318
Parkkinen et al.
J. Opt. Soc. Am. A/Vol. 6, No. 2/February 1989
Characteristic spectra of Munsell colors
J. P. S. Parkkinen
Department of Computer Science and Mathematics, University of Kuopio, P.O. Box 6, SF-70 211 Kuopio, Finland
J. Hallikainen
Department of Physics, University of Kuopio, P.O. Box 6, SF-70 211 Kuopio, Finland
T. Jaaskelainen*
Faculty of Engineering, Saitama University, 255 Shimo-Okubo, Urawa, Japan
Received January 8, 1988; accepted October 3, 1988
The 1257 reflectance spectra of the chips in the Munsell Book of Color-Matte Finish Collection (Munsell Color,
Baltimore, Md., 1976) were measured with a rapid acousto-optic spectrophotometer. Measured spectra were
sampled from 400 to 700 nm at 5-nm intervals. The correlation matrix of this sample set was formed, and the
characteristic vectors of this matrix were computed. It is shown, contradictory to earlier recommendations
[Psychon. Sci. 1, 369 (1964)], that as many as eight characteristic spectra are needed to achieve good representation
for all spectra.
INTRODUCTION
Color is an important factor in many areas of human life. It
forms the basic parameter in a wide variety of physical,
chemical, and other measurements. The importance of color is growing also in image analysis, because of the development of color imaging systems. In some applications the
needed accuracy can be achieved by measuring the color
through a few filters, but the whole spectrum is needed for
accurate results, e.g., in spectrophotometry.
When whole spectra are used, some problems arise in the
computational analysis of the measurements. Much computer memory and time are needed for such an analysis. In
this paper the statistical structure of the color spectra is
analyzed, and a new compression method for color spectra is
described.
A total of 1257 reflectance spectra from the visible region
were measured from the Munsell color chips.' Our analysis
is based on the Karhunen-Loeve expansion, which is closely
related to the principal-component analysis. Chromatic
data are represented by the sum of some basis functions in
several papers in the literature, such as those by D'Zmura
and Lennie, 2 Maloney and Wandell,3 Sobagaki, 4 and
Young. 5 However, the emphasis in most of these papers was
on the trichromatic representation of color, based on the
tristimulus values.
To our knowledge, there are only two papers 6' 7 in which
the reflectance curves of the Munsell color chips were been
measured and analyzed by using principal-component analysis. Cohen used the correlation matrix of 433 spectra, and
he concluded that the first four principal components are
enough to reconstruct the measured spectra. Our sample
set was significantly larger than his set. Therefore some
differences in results were obtained. These are discussed
later in the present paper. Parkkinen and co-workers used
0740/3232/89/020318-05$02.00
similar methods previously for color discrimination of transparencies.8 9,
METHODS
Measurements
We measured the reflectance spectra of 1257 Munsell color
chips with different hues, saturations, and values,7 using a
new acousto-optic spectrophotometer.10 The objects were
illuminated through optical fibers by a standard halogen
lamp. The reflected light was collected through optical
fibers to the acousto-optic crystal. The 0/0 geometry was
used. A photomultiplier was used to measure the intensities, which were digitized and stored in the memory of a
personal computer. The personal computer was connected
to a VAX 11/785 minicomputer with a local area network.
The data were analyzed with both computers. The measured spectra were sampled from 400 to 700 nm at 5-nm
intervals. The wavelength error of the measurements was
less than 2 nm, and the repetition accuracy was 0.5% on the
0-100% reflectance scale. The light intensity of the illuminator at wavelengths below 420 nm was so low that it caused
extra noise at the blue end of the measured spectra.
Data Analysis
The data were analyzed by using the Karhunen-Loeve
transformation to find the characteristic spectra of the color
set. In this method, which is used widely in pattern recognition, we assume the spectra to be a realization of a continuous stochastic process. In practice the measured spectra are
not continuous functions but finite dimensional vectors.
The method is also known as principal component analysis.
Let us denote the measured spectrum by
© 1989 Optical Society of America
Parkkinen et al.
