Abstract

Using, as an example, the color matching functions of the StilesBurch1955 2° pilot group, we show how an XYZ representation of color space can be developed from a given set of color matching data. In doing this, we present a set of criteria that unequivocally defines the representation. The method outlined is general and can be applied to any set of color matching data, in particular those sets that are related to the physiological fundamentals.

© 1999 Optical Society of America

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References

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  1. F. Viénot, “Fundamental chromaticity diagram with physiologically significant axes,” in Proceedings of the Symposium ’96 on Colour Standards for Image Technology, Vienna, 1996 (Central Bureau of the CIE, Vienna, 1996), pp. 35–40.
  2. The CIE committee TC 1-36 distinguishes between the “fundamental response curves,” representing the fundamental spectral sensitivity functions of the three types of cones at the corneal level, and the “cone absorption curves,” which are derived by correcting the fundamentals by the absorption of the lens and ocular media and the macular pigment. Hence, if Tmedia(λ) denotes the spectral transmittance of the lens and ocular media and Tmacula(λ) denotes the spectral transmittance of the macular pigment, each cone absorption curve is given in terms of the corresponding fundamental response curve by the equation cone absorption curve=fundamental response curveTmedia(λ)Tmacula(λ).
  3. CIE, Proceedings of the 8th Session of the CIE, Cambridge, 1931 (Cambridge U. Press, Cambridge, UK, 1932), pp. 19–29.
  4. ISO/CIE, CIE Standard Colorimetric Observers, Publication ISO/CIE 10527 (Central Bureau of the CIE, Vienna, 1991).
  5. W. S. Stiles, J. M. Burch, “NPL colour-matching investigation: final report (1958),” Opt. Acta 6, 1–26 (1959).
    [CrossRef]
  6. R. Luther, “Aus dem Gebiet der Farbreizmetrik,” Z. Tech. Phys. 8, 540–558 (1927).
  7. D. I. A. MacLeod, R. M. Boynton, “Chromaticity diagram showing cone excitation by stimuli of equal luminance,” J. Opt. Soc. Am. 69, 1183–1186 (1979).
    [CrossRef] [PubMed]
  8. R. M. Boynton, “A system of photometry and colorimetry based on cone excitation,” Color Res. Appl. 11, 244–252 (1986).
    [CrossRef]
  9. CIE, , “Fundamental chromaticity diagram with physiological axes” (Central Bureau of the CIE, Vienna, 1998).
  10. W. S. Stiles, J. M. Burch, “Interim report to the Commission Internationale de l’Eclairage, Zürich, 1955, on the National Physical Laboratory’s investigation of colour-matching,” Opt. Acta 2, 168–181 (1955).
    [CrossRef]
  11. J. H. Wold, A. Valberg, “Cones for colorimetry,” in Proceedings of the 23rd Session of the CIE, New Delhi, 1995 (Central Bureau of the CIE, Vienna, 1995), Vol. 1, pp. 24–27.
  12. T. Smith, J. Guild, “The C.I.E. colorimetric standards and their use,” Trans. Opt. Soc. 33, 73–134 (1931–1932).
    [CrossRef]
  13. H. S. Fairman, M. H. Brill, H. Hemmendinger, “How the CIE 1931 color matching functions were derived from Wright-Guild data,” Color Res. Appl. 22, 11–23 (1997).
    [CrossRef]
  14. D. B. Judd, “Reduction of data on mixture of color stimuli,” Bur. Stand. J. Res. 4, 515–548 (1930).
    [CrossRef]
  15. D. L. MacAdam, Color Measurement: Theme and Variations (Springer-Verlag, Berlin, 1981).
  16. The CIE RGB representation refers to Wright primaries R, G, and B, representing monochromatic stimuli of wavelengths 700.0, 546.1, and 435.8 nm, respectively (see Refs. 3 and 17). The norms of R, G, and B are defined so as to ensure that the chromaticity coordinates of Illuminant E are rE=gE=bE=1/3.
  17. W. D. Wright, “A re-determination of the trichromatic coefficients of the spectral colours,” Trans. Opt. Soc. 30, 141–164 (1928–1929).
    [CrossRef]
  18. E. Schrödinger, “Über das Verhältnis der Vierfarben- zur Dreifarbentheorie,” Sitzungber. Kaiserl. Wien. Akad. Wiss. Math.-Naturwiss. Kl. 134, Abt. IIa, 471–490 (1925).
  19. CIE, Proceedings of the 6th Session of the CIE, Geneva, July 1924 (Cambridge U. Press, Cambridge, UK, 1926), pp. 67–70.
