Ville Heikkinen, Reiner Lenz, Tuija Jetsu, Jussi Parkkinen, Markku Hauta-Kasari, and Timo Jääskeläinen, "Evaluation and unification of some methods for estimating reflectance spectra from RGB images," J. Opt. Soc. Am. A 25, 2444-2458 (2008)
The problem of estimating spectral reflectances from the responses of a digital camera has received considerable attention recently. This problem can be cast as a regularized regression problem or as a statistical inversion problem. We discuss some previously suggested estimation methods based on critically undersampled RGB measurements and describe some relations between them. We concentrate mainly on those models that are using a priori information in the form of high-resolution measurements. We use the “kernel machine” framework in our evaluations and concentrate on the use of multiple illuminations and on the investigation of the performance of global and locally adapted estimation methods. We also introduce a nonlinear transformation of reflectance values to ensure that the estimated reflection spectra fulfill physically motivated boundary conditions. The reported experimental results are derived from measured and simulated camera responses from the Munsell Matte, NCS, and Pantone data sets.
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Similarities of Subspaces of Used Sets, Where k Denotes the Dimension of the Subspacea
k
1
0.9910
0.9983
1.0000
2
0.9918
0.9951
0.9932
3
0.9920
0.9928
0.9998
4
0.9627
0.9878
0.9986
5
0.9517
0.9766
0.9981
6
0.8787
0.9032
0.8698
7
0.8986
0.9333
0.9977
8
0.8557
0.9121
0.9969
9
0.8780
0.9731
0.9989
10
0.8766
0.9821
0.9989
11
0.8819
0.9728
0.9983
Munsell (M), Pantone (P), and NCS (N) are full sets. Munsell I (MI) corresponds to the subset of 669 samples; Munsell II (MII) corresponds to the subset of 600 samples.
Table 2
Overview of Training and Test Sets
Experiment
Training Set
Test Set
1
Munsell I (669 samples)
Munsell II (600 samples)
2
NCS (1750 samples)
Munsell (1269 samples)
3
Pantone (922 samples)
Munsell (1269 samples)
Table 3
Average RMSE Values for the Simulated Munsell I/Munsell II Settinga
Method
Avg.
Std.
Max.
Fluorescent
Ideal Wiener
0.0230
0.0186
0.1534
Ideal Kernel 1,
0.0142
0.0117
0.1013
Kernel 1 (n, t, g)
0.0148
0.0116
0.1006
Kernel 1 (n, t, l)
0.0121
0.0105
0.0948
Kernel 2 (n, t, l)
0.0116
0.0100
0.0870
Tungsten
Ideal Wiener
0.0211
0.0154
0.1063
Ideal Kernel 1,
0.0122
0.0092
0.0725
Kernel 1 (n, t, g)
0.0128
0.0088
0.0720
Kernel 1 (n, t, l)
0.0107
0.0088
0.0672
Kernel 2 (n, t, l)
0.0111
0.0087
0.0686
Fluorescent and Tungsten
Ideal Wiener
0.0090
0.0052
0.0422
Ideal Kernel 1,
0.0055
0.0039
0.0370
Kernel 1 (n, t, g)
0.0114
0.0055
0.0380
Kernel 1 (n, t, l)
0.0079
0.0048
0.0389
Kernel 2 (n, t, l)
0.0085
0.0055
0.0461
Average of ten randomizations for noise and sets. Illumination sources: fluorescent and/or tungsten.
Table 4
Average RMSE Values for the Simulated NCS/Munsell Settinga
Method
Avg.
Std.
Max.
Fluorescent
Ideal Wiener
0.0269
0.0164
0.1456
Ideal Kernel 1,
0.0200
0.0141
0.0905
Kernel 1 (n, t, g)
0.0204
0.0155
0.1063
Kernel 1 (n, t, l)
0.0209
0.0172
0.1200
Kernel 2 (n, t, l)
0.0209
0.0165
0.0997
Tungsten
Ideal Wiener
0.0237
0.0139
0.1078
Ideal Kernel 1,
0.0167
0.0113
0.0679
Kernel 1 (n, t, g)
0.0176
0.0116
0.0777
Kernel 1 (n, t, l)
0.0173
0.0125
0.0751
Kernel 2 (n, t, l)
0.0179
0.0124
0.0751
Fluorescent and Tungsten
Ideal Wiener
0.0119
0.0056
0.0363
Ideal Kernel 1,
0.0085
0.0055
0.0462
Kernel 1 (n, t, g)
0.0159
0.0080
0.0563
Kernel 1 (n, t, l)
0.0128
0.0076
0.0520
Kernel 2 (n, t, l)
0.0138
0.0081
0.0531
Average of ten randomizations for noise. Illumination sources: fluorescent and/or tungsten.
Table 5
Average RMSE Values for the Simulated Pantone/Munsell Settinga
Method
Avg.
Std.
Max.
Fluorescent
Ideal Wiener
0.0357
0.0172
0.1487
Ideal Kernel 1,
0.0325
0.0205
0.1667
Kernel 1 (n, t, g)
0.0333
0.0213
0.1736
Kernel 1 (n, t, l)
0.0338
0.0227
0.1657
Kernel 2 (n, t, l)
0.0358
0.0207
0.1472
Tungsten
Ideal Wiener
0.0301
0.0145
0.1158
Ideal Kernel 1,
0.0269
0.0160
0.1111
Kernel 1 (n, t, g)
0.0278
0.0157
0.1131
Kernel 1 (n, t, l)
0.0287
0.0171
0.1197
Kernel 2 (n, t, l)
0.0305
0.0148
0.1046
Fluorescent and Tungsten
Ideal Wiener
0.0133
0.0090
0.0503
Ideal Kernel 1,
0.0160
0.0096
0.0508
Kernel 1 (n, t, g)
0.0219
0.0106
0.0727
Kernel 1 (n, t, l)
0.0220
0.0124
0.1098
Kernel 2 (n, t, l)
0.0236
0.0112
0.1101
Average of ten randomizations for noise. Illumination sources: fluorescent and/or tungsten.
