Hannu Laamanen, Tuija Jetsu, Timo Jaaskelainen, and Jussi Parkkinen, "Weighted compression of spectral color information," J. Opt. Soc. Am. A 25, 1383-1388 (2008)
Spectral color information is used nowadays in many different applications. Accurate spectral images are usually very large files, but a proper compression method can reduce needed storage space remarkably with a minimum loss of information. In this paper we introduce a principal component analysis (PCA) -based compression method of spectral color information. In this approach spectral data is weighted with a proper weight function before forming the correlation matrix and calculating the eigenvector basis. First we give a general framework for how to use weight functions in compression of relevant color information. Then we compare the weighted compression method with the traditional PCA compression method by compressing and reconstructing the Munsell data set consisting of 1269 reflectance spectra and the Pantone data set consisting of 922 reflectance spectra. Two different weight functions are proposed and tested. We show that weighting clearly improves retention of color information in the PCA-based compression process.
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Munsell Data Reconstructed with Three Different Vector Bases, One Calculated for Nonweighted (NW) Munsell Data and Two Calculated for Weighted (WF1 and WF2) Munsell Dataa
Mean
Max.
Samples Where
Components
NW
WF1
WF2
NW
WF1
WF2
NW
WF1
WF2
3
3.171
2.092
0.732
20.37
15.88
4.138
7.41%
24.0%
64.6%
4
1.194
0.418
0.565
5.856
5.771
3.203
30.3%
90.6%
74.9%
5
1.059
0.355
0.316
10.47
4.427
2.807
39.2%
94.6%
93.6%
6
0.728
0.278
0.233
11.13
4.285
1.851
68.3%
95.9%
97.3%
7
0.333
0.215
0.194
1.923
3.304
1.613
96.3%
97.7%
99.4%
8
0.244
0.204
0.134
3.148
3.200
1.261
98.8%
98.2%
99.5%
9
0.208
0.083
0.103
2.080
1.124
0.932
99.6%
99.9%
99.9%
10
0.134
0.051
0.095
1.074
0.475
0.993
99.8%
100%
99.9%
11
0.077
0.025
0.039
0.864
0.230
0.304
99.8%
100%
100%
12
0.059
0.017
0.024
0.715
0.149
0.193
100%
100%
100%
13
0.043
0.005
0.024
0.612
0.098
0.211
100%
100%
100%
14
0.037
0.004
0.020
0.518
0.045
0.159
100%
100%
100%
15
0.037
0.003
0.019
0.518
0.038
0.121
100%
100%
100%
16
0.032
0.003
0.016
0.482
0.034
0.132
100%
100%
100%
17
0.020
0.002
0.016
0.300
0.025
0.135
100%
100%
100%
18
0.010
0.002
0.016
0.111
0.014
0.131
100%
100%
100%
19
0.008
0.001
0.015
0.068
0.014
0.119
100%
100%
100%
20
0.006
0.001
0.011
0.055
0.015
0.107
100%
100%
100%
Color differences are estimated by using the CIE color difference formula and the CIE standard source D65. In comparison to with the results above, we can state that if spectral Munsell data are compressed and reconstructed using the normalized and orthogonalized CIE 1931 standard color-matching functions (three components), maximum color difference is 4.3330 and mean value of all differences is 1.8134 (D65 source).
Table 3
Munsell Data Reconstructed with Three Different Vector Bases, One Calculated for Nonweighted (NW) Munsell Data and Two Calculated for Weighted (WF1 and WF2) Munsell Dataa
Mean
Max
Samples Where
Components
NW
WF1
WF2
NW
WF1
WF2
NW
WF1
WF2
3
2.672
1.353
1.066
19.98
15.04
7.487
6.30%
30.3%
51.5%
4
1.509
0.854
0.688
11.14
10.35
5.554
20.5%
63.8%
67.5%
5
0.857
0.407
0.375
7.020
7.624
4.116
46.2%
91.7%
88.3%
6
0.576
0.255
0.321
7.174
2.848
3.452
78.9%
96.9%
92.1%
7
0.374
0.193
0.239
3.263
2.723
3.088
93.4%
98.5%
97.3%
8
0.242
0.205
0.181
3.878
2.509
2.816
98.1%
98.0%
97.5%
9
0.182
0.082
0.112
2.089
1.031
0.968
99.6%
99.9%
99.9%
10
0.102
0.047
0.090
1.193
0.494
0.853
99.7%
100%
99.8%
11
0.066
0.030
0.033
1.120
0.503
0.294
99.7%
100%
100%
12
0.052
0.012
0.032
0.920
0.087
0.301
99.9%
100%
100%
13
0.031
0.005
0.026
0.195
0.063
0.290
100%
100%
100%
14
0.025
0.004
0.021
0.201
0.061
0.155
100%
100%
100%
15
0.024
0.004
0.020
0.222
0.059
0.155
100%
100%
100%
16
0.019
0.003
0.019
0.187
0.053
0.154
100%
100%
100%
17
0.013
0.003
0.019
0.196
0.044
0.152
100%
100%
100%
18
0.006
0.002
0.019
0.055
0.017
0.149
100%
100%
100%
19
0.004
0.002
0.019
0.036
0.015
0.140
100%
100%
100%
20
0.004
0.001
0.014
0.044
0.012
0.117
100%
100%
100%
Color differences are estimated by using the CIE color difference formula and the CIE standard source A.
