I consider numerical methods for evaluating the tristimulus values of Pλ, an arbitrary spectral power distribution. Various classical quadrature rules are discussed, and numerically stable algorithms for their construction are presented. I introduce a new quadrature rule developed specifically for evaluating tristimulus values. The method involves the simultaneous generation of three quadrature rules sharing a common set of nodes that is optimal in a sense much like the optimality of the Gauss rules.
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Gauss Rules for CIE XYZ Matching Functions Biased with CIE Standard Illuminant A
X
Y
Z
Order
λ
Weight
λ
Weight
λ
Weight
3
453.6
0.065261
0.503874
0.177165
0.431958
0.127213
585.4
0.730193
0.573064
0.663361
0.474276
0.213467
650.5
0.303221
0.644716
0.159474
0.540278
0.015236
4
440.7
0.043219
0.478896
0.047905
0.421835
0.055088
553.5
0.297236
0.544182
0.489976
0.457523
0.225467
614.9
0.687473
0.606651
0.428121
0.502893
0.073651
678.7
0.070747
0.674579
0.033998
0.579973
0.001709
5
433.4
0.030186
0.459534
0.013939
0.410887
0.016586
500.2
0.063214
0.523022
0.275672
0.443499
0.169740
584.1
0.607578
0.578274
0.527233
0.479739
0.150248
638.4
0.385966
0.635738
0.178715
0.528279
0.018902
708.2
0.011731
0.707747
0.004441
0.601692
0.000441
6
423.6
0.014005
0.445510
0.004843
0.399538
0.003869
462.8
0.042873
0.504633
0.126493
0.432846
0.103662
562.4
0.326747
0.554856
0.461960
0.464566
0.186371
613.3
0.596717
0.607336
0.354234
0.503312
0.058073
664.8
0.117059
0.662150
0.051969
0.555295
0.003767
741.7
0.001273
0.740704
0.000500
0.613557
0.000175
Table 2
Gauss Rules for CIE XYZ Matching Functions with No Illuminant Bias
X
Y
Z
Order
λ
Weight
λ
Weight
λ
Weight
3
441.1
0.158710
0.487025
0.161582
0.424071
0.319491
573.6
0.555942
0.559653
0.672840
0.463665
0.628227
640.9
0.285348
0.633785
0.165577
0.521996
0.052282
4
433.0
0.119848
0.462179
0.046978
0.412796
0.119811
518.6
0.174684
0.531409
0.487040
0.447986
0.645877
600.2
0.614486
0.594567
0.427747
0.490538
0.229805
664.3
0.090982
0.662693
0.038235
0.562460
0.004507
5
422.6
0.061798
0.445422
0.016657
0.399833
0.029414
465.3
0.120348
0.509732
0.264278
0.434952
0.465719
572.7
0.453971
0.565290
0.529288
0.469779
0.450593
628.6
0.350250
0.624277
0.184228
0.516066
0.053619
695.3
0.013634
0.693947
0.005550
0.592562
0.000655
6
412.7
0.025929
0.432699
0.006338
0.388772
0.007640
449.7
0.131914
0.489058
0.114729
0.425429
0.272119
546.2
0.207412
0.541518
0.465343
0.455615
0.537070
601.2
0.504725
0.594973
0.355536
0.491915
0.172771
653.0
0.128419
0.650571
0.057452
0.541050
0.010189
727.3
0.001600
0.726910
0.000602
0.607877
0.000210
Table 3
Shared-Node Rules for the XYZ Matching Functions with No Illuminant Bias
Weights
Order
λ
X
Y
Z
3
445.4
0.154337
0.035122
0.892225
540.2
0.257263
0.668486
0.123094
618.7
0.588400
0.296392
−0.015320
6
424.6
0.060565
0.001520
0.292508
460.7
0.107723
0.031086
0.629756
517.5
0.017953
0.302792
0.082737
567.8
0.367156
0.462950
−0.006311
618.5
0.399579
0.184717
0.001433
669.2
0.047024
0.016935
−0.000123
9
412.8
0.014184
0.000401
0.067868
437.5
0.085663
0.004702
0.427231
467.6
0.066322
0.027289
0.421487
505.9
0.001839
0.148927
0.079127
542.0
0.107973
0.330735
0.003143
578.2
0.298333
0.295194
0.001145
613.2
0.315996
