Abstract

The generalized spectral decomposition (GSD) theorem is introduced, and the generalized fundamental stimulus and metameric black are analyzed to show how they convey the valuable features in terms of color information. The suggestion would be considered as the generalization of Cohen and Kappauf’s matrix R theory and its later application in parameric correction by Fairman. The GSD theorem provides a modular model whose arguments can be elaborately set up for high-performance spectral recovery. It is also shown that the suggested methods for spectral decomposition and/or spectral reconstruction proposed by different researchers could be considered as special cases of GSD.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Wyszecki, “Valenzmetrische Untersuchung des Zusammenhanges zwischen normaler und anomaler Trichromasie(Psychophysical investigation of relationship between normal and abnormal trichromatic vision),” Farbe 2, 39–52 (1953).
  2. J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
    [CrossRef] [PubMed]
  3. J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
    [CrossRef]
  4. J. B. Cohen, “Color and color mixture: scalar and vector fundamentals,” Color Res. Appl. 13, 5–39 (1988).
    [CrossRef]
  5. J. B. Cohen, Visual Color and Color Mixture: The Fundamental Color Space (University of Illinois Press, 2001).
  6. Y. Zhao and R. S. Berns, “Image-based spectral reflectance reconstruction using the matrix R method,” Color Res. Appl. 32, 343–351 (2007).
    [CrossRef]
  7. H. S. Fairman, “Metameric correction using parameric decomposition,” Color Res. Appl. 12, 261–265 (1987).
    [CrossRef]
  8. H. S. Fairman, “Recommended terminology for matrix R and metamerism,” Color Res. Appl. 16, 337–341 (1991).
    [CrossRef]
  9. Z. Li and R. S. Berns, “Comparison of methods of parameric correction for evaluating metamerism,” Color Res. Appl. 32, 293–303 (2007).
    [CrossRef]
  10. J. A. Worthey, “Calculation of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988).
    [CrossRef]
  11. S. A. Burns, J. B. Cohen, and E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
    [CrossRef]
  12. H. J. Trussell, “Applications of set theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
    [CrossRef]
  13. G. D. Finlayson and P. Morovic, “Metamer sets,” J. Opt. Soc. Am. A 22, 810–819 (2005).
    [CrossRef]
  14. P. Urban and R. R. Grigat, “Metamer density estimated color correction,” Signal Image Video Process. 3, 171–182 (2009).
    [CrossRef]
  15. P. Urban, R. S. Berns, and R. R. Grigat, “Color correction by considering the distribution of metamers within the mismatch gamut,” in Fifteenth Color Imaging Conference: Color Science and Engineering Systems, Technologies, and Applications (Society for Imaging Sciences and Technology, 2007), pp. 222–227.
    [PubMed]
  16. M. W. Derhak and M. R. Rosen, “Spectral colorimetry using LabPQR: an interim connection space,” J. Imaging Sci. Technol. 50, 53–63 (2006).
    [CrossRef]
  17. S. Tsutsumi, M. R. Rosen, and R. S. Berns, “Spectral color reproduction using an interim connection space-based lookup table,” J. Imaging Sci. Technol. 52, 040201–040213 (2008).
    [CrossRef]
  18. S. Tsutsumi, M. R. Rosen, and R. S. Berns, “Spectral color management using interim connection spaces based on spectral decomposition,” Color Res. Appl. 33, 282–299 (2008).
    [CrossRef]
  19. H. Kotera, “FCS-based prediction in color changes under different illuminants,” in Proceedings of AIC 2007, Color Science for Industry, Midterm Meeting of the International Color Association (International Color Association, 2007), pp. 134–137.
  20. H. Kotera, “Geometrical structures of fundamental color space,” in Proceedings of AIC 2007, Color Science for Industry, Midterm Meeting of the International Color Association, (International Color Association, 2007), pp. 130–133.
  21. H. Abdi, “Eigen-decomposition: eigenvalues and eigenvecteurs,” in Encyclopedia of Measurement and Statistics, N.J.Salkind, ed. (Sage, 2007), pp. 304–308.
  22. R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, 5th ed. (Prentice-Hall, 2002).
  23. D.-Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
    [CrossRef]
  24. H. S. Fairman and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
    [CrossRef]
  25. J. E. Gentle, Matrix Algebra: Theory, Computations, and Applications in Statistics, Springer Texts in Statistics(Springer, 2007).
    [PubMed]
  26. C. D. Meyer, Matrix Analysis and Applied Linear Algebra, 1st ed. (SIAM, 2000).
    [CrossRef]
  27. J. A. Worthey and M. H. Brill, “Principal components applied to modeling: dealing with the mean vector,” Color Res. Appl. 29, 261–266 (2004).
    [CrossRef]
  28. S. Peyvandi and S. H. Amirshahi, “Paramerism and reliable parameric correction,” Color Res. Appl., doi:10.1002/col.20642 (to be published).
    [CrossRef]
  29. Spectral Database, University of Joensuu Color Group, http://spectral.joensuu.fi/.
  30. A. Hard and L. Sivik, “NCS—Natural Color System: a Swedish standard for coloer notation,” Color Res. Appl. 6, 129–138(1981).
    [CrossRef]
  31. A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andrés, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
    [CrossRef]
  32. M. S. Drew and B. V. Funt, “Natural metamers,” CVGIP Image Underst. 56, 139–151 (1992).
    [CrossRef]
  33. S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
    [CrossRef]
  34. C. Li and M. R. Luo, “The estimation of spectral reflectances using the smoothest constraint condition,” in Ninth Color Imaging Conference: Color Science and Engineering: Systems, Technologies, and Applications (Imaging Sciences and Technology, 2001), pp. 62–67.
    [PubMed]
  35. C. van Trigt, “Smoothest reflectance functions. I. Definition and main results,” J. Opt. Soc. Am. A 7, 1891–1904 (1990).
    [CrossRef]
  36. C. van Trigt, “Smoothest reflectance functions. II. Complete results,” J. Opt. Soc. Am. A 7, 2208–2222 (1990).
    [CrossRef]

