Abstract

The dichromatic color appearance of a chromatic stimulus T can be described if a stimulus S is found that verifies that a normal observer experiences the same sensation viewing S as a dichromat viewing T. If dichromatic and normal versions of the same color vision model are available, S can be computed by applying the inverse of the normal model to the descriptors of T obtained with the dichromatic model. We give analytical form to this algorithm, which we call the corresponding-pair procedure. The analytical form highlights the requisites that a color vision model must verify for this procedure to be used. To show the capabilities of the method, we apply the algorithm to different color vision models that verify such requisites. This algorithm avoids the need to introduce empirical information alien to the color model used, as was the case with previous methods. The relative simplicity of the procedure and its generality makes the prediction of dichromatic color appearance an additional test of the validity of color vision models.

© 2004 Optical Society of America

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References

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  6. A. Stockman, D. I. A. MacLeod, N. E. Johnson, “Spectral sensitivities of the human cones,” J. Opt. Soc. Am. A 10, 2491–2521 (1993).
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  7. D. B. Judd, “Color perceptions of deuteranopic and protanopic observers,” J. Res. Natl. Bur. Stand. 41, 247–271 (1948).
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  8. E. C. de Vries-de Mol, L. N. Went, “Unilateral colour vision disturbance: a family study,” Clin. Genet. 2, 15–27 (1971).
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  10. M. Alpern, K. Kitahara, D. H. Krantz, “Perception of colour in unilateral tritanopia,” J. Physiol. (London) 335, 683–697 (1983).
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  30. K. H. Ruddock, “Psychophysics of inherited colour vision deficiencies,” in Inherited and Acquired Colour Vision Deficiencies, D. H. Foster, ed., Vol 7 of Vision and Visual Dysfunction (Macmillan, New York, 1991), pp. 4–37.
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1997 (2)

R. L. De Valois, K. K. De Valois, E. Switkes, M. Luke, “Hue scaling of isoluminant and cone-specific lights,” Vision Res. 37, 885–897 (1997).
[CrossRef] [PubMed]

H. Brettel, F. Viénot, J. D. Mollon, “Computerized simulation of color appearance for dichromats,” J. Opt. Soc. Am. A 14, 2647–2655 (1997).
[CrossRef]

1995 (1)

F. Viénot, H. Brettel, L. Ott, A. Ben M’Barek, J. D. Mollon, “What do color blind people see?” Nature (London) 376, 127–128 (1995).
[CrossRef]

1993 (2)

1991 (1)

1986 (2)

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

T. Seim, A. Valberg, “Towards a uniform color space. A better formula to describe the Munsell and OSA color scales,” Color Res. Appl. 11, 11–24 (1986).
[CrossRef]

1984 (1)

S. A. Burns, A. E. Elsner, J. Pokorny, V. C. Smith, “The Abney effect: chromaticity coordinates of unique and other constant hues,” Vision Res. 24, 479–489 (1984).
[CrossRef] [PubMed]

1983 (1)

M. Alpern, K. Kitahara, D. H. Krantz, “Perception of colour in unilateral tritanopia,” J. Physiol. (London) 335, 683–697 (1983).

1982 (1)

M. Hendricks, K. H. Ruddock, W. A. Waterfield, “Spectral sensitivity functions of post-receptoral responses in human vision,” J. Physiol. (London) 331, 17–33 (1982).

1980 (1)

1978 (1)

M. Romeskie, D. Yager, “Psychopysical measures and theoretical analysis of dichromatic opponent response functions,” Mod. Probl. Ophthalmol. 19, 212–217 (1978).

1977 (1)

C. R. Ingling, B. H. Tsou, “Orthogonal combinations of three visual channels,” Vision Res. 17, 1075–1082 (1977).
[CrossRef]

1976 (1)

D. I. A. MacLeod, P. Lennie, “Red–green blindness confined to one eye,” Vision Res. 16, 691–702 (1976).
[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]

1972 (1)

B. G. Bender, K. H. Ruddock, E. C. de Vries-de Mol, L. N. Went, “The colour vision characteristics of an observer with unilateral defective colour vision: results and analysis,” Vision Res. 12, 2035–2057 (1972).
[CrossRef] [PubMed]

1971 (1)

E. C. de Vries-de Mol, L. N. Went, “Unilateral colour vision disturbance: a family study,” Clin. Genet. 2, 15–27 (1971).

1967 (1)