0.151496
−0.000015
649.5
0.100473
0.037923
0.000016
685.7
0.009218
0.003334
−0.000001
Table 4
Shared-Node Rules for the XYZ Matching Functions with No Illuminant Bias
Weights
Order
λ
X
Y
Z
12
405.9
0.003545
0.000097
0.016898
423.4
0.031729
0.001038
0.153443
442.8
0.070672
0.005311
0.359626
465.5
0.052995
0.016731
0.325076
490.4
0.007371
0.053570
0.116537
518.2
0.013876
0.175890
0.024759
547.3
0.111094
0.279217
0.003038
577.5
0.247813
0.250495
0.000481
607.2
0.284907
0.149095
0.000135
636.9
0.142553
0.056324
0.000006
666.7
0.029961
0.010976
0.000000
692.2
0.003483
0.001257
0.000000
Table 5
Shared-Node Rules for the XYZ Matching Functions with No Illuminant Bias
Weights
Order
λ
X
Y
Z
15
391.4
0.000013
0.000000
0.000061
406.6
0.003830
0.000105
0.018258
422.9
0.028008
0.000900
0.135400
440.4
0.062538
0.004172
0.315041
460.7
0.055098
0.012089
0.319006
481.7
0.016841
0.029311
0.152055
503.9
0.000342
0.083464
0.046273
527.2
0.029561
0.185479
0.011581
552.0
0.110422
0.235785
0.001743
577.1
0.203962
0.209324
0.000405
602.0
0.248496
0.141796
0.000158
627.2
0.168892
0.070770
0.000019
652.8
0.058476
0.021897
0.000000
677.1
0.011701
0.004250
0.000000
695.1
0.001819
0.000656
0.000000
Table 6
Shared-Node Rules for the XYZ Matching Functions with No Illuminant Bias
Weights
Order
λ
X
Y
Z
18
401.8
0.000699
0.000019
0.003325
408.5
0.002870
0.000080
0.013695
418.8
0.013167
0.000382
0.063340
431.6
0.039731
0.001743
0.195009
447.2
0.055318
0.005309
0.287454
465.1
0.042716
0.012941
0.261283
484.5
0.012002
0.031483
0.122632
505.3
0.000429
0.080684
0.039522
526.0
0.023099
0.157015
0.010674
547.2
0.079467
0.199378
0.002298
568.9
0.151325
0.195466
0.000454
590.7
0.212110
0.153464
0.000242
612.6
0.198881
0.096310
0.000065
634.1
0.111990
0.044921
0.000007
654.8
0.040953
0.015261
0.000000
673.2
0.011358
0.004139
0.000000
687.0
0.003115
0.001125
0.000000
697.6
0.000774
0.000279
0.000000
Table 7
Shared-Node Rules for the XYZ Matching Functions with No Illuminant Bias
Weights
Order
λ
X
Y
Z
21
401.7
0.000675
0.000019
0.003210
408.6
0.003124
0.000086
0.014912
419.1
0.013096
0.000385
0.063017
430.9
0.034347
0.001453
0.168337
444.2
0.046671
0.003848
0.238865
459.1
0.042222
0.008154
0.240366
474.6
0.022126
0.016571
0.160981
491.3
0.004291
0.035610
0.069958
507.6
0.000617
0.063483
0.025332
522.1
0.011839
0.106794
0.009947
539.3
0.049471
0.167106
0.003787
558.9
0.108547
0.187389
0.000793
578.9
0.167095
0.162975
0.000301
598.3
0.188323
0.115495
0.000149
616.4
0.146667
0.067719
0.000036
631.5
0.074237
0.030302
0.000006
644.2
0.049103
0.018814
0.000001
659.8
0.025734
0.009510
0.000000
675.9
0.008616
0.003134
0.000000
689.4
0.002548
0.000920
0.000000
697.9
0.000651
0.000235
0.000000
Table 8
Means and Standard Deviations of CIE L*u*v* Color Differences between Tristimulus Values Computed with 1-nm Riemann Summation and Shared-Node Rulesa
Nickerson
Macbeth
Order
μ
σ
μ
σ
3
6.0421
4.3969
7.5159
6.3821
6
2.0323
1.7020
1.8732
1.6587
9
0.6693
0.7503
0.5741
0.5146
12
0.2937
0.2782
0.3417
0.3404
15
0.1568
0.1033
0.1939
0.1345
18
0.0819
0.0571
0.1599
0.0905
21
0.0615
0.0429
0.1183
0.0825
Data are for 462 spectral reflectances of Munsell chips collected by Nickerson and for 24 spectral reflectances from the Macbeth Color Checker: μ, mean; σ, standard deviation.
