Equations are derived to fit the cone sensitivity values of Smith and Pokorny, which are based mainly upon Judd’s modified color matching functions and the spectral sensitivities of normal and dichromatic observers. The equations, which fit Smith and Pokorny’s values to within 1 to 3%, are smooth and differentiable. Derivatives of cone sensitivities for an equal luminance spectrum, which are of particular importance for models of chromatic discrimination, have been calculated and are shown graphically.
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For SR and SB, these are coefficients of frequency defined as ν = 2.998/λ, with λ in μm. For SG, these are coeffients of wavelength in μm.
TABLE II
Values for B, G, and R functions compared to SR, SG, and SB as given by Smith and Pokorny. Differences between the two are shown at 10-nm intervals, from which the percentages of differences have been calculated.
Wavelength (nm)
SB
B
Diff.
Pct.
SG
G
Diff.
Pct.
SR
R
Diff.
Pct.
SR + SG
R + G
Diff.
Pct.
400
0.2799
0.2793
0.0006
00.21
0.0018
0.0017
0.0001
05.56
0.0027
0.0027
0.0000
00.00
0.0045
0.0044
0.0001
02.22
10
0.5835
0.5922
−0.0087
01.49
0.0038
0.0040
−0.0002
05.26
0.0055
0.0057
−0.0002
03.64
0.0093
0.0098
−0.0005
05.38
20
1.0622
1.0357
0.0265
02.49
0.0074
0.0075
−0.0001
01.35
0.0101
0.0100
0.0001
00.99
0.0175
0.0175
−0.0000
00.00
30
1.4526
1.4606
−0.0080
00.55
0.0124
0.0121
0.0003
02.42
0.0149
0.0148
0.0001
00.67
0.0273
0.0269
0.0004
01.47
40
1.6064
1.6430
−0.0376
02.28
0.0187
0.0178
0.0009
04.81
0.0192
0.0190
0.0002
01.04
0.0379
0.0368
0.0011
02.90
50
1.4717
1.5063
−0.0346
02.35
0.0249
0.0251
−0.0002
00.80
0.0219
0.0223
−0.0004
01.83
0.0468
0.0474
−0.0006
01.28
60
1.2880
1.2886
−0.0006
00.05
0.0337
0.0353
−0.0016
04.75
0.0263
0.0262
0.0001
00.38
0.0600
0.0614
−0.0014
02.33
70
1.1133
1.0500
0.0633
05.69
0.0511
0.0504
−0.0007
01.37
0.0399
0.0368
0.0031
07.77
0.0910
0.0871
0.0039
04.29
80
0.7552
0.7527
0.0025
00.33
0.0740
0.0734
0.0006
00.81
0.0650
0.0619
0.0031
04.77
0.1390
0.1353
0.0037
02.66
90
0.4461
0.4804
−0.0343
07.69
0.1047
0.1087
−0.0040
03.82
0.1032
0.1066
−0.0034
03.29
0.2080
0.2153
−0.0073
03.51
500
0.2644
0.2798
−0.0154
05.78
0.1555
0.1610
−0.0055
03.54
0.1675
0.1742
−0.0067
04.00
0.3230
0.3352
−0.0122
03.78
10
0.1541
0.1518
0.0023
01.49
0.2334
0.2315
0.0019
00.81
0.2696
0.2649
0.0047
01.74
0.5030
0.4964
0.0066
01.31
20
0.0763
0.0781
−0.0018
02.36
0.3170
0.3092
0.0078
02.46
0.3930
0.3718
0.0212
05.39
0.7100
0.6810
0.0290
04.08
30
0.0412
0.0387
0.0025
06.07
0.3694
0.3684
0.0010
00.27
0.4926
0.4804
0.0122
02.48
0.8620
0.8488
0.0132
01.53
40
0.0200
0.0187
0.0013
06.50
0.3911
0.3906
0.0005
00.13
0.5629
0.5709
−0.0080
01.42
0.9540
0.9616
−0.0076
00.80
50
0.0088
0.0089
−0.0001
01.14
0.3871
0.3860
0.0011
00.28
0.6078
0.6264
−0.0186
03.06
0.9950
1.0124
−0.0174
01.75
60
0.0039
0.0042
−0.0003
07.69