C. H. Graham, Y. Hsia, “Visual discriminations of a subject with acquired unilateral tritanopia,” Vision Res. 7, 469–479 (1967).
[CrossRef] [PubMed]

1958 (2)

C. H. Graham, Y. Hsia, “Color defect and color theory: studies in normal and blind persons, including a subject color-blind in one eye but not in the other,” Science 127, 675–682 (1958).
[CrossRef] [PubMed]

C. H. Graham, Y. Hsia, “The spectral luminosity curves for a dichromatic eye and a normal eye in the same person,” Proc. Natl. Acad. Sci. USA 44, 46–49 (1958).
[CrossRef] [PubMed]

1948 (2)

D. B. Judd, “Color perceptions of deuteranopic and protanopic observers,” J. Res. Natl. Bur. Stand. 41, 247–271 (1948).
[CrossRef]

L. L. Sloan, L. Wollach, “A case of unilateral deuteranopia,” J. Opt. Soc. Am. 38, 502–509 (1948).
[CrossRef] [PubMed]

1941 (1)

W. Trendelenburg, “Ueber Vererbung bei einen Fall von anomalen Farbensinn des einen und normalen Farbensinn des anderen Auges beim Mann,” Klin. Mbl. Augenheilk. 107, 280–293 (1941).

Alpern, M.

M. Alpern, K. Kitahara, D. H. Krantz, “Perception of colour in unilateral tritanopia,” J. Physiol. (London) 335, 683–697 (1983).

Ben M’Barek, A.

F. Viénot, H. Brettel, L. Ott, A. Ben M’Barek, J. D. Mollon, “What do color blind people see?” Nature (London) 376, 127–128 (1995).
[CrossRef]

Bender, B. G.

B. G. Bender, K. H. Ruddock, E. C. de Vries-de Mol, L. N. Went, “The colour vision characteristics of an observer with unilateral defective colour vision: results and analysis,” Vision Res. 12, 2035–2057 (1972).
[CrossRef] [PubMed]

Benzschawel, T.

Boynton, R. M.

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

Brettel, H.

H. Brettel, F. Viénot, J. D. Mollon, “Computerized simulation of color appearance for dichromats,” J. Opt. Soc. Am. A 14, 2647–2655 (1997).
[CrossRef]

F. Viénot, H. Brettel, L. Ott, A. Ben M’Barek, J. D. Mollon, “What do color blind people see?” Nature (London) 376, 127–128 (1995).
[CrossRef]

Burns, S. A.

S. A. Burns, A. E. Elsner, J. Pokorny, V. C. Smith, “The Abney effect: chromaticity coordinates of unique and other constant hues,” Vision Res. 24, 479–489 (1984).
[CrossRef] [PubMed]

De Valois, K. K.

R. L. De Valois, K. K. De Valois, E. Switkes, M. Luke, “Hue scaling of isoluminant and cone-specific lights,” Vision Res. 37, 885–897 (1997).
[CrossRef] [PubMed]

R. L. De Valois, K. K. De Valois, “A multi-stage color model,” Vision Res. 33, 1053–1065 (1993).
[CrossRef] [PubMed]

De Valois, R. L.

R. L. De Valois, K. K. De Valois, E. Switkes, M. Luke, “Hue scaling of isoluminant and cone-specific lights,” Vision Res. 37, 885–897 (1997).
[CrossRef] [PubMed]

R. L. De Valois, K. K. De Valois, “A multi-stage color model,” Vision Res. 33, 1053–1065 (1993).
[CrossRef] [PubMed]

de Vries-de Mol, E. C.

B. G. Bender, K. H. Ruddock, E. C. de Vries-de Mol, L. N. Went, “The colour vision characteristics of an observer with unilateral defective colour vision: results and analysis,” Vision Res. 12, 2035–2057 (1972).
[CrossRef] [PubMed]

E. C. de Vries-de Mol, L. N. Went, “Unilateral colour vision disturbance: a family study,” Clin. Genet. 2, 15–27 (1971).

Elsner, A. E.

S. A. Burns, A. E. Elsner, J. Pokorny, V. C. Smith, “The Abney effect: chromaticity coordinates of unique and other constant hues,” Vision Res. 24, 479–489 (1984).
[CrossRef] [PubMed]

Evans, R. M.

R. M. Evans, An Introduction to Color (Wiley, New York, 1948).

Graham, C. H.

