Abstract

For evaluating the performance of color-difference equations, several goodness-of-fit measures were proposed in the past, such as Pearson’s correlation coefficient (r), the performance factor PF/3, and the recently proposed standardized residual sum of squares (STRESS) measure. The STRESS shares its main advantage, which is the possibility to statistically test performance differences, with the correlation coefficient. We show, by mathematical analysis supported by instructive numerical examples, that the STRESS has no meaningful interpretation in this regression analysis context. In addition, we present objections to the use of the STRESS for evaluating color- difference equations. Therefore, we recommend using the correlation coefficient in combination with a graphical and diagnostics analysis to ensure proper application as with any statistical technique.

© 2011 Optical Society of America

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    [CrossRef]
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2011 (2)

S. G. Lee, R. Shamey, D. Hinks, and W. Jasper, “Development of a comprehensive visual dataset based on a CIE blue color center: assessment of color difference formulae using various statistical methods,” Color Res. Appl. 36, 27–41 (2011).
[CrossRef]

S. Shen and R. S. Berns, “Color-difference formula performance for several datasets of small color differences based on visual uncertainty,” Color Res. Appl. 36, 15–26 (2011).
[CrossRef]

2010 (5)

2009 (3)

W.-G. Kuo, Y.-C. Wei, A. Liu, S.-M. Lin, and Y.-T. Shu, “The new deviate visual functions improving the instrumental estimation on the visual color difference for the metamers with about 1° field size,” Color Res. Appl. 34, 115–127 (2009).
[CrossRef]

S. Shen and R. S. Berns, “Evaluating color difference equation performance incorporating visual uncertainty,” Color Res. Appl. 34, 375–390 (2009).
[CrossRef]

E. Chorro, F. M. Martínez-Verdú, D. de Fez, P. Capilla, and M. J. Luque, “Analyzing the metrics of the perceptual space in a new multistage physiological colour vision model,” Color Res. Appl. 34, 359–366 (2009).
[CrossRef]

2008 (2)

2007 (1)

2006 (1)

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[CrossRef]

2005 (3)

H. Xu and H. Yaguchi, “Visual evaluation at scale of threshold to suprathreshold color difference,” Color Res. Appl. 30, 198–208(2005).
[CrossRef]

J. H. Xin, H. L. Shen, and C. C. Lam, “Investigation of texture effect on visual colour difference evaluation,” Color Res. Appl. 30, 341–347 (2005).
[CrossRef]

J. M. Gibert, J. M. Daga, E. J. Gilabert, and J. Valldeperas, “Evaluation of colour difference formulae,” Color Technol. 121, 147–152 (2005).
[CrossRef]

2004 (3)

J. H. Xin, C. C. Lam, and M. R. Luo, “Evaluation of the crispening effect using CRT-displayed colour samples,” Color Res. Appl. 29, 374–380 (2004).
[CrossRef]

M. Melgosa, R. Huertas, and R. S. Berns, “Relative significance of the terms in the CIEDE2000 and CIE94 color-difference formulas,” J. Opt. Soc. Am. A 21, 2269–2275 (2004).
[CrossRef]

M. Melgosa, R. Huertas, A. Yebra, and M. M. Pérez, “Are chroma tolerances dependent on hue-angle?” Color Res. Appl. 29, 420–427 (2004).
[CrossRef]

2003 (1)

E. D. Montag and D. C. Wilber, “A comparison of constant stimuli and gray-scale methods of color difference scaling,” Color Res. Appl. 28, 36–44 (2003).
[CrossRef]

2002 (2)

M. R. Luo, “Development of colour-difference formulae,” Rev. Prog. Color. Relat. Top. 32, 28–39 (2002).
[CrossRef]

Y.-M. Lam and J. H. Xin, “Evaluation of the quality of different D65 simulators for visual assessment,” Color Res. Appl. 27, 243–251 (2002).
[CrossRef]

2001 (2)

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE colour-difference formula: CIEDE2000,” Color Res. Appl. 26, 340–350 (2001).
[CrossRef]

J. H. Xin, C. C. Lam, and M. R. Luo, “Investigation of parametric effects using medium colour differences,” Color Res. Appl. 26, 376–383 (2001).
[CrossRef]

2000 (1)

G. G. Attridge and M. R. Pointer, “Some aspects of the visual scaling of large colour differences—II,” Color Res. Appl. 25, 116–122 (2000).
[CrossRef]

1999 (2)

S. S. Guan and M. R. Luo, “Investigation of parametric effects using small colour differences,” Color Res. Appl. 24, 331–343 (1999).
[CrossRef]

S. S. Guan and M. R. Luo, “A colour-difference formula for assessing large colour differences,” Color Res. Appl. 24, 344–355(1999).
[CrossRef]

1996 (1)

M. R. Luo, M.-C. Lo, and W.-G. Kuo, “The LLAB (l:c) colour model,” Color Res. Appl. 21, 412–429 (1996).
[CrossRef]

1992 (1)

A. F. Bissell, “Lines through the origin—is NO INT the answer?” J. Appl. Stat. 19, 193–210 (1992).
[CrossRef]

1987 (2)

M. R. Luo and B. Rigg, “BFD (l:c) colour-difference formula. II. Performance of the formula,” J. Soc. Dyers Colour. 103, 126–132 (1987).
[CrossRef]