Vol. 6, No. 2/February 1989/J. Opt. Soc. Am. A
Table 1. Eight Characteristic Vectors and Corresponding Eigenvalues for Munsell Colorsa
Normalized Amplitude of the Following Eigenvector
Third
Fourth
Fifth
Sixth
Wavelength (nm)
First
Second
400
0.1673
0.1474
0.1369
0.1298
0.1264
0.1245
0.1234
0.1235
0.1237
0.1228
0.1228
0.1232
0.1236
0.1241
0.1248
0.1248
0.1256
0.1254
0.1251
0.1252
-0.0479
-0.0863
-0.0974
-0.1035
-0.1068
-0.1188
-0.1184
-0.1234
-0.1302
-0.1348
-0.1406
-0.1468
-0.1514
-0.1572
-0.1613
-0.1665
-0.1717
-0.1722
-0.1681
-0.1648
0.2434
0.2267
0.1742
0.1646
0.1702
0.1666
0.1556
0.1495
0.1445
0.1376
0.1272
0.1132
0.0948
0.0806
0.0657
0.0406
0.0082
-0.0203
-0.0472
-0.0632
-0.8923
-0.1868
-0.011
0.0038
0.0285
0.0417
0.0623
0.073
0.0787
0.0829
0.0823
0.0837
0.0823
0.0819
0.0816
0.0789
0.0755
0.0689
0.0627
0.0568
0.2769
-0.3863
-0.1933
-0.1627
-0.1663
-0.1395
-0.1225
-0.1187
-0.1064
-0.0937
-0.0739
-0.0466
-0.0089
0.0216
0.0488
0.0924
0.1419
0.183
0.1981
0.2029
500
0.1256
0.1261
0.1263
0.1266
0.1261
0.125
0.1244
0.1245
0.1244
0.1244
0.125
0.1259
0.1263
0.1266
0.127
0.1277
0.1287
0.1296
0.1301
0.1304
-0.1588
-0.1498
-0.1346
-0.1219
-0.1098
-0.0954
-0.0788
-0.0663
-0.0511
-0.0338
-0.015
0.0002
0.0115
0.0236
0.0359
0.0512
0.0693
0.0838
0.0961
0.1113
-0.0773
-0.1001
-0.1287
-0.1542
-0.1726
-0.1885
-0.1999
-0.2056
-0.2045
-0.2029
-0.1992
-0.1977
-0.1945
-0.1892
-0.1815
-0.1652
-0.1441
-0.1224
-0.1023
-0.0773
0.0494
0.0373
0.0198
0.0049
-0.0077
-0.0196
-0.032
-0.0392
-0.0448
-0.0483
-0.0519
-0.0552
-0.0565
-0.0577
-0.0567
-0.0541
-0.05
-0.0422
-0.0355
-0.0256
600
0.1301
0.1301
0.1297
0.1298
0.1291
0.1288
0.1283
0.1284
0.1284
0.1283
0.1282
0.1282
0.1281
0.1284
0.1282
0.1284
0.1285
0.1283
0.1284
0.1277
0.1267
0.1256
0.1359
0.1422
0.1463
0.1482
0.1507
0.1524
0.1538
0.1539
0.1552
0.1544
0.1547
0.1533
0.1533
0.1535
0.153
0.1529
0.1519
0.1524
0.1481
0.1413
-0.0495
-0.0279
-0.0116
-0.0041
0.0032
0.0129
0.0248
0.0363
0.0435
0.0512
0.0584
0.0661
0.0709
0.0779
0.0835
0.0859
0.0876
0.0876
0.087
0.0855
0.0808
-0.0146
-0.0035
0.0058
0.0103
0.0129
0.0206
0.0263
0.0321
0.0371
0.0422
0.0463
0.0508
0.0541
0.0599
0.0639
0.0655
0.0674
0.0699
0.0703
0.0722
0.07
700
a
Seventh
Eighth
0.1513
-0.6154
-0.148
0.0098
0.0201
0.0455
0.0548
0.0771
0.0936
0.1067
0.1169
0.1221
0.1243
0.1174
0.1112
0.0945
0.0626
0.0363
0.0053
-0.014
-0.1091
0.5428
-0.0789
-0.1436
-0.1216
-0.132
-0.1136
-0.1266
-0.1107
-0.0986
-0.0806
-0.0444
-0.0065
0.0256
0.0499
0.0823
0.1217
0.1481
0.1571
0.1575
-0.0392
-0.2004
0.5319
0.3421
0.1184
0.0289
-0.0101
-0.0389
-0.0488
-0.0733
-0.0982
-0.1127
-0.114
-0.1291
-0.1285
-0.123
-0.0919
-0.0616
-0.0228
0.0055
0.2039
0.1895
0.1599
0.1313
0.1022
0.0633
0.0226
-0.009
-0.0382
-0.0683
-0.0971
-0.1186
-0.1324
-0.1461
-0.1561
-0.1607
-0.1589
-0.1518
-0.1378
-0.1154
-0.025
-0.045
-0.0739
-0.0945
-0.1141
-0.1281
-0.1357
-0.1294
-0.1139
-0.0892
-0.0557