2009

P. Urban and R. R. Grigat, “Metamer density estimated color correction,” Signal Image Video Process. 3, 171–182 (2009).
[CrossRef]

2008

S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
[CrossRef]

S. Tsutsumi, M. R. Rosen, and R. S. Berns, “Spectral color reproduction using an interim connection space-based lookup table,” J. Imaging Sci. Technol. 52, 040201–040213 (2008).
[CrossRef]

S. Tsutsumi, M. R. Rosen, and R. S. Berns, “Spectral color management using interim connection spaces based on spectral decomposition,” Color Res. Appl. 33, 282–299 (2008).
[CrossRef]

2007

Y. Zhao and R. S. Berns, “Image-based spectral reflectance reconstruction using the matrix R method,” Color Res. Appl. 32, 343–351 (2007).
[CrossRef]

Z. Li and R. S. Berns, “Comparison of methods of parameric correction for evaluating metamerism,” Color Res. Appl. 32, 293–303 (2007).
[CrossRef]

2006

M. W. Derhak and M. R. Rosen, “Spectral colorimetry using LabPQR: an interim connection space,” J. Imaging Sci. Technol. 50, 53–63 (2006).
[CrossRef]

2005

D.-Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

G. D. Finlayson and P. Morovic, “Metamer sets,” J. Opt. Soc. Am. A 22, 810–819 (2005).
[CrossRef]

2004

H. S. Fairman and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

J. A. Worthey and M. H. Brill, “Principal components applied to modeling: dealing with the mean vector,” Color Res. Appl. 29, 261–266 (2004).
[CrossRef]

1998

A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andrés, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