  20. Other constraints have also been considered. For instance, D. Brainard (topical editor, 1999, personal communication) has suggested minimizing the difference between CMF’s (Judd–Vos versus Stiles–Burch1955) rather than between the spectral loci of the chromaticity diagrams. This alternative seemed most attractive because of its simplicity. However, it turns out that such a minimization of differences of tristimulus values in three dimensions leads to severe and considerable distortions of the chromaticity diagram (see Refs. 21 and 22). Yet another alternative would be to let the tristimulus vector to which the fundamental S-response curve (S fundamental) refers serve as the Z primary. (See Section 8 for further details.)
  21. J. H. Wold, A. Valberg, “A comparison of two principles for deriving XYZ tristimulus spaces,” in The 15th Symposium of the International Colour Vision Society, Göttingen, July 1999, Abstract P40.
  22. J. H. Wold, A. Valberg, “The derivation of XYZ tristimulus spaces: a comparison of two alternative methods,” Color Res. Appl. (to be published).
  23. D. B. Judd, “CIE Technical Committee No. 7 “Colorimetry and artificial daylight,” Report of Secretariat United States Committee,” in Proceedings of the 12th Session of the CIE, Stockholm, 1951 (Bureau Central de la CIE, Paris, 1951), Vol. 1, Part 7, pp. 1–60.
  24. J. J. Vos, “Colorimetric and photometric properties of a 2° fundamental observer,” Color Res. Appl. 3, 125–128 (1978).
    [CrossRef]
  25. S. L. Guth, J. V. Alexander, J. I. Chumbly, C. B. Gillman, M. M. Patterson, “Factors affecting luminance additivity at threshold among normal and color-blind subjects and elaborations of a trichromatic-opponent colors theory,” Vision Res. 8, 913–928 (1968).
    [CrossRef] [PubMed]
  26. V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
    [CrossRef] [PubMed]
  27. A. Eisner, D. I. A. MacLeod, “Blue-sensitive cones do not contribute to luminance,” J. Opt. Soc. Am. 70, 121–123 (1980).
    [CrossRef] [PubMed]
  28. W. Verdon, A. J. Adams, “Short-wavelength-sensitive cones do not contribute to mesopic luminosity,” J. Opt. Soc. Am. A 4, 91–95 (1987).
    [CrossRef] [PubMed]
  29. The contribution of the S cones to luminance has been somewhat contentious. Some authors claim that S cones do make a small contribution, whereas others maintain that they do not. However, given that the contribution, if any, is minor, it is convenient to assume that the S-cone contribution is zero. See, for instance, A. Stockman, L. T. Sharpe, “Cone spectral sensitivities and color matching,” in Color Vision: From Genes to Perception, K. R. Gegenfurtner, L. T. Sharpe, eds. (Cambridge U. Press, Cambridge, U.K., 1999).
  30. A. Stockman, D. I. A. MacLeod, N. E. Johnson, “Spectral sensitivities of the human cones,” J. Opt. Soc. Am. A 10, 2491–2521 (1993).
    [CrossRef]
  31. G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).
  32. In the paper by Stockman et al.,30 the proposed values are 0.68237 and 0.35235. If we apply these values, the maximum of V⋅(λ) turns out to be 0.999777 (at 553 nm). To normalize V⋅(λ) to a maximum value of 1.000000, the coefficients are multiplied by the factor 1/0.999777.
  33. O. Estévez, “On the fundamental data-base of normal and dichromatic color vision,” Ph.D. dissertation (Amsterdam University, Krips Repro Meppel, Amsterdam, 1979).
  34. J. H. Wold, A. Valberg, “Mathematical description of a method for deriving an XYZ tristimulus space,” Department of Physics Report Series (University of Oslo, Oslo, 1999).
  35. CIE, CIE 1988 2° Spectral Luminous Efficiency Function for Photopic Vision, 1st ed., Publication CIE No. 86 (Central Bureau of the CIE, Vienna, 1990).
  36. H. G. Sperling, “An experimental investigation of the relationship between colour mixture and luminous efficiency,” in Visual Problems of Colour (Her Majesty’s Stationery Office, London, 1958), Vol. 1, pp. 249–247.
  37. J. H. Wold, “Colorimetric transformation equations allowing flexible normalization,” Die Farbe (to be published).
  38. A computer program (written in Mathematica) that determines the coefficient of the transformations that convert a given set of color matching data into a corresponding XYZ representation may be obtained from the authors on request.

