Xiandou Zhang and Haisong Xu, "Reconstructing spectral reflectance by dividing spectral space and extending the principal components in principal component analysis," J. Opt. Soc. Am. A 25, 371-378 (2008)
Principal component analysis (PCA) is widely used to reconstruct the spectral reflectance of surface colors. However, the estimated spectral accuracy is low when using only one set of three principal components for three-channel color-acquisition devices. In this study, the spectral space was first divided into 11 subgroups, and the principal components were calculated for individual subgroups. Then the principal components were further extended from three to nine through the residual spectral error of the reflectance in each subgroup. For each target sample, the extended principal components of the corresponding subgroup samples were used in the common PCA method to reconstruct the spectral reflectance. The results show that this proposed method is quite accurate and outperforms other related methods.
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CIELAB color difference under testing illuminant D75
Mean
0.2101
0.2253
0.1001
0.0767
0.2158
0.2125
0.1443
0.1361
0.3242
0.3208
0.1508
0.1264
Var.
0.0483
0.0466
0.0096
0.0049
0.1022
0.0889
0.0142
0.0107
0.1665
0.1340
0.0192
0.0090
Max.
1.4181
1.4785
0.7061
0.5117
3.9684
3.6353
1.1989
0.9722
3.4570
2.3984
0.7225
0.5494
%
0.0000
0.0000
0.0000
0.0000
0.2051
0.1026
0.0000
0.0000
0.5814
0.0000
0.0000
0.0000
CIELAB color difference under testing illuminant F11
Mean
2.1224
2.1809
1.0834
0.9389
2.0599
2.0493
1.5907
1.5278
2.6588
2.6239
1.4976
1.3055
Var.
4.3802
4.2396
1.3164
0.8373
5.0103
4.8548
2.1123
1.6200
6.8337
6.6153
2.2604
1.2383
Max.
11.5850
11.3470
9.2257
7.9506
17.7330
17.6280
17.9100
14.5540
13.5630
13.4660
10.6000
9.0457
%
23.6590
24.6060
6.7823
4.2587
21.7950
21.5900
15.8970
14.7180
34.8840
32.5580
12.7910
9.8837
‘Var.’ means the variance of the CIELAB color difference.
% means the percentage of testing samples with color differences greater than 3 CIELAB units.
Boldface values indicate that the CIELAB color differences of method ii are lower than those of method iii.
Table 2
ANOVA Results of RMSE among Methods pi, i, ii, and iii under the Condition of CIE Illuminant D65
Method
Testing Samples
Half of Munsell Chips
All the NCS Chips
All the DC Chips
i
ii
iii
i
ii
iii
i
ii
iii
pi
F-Value
0.84
62.88
176.46
0.76
4.74
20.43
0.01
14.97
20.93
P-Value
0.3584
0.0000
0.0000
0.3836
0.0295
0.0000
0.9049
0.0001
0.0000
i
F-Value
87.12
232.87
1.88
14.52
16.98
24.18
P-Value
0.0000
0.0000
0.1704
0.0001
0.0000
0.0000
ii
F-Value
26.02
5.60
0.80
P-Value
0.0000
0.0180
0.3714
Table 3
Comparisons of Performances among Methods pi, i, ii, and iii under the Conditions of CIE Illuminant D65 with Odd Chips of NCS and Munsell Atlas As Training and Testing Samples, Respectively
Method
RMSE
under Testing Illuminant A
under Testing Illuminant F11
Mean
Max.
Mean
Var.
Max.
%
Mean
Var.
Max.
%
pi
0.0263
0.1622
1.6796
2.9948
10.8030
13.3860
2.3788
5.0830
13.7750
28.5040
i
0.0293
0.1441
1.9898
2.6290
10.6980
15.7480
2.5544
5.3695
13.7690
29.1340
ii
0.0270
0.2562
1.0931
0.6920
5.7022
6.9291
1.5081
1.4949
9.9385
14.8030
iii
0.0199
0.1082
0.9201
0.4388
4.3471
3.9370
1.3314
1.1176
7.4566
10.0790
Table 4
Comparisons of Performances between Ayala Method[13] and Proposed Method iii
See footnotes a, b to Table 1 for definitions.
‘’ means both the training and testing samples are from Munsell Atlas.
‘’ means the training samples are from Munsell Atlas and the testing samples from NCS Atlas.