Table 4
Munsell Data Reconstructed with Three Different Vector Bases, One Calculated for Nonweighted (NW) Munsell Data and Two Calculated for Weighted (WF1 and WF2) Munsell Dataa
Mean GFC
Samples Where
Components
NW
WF1
WF2
NW
WF1
WF2
3
0.9938
0.9933
0.9916
12.3 %
11.5 %
10.2 %
4
0.9964
0.9960
0.9934
28.0 %
26.1 %
19.9 %
5
0.9976
0.9974
0.9956
38.6 %
36.3 %
27.2 %
6
0.9989
0.9979
0.9960
70.5 %
46.1 %
33.5 %
7
0.9993
0.9989
0.9964
82.4 %
68.6 %
36.8 %
8
0.9996
0.9994
0.9966
92.2 %
83.9 %
39.6 %
9
0.9997
0.9996
0.9977
94.6 %
92.0 %
45.9 %
10
0.9998
0.9998
0.9980
98.0 %
96.9 %
50.7 %
11
0.9999
0.9998
0.9983
98.7 %
97.7 %
61.4 %
12
0.9999
0.9998
0.9984
99.3 %
98.0 %
62.9 %
13
0.9999
0.9999
0.9994
99.6 %
99.5 %
85.8 %
14
0.9999
0.9999
0.9995
99.8 %
99.5 %
85.9 %
15
0.9999
0.9999
0.9996
99.8 %
99.6 %
90.1 %
16
1.0000
1.0000
0.9996
99.8 %
99.8 %
90.5 %
17
1.0000
1.0000
0.9996
99.9 %
99.9 %
90.6 %
18
1.0000
1.0000
0.9996
99.9 %
99.9 %
91.1 %
19
1.0000
1.0000
0.9996
99.9 %
99.9 %
91.5 %
20
1.0000
1.0000
0.9997
99.9 %
99.9 %
94.2 %
Spectral differences are estimated by using the GFC formula, Eq. (4).
Tables (4)
Table 1
Values of the Second Weight Function (WF2)
λ
λ
λ
λ
380
0.0229
485
0.2372
590
0.4799
695
0.0336
385
0.0247
490
0.2725
595
0.5678
700
0.0297
390
0.0289
495
0.3478
600
0.6395
705
0.0270
395
0.0361
500
0.4455
605
0.6897
710
0.0250
400
0.0501
505
0.5654
610
0.7086
715
0.0235
405
0.0689
510
0.6909
615
0.6994
720
0.0225
410
0.1118
515
0.8138
620
0.6634
725
0.0218
415
0.1842
520
0.9135
625
0.6029
730
0.0212
420
0.3047
525
0.9714
630
0.5298
735
0.0209
425
0.4761
530
0.9973
635
0.4581
740
0.0206
430
0.6242
535
0.9963
640
0.3875
745
0.0204
435
0.7210
540
0.9713
645
0.3195
750
0.0204
440
0.7652
545
0.9237
650
0.2573
755
0.0204
445
0.7681
550
0.8560
655
0.2042
760
0.0204
450
0.7478
555
0.7719
660
0.1596
765
0.0204
455
0.7170
560
0.6740
665
0.1229
770
0.0204
460
0.6641
565
0.5676
670
0.0944
775
0.0204
465
0.5833
570
0.4647
675
0.0742
780
0.0204
470
0.4633
575
0.3860
680
0.0600
475
0.3494
580
0.3612
685
0.0482
480
0.2645
585
0.4008
690
0.0394
Table 2
Munsell Data Reconstructed with Three Different Vector Bases, One Calculated for Nonweighted (NW) Munsell Data and Two Calculated for Weighted (WF1 and WF2) Munsell Dataa
Mean
Max.
Samples Where
Components
NW
WF1
WF2
NW
WF1
WF2
NW
WF1
WF2
3
3.171
2.092
0.732
20.37
15.88
4.138
7.41%
24.0%
64.6%
4
1.194
0.418
0.565
5.856
5.771
3.203
30.3%
90.6%
74.9%
5
1.059
0.355
0.316
10.47
4.427
2.807
39.2%
94.6%
93.6%
6
0.728
0.278
0.233
11.13
4.285
1.851
68.3%
95.9%
97.3%
7
0.333
0.215
0.194
1.923
3.304
1.613
96.3%
97.7%
99.4%
8
0.244
0.204
0.134
3.148
3.200
1.261
98.8%
98.2%
99.5%
9
0.208
0.083
0.103
2.080
1.124
0.932
99.6%
99.9%
99.9%
10
0.134
0.051
0.095
1.074
0.475
0.993
99.8%
100%
99.9%
11
0.077
0.025
0.039
0.864
0.230
0.304
99.8%
100%
100%
12
0.059
0.017
0.024
0.715
0.149
0.193
100%
100%
100%
13
0.043
0.005
0.024
0.612
0.098
0.211
100%
100%
100%
14
0.037
0.004
0.020
0.518
0.045
0.159
100%
100%
100%
15
0.037
0.003
0.019
0.518
0.038
0.121
100%
100%
100%
16
0.032
0.003
0.016
0.482
0.034
0.132
100%
100%
100%
17
0.020
0.002
0.016
0.300
0.025
0.135
100%
100%
100%
18
0.010
0.002
0.016
0.111
0.014
0.131
100%
100%
100%
19
0.008
0.001
0.015
0.068
0.014
0.119
100%
100%
100%
20
0.006
0.001
0.011
0.055
0.015
0.107
100%
100%
100%
Color differences are estimated by using the CIE color difference formula and the CIE standard source D65. In comparison to with the results above, we can state that if spectral Munsell data are compressed and reconstructed using the normalized and orthogonalized CIE 1931 standard color-matching functions (three components), maximum color difference is 4.3330 and mean value of all differences is 1.8134 (D65 source).