1992

M. S. Drew and B. V. Funt, “Natural metamers,” CVGIP Image Underst. 56, 139–151 (1992).
[CrossRef]

1991

H. J. Trussell, “Applications of set theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
[CrossRef]

H. S. Fairman, “Recommended terminology for matrix R and metamerism,” Color Res. Appl. 16, 337–341 (1991).
[CrossRef]

1990

1989

S. A. Burns, J. B. Cohen, and E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

1988

J. A. Worthey, “Calculation of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988).
[CrossRef]

J. B. Cohen, “Color and color mixture: scalar and vector fundamentals,” Color Res. Appl. 13, 5–39 (1988).
[CrossRef]

1987

H. S. Fairman, “Metameric correction using parameric decomposition,” Color Res. Appl. 12, 261–265 (1987).
[CrossRef]

1985

J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

1982

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef] [PubMed]

1981

A. Hard and L. Sivik, “NCS—Natural Color System: a Swedish standard for coloer notation,” Color Res. Appl. 6, 129–138(1981).
[CrossRef]

1953

G. Wyszecki, “Valenzmetrische Untersuchung des Zusammenhanges zwischen normaler und anomaler Trichromasie(Psychophysical investigation of relationship between normal and abnormal trichromatic vision),” Farbe 2, 39–52 (1953).

Abdi, H.

H. Abdi, “Eigen-decomposition: eigenvalues and eigenvecteurs,” in Encyclopedia of Measurement and Statistics, N.J.Salkind, ed. (Sage, 2007), pp. 304–308.

Amirshahi, S. H.

S. Peyvandi and S. H. Amirshahi, “Paramerism and reliable parameric correction,” Color Res. Appl., doi:10.1002/col.20642 (to be published).
[CrossRef]

Berns, R. S.

S. Tsutsumi, M. R. Rosen, and R. S. Berns, “Spectral color management using interim connection spaces based on spectral decomposition,” Color Res. Appl. 33, 282–299 (2008).
[CrossRef]

S. Tsutsumi, M. R. Rosen, and R. S. Berns, “Spectral color reproduction using an interim connection space-based lookup table,” J. Imaging Sci. Technol. 52, 040201–040213 (2008).
[CrossRef]

Y. Zhao and R. S. Berns, “Image-based spectral reflectance reconstruction using the matrix R method,” Color Res. Appl. 32, 343–351 (2007).
[CrossRef]

Z. Li and R. S. Berns, “Comparison of methods of parameric correction for evaluating metamerism,” Color Res. Appl. 32, 293–303 (2007).
[CrossRef]

D.-Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

P. Urban, R. S. Berns, and R. R. Grigat, “Color correction by considering the distribution of metamers within the mismatch gamut,” in Fifteenth Color Imaging Conference: Color Science and Engineering Systems, Technologies, and Applications (Society for Imaging Sciences and Technology, 2007), pp. 222–227.
[PubMed]

Brill, M. H.

H. S. Fairman and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

J. A. Worthey and M. H. Brill, “Principal components applied to modeling: dealing with the mean vector,” Color Res. Appl. 29, 261–266 (2004).
[CrossRef]

Burns, S. A.

S. A. Burns, J. B. Cohen, and E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

Cohen, J. B.

S. A. Burns, J. B. Cohen, and E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

J. B. Cohen, “Color and color mixture: scalar and vector fundamentals,” Color Res. Appl. 13, 5–39 (1988).
[CrossRef]

J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef] [PubMed]

J. B. Cohen, Visual Color and Color Mixture: The Fundamental Color Space (University of Illinois Press, 2001).

Derhak, M. W.

M. W. Derhak and M. R. Rosen, “Spectral colorimetry using LabPQR: an interim connection space,” J. Imaging Sci. Technol. 50, 53–63 (2006).
[CrossRef]

Drew, M. S.

M. S. Drew and B. V. Funt, “Natural metamers,” CVGIP Image Underst. 56, 139–151 (1992).
[CrossRef]

Fairman, H. S.

H. S. Fairman and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

H. S. Fairman, “Recommended terminology for matrix R and metamerism,” Color Res. Appl. 16, 337–341 (1991).
[CrossRef]

H. S. Fairman, “Metameric correction using parameric decomposition,” Color Res. Appl. 12, 261–265 (1987).
[CrossRef]

Finlayson, G. D.