M. R. Luo and B. Rigg, “BFD (l:c) colour difference formula. I. Development of the formula,” J. Soc. Dyers Colour. 103, 86–94 (1987).
[CrossRef]

1986 (1)

M. R. Luo and B. Rigg, “Chromaticity-discrimination ellipses for surface colours,” Color Res. Appl. 11, 25–42 (1986).
[CrossRef]

1981 (2)

H. A. Gordon, “Errors in computer packages: least squares regression through the origin,” Statistician 30, 23–29 (1981).
[CrossRef]

E. Coates, K. Y. Fong, and B. Rigg, “Uniform lightness scales,” J. Soc. Dyers Colour. 97, 179–183 (1981).
[CrossRef]

1979 (1)

E. C. Poulton, “Models for biases in judging sensory magnitude,” Psychol. Bull. 86, 777–803 (1979).
[CrossRef] [PubMed]

1977 (1)

D. R. Cook, “Detection of influential observations in linear regression,” Technometrics 19, 15–18 (1977).
[CrossRef]

1976 (1)

R. T. Marcus, M. L. Utter, and J. W. Wilkinson, “Regression analysis and the evaluation of color-difference metrics,” Color Res. Appl. 1, 87–90 (1976).

1972 (1)

E. Coates, S. Day, J. R. Provost, and B. Rigg, “The measurement and assessment of colour differences for industrial use. II. The accuracy of colour-difference equations,” J. Soc. Dyers Colour. 88, 69–75 (1972).
[CrossRef]

1970 (1)

K. McLaren, “Colour passing—visual or instrumental?” J. Soc. Dyers Colour. 86, 389–392 (1970).
[CrossRef]

1964 (2)

J. B. Kruskal, “Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis,” Psychometrika 29, 1–27 (1964).
[CrossRef]

J. B. Kruskal, “Nonmetric multidimensional scaling: a numerical method,” Psychometrika 29, 115–129 (1964).
[CrossRef]

Attridge, G. G.

G. G. Attridge and M. R. Pointer, “Some aspects of the visual scaling of large colour differences—II,” Color Res. Appl. 25, 116–122 (2000).
[CrossRef]

Badu, A. S.

A. S. Badu, “Large colour differences between surface colours,” Ph.D. dissertation (University of Bradford, 1986).

Berns, R. S.

S. Shen and R. S. Berns, “Color-difference formula performance for several datasets of small color differences based on visual uncertainty,” Color Res. Appl. 36, 15–26 (2011).
[CrossRef]

R. S. Berns and B. Hou, “RIT-DuPont supra-threshold color-tolerance individual color-difference pair dataset,” Color Res. Appl. 35, 274–283 (2010).
[CrossRef]

S. Shen and R. S. Berns, “Evaluating color difference equation performance incorporating visual uncertainty,” Color Res. Appl. 34, 375–390 (2009).
[CrossRef]

M. Melgosa, R. Huertas, and R. S. Berns, “Performance of recent advanced color-difference formulae using the standardized residual sum of squares index,” J. Opt. Soc. Am. A 25, 1828–1834(2008).
[CrossRef]

M. Melgosa, R. Huertas, and R. S. Berns, “Relative significance of the terms in the CIEDE2000 and CIE94 color-difference formulas,” J. Opt. Soc. Am. A 21, 2269–2275 (2004).
[CrossRef]

Bismillah, J. K.

M. Melgosa, J. Romero, R. Huertas, L. Gomez-Robledo, P. A. García, D. Khaustova, S. Q. Bukhari, J. K. Bismillah, P. Wang, and A. Tremeau, “Color-difference measurements using 9-steps gray scales,” in Proceedings of the Interim Meeting of the International Color Association (AIC2010), p. 131.

Bissell, A. F.

A. F. Bissell, “Lines through the origin—is NO INT the answer?” J. Appl. Stat. 19, 193–210 (1992).
[CrossRef]

Bukhari, S. Q.

M. Melgosa, J. Romero, R. Huertas, L. Gomez-Robledo, P. A. García, D. Khaustova, S. Q. Bukhari, J. K. Bismillah, P. Wang, and A. Tremeau, “Color-difference measurements using 9-steps gray scales,” in Proceedings of the Interim Meeting of the International Color Association (AIC2010), p. 131.

Capilla, P.

E. Chorro, F. M. Martínez-Verdú, D. de Fez, P. Capilla, and M. J. Luque, “Analyzing the metrics of the perceptual space in a new multistage physiological colour vision model,” Color Res. Appl. 34, 359–366 (2009).
[CrossRef]

Cardenas, L.

R. Shamey, D. Hinks, M. Melgosa, M. R. Luo, G. Cui, R. Huertas, L. Cardenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two NCSU experimental datasets,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging, and Vision (CGIV, 2010), pp. 423–428.

Cárdenas, L. M.

Chorro, E.

E. Chorro, F. M. Martínez-Verdú, D. de Fez, P. Capilla, and M. J. Luque, “Analyzing the metrics of the perceptual space in a new multistage physiological colour vision model,” Color Res. Appl. 34, 359–366 (2009).
[CrossRef]

Coates, E.

E. Coates, K. Y. Fong, and B. Rigg, “Uniform lightness scales,” J. Soc. Dyers Colour. 97, 179–183 (1981).
[CrossRef]

E. Coates, S. Day, J. R. Provost, and B. Rigg, “The measurement and assessment of colour differences for industrial use. II. The accuracy of colour-difference equations,” J. Soc. Dyers Colour. 88, 69–75 (1972).
[CrossRef]

Cook, D. R.