C. H. Graham, Y. Hsia, “Visual discriminations of a subject with acquired unilateral tritanopia,” Vision Res. 7, 469–479 (1967).
[CrossRef] [PubMed]

C. H. Graham, Y. Hsia, “Color defect and color theory: studies in normal and blind persons, including a subject color-blind in one eye but not in the other,” Science 127, 675–682 (1958).
[CrossRef] [PubMed]

C. H. Graham, Y. Hsia, “The spectral luminosity curves for a dichromatic eye and a normal eye in the same person,” Proc. Natl. Acad. Sci. USA 44, 46–49 (1958).
[CrossRef] [PubMed]

Guth, S. L.

Hendricks, M.

M. Hendricks, K. H. Ruddock, W. A. Waterfield, “Spectral sensitivity functions of post-receptoral responses in human vision,” J. Physiol. (London) 331, 17–33 (1982).

Hsia, Y.

C. H. Graham, Y. Hsia, “Visual discriminations of a subject with acquired unilateral tritanopia,” Vision Res. 7, 469–479 (1967).
[CrossRef] [PubMed]

C. H. Graham, Y. Hsia, “The spectral luminosity curves for a dichromatic eye and a normal eye in the same person,” Proc. Natl. Acad. Sci. USA 44, 46–49 (1958).
[CrossRef] [PubMed]

C. H. Graham, Y. Hsia, “Color defect and color theory: studies in normal and blind persons, including a subject color-blind in one eye but not in the other,” Science 127, 675–682 (1958).
[CrossRef] [PubMed]

Ingling, C. R.

C. R. Ingling, B. H. Tsou, “Orthogonal combinations of three visual channels,” Vision Res. 17, 1075–1082 (1977).
[CrossRef]

Jägle, H.

L. T. Sharpe, A. Stockman, H. Jägle, J. Nathans, “Opsin genes, cone photopigments, color vision, and color blindness,” in K. R. Gegenfurtner, L. T. Sharpe, eds., Color Vision: From Genes to Perception (Cambridge U. Press, 1999), Part 1, Chap. 1, pp. 3–52.

Johnson, N. E.

Judd, D. B.

D. B. Judd, “Color perceptions of deuteranopic and protanopic observers,” J. Res. Natl. Bur. Stand. 41, 247–271 (1948).
[CrossRef]

Kitahara, K.

M. Alpern, K. Kitahara, D. H. Krantz, “Perception of colour in unilateral tritanopia,” J. Physiol. (London) 335, 683–697 (1983).

Krantz, D. H.

M. Alpern, K. Kitahara, D. H. Krantz, “Perception of colour in unilateral tritanopia,” J. Physiol. (London) 335, 683–697 (1983).

Ladd-Franklin, C.

C. Ladd-Franklin, Colour and Colour Theories (Kegan Paul, London, 1932).

Lennie, P.

D. I. A. MacLeod, P. Lennie, “Red–green blindness confined to one eye,” Vision Res. 16, 691–702 (1976).
[CrossRef]

Luke, M.

R. L. De Valois, K. K. De Valois, E. Switkes, M. Luke, “Hue scaling of isoluminant and cone-specific lights,” Vision Res. 37, 885–897 (1997).
[CrossRef] [PubMed]

MacLeod, D. I. 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]

D. I. A. MacLeod, P. Lennie, “Red–green blindness confined to one eye,” Vision Res. 16, 691–702 (1976).
[CrossRef]

Massof, R. W.

Mollon, J. D.

H. Brettel, F. Viénot, J. D. Mollon, “Computerized simulation of color appearance for dichromats,” J. Opt. Soc. Am. A 14, 2647–2655 (1997).
[CrossRef]

F. Viénot, H. Brettel, L. Ott, A. Ben M’Barek, J. D. Mollon, “What do color blind people see?” Nature (London) 376, 127–128 (1995).
[CrossRef]

Nathans, J.

L. T. Sharpe, A. Stockman, H. Jägle, J. Nathans, “Opsin genes, cone photopigments, color vision, and color blindness,” in K. R. Gegenfurtner, L. T. Sharpe, eds., Color Vision: From Genes to Perception (Cambridge U. Press, 1999), Part 1, Chap. 1, pp. 3–52.

Ott, L.

F. Viénot, H. Brettel, L. Ott, A. Ben M’Barek, J. D. Mollon, “What do color blind people see?” Nature (London) 376, 127–128 (1995).
[CrossRef]

Pokorny, J.

