value of the CIE X color-m
30、atching 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
31、a 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
32、 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:TABLE 1 The Lagrange Quadratic Interpolation CoefficientsApplicable to the First and Last Missing Inte
33、rval for Calculationof 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 C
34、ubic Interpolation 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.
35、0625 0.5625 0.5625 .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 Tristim
36、ulus Integration.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 0.42675 0.6975 0.1237510 0.37500 0.7500 0.1250011 0.3262
37、5 0.7975 0.1237512 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 Miss
38、ing Intervals for 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 0.27
39、34375 0.03906256 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 0.051187
40、5 0.3898125 0.7239375 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 0.0154375E2022063x l0!Sl0!L0! 1 x l1!S
41、l1!L0!1 . m05 wt0m0(13)6.7 Normalize the weighting factors by calculating thefollowing 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 normali
42、ze the setto Y = 100 for the perfect reflecting diffuser.6.9 Correction for Bandpass 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 W
43、Mi! 0.083 WMi 1 1!(15)whereW = the indexed weight,c = a corrected weight, andm = 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
44、.083 WMi! 0.083 WMi 6 1! (16)where the symbols are the same as those of Eq 15 and 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.
45、 The CIE color-matching functions are precise tosix digits by definition. The Lagrange 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 C
46、ALCULATIONSTABLE X1.1 Spectral Power Distribution of Typical 3-Band Fluorescent Lamp with Correlated Color Temperature of 3000 K (1-nmmeasurement interval)l SPD l SPD l SPD l SPD l SPD l SPD360 0.004880 450 0.014870 540 0.162400 630 0.111200 720 0.004410 810 0.000000361 0.004595 451 0.015040 541 0.2
47、77600 631 0.102900 721 0.003505 811 0.000000362 0.004310 452 0.015210 542 0.392800 632 0.094620 722 0.002600 812 0.000000363 0.020290 453 0.014980 543 0.353900 633 0.062350 723 0.002470 813 0.000000364 0.036270 454 0.014750 544 0.315100 634 0.030080 724 0.002340 814 0.000000365 0.047350 455 0.014370
48、 545 0.429800 635 0.027420 725 0.002375 815 0.000000366 0.058440 456 0.014000 546 0.544600 636 0.024770 726 0.002410 816 0.000000367 0.031870 457 0.014060 547 0.383500 637 0.023050 727 0.002450 817 0.000000368 0.005300 458 0.014110 548 0.222500 638 0.021330 728 0.002490 818 0.000000369 0.004700 459
49、0.013930 549 0.182100 639 0.020750 729 0.001795 819 0.000000370 0.004100 460 0.013760 550 0.141700 640 0.020170 730 0.001100 820 0.000000371 0.003785 461 0.013470 551 0.113500 641 0.019920 731 0.001120 821 0.000000372 0.003470 462 0.013180 552 0.085290 642 0.019660 732 0.001140 822 0.000000373 0.003540 463 0.013470 553 0.070050 643 0.019740 733 0.001750 823 0.000000374 0.003610 464 0.013750 5
