Zsolt T. Kosztyán, George P. Eppeldauer, and János D. Schanda, "Matrix-based color measurement corrections of tristimulus colorimeters," Appl. Opt. 49, 2288-2301 (2010)
For colorimetric imaging the tristimulus technique is still the best practical method to keep the measurement time within reasonable limits. However, the achievable color measurement uncertainties for special sources can be large. It is described how the systematic errors can be significantly reduced by using matrix-based color corrections and how the matrix elements can be optimized to obtain the smallest spectral mismatch errors for different light-source distributions. An approach for decreasing the systematic errors is to increase the number of the colorimeter channels (or filters) used for a measurement. Using five channels in a colorimeter is an optimum choice. Determination of the optimum matrices for the five channels is discussed. The correction matrices are designed such that the spectral mismatch errors of the realized functions are minimized relative to the CIE standard color matching functions for several selected test-source distributions. The optimum matrix depends on the (test) light source to be measured. Adaptive matrix values are determined by using the channel outputs and the spectral power distribution of color LEDs approximated with a simple approximation function. The systematic errors are evaluated for a number of colored and white LEDs. The noise propagation with the applied matrix corrections is investigated.
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Here the reference white is , , . LEDs: CW, cold white; WW, warm white; B, blue; C, cyan; G, green; Y, yellow; O, orange; R, red. The sensitivity is determined by the 1% noncorrelated noise (see the definition in Subsection 5E).
Table 2
Global Matrices for Different Kinds of Tristimulus Colorimetric Systems
Calibrated to CIE-A
Optimized to All Sources, 4 Filters
Optimized to All Sources,
Optimized to All Sources, 5 Filters
0.6013
0
0
0.806
0.1221
2.81962
5.67519
13.953
0
0
14.188
0.0978
0.03237
13.4261
13.6225
0.04983
0
1
0
1
0.02701
1
0.00302
0.02311
1
0
0
3.0601
3.25214
2.04086
0.66784
1.91104
0.14621
0
0
0
0
0
0
0.05164
7.44134
Table 3
Results of Unique Matrix Transformations for Cyan LEDsa
Optimal Matrix Transformation, Cyan LEDs
4 Filters
5 Filters
Avg.
SD
Avg.
SD
Avg.
SD
Avg.
SD
All
Colorimetric error
27.87
20.10
32.00
28.62
8.71
8.3
10.75
10.54
Sensitivity
0.88
0.25
1.39
0.99
1.06
0.56
1.40
0.79
Planck
Colorimetric error
27.38
2.22
38.86
4.92
9.1
0.75
12.69
1.91
Sensitivity
0.86
0.00
1.49
0.14
0.85
0.01
1.39
0.10
ΣLED
Colorimetric error
28.10
21.88
31.01
31.01
8.67
9.04
10.48
11.44
Sensitivity
0.89
0.27
1.38
1.08
1.1
0.6
1.40
0.86
CW
Colorimetric error
19.28
0.65
21.89
1.12
6.6
0.77
6.87
0.85
Sensitivity
0.9
0.02
1.05
0.04
1.01
0.04
1.21
0.03
WW
Colorimetric error
24.8
0.42
33.66
0.96
7.91
0.29
10.24
0.30
Sensitivity
0.85
0.01
1.38
0.03
0.84
0.02
1.31
0.02
B
Colorimetric error
17.35
20.54
6.77
5.50
12.30
17.69
4.52
4.07
Sensitivity
1.33
0.34
0.70
0.22
2.31
0.48
1.20
0.11
C
Colorimetric error
0.25
0.33
0.15
0.16
0.04
0.04
0.03
0.04
Sensitivity
0.67
0.05
0.23
0.02
1.32
0.34
0.56
0.16
G
Colorimetric error
11.77
15.39
5.43
7.63
3.16
6.03
1.49
2.89
Sensitivity
0.6
0.04
0.45
0.25
0.61
0.04
0.45
0.19
Y
Colorimetric error
49.21
0.22
39.59
5.48
13.59
2.34
13.63
1.54
Sensitivity
0.76
0.02
1.51
0.19
0.73
0.02
1.32
0.17
O
Colorimetric error
48.26
3.36
52.00
4.08
9.69
0.59
16.28
0.32
Sensitivity
0.82
0.03
2.01
0.25
0.78
0.04
1.77
0.22
R
Colorimetric error
57.83
5.06
86.16
17.52
14.46
5.82
28.95
13.41
Sensitivity
0.96
0.03
3.40
0.33
0.91
0.03
3.00
0.30
Avg., average; SD, standard deviation.
