Claudio Oleari, Manuel Melgosa, and Rafael Huertas, "Euclidean color-difference formula for small-medium color differences in log-compressed OSA-UCS space," J. Opt. Soc. Am. A 26, 121-134 (2009)
This work continues previous research by the same authors [J. Opt. Soc. Am. A 23, 2077 (2006)
], where empirical small–medium color differences were represented by an ellipsoidal equation in the Uniform Color System of the Optical Society of America. Now logarithmic compressions on chroma and lightness are introduced to produce a new space with Euclidean color-difference formulas . The CIEDE2000, , and formulas are found statistically equivalent in the prediction of many available empirical datasets. However, is the simplest formula providing relationships with visual processing. These analyses hold true for CIE 1964 Supplementary Standard Observer and D65 illuminant.
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STRESS of Four Different Color-Difference Formulas on the Empirical Basis of the Constant Luminance BFD Ellipsesa
Dataset
Chroma Log Compressed
DIN99d
Parameters
STRESS
RMS
Parameters
STRESS
RMS
STRESS
STRESS
BFD ellipses
21.16
0.2116
21.93
0.2195
21.27
23.57
BFD-P ellipses
21.21
0.2121
22.42
0.2246
21.64
24.27
BFD-A ellipses
20.65
0.2065
20.82
0.2083
20.68
22.45
The fitted parameters and the RMS values for the and formulas are also given.
Table 2
F-Test among All the Color-Difference Formulas Considered, , , CIEDE2000, and DIN99da
Dataset
BFD ellipses
1.074
1.063
BFD-P ellipses
1.073
BFD-A ellipses
1.017
1.014
The STRESS values are given in Table 1. Underlined values indicate statistically significant differences at a 5% significance level.
Table 3
Equation Parameters , RMS, and STRESS Values for and for the Two Formulas, Computed for Each One of the Datasets Constituting the COM Datasets, Using Their Own Empirical Background Luminance (Second Column)
Equation Parameters , RMS, and STRESS Values for and for the Two Formulas, Computed for the Two COM Datasets, Using the Background Luminance , as in the OSA-UCS Space
The parameters of the and ’s formulas are given in Table 4.
Table 6
Comparison between and ’s Formulas Based on F-Test of the STRESS (Table 5) and on the Empirical Data Constituted by BFD Ellipses and by Individual and COM Datasetsa
The parameters of the formulas are given in Table 4. Underlined values indicate statistically significant differences at 5% significance level.
Table 7
Comparison between and All the Other Considered Formulas (, both ’s and DIN99d) Based on F-test of the STRESS (Table 5) and the Empirical Data Constituted by BFD Ellipses and by Individual and COM Datasetsa
Dataset
for Fitted on W-COM
for Fitted on unW-COM
for with Log-Compressed Lightness (15) Fitted on W-COM
for with Log-Compressed Lightness (15) Fitted on unW-COM
for with Rescaled Lightness (17) Fitted on W-COM
for with Rescaled Lightness (17) Fitted on unW-COM
for DIN99d Fitted on W-COM
BFD ellipses
1.000
1.068
1.018
1.015
1.009
1.007
1.228
BFD-P ellipses
0.972
1.006
0.975
0.973
0.971
0.969
1.258
BFD-A ellipses
1.022
1.134
1.059
1.055
1.046
1.040
1.179
BFD-D65
1.316
1.067
1.135
1.062
1.311
1.197
1.192
BFD-M
0.885
0.833
0.839
0.842
0.848
0.854
1.125
BFD-C
1.071
1.231
1.164
1.229
1.186
1.268
0.980
BFD-P
1.040
0.916
0.945
0.919
1.020
0.979
1.151
WITT
1.135
1.456
1.348
1.436
1.128
1.233
0.989
RIT-DuPont
1.555
1.879
1.689
1.859
1.849
1.990
1.152
Leeds
2.027
2.020
1.855
1.940
2.274
2.221
1.398
unW-COM
1.062
0.966
0.983
0.967
1.050
1.018
1.145
W-COM
1.167
1.182
1.150
1.170
1.190
1.193
1.131
The parameters of and both formulas are given in Table 4. Underlined values indicate statistically significant differences at a 5% significance level.
Table 8
Number of Independent Color Pairs N (in Brackets is the Number of Nonindependent Color Pairs Obtained by Weighting Factors and Employed in the Computation) and Critical Values of the Two-Tailed F-Distribution at a 5% Significance Level for Each One of the Individual Datasets Constituting the COM Datasets and for the W-COM and unW-COM Datasetsa
Dataset
BFD Ellipses
BFD-P Ellipses
BFD-A Ellipses
BFD-D65
BFD-M
BFD-C
BFD-P
WITT
RIT-DuPont
Leeds
unW-COM
W-COM
N
2028
548
200
2776
418
156 (312)
307
3657 (3813)
3657 (11273)
0.927
0.907
0.885
0.917
0.846
0.757
0.928
0.825
0.729
0.799
0.937
0.937
1.079
1.103
1.129
1.091
1.183
1.321
1.077
1.212
1.372
1.252
1.067
1.067
The critical values are computed taking into account the number of independent color pairs.