Funt, B. V.

M. S. Drew and B. V. Funt, “Natural metamers,” CVGIP Image Underst. 56, 139–151 (1992).
[CrossRef]

Garcia-Beltran, A.

A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andrés, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

Gentle, J. E.

J. E. Gentle, Matrix Algebra: Theory, Computations, and Applications in Statistics, Springer Texts in Statistics(Springer, 2007).
[PubMed]

Grigat, R. R.

P. Urban and R. R. Grigat, “Metamer density estimated color correction,” Signal Image Video Process. 3, 171–182 (2009).
[CrossRef]

P. Urban, R. S. Berns, and R. R. Grigat, “Color correction by considering the distribution of metamers within the mismatch gamut,” in Fifteenth Color Imaging Conference: Color Science and Engineering Systems, Technologies, and Applications (Society for Imaging Sciences and Technology, 2007), pp. 222–227.
[PubMed]

Hard, A.

A. Hard and L. Sivik, “NCS—Natural Color System: a Swedish standard for coloer notation,” Color Res. Appl. 6, 129–138(1981).
[CrossRef]

Hernandez-Andrés, J.

A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andrés, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

Johnson, R. A.

R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, 5th ed. (Prentice-Hall, 2002).

Kappauf, W. E.

J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef] [PubMed]

Kotera, H.

H. Kotera, “FCS-based prediction in color changes under different illuminants,” in Proceedings of AIC 2007, Color Science for Industry, Midterm Meeting of the International Color Association (International Color Association, 2007), pp. 134–137.

H. Kotera, “Geometrical structures of fundamental color space,” in Proceedings of AIC 2007, Color Science for Industry, Midterm Meeting of the International Color Association, (International Color Association, 2007), pp. 130–133.

Kuznetsov, E. N.

S. A. Burns, J. B. Cohen, and E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

Li, C.

C. Li and M. R. Luo, “The estimation of spectral reflectances using the smoothest constraint condition,” in Ninth Color Imaging Conference: Color Science and Engineering: Systems, Technologies, and Applications (Imaging Sciences and Technology, 2001), pp. 62–67.
[PubMed]

Li, Z.

Z. Li and R. S. Berns, “Comparison of methods of parameric correction for evaluating metamerism,” Color Res. Appl. 32, 293–303 (2007).
[CrossRef]

Luo, M. R.

C. Li and M. R. Luo, “The estimation of spectral reflectances using the smoothest constraint condition,” in Ninth Color Imaging Conference: Color Science and Engineering: Systems, Technologies, and Applications (Imaging Sciences and Technology, 2001), pp. 62–67.
[PubMed]

Meyer, C. D.

C. D. Meyer, Matrix Analysis and Applied Linear Algebra, 1st ed. (SIAM, 2000).
[CrossRef]

Morovic, P.

Nieves, J. L.

A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andrés, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

Peyvandi, S.

S. Peyvandi and S. H. Amirshahi, “Paramerism and reliable parameric correction,” Color Res. Appl., doi:10.1002/col.20642 (to be published).
[CrossRef]

Romero, J.

A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andrés, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

Rosen, M. R.

S. Tsutsumi, M. R. Rosen, and R. S. Berns, “Spectral color management using interim connection spaces based on spectral decomposition,” Color Res. Appl. 33, 282–299 (2008).
[CrossRef]

S. Tsutsumi, M. R. Rosen, and R. S. Berns, “Spectral color reproduction using an interim connection space-based lookup table,” J. Imaging Sci. Technol. 52, 040201–040213 (2008).
[CrossRef]

M. W. Derhak and M. R. Rosen, “Spectral colorimetry using LabPQR: an interim connection space,” J. Imaging Sci. Technol. 50, 53–63 (2006).
[CrossRef]

Santini, S.

S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
[CrossRef]

Schettini, R.

S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
[CrossRef]

Sivik, L.

A. Hard and L. Sivik, “NCS—Natural Color System: a Swedish standard for coloer notation,” Color Res. Appl. 6, 129–138(1981).
[CrossRef]

Trussell, H. J.