Table 3
Munsell Data Reconstructed with Three Different Vector Bases, One Calculated for Nonweighted (NW) Munsell Data and Two Calculated for Weighted (WF1 and WF2) Munsell Dataa
Mean
Max
Samples Where
Components
NW
WF1
WF2
NW
WF1
WF2
NW
WF1
WF2
3
2.672
1.353
1.066
19.98
15.04
7.487
6.30%
30.3%
51.5%
4
1.509
0.854
0.688
11.14
10.35
5.554
20.5%
63.8%
67.5%
5
0.857
0.407
0.375
7.020
7.624
4.116
46.2%
91.7%
88.3%
6
0.576
0.255
0.321
7.174
2.848
3.452
78.9%
96.9%
92.1%
7
0.374
0.193
0.239
3.263
2.723
3.088
93.4%
98.5%
97.3%
8
0.242
0.205
0.181
3.878
2.509
2.816
98.1%
98.0%
97.5%
9
0.182
0.082
0.112
2.089
1.031
0.968
99.6%
99.9%
99.9%
10
0.102
0.047
0.090
1.193
0.494
0.853
99.7%
100%
99.8%
11
0.066
0.030
0.033
1.120
0.503
0.294
99.7%
100%
100%
12
0.052
0.012
0.032
0.920
0.087
0.301
99.9%
100%
100%
13
0.031
0.005
0.026
0.195
0.063
0.290
100%
100%
100%
14
0.025
0.004
0.021
0.201
0.061
0.155
100%
100%
100%
15
0.024
0.004
0.020
0.222
0.059
0.155
100%
100%
100%
16
0.019
0.003
0.019
0.187
0.053
0.154
100%
100%
100%
17
0.013
0.003
0.019
0.196
0.044
0.152
100%
100%
100%
18
0.006
0.002
0.019
0.055
0.017
0.149
100%
100%
100%
19
0.004
0.002
0.019
0.036
0.015
0.140
100%
100%
100%
20
0.004
0.001
0.014
0.044
0.012
0.117
100%
100%
100%
Color differences are estimated by using the CIE color difference formula and the CIE standard source A.
Table 4
Munsell Data Reconstructed with Three Different Vector Bases, One Calculated for Nonweighted (NW) Munsell Data and Two Calculated for Weighted (WF1 and WF2) Munsell Dataa
Mean GFC
Samples Where
Components
NW
WF1
WF2
NW
WF1
WF2
3
0.9938
0.9933
0.9916
12.3 %
11.5 %
10.2 %
4
0.9964
0.9960
0.9934
28.0 %
26.1 %
19.9 %
5
0.9976
0.9974
0.9956
38.6 %
36.3 %
27.2 %
6
0.9989
0.9979
0.9960
70.5 %
46.1 %
33.5 %
7
0.9993
0.9989
0.9964
82.4 %
68.6 %
36.8 %
8
0.9996
0.9994
0.9966
92.2 %
83.9 %
39.6 %
9
0.9997
0.9996
0.9977
94.6 %
92.0 %
45.9 %
10
0.9998
0.9998
0.9980
98.0 %
96.9 %
50.7 %
11
0.9999
0.9998
0.9983
98.7 %
97.7 %
61.4 %
12
0.9999
0.9998
0.9984
99.3 %
98.0 %
62.9 %
13
0.9999
0.9999
0.9994
99.6 %
99.5 %
85.8 %
14
0.9999
0.9999
0.9995
99.8 %
99.5 %
85.9 %
15
0.9999
0.9999
0.9996
99.8 %
99.6 %
90.1 %
16
1.0000
1.0000
0.9996
99.8 %
99.8 %
90.5 %
17
1.0000
1.0000
0.9996
99.9 %
99.9 %
90.6 %
18
1.0000
1.0000
0.9996
99.9 %
99.9 %
91.1 %
19
1.0000
1.0000
0.9996
99.9 %
99.9 %
91.5 %
20
1.0000
1.0000
0.9997
99.9 %
99.9 %
94.2 %
Spectral differences are estimated by using the GFC formula, Eq. (4).