-0.0282
-0.0076
0.0117
0.0383
0.0713
0.108
0.1348
0.1526
0.1624
0.1481
0.1238
0.0774
0.0262
-0.0188
-0.0685
-0.1133
-0.1376
-0.1462
-0.1401
-0.1226
-0.1122
-0.0995
-0.0798
-0.0632
-0.0227
0.0193
0.0606
0.0897
0.1182
0.0297
0.0569
0.0885
0.1002
0.1113
0.1163
0.1185
0.1085
0.0914
0.0454
-0.017
-0.0656
-0.1002
-0.1322
-0.1525
-0.1553
-0.1381
-0.1112
-0.0787
-0.0258
-0.0829
-0.0573
-0.03
-0.0139
-0.0033
0.0117
0.0287
0.0412
0.051
0.0596
0.0635
0.0683
0.0778
0.0834
0.0908
0.099
0.1078
0.1131
0.1158
0.1197
0.1193
0.1611
0.1508
0.1336
0.1204
0.1129
0.0971
0.0791
0.0553
0.0329
0.0192
0.0019
-0.0213
-0.0414
-0.0657
-0.0888
-0.1103
-0.1327
-0.1502
-0.1643
-0.1711
-0.175
0.1394
0.1498
0.1451
0.1446
0.1392
0.1282
0.1173
0.0964
0.0736
0.055
0.0353
0.0108
-0.0169
-0.0472
-0.0716
-0.0976
-0.1193
-0.1344
-0.1532
-0.1686
-0.1697
0.024
0.0563
0.0889
0.111
0.1219
0.1282
0.1328
0.1292
0.1158
0.1031
0.0828
0.0564
0.0266
-0.0005
-0.0423
-0.0694
-0.1014
-0.1221
-0.1463
-0.1637
-0.1772
Eigenvalues for the first, second, third, fourth, fifth, sixth, seventh, and eighth eigenvectors are 1129.2, 72.7, 28.8, 12.7, 5.0, 3.4, 2.2, and 0.8, respectively.
319
320
Parkkinen et al.
J. Opt. Soc. Am. A/Vol. 6, No. 2/February 1989
S(X) = [S(X1), S(X2 ), . ..
,S(X")]T,
(1)
where Xk is the kth wavelength in the spectrum and T denotes the transpose.
To compute the characteristic vectors, we used the correlation matrix R,
p
R=
E
T
S,(X)S(X) ,
ae
(2)
2
14
i=l
I
where the index i represents the ith spectrum in the measured spectra set.
We used the eigenvectors I of this matrix, i.e., the solutions of the equation
R(P = Hib,
(3)
400
where a is the eigenvalue of R. The matrix R is an n X n
matrix, and it has n eigenvalues and eigenvectors. We can
represent an arbitrary color spectrum by using only few of
these eigenvectors. This is true if the sample set leading to
500
600
700
WAVMELI (nm)
Fig. 1. The four first eigenvectors of the set of 1257 reflectance
spectra measured from the Munsell book of colors.
matrix R is large enough and covers the whole color space.
The representation is based on the reconstruction equa-
0.3
tion,
028
026
n
SA(X) =
E [Sa(X)4'j(X)]4'j(X)X
024
(4)
0.22
j=1
02
where sa(X) is an arbitrary color spectrum, Gj(X)
is the jth
eigenvector of R, and [] denotes the inner product.
We must point out that the sum in Eq. (4) gives an exact
reconstruction of the spectrum Sa(X), provided that it belongs to the original data set. However, the components of a
color spectrum are highly correlated, and noise exists in the
measurements. Therefore sufficient reconstruction is
achieved with only a few (4-8) eigenvectors. This method is
described in more detail in other papers."",12
W
0
0.18
0.16
0 .14
cc
0.12
0.1
0.08
Om
0.04
0.02
0
400
500
RESULTS
The eigenvalues and eigenvectors of the correlation matrix R
that contained the whole sample set were determined.