Tables (8)
Table 1
Gauss Rules for CIE XYZ Matching Functions Biased with CIE Standard Illuminant A
X
Y
Z
Order
λ
Weight
λ
Weight
λ
Weight
3
453.6
0.065261
0.503874
0.177165
0.431958
0.127213
585.4
0.730193
0.573064
0.663361
0.474276
0.213467
650.5
0.303221
0.644716
0.159474
0.540278
0.015236
4
440.7
0.043219
0.478896
0.047905
0.421835
0.055088
553.5
0.297236
0.544182
0.489976
0.457523
0.225467
614.9
0.687473
0.606651
0.428121
0.502893
0.073651
678.7
0.070747
0.674579
0.033998
0.579973
0.001709
5
433.4
0.030186
0.459534
0.013939
0.410887
0.016586
500.2
0.063214
0.523022
0.275672
0.443499
0.169740
584.1
0.607578
0.578274
0.527233
0.479739
0.150248
638.4
0.385966
0.635738
0.178715
0.528279
0.018902
708.2
0.011731
0.707747
0.004441
0.601692
0.000441
6
423.6
0.014005
0.445510
0.004843
0.399538
0.003869
462.8
0.042873
0.504633
0.126493
0.432846
0.103662
562.4
0.326747
0.554856
0.461960
0.464566
0.186371
613.3
0.596717
0.607336
0.354234
0.503312
0.058073
664.8
0.117059
0.662150
0.051969
0.555295
0.003767
741.7
0.001273
0.740704
0.000500
0.613557
0.000175
Table 2
Gauss Rules for CIE XYZ Matching Functions with No Illuminant Bias
X
Y
Z
Order
λ
Weight
λ
Weight
λ
Weight
3
441.1
0.158710
0.487025
0.161582
0.424071
0.319491
573.6
0.555942
0.559653
0.672840
0.463665
0.628227
640.9
0.285348
0.633785
0.165577
0.521996
0.052282
4
433.0
0.119848
0.462179
0.046978
0.412796
0.119811
518.6
0.174684
0.531409
0.487040
0.447986
0.645877
600.2
0.614486
0.594567
0.427747
0.490538
0.229805
664.3
0.090982
0.662693
0.038235
0.562460
0.004507
5
422.6
0.061798
0.445422
0.016657
0.399833
0.029414
465.3
0.120348
0.509732
0.264278
0.434952
0.465719
572.7
0.453971
0.565290
0.529288
0.469779
0.450593
628.6
0.350250
0.624277
0.184228
0.516066
0.053619
695.3
0.013634
0.693947
0.005550
0.592562
0.000655
6
412.7
0.025929
0.432699
0.006338
0.388772
0.007640
449.7
0.131914
0.489058
0.114729
0.425429
0.272119
546.2
0.207412
0.541518
0.465343
0.455615
0.537070
601.2
0.504725
0.594973
0.355536
0.491915
0.172771
653.0
0.128419
0.650571
0.057452
0.541050
0.010189
727.3
0.001600
0.726910
0.000602
0.607877
0.000210
Table 3
Shared-Node Rules for the XYZ Matching Functions with No Illuminant Bias
Weights
Order
λ
X
Y
Z
3
445.4
0.154337
0.035122
0.892225
540.2
0.257263
0.668486
0.123094
618.7
0.588400
0.296392
−0.015320
6
424.6
0.060565
0.001520
0.292508
460.7
0.107723
0.031086
0.629756
517.5
0.017953
0.302792
0.082737
567.8
0.367156
0.462950
−0.006311
618.5
0.399579
0.184717
0.001433
669.2
0.047024
0.016935
−0.000123
9
412.8
0.014184
0.000401
0.067868
437.5
0.085663
0.004702
0.427231
467.6
0.066322
0.027289
0.421487
505.9
0.001839
0.148927
0.079127
542.0
0.107973
0.330735
0.003143
578.2
0.298333
0.295194
0.001145
613.2
0.315996
0.151496
−0.000015
649.5
0.100473
0.037923
0.000016
685.7
0.009218
0.003334
−0.000001
Table 4
Shared-Node Rules for the XYZ Matching Functions with No Illuminant Bias
Weights
Order
λ
X
Y
Z
12
405.9
0.003545
0.000097
0.016898
423.4
0.031729
0.001038
0.153443
442.8
0.070672