D. R. Cook, “Detection of influential observations in linear regression,” Technometrics 19, 15–18 (1977).
[CrossRef]

Cui, G.

Z. Huang, H. Xu, M. R. Luo, G. Cui, and H. Feng, “Assessing total differences for effective samples having variations in color, coarseness, and glint,” Chin. Opt. Lett. 8, 717–720 (2010).
[CrossRef]

P. A. García, R. Huertas, M. Melgosa, and G. Cui, “Measurement of the relationship between perceived and computed color differences,” J. Opt. Soc. Am. A 24, 1823–1829 (2007).
[CrossRef]

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[CrossRef]

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE colour-difference formula: CIEDE2000,” Color Res. Appl. 26, 340–350 (2001).
[CrossRef]

R. Shamey, D. Hinks, M. Melgosa, M. R. Luo, G. Cui, R. Huertas, L. Cardenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two NCSU experimental datasets,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging, and Vision (CGIV, 2010), pp. 423–428.

Y. Pan, G. Cui, and R. Luo, “Gloss effect on colour difference evaluation,” in Proceedings of the Interim Meeting of the International Color Association (AIC, 2007), pp. 199–202.

G. Cui, M. R. Luo, and B. Rigg, “Colour difference evaluation under illuminant A,” in Proceedings of the 10th Congress of the International Colour Association (AIC Colour 05) (AIC, 2005), pp. 575–578.

Daga, J. M.

J. M. Gibert, J. M. Daga, E. J. Gilabert, and J. Valldeperas, “Evaluation of colour difference formulae,” Color Technol. 121, 147–152 (2005).
[CrossRef]

Day, S.

E. Coates, S. Day, J. R. Provost, and B. Rigg, “The measurement and assessment of colour differences for industrial use. II. The accuracy of colour-difference equations,” J. Soc. Dyers Colour. 88, 69–75 (1972).
[CrossRef]

de Fez, D.

E. Chorro, F. M. Martínez-Verdú, D. de Fez, P. Capilla, and M. J. Luque, “Analyzing the metrics of the perceptual space in a new multistage physiological colour vision model,” Color Res. Appl. 34, 359–366 (2009).
[CrossRef]

Domino, G.

G. Domino and M. L. Domino, Psychological Testing—An Introduction, 2nd ed. (Cambridge University, 2006), p. 139.

Domino, M. L.

G. Domino and M. L. Domino, Psychological Testing—An Introduction, 2nd ed. (Cambridge University, 2006), p. 139.

Fairchild, M. D.

M. D. Fairchild, Color Appearance Models, 2nd ed. (Wiley, 2005), pp. 49–50.

Feng, H.

Fong, K. Y.

E. Coates, K. Y. Fong, and B. Rigg, “Uniform lightness scales,” J. Soc. Dyers Colour. 97, 179–183 (1981).
[CrossRef]

García, P. A.

P. A. García, R. Huertas, M. Melgosa, and G. Cui, “Measurement of the relationship between perceived and computed color differences,” J. Opt. Soc. Am. A 24, 1823–1829 (2007).
[CrossRef]

M. Melgosa, R. Huertas, and P. A. García, “Performance of CIEDE2000 color difference models for the RI-DuPont dataset,” in Proceedings of the Interim Meeting of the International Color Association (AIC, 2007), pp. 293–295.

M. Melgosa, J. Romero, R. Huertas, L. Gomez-Robledo, P. A. García, D. Khaustova, S. Q. Bukhari, J. K. Bismillah, P. Wang, and A. Tremeau, “Color-difference measurements using 9-steps gray scales,” in Proceedings of the Interim Meeting of the International Color Association (AIC2010), p. 131.

Gibert, J. M.

J. M. Gibert, J. M. Daga, E. J. Gilabert, and J. Valldeperas, “Evaluation of colour difference formulae,” Color Technol. 121, 147–152 (2005).
[CrossRef]

Gilabert, E. J.

J. M. Gibert, J. M. Daga, E. J. Gilabert, and J. Valldeperas, “Evaluation of colour difference formulae,” Color Technol. 121, 147–152 (2005).
[CrossRef]

Gomez-Robledo, L.

M. Melgosa, J. Romero, R. Huertas, L. Gomez-Robledo, P. A. García, D. Khaustova, S. Q. Bukhari, J. K. Bismillah, P. Wang, and A. Tremeau, “Color-difference measurements using 9-steps gray scales,” in Proceedings of the Interim Meeting of the International Color Association (AIC2010), p. 131.

Gordon, H. A.

H. A. Gordon, “Errors in computer packages: least squares regression through the origin,” Statistician 30, 23–29 (1981).
[CrossRef]

Guan, S. S.

S. S. Guan and M. R. Luo, “A colour-difference formula for assessing large colour differences,” Color Res. Appl. 24, 344–355(1999).
[CrossRef]

S. S. Guan and M. R. Luo, “Investigation of parametric effects using small colour differences,” Color Res. Appl. 24, 331–343 (1999).
[CrossRef]

Guilford, J. P.

J. P. Guilford, Psychometric Methods (McGraw-Hill, 1954), p. 265.

Hinks, D.

