Abstract

Many theories of color discrimination predict a discontinuity in the wavelength-discrimination function of a tritanope at the point in the spectrum at which the rate of change of the visual signal constrained to an equiluminant plane passes through zero (near 460 nm). The predicted discontinuity follows from the use of a first-order approximation for which the reciprocal of the slope of the response function that generates the visual signal is proportional to the discrimination limen. In view of the good discrimination shown by such observers elsewhere in the spectrum, however, such a singularity is impossible. I show that the inclusion of the higher-order terms produces a finite value in the 460-nm region that falls in the range of values from the literature that have been obtained experimentally.

© 1993 Optical Society of America

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References

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  1. E. N. Wilmer, W. D. Wright, “Colour sensitivity of the fovea centralis,” Nature (London) 156, 119–121 (1945).
    [Crossref]
  2. F. P. Fischer, M. A. Bouman, J. ten Doesschate, “A case of tritanopy,” Doc. Ophthalmol. 5/6, 73–87 (1951).
    [Crossref]
  3. W. D. Wright, “The characteristics of tritanopia,” J. Opt. Soc. Am. 42, 509–521 (1952).
    [Crossref] [PubMed]
  4. A. E. Krill, V C. Smith, J. Pokorny, “Further studies supporting the identity of congenital tritanopia and hereditary dominant optic atrophy,” Invest. Ophthalmol. 10, 457–465 (1971).
    [PubMed]
  5. D. Smith, “Color naming and hue discrimination in congenital tritanopia and tritanomaly,” Vision Res. 13, 209–218 (1973).
    [Crossref] [PubMed]
  6. J. Voke-Fletcher, R. J. Fletcher, “A case of tritanopia,” Mod. Probl. Ophthalmol. 19, 229–231 (1978).
    [PubMed]
  7. C. R. Cavonius, O. Estévez, “π-mechanisms and cone fundamentals,” in Visual Psychophysics and Physiology, J. C. Armington, J. Krauskopf, B. R. Wooten, eds. (Academic, New York, 1978), pp. 221–231.
    [Crossref]
  8. M. Alpern, K. Kitahara, D. H. Krantz, “Classical tritanopia,” J. Physiol. 335, 655–681 (1983).
    [PubMed]
  9. P. K. Kaiser, R. M. Boynton, “Role of the blue mechanism in wavelength discrimination,” Vision Res. 25, 523–529 (1985).
    [Crossref] [PubMed]
  10. P. K. Kaiser, M. Ayama, “Just noticeable inhomogeneity criterion for determining wavelength discrimination function,” Vision Res. 25, 1327–1330 (1985).
    [Crossref]
  11. J. D. Mollon, O. Estévez, “Tyndall’s paradox of hue discrimination,” J. Opt. Soc. Am. A 5, 151–159 (1988).
    [Crossref] [PubMed]
  12. J. D. Mollon, O. Estévez, C. R. Cavonius, “The two subsystems of colour vision and their rôles in wavelength discrimination,” in Vision: Coding and Efficiency, C. Blakemore, ed. (Cambridge U. Press, Cambridge, 1990), pp. 119–131.
  13. R. M. Boynton, Human Color Vision (Holt, Rinehart & Winston, New York, 1979).
  14. J. Pokorny, V. C. Smith, G. Verriest, A. J. L. G. Pinckers, Congenital and Acquired Color Vision Defects (Grune & Stratton, New York, 1979).
  15. R. M. Boynton, N. Kambe, “Chromatic difference steps of moderate size measured along theoretically critical axes,” Color Res. Appl. 5, 13–23 (1980).
    [Crossref]
  16. A. L. Nagy, R. T. Eskew, R. M. Boynton, “Analysis of color-matching ellipses in cone-excitation space,” J. Opt. Soc. Am. A 4, 756–768 (1987).
    [Crossref] [PubMed]
  17. The differentiable approximations of L and M cone sensitivities published by R. M. Boynton and J. J. Wisowaty [“Equations for chromatic discrimination models,” J. Opt. Soc. Am. 70, 1471–1476 (1980)] could have been used here, although their use would be unlikely to have affected the results. Contrary to the authors′ assertion, the cubic spline curve is continuous [W H. Press, B. P. Flannery, S. A. Teukolsky, and W T. Vetterling, Numerical Recipes in Pascal, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Sec. 3.3] and should suffice for the purposes here.
  18. L. M. Hurvich, D. Jameson, “Some quantitative aspects of an opponent-colors theory. II. Brightness, saturation, and hue in normal and dichromatic vision,” J. Opt. Soc. Am. 45, 602–616 (1955).
    [Crossref] [PubMed]
  19. W. S. Stiles, “A modified Helmholtz line-element in brightness-colour space,” Proc. Phys. Soc. 58, 41–65 (1946).
    [Crossref]
  20. M. A. Bouman, P. L. Walraven, “Quantum theory of discrimination of dichromates,” Vision Res. 2, 177–187 (1962).
    [Crossref]
  21. J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision—I. Basic concepts,” Vision Res. 12, 1327–1344 (1972).
    [Crossref] [PubMed]
  22. J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision—II. The derivation of the line element,” Vision Res. 12, 1345–1365 (1972).
    [Crossref] [PubMed]
  23. D. B. Judd, G. T. Yonemura, “CIE 1960 UCS diagram and the Müller theory of color vision,” J. Res. Nat. Bur. Stand. Sect. A 74, 23–30 (1970).
    [Crossref]
  24. S. L. Guth, R. Massof, T. Benzschawel, “Vector model for normal and dichromatic vision,” J. Opt. Soc. Am. 70, 187–211 (1980).
    [Crossref]
  25. Given that the slope is zero, such metameric pairs need not exist; if the second derivative vanishes also, then the point at which the slope becomes zero is an inflection rather than an extremum, and the spectral curve would be monotonic.