S. A. Burns, A. E. Elsner, J. Pokorny, V. C. Smith, “The Abney effect: chromaticity coordinates of unique and other constant hues,” Vision Res. 24, 479–489 (1984).
[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]

Rayleigh, Lord

Lord Rayleigh, Report of Committee on Colour-Vision (The Royal Society, London, 1890).

Romeskie, M.

M. Romeskie, D. Yager, “Psychopysical measures and theoretical analysis of dichromatic opponent response functions,” Mod. Probl. Ophthalmol. 19, 212–217 (1978).

Ruddock, K. H.

M. Hendricks, K. H. Ruddock, W. A. Waterfield, “Spectral sensitivity functions of post-receptoral responses in human vision,” J. Physiol. (London) 331, 17–33 (1982).

B. G. Bender, K. H. Ruddock, E. C. de Vries-de Mol, L. N. Went, “The colour vision characteristics of an observer with unilateral defective colour vision: results and analysis,” Vision Res. 12, 2035–2057 (1972).
[CrossRef] [PubMed]

K. H. Ruddock, “Psychophysics of inherited colour vision deficiencies,” in Inherited and Acquired Colour Vision Deficiencies, D. H. Foster, ed., Vol 7 of Vision and Visual Dysfunction (Macmillan, New York, 1991), pp. 4–37.

K. H. Ruddock, “Psychophysics of inherited colour vision deficiencies,” in Inherited and Acquired Colour Vision Deficiencies, D. H. Foster, ed., Vol 7 of Vision and Visual Dysfunction (Macmillan, New York, 1991), pp. 4–37.

Seim, T.

T. Seim, A. Valberg, “Towards a uniform color space. A better formula to describe the Munsell and OSA color scales,” Color Res. Appl. 11, 11–24 (1986).
[CrossRef]

Sharpe, L. T.

L. T. Sharpe, A. Stockman, H. Jägle, J. Nathans, “Opsin genes, cone photopigments, color vision, and color blindness,” in K. R. Gegenfurtner, L. T. Sharpe, eds., Color Vision: From Genes to Perception (Cambridge U. Press, 1999), Part 1, Chap. 1, pp. 3–52.

Sloan, L. L.

Smith, V. C.

S. A. Burns, A. E. Elsner, J. Pokorny, V. C. Smith, “The Abney effect: chromaticity coordinates of unique and other constant hues,” Vision Res. 24, 479–489 (1984).
[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]

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]

L. T. Sharpe, A. Stockman, H. Jägle, J. Nathans, “Opsin genes, cone photopigments, color vision, and color blindness,” in K. R. Gegenfurtner, L. T. Sharpe, eds., Color Vision: From Genes to Perception (Cambridge U. Press, 1999), Part 1, Chap. 1, pp. 3–52.

Switkes, E.

R. L. De Valois, K. K. De Valois, E. Switkes, M. Luke, “Hue scaling of isoluminant and cone-specific lights,” Vision Res. 37, 885–897 (1997).
[CrossRef] [PubMed]

Trendelenburg, W.

W. Trendelenburg, “Ueber Vererbung bei einen Fall von anomalen Farbensinn des einen und normalen Farbensinn des anderen Auges beim Mann,” Klin. Mbl. Augenheilk. 107, 280–293 (1941).

Tsou, B. H.

C. R. Ingling, B. H. Tsou, “Orthogonal combinations of three visual channels,” Vision Res. 17, 1075–1082 (1977).
[CrossRef]

Valberg, A.

T. Seim, A. Valberg, “Towards a uniform color space. A better formula to describe the Munsell and OSA color scales,” Color Res. Appl. 11, 11–24 (1986).
[CrossRef]

Viénot, F.

H. Brettel, F. Viénot, J. D. Mollon, “Computerized simulation of color appearance for dichromats,” J. Opt. Soc. Am. A 14, 2647–2655 (1997).
[CrossRef]

F. Viénot, H. Brettel, L. Ott, A. Ben M’Barek, J. D. Mollon, “What do color blind people see?” Nature (London) 376, 127–128 (1995).
[CrossRef]

Waterfield, W. A.

M. Hendricks, K. H. Ruddock, W. A. Waterfield, “Spectral sensitivity functions of post-receptoral responses in human vision,” J. Physiol. (London) 331, 17–33 (1982).

Went, L. N.

B. G. Bender, K. H. Ruddock, E. C. