Mika Flinkman, Hannu Laamanen, Jukka Tuomela, Pasi Vahimaa, and Markku Hauta-Kasari, "Eigenvectors of optimal color spectra," J. Opt. Soc. Am. A 30, 1806-1813 (2013)
Principal component analysis (PCA) and weighted PCA were applied to spectra of optimal colors belonging to the outer surface of the object-color solid or to so-called MacAdam limits. The correlation matrix formed from this data is a circulant matrix whose biggest eigenvalue is simple and the corresponding eigenvector is constant. All other eigenvalues are double, and the eigenvectors can be expressed with trigonometric functions. Found trigonometric functions can be used as a general basis to reconstruct all possible smooth reflectance spectra. When the spectral data are weighted with an appropriate weight function, the essential part of the color information is compressed to the first three components and the shapes of the first three eigenvectors correspond to one achromatic response function and to two chromatic response functions, the latter corresponding approximately to Munsell opponent-hue directions 9YR-9B and 2BG-2R.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
Munsell Data Reconstructed by Using the Eigenvectors Solved from the Munsell Data and Optimal Color Spectraa
Spectral Difference
Color Difference
Munsell
Optimal Colors
Munsell
Optimal Colors
Components
Mean
Max
Mean
Max
Mean
Max
Mean
Max
3
0.4639
2.9050
0.9640
3.3164
3.1588
20.3349
6.6523
28.6582
4
0.3373
1.5937
0.8979
3.2432
1.1843
5.7921
6.1442
27.6906
5
0.2682
1.0463
0.7461
2.2775
1.0474
10.2691
2.8635
30.0885
6
0.1915
1.0402
0.7099
2.2237
0.7157
10.9162
2.0161
24.1403
7
0.1463
0.8227
0.6434
1.9411
0.3280
1.9232
0.6328
9.7317
8
0.1102
0.5579
0.6264
1.8607
0.2404
3.1293
0.5519
9.1520
9
0.0908
0.4056
0.5871
1.7546
0.2037
2.0799
0.2509
1.1296
10
0.0712
0.3831
0.5678
1.6936
0.1311
0.9899
0.1635
1.2132
Spectral differences are calculated using Eq. (8), and color differences are estimated using the CIEDE2000 total color difference formula and CIE standard source D65.
Table 2.
Pantone Data Reconstructed by Using the Eigenvectors Solved from the Pantone Data and Optimal Color Spectraa
Spectral Difference
Color Difference
Pantone
Optimal Colors
Pantone
Optimal Colors
Components
Mean
Max
Mean
Max
Mean
Max
Mean
Max
3
0.7728
2.6475
1.4671
3.9651
3.6032
28.7822
8.7267
42.8269
4
0.5380
1.7211
1.4044
3.7215
1.8291
14.0696
8.0117
40.7998
5
0.4176
1.5353
1.1497
2.7520
1.7935
13.6086
3.6575
18.1470
6
0.3055
1.1642
1.0961
2.7471
0.6310
2.8511
3.8646
21.8081
7
0.2311
1.0992
0.9155
2.3370
0.5250
2.3311
0.9272
6.1358
8
0.1752
0.7080
0.8964
2.3190
0.2746
2.1998
0.9002
6.1327
9
0.1410
0.5059
0.8374
1.9876
0.1886
1.6740
0.3772
2.6780
10
0.1180
0.4224
0.8299
1.9487
0.1672
1.6334
0.3797
2.3883
Spectral differences are calculated using Eq. (8), and color differences are estimated using the CIEDE2000 total color difference formula and CIE standard source D65.
Table 3.
Munsell Data Reconstructed by Using the Eigenvectors Solved from the Munsell Data and Optimal Color Spectraa
Munsell
Optimal Colors
Components
Mean
Max
Mean
Max
3
0.3853
2.6698
0.5946
2.6366
4
0.2609
1.6973
0.4124
1.6459
5
0.1865
1.1184
0.3361
1.4851
6
0.1475
0.5348
0.2634
1.0672
7
0.1173
0.5246
0.2319
0.9897
8
0.0949
0.4405
0.1993
0.6889
9
0.0749
0.3473
0.1812
0.6155
10
0.0576
0.3462
0.1541
0.5603
Spectral differences are calculated using Eq. (8) in the wavelength range from 400 to 760 nm.
Table 4.
Munsell and Pantone Data Reconstructed by Using the Weighted Eigenvectors Solved from the Optimal Color Spectraa
Spectral Difference
Color Difference
D65
A
Components
Munsell
Pantone
Munsell
Pantone
Munsell
Pantone
3
1.2996
2.4742
0.8464
1.2547
1.4731
2.6269
4
1.2596
2.3816
0.7334
1.0784
0.9791
1.5272
5
1.1998
2.2352
0.5376
0.8264
0.7311
1.2205
6
1.1941
2.2191
0.4719
0.6992
0.6008
0.9986
7
1.1541
2.1857
0.4359
0.6389
0.5860
0.9427
8
1.1400
2.1717
0.3184
0.5079
0.4680
0.7884
9
1.1197
2.1398
0.2677
0.4541
0.3828
0.7101
10
1.1174
2.1323
0.2417
0.3711
0.3483
0.5721
Spectral differences are calculated using Eq. (8), and color differences are estimated using the CIEDE2000 total color difference formula and CIE standard sources D65 and A. Table includes only the mean values of the differences.