Tables (8)
Table 1
STRESS of Four Different Color-Difference Formulas on the Empirical Basis of the Constant Luminance BFD Ellipsesa
Dataset
Chroma Log Compressed
DIN99d
Parameters
STRESS
RMS
Parameters
STRESS
RMS
STRESS
STRESS
BFD ellipses
21.16
0.2116
21.93
0.2195
21.27
23.57
BFD-P ellipses
21.21
0.2121
22.42
0.2246
21.64
24.27
BFD-A ellipses
20.65
0.2065
20.82
0.2083
20.68
22.45
The fitted parameters and the RMS values for the and formulas are also given.
Table 2
F-Test among All the Color-Difference Formulas Considered, , , CIEDE2000, and DIN99da
Dataset
BFD ellipses
1.074
1.063
BFD-P ellipses
1.073
BFD-A ellipses
1.017
1.014
The STRESS values are given in Table 1. Underlined values indicate statistically significant differences at a 5% significance level.
Table 3
Equation Parameters , RMS, and STRESS Values for and for the Two Formulas, Computed for Each One of the Datasets Constituting the COM Datasets, Using Their Own Empirical Background Luminance (Second Column)
Equation Parameters , RMS, and STRESS Values for and for the Two Formulas, Computed for the Two COM Datasets, Using the Background Luminance , as in the OSA-UCS Space
The parameters of the and ’s formulas are given in Table 4.
Table 6
Comparison between and ’s Formulas Based on F-Test of the STRESS (Table 5) and on the Empirical Data Constituted by BFD Ellipses and by Individual and COM Datasetsa
The parameters of the formulas are given in Table 4. Underlined values indicate statistically significant differences at 5% significance level.
Table 7
Comparison between and All the Other Considered Formulas (, both ’s and DIN99d) Based on F-test of the STRESS (Table 5) and the Empirical Data Constituted by BFD Ellipses and by Individual and COM Datasetsa
Dataset
for Fitted on W-COM
for Fitted on unW-COM
for with Log-Compressed Lightness (15) Fitted on W-COM
for with Log-Compressed Lightness (15) Fitted on unW-COM
for with Rescaled Lightness (17) Fitted on W-COM
for with Rescaled Lightness (17) Fitted on unW-COM
for DIN99d Fitted on W-COM
BFD ellipses
1.000
1.068
1.018
1.015
1.009
1.007
1.228
BFD-P ellipses
0.972
1.006
0.975
0.973
0.971
0.969
1.258
BFD-A ellipses
1.022
1.134
1.059
1.055
1.046
1.040
1.179
BFD-D65
1.316
1.067
1.135
1.062
1.311
1.197
1.192
BFD-M
0.885
0.833
0.839
0.842
0.848
0.854
1.125
BFD-C
1.071
1.231
1.164
1.229
1.186
1.268
0.980
BFD-P
1.040
0.916
0.945
0.919
1.020
0.979
1.151
WITT
1.135
1.456
1.348
1.436
1.128
1.233
0.989
RIT-DuPont
1.555
1.879
1.689
1.859
1.849
1.990
1.152
Leeds
2.027
2.020
1.855
1.940
2.274
2.221
1.398
unW-COM
1.062
0.966
0.983
0.967
1.050
1.018
1.145
W-COM
1.167
1.182
1.150
1.170
1.190
1.193
1.131
The parameters of and both formulas are given in Table 4. Underlined values indicate statistically significant differences at a 5% significance level.
Table 8
Number of Independent Color Pairs N (in Brackets is the Number of Nonindependent Color Pairs Obtained by Weighting Factors and Employed in the Computation) and Critical Values of the Two-Tailed F-Distribution at a 5% Significance Level for Each One of the Individual Datasets Constituting the COM Datasets and for the W-COM and unW-COM Datasetsa
Dataset
BFD Ellipses
BFD-P Ellipses
BFD-A Ellipses
BFD-D65
BFD-M
BFD-C
BFD-P
WITT
RIT-DuPont
Leeds
unW-COM
W-COM
N
2028
548
200
2776
418
156 (312)
307
3657 (3813)
3657 (11273)
0.927
0.907
0.885
0.917
0.846
0.757
0.928
0.825
0.729
0.799
0.937
0.937
1.079
1.103
1.129
1.091
1.183
1.321
1.077
1.212
1.372
1.252
1.067
1.067
The critical values are computed taking into account the number of independent color pairs.