Table 4
Optimal Unique Matrices Optimized to Cyan LED Sources
4 Filters
5 Filters
0.6489
2.78267
16.967
0.0005
0.04162
14.6454
0.04459
1
0.03596
0.00018
1
3.08723
1.5321
0.26799
0.04434
0
0
0
3.77237
Table 5
Comparison of Results for Detector-Based and Matrix-Corrected Calibration Methods of Tristimulus Colorimeters, Measuring Color LEDs (Four Filters)
LEDs (DWs, nm)
Colorimetric errors
Detector-Based Calibration Using CIE-A
Using Global Matrix Transformation for All Sources
Using Unique Matrix Transformation
Using Adaptive Matrix Transformation
Gaussian
Generalized Lorentzian
Triangle
β
450
1.17
5.35
0.74
0.74
0.79
0.56
0,94
Sensitivity
0.85
0.88
0.87
0.89
0.73
0.83
0,86
490
6.83
1.82
0.13
1.31
1.50
1.32
1,14
Sensitivity
0.86
0.24
0.23
0.23
0.23
0.23
0,23
522
2.10
2.49
0.45
1.37
1.26
1.55
1,13
Sensitivity
0.27
0.28
0.28
0.28
0.28
0.28
0,28
582
0.42
1.36
0.14
2.54
0.45
0.35
0,51
Sensitivity
1.41
1.41
1.41
1.43
1.42
1.42
1,42
592
0.52
1.73
0.82
1.93
1.50
1.54
1,21
Sensitivity
1.53
1.53
1.54
1.59
1.54
1.55
1,61
628
7.70
7.95
2.53
2.40
2.00
0.62
0,24
Sensitivity
2.57
2.53
1.99
2.43
1.92
1.91
1,29
Avg.
3.12
3.45
0.80
1.72
1.25
0.99
0.86
SD
3.28
2.64
0.89
0.70
0.56
0.54
0.40
Means of sensitivities
1.25
1.15
1.05
1.14
1.02
1.04
0.95
Standard deviations of sensitivities
0.79
0.87
0.71
0.85
0.71
0.70
0.59
Table 6
Comparison of Results of Detector-Based and Matrix-Corrected Calibration Methods of Tristimulus Colorimeters, Measuring Color LEDs (Five Filters)
LEDs (DWs, nm)
Colorimetric errors
Detector-Based Calibration Using CIE-A
Using Global Matrix Transformation for All Sources
Using Unique Matrix Transformation
Using Adaptive Matrix Transformation
Gaussian
Generalized Lorentzian
Triangle
β
450
1.17
1.93
0.65
0.36
0.45
0.43
0.20
Sensitivity
0.85
1.85
2.48
2.48
0.86
2.49
2.48
490
6.83
2.20
0.10
0.89
0.53
0.47
0.14
Sensitivity
0.20
0.85
0.67
0.67
0.87
0.67
0.67
522
2.10
0.15
0.40
0.28
0.83
0.70
0.38
Sensitivity
0.27
0.34
0.42
0.42
0.42
0.42
0.42
582
0.42
1.13
0.10
1.38
0.49
0.34
0.05
Sensitivity
0.42
1.41
1.49
1.50
1.38
1.50
1.46
592
0.52
1.33
0.80
1.75
1.26
1.23
1.07
Sensitivity
0.52
1.53
1.63
1.61
1.62
1.63
1.60
628
7.70
3.83
2.12
1.29
1.10
0.41
0.24
Sensitivity
2.56
0.88
0.66
0.81
1.97
0.95
0.85
Avg.
3.12
1.76
0.70
0.99
0.78
0.60
0.45
SD
3.28
1.24
0.75
0.59
0.34
0.33
0.54
Means of sensitivities
0.80
1.14
1.23
1.25
1.19
1.28
0.76
Standard deviations of sensitivities
0.89
0.55
0.79
0.76
0.57
0.76
0.13
Tables (6)
Table 1
Average Colorimetric Errors for Planckian Sources and a Number of LEDsa
Detector-Based Calibration Using CIE Illuminant A Distribution
Here the reference white is , , . LEDs: CW, cold white; WW, warm white; B, blue; C, cyan; G, green; Y, yellow; O, orange; R, red. The sensitivity is determined by the 1% noncorrelated noise (see the definition in Subsection 5E).
Table 2
Global Matrices for Different Kinds of Tristimulus Colorimetric Systems
Calibrated to CIE-A
Optimized to All Sources, 4 Filters
Optimized to All Sources,
Optimized to All Sources, 5 Filters
0.6013
0
0
0.806
0.1221
2.81962
5.67519
13.953
0
0
14.188
0.0978
0.03237
13.4261
13.6225
0.04983
0
1
0
1
0.02701
1
0.00302
0.02311
1
0
0
3.0601
3.25214
2.04086
0.66784
1.91104
0.14621
0
0
0
0
0
0
0.05164
7.44134
Table 3
Results of Unique Matrix Transformations for Cyan LEDsa
Optimal Matrix Transformation, Cyan LEDs
4 Filters
5 Filters
Avg.