1997 (1)

H. S. Fairman, M. H. Brill, H. Hemmendinger, “How the CIE 1931 color matching functions were derived from Wright-Guild data,” Color Res. Appl. 22, 11–23 (1997).
[CrossRef]

1993 (1)

1987 (1)

1986 (1)

R. M. Boynton, “A system of photometry and colorimetry based on cone excitation,” Color Res. Appl. 11, 244–252 (1986).
[CrossRef]

1980 (1)

1979 (1)

1978 (1)

J. J. Vos, “Colorimetric and photometric properties of a 2° fundamental observer,” Color Res. Appl. 3, 125–128 (1978).
[CrossRef]

1975 (1)

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

1968 (1)

S. L. Guth, J. V. Alexander, J. I. Chumbly, C. B. Gillman, M. M. Patterson, “Factors affecting luminance additivity at threshold among normal and color-blind subjects and elaborations of a trichromatic-opponent colors theory,” Vision Res. 8, 913–928 (1968).
[CrossRef] [PubMed]

1959 (1)

W. S. Stiles, J. M. Burch, “NPL colour-matching investigation: final report (1958),” Opt. Acta 6, 1–26 (1959).
[CrossRef]

1955 (1)

W. S. Stiles, J. M. Burch, “Interim report to the Commission Internationale de l’Eclairage, Zürich, 1955, on the National Physical Laboratory’s investigation of colour-matching,” Opt. Acta 2, 168–181 (1955).
[CrossRef]

1930 (1)

D. B. Judd, “Reduction of data on mixture of color stimuli,” Bur. Stand. J. Res. 4, 515–548 (1930).
[CrossRef]

1927 (1)

R. Luther, “Aus dem Gebiet der Farbreizmetrik,” Z. Tech. Phys. 8, 540–558 (1927).

1925 (1)

E. Schrödinger, “Über das Verhältnis der Vierfarben- zur Dreifarbentheorie,” Sitzungber. Kaiserl. Wien. Akad. Wiss. Math.-Naturwiss. Kl. 134, Abt. IIa, 471–490 (1925).

Adams, A. J.

Alexander, J. V.

S. L. Guth, J. V. Alexander, J. I. Chumbly, C. B. Gillman, M. M. Patterson, “Factors affecting luminance additivity at threshold among normal and color-blind subjects and elaborations of a trichromatic-opponent colors theory,” Vision Res. 8, 913–928 (1968).
[CrossRef] [PubMed]

Boynton, R. M.

R. M. Boynton, “A system of photometry and colorimetry based on cone excitation,” Color Res. Appl. 11, 244–252 (1986).
[CrossRef]

D. I. A. MacLeod, R. M. Boynton, “Chromaticity diagram showing cone excitation by stimuli of equal luminance,” J. Opt. Soc. Am. 69, 1183–1186 (1979).
[CrossRef] [PubMed]

Brill, M. H.

H. S. Fairman, M. H. Brill, H. Hemmendinger, “How the CIE 1931 color matching functions were derived from Wright-Guild data,” Color Res. Appl. 22, 11–23 (1997).
[CrossRef]

Burch, J. M.

W. S. Stiles, J. M. Burch, “NPL colour-matching investigation: final report (1958),” Opt. Acta 6, 1–26 (1959).
[CrossRef]

W. S. Stiles, J. M. Burch, “Interim report to the Commission Internationale de l’Eclairage, Zürich, 1955, on the National Physical Laboratory’s investigation of colour-matching,” Opt. Acta 2, 168–181 (1955).
[CrossRef]

Chumbly, J. I.

S. L. Guth, J. V. Alexander, J. I. Chumbly, C. B. Gillman, M. M. Patterson, “Factors affecting luminance additivity at threshold among normal and color-blind subjects and elaborations of a trichromatic-opponent colors theory,” Vision Res. 8, 913–928 (1968).
[CrossRef] [PubMed]

Eisner, A.

Estévez, O.

O. Estévez, “On the fundamental data-base of normal and dichromatic color vision,” Ph.D. dissertation (Amsterdam University, Krips Repro Meppel, Amsterdam, 1979).

Fairman, H. S.

H. S. Fairman, M. H. Brill, H. Hemmendinger, “How the CIE 1931 color matching functions were derived from Wright-Guild data,” Color Res. Appl. 22, 11–23 (1997).
[CrossRef]

Gillman, C. B.

S. L. Guth, J. V. Alexander, J. I. Chumbly, C. B. Gillman, M. M. Patterson, “Factors affecting luminance additivity at threshold among normal and color-blind subjects and elaborations of a trichromatic-opponent colors theory,” Vision Res. 8, 913–928 (1968).
[CrossRef] [PubMed]

Guild, J.

T. Smith, J. Guild, “The C.I.E. colorimetric standards and their use,” Trans. Opt. Soc. 33, 73–134 (1931–1932).
[CrossRef]

Guth, S. L.

S. L. Guth, J. V. Alexander, J. I. Chumbly, C. B. Gillman, M. M. Patterson, “Factors affecting luminance additivity at threshold among normal and color-blind subjects and elaborations of a trichromatic-opponent colors theory,” Vision Res. 8, 913–928 (1968).
[CrossRef] [PubMed]

Hemmendinger, H.