H. J. Trussell, “Applications of set theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
[CrossRef]

Tsutsumi, S.

S. Tsutsumi, M. R. Rosen, and R. S. Berns, “Spectral color reproduction using an interim connection space-based lookup table,” J. Imaging Sci. Technol. 52, 040201–040213 (2008).
[CrossRef]

S. Tsutsumi, M. R. Rosen, and R. S. Berns, “Spectral color management using interim connection spaces based on spectral decomposition,” Color Res. Appl. 33, 282–299 (2008).
[CrossRef]

Tzeng, D.-Y.

D.-Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

Urban, P.

P. Urban and R. R. Grigat, “Metamer density estimated color correction,” Signal Image Video Process. 3, 171–182 (2009).
[CrossRef]

P. Urban, R. S. Berns, and R. R. Grigat, “Color correction by considering the distribution of metamers within the mismatch gamut,” in Fifteenth Color Imaging Conference: Color Science and Engineering Systems, Technologies, and Applications (Society for Imaging Sciences and Technology, 2007), pp. 222–227.
[PubMed]

van Trigt, C.

Wichern, D. W.

R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, 5th ed. (Prentice-Hall, 2002).

Worthey, J. A.

J. A. Worthey and M. H. Brill, “Principal components applied to modeling: dealing with the mean vector,” Color Res. Appl. 29, 261–266 (2004).
[CrossRef]

J. A. Worthey, “Calculation of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988).
[CrossRef]

Wyszecki, G.

G. Wyszecki, “Valenzmetrische Untersuchung des Zusammenhanges zwischen normaler und anomaler Trichromasie(Psychophysical investigation of relationship between normal and abnormal trichromatic vision),” Farbe 2, 39–52 (1953).

Zhao, Y.

Y. Zhao and R. S. Berns, “Image-based spectral reflectance reconstruction using the matrix R method,” Color Res. Appl. 32, 343–351 (2007).
[CrossRef]

Zuffi, S.

S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
[CrossRef]

Am. J. Psychol.

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982).
[CrossRef] [PubMed]

J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985).
[CrossRef]

Color Res. Appl.

J. B. Cohen, “Color and color mixture: scalar and vector fundamentals,” Color Res. Appl. 13, 5–39 (1988).
[CrossRef]

S. Tsutsumi, M. R. Rosen, and R. S. Berns, “Spectral color management using interim connection spaces based on spectral decomposition,” Color Res. Appl. 33, 282–299 (2008).
[CrossRef]

Y. Zhao and R. S. Berns, “Image-based spectral reflectance reconstruction using the matrix R method,” Color Res. Appl. 32, 343–351 (2007).
[CrossRef]

H. S. Fairman, “Metameric correction using parameric decomposition,” Color Res. Appl. 12, 261–265 (1987).
[CrossRef]

H. S. Fairman, “Recommended terminology for matrix R and metamerism,” Color Res. Appl. 16, 337–341 (1991).
[CrossRef]

Z. Li and R. S. Berns, “Comparison of methods of parameric correction for evaluating metamerism,” Color Res. Appl. 32, 293–303 (2007).
[CrossRef]

J. A. Worthey, “Calculation of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988).
[CrossRef]

S. A. Burns, J. B. Cohen, and E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989).
[CrossRef]

H. J. Trussell, “Applications of set theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991).
[CrossRef]

D.-Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

H. S. Fairman and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

J. A. Worthey and M. H. Brill, “Principal components applied to modeling: dealing with the mean vector,” Color Res. Appl. 29, 261–266 (2004).
[CrossRef]

A. Hard and L. Sivik, “NCS—Natural Color System: a Swedish standard for coloer notation,” Color Res. Appl. 6, 129–138(1981).
[CrossRef]

A. Garcia-Beltran, J. L. Nieves, J. Hernandez-Andrés, and J. Romero, “Linear bases for spectral reflectance functions of acrylic paints,” Color Res. Appl. 23, 39–45 (1998).
[CrossRef]

CVGIP Image Underst.