S. G. Lee, R. Shamey, D. Hinks, and W. Jasper, “Development of a comprehensive visual dataset based on a CIE blue color center: assessment of color difference formulae using various statistical methods,” Color Res. Appl. 36, 27–41 (2011).
[CrossRef]

R. Shamey, L. M. Cárdenas, D. Hinks, and R. Woodard, “Comparison of naïve and expert subjects in the assessment of small color differences,” J. Opt. Soc. Am. A 27, 1482–1489(2010).
[CrossRef]

R. Shamey, D. Hinks, M. Melgosa, M. R. Luo, G. Cui, R. Huertas, L. Cardenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two NCSU experimental datasets,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging, and Vision (CGIV, 2010), pp. 423–428.

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R. S. Berns and B. Hou, “RIT-DuPont supra-threshold color-tolerance individual color-difference pair dataset,” Color Res. Appl. 35, 274–283 (2010).
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Huertas, R.

M. Melgosa, R. Huertas, and R. S. Berns, “Performance of recent advanced color-difference formulae using the standardized residual sum of squares index,” J. Opt. Soc. Am. A 25, 1828–1834(2008).
[CrossRef]

P. A. García, R. Huertas, M. Melgosa, and G. Cui, “Measurement of the relationship between perceived and computed color differences,” J. Opt. Soc. Am. A 24, 1823–1829 (2007).
[CrossRef]

M. Melgosa, R. Huertas, and R. S. Berns, “Relative significance of the terms in the CIEDE2000 and CIE94 color-difference formulas,” J. Opt. Soc. Am. A 21, 2269–2275 (2004).
[CrossRef]

M. Melgosa, R. Huertas, A. Yebra, and M. M. Pérez, “Are chroma tolerances dependent on hue-angle?” Color Res. Appl. 29, 420–427 (2004).
[CrossRef]

M. Melgosa, J. Romero, R. Huertas, L. Gomez-Robledo, P. A. García, D. Khaustova, S. Q. Bukhari, J. K. Bismillah, P. Wang, and A. Tremeau, “Color-difference measurements using 9-steps gray scales,” in Proceedings of the Interim Meeting of the International Color Association (AIC2010), p. 131.

R. Huertas, M. J. Rivas, and M. Melgosa, “Assessing large color differences using 3 step color series,” in Proceedings of the Fourth European Conference on Colour in Graphics, Imaging, and Vision (CGIV, 2008), pp. 85–88.

M. Melgosa, R. Huertas, and P. A. García, “Performance of CIEDE2000 color difference models for the RI-DuPont dataset,” in Proceedings of the Interim Meeting of the International Color Association (AIC, 2007), pp. 293–295.

R. Shamey, D. Hinks, M. Melgosa, M. R. Luo, G. Cui, R. Huertas, L. Cardenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two NCSU experimental datasets,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging, and Vision (CGIV, 2010), pp. 423–428.

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S. G. Lee, R. Shamey, D. Hinks, and W. Jasper, “Development of a comprehensive visual dataset based on a CIE blue color center: assessment of color difference formulae using various statistical methods,” Color Res. Appl. 36, 27–41 (2011).
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S. G. Kandi and M. A. Tehran, “Investigating the effect of texture on the performance of color difference formulae,” Color Res. Appl. 35, 94–100 (2010).

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M. Melgosa, J. Romero, R. Huertas, L. Gomez-Robledo, P. A. García, D. Khaustova, S. Q. Bukhari, J. K. Bismillah, P. Wang, and A. Tremeau, “Color-difference measurements using 9-steps gray scales,” in Proceedings of the Interim Meeting of the International Color Association (AIC2010), p. 131.

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W.-G. Kuo, Y.-C. Wei, A. Liu, S.-M. Lin, and Y.-T. Shu, “The new deviate visual functions improving the instrumental estimation on the visual color difference for the metamers with about 1° field size,” Color Res. Appl. 34, 115–127 (2009).
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J. H. Xin, H. L. Shen, and C. C. Lam, “Investigation of texture effect on visual colour difference evaluation,” Color Res. Appl. 30, 341–347 (2005).
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J. H. Xin, C. C. Lam, and M. R. Luo, “Evaluation of the crispening effect using CRT-displayed colour samples,” Color Res. Appl. 29, 374–380 (2004).
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J. H. Xin, C. C. Lam, and M. R. Luo, “Investigation of parametric effects using medium colour differences,” Color Res. Appl. 26, 376–383 (2001).
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Y.-M. Lam and J. H. Xin, “Evaluation of the quality of different D65 simulators for visual assessment,” Color Res. Appl. 27, 243–251 (2002).
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S. G. Lee, R. Shamey, D. Hinks, and W. Jasper, “Development of a comprehensive visual dataset based on a CIE blue color center: assessment of color difference formulae using various statistical methods,” Color Res. Appl. 36, 27–41 (2011).
[CrossRef]

R. Shamey, D. Hinks, M. Melgosa, M. R. Luo, G. Cui, R. Huertas, L. Cardenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two NCSU experimental datasets,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging, and Vision (CGIV, 2010), pp. 423–428.

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M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
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W.-G. Kuo, Y.-C. Wei, A. Liu, S.-M. Lin, and Y.-T. Shu, “The new deviate visual functions improving the instrumental estimation on the visual color difference for the metamers with about 1° field size,” Color Res. Appl. 34, 115–127 (2009).
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Liu, A.

