1、Designation: E2022 11Standard Practice forCalculation of Weighting Factors for Tristimulus Integration1This standard is issued under the fixed designation E2022; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revi
2、sion. 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 cus
3、tom 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 practic
4、e 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 stat
5、ed 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 and
6、 health practices and determine the applica-bility of regulatory limitations prior to its use.2. Referenced Documents2.1 ASTM Standards:2E284 Terminology of AppearanceE308 Practice for Computing the Colors of Objects byUsing the CIE SystemE2729 Practice for Rectification of SpectrophotometricBandpas
7、s Differences2.2 CIE Standard:CIE Standard S 002 Colorimetric Observers33. Terminology3.1 DefinitionsAppearance terms in this practice are inaccordance with Terminology E284.3.2 Definitions of Terms Specific to This Standard:3.2.1 illuminant, nreal or ideal radiant flux, specified byits spectral dis
8、tribution 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 distribution of that light.3.2.2.1 DiscussionA source is an emitter of visible radia-tion. An illuminant is a tabl
9、e 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. Illuminant D65 is astandard illuminant that represents average north sky daylightbut has no representative
10、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 wavelength.4. Summary of Practice4.1 CIE color-matching functions are standardized at 1-nmwavelength intervals.
11、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 spectral data are missing, but needed.4.2 Lagrange interpolating coefficients are calculated for themissing wa
12、velengths. 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 multiplied into the CIE 1-nm color-matchingdata, and into the source spectral power distribution. Eachseparate term
13、 of this multiplication is collected into a valueassociated with a measured spectral wavelength, thus formingweighting factors for tristimulus integration.5. Significance and Use5.1 This practice is intended to provide a method that willyield uniformity of calculations used in making, matching, orco
14、ntrolling colors of objects. This uniformity is 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 responsibil
15、ity of Subcommittee E12.04 on Color andAppearance Analysis.Current edition approved June 1, 2011. Published June 2011. Originallyapproved in 1999. Last previous edition approved in 2008 as E2022 - 08. DOI:10.1520/E2022-11.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontac
16、t 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, TLA-Lighting Consultants, Inc
17、., 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.obtain the weighting factors for common illuminant-observercombinations contained in Practice E308.5.2 This practice should be util
18、ized 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 measurement interval isequal to the spectral bandwidth integral whe
19、n 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 should be quadratic (three-point) inthe first and last missing
20、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)i50 ifijnr ri!rj ri!, for j 5 0,1,.n (1)where:n = degree of
21、 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 and j.6.1.2.1 Fig. 1 assist the user in selecting the values
22、 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 or constant interval, theequation reduces to the following:
23、L05r 1!r 2!r 3!6(2)L15r!r 2!r 3!2(3)L25r 1!r!r 3!2(4)4Hildebrand, F. B., Introduction to Numerical Analysis, Second Edition, 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.FI
24、G. 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, C) the Last Measurement Interval Being InterpolatedE2022 112L35r 1!r 2!r!6(5)for the cubic case, and toL05r 1!r 2!2(
25、6)L15r!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 10-nm or 20-nm intervals, and the interpolatedinterval is
26、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 spectral data may be predicted as follows:Pl! 5(i50nLimi(9)wher
27、e: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 form:Pl! 5 L0m01 L1m11 L2m21 L3m3(10)6.3 Multiply each P(l) by
28、 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 intervals.6.3.2 Doing so results in the following equation:S
29、l!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 CIE 1-nm data from CIE Standard S 002,Colorimetric Observer
30、s. Doing so results in the followingequation:TABLE 1 The Lagrange Quadratic Interpolation CoefficientsApplicable to the First and Last Missing Interval 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
31、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 Interpolation CoefficientsApplicable to the Interior Missing Intervals for Calculation of10-nm Weighting Factors for Tristimulus IntegrationIndex of
32、 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 0.5625 .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 L
33、agrange Quadratic Interpolating CoefficientsApplicable to the First and Last Missing Interval for Calculationof 20-nm Weighting Factors for Tristimulus 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 0.656
34、25 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 0.1237512 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
35、.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 Calculation of20-nm Weighting Factors for Tristimulus IntegrationIndex of MissingWavelength L0L1L2L31 0.0154375 0.9725625 0.0511875 0.0
36、0831252 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 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
37、.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 0.7239375 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
38、.0244375 0.1595625 0.9041875 0.039312518 0.016500 0.104500 0.940500 0.02850019 0.0083125 0.0511875 0.9725625 0.0154375E2022 113x 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-matching function atwavelength l, and the calculations are c
39、arried 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 single coefficient. Thefourth factor, mi, in each of the
40、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 set ofweighting factors that then will contain one value f
41、or 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 thefollowing normalizing coefficient:k 5100(Sl!y l!(14)where:k = t
42、he 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 Beginning in January of 2010, rectification of bandpassdifferences is no longer
43、 accomplished by building the correc-tion factors into a weight set for tristimulus integration. This isbecause to do so fails to correct the spectrum itself and correctsonly the tristimulus values. Bandpass rectification is now underthe jurisdiction of Practice E2729.7. Precision7.1 The precision o
44、f 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 Lagrange coefficients are precise toseven digits.8. Keywords8.1 color-matching functions; illuminant; illuminant
45、-observer weights; source; tristimulus weighting factorsAPPENDIXES(Nonmandatory Information)X1. EXAMPLE OF THE CALCULATIONSX1.1 Table X1.1 gives the spectral power distribution(SPD) of a typical 3-band fluorescent lamp with a correlatedcolor temperature of about 3000K. The first step is to multiplye
46、ach 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 830 nm. Table X1.3 shows a typical interpolationof a measured reflectance curve from a 10-nm reported interva
47、lto 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 in several different intervals, can be combinedwith the illuminant-color matching function product to form a
48、single 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 E308.E2022 114TABLE X1.1 Spectral Power Distribution of Typical 3-Band Fluorescent Lamp wi
49、th 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 0.277600 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 0.00000