1988 (1)

1987 (1)

1985 (2)

P. K. Kaiser, R. M. Boynton, “Role of the blue mechanism in wavelength discrimination,” Vision Res. 25, 523–529 (1985).
[Crossref] [PubMed]

P. K. Kaiser, M. Ayama, “Just noticeable inhomogeneity criterion for determining wavelength discrimination function,” Vision Res. 25, 1327–1330 (1985).
[Crossref]

1983 (1)

M. Alpern, K. Kitahara, D. H. Krantz, “Classical tritanopia,” J. Physiol. 335, 655–681 (1983).
[PubMed]

1980 (2)

R. M. Boynton, N. Kambe, “Chromatic difference steps of moderate size measured along theoretically critical axes,” Color Res. Appl. 5, 13–23 (1980).
[Crossref]

S. L. Guth, R. Massof, T. Benzschawel, “Vector model for normal and dichromatic vision,” J. Opt. Soc. Am. 70, 187–211 (1980).
[Crossref]

1978 (1)

J. Voke-Fletcher, R. J. Fletcher, “A case of tritanopia,” Mod. Probl. Ophthalmol. 19, 229–231 (1978).
[PubMed]

1973 (1)

D. Smith, “Color naming and hue discrimination in congenital tritanopia and tritanomaly,” Vision Res. 13, 209–218 (1973).
[Crossref] [PubMed]

1972 (2)

J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision—I. Basic concepts,” Vision Res. 12, 1327–1344 (1972).
[Crossref] [PubMed]

J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision—II. The derivation of the line element,” Vision Res. 12, 1345–1365 (1972).
[Crossref] [PubMed]

1971 (1)

A. E. Krill, V C. Smith, J. Pokorny, “Further studies supporting the identity of congenital tritanopia and hereditary dominant optic atrophy,” Invest. Ophthalmol. 10, 457–465 (1971).
[PubMed]

1970 (1)

D. B. Judd, G. T. Yonemura, “CIE 1960 UCS diagram and the Müller theory of color vision,” J. Res. Nat. Bur. Stand. Sect. A 74, 23–30 (1970).
[Crossref]