Results are average of ten randomizations of 600 samples and for the whole set of 1269 samples using rank-k PCA approximations.
Table 10
RMSE Errors for the 15-Sample Munsell Clusters Corresponding to the Test Samplesa
Rank
15 Samples
Avg.
Std.
Max.
0.0058
0.0027
0.0116
0.0014
0.0005
0.0023
Results are the average of 600 clusters using rank-k PCA approximations.
Tables (10)
Table 1
Similarities of Subspaces of Used Sets, Where k Denotes the Dimension of the Subspacea
k
1
0.9910
0.9983
1.0000
2
0.9918
0.9951
0.9932
3
0.9920
0.9928
0.9998
4
0.9627
0.9878
0.9986
5
0.9517
0.9766
0.9981
6
0.8787
0.9032
0.8698
7
0.8986
0.9333
0.9977
8
0.8557
0.9121
0.9969
9
0.8780
0.9731
0.9989
10
0.8766
0.9821
0.9989
11
0.8819
0.9728
0.9983
Munsell (M), Pantone (P), and NCS (N) are full sets. Munsell I (MI) corresponds to the subset of 669 samples; Munsell II (MII) corresponds to the subset of 600 samples.
Table 2
Overview of Training and Test Sets
Experiment
Training Set
Test Set
1
Munsell I (669 samples)
Munsell II (600 samples)
2
NCS (1750 samples)
Munsell (1269 samples)
3
Pantone (922 samples)
Munsell (1269 samples)
Table 3
Average RMSE Values for the Simulated Munsell I/Munsell II Settinga
Method
Avg.
Std.
Max.
Fluorescent
Ideal Wiener
0.0230
0.0186
0.1534
Ideal Kernel 1,
0.0142
0.0117
0.1013
Kernel 1 (n, t, g)
0.0148
0.0116
0.1006
Kernel 1 (n, t, l)
0.0121
0.0105
0.0948
Kernel 2 (n, t, l)
0.0116
0.0100
0.0870
Tungsten
Ideal Wiener
0.0211
0.0154
0.1063
Ideal Kernel 1,
0.0122
0.0092
0.0725
Kernel 1 (n, t, g)
0.0128
0.0088
0.0720
Kernel 1 (n, t, l)
0.0107
0.0088
0.0672
Kernel 2 (n, t, l)
0.0111
0.0087
0.0686
Fluorescent and Tungsten
Ideal Wiener
0.0090
0.0052
0.0422
Ideal Kernel 1,
0.0055
0.0039
0.0370
Kernel 1 (n, t, g)
0.0114
0.0055
0.0380
Kernel 1 (n, t, l)
0.0079
0.0048
0.0389
Kernel 2 (n, t, l)
0.0085
0.0055
0.0461
Average of ten randomizations for noise and sets. Illumination sources: fluorescent and/or tungsten.
Table 4
Average RMSE Values for the Simulated NCS/Munsell Settinga
Method
Avg.
Std.
Max.
Fluorescent
Ideal Wiener
0.0269
0.0164
0.1456
Ideal Kernel 1,
0.0200
0.0141
0.0905
Kernel 1 (n, t, g)
0.0204
0.0155
0.1063
Kernel 1 (n, t, l)
0.0209
0.0172
0.1200
Kernel 2 (n, t, l)
0.0209
0.0165
0.0997
Tungsten
Ideal Wiener
0.0237
0.0139
0.1078
Ideal Kernel 1,
0.0167
0.0113
0.0679
Kernel 1 (n, t, g)
0.0176
0.0116
0.0777
Kernel 1 (n, t, l)
0.0173
0.0125
0.0751
Kernel 2 (n, t, l)
0.0179
0.0124
0.0751
Fluorescent and Tungsten
Ideal Wiener
0.0119
0.0056
0.0363
Ideal Kernel 1,
0.0085
0.0055
0.0462
Kernel 1 (n, t, g)
0.0159
0.0080
0.0563
Kernel 1 (n, t, l)
0.0128
0.0076
0.0520
Kernel 2 (n, t, l)
0.0138
0.0081
0.0531
Average of ten randomizations for noise. Illumination sources: fluorescent and/or tungsten.
Table 5
Average RMSE Values for the Simulated Pantone/Munsell Settinga
Method
Avg.
Std.
Max.
Fluorescent
Ideal Wiener
0.0357
0.0172
0.1487
Ideal Kernel 1,
0.0325
0.0205
0.1667
Kernel 1 (n, t, g)
0.0333
0.0213
0.1736
Kernel 1 (n, t, l)
0.0338
0.0227
0.1657
Kernel 2 (n, t, l)
0.0358
0.0207
0.1472
Tungsten
Ideal Wiener
0.0301
0.0145
0.1158
Ideal Kernel 1,
0.0269
0.0160
0.1111
Kernel 1 (n, t, g)
0.0278
0.0157
0.1131
Kernel 1 (n, t, l)
0.0287
0.0171
0.1197
Kernel 2 (n, t, l)
0.0305
0.0148
0.1046
Fluorescent and Tungsten
Ideal Wiener
0.0133
0.0090
0.0503
Ideal Kernel 1,
0.0160
0.0096
0.0508
Kernel 1 (n, t, g)
0.0219
0.0106
0.0727
Kernel 1 (n, t, l)
0.0220
0.0124
0.1098
Kernel 2 (n, t, l)
0.0236
0.0112
0.1101
Average of ten randomizations for noise. Illumination sources: fluorescent and/or tungsten.