Abstract

Using simulated data sets, we have analyzed some mathematical properties of different statistical measurements that have been employed in previous literature to test the performance of different color-difference formulas. Specifically, the properties of the combined index PF3 (performance factor obtained as average of three terms), widely employed in current literature, have been considered. A new index named standardized residual sum of squares (STRESS), employed in multidimensional scaling techniques, is recommended. The main difference between PF3 and STRESS is that the latter is simpler and allows inferences on the statistical significance of two color-difference formulas with respect to a given set of visual data.

© 2007 Optical Society of America

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  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]
  2. M. R. Luo, "Development of colour-difference formulae," Rev. Prog. Color. Relat. Top. 32, 28-39 (2002).
    [CrossRef]
  3. 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]
  4. 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]
  5. J. M. Gibert, J. M. Dagà, E. J. Gilabert, J. Valldeperas, and the Colorimetry Group, "Evaluation of colour difference formulae," Color Technol. 121, 147-152 (2005).
    [CrossRef]
  6. H. Xu and H. Yaguchi, "Visual evaluation at scale of threshold to suprathreshold color difference," Color Res. Appl. 30, 198-208 (2005).
    [CrossRef]
  7. 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]
  8. 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]
  9. D. H. Alman, DuPont, 945 Stephenson Highway, P.O. Box 2802, Troy, Michigan, USA 48007-2802 (personal communication, 2000).
  10. 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.
  11. D. H. Alman, R. S. Berns, G. D. Snyder, and W. A. Larsen, "Performance testing of color-difference metrics using a color tolerance dataset," Color Res. Appl. 14, 139-151 (1989).
    [CrossRef]
  12. J. B. Kruskal, "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis," Pychometrika 29, 1-27 (1964).
    [CrossRef]
  13. A. P. M. Coxon, The User's Guide to Multidimensional Scaling (Heinemann, 1982).
  14. S. S. Guan and M. R. Luo, "Investigation of parametric effects using small colour differences," Color Res. Appl. 24, 331-343 (1999).
    [CrossRef]
  15. M. R. Luo and B. Rigg, "BFD (l:c) colour-difference formula. Part 1—Development of the formula," J. Soc. Dyers Colour. 103, 86-94 (1987).
    [CrossRef]
  16. C. Alder, K. P. Chaing, T. F. Chong, E. Coates, A. A. Khalili, and B. Rigg, "Uniform chromaticity scales—new experimental data," J. Soc. Dyers Colour. 98, 14-20 (1982).
    [CrossRef]
  17. W. Schultze, "The usefulness of colour-difference formulae for fixing colour tolerances," in Proceedings AIC/Holland (Soesterberg, 1972), pp. 254-265.
  18. F. J. J. Clarke, R. McDonald, and B. Rigg, "Modification to the JPC79 colour-difference formula," J. Soc. Dyers Colour. 100, 128-132 (1984).
    [CrossRef]
  19. S. Y. Zhu, "Uniform colour spaces for image applications," Ph.D. dissertation (University of Derby, 2002), p. 24.
  20. M. A. López-Alvarez, J. Hernández-Andrés, J. Romero, and R. L. Lee, Jr., "Designing a practical system for spectral imaging of skylight," Appl. Opt. 44, 5688-5695 (2005).
    [CrossRef] [PubMed]
  21. K. McLaren, "Colour passing—visual or instrumental," J. Soc. Dyers Colour. 86, 389-392 (1970).
    [CrossRef]
  22. 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]
  23. R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, "Visual determination of suprathreshold color difference tolerances using probit analysis," Color Res. Appl. 16, 297-316 (1991).
    [CrossRef]
  24. F. H. Imai, M. R. Rosen, and R. S. Berns, "Comparative study of metrics for spectral match quality," in Proceedings of the 1st European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 492-496.

2006

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

J. M. Gibert, J. M. Dagà, E. J. Gilabert, J. Valldeperas, and the Colorimetry Group, "Evaluation of colour difference formulae," Color Technol. 121, 147-152 (2005).
[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]

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.