1962 (1)

M. A. Bouman, P. L. Walraven, “Quantum theory of discrimination of dichromates,” Vision Res. 2, 177–187 (1962).
[Crossref]

1955 (1)

1952 (1)

1951 (1)

F. P. Fischer, M. A. Bouman, J. ten Doesschate, “A case of tritanopy,” Doc. Ophthalmol. 5/6, 73–87 (1951).
[Crossref]

1946 (1)

W. S. Stiles, “A modified Helmholtz line-element in brightness-colour space,” Proc. Phys. Soc. 58, 41–65 (1946).
[Crossref]

1945 (1)

E. N. Wilmer, W. D. Wright, “Colour sensitivity of the fovea centralis,” Nature (London) 156, 119–121 (1945).
[Crossref]

Alpern, M.

M. Alpern, K. Kitahara, D. H. Krantz, “Classical tritanopia,” J. Physiol. 335, 655–681 (1983).
[PubMed]

Ayama, M.

P. K. Kaiser, M. Ayama, “Just noticeable inhomogeneity criterion for determining wavelength discrimination function,” Vision Res. 25, 1327–1330 (1985).
[Crossref]

Benzschawel, T.

S. L. Guth, R. Massof, T. Benzschawel, “Vector model for normal and dichromatic vision,” J. Opt. Soc. Am. 70, 187–211 (1980).
[Crossref]

Bouman, M. A.

M. A. Bouman, P. L. Walraven, “Quantum theory of discrimination of dichromates,” Vision Res. 2, 177–187 (1962).
[Crossref]

F. P. Fischer, M. A. Bouman, J. ten Doesschate, “A case of tritanopy,” Doc. Ophthalmol. 5/6, 73–87 (1951).
[Crossref]

Boynton, R. M.

A. L. Nagy, R. T. Eskew, R. M. Boynton, “Analysis of color-matching ellipses in cone-excitation space,” J. Opt. Soc. Am. A 4, 756–768 (1987).
[Crossref] [PubMed]

P. K. Kaiser, R. M. Boynton, “Role of the blue mechanism in wavelength discrimination,” Vision Res. 25, 523–529 (1985).
[Crossref] [PubMed]

R. M. Boynton, N. Kambe, “Chromatic difference steps of moderate size measured along theoretically critical axes,” Color Res. Appl. 5, 13–23 (1980).
[Crossref]

R. M. Boynton, Human Color Vision (Holt, Rinehart & Winston, New York, 1979).

Cavonius, C. R.

J. D. Mollon, O. Estévez, C. R. Cavonius, “The two subsystems of colour vision and their rôles in wavelength discrimination,” in Vision: Coding and Efficiency, C. Blakemore, ed. (Cambridge U. Press, Cambridge, 1990), pp. 119–131.

C. R. Cavonius, O. Estévez, “π-mechanisms and cone fundamentals,” in Visual Psychophysics and Physiology, J. C. Armington, J. Krauskopf, B. R. Wooten, eds. (Academic, New York, 1978), pp. 221–231.
[Crossref]

Doesschate, J. ten

F. P. Fischer, M. A. Bouman, J. ten Doesschate, “A case of tritanopy,” Doc. Ophthalmol. 5/6, 73–87 (1951).
[Crossref]

Eskew, R. T.

Estévez, O.

J. D. Mollon, O. Estévez, “Tyndall’s paradox of hue discrimination,” J. Opt. Soc. Am. A 5, 151–159 (1988).
[Crossref] [PubMed]

C. R. Cavonius, O. Estévez, “π-mechanisms and cone fundamentals,” in Visual Psychophysics and Physiology, J. C. Armington, J. Krauskopf, B. R. Wooten, eds. (Academic, New York, 1978), pp. 221–231.
[Crossref]

J. D. Mollon, O. Estévez, C. R. Cavonius, “The two subsystems of colour vision and their rôles in wavelength discrimination,” in Vision: Coding and Efficiency, C. Blakemore, ed. (Cambridge U. Press, Cambridge, 1990), pp. 119–131.