SD
Avg.
SD
Avg.
SD
Avg.
SD
All
Colorimetric error
27.87
20.10
32.00
28.62
8.71
8.3
10.75
10.54
Sensitivity
0.88
0.25
1.39
0.99
1.06
0.56
1.40
0.79
Planck
Colorimetric error
27.38
2.22
38.86
4.92
9.1
0.75
12.69
1.91
Sensitivity
0.86
0.00
1.49
0.14
0.85
0.01
1.39
0.10
ΣLED
Colorimetric error
28.10
21.88
31.01
31.01
8.67
9.04
10.48
11.44
Sensitivity
0.89
0.27
1.38
1.08
1.1
0.6
1.40
0.86
CW
Colorimetric error
19.28
0.65
21.89
1.12
6.6
0.77
6.87
0.85
Sensitivity
0.9
0.02
1.05
0.04
1.01
0.04
1.21
0.03
WW
Colorimetric error
24.8
0.42
33.66
0.96
7.91
0.29
10.24
0.30
Sensitivity
0.85
0.01
1.38
0.03
0.84
0.02
1.31
0.02
B
Colorimetric error
17.35
20.54
6.77
5.50
12.30
17.69
4.52
4.07
Sensitivity
1.33
0.34
0.70
0.22
2.31
0.48
1.20
0.11
C
Colorimetric error
0.25
0.33
0.15
0.16
0.04
0.04
0.03
0.04
Sensitivity
0.67
0.05
0.23
0.02
1.32
0.34
0.56
0.16
G
Colorimetric error
11.77
15.39
5.43
7.63
3.16
6.03
1.49
2.89
Sensitivity
0.6
0.04
0.45
0.25
0.61
0.04
0.45
0.19
Y
Colorimetric error
49.21
0.22
39.59
5.48
13.59
2.34
13.63
1.54
Sensitivity
0.76
0.02
1.51
0.19
0.73
0.02
1.32
0.17
O
Colorimetric error
48.26
3.36
52.00
4.08
9.69
0.59
16.28
0.32
Sensitivity
0.82
0.03
2.01
0.25
0.78
0.04
1.77
0.22
R
Colorimetric error
57.83
5.06
86.16
17.52
14.46
5.82
28.95
13.41
Sensitivity
0.96
0.03
3.40
0.33
0.91
0.03
3.00
0.30
Avg., average; SD, standard deviation.
Table 4
Optimal Unique Matrices Optimized to Cyan LED Sources
4 Filters
5 Filters
0.6489
2.78267
16.967
0.0005
0.04162
14.6454
0.04459
1
0.03596
0.00018
1
3.08723
1.5321
0.26799
0.04434
0
0
0
3.77237
Table 5
Comparison of Results for Detector-Based and Matrix-Corrected Calibration Methods of Tristimulus Colorimeters, Measuring Color LEDs (Four Filters)
LEDs (DWs, nm)
Colorimetric errors
Detector-Based Calibration Using CIE-A
Using Global Matrix Transformation for All Sources
Using Unique Matrix Transformation
Using Adaptive Matrix Transformation
Gaussian
Generalized Lorentzian
Triangle
β
450
1.17
5.35
0.74
0.74
0.79
0.56
0,94
Sensitivity
0.85
0.88
0.87
0.89
0.73
0.83
0,86
490
6.83
1.82
0.13
1.31
1.50
1.32
1,14
Sensitivity
0.86
0.24
0.23
0.23
0.23
0.23
0,23
522
2.10
2.49
0.45
1.37
1.26
1.55
1,13
Sensitivity
0.27
0.28
0.28
0.28
0.28
0.28
0,28
582
0.42
1.36
0.14
2.54
0.45
0.35
0,51
Sensitivity
1.41
1.41
1.41
1.43
1.42
1.42
1,42
592
0.52
1.73
0.82
1.93
1.50
1.54
1,21
Sensitivity
1.53
1.53
1.54
1.59
1.54
1.55
1,61
628
7.70
7.95
2.53
2.40
2.00
0.62
0,24
Sensitivity
2.57
2.53
1.99
2.43
1.92
1.91
1,29
Avg.
3.12
3.45
0.80
1.72
1.25
0.99
0.86
SD
3.28
2.64
0.89
0.70
0.56
0.54
0.40
Means of sensitivities
1.25
1.15
1.05
1.14
1.02
1.04
0.95
Standard deviations of sensitivities
0.79
0.87
0.71
0.85
0.71
0.70
0.59
Table 6
Comparison of Results of Detector-Based and Matrix-Corrected Calibration Methods of Tristimulus Colorimeters, Measuring Color LEDs (Five Filters)
LEDs (DWs, nm)
Colorimetric errors
Detector-Based Calibration Using CIE-A
Using Global Matrix Transformation for All Sources