M. A. López-Alvarez, J. Hernández-Andrés, J. Romero, and R. L. Lee, Jr., "Designing a practical system for spectral imaging of skylight," Appl. Opt. 44, 5688-5695 (2005).
[CrossRef] [PubMed]

2004

2003

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

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

S. Y. Zhu, "Uniform colour spaces for image applications," Ph.D. dissertation (University of Derby, 2002), p. 24.

F. H. Imai, M. R. Rosen, and R. S. Berns, "Comparative study of metrics for spectral match quality," in Proceedings of the 1st European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 492-496.

2001

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]

2000

D. H. Alman, DuPont, 945 Stephenson Highway, P.O. Box 2802, Troy, Michigan, USA 48007-2802 (personal communication, 2000).

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

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

1991

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, "Visual determination of suprathreshold color difference tolerances using probit analysis," Color Res. Appl. 16, 297-316 (1991).
[CrossRef]

1989

D. H. Alman, R. S. Berns, G. D. Snyder, and W. A. Larsen, "Performance testing of color-difference metrics using a color tolerance dataset," Color Res. Appl. 14, 139-151 (1989).
[CrossRef]

1987

M. R. Luo and B. Rigg, "BFD (l:c) colour-difference formula. Part 1—Development of the formula," J. Soc. Dyers Colour. 103, 86-94 (1987).
[CrossRef]

1984

F. J. J. Clarke, R. McDonald, and B. Rigg, "Modification to the JPC79 colour-difference formula," J. Soc. Dyers Colour. 100, 128-132 (1984).
[CrossRef]

1982

A. P. M. Coxon, The User's Guide to Multidimensional Scaling (Heinemann, 1982).

C. Alder, K. P. Chaing, T. F. Chong, E. Coates, A. A. Khalili, and B. Rigg, "Uniform chromaticity scales—new experimental data," J. Soc. Dyers Colour. 98, 14-20 (1982).
[CrossRef]

1972

W. Schultze, "The usefulness of colour-difference formulae for fixing colour tolerances," in Proceedings AIC/Holland (Soesterberg, 1972), pp. 254-265.

1970

K. McLaren, "Colour passing—visual or instrumental," J. Soc. Dyers Colour. 86, 389-392 (1970).
[CrossRef]

1964

J. B. Kruskal, "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis," Pychometrika 29, 1-27 (1964).
[CrossRef]

Alder, C.

C. Alder, K. P. Chaing, T. F. Chong, E. Coates, A. A. Khalili, and B. Rigg, "Uniform chromaticity scales—new experimental data," J. Soc. Dyers Colour. 98, 14-20 (1982).
[CrossRef]

Alman, D. H.

D. H. Alman, DuPont, 945 Stephenson Highway, P.O. Box 2802, Troy, Michigan, USA 48007-2802 (personal communication, 2000).

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, "Visual determination of suprathreshold color difference tolerances using probit analysis," Color Res. Appl. 16, 297-316 (1991).
[CrossRef]

D. H. Alman, R. S. Berns, G. D. Snyder, and W. A. Larsen, "Performance testing of color-difference metrics using a color tolerance dataset," Color Res. Appl. 14, 139-151 (1989).
[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]

Balonon-Rosen, M. R.

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, "Visual determination of suprathreshold color difference tolerances using probit analysis," Color Res. Appl. 16, 297-316 (1991).
[CrossRef]

Berns, R. S.

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]

F. H. Imai, M. R. Rosen, and R. S. Berns, "Comparative study of metrics for spectral match quality," in Proceedings of the 1st European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 492-496.

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, "Visual determination of suprathreshold color difference tolerances using probit analysis," Color Res. Appl. 16, 297-316 (1991).
[CrossRef]

D. H. Alman, R. S. Berns, G. D. Snyder, and W. A. Larsen, "Performance testing of color-difference metrics using a color tolerance dataset," Color Res. Appl. 14, 139-151 (1989).
[CrossRef]

Chaing, K. P.

C. Alder, K. P. Chaing, T. F. Chong, E. Coates, A. A. Khalili, and B. Rigg, "Uniform chromaticity scales—new experimental data," J. Soc. Dyers Colour. 98, 14-20 (1982).
[CrossRef]

Chong, T. F.

C. Alder, K. P. Chaing, T. F. Chong, E. Coates, A. A. Khalili, and B. Rigg, "Uniform chromaticity scales—new experimental data," J. Soc. Dyers Colour. 98, 14-20 (1982).
[CrossRef]

Clarke, F. J. J.