0.3621
0.3664
−0.0043
01.19
0.6329
0.6439
−0.0110
01.74
0.9950
1.0103
−0.0153
01.54
70
0.0020
0.0021
−0.0001
05.00
0.3164
0.3233
−0.0069
02.18
0.6355
0.6409
−0.0054
00.85
0.9520
0.9642
−0.0122
01.28
80
0.0016
0.0016
0.0000
00.00
0.2554
0.2573
−0.0019
00.74
0.6146
0.6274
−0.0128
02.08
0.8700
0.8847
−0.0147
01.69
90
0.0011
0.0011
0.0000
00.00
0.1872
0.1851
0.0021
01.12
0.5697
0.5877
−0.0180
03.16
0.7570
0.7728
−0.0158
02.09
600
0.0007
0.0006
0.0001
14.29
0.1247
0.1219
0.0028
02.25
0.5063
0.5155
−0.0092
01.82
0.6310
0.6374
−0.0064
01.01
10
0.0003
0.0003
0.0000
00.00
0.0759
0.0743
0.0016
02.11
0.4271
0.4214
0.0057
01.33
0.5030
0.4956
0.0074
01.47
20
0.O002
0.0002
0.0000
00.00
0.0432
0.0424
0.0008
01.85
0.3377
0.3211
0.0166
04.92
0.3810
0.3635
0.0175
04.59
30
0.0001
0.0001
0.0000
00.00
0.0229
0.0229
0.0000
00.00
0.2421
0.2287
0.0134
05.53
0.2650
0.2516
0.0134
05.06
40
0.0116
0.0118
−0.0002
01.72
0.1634
0.1527
0.0107
06.55
0.1750
0.1645
0.0105
06.00
50
0.0056
0.0059
−0.0003
05.36
0.1014
0.0958
0.0056
05.52
0.1070
0.1016
0.0054
05.05
60
0.0028
0.0028
0.0000
0
0.0582
0.0566
0.0016
02.75
0.0610
0.0595
0.0015
02.46
70
0.0013
0.0013
0.0000
0
0.0307
0.0316
−0.0009
02.93
0.0320
0.0330
−0.0010
03.13
80
0.0006
0.0006
0.0000
0
0.0164
0.0168
−0.0004
02.44
0.0170
0.0174
−0.0004
02.35
90
0.0003
0.0003
0.0000
0
0.0079
0.0084
−0.0005
06.33
0.0082
0.0087
−0.0005
06.10
700
0.0001
0.0001
0.0000
0
0.0040
0.0041
−0.0001
02.50
0.0041
0.0042
−0.0001
02.44
AVE% DIFF = 3.06%
AVE% DIFF = 1.84%
AVE% DIFF = 3.00%
AVE% DIFF = 2.76%
S.D. = 3.59
S.D. = 1.80
S.D. = 2.00
S.D. = 1.62
TABLE III
Values of n and the coefficients for Equations (9)–(13), to be used in the calculation of lens and macular pigment density.
For SR and SB, these are coefficients of frequency defined as ν = 2.998/λ, with λ in μm. For SG, these are coeffients of wavelength in μm.
TABLE II
Values for B, G, and R functions compared to SR, SG, and SB as given by Smith and Pokorny. Differences between the two are shown at 10-nm intervals, from which the percentages of differences have been calculated.
Wavelength (nm)
SB
B
Diff.
Pct.
SG
G
Diff.
Pct.
SR
R
Diff.
Pct.
SR + SG
R + G
Diff.
Pct.
400
0.2799
0.2793
0.0006
00.21
0.0018
0.0017
0.0001
05.56
0.0027
0.0027
0.0000
00.00
0.0045
0.0044
0.0001
02.22
10
0.5835
0.5922
−0.0087
01.49
0.0038
0.0040
−0.0002
05.26
0.0055
0.0057
−0.0002
03.64
0.0093
0.0098
−0.0005
05.38
20
1.0622
1.0357
0.0265
02.49
0.0074
0.0075
−0.0001
01.35
0.0101
0.0100
0.0001
00.99
0.0175
0.0175
−0.0000
00.00
30
1.4526
1.4606
−0.0080
00.55
0.0124
0.0121
0.0003
02.42
0.0149
0.0148
0.0001
00.67
0.0273
0.0269
0.0004
01.47
40
1.6064
1.6430
−0.0376
02.28
0.0187
0.0178
0.0009
04.81
0.0192
0.0190
0.0002
01.04
0.0379
0.0368
0.0011
02.90
50
1.4717
1.5063
−0.0346
02.35
0.0249
0.0251
−0.0002
00.80
0.0219
0.0223
−0.0004
01.83
0.0468
0.0474
−0.0006
01.28
60
1.2880
1.2886