W.-G. Kuo, Y.-C. Wei, A. Liu, S.-M. Lin, and Y.-T. Shu, “The new deviate visual functions improving the instrumental estimation on the visual color difference for the metamers with about 1° field size,” Color Res. Appl. 34, 115–127 (2009).
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M. R. Luo, M.-C. Lo, and W.-G. Kuo, “The LLAB (l:c) colour model,” Color Res. Appl. 21, 412–429 (1996).
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Z. Huang, H. Xu, M. R. Luo, G. Cui, and H. Feng, “Assessing total differences for effective samples having variations in color, coarseness, and glint,” Chin. Opt. Lett. 8, 717–720 (2010).
[CrossRef]

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
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J. H. Xin, C. C. Lam, and M. R. Luo, “Evaluation of the crispening effect using CRT-displayed colour samples,” Color Res. Appl. 29, 374–380 (2004).
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S. S. Guan and M. R. Luo, “A colour-difference formula for assessing large colour differences,” Color Res. Appl. 24, 344–355(1999).
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S. S. Guan and M. R. Luo, “Investigation of parametric effects using small colour differences,” Color Res. Appl. 24, 331–343 (1999).
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M. R. Luo, M.-C. Lo, and W.-G. Kuo, “The LLAB (l:c) colour model,” Color Res. Appl. 21, 412–429 (1996).
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G. Cui, M. R. Luo, and B. Rigg, “Colour difference evaluation under illuminant A,” in Proceedings of the 10th Congress of the International Colour Association (AIC Colour 05) (AIC, 2005), pp. 575–578.

Luo, R.

Y. Pan, G. Cui, and R. Luo, “Gloss effect on colour difference evaluation,” in Proceedings of the Interim Meeting of the International Color Association (AIC, 2007), pp. 199–202.

Luque, M. J.

E. Chorro, F. M. Martínez-Verdú, D. de Fez, P. Capilla, and M. J. Luque, “Analyzing the metrics of the perceptual space in a new multistage physiological colour vision model,” Color Res. Appl. 34, 359–366 (2009).
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R. T. Marcus, M. L. Utter, and J. W. Wilkinson, “Regression analysis and the evaluation of color-difference metrics,” Color Res. Appl. 1, 87–90 (1976).

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E. Chorro, F. M. Martínez-Verdú, D. de Fez, P. Capilla, and M. J. Luque, “Analyzing the metrics of the perceptual space in a new multistage physiological colour vision model,” Color Res. Appl. 34, 359–366 (2009).
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K. McLaren, “Colour passing—visual or instrumental?” J. Soc. Dyers Colour. 86, 389–392 (1970).
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M. Melgosa, R. Huertas, and R. S. Berns, “Performance of recent advanced color-difference formulae using the standardized residual sum of squares index,” J. Opt. Soc. Am. A 25, 1828–1834(2008).
[CrossRef]

P. A. García, R. Huertas, M. Melgosa, and G. Cui, “Measurement of the relationship between perceived and computed color differences,” J. Opt. Soc. Am. A 24, 1823–1829 (2007).
[CrossRef]

M. Melgosa, R. Huertas, and R. S. Berns, “Relative significance of the terms in the CIEDE2000 and CIE94 color-difference formulas,” J. Opt. Soc. Am. A 21, 2269–2275 (2004).
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M. Melgosa, R. Huertas, A. Yebra, and M. M. Pérez, “Are chroma tolerances dependent on hue-angle?” Color Res. Appl. 29, 420–427 (2004).
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M. Melgosa, J. Romero, R. Huertas, L. Gomez-Robledo, P. A. García, D. Khaustova, S. Q. Bukhari, J. K. Bismillah, P. Wang, and A. Tremeau, “Color-difference measurements using 9-steps gray scales,” in Proceedings of the Interim Meeting of the International Color Association (AIC2010), p. 131.

R. Huertas, M. J. Rivas, and M. Melgosa, “Assessing large color differences using 3 step color series,” in Proceedings of the Fourth European Conference on Colour in Graphics, Imaging, and Vision (CGIV, 2008), pp. 85–88.

M. Melgosa, R. Huertas, and P. A. García, “Performance of CIEDE2000 color difference models for the RI-DuPont dataset,” in Proceedings of the Interim Meeting of the International Color Association (AIC, 2007), pp. 293–295.

R. Shamey, D. Hinks, M. Melgosa, M. R. Luo, G. Cui, R. Huertas, L. Cardenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two NCSU experimental datasets,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging, and Vision (CGIV, 2010), pp. 423–428.

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E. D. Montag and D. C. Wilber, “A comparison of constant stimuli and gray-scale methods of color difference scaling,” Color Res. Appl. 28, 36–44 (2003).
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Y. Pan, G. Cui, and R. Luo, “Gloss effect on colour difference evaluation,” in Proceedings of the Interim Meeting of the International Color Association (AIC, 2007), pp. 199–202.