H. S. Fairman, M. H. Brill, H. Hemmendinger, “How the CIE 1931 color matching functions were derived from Wright-Guild data,” Color Res. Appl. 22, 11–23 (1997).
[CrossRef]

Johnson, N. E.

Judd, D. B.

D. B. Judd, “Reduction of data on mixture of color stimuli,” Bur. Stand. J. Res. 4, 515–548 (1930).
[CrossRef]

D. B. Judd, “CIE Technical Committee No. 7 “Colorimetry and artificial daylight,” Report of Secretariat United States Committee,” in Proceedings of the 12th Session of the CIE, Stockholm, 1951 (Bureau Central de la CIE, Paris, 1951), Vol. 1, Part 7, pp. 1–60.

Luther, R.

R. Luther, “Aus dem Gebiet der Farbreizmetrik,” Z. Tech. Phys. 8, 540–558 (1927).

MacAdam, D. L.

D. L. MacAdam, Color Measurement: Theme and Variations (Springer-Verlag, Berlin, 1981).

MacLeod, D. I. A.

Patterson, M. M.

S. L. Guth, J. V. Alexander, J. I. Chumbly, C. B. Gillman, M. M. Patterson, “Factors affecting luminance additivity at threshold among normal and color-blind subjects and elaborations of a trichromatic-opponent colors theory,” Vision Res. 8, 913–928 (1968).
[CrossRef] [PubMed]

Pokorny, J.

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Schrödinger, E.

E. Schrödinger, “Über das Verhältnis der Vierfarben- zur Dreifarbentheorie,” Sitzungber. Kaiserl. Wien. Akad. Wiss. Math.-Naturwiss. Kl. 134, Abt. IIa, 471–490 (1925).

Sharpe, L. T.

The contribution of the S cones to luminance has been somewhat contentious. Some authors claim that S cones do make a small contribution, whereas others maintain that they do not. However, given that the contribution, if any, is minor, it is convenient to assume that the S-cone contribution is zero. See, for instance, A. Stockman, L. T. Sharpe, “Cone spectral sensitivities and color matching,” in Color Vision: From Genes to Perception, K. R. Gegenfurtner, L. T. Sharpe, eds. (Cambridge U. Press, Cambridge, U.K., 1999).

Smith, T.

T. Smith, J. Guild, “The C.I.E. colorimetric standards and their use,” Trans. Opt. Soc. 33, 73–134 (1931–1932).
[CrossRef]

Smith, V. C.

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Sperling, H. G.

H. G. Sperling, “An experimental investigation of the relationship between colour mixture and luminous efficiency,” in Visual Problems of Colour (Her Majesty’s Stationery Office, London, 1958), Vol. 1, pp. 249–247.

Stiles, W. S.

W. S. Stiles, J. M. Burch, “NPL colour-matching investigation: final report (1958),” Opt. Acta 6, 1–26 (1959).
[CrossRef]

W. S. Stiles, J. M. Burch, “Interim report to the Commission Internationale de l’Eclairage, Zürich, 1955, on the National Physical Laboratory’s investigation of colour-matching,” Opt. Acta 2, 168–181 (1955).
[CrossRef]

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

Stockman, A.

A. Stockman, D. I. A. MacLeod, N. E. Johnson, “Spectral sensitivities of the human cones,” J. Opt. Soc. Am. A 10, 2491–2521 (1993).
[CrossRef]

The contribution of the S cones to luminance has been somewhat contentious. Some authors claim that S cones do make a small contribution, whereas others maintain that they do not. However, given that the contribution, if any, is minor, it is convenient to assume that the S-cone contribution is zero. See, for instance, A. Stockman, L. T. Sharpe, “Cone spectral sensitivities and color matching,” in Color Vision: From Genes to Perception, K. R. Gegenfurtner, L. T. Sharpe, eds. (Cambridge U. Press, Cambridge, U.K., 1999).

Valberg, A.

J. H. Wold, A. Valberg, “Cones for colorimetry,” in Proceedings of the 23rd Session of the CIE, New Delhi, 1995 (Central Bureau of the CIE, Vienna, 1995), Vol. 1, pp. 24–27.

J. H. Wold, A. Valberg, “A comparison of two principles for deriving XYZ tristimulus spaces,” in The 15th Symposium of the International Colour Vision Society, Göttingen, July 1999, Abstract P40.

J. H. Wold, A. Valberg, “Mathematical description of a method for deriving an XYZ tristimulus space,” Department of Physics Report Series (University of Oslo, Oslo, 1999).

J. H. Wold, A. Valberg, “The derivation of XYZ tristimulus spaces: a comparison of two alternative methods,” Color Res. Appl. (to be published).

Verdon, W.

Viénot, F.