M. S. Drew and B. V. Funt, “Natural metamers,” CVGIP Image Underst. 56, 139–151 (1992).
[CrossRef]

Farbe

G. Wyszecki, “Valenzmetrische Untersuchung des Zusammenhanges zwischen normaler und anomaler Trichromasie(Psychophysical investigation of relationship between normal and abnormal trichromatic vision),” Farbe 2, 39–52 (1953).

IEEE Trans. Signal Process.

S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
[CrossRef]

J. Imaging Sci. Technol.

M. W. Derhak and M. R. Rosen, “Spectral colorimetry using LabPQR: an interim connection space,” J. Imaging Sci. Technol. 50, 53–63 (2006).
[CrossRef]

S. Tsutsumi, M. R. Rosen, and R. S. Berns, “Spectral color reproduction using an interim connection space-based lookup table,” J. Imaging Sci. Technol. 52, 040201–040213 (2008).
[CrossRef]

J. Opt. Soc. Am. A

Signal Image Video Process.

P. Urban and R. R. Grigat, “Metamer density estimated color correction,” Signal Image Video Process. 3, 171–182 (2009).
[CrossRef]

Other

P. Urban, R. S. Berns, and R. R. Grigat, “Color correction by considering the distribution of metamers within the mismatch gamut,” in Fifteenth Color Imaging Conference: Color Science and Engineering Systems, Technologies, and Applications (Society for Imaging Sciences and Technology, 2007), pp. 222–227.
[PubMed]

H. Kotera, “FCS-based prediction in color changes under different illuminants,” in Proceedings of AIC 2007, Color Science for Industry, Midterm Meeting of the International Color Association (International Color Association, 2007), pp. 134–137.

H. Kotera, “Geometrical structures of fundamental color space,” in Proceedings of AIC 2007, Color Science for Industry, Midterm Meeting of the International Color Association, (International Color Association, 2007), pp. 130–133.

H. Abdi, “Eigen-decomposition: eigenvalues and eigenvecteurs,” in Encyclopedia of Measurement and Statistics, N.J.Salkind, ed. (Sage, 2007), pp. 304–308.

R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, 5th ed. (Prentice-Hall, 2002).

J. B. Cohen, Visual Color and Color Mixture: The Fundamental Color Space (University of Illinois Press, 2001).

C. Li and M. R. Luo, “The estimation of spectral reflectances using the smoothest constraint condition,” in Ninth Color Imaging Conference: Color Science and Engineering: Systems, Technologies, and Applications (Imaging Sciences and Technology, 2001), pp. 62–67.
[PubMed]

S. Peyvandi and S. H. Amirshahi, “Paramerism and reliable parameric correction,” Color Res. Appl., doi:10.1002/col.20642 (to be published).
[CrossRef]

Spectral Database, University of Joensuu Color Group, http://spectral.joensuu.fi/.

J. E. Gentle, Matrix Algebra: Theory, Computations, and Applications in Statistics, Springer Texts in Statistics(Springer, 2007).
[PubMed]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra, 1st ed. (SIAM, 2000).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

First six basis vectors of the employed spectral data set.

Fig. 2
Fig. 2

Spectra of four selected reflectance r ̲ (solid curve) are approximated by the FB approach (dotted curve) and the UG method (crossed curve). The CIELCH color coordinates of the selected spectra under illuminant D65 and the 10 ° 1964 standard observer are also presented.

