1、Designation: E2022 16Standard 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 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.3 This standard does not purport to address all of thesafety concerns, if any, a
4、ssociated 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 ASTM Standards:2E284 Terminology of AppearanceE308 Practice for Co
5、mputing the Colors of Objects by Usingthe CIE SystemE2729 Practice for Rectification of SpectrophotometricBandpass Differences2.2 CIE Standard:CIE Standard S 002 Colorimetric Observers33. Terminology3.1 DefinitionsAppearance terms in this practice are inaccordance with Terminology E284.3.2 Definitio
6、ns 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 affect their perceived colors.3.2.2 source, nan object that produces light or otherradiant flux, or the spectral power d
7、istribution 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 distribu
8、tion. Illuminant D65 is astandard illuminant that represents average north sky daylightbut has no representative source.3.2.3 spectral power distribution, SPD, S(),nspecification of an illuminant by the spectral composition ofa radiometric quantity, such as radiance or radiant flux, as afunction of
9、wavelength.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; therefore, intermediate1-nm interval spectral
10、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 multiplied int
11、o 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.5. Significance and Use5.1 This practice is
12、intended to provide a method that willyield uniformity of 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 wei
13、ghting factors for common illuminant-observercombinations contained in Practice E308.5.2 This practice should be utilized by persons desiring tocalculate a set of weighting factors for tristimulus integrationwho have custom source, or illuminant spectral powerdistributions, or custom observer respon
14、se functions.1This practice is under the jurisdiction of ASTM Committee E12 on Color andAppearance and is the direct responsibility of Subcommittee E12.04 on Color andAppearance Analysis.Current edition approved Aug. 1, 2016. Published August 2016. Originallyapproved in 1999. Last previous edition a
15、pproved in 2011 as E2022 11. DOI:10.1520/E2022-16.2For referenced ASTM standards, visit the ASTM website, 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
16、from CIE (International Commission on Illumination), http:/www.cie.co.at or http:/.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States16. Procedure6.1 Calculation of Lagrange CoeffcientsObtain bycalculation, or by table look-up, a set of
17、Lagrange interpolatingcoefficients 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 CoeffcientsLagrang
18、e coeffi-cients may be calculated for any interval and number ofmissing wavelengths by Eq 1:Ljr! 5)i50 ifijnr 2 ri!rj2 ri!, for j 5 0,1,n (1)where:n = degree of coefficients beingcalculated,5i and j = indices denoting the locationalong the abscissa, = repetitive multiplication ofthe terms in the num
19、eratorand the denominator, andindices of the 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, regu
20、lar or irregular.6.1.3 10-nm Lagrange CoeffcientsWhere the measuredspectral data have a regular or constant interval, the equationreduces to the following:L05r 2 1!r 2 2!r 2 3!26(2)L15r!r 2 2!r 2 3!2(3)L25r 2 1!r!r 2 3!22(4)L35r 2 1!r 2 2!r!6(5)for the cubic case, and toL05r 2 1!r 2 2!2(6)L15r!r 2 2
21、!21(7)L25r 2 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 intervals, and the interpolated interval isregular with res
22、pect to the measurement interval, presumably1-nm.6.1.4 Tables 1 and 2 provide both quadratic and cubicLagrange coefficients for 10-nm intervals.6.2 With the Lagrange coefficients provided, the intermedi-ate missing spectral data may be predicted as follows:P! 5(i50nLimi(9)where:P = the value being i
23、nterpolated at interval ,L = the Lagrange coefficients, andm = the measured abridged spectral values.4Hildebrand, F. B., Introduction to Numerical Analysis, Second Edition, Dover,New York, 1974, Chapter 3.5Fairman, H. S., “The Calculation of Weight Factors for TristimulusIntegration,” Color Research
24、 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 Measurement Inter-val; B) the Intermediate Measurement Intervals; and, C) the Last Measurement Interval Being InterpolatedTABLE 1 The Lagran
25、ge 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 0.510 0.1054 0.480 0.640 0.1205 0.375 0.750 0.1256 0.280 0.840
26、0.1207 0.195 0.910 0.1058 0.120 0.960 0.0809 0.055 0.990 0.045E2022 162Because the measured spectral values are as yet unknown, itmay be best to consider this equation in its expanded form:P! 5 L0m01L1m11L2m21L3m3(10)6.3 Multiply each P() by the 1-nm interval relative spectralpower of the source or
27、illuminant being considered.6.3.1 It may be necessary to interpolate missing values ofthe source spectral power distribution S(), if the source hasbeen measured at other than 1-nm intervals.6.3.2 Doing so results in the following equation:S!P! 5 S!L0m01S!L1m11S!L2m21S!L3m3(11)6.4 Multiply the weight
28、ed 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 Observers. Doing so results in the followingequation:x!S!P! 5 x!S!L0#m01x!S!L
29、1#m11x!S!L2#m21x!S!L3#m3(12)where:x() = the value of the CIE X color-matching function atwavelength , and the calculations are carried out foreach of the three CIE color-matching functions, x(),y(), and z().6.5 In the four terms on the right-hand side of this equation,the numerical values of the thr
30、ee factors in the brackets areknown and should be multiplied into 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 co
31、efficient appli-cable to that wavelength. This results in a single 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:x0!S0!L0!1x1!S1!L0!1 #m05 wt0m0(13)6
32、.7 Normalize the weighting factors by calculating thefollowing normalizing coefficient:k 5100(S!y!(14)where:k = the normalizing coefficient,S() = the power in the 1-nm spectrum, andy() = the CIE Y color-matching function.6.8 Multiply the weighting factors by k to normalize the setto Y = 100 for the
33、perfect reflecting diffuser.6.9 Beginning in January of 2010, rectification of bandpassdifferences is no longer 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 tristimulu
34、s values. Bandpass rectification is now underthe jurisdiction of Practice E2729.7. 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 L
35、agrange coefficients are precise toseven digits.8. Keywords8.1 color-matching functions; illuminant; illuminant-observer weights; source; tristimulus weighting factorsTABLE 2 The Lagrange Cubic Interpolation CoefficientsApplicable to the Interior Missing Intervals for Calculation of10-nm Weighting F
36、actors 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 0.5625 0.06256 0.0560 0.4480 0.6720 0.06407 0.0455 0.3315 0.7735 0.05958 0.0320 0.2160 0.8640 0.048
37、09 0.0165 0.1045 0.9405 0.0285E2022 163APPENDIXES(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 correlated colortemperature of about 3000K. The first step is to multiply eachvalue of the
38、SPD by the appropriate CIE color matchingfunction (y in this case), wavelength by wavelength, which isshown inTable 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 intervalto the 1-nm inter
39、val 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 asingle weight at a
40、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 164TABLE X1.1 Spectral Power Distribution of Typical 3-Band Fluorescent Lamp with Correlated Color
41、 Temperature of 3000 K (1-nmmeasurement interval) SPD SPD SPD SPD SPD 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 0.00
42、0000363 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 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
43、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 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
44、.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 554 0.054810 644 0.0198
45、10 734 0.002360 824 0.000000375 0.003615 465 0.014000 555 0.046030 645 0.019550 735 0.002190 825 0.000000376 0.003620 466 0.014250 556 0.037250 646 0.019280 736 0.002020 826 0.000000377 0.004210 467 0.013810 557 0.034310 647 0.019080 737 0.003930 827 0.000000378 0.004800 468 0.013370 558 0.031360 64
46、8 0.018880 738 0.005840 828 0.000000379 0.005170 469 0.012870 559 0.030480 649 0.030460 739 0.003355 829 0.000000380 0.005540 470 0.012370 560 0.029590 650 0.042050 740 0.000870 830 0.000000381 0.005240 471 0.012640 561 0.029650 651 0.034870 741 0.002235382 0.004940 472 0.012900 562 0.029700 652 0.0
47、27690 742 0.003600383 0.004615 473 0.012640 563 0.029530 653 0.024990 743 0.002500384 0.004290 474 0.012380 564 0.029360 654 0.022290 744 0.001400385 0.003750 475 0.011680 565 0.029200 655 0.020120 745 0.002155386 0.003210 476 0.010970 566 0.029040 656 0.017950 746 0.002910387 0.003050 477 0.011050
48、567 0.029500 657 0.019130 747 0.002970388 0.002890 478 0.011130 568 0.029960 658 0.020320 748 0.003030389 0.002980 479 0.012680 569 0.029480 659 0.017400 749 0.003615390 0.003070 480 0.014240 570 0.029000 660 0.014470 750 0.004200391 0.002795 481 0.019080 571 0.029140 661 0.020750 751 0.003470392 0.
49、002520 482 0.023910 572 0.029280 662 0.027030 752 0.002740393 0.002395 483 0.035600 573 0.029390 663 0.022910 753 0.002225394 0.002270 484 0.047290 574 0.029500 664 0.018790 754 0.001710395 0.002285 485 0.064030 575 0.040510 665 0.015270 755 0.000855396 0.002300 486 0.080770 576 0.051530 666 0.011740 756 0.000000397 0.002420 487 0.082540 577 0.060840 667 0.012890 757 0.000310398 0.002540 488 0.084310 578