Fischer, F. P.

F. P. Fischer, M. A. Bouman, J. ten Doesschate, “A case of tritanopy,” Doc. Ophthalmol. 5/6, 73–87 (1951).
[Crossref]

Fletcher, R. J.

J. Voke-Fletcher, R. J. Fletcher, “A case of tritanopia,” Mod. Probl. Ophthalmol. 19, 229–231 (1978).
[PubMed]

Guth, S. L.

S. L. Guth, R. Massof, T. Benzschawel, “Vector model for normal and dichromatic vision,” J. Opt. Soc. Am. 70, 187–211 (1980).
[Crossref]

Hurvich, L. M.

Jameson, D.

Judd, D. B.

D. B. Judd, G. T. Yonemura, “CIE 1960 UCS diagram and the Müller theory of color vision,” J. Res. Nat. Bur. Stand. Sect. A 74, 23–30 (1970).
[Crossref]

Kaiser, P. K.

P. K. Kaiser, R. M. Boynton, “Role of the blue mechanism in wavelength discrimination,” Vision Res. 25, 523–529 (1985).
[Crossref] [PubMed]

P. K. Kaiser, M. Ayama, “Just noticeable inhomogeneity criterion for determining wavelength discrimination function,” Vision Res. 25, 1327–1330 (1985).
[Crossref]

Kambe, N.

R. M. Boynton, N. Kambe, “Chromatic difference steps of moderate size measured along theoretically critical axes,” Color Res. Appl. 5, 13–23 (1980).
[Crossref]

Kitahara, K.

M. Alpern, K. Kitahara, D. H. Krantz, “Classical tritanopia,” J. Physiol. 335, 655–681 (1983).
[PubMed]

Krantz, D. H.

M. Alpern, K. Kitahara, D. H. Krantz, “Classical tritanopia,” J. Physiol. 335, 655–681 (1983).
[PubMed]

Krill, A. E.

A. E. Krill, V C. Smith, J. Pokorny, “Further studies supporting the identity of congenital tritanopia and hereditary dominant optic atrophy,” Invest. Ophthalmol. 10, 457–465 (1971).
[PubMed]

Massof, R.

S. L. Guth, R. Massof, T. Benzschawel, “Vector model for normal and dichromatic vision,” J. Opt. Soc. Am. 70, 187–211 (1980).
[Crossref]

Mollon, J. D.

J. D. Mollon, O. Estévez, “Tyndall’s paradox of hue discrimination,” J. Opt. Soc. Am. A 5, 151–159 (1988).
[Crossref] [PubMed]

J. D. Mollon, O. Estévez, C. R. Cavonius, “The two subsystems of colour vision and their rôles in wavelength discrimination,” in Vision: Coding and Efficiency, C. Blakemore, ed. (Cambridge U. Press, Cambridge, 1990), pp. 119–131.

Nagy, A. L.

Pinckers, A. J. L. G.

J. Pokorny, V. C. Smith, G. Verriest, A. J. L. G. Pinckers, Congenital and Acquired Color Vision Defects (Grune & Stratton, New York, 1979).

Pokorny, J.

A. E. Krill, V C. Smith, J. Pokorny, “Further studies supporting the identity of congenital tritanopia and hereditary dominant optic atrophy,” Invest. Ophthalmol. 10, 457–465 (1971).
[PubMed]

J. Pokorny, V. C. Smith, G. Verriest, A. J. L. G. Pinckers, Congenital and Acquired Color Vision Defects (Grune & Stratton, New York, 1979).

Smith, D.

D. Smith, “Color naming and hue discrimination in congenital tritanopia and tritanomaly,” Vision Res. 13, 209–218 (1973).
[Crossref] [PubMed]

Smith, V C.

A. E. Krill, V C. Smith, J. Pokorny, “Further studies supporting the identity of congenital tritanopia and hereditary dominant optic atrophy,” Invest. Ophthalmol. 10, 457–465 (1971).
[PubMed]

Smith, V. C.