F. J. J. Clarke, R. McDonald, and B. Rigg, "Modification to the JPC79 colour-difference formula," J. Soc. Dyers Colour. 100, 128-132 (1984).
[CrossRef]

Coates, E.

C. Alder, K. P. Chaing, T. F. Chong, E. Coates, A. A. Khalili, and B. Rigg, "Uniform chromaticity scales—new experimental data," J. Soc. Dyers Colour. 98, 14-20 (1982).
[CrossRef]

Coxon, A. P. M.

A. P. M. Coxon, The User's Guide to Multidimensional Scaling (Heinemann, 1982).

Cui, G.

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]

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.

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]

Dagà, J. M.

J. M. Gibert, J. M. Dagà, E. J. Gilabert, J. Valldeperas, and the Colorimetry Group, "Evaluation of colour difference formulae," Color Technol. 121, 147-152 (2005).
[CrossRef]

Gibert, J. M.

J. M. Gibert, J. M. Dagà, E. J. Gilabert, J. Valldeperas, and the Colorimetry Group, "Evaluation of colour difference formulae," Color Technol. 121, 147-152 (2005).
[CrossRef]

Gilabert, E. J.

J. M. Gibert, J. M. Dagà, E. J. Gilabert, J. Valldeperas, and the Colorimetry Group, "Evaluation of colour difference formulae," Color Technol. 121, 147-152 (2005).
[CrossRef]

Guan, S. S.

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

Hernández-Andrés, J.

Huertas, R.

Imai, F. H.

F. H. Imai, M. R. Rosen, and R. S. Berns, "Comparative study of metrics for spectral match quality," in Proceedings of the 1st European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 492-496.

Khalili, A. A.

C. Alder, K. P. Chaing, T. F. Chong, E. Coates, A. A. Khalili, and B. Rigg, "Uniform chromaticity scales—new experimental data," J. Soc. Dyers Colour. 98, 14-20 (1982).
[CrossRef]

Kruskal, J. B.

J. B. Kruskal, "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis," Pychometrika 29, 1-27 (1964).
[CrossRef]

Lam, C. C.

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]

Larsen, W. A.

D. H. Alman, R. S. Berns, G. D. Snyder, and W. A. Larsen, "Performance testing of color-difference metrics using a color tolerance dataset," Color Res. Appl. 14, 139-151 (1989).
[CrossRef]

Lee, R. L.

Li, C.

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]

López-Alvarez, M. A.

Luo, M. R.

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]

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.

M. R. Luo, "Development of colour-difference formulae," Rev. Prog. Color. Relat. Top. 32, 28-39 (2002).
[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. S. Guan and M. R. Luo, "Investigation of parametric effects using small colour differences," Color Res. Appl. 24, 331-343 (1999).
[CrossRef]

M. R. Luo and B. Rigg, "BFD (l:c) colour-difference formula. Part 1—Development of the formula," J. Soc. Dyers Colour. 103, 86-94 (1987).
[CrossRef]

McDonald, R.

F. J. J. Clarke, R. McDonald, and B. Rigg, "Modification to the JPC79 colour-difference formula," J. Soc. Dyers Colour. 100, 128-132 (1984).
[CrossRef]

McLaren, K.

K. McLaren, "Colour passing—visual or instrumental," J. Soc. Dyers Colour. 86, 389-392 (1970).
[CrossRef]

Melgosa, M.

Montag, E. D.

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]

Pointer, M. R.

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]

Reniff, L.

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, "Visual determination of suprathreshold color difference tolerances using probit analysis," Color Res. Appl. 16, 297-316 (1991).
[CrossRef]

Rigg, B.

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.

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]

M. R. Luo and B. Rigg, "BFD (l:c) colour-difference formula. Part 1—Development of the formula," J. Soc. Dyers Colour. 103, 86-94 (1987).
[CrossRef]

F. J. J. Clarke, R. McDonald, and B. Rigg, "Modification to the JPC79 colour-difference formula," J. Soc. Dyers Colour. 100, 128-132 (1984).
[CrossRef]

C. Alder, K. P. Chaing, T. F. Chong, E. Coates, A. A. Khalili, and B. Rigg, "Uniform chromaticity scales—new experimental data," J. Soc. Dyers Colour. 98, 14-20 (1982).
[CrossRef]

Romero, J.