Table 5
Comparisons of Performances among Methods pi, i, ii, and iii under the Conditions of Illuminants A and F11
Method
pi (A)
pi (A)
ii (A)
iii (A)
pi (F11)
i (F11)
ii (F11)
iii (F11)
Odd chips from Munsell Atlas as training samples, even chips as testing samples
Odd chips from Munsell Atlas as training samples, all the NCS chips as testing samples
Number of training samples
635
635
635
635
635
635
635
635
Number of testing samples
1950
1950
1950
1950
1950
1950
1950
1950
Mean RMSE
0.0238
0.0234
0.0231
0.0217
0.0254
0.0254
0.0271
0.0240
Max. RMSE
0.1907
0.1797
0.1651
0.1401
0.2348
0.2218
0.2490
0.1623
Mean
0.0414
0.0345
0.0032
0.0003
0.0157
0.0199
0.0166
0.0002
Var.
0.1475
0.1104
0.0039
0.0000
0.0256
0.0376
0.0510
0.0000
Max.
6.5973
5.9281
1.7469
0.1555
2.9329
3.5129
5.7420
0.1265
%
0.4615
0.4615
0.0000
0.0000
0.0000
0.1026
0.1539
0.0000
‘Var.’ means the variance of the CIELAB color difference.
‘% ’ means the percentage of testing samples with color differences greater than 3 CIELAB units.
Table 6
ANOVA Results of RMSE between Method iii and Methods pi, i, and ii under the Conditions of Illuminants A and F11
Testing Samples
Illuminant
A
F11
iii/pi
iii/i
iii/ii
iii/pi
iii/i
iii/ii
Munsell
F-Value
189.26
217.30
29.82
124.93
155.60
32.55
P-Value
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
NCS
F-Value
10.63
7.11
4.88
3.42
3.80
15.02
P-Value
0.0011
0.0770
0.0271
0.0645
0.0514
0.0001
Table 7
ANOVA Results of RMSE for Methods pi, i, ii, and iii between Illuminants A and F11
Testing Samples
Illuminant A/F11
pi
i
ii
iii
Munsell
F-Value
2.18
4.30
13.65
8.62
P-Value
0.1404
0.0383
0.0002
0.0034
NCS
F-Value
4.85
9.18
26.70
13.36
P-Value
0.0278
0.0025
0.0000
0.0003
Tables (7)
Table 1
Comparisons of Performances among Methods pi, i, ii, and iii under the Condition of CIE Illuminant D65
Method
Testing Samples
Half of Munsell Chips
All the NCS Chips
All the DC Chips
pi
i
ii
iii
pi
i
ii
iii
pi
i
ii
iii
Number of training samples
635
635
635
635
635
635
635
635
635
635
635
635
Number of testing samples
634
634
634
634
1950
1950
1950
1950
172
172
172
172
Time (s)
1.984
2.672
3.156
4.969
4.063
6.516
6.953
10.375
1.313
1.500
1.813
2.906
Mean RMSE
0.0234
0.0244
0.0152
0.0112
0.0258
0.0252
0.0242
0.0226
0.0321
0.0317
0.0213
0.0195
Max RMSE
0.1637
0.1489
0.1286
0.1014
0.2258
0.2107
0.1918
0.1339
0.2060
0.1889
0.1067
0.0897
CIELAB color difference under testing illuminant A
CIELAB color difference under testing illuminant D75
Mean
0.2101
0.2253
0.1001
0.0767
0.2158
0.2125
0.1443
0.1361
0.3242
0.3208
0.1508
0.1264
Var.
0.0483
0.0466
0.0096
0.0049
0.1022
0.0889
0.0142
0.0107
0.1665
0.1340
0.0192
0.0090
Max.
1.4181
1.4785
0.7061
0.5117
3.9684
3.6353
1.1989
0.9722
3.4570
2.3984
0.7225
0.5494
%
0.0000
0.0000
0.0000
0.0000
0.2051
0.1026
0.0000
0.0000
0.5814
0.0000
0.0000
0.0000
CIELAB color difference under testing illuminant F11
Mean
2.1224
2.1809
1.0834
0.9389
2.0599
2.0493
1.5907
1.5278
2.6588
2.6239
1.4976
1.3055
Var.
4.3802
4.2396
1.3164
0.8373
5.0103
4.8548
2.1123
1.6200
6.8337
6.6153
2.2604
1.2383
Max.