Tables (4)
Table 1.
Munsell Data Reconstructed by Using the Eigenvectors Solved from the Munsell Data and Optimal Color Spectraa
Spectral Difference
Color Difference
Munsell
Optimal Colors
Munsell
Optimal Colors
Components
Mean
Max
Mean
Max
Mean
Max
Mean
Max
3
0.4639
2.9050
0.9640
3.3164
3.1588
20.3349
6.6523
28.6582
4
0.3373
1.5937
0.8979
3.2432
1.1843
5.7921
6.1442
27.6906
5
0.2682
1.0463
0.7461
2.2775
1.0474
10.2691
2.8635
30.0885
6
0.1915
1.0402
0.7099
2.2237
0.7157
10.9162
2.0161
24.1403
7
0.1463
0.8227
0.6434
1.9411
0.3280
1.9232
0.6328
9.7317
8
0.1102
0.5579
0.6264
1.8607
0.2404
3.1293
0.5519
9.1520
9
0.0908
0.4056
0.5871
1.7546
0.2037
2.0799
0.2509
1.1296
10
0.0712
0.3831
0.5678
1.6936
0.1311
0.9899
0.1635
1.2132
Spectral differences are calculated using Eq. (8), and color differences are estimated using the CIEDE2000 total color difference formula and CIE standard source D65.
Table 2.
Pantone Data Reconstructed by Using the Eigenvectors Solved from the Pantone Data and Optimal Color Spectraa
Spectral Difference
Color Difference
Pantone
Optimal Colors
Pantone
Optimal Colors
Components
Mean
Max
Mean
Max
Mean
Max
Mean
Max
3
0.7728
2.6475
1.4671
3.9651
3.6032
28.7822
8.7267
42.8269
4
0.5380
1.7211
1.4044
3.7215
1.8291
14.0696
8.0117
40.7998
5
0.4176
1.5353
1.1497
2.7520
1.7935
13.6086
3.6575
18.1470
6
0.3055
1.1642
1.0961
2.7471
0.6310
2.8511
3.8646
21.8081
7
0.2311
1.0992
0.9155
2.3370
0.5250
2.3311
0.9272
6.1358
8
0.1752
0.7080
0.8964
2.3190
0.2746
2.1998
0.9002
6.1327
9
0.1410
0.5059
0.8374
1.9876
0.1886
1.6740
0.3772
2.6780
10
0.1180
0.4224
0.8299
1.9487
0.1672
1.6334
0.3797
2.3883
Spectral differences are calculated using Eq. (8), and color differences are estimated using the CIEDE2000 total color difference formula and CIE standard source D65.
Table 3.
Munsell Data Reconstructed by Using the Eigenvectors Solved from the Munsell Data and Optimal Color Spectraa
Munsell
Optimal Colors
Components
Mean
Max
Mean
Max
3
0.3853
2.6698
0.5946
2.6366
4
0.2609
1.6973
0.4124
1.6459
5
0.1865
1.1184
0.3361
1.4851
6
0.1475
0.5348
0.2634
1.0672
7
0.1173
0.5246
0.2319
0.9897
8
0.0949
0.4405
0.1993
0.6889
9
0.0749
0.3473
0.1812
0.6155
10
0.0576
0.3462
0.1541
0.5603
Spectral differences are calculated using Eq. (8) in the wavelength range from 400 to 760 nm.
Table 4.
Munsell and Pantone Data Reconstructed by Using the Weighted Eigenvectors Solved from the Optimal Color Spectraa
Spectral Difference
Color Difference
D65
A
Components
Munsell
Pantone
Munsell
Pantone
Munsell
Pantone
3
1.2996
2.4742
0.8464
1.2547
1.4731
2.6269
4
1.2596
2.3816
0.7334
1.0784
0.9791
1.5272
5
1.1998
2.2352
0.5376
0.8264
0.7311
1.2205
6
1.1941
2.2191
0.4719
0.6992
0.6008
0.9986
7
1.1541
2.1857
0.4359
0.6389
0.5860
0.9427
8
1.1400
2.1717
0.3184
0.5079
0.4680
0.7884
9
1.1197
2.1398
0.2677
0.4541
0.3828
0.7101
10
1.1174
2.1323
0.2417
0.3711
0.3483
0.5721
Spectral differences are calculated using Eq. (8), and color differences are estimated using the CIEDE2000 total color difference formula and CIE standard sources D65 and A. Table includes only the mean values of the differences.