Rosen, M. R.

F. H. Imai, M. R. Rosen, and R. S. Berns, "Comparative study of metrics for spectral match quality," in Proceedings of the 1st European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 492-496.

Schultze, W.

W. Schultze, "The usefulness of colour-difference formulae for fixing colour tolerances," in Proceedings AIC/Holland (Soesterberg, 1972), pp. 254-265.

Shen, H.-L.

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]

Snyder, G. D.

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, "Visual determination of suprathreshold color difference tolerances using probit analysis," Color Res. Appl. 16, 297-316 (1991).
[CrossRef]

D. H. Alman, R. S. Berns, G. D. Snyder, and W. A. Larsen, "Performance testing of color-difference metrics using a color tolerance dataset," Color Res. Appl. 14, 139-151 (1989).
[CrossRef]

Valldeperas, J.

J. M. Gibert, J. M. Dagà, E. J. Gilabert, J. Valldeperas, and the Colorimetry Group, "Evaluation of colour difference formulae," Color Technol. 121, 147-152 (2005).
[CrossRef]

Wilber, D. C.

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]

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]

Xu, H.

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

Yaguchi, H.

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

Zhu, S. Y.

S. Y. Zhu, "Uniform colour spaces for image applications," Ph.D. dissertation (University of Derby, 2002), p. 24.

Appl. Opt.

Color Res. Appl.

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]

R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, "Visual determination of suprathreshold color difference tolerances using probit analysis," Color Res. Appl. 16, 297-316 (1991).
[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]

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

D. H. Alman, R. S. Berns, G. D. Snyder, and W. A. Larsen, "Performance testing of color-difference metrics using a color tolerance dataset," Color Res. Appl. 14, 139-151 (1989).
[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]

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]

Color Technol.

J. M. Gibert, J. M. Dagà, E. J. Gilabert, J. Valldeperas, and the Colorimetry Group, "Evaluation of colour difference formulae," Color Technol. 121, 147-152 (2005).
[CrossRef]

J. Opt. Soc. Am. A

J. Soc. Dyers Colour.

K. McLaren, "Colour passing—visual or instrumental," J. Soc. Dyers Colour. 86, 389-392 (1970).
[CrossRef]

F. J. J. Clarke, R. McDonald, and B. Rigg, "Modification to the JPC79 colour-difference formula," J. Soc. Dyers Colour. 100, 128-132 (1984).
[CrossRef]

M. R. Luo and B. Rigg, "BFD (l:c) colour-difference formula. Part 1—Development of the formula," J. Soc. Dyers Colour. 103, 86-94 (1987).
[CrossRef]

C. Alder, K. P. Chaing, T. F. Chong, E. Coates, A. A. Khalili, and B. Rigg, "Uniform chromaticity scales—new experimental data," J. Soc. Dyers Colour. 98, 14-20 (1982).
[CrossRef]

Pychometrika

J. B. Kruskal, "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis," Pychometrika 29, 1-27 (1964).
[CrossRef]

Rev. Prog. Color. Relat. Top.

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

Other

A. P. M. Coxon, The User's Guide to Multidimensional Scaling (Heinemann, 1982).

S. Y. Zhu, "Uniform colour spaces for image applications," Ph.D. dissertation (University of Derby, 2002), p. 24.

W. Schultze, "The usefulness of colour-difference formulae for fixing colour tolerances," in Proceedings AIC/Holland (Soesterberg, 1972), pp. 254-265.

D. H. Alman, DuPont, 945 Stephenson Highway, P.O. Box 2802, Troy, Michigan, USA 48007-2802 (personal communication, 2000).

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.

F. H. Imai, M. R. Rosen, and R. S. Berns, "Comparative study of metrics for spectral match quality," in Proceedings of the 1st European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 492-496.

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Figures (4)

Fig. 1
Fig. 1

Changes in six different indices when all Δ E i ( i = 1 , , N ) values approach Δ V i ( i = 1 , , N ) values. Each curve corresponds to a different set of 500 pairs of random Δ E i and Δ V i ( i = 1 , , N ) values in the range 0 to 5.

Fig. 2
Fig. 2

Results of PCA from the 50 simulated data sets employed in Fig. 1.