11.5850
11.3470
9.2257
7.9506
17.7330
17.6280
17.9100
14.5540
13.5630
13.4660
10.6000
9.0457
%
23.6590
24.6060
6.7823
4.2587
21.7950
21.5900
15.8970
14.7180
34.8840
32.5580
12.7910
9.8837
‘Var.’ means the variance of the CIELAB color difference.
% means the percentage of testing samples with color differences greater than 3 CIELAB units.
Boldface values indicate that the CIELAB color differences of method ii are lower than those of method iii.
Table 2
ANOVA Results of RMSE among Methods pi, i, ii, and iii under the Condition of CIE Illuminant D65
Method
Testing Samples
Half of Munsell Chips
All the NCS Chips
All the DC Chips
i
ii
iii
i
ii
iii
i
ii
iii
pi
F-Value
0.84
62.88
176.46
0.76
4.74
20.43
0.01
14.97
20.93
P-Value
0.3584
0.0000
0.0000
0.3836
0.0295
0.0000
0.9049
0.0001
0.0000
i
F-Value
87.12
232.87
1.88
14.52
16.98
24.18
P-Value
0.0000
0.0000
0.1704
0.0001
0.0000
0.0000
ii
F-Value
26.02
5.60
0.80
P-Value
0.0000
0.0180
0.3714
Table 3
Comparisons of Performances among Methods pi, i, ii, and iii under the Conditions of CIE Illuminant D65 with Odd Chips of NCS and Munsell Atlas As Training and Testing Samples, Respectively
Method
RMSE
under Testing Illuminant A
under Testing Illuminant F11
Mean
Max.
Mean
Var.
Max.
%
Mean
Var.
Max.
%
pi
0.0263
0.1622
1.6796
2.9948
10.8030
13.3860
2.3788
5.0830
13.7750
28.5040
i
0.0293
0.1441
1.9898
2.6290
10.6980
15.7480
2.5544
5.3695
13.7690
29.1340
ii
0.0270
0.2562
1.0931
0.6920
5.7022
6.9291
1.5081
1.4949
9.9385
14.8030
iii
0.0199
0.1082
0.9201
0.4388
4.3471
3.9370
1.3314
1.1176
7.4566
10.0790
Table 4
Comparisons of Performances between Ayala Method[13] and Proposed Method iii
See footnotes a, b to Table 1 for definitions.
‘’ means both the training and testing samples are from Munsell Atlas.
‘’ means the training samples are from Munsell Atlas and the testing samples from NCS Atlas.
Table 5
Comparisons of Performances among Methods pi, i, ii, and iii under the Conditions of Illuminants A and F11
Method
pi (A)
pi (A)
ii (A)
iii (A)
pi (F11)
i (F11)
ii (F11)
iii (F11)
Odd chips from Munsell Atlas as training samples, even chips as testing samples
Odd chips from Munsell Atlas as training samples, all the NCS chips as testing samples
Number of training samples
635
635
635
635
635
635
635
635
Number of testing samples
1950
1950
1950
1950
1950
1950
1950
1950
Mean RMSE
0.0238
0.0234
0.0231
0.0217
0.0254
0.0254
0.0271
0.0240
Max. RMSE
0.1907
0.1797
0.1651
0.1401
0.2348
0.2218
0.2490
0.1623
Mean
0.0414
0.0345
0.0032
0.0003
0.0157
0.0199
0.0166
0.0002
Var.
0.1475
0.1104
0.0039
0.0000
0.0256
0.0376
0.0510
0.0000
Max.
6.5973
5.9281
1.7469
0.1555
2.9329
3.5129
5.7420
0.1265
%
0.4615
0.4615
0.0000
0.0000
0.0000
0.1026
0.1539
0.0000
‘Var.’ means the variance of the CIELAB color difference.
‘% ’ means the percentage of testing samples with color differences greater than 3 CIELAB units.
Table 6
ANOVA Results of RMSE between Method iii and Methods pi, i, and ii under the Conditions of Illuminants A and F11
Testing Samples
Illuminant
A
F11
iii/pi
iii/i
iii/ii
iii/pi
iii/i
iii/ii
Munsell
F-Value
189.26
217.30
29.82
124.93
155.60
32.55
P-Value
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
NCS
F-Value
10.63
7.11
4.88
3.42
3.80
15.02
P-Value
0.0011
0.0770
0.0271
0.0645
0.0514
0.0001
Table 7
ANOVA Results of RMSE for Methods pi, i, ii, and iii between Illuminants A and F11