0.005311
0.359626
465.5
0.052995
0.016731
0.325076
490.4
0.007371
0.053570
0.116537
518.2
0.013876
0.175890
0.024759
547.3
0.111094
0.279217
0.003038
577.5
0.247813
0.250495
0.000481
607.2
0.284907
0.149095
0.000135
636.9
0.142553
0.056324
0.000006
666.7
0.029961
0.010976
0.000000
692.2
0.003483
0.001257
0.000000
Table 5
Shared-Node Rules for the XYZ Matching Functions with No Illuminant Bias
Weights
Order
λ
X
Y
Z
15
391.4
0.000013
0.000000
0.000061
406.6
0.003830
0.000105
0.018258
422.9
0.028008
0.000900
0.135400
440.4
0.062538
0.004172
0.315041
460.7
0.055098
0.012089
0.319006
481.7
0.016841
0.029311
0.152055
503.9
0.000342
0.083464
0.046273
527.2
0.029561
0.185479
0.011581
552.0
0.110422
0.235785
0.001743
577.1
0.203962
0.209324
0.000405
602.0
0.248496
0.141796
0.000158
627.2
0.168892
0.070770
0.000019
652.8
0.058476
0.021897
0.000000
677.1
0.011701
0.004250
0.000000
695.1
0.001819
0.000656
0.000000
Table 6
Shared-Node Rules for the XYZ Matching Functions with No Illuminant Bias
Weights
Order
λ
X
Y
Z
18
401.8
0.000699
0.000019
0.003325
408.5
0.002870
0.000080
0.013695
418.8
0.013167
0.000382
0.063340
431.6
0.039731
0.001743
0.195009
447.2
0.055318
0.005309
0.287454
465.1
0.042716
0.012941
0.261283
484.5
0.012002
0.031483
0.122632
505.3
0.000429
0.080684
0.039522
526.0
0.023099
0.157015
0.010674
547.2
0.079467
0.199378
0.002298
568.9
0.151325
0.195466
0.000454
590.7
0.212110
0.153464
0.000242
612.6
0.198881
0.096310
0.000065
634.1
0.111990
0.044921
0.000007
654.8
0.040953
0.015261
0.000000
673.2
0.011358
0.004139
0.000000
687.0
0.003115
0.001125
0.000000
697.6
0.000774
0.000279
0.000000
Table 7
Shared-Node Rules for the XYZ Matching Functions with No Illuminant Bias
Weights
Order
λ
X
Y
Z
21
401.7
0.000675
0.000019
0.003210
408.6
0.003124
0.000086
0.014912
419.1
0.013096
0.000385
0.063017
430.9
0.034347
0.001453
0.168337
444.2
0.046671
0.003848
0.238865
459.1
0.042222
0.008154
0.240366
474.6
0.022126
0.016571
0.160981
491.3
0.004291
0.035610
0.069958
507.6
0.000617
0.063483
0.025332
522.1
0.011839
0.106794
0.009947
539.3
0.049471
0.167106
0.003787
558.9
0.108547
0.187389
0.000793
578.9
0.167095
0.162975
0.000301
598.3
0.188323
0.115495
0.000149
616.4
0.146667
0.067719
0.000036
631.5
0.074237
0.030302
0.000006
644.2
0.049103
0.018814
0.000001
659.8
0.025734
0.009510
0.000000
675.9
0.008616
0.003134
0.000000
689.4
0.002548
0.000920
0.000000
697.9
0.000651
0.000235
0.000000
Table 8
Means and Standard Deviations of CIE L*u*v* Color Differences between Tristimulus Values Computed with 1-nm Riemann Summation and Shared-Node Rulesa
Nickerson
Macbeth
Order
μ
σ
μ
σ
3
6.0421
4.3969
7.5159
6.3821
6
2.0323
1.7020
1.8732
1.6587
9
0.6693
0.7503
0.5741
0.5146
12
0.2937
0.2782
0.3417
0.3404
15
0.1568
0.1033
0.1939
0.1345
18
0.0819
0.0571
0.1599
0.0905
21
0.0615
0.0429
0.1183
0.0825
Data are for 462 spectral reflectances of Munsell chips collected by Nickerson and for 24 spectral reflectances from the Macbeth Color Checker: μ, mean; σ, standard deviation.