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M. Melgosa, R. Huertas, A. Yebra, and M. M. Pérez, “Are chroma tolerances dependent on hue-angle?” Color Res. Appl. 29, 420–427 (2004).
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G. G. Attridge and M. R. Pointer, “Some aspects of the visual scaling of large colour differences—II,” Color Res. Appl. 25, 116–122 (2000).
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M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE colour-difference formula: CIEDE2000,” Color Res. Appl. 26, 340–350 (2001).
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M. R. Luo and B. Rigg, “BFD (l:c) colour-difference formula. II. Performance of the formula,” J. Soc. Dyers Colour. 103, 126–132 (1987).
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M. R. Luo and B. Rigg, “BFD (l:c) colour difference formula. I. Development of the formula,” J. Soc. Dyers Colour. 103, 86–94 (1987).
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M. R. Luo and B. Rigg, “Chromaticity-discrimination ellipses for surface colours,” Color Res. Appl. 11, 25–42 (1986).
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E. Coates, K. Y. Fong, and B. Rigg, “Uniform lightness scales,” J. Soc. Dyers Colour. 97, 179–183 (1981).
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E. Coates, S. Day, J. R. Provost, and B. Rigg, “The measurement and assessment of colour differences for industrial use. II. The accuracy of colour-difference equations,” J. Soc. Dyers Colour. 88, 69–75 (1972).
[CrossRef]

G. Cui, M. R. Luo, and B. Rigg, “Colour difference evaluation under illuminant A,” in Proceedings of the 10th Congress of the International Colour Association (AIC Colour 05) (AIC, 2005), pp. 575–578.

Rivas, M. J.

R. Huertas, M. J. Rivas, and M. Melgosa, “Assessing large color differences using 3 step color series,” in Proceedings of the Fourth European Conference on Colour in Graphics, Imaging, and Vision (CGIV, 2008), pp. 85–88.

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M. Melgosa, J. Romero, R. Huertas, L. Gomez-Robledo, P. A. García, D. Khaustova, S. Q. Bukhari, J. K. Bismillah, P. Wang, and A. Tremeau, “Color-difference measurements using 9-steps gray scales,” in Proceedings of the Interim Meeting of the International Color Association (AIC2010), p. 131.

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S. G. Lee, R. Shamey, D. Hinks, and W. Jasper, “Development of a comprehensive visual dataset based on a CIE blue color center: assessment of color difference formulae using various statistical methods,” Color Res. Appl. 36, 27–41 (2011).
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R. Shamey, L. M. Cárdenas, D. Hinks, and R. Woodard, “Comparison of naïve and expert subjects in the assessment of small color differences,” J. Opt. Soc. Am. A 27, 1482–1489(2010).
[CrossRef]

R. Shamey, D. Hinks, M. Melgosa, M. R. Luo, G. Cui, R. Huertas, L. Cardenas, and S. G. Lee, “Evaluation of performance of twelve color-difference formulae using two NCSU experimental datasets,” in Proceedings of the Fifth European Conference on Colour in Graphics, Imaging, and Vision (CGIV, 2010), pp. 423–428.

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J. H. Xin, H. L. Shen, and C. C. Lam, “Investigation of texture effect on visual colour difference evaluation,” Color Res. Appl. 30, 341–347 (2005).
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S. Shen and R. S. Berns, “Color-difference formula performance for several datasets of small color differences based on visual uncertainty,” Color Res. Appl. 36, 15–26 (2011).
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S. Shen and R. S. Berns, “Evaluating color difference equation performance incorporating visual uncertainty,” Color Res. Appl. 34, 375–390 (2009).
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W.-G. Kuo, Y.-C. Wei, A. Liu, S.-M. Lin, and Y.-T. Shu, “The new deviate visual functions improving the instrumental estimation on the visual color difference for the metamers with about 1° field size,” Color Res. Appl. 34, 115–127 (2009).
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S. G. Kandi and M. A. Tehran, “Investigating the effect of texture on the performance of color difference formulae,” Color Res. Appl. 35, 94–100 (2010).

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M. Melgosa, J. Romero, R. Huertas, L. Gomez-Robledo, P. A. García, D. Khaustova, S. Q. Bukhari, J. K. Bismillah, P. Wang, and A. Tremeau, “Color-difference measurements using 9-steps gray scales,” in Proceedings of the Interim Meeting of the International Color Association (AIC2010), p. 131.

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J. M. Gibert, J. M. Daga, E. J. Gilabert, and J. Valldeperas, “Evaluation of colour difference formulae,” Color Technol. 121, 147–152 (2005).
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Wang, P.

M. Melgosa, J. Romero, R. Huertas, L. Gomez-Robledo, P. A. García, D. Khaustova, S. Q. Bukhari, J. K. Bismillah, P. Wang, and A. Tremeau, “Color-difference measurements using 9-steps gray scales,” in Proceedings of the Interim Meeting of the International Color Association (AIC2010), p. 131.

Wei, Y.-C.

W.-G. Kuo, Y.-C. Wei, A. Liu, S.-M. Lin, and Y.-T. Shu, “The new deviate visual functions improving the instrumental estimation on the visual color difference for the metamers with about 1° field size,” Color Res. Appl. 34, 115–127 (2009).
[CrossRef]

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E. D. Montag and D. C. Wilber, “A comparison of constant stimuli and gray-scale methods of color difference scaling,” Color Res. Appl. 28, 36–44 (2003).
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R. T. Marcus, M. L. Utter, and J. W. Wilkinson, “Regression analysis and the evaluation of color-difference metrics,” Color Res. Appl. 1, 87–90 (1976).

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Xin, J. H.