Fig. 3
Fig. 3

Relationship among STRESS and the other five indices from the 50 simulated data sets used in Fig. 1.

Fig. 4
Fig. 4

Relative changes in the different indices when different numbers of outliers (arbitrarily defined as a factor of 10) were randomly introduced in each of the five initial data sets employed in Fig. 1. On the y axis, Index and Index 0 mean the values of any of the indices with and without the presence of outliers, respectively.

Tables (1)

Tables Icon

Table 1 Summary of Properties of Different Indices Considered in This Paper

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

log 10 ( γ ) = 1 N i = 1 N [ log 10 ( Δ E i Δ V i ) log 10 ( Δ E i Δ V i ) ¯ ] 2 ,
V A B = 1 N i = 1 N ( Δ E i F Δ V i ) 2 Δ E i F Δ V i , with F = i = 1 N Δ E i Δ V i i = 1 N Δ V i Δ E i ,
C V = 100 1 N i = 1 N ( Δ E i f Δ V i ) 2 ( Δ E ¯ ) 2 , with f = i = 1 N Δ E i Δ V i i = 1 N Δ V i 2 ,
P F 3 = 100 [ ( γ 1 ) + V A B + C V 100 ] 3 ,
V A = ( Δ V i a A Δ E A , i ) 2 ( N 1 ) , with a A = ( Δ E A , i Δ V i ) ( Δ E A , i ) 2 ,
V B = ( Δ V i a B Δ E B , i ) 2 ( N 1 ) , with a B = ( Δ E B , i Δ V i ) ( Δ E B , i ) 2 ,
F = V A V B = ( Δ V i a A Δ E A , i ) 2 ( Δ V i a B Δ E B , i ) 2 ,
STRESS = ( ( Δ E i F 1 Δ V i ) 2 F 1 2 Δ V i 2 ) 1 2 , with F 1 = Δ E i 2 Δ E i Δ V i .
STRESS = ( ( F 2 Δ E i Δ V i ) 2 Δ V i 2 ) 1 2 , with F 2 = Δ E i Δ V i Δ E i 2 = 1 F 1 ,
STRESS = ( ( Δ E i F 3 Δ V i ) 2 Δ E i 2 ) 1 2 , with F 3 = Δ E i Δ V i Δ V i 2 ,
Weighted normalized STRESS = ( w i ( Δ E i f Δ V i ) 2 w i ( f Δ V i ) 2 ) 1 2 ,
F = V A V B = ( Δ V i a A Δ E A , i ) 2 ( Δ V i a B Δ E B , i ) 2 = ( F 2 Δ E A , i Δ V i ) 2 ( F 2 Δ E B , i Δ V i ) 2 = STRESS A 2 STRESS B 2 ,
STRESS 2 = ( Δ E i F 1 Δ V i ) 2 F 1 2 Δ V i 2 = ( Δ E i ( 1 F 2 ) Δ V i ) 2 ( 1 F 2 2 ) Δ V i 2 = ( F 2 Δ E i Δ V i ) 2 Δ V i 2 .
STRESS 2 = ( Δ E i F 1 Δ V i ) 2 F 1 2 Δ V i 2 = ( Δ E i 2 2 F 1 Δ E i Δ V i + F 1 2 Δ V 1 2 ) F 1 2 Δ V i 2 = ( Δ E i Δ V i Δ E i 2 ) 2 Δ E i 2 Δ V i 2 2 Δ E i Δ V i Δ E i 2 Δ E i Δ V i Δ V i 2 + 1 = 1 ( Δ E i Δ V i ) 2 Δ E i 2 Δ V i 2 ,
STRESS 2 = ( Δ E i F 3 Δ V i ) 2 Δ E i 2 = ( Δ E i 2 2 F 3 Δ E i Δ V i + F 3 2 Δ V i 2 ) Δ E i 2 = 1 2 F 3 Δ E i Δ V i Δ E i 2 + F 3 2 Δ V i 2 Δ E i 2 = 1 2 Δ E i Δ V i Δ V i 2 Δ E i Δ V i Δ E i 2 + ( Δ E i Δ V i Δ V i 2 ) 2 Δ V i 2 Δ E i 2 = 1 ( Δ E i Δ V i ) 2 Δ E i 2 Δ V i 2 .

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