Use of three tristimulus values from surface reflectance spectra to calculate the principal components for reconstructing these spectra by using only three eigenvectors
Fernando Ayala, José F. Echávarri, Pilar Renet, and Angel I. Negueruela
Fernando Ayala, José F. Echávarri, Pilar Renet, and Angel I. Negueruela, "Use of three tristimulus values from surface reflectance spectra to calculate the principal components for reconstructing these spectra by using only three eigenvectors," J. Opt. Soc. Am. A 23, 2020-2026 (2006)
The division of Color Space into ten zones, corresponding to the ten Munsell hues, allows a good reconstruction of surface reflectance spectra using just three eigenvectors, obtained by applying principal components analysis to the reflectance spectra of the Munsell Atlas specimens (model group), although the basis vectors obtained are different for each subspace. The use of the tristimulus values from each measured spectrum, calculated with the Illuminant D65 and the Standard Observer CIE64 to obtain the principal components necessary to reconstruct the spectrum, allows a very high degree of metamerism to be attained between the two spectra (measured and reconstructed). Furthermore, this method of calculating the principal components allows reconstruction of the spectra of specimens from other sample sets that differ from the model group used in the PCA. The colorimetric accuracy obtained in the new sample sets is similar to that obtained for the model group.
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Percentage of Variance Accounted for by a Linear Model with 1–4 Eigenvectors for the Ten Munsell Hue Zones
R
YR
Y
GY
G
BG
B
PB
P
RP
Vector 1
80.35
83.69
87.44
90.94
94.59
94.40
91.85
93.87
92.85
87.37
Vector 2
17.77
15.08
10.88
5.83
4.60
4.93
7.36
4.85
5.51
11.40
Vector 3
1.36
0.88
0.94
2.15
0.39
0.31
0.37
0.99
0.86
0.56
Vector 4
0.28
0.16
0.54
0.71
0.21
0.20
0.26
0.14
0.42
0.40
Table 2
Statistics of Color Differences (in CIELAB Units) between Colors Obtained from Experimental Spectra and Spectra Reconstructed Directly from PCA for Each Specimen of the Munsell Atlas ( Specimens)
Illuminant
D50
D65
D75
A
F11
Mean
2.0
2.6
2.5
1.9
2.6
Variance
9.3
11.1
11.0
6.6
9.8
Max.
45.2
49.8
49.4
36.8
42.5
Color coordinates calculated with the CIE64 Observer and CIE Illuminants D50, D65, D75, A, and F 11.
Table 3
Statistics of Color Differences (in CIELAB Units) between Colors Obtained from Experimental and Reconstructed Spectra for Each Specimen of the Three Atlases: Munsell ( Specimens), NCS ( Specimens) and Munsell Measured by Parkkinen and Colleagues ( Specimens)
Illuminant
D50
D65
D75
Munsell:
Mean
0.3
0.0
0.1
Variance
0.1
0.0
0.0
Max.
2.3
0.1
1.0
NCS:
Mean
0.3
0.0
0.2
Variance
0.1
0.0
0.0
Max.
1.0
0.1
0.9
Munsell:
Mean
0.3
0.0
0.1
Variance
0.1
0.0
0.0
Max.
1.9
0.1
0.9
Color coordinates calculated with the CIE64 Observer and CIE Illuminants D50, D65 and D75.