F. Viénot, “Fundamental chromaticity diagram with physiologically significant axes,” in Proceedings of the Symposium ’96 on Colour Standards for Image Technology, Vienna, 1996 (Central Bureau of the CIE, Vienna, 1996), pp. 35–40.

Vos, J. J.

J. J. Vos, “Colorimetric and photometric properties of a 2° fundamental observer,” Color Res. Appl. 3, 125–128 (1978).
[CrossRef]

Wold, J. H.

J. H. Wold, “Colorimetric transformation equations allowing flexible normalization,” Die Farbe (to be published).

J. H. Wold, A. Valberg, “The derivation of XYZ tristimulus spaces: a comparison of two alternative methods,” Color Res. Appl. (to be published).

J. H. Wold, A. Valberg, “Mathematical description of a method for deriving an XYZ tristimulus space,” Department of Physics Report Series (University of Oslo, Oslo, 1999).

J. H. Wold, A. Valberg, “Cones for colorimetry,” in Proceedings of the 23rd Session of the CIE, New Delhi, 1995 (Central Bureau of the CIE, Vienna, 1995), Vol. 1, pp. 24–27.

J. H. Wold, A. Valberg, “A comparison of two principles for deriving XYZ tristimulus spaces,” in The 15th Symposium of the International Colour Vision Society, Göttingen, July 1999, Abstract P40.

Wright, W. D.

W. D. Wright, “A re-determination of the trichromatic coefficients of the spectral colours,” Trans. Opt. Soc. 30, 141–164 (1928–1929).
[CrossRef]

Wyszecki, G.

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

Bur. Stand. J. Res. (1)

D. B. Judd, “Reduction of data on mixture of color stimuli,” Bur. Stand. J. Res. 4, 515–548 (1930).
[CrossRef]

Color Res. Appl. (3)

H. S. Fairman, M. H. Brill, H. Hemmendinger, “How the CIE 1931 color matching functions were derived from Wright-Guild data,” Color Res. Appl. 22, 11–23 (1997).
[CrossRef]

R. M. Boynton, “A system of photometry and colorimetry based on cone excitation,” Color Res. Appl. 11, 244–252 (1986).
[CrossRef]

J. J. Vos, “Colorimetric and photometric properties of a 2° fundamental observer,” Color Res. Appl. 3, 125–128 (1978).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Acta (2)

W. S. Stiles, J. M. Burch, “NPL colour-matching investigation: final report (1958),” Opt. Acta 6, 1–26 (1959).
[CrossRef]

W. S. Stiles, J. M. Burch, “Interim report to the Commission Internationale de l’Eclairage, Zürich, 1955, on the National Physical Laboratory’s investigation of colour-matching,” Opt. Acta 2, 168–181 (1955).
[CrossRef]

Sitzungber. Kaiserl. Wien. Akad. Wiss. Math.-Naturwiss. Kl. (1)

E. Schrödinger, “Über das Verhältnis der Vierfarben- zur Dreifarbentheorie,” Sitzungber. Kaiserl. Wien. Akad. Wiss. Math.-Naturwiss. Kl. 134, Abt. IIa, 471–490 (1925).

Trans. Opt. Soc. (2)

W. D. Wright, “A re-determination of the trichromatic coefficients of the spectral colours,” Trans. Opt. Soc. 30, 141–164 (1928–1929).
[CrossRef]

T. Smith, J. Guild, “The C.I.E. colorimetric standards and their use,” Trans. Opt. Soc. 33, 73–134 (1931–1932).
[CrossRef]

Vision Res. (2)

S. L. Guth, J. V. Alexander, J. I. Chumbly, C. B. Gillman, M. M. Patterson, “Factors affecting luminance additivity at threshold among normal and color-blind subjects and elaborations of a trichromatic-opponent colors theory,” Vision Res. 8, 913–928 (1968).
[CrossRef] [PubMed]

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Z. Tech. Phys. (1)

R. Luther, “Aus dem Gebiet der Farbreizmetrik,” Z. Tech. Phys. 8, 540–558 (1927).

Other (22)

The contribution of the S cones to luminance has been somewhat contentious. Some authors claim that S cones do make a small contribution, whereas others maintain that they do not. However, given that the contribution, if any, is minor, it is convenient to assume that the S-cone contribution is zero. See, for instance, A. Stockman, L. T. Sharpe, “Cone spectral sensitivities and color matching,” in Color Vision: From Genes to Perception, K. R. Gegenfurtner, L. T. Sharpe, eds. (Cambridge U. Press, Cambridge, U.K., 1999).

CIE, Proceedings of the 6th Session of the CIE, Geneva, July 1924 (Cambridge U. Press, Cambridge, UK, 1926), pp. 67–70.