J. H. Xin, H. L. Shen, and C. C. Lam, “Investigation of texture effect on visual colour difference evaluation,” Color Res. Appl. 30, 341–347 (2005).
[CrossRef]

J. H. Xin, C. C. Lam, and M. R. Luo, “Evaluation of the crispening effect using CRT-displayed colour samples,” Color Res. Appl. 29, 374–380 (2004).
[CrossRef]

Y.-M. Lam and J. H. Xin, “Evaluation of the quality of different D65 simulators for visual assessment,” Color Res. Appl. 27, 243–251 (2002).
[CrossRef]

J. H. Xin, C. C. Lam, and M. R. Luo, “Investigation of parametric effects using medium colour differences,” Color Res. Appl. 26, 376–383 (2001).
[CrossRef]

Xu, H.

Z. Huang, H. Xu, M. R. Luo, G. Cui, and H. Feng, “Assessing total differences for effective samples having variations in color, coarseness, and glint,” Chin. Opt. Lett. 8, 717–720 (2010).
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H. Xu and H. Yaguchi, “Visual evaluation at scale of threshold to suprathreshold color difference,” Color Res. Appl. 30, 198–208(2005).
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M. Melgosa, R. Huertas, A. Yebra, and M. M. Pérez, “Are chroma tolerances dependent on hue-angle?” Color Res. Appl. 29, 420–427 (2004).
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Chin. Opt. Lett. (1)

Color Res. Appl. (23)

R. T. Marcus, M. L. Utter, and J. W. Wilkinson, “Regression analysis and the evaluation of color-difference metrics,” Color Res. Appl. 1, 87–90 (1976).

M. R. Luo and B. Rigg, “Chromaticity-discrimination ellipses for surface colours,” Color Res. Appl. 11, 25–42 (1986).
[CrossRef]

S. G. Kandi and M. A. Tehran, “Investigating the effect of texture on the performance of color difference formulae,” Color Res. Appl. 35, 94–100 (2010).

S. G. Lee, R. Shamey, D. Hinks, and W. Jasper, “Development of a comprehensive visual dataset based on a CIE blue color center: assessment of color difference formulae using various statistical methods,” Color Res. Appl. 36, 27–41 (2011).
[CrossRef]

J. H. Xin, C. C. Lam, and M. R. Luo, “Evaluation of the crispening effect using CRT-displayed colour samples,” Color Res. Appl. 29, 374–380 (2004).
[CrossRef]

J. H. Xin, C. C. Lam, and M. R. Luo, “Investigation of parametric effects using medium colour differences,” Color Res. Appl. 26, 376–383 (2001).
[CrossRef]

G. G. Attridge and M. R. Pointer, “Some aspects of the visual scaling of large colour differences—II,” Color Res. Appl. 25, 116–122 (2000).
[CrossRef]

E. D. Montag and D. C. Wilber, “A comparison of constant stimuli and gray-scale methods of color difference scaling,” Color Res. Appl. 28, 36–44 (2003).
[CrossRef]

W.-G. Kuo, Y.-C. Wei, A. Liu, S.-M. Lin, and Y.-T. Shu, “The new deviate visual functions improving the instrumental estimation on the visual color difference for the metamers with about 1° field size,” Color Res. Appl. 34, 115–127 (2009).
[CrossRef]

M. R. Luo, M.-C. Lo, and W.-G. Kuo, “The LLAB (l:c) colour model,” Color Res. Appl. 21, 412–429 (1996).
[CrossRef]

S. S. Guan and M. R. Luo, “Investigation of parametric effects using small colour differences,” Color Res. Appl. 24, 331–343 (1999).
[CrossRef]

H. Xu and H. Yaguchi, “Visual evaluation at scale of threshold to suprathreshold color difference,” Color Res. Appl. 30, 198–208(2005).
[CrossRef]

J. H. Xin, H. L. Shen, and C. C. Lam, “Investigation of texture effect on visual colour difference evaluation,” Color Res. Appl. 30, 341–347 (2005).
[CrossRef]

M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31, 320–330 (2006).
[CrossRef]

E. Chorro, F. M. Martínez-Verdú, D. de Fez, P. Capilla, and M. J. Luque, “Analyzing the metrics of the perceptual space in a new multistage physiological colour vision model,” Color Res. Appl. 34, 359–366 (2009).
[CrossRef]

M. Melgosa, R. Huertas, A. Yebra, and M. M. Pérez, “Are chroma tolerances dependent on hue-angle?” Color Res. Appl. 29, 420–427 (2004).
[CrossRef]

Y.-M. Lam and J. H. Xin, “Evaluation of the quality of different D65 simulators for visual assessment,” Color Res. Appl. 27, 243–251 (2002).
[CrossRef]

R. G. Kuehni, “Color difference formulas: an unsatisfactory state of affairs,” Color Res. Appl. 33, 324–326 (2008).
[CrossRef]

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE colour-difference formula: CIEDE2000,” Color Res. Appl. 26, 340–350 (2001).
[CrossRef]

S. Shen and R. S. Berns, “Evaluating color difference equation performance incorporating visual uncertainty,” Color Res. Appl. 34, 375–390 (2009).
[CrossRef]

S. Shen and R. S. Berns, “Color-difference formula performance for several datasets of small color differences based on visual uncertainty,” Color Res. Appl. 36, 15–26 (2011).
[CrossRef]

R. S. Berns and B. Hou, “RIT-DuPont supra-threshold color-tolerance individual color-difference pair dataset,” Color Res. Appl. 35, 274–283 (2010).
[CrossRef]