Eight of these vectors, normalized to unity, are represented
in Table 1, and the first four are shown in Fig. 1. The first
eigenvector corresponds to the mean of the measured vectors, and its flatness indicates uniform covering of the color
space.
In the second phase we tested the reproducibility of the
spectra by the eigenvectors. Two examples are shown in
Figs. 2 and 3. Table 2 shows the xy color coordinates of the
reproductions obtained by using two, four, six, and eight
eigenvectors. Figure 2 represents the Munsell chip 5 B 7/8.
It shows that the spectrum can be reconstructed almost
exactly by using four eigenvectors. This is also confirmed
by Table 2.
Figure 3 shows an example of a poorly reconstructed spectrum. With more than six eigenvectors, the changes in the
reconstruction are minimal. This is caused by the fact that
the seventh and eighth eigenvectors and all eigenvectors
after them are almost orthogonal to the measured spectrum,
and so the coefficients in Eq. (4) are small. Thus, as Table 2
shows, the accuracy of the xy coordinates is not monotonically increasing with an increasing number of eigenvectors.
600
700
WA-ELENGT
(n)
Fig. 2. A measured spectrum (5 B 7/8) and its reconstruction with
four eigenvectors.
0.3
0.28
026 0.2 -,
022
02 -
0.16
Oo0:
0
400
500
600
700
WAVEGh (nm)
Fig. 3. A measured spectrum (5 R 6/14) and its reconstruction with
eight eigenvectors.
Parkkinen et al.
Vol. 6, No. 2/February 1989/J. Opt. Soc. Am. A
321
Table 2. xy Color Coordinates of Two Munsell Samples and Their Reconstructions with Two, Four, Six, and Eight
Eigenvectorsa
Sample
Original
Two Vectors
Coordinate Spectrum Reconstruction
Error
Four Vectors
Reconstruction
Error
Six Vectors
Reconstruction
Error
Eight Vectors
Reconstruction Error
5 B 7/8
x
y
5R 6/14
x
y
a The
0.226
0.231
0.005
0.226
-0.000
0.225
-0.001
0.226
0.000
0.260
0.278
0.018
0.259
-0.001
0.259
-0.001
0.260
0.000
0.508
0.317
0.538
0.429
0.029
0.111
0.504
0.285
-0.005
-0.033
0.516
0.318
0.008
0.001
0.520
0.322
0.012
0.005
spectra of the sample are shown in Figs. 2 and 3.
Table 3. Maximum and Average Errors of xy Color
Coordinates
Number of
Eigenvectors
Average Errora
x
y
3
4
5
6
7
8
9
10
a The
b The
0.0055
0.0040
0.0028
0.0027
0.0011
0.0009
0.0005
0.0002
Maximum Errorb
x
y
0.0100
0.0054
0.0053
0.0016
0.0016
0.0011
0.0007
0.0003
0.0775
0.0602
0.0503
0.0505
0.0124
0.0190
0.0090
0.0044
0.0582
0.0480
0.0534
0.0121
0.0120
0.0121
0.0056
0.0040
average of absolute values of errors.
maximum of absolute values or errors.
-
0.E
Cr
0
uJ
C
- 0.5 400
I
I
l
500
600
WAVELENGTH (nm)
700
Fig. 4. Error bands for four-dimensional (area between dotted
curves) and eight-dimensional (area between solid curves) reconstructions. The upper and lower limits of the bands represent the
maximum positive and the maximum negative differences in the
absolute reflectance scale.
are used. Furthermore, the width of the error band does not
depend significantly on the wavelength.
All reconstruction errors belong to bands such as those in
Fig. 4, but error distribution inside the band is not uniform.
Figure 5 shows a typical error distribution. According to
Fig. 5, 71.2% of the errors are within the range +0.01 although the boundaries of the whole error band are +0.045.
Because the reconstruction error is almost independent of
the wavelength, we consider here the error averaged over the
wavelength region. All 1257 spectra were reconstructed
with 4-10 eigenvectors. The absolute value of the reconstruction error averaged over the wavelength region was
determined for each spectrum. The results are shown in
Table 4. Note, that, when eight eigenvectors are used, almost all spectra can be reconstructed so that the error is
<0.02, and the mean error is only 0.008.