Tables (3)

Tables Icon

Table 1 Total Number of Feasible Approximated Spectra (Out of 8793) from the Tristimulus Values Vectors Created under Different Illumination–Observer Matrices

Tables Icon

Table 2 Analysis of the Spectral Performance Calculated by RMS of the Spectrum r ̲ and the Recovered Reflectance r ̲ ^ by the FB and UG Approaches a

Tables Icon

Table 3 Analysis of the Colorimetric Performance Calculated by CIELAB Color Difference of the Spectrum and the Approximated One Recovered Separately by the FB and UG Approaches a

Equations (37)

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

Σ r = U Λ U T ,
l ̲ = U T r ̲ ,
l j = u ̲ j T r ̲ .
r ̲ = U l ̲ = j l j u ̲ j .
r ̲ = V α ̲ + ε ̲ ,
l j = u ̲ j T V α ̲ + u ̲ j T ε ̲ .
l j = ω j u ̲ j T V α ̲ ,
l ̲ = Ω U T V α ̲ ,
r ̲ = U Ω U T V α ̲ .
c ̲ = A T r ̲ .
c ̲ = A T U Ω U T V α ̲ .
α ̲ ^ = ( A T U Ω U T V ) 1 c ̲ .
r ^ ̲ = U Ω U T V ( A T U Ω U T V ) 1 c ̲
r ̲ c = E ( A T E ) 1 c ̲ ,
r ̲ i o = r ̲ i E ( A T E ) 1 A T r ̲ i ,
Σ ^ r = ( I E ( A T E ) 1 A T ) Σ r ( I E ( A T E ) 1 A T ) T ,
μ ̲ ^ r = r ̲ c + ( I E ( A T E ) 1 A T ) μ ̲ r ,
A T r ̲ ^ i = A T r ̲ c + A T r ̲ i o .
A T r ̲ ^ i = A T E ( A T E ) 1 c ̲ + A T [ r ̲ i E ( A T E ) 1 A T r ̲ i ] = c ̲ + A T r ̲ i A T E ( A T E ) 1 A T r ̲ i = c ̲ + A T r ̲ i A T r ̲ i = c ̲ .
Σ ^ r = Σ r o = ( I E ( A T E ) 1 A T ) Σ r ( I E ( A T E ) 1 A T ) T ,
μ ̲ ^ r = r ¯ c + μ ¯ r o = r ̲ c + ( I E ( A T E ) 1 A T ) μ ̲ r ,
r ̲ c = G r ̲ = E ( A T E ) 1 c ̲ ,
r ̲ o = r ̲ r ̲ c = r ̲ G r ̲ ,
A T r ̲ c = A T E ( A T E ) 1 c ̲ = c ̲ ,
A T r ̲ o = A T r ̲ A T r ̲ c = c ̲ c ̲ = 0 ̲ .
E = U Ω U T V = I n A = A ,
ω j = l j ( l j u ̲ j T ε ̲ ) 1 = 1 l j = ( l j u ̲ j T ε ̲ ) ε ̲ = 0 ̲ .
r ̲ c = A ( A T A ) 1 A T r ̲ r ̲ o = [ I A ( A T A ) 1 A T ] r ̲ ,
A = [ A 1 ̲ A 2 ̲ A ¯ t ]
r ̲ ^ = r ̲ avg + V ( A T V ) 1 ( c ̲ c ̲ avg ) ,
r ̲ ^ = r ̲ avg + V ( A T V ) 1 c ̲ V ( A T V ) 1 c ̲ avg = r ̲ avg V ( A T V ) 1 A T r ̲ avg + V ( A T V ) 1 c ̲ = [ I V ( A T V ) 1 A T ] r ̲ avg + V ( A T V ) 1 c ̲ = r ̲ avg o + r ̲ c .
E = U Ω U T V = U Λ U T A = Σ r A .
Σ ^ r = ( I Σ r A ( A T Σ r A ) 1 A T ) Σ r ,
μ ̲ ^ r = r ̲ c + ( I Σ r A ( A T Σ r A ) 1 A T ) μ ̲ r ,
H = { H | H = U diag ( ω ̲ ) U T V ( A T U diag ( ω ̲ ) U T V ) 1 ; ω ̲ q } ,
R ( c ) = { r ̲ c | r ̲ c = H c ̲ } .
r ̲ : = { r ̲ c | ω ̲ R q : r ̲ c = U diag ( ω ̲ ) U T A ( A T U diag ( ω ̲ ) U T A ) 1 c ̲ r ̲ c = r ̲ } .

Metrics