J. Pokorny, V. C. Smith, G. Verriest, A. J. L. G. Pinckers, Congenital and Acquired Color Vision Defects (Grune & Stratton, New York, 1979).

Stiles, W. S.

W. S. Stiles, “A modified Helmholtz line-element in brightness-colour space,” Proc. Phys. Soc. 58, 41–65 (1946).
[Crossref]

Verriest, G.

J. Pokorny, V. C. Smith, G. Verriest, A. J. L. G. Pinckers, Congenital and Acquired Color Vision Defects (Grune & Stratton, New York, 1979).

Voke-Fletcher, J.

J. Voke-Fletcher, R. J. Fletcher, “A case of tritanopia,” Mod. Probl. Ophthalmol. 19, 229–231 (1978).
[PubMed]

Vos, J. J.

J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision—I. Basic concepts,” Vision Res. 12, 1327–1344 (1972).
[Crossref] [PubMed]

J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision—II. The derivation of the line element,” Vision Res. 12, 1345–1365 (1972).
[Crossref] [PubMed]

Walraven, P. L.

J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision—II. The derivation of the line element,” Vision Res. 12, 1345–1365 (1972).
[Crossref] [PubMed]

J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision—I. Basic concepts,” Vision Res. 12, 1327–1344 (1972).
[Crossref] [PubMed]

M. A. Bouman, P. L. Walraven, “Quantum theory of discrimination of dichromates,” Vision Res. 2, 177–187 (1962).
[Crossref]

Wilmer, E. N.

E. N. Wilmer, W. D. Wright, “Colour sensitivity of the fovea centralis,” Nature (London) 156, 119–121 (1945).
[Crossref]

Wright, W. D.

W. D. Wright, “The characteristics of tritanopia,” J. Opt. Soc. Am. 42, 509–521 (1952).
[Crossref] [PubMed]

E. N. Wilmer, W. D. Wright, “Colour sensitivity of the fovea centralis,” Nature (London) 156, 119–121 (1945).
[Crossref]

Yonemura, G. T.

D. B. Judd, G. T. Yonemura, “CIE 1960 UCS diagram and the Müller theory of color vision,” J. Res. Nat. Bur. Stand. Sect. A 74, 23–30 (1970).
[Crossref]

Color Res. Appl. (1)

R. M. Boynton, N. Kambe, “Chromatic difference steps of moderate size measured along theoretically critical axes,” Color Res. Appl. 5, 13–23 (1980).
[Crossref]

Doc. Ophthalmol. (1)

F. P. Fischer, M. A. Bouman, J. ten Doesschate, “A case of tritanopy,” Doc. Ophthalmol. 5/6, 73–87 (1951).
[Crossref]

Invest. Ophthalmol. (1)

A. E. Krill, V C. Smith, J. Pokorny, “Further studies supporting the identity of congenital tritanopia and hereditary dominant optic atrophy,” Invest. Ophthalmol. 10, 457–465 (1971).
[PubMed]

J. Opt. Soc. Am. (3)

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

J. Physiol. (1)

M. Alpern, K. Kitahara, D. H. Krantz, “Classical tritanopia,” J. Physiol. 335, 655–681 (1983).
[PubMed]

J. Res. Nat. Bur. Stand. Sect. A (1)

D. B. Judd, G. T. Yonemura, “CIE 1960 UCS diagram and the Müller theory of color vision,” J. Res. Nat. Bur. Stand. Sect. A 74, 23–30 (1970).
[Crossref]

Mod. Probl. Ophthalmol. (1)

J. Voke-Fletcher, R. J. Fletcher, “A case of tritanopia,” Mod. Probl. Ophthalmol. 19, 229–231 (1978).
[PubMed]

Nature (London) (1)

E. N. Wilmer, W. D. Wright, “Colour sensitivity of the fovea centralis,” Nature (London) 156, 119–121 (1945).
[Crossref]

Proc. Phys. Soc. (1)