−0.0006
00.05
0.0337
0.0353
−0.0016
04.75
0.0263
0.0262
0.0001
00.38
0.0600
0.0614
−0.0014
02.33
70
1.1133
1.0500
0.0633
05.69
0.0511
0.0504
−0.0007
01.37
0.0399
0.0368
0.0031
07.77
0.0910
0.0871
0.0039
04.29
80
0.7552
0.7527
0.0025
00.33
0.0740
0.0734
0.0006
00.81
0.0650
0.0619
0.0031
04.77
0.1390
0.1353
0.0037
02.66
90
0.4461
0.4804
−0.0343
07.69
0.1047
0.1087
−0.0040
03.82
0.1032
0.1066
−0.0034
03.29
0.2080
0.2153
−0.0073
03.51
500
0.2644
0.2798
−0.0154
05.78
0.1555
0.1610
−0.0055
03.54
0.1675
0.1742
−0.0067
04.00
0.3230
0.3352
−0.0122
03.78
10
0.1541
0.1518
0.0023
01.49
0.2334
0.2315
0.0019
00.81
0.2696
0.2649
0.0047
01.74
0.5030
0.4964
0.0066
01.31
20
0.0763
0.0781
−0.0018
02.36
0.3170
0.3092
0.0078
02.46
0.3930
0.3718
0.0212
05.39
0.7100
0.6810
0.0290
04.08
30
0.0412
0.0387
0.0025
06.07
0.3694
0.3684
0.0010
00.27
0.4926
0.4804
0.0122
02.48
0.8620
0.8488
0.0132
01.53
40
0.0200
0.0187
0.0013
06.50
0.3911
0.3906
0.0005
00.13
0.5629
0.5709
−0.0080
01.42
0.9540
0.9616
−0.0076
00.80
50
0.0088
0.0089
−0.0001
01.14
0.3871
0.3860
0.0011
00.28
0.6078
0.6264
−0.0186
03.06
0.9950
1.0124
−0.0174
01.75
60
0.0039
0.0042
−0.0003
07.69
0.3621
0.3664
−0.0043
01.19
0.6329
0.6439
−0.0110
01.74
0.9950
1.0103
−0.0153
01.54
70
0.0020
0.0021
−0.0001
05.00
0.3164
0.3233
−0.0069
02.18
0.6355
0.6409
−0.0054
00.85
0.9520
0.9642
−0.0122
01.28
80
0.0016
0.0016
0.0000
00.00
0.2554
0.2573
−0.0019
00.74
0.6146
0.6274
−0.0128
02.08
0.8700
0.8847
−0.0147
01.69
90
0.0011
0.0011
0.0000
00.00
0.1872
0.1851
0.0021
01.12
0.5697
0.5877
−0.0180
03.16
0.7570
0.7728
−0.0158
02.09
600
0.0007
0.0006
0.0001
14.29
0.1247
0.1219
0.0028
02.25
0.5063
0.5155
−0.0092
01.82
0.6310
0.6374
−0.0064
01.01
10
0.0003
0.0003
0.0000
00.00
0.0759
0.0743
0.0016
02.11
0.4271
0.4214
0.0057
01.33
0.5030
0.4956
0.0074
01.47
20
0.O002
0.0002
0.0000
00.00
0.0432
0.0424
0.0008
01.85
0.3377
0.3211
0.0166
04.92
0.3810
0.3635
0.0175
04.59
30
0.0001
0.0001
0.0000
00.00
0.0229
0.0229
0.0000
00.00
0.2421
0.2287
0.0134
05.53
0.2650
0.2516
0.0134
05.06
40
0.0116
0.0118
−0.0002
01.72
0.1634
0.1527
0.0107
06.55
0.1750
0.1645
0.0105
06.00
50
0.0056
0.0059
−0.0003
05.36
0.1014
0.0958
0.0056
05.52
0.1070
0.1016
0.0054
05.05
60
0.0028
0.0028
0.0000
0
0.0582
0.0566
0.0016
02.75
0.0610
0.0595
0.0015
02.46
70
0.0013
0.0013
0.0000
0
0.0307
0.0316
−0.0009
02.93
0.0320
0.0330
−0.0010
03.13
80
0.0006
0.0006
0.0000
0
0.0164
0.0168
−0.0004
02.44
0.0170
0.0174
−0.0004
02.35
90
0.0003
0.0003
0.0000
0
0.0079
0.0084
−0.0005
06.33
0.0082
0.0087
−0.0005
06.10
700
0.0001
0.0001
0.0000
0
0.0040
0.0041
−0.0001
02.50
0.0041
0.0042
−0.0001
02.44
AVE% DIFF = 3.06%
AVE% DIFF = 1.84%
AVE% DIFF = 3.00%
AVE% DIFF = 2.76%
S.D. = 3.59
S.D. = 1.80
S.D. = 2.00
S.D. = 1.62
TABLE III
Values of n and the coefficients for Equations (9)–(13), to be used in the calculation of lens and macular pigment density.