It is difficult to say exactly how many eigenvectors should
be used. The number depends on the accuracy and on the
application. However, according to Tables 3 and 4, the use
of seven or eight eigenvectors is shown to lead to good reconstructions in the whole data set. This is also confirmed by
Fig. 6, which shows the worst case of four-dimensional reconstructions. Figure 6 also shows that the eight-dimensional
reconstruction of this worst case of the whole data set is not
bad.
DISCUSSION
Reflectance spectra of the 1257 Munsell color chips were
measured, and the characteristic spectra of this sample set
General information about the color coordinate errors is
given in Table 3. The absolute values of the differences
between the color coordinates calculated from the original
spectra and the 3-10-dimensional reconstructions were determined for the whole data set. The average and maximum
values of these differences are shown in Table 3.
To determine the overall goodness of fit, we compared the
reconstructions directly with the original data without computing any color coordinates. First, we determined the reconstruction errors in the wavelength region. Figure 4
shows the error bands for four- and eight-dimensional recon-
structions. The unit in the coordinate axis is the reflectance
difference between the originals and the reconstructions.
The error band becomes narrower when more eigenvectors
.
-9
.
.
-7
.
.
-5
.
.
-3
.
.
I
.
.
-1 1 3
INTERVAL
.
.
5
.
.
7
.
.
9
Fig. 5. Error distribution in the band of eight-dimensional reconstructions at 500 nm. Here No is the number of samples, and error
intervals are 0.005. The ith bar represents the error interval, which
is i.005 apart from the error 0.
322
Parkkinen et al.
J. Opt. Soc. Am. A/Vol. 6, No. 2/February 1989
Table 4. Cumulative Error Distributions for 4-10-Dimensional Reconstructions
Percentage of Samples for the Following Number of Eigenvectors
Error Limita
4
5
6
7
8
9
10
<0.01
<0.02
<0.03
22.8
55.4
80.9
34.5
72.2
92.5
44.8
83.6
97.0
63.9
95.5
99.9
72.0
98.4
100
85.7
100
91.3
100
•0.04
92.7
97.9
<0.05
<0.06
97.2
98.7
99.4
100
<0.07
99.5
<0.08
•0.09
•0.10
99.8
99.8
100
99.9
100
100
aThe error limit is defined as the difference between the original and the reconstructed spectral reflectances averaged over the wavelength band.
sign of the vector. The decision of the sign fixes the form of
the principal colors corresponding to the eigenvectors. For
example, an eigenvector can thought to be bluish-green or
reddish-yellow, depending on the sign of the vector. Some
examples for sign decision are as follows:
(1) The sum of the inner products in Eq. (4) over all
spectra should be nonnegative for each eigenvector.
(2) The sum of the eigenvector components should be
nonegative.
0.0
400
Fig. 6.
500
600
WAVELENGTH (nm)
700
* Permanent address, Department of Physics, University
of Kuopio, P.O. Box 6, SF-70 211 Kuopio, Finalnd.
The worst fit obtained using four eigenvectors (dotted
curve). The dashed curve represents the worst-case eight-dimensional reconstruction, and the solid curve represents the original
spectrum.
were computed. As mentioned above, Cohen was the first
(to our knowledge) to investigate this problem. 6 He calculated the characteristic vectors of a set containing only 150
reflectance curves. Maloney repeated these computations,
7
using 462 samples, but did not publish his vectors. Maloney's and Cohen's sets of measured spectra were significantly
smaller than those in our study. Furthermore, the sampling
REFERENCES
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for recovering surface spectral reflectance," J. Opt. Soc. Am. A
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4. H. Sobagaki, "New approach to the calorimetric standardization for object colors," Bull. Electrotech. Lab. Jpn. 48, 875-792
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than ours. Thus our characteristic vectors offer a more
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5. R. A.Young, "Principal-component analysis of macaque lateral
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space. In contrast, our first eigenvector is nearly flat. Furthermore, there are some differences in the locations of local
minimas and maximas between our results and Cohen's results.
All the relevant spectral information of the Munsell color
chips can be compressed into few characteristic vectors. We
believe that the eigenvectors given in this paper lead to
accurate representations of colors in a natural environment,
too.
The negative values of the eigenvector components cause
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