W. S. Stiles, “A modified Helmholtz line-element in brightness-colour space,” Proc. Phys. Soc. 58, 41–65 (1946).
[Crossref]

Vision Res. (6)

M. A. Bouman, P. L. Walraven, “Quantum theory of discrimination of dichromates,” Vision Res. 2, 177–187 (1962).
[Crossref]

J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision—I. Basic concepts,” Vision Res. 12, 1327–1344 (1972).
[Crossref] [PubMed]

J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision—II. The derivation of the line element,” Vision Res. 12, 1345–1365 (1972).
[Crossref] [PubMed]

P. K. Kaiser, R. M. Boynton, “Role of the blue mechanism in wavelength discrimination,” Vision Res. 25, 523–529 (1985).
[Crossref] [PubMed]

P. K. Kaiser, M. Ayama, “Just noticeable inhomogeneity criterion for determining wavelength discrimination function,” Vision Res. 25, 1327–1330 (1985).
[Crossref]

D. Smith, “Color naming and hue discrimination in congenital tritanopia and tritanomaly,” Vision Res. 13, 209–218 (1973).
[Crossref] [PubMed]

Other (6)

C. R. Cavonius, O. Estévez, “π-mechanisms and cone fundamentals,” in Visual Psychophysics and Physiology, J. C. Armington, J. Krauskopf, B. R. Wooten, eds. (Academic, New York, 1978), pp. 221–231.
[Crossref]

J. D. Mollon, O. Estévez, C. R. Cavonius, “The two subsystems of colour vision and their rôles in wavelength discrimination,” in Vision: Coding and Efficiency, C. Blakemore, ed. (Cambridge U. Press, Cambridge, 1990), pp. 119–131.

R. M. Boynton, Human Color Vision (Holt, Rinehart & Winston, New York, 1979).

J. Pokorny, V. C. Smith, G. Verriest, A. J. L. G. Pinckers, Congenital and Acquired Color Vision Defects (Grune & Stratton, New York, 1979).

The differentiable approximations of L and M cone sensitivities published by R. M. Boynton and J. J. Wisowaty [“Equations for chromatic discrimination models,” J. Opt. Soc. Am. 70, 1471–1476 (1980)] could have been used here, although their use would be unlikely to have affected the results. Contrary to the authors′ assertion, the cubic spline curve is continuous [W H. Press, B. P. Flannery, S. A. Teukolsky, and W T. Vetterling, Numerical Recipes in Pascal, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Sec. 3.3] and should suffice for the purposes here.

Given that the slope is zero, such metameric pairs need not exist; if the second derivative vanishes also, then the point at which the slope becomes zero is an inflection rather than an extremum, and the spectral curve would be monotonic.

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Figures (1)

Fig. 1
Fig. 1

Theoretical curves of the just-discriminable wavelength difference Δλ as a function of wavelength for a tritanope. The solid curve is based on the assumption that discrimination is proportional to the slope of the visual response curve. The dashed and dotted curves were calculated by interpolation of criterion values of 0.01 and 0.03, respectively, along the visual response curve.

Tables (1)

Tables Icon

Table 1 Summary of Characteristics of Discrimination at Short-Wavelength Pessimum Found under Tritan Conditions

Equations (7)

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

f ( λ , Δ λ ) = | Γ ( λ ) Γ ( λ + Δ λ ) | = c ,
| Γ ( λ ) Γ ( λ + Δ λ ) | = | λ λ + Δ λ Γ ( λ ) d λ | = c .
Δ λ | Γ ( λ ) | = c ,
Δ λ = c | Γ ( λ ) | .
Γ ( λ + Δ λ ) = Γ ( λ ) + Γ ( λ ) Δ λ + Γ ( λ ) Δ λ 2 / 2 ! + .
Γ ( λ ) = ( L 2 M ) / ( L + M ) ,
Γ ( λ ) = 2 + 3 I λ L .

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