Other constraints have also been considered. For instance, D. Brainard (topical editor, 1999, personal communication) has suggested minimizing the difference between CMF’s (Judd–Vos versus Stiles–Burch1955) rather than between the spectral loci of the chromaticity diagrams. This alternative seemed most attractive because of its simplicity. However, it turns out that such a minimization of differences of tristimulus values in three dimensions leads to severe and considerable distortions of the chromaticity diagram (see Refs. 21 and 22). Yet another alternative would be to let the tristimulus vector to which the fundamental S-response curve (S fundamental) refers serve as the Z primary. (See Section 8 for further details.)

J. H. Wold, A. Valberg, “A comparison of two principles for deriving XYZ tristimulus spaces,” in The 15th Symposium of the International Colour Vision Society, Göttingen, July 1999, Abstract P40.

J. H. Wold, A. Valberg, “The derivation of XYZ tristimulus spaces: a comparison of two alternative methods,” Color Res. Appl. (to be published).

D. B. Judd, “CIE Technical Committee No. 7 “Colorimetry and artificial daylight,” Report of Secretariat United States Committee,” in Proceedings of the 12th Session of the CIE, Stockholm, 1951 (Bureau Central de la CIE, Paris, 1951), Vol. 1, Part 7, pp. 1–60.

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

In the paper by Stockman et al.,30 the proposed values are 0.68237 and 0.35235. If we apply these values, the maximum of V⋅(λ) turns out to be 0.999777 (at 553 nm). To normalize V⋅(λ) to a maximum value of 1.000000, the coefficients are multiplied by the factor 1/0.999777.

O. Estévez, “On the fundamental data-base of normal and dichromatic color vision,” Ph.D. dissertation (Amsterdam University, Krips Repro Meppel, Amsterdam, 1979).

J. H. Wold, A. Valberg, “Mathematical description of a method for deriving an XYZ tristimulus space,” Department of Physics Report Series (University of Oslo, Oslo, 1999).

CIE, CIE 1988 2° Spectral Luminous Efficiency Function for Photopic Vision, 1st ed., Publication CIE No. 86 (Central Bureau of the CIE, Vienna, 1990).

H. G. Sperling, “An experimental investigation of the relationship between colour mixture and luminous efficiency,” in Visual Problems of Colour (Her Majesty’s Stationery Office, London, 1958), Vol. 1, pp. 249–247.

J. H. Wold, “Colorimetric transformation equations allowing flexible normalization,” Die Farbe (to be published).

A computer program (written in Mathematica) that determines the coefficient of the transformations that convert a given set of color matching data into a corresponding XYZ representation may be obtained from the authors on request.

F. Viénot, “Fundamental chromaticity diagram with physiologically significant axes,” in Proceedings of the Symposium ’96 on Colour Standards for Image Technology, Vienna, 1996 (Central Bureau of the CIE, Vienna, 1996), pp. 35–40.

The CIE committee TC 1-36 distinguishes between the “fundamental response curves,” representing the fundamental spectral sensitivity functions of the three types of cones at the corneal level, and the “cone absorption curves,” which are derived by correcting the fundamentals by the absorption of the lens and ocular media and the macular pigment. Hence, if Tmedia(λ) denotes the spectral transmittance of the lens and ocular media and Tmacula(λ) denotes the spectral transmittance of the macular pigment, each cone absorption curve is given in terms of the corresponding fundamental response curve by the equation cone absorption curve=fundamental response curveTmedia(λ)Tmacula(λ).

CIE, Proceedings of the 8th Session of the CIE, Cambridge, 1931 (Cambridge U. Press, Cambridge, UK, 1932), pp. 19–29.

ISO/CIE, CIE Standard Colorimetric Observers, Publication ISO/CIE 10527 (Central Bureau of the CIE, Vienna, 1991).

D. L. MacAdam, Color Measurement: Theme and Variations (Springer-Verlag, Berlin, 1981).

The CIE RGB representation refers to Wright primaries R, G, and B, representing monochromatic stimuli of wavelengths 700.0, 546.1, and 435.8 nm, respectively (see Refs. 3 and 17). The norms of R, G, and B are defined so as to ensure that the chromaticity coordinates of Illuminant E are rE=gE=bE=1/3.

J. H. Wold, A. Valberg, “Cones for colorimetry,” in Proceedings of the 23rd Session of the CIE, New Delhi, 1995 (Central Bureau of the CIE, Vienna, 1995), Vol. 1, pp. 24–27.

CIE, , “Fundamental chromaticity diagram with physiological axes” (Central Bureau of the CIE, Vienna, 1998).