S. S. Guan and M. R. Luo, “A colour-difference formula for assessing large colour differences,” Color Res. Appl. 24, 344–355(1999).
[CrossRef]

Color Technol. (1)

J. M. Gibert, J. M. Daga, E. J. Gilabert, and J. Valldeperas, “Evaluation of colour difference formulae,” Color Technol. 121, 147–152 (2005).
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J. Opt. Soc. Am. A (5)

J. Soc. Dyers Colour. (5)

E. Coates, S. Day, J. R. Provost, and B. Rigg, “The measurement and assessment of colour differences for industrial use. II. The accuracy of colour-difference equations,” J. Soc. Dyers Colour. 88, 69–75 (1972).
[CrossRef]

E. Coates, K. Y. Fong, and B. Rigg, “Uniform lightness scales,” J. Soc. Dyers Colour. 97, 179–183 (1981).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Artificial set of data points, as used by Marcus et al. [55], comprising (a) 32 randomly chosen data points with correlation coefficient r = 0.015 and S T R E S S = 0.382 ; (b) position of the location of a spurious data point as explained in Subsection 4D. See also Table 2.

Fig. 2
Fig. 2

Artificial set of data points, generated as discussed in Subsection 4D. The resulting values for the correlation coefficient r = 0.79 and S T R E S S = 0.36 .

Fig. 3
Fig. 3

Plots showing 100 data points generated around the same trend line as in Fig. 2 but normally distributed around (a) a center with x = 40 and standard deviation 30 and (b) a center with x = 60 and standard deviation 30.

Tables (4)

Tables Icon

Table 1 Measures That Have Been Introduced to Test the Performance of Color-Difference Equations

Tables Icon

Table 2 Value of Correlation Coefficient and STRESS for the Data Points Shown in Fig. 1 a

Tables Icon

Table 3 Statistical Results for Different Distributions of Data Points a

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Table 4 Summarizing the Relative Merits of STRESS and the Correlation Coefficient

Equations (20)

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STRESS 1 = i = 1 n ( y i y ^ i ) 2 / i = 1 n y i 2 ,
STRESS 2 = i = 1 n ( y i y ^ i ) 2 / i = 1 n ( y i y ¯ ) 2 .
c 2 = ( i = 1 n x i y i ) 2 / [ ( i = 1 n y i 2 ) ( i = 1 n x i 2 ) ] .
STRESS 2 = i = 1 n ( y i F 1 x i ) 2 / [ F 1 2 i = 1 n x i 2 ] , with   F 1 = i = 1 n y i 2 / i = 1 n y i x i .
STRESS 2 = i = 1 n y i 2 F 1 2 i = 1 n x i 2 2 F 1 i = 1 n x i y i i = 1 n x i 2 + 1 = 1 ( i = 1 n x i y i ) 2 ( i = 1 n y i 2 ) ( i = 1 n x i 2 ) = 1 c 2 ,
r 2 = [ i = 1 n ( x i x ¯ ) ( y i y ¯ ) ] 2 [ i = 1 n ( x i x ¯ ) 2 ] [ i = 1 n ( y i y ¯ ) 2 ] = [ i = 1 n x i y i ] 2 [ i = 1 n x i 2 ] [ i = 1 n y i 2 ] = 1 STRESS 2
S S R = i = 1 n ( y i y ^ i ) 2 .
S S R = i = 1 n ( y i a b x i ) 2 .
d S S R d a = 0 and d S S R d b = 0.
b = Covar ( x , y ) Var ( x ) and a = y ¯ b x ¯ = y ¯ x ¯  Covar ( x , y ) Var ( x ) ,
Var ( x ) = 1 n 1 i = 1 n ( x i x ¯ ) 2 and Covar ( x , y ) = 1 n 1 i = 1 n ( x i x ¯ ) ( y i y ¯ ) .
r 2 = 1 S S R ( n 1 ) Var ( y ) = 1 1 ( n 1 ) i = 1 n ( y i a b x i ) 2 i = 1 n ( y i y ¯ ) 2 = [ i = 1 n ( x i x ¯ ) ( y i y ¯ ) ] 2 [ i = 1 n ( x i x ¯ ) 2 ] [ i = 1 n ( y i y ¯ ) 2 ] ,
S S R = i = 1 n ( y i b C x i ) 2 .
d S S R d b c = 0.
b C = i = 1 n x i y i / i = 1 n x i 2 .
r C 2 = 1 S S R i = 1 n ( y i 0 ) 2 = 1 i = 1 n ( y i b C x i ) 2 i = 1 n y i 2 ,
r C 2 = 1 i = 1 n y i 2 i = 1 n y i 2 + 2 ( i = 1 n x i y i ) 2 i = 1 n x i 2 i = 1 n y i 2 ( i = 1 n x i y i ) 2 ( i = 1 n x i 2 ) 2 i = 1 n x i 2 i = 1 n y i 2 ,
r C 2 = 1 i = 1 n b 1 2 x i 2 b 1 2 i = 1 n x i 2 + 2 b 1 b 1 2 i = 1 n x i y i i = 1 n x i 2 1 b 1 2 i = 1 n y i 2 i = 1 n x i 2 ,
r C 2 = 1 i = 1 n ( y i b 1 x i ) 2 b 1 2 i = 1 n x i 2 ,
with     b 1 = i = 1 n y i 2 / i = 1 n x i y i .

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