Table 4
Statistics of Color Differences, (in CIELAB Units) between Colors Obtained from Experimental and Reconstructed Spectra for Each Specimen, Calculated with the CIE64 Observer and CIE Illuminant A or Illuminant F11a
Hue
R
YR
Y
GY
G
BG
B
PB
P
RP
All
Munsell Atlas
Illuminant A
Mean
1.6
0.6
1.2
1.5
0.9
1.0
1.1
0.5
0.5
1.0
1.0
Variance
1.2
0.3
1.2
1.4
0.6
1.2
2.0
0.4
0.1
0.6
1.0
Maximum
5.3
3.1
5.7
5.3
4.4
6.6
8.7
5.4
2.5
4.6
8.7
10.3
0.6
9.0
13.0
2.7
6.6
7.4
0.7
0.0
2.2
5.2
Illuminant F11
Mean
1.2
1.0
2.5
2.5
1.1
1.4
1.5
0.8
1.4
1.3
1.5
Variance
0.7
0.6
3.8
4.9
1.4
2.1
2.9
0.7
1.8
1.8
2.3
Maximum
4.8
4.6
9.1
10.7
6.6
8.3
10.6
7.6
7.7
11.2
11.2
2.9
4.2
26.9
32.1
6.8
10.2
10.7
2.2
10.2
5.5
10.6
N
174
166
145
131
146
137
121
137
147
181
1485
NCS Atlas
Illuminant A
Mean
2.2
1.0
1.0
1.2
1.2
1.3
0.9
0.8
0.6
1.1
1.2
Variance
1.9
0.2
0.6
0.9
0.6
1.4
1.5
0.4
0.2
0.8
1.0
Maximum
6.0
2.8
5.1
6.3
4.4
5.4
6.5
3.2
2.3
3.8
6.5
28.0
0.0
2.2
3.9
1.3
10.4
7.0
1.4
0.0
4.9
6.0
Illuminant F11
Mean
1.8
1.7
2.3
2.2
1.9
1.8
1.3
0.8
1.2
1.2
1.7
Variance
2.1
1.9
2.6
3.5
2.7
2.7
1.9
1.0
0.6
1.1
2.4
Maximum
9.1
8.7.
8.9
13.8
7.4
7.9
7.1
5.7
4.1
5.4
13.8
15.3
15.5
28.1
24.2
17.4
15.3
9.6
4.3
4.9
5.7
15.6
N
189
283
178
256
149
144
114
138
102
122
1675
Munsell Atlas measured by Parkkinen et al.
Illuminant A
Mean
1.7
0.6
1.5
1.8
1.0
1.1
1.3
0.7
0.6
1.2
1.1
Variance
1.1
0.1
0.9
1.1
0.4
1.5
2.0
0.5
0.3
0.9
1.0
Maximum
4.4
1.5
4.0
4.2
4.4
5.6
6.4
4.5
2.9
4.8
6.4
12.3
0.0
8.7
15.4
0.9
8.5
11.8
2.7
0.0
5.6
6.2
Illuminant F11
Mean
1.1
0.8
2.8
2.7
1.2
1.4
1.5
1.0
1.3
1.3
1.5
Variance
0.5
0.2
3.4
3.4
0.9
2.5
2.1
1.0
1.8
1.2
2.0
Maximum
4.0
3.0
7.3
8.7
7.8
7.7
6.7
5.6
9.1
9.6
9.6
2.3
0.0
40.9
41.3
2.7
15.3
12.7
6.7
8.1
4.3
12.7
N
130
117
127
104
112
118
102
149
148
162
1269
( is the percentage of specimens with color differences greater that 3 CIELAB units).
Table 5
Correlations among the Color Differences Obtained with Different CIE Standard IIluminants
Munsell (1485 specimens)
Correlation
NCS(1675 specimens)
Munsell measured by Parkkinen et al.[23] (1269 specimens.)
Tables (5)
Table 1
Percentage of Variance Accounted for by a Linear Model with 1–4 Eigenvectors for the Ten Munsell Hue Zones
R
YR
Y
GY
G
BG
B
PB
P
RP
Vector 1
80.35
83.69
87.44
90.94
94.59
94.40
91.85
93.87
92.85
87.37
Vector 2
17.77
15.08
10.88
5.83
4.60
4.93
7.36
4.85
5.51
11.40
Vector 3
1.36
0.88
0.94
2.15
0.39
0.31
0.37
0.99
0.86
0.56
Vector 4
0.28
0.16
0.54
0.71
0.21
0.20
0.26
0.14
0.42
0.40
Table 2
Statistics of Color Differences (in CIELAB Units) between Colors Obtained from Experimental Spectra and Spectra Reconstructed Directly from PCA for Each Specimen of the Munsell Atlas ( Specimens)
Illuminant
D50
D65
D75
A
F11
Mean
2.0
2.6
2.5
1.9
2.6
Variance
9.3
11.1
11.0
6.6
9.8
Max.
45.2
49.8
49.4
36.8
42.5
Color coordinates calculated with the CIE64 Observer and CIE Illuminants D50, D65, D75, A, and F 11.
Table 3
Statistics of Color Differences (in CIELAB Units) between Colors Obtained from Experimental and Reconstructed Spectra for Each Specimen of the Three Atlases: Munsell ( Specimens), NCS ( Specimens) and Munsell Measured by Parkkinen and Colleagues ( Specimens)
Illuminant
D50
D65
D75
Munsell:
Mean
0.3
0.0
0.1
Variance
0.1
0.0
0.0
Max.