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Figures (7)

Fig. 1
Fig. 1

(a) (x, y) chromaticity diagram of the Judd–Vos modified 2° observer. Filled circles on the spectrum locus mark the chromaticity points (R), (G), and (B) of the new Wright primaries. The chromaticity point (E) of Illuminant E is positioned at (xE, yE)=(0.33499, 0.33618), i.e., slightly displaced from the ideal point (1/3, 1/3). (b) (r, g) chromaticity diagram of the Judd–Vos modified 2° observer resulting from transformation Eqs. (1). The diagram refers to Wright primaries R, G, and B representing monochromatic stimuli of wavelengths 700.0, 546.1, and 435.8 nm, normalized so that the chromaticity point (E) of Illuminant E is positioned at (rE, gE)=(1/3, 1/3). Lines L1, L2, and L3 (dashed) constitute a circumscribing triangle with vertices at the chromaticity points (X), (Y), and (Z) of the old primaries. Ordinates d1 and d3 of the inset magnifications give the Euclidean distances between the points on the locus segments (framed) and their respective closest points on the lines L1 and L3. Point (D) marks the locus point of shortest distance to line L1. Line L3 intersects the abscissa axis in point (P).

Fig. 2
Fig. 2

(a) (rˆ, gˆ) chromaticity diagram of the StilesBurch1955 2° pilot group. The diagram refers to primaries R^, G^, and B^ representing unit radiance, monochromatic stimuli with wave numbers 15,500, 19,000, and 22,500 cm-1. Filled circles on the spectrum locus mark the chromaticity points (R), (G), and (B) of the new Wright primaries. The chromaticity point (E) of Illuminant E is positioned at (r^E, g^E)=(0.579468, 0.265210). (b) (r, g) chromaticity diagram of the StilesBurch1955 2° pilot group resulting from transformation Eqs. (4). The diagram refers to Wright primaries R, G, and B representing monochromatic stimuli of wavelengths 700.0, 546.1, and 435.8 nm, normalized so that the chromaticity point (E) of Illuminant E is positioned at (rE, gE)=(1/3, 1/3). Filled squares on the spectrum locus mark the chromaticity points (R^), (G^), and (B^) of the original Stiles–Burch primaries (joined by dashed lines).

Fig. 3
Fig. 3

(a) (r, g) chromaticity diagram of the StilesBurch1955 2° pilot group [same as the diagram of Fig. 2(b)] with lines L1, L2, and L3 (dashed) constituting a circumscribing triangle with vertices at the chromaticity points (X), (Y), and (Z) of the new primaries X, Y, and Z. Ordinates d1 and d3 of the inset magnifications give the Euclidean distances between the points on the locus segments (framed) and their respective closest points on the lines L1 and L3. Point (D) marks the locus point of shortest distance to line L1. Line L3 is tangent to the spectrum locus in point (T) and intersects the abscissa axis in (P). (b) (x, y) chromaticity diagram of the StilesBurch1955 2° pilot group resulting from transformation Eqs. (19). Filled circles on the spectrum locus mark the chromaticity points (R), (G), and (B) of the Wright primaries underlying the (r, g) diagram. The chromaticity point (E) of Illuminant E is positioned at (xE, yE)=(1/3, 1/3).

Fig. 4
Fig. 4

(a) Color-matching functions defining the new StilesBurch1955 XYZ tristimulus space. (b) Color-matching functions defining the Judd–Vos XYZ tristimulus space.  

Fig. 5
Fig. 5

Joint plot displaying the spectrum locus of the new (x, y) diagram (solid curve) together with the spectrum locus of the Judd–Vos (x, y) diagram used as reference (dashed curve). The corresponding two coordinate points of Illuminant E are shown to be slightly separated. In the inset diagram the Euclidean distances sλ[(xλ-xλ)2+(yλ-yλ)2]1/2 between corresponding points on the spectrum loci are displayed as a function of the wavelength parameter λ.

Fig. 6
Fig. 6

Complementary wavelengths of the StilesBurch1955 2° pilot group (solid curve) and the Judd–Vos modified 2° observer (dashed curve).

Fig. 7
Fig. 7

The (x, y) chromaticity diagram of the StilesBurch1955 2° pilot group showing (estimated) locations of the dichromatic confusion points (filled circles). Point (S) is the tritanopic confusion point as determined relative to the original cone fundamentals of Stockman et al.30 Points (Llin) and (Mlin) represent the primaries that the L and M fundamentals of Stockman et al. would be referring to if the domain of their S fundamental’s linear relationship to the CMF’s of the StilesBurch1955 2° pilot group were extended to include the entire visible spectrum. The lines marked L=0 and M=0 (solid) and the line marked Slin=0 (dashed) represent the (virtual) stimuli for which respectively the fundamental L, M, and (approximate) S response is zero.    

Tables (2)

Tables Icon

Table 1 Chromaticity Coordinates of the Reference Stimuli X, Y, Z and R, G, B As Well As of the CIE Illuminant E, in the Respective XYZ and RGB Representations of the Judd–Vos Modified 2° Observer

Tables Icon

Table 2 Chromaticity Coordinates of the Reference Stimuli R^, G^, B^; R, G, B and X, Y, Z As Well As of the CIE Illuminant E, in the Respective RˆGˆBˆ, RGB, and XYZ Representations of the StilesBurch1955 2° Pilot Group

Equations (48)

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rQ=4.532604xQ-0.682602yQ-0.7582201.933206xQ-0.165284yQ+1,
gQ=-0.978903xQ+2.169755yQ+0.1291751.933206xQ-0.165284yQ+1,
d1D=0.020365.
α3=gY-gXrY-rX=-1.010269.
rQ=0.403966r^Q-0.217818g^Q-0.016470-0.607510r^Q-0.635054g^Q+1,
gQ=-0.003951r^Q+0.575889g^Q+0.009406-0.607510r^Q-0.635054g^Q+1,
V(λ)=cLL(λ)+cMM(λ).
L(λ)=aL1r^¯(λ)+aL2g^¯(λ)+aL3b^¯(λ),
M(λ)=aM1r^¯(λ)+aM2g^¯(λ)+aM3b^¯(λ),
V(λ)=(cLaL1+cMaM1)r^¯(λ)+(cLaL2+cMaM2)g^¯(λ)+(cLaL3+cMaM3)b^¯(λ).
[cL(aL1-aL3)+cM(aM1-aM3)]rˆ+[cL(aL2-aL3)
+cM(aM2-aM3)]gˆ+(cLaL3+cMaM3)=0,
aL1=0.214808,aM1=0.022882,aL2=0.751035,aM2=0.940534,aL3=0.045156,aM3=0.076827.
cL=0.682882,cM=0.352429.
rˆ=2.510421r+0.984891g+0.0320821.520274r+1.677748g+1,
gˆ=0.007609r+1.699724g-0.0161131.520274r+1.677748g+1,
L2:g=α2r+β2,α2=-0.212634,
β2=-0.031615.
α3=α3=-1.010269,
L3:g=α3r+β3,α3=-1.010269,
β3=1.011432.
d1D=d1D=0.020365.
RMS=1341 λ=390730[(xλ-xλ)2+(yλ-yλ)2]1/2,
L1:g=α1r+β1,α1=-2.629242,
β1=-1.946861.
rX=β3-β2α2-α3,gX=α2β3-α3β2α2-α3,
rY=β1-β3α3-α1,gY=α3β1-α1β3α3-α1,
rZ=β2-β1α1-α2,gZ=α1β2-α2β1α1-α2.
xQ=0.228051rQ+0.086736gQ+0.168864-0.448711rQ-0.087153gQ+1,
yQ=0.133580rQ+0.628216gQ+0.019861-0.448711rQ-0.087153gQ+1.
xQ=-0.010734r^Q-0.106262g^Q+0.164841-0.783283r^Q-0.583672g^Q+1,
yQ=0.039157r^Q+0.317985g^Q+0.023416-0.783283r^Q-0.583672g^Q+1.
r^Q=443.553255xQ+13.440336yQ-73.430328302.544303xQ+102.938188yQ+1,
g^Q=-76.898538xQ+158.326203yQ+8.968607302.544303xQ+102.938188yQ+1.
x¯(λ)y¯(λ)z¯(λ)=0.3811300.1448730.4076770.1547530.8443390.0579120.0000910.0404332.007571×r^¯(λ)g^¯(λ)b^¯(λ).
Txr : rjQ=i=13XijRxiQl,i=13XilRxiQ,XijRrjExijRρ=13xρjRxρE
(j=1, 2, 3),
Trr :rjQ=i=13R^ijRr^iQl,i=13R^ilRr^iQ,R^ijRrjE r^ijRρ=13r^ρjRr^ρE
(j=1, 2, 3),
Trx :xkQ=j=13RjkXrjQm,j=13RjmX rjQ,RjkXxkE rjkXμ=13rμkX rμE
(k=1, 2, 3),
TrˆxTrx ° Trˆr :xkQ=i=13R^ikXr^iQn,i=13R^inXr^iQ,
R^ikXj=13 (rjE xkE)(r^ijRrjkX)ρ,μ=13(r^ρjRrμkX)(r^ρE rμE)(k=1, 2, 3),
Tr^¯x¯ :x¯k(λ)=κi=13R^ikXr^¯i(λ),
R^ikXj=13 (rjE xkE)(r^ijRrjkX)ρ,μ=13(r^ρjRrμkX)(r^ρE rμE),
κ0(k=1, 2, 3),
κ=cLaLi+cMaMiR^i2X,i{1, 2, 3}.
coneabsorptioncurve=fundamentalresponsecurveTmedia(λ)Tmacula(λ).

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