2.3
0.1
1.0
NCS:
Mean
0.3
0.0
0.2
Variance
0.1
0.0
0.0
Max.
1.0
0.1
0.9
Munsell:
Mean
0.3
0.0
0.1
Variance
0.1
0.0
0.0
Max.
1.9
0.1
0.9
Color coordinates calculated with the CIE64 Observer and CIE Illuminants D50, D65 and D75.
Table 4
Statistics of Color Differences, (in CIELAB Units) between Colors Obtained from Experimental and Reconstructed Spectra for Each Specimen, Calculated with the CIE64 Observer and CIE Illuminant A or Illuminant F11a
Hue
R
YR
Y
GY
G
BG
B
PB
P
RP
All
Munsell Atlas
Illuminant A
Mean
1.6
0.6
1.2
1.5
0.9
1.0
1.1
0.5
0.5
1.0
1.0
Variance
1.2
0.3
1.2
1.4
0.6
1.2
2.0
0.4
0.1
0.6
1.0
Maximum
5.3
3.1
5.7
5.3
4.4
6.6
8.7
5.4
2.5
4.6
8.7
10.3
0.6
9.0
13.0
2.7
6.6
7.4
0.7
0.0
2.2
5.2
Illuminant F11
Mean
1.2
1.0
2.5
2.5
1.1
1.4
1.5
0.8
1.4
1.3
1.5
Variance
0.7
0.6
3.8
4.9
1.4
2.1
2.9
0.7
1.8
1.8
2.3
Maximum
4.8
4.6
9.1
10.7
6.6
8.3
10.6
7.6
7.7
11.2
11.2
2.9
4.2
26.9
32.1
6.8
10.2
10.7
2.2
10.2
5.5
10.6
N
174
166
145
131
146
137
121
137
147
181
1485
NCS Atlas
Illuminant A
Mean
2.2
1.0
1.0
1.2
1.2
1.3
0.9
0.8
0.6
1.1
1.2
Variance
1.9
0.2
0.6
0.9
0.6
1.4
1.5
0.4
0.2
0.8
1.0
Maximum
6.0
2.8
5.1
6.3
4.4
5.4
6.5
3.2
2.3
3.8
6.5
28.0
0.0
2.2
3.9
1.3
10.4
7.0
1.4
0.0
4.9
6.0
Illuminant F11
Mean
1.8
1.7
2.3
2.2
1.9
1.8
1.3
0.8
1.2
1.2
1.7
Variance
2.1
1.9
2.6
3.5
2.7
2.7
1.9
1.0
0.6
1.1
2.4
Maximum
9.1
8.7.
8.9
13.8
7.4
7.9
7.1
5.7
4.1
5.4
13.8
15.3
15.5
28.1
24.2
17.4
15.3
9.6
4.3
4.9
5.7
15.6
N
189
283
178
256
149
144
114
138
102
122
1675
Munsell Atlas measured by Parkkinen et al.
Illuminant A
Mean
1.7
0.6
1.5
1.8
1.0
1.1
1.3
0.7
0.6
1.2
1.1
Variance
1.1
0.1
0.9
1.1
0.4
1.5
2.0
0.5
0.3
0.9
1.0
Maximum
4.4
1.5
4.0
4.2
4.4
5.6
6.4
4.5
2.9
4.8
6.4
12.3
0.0
8.7
15.4
0.9
8.5
11.8
2.7
0.0
5.6
6.2
Illuminant F11
Mean
1.1
0.8
2.8
2.7
1.2
1.4
1.5
1.0
1.3
1.3
1.5
Variance
0.5
0.2
3.4
3.4
0.9
2.5
2.1
1.0
1.8
1.2
2.0
Maximum
4.0
3.0
7.3
8.7
7.8
7.7
6.7
5.6
9.1
9.6
9.6
2.3
0.0
40.9
41.3
2.7
15.3
12.7
6.7
8.1
4.3
12.7
N
130
117
127
104
112
118
102
149
148
162
1269
( is the percentage of specimens with color differences greater that 3 CIELAB units).
Table 5
Correlations among the Color Differences Obtained with Different CIE Standard IIluminants
Munsell (1485 specimens)
Correlation
NCS(1675 specimens)
Munsell measured by Parkkinen et al.[23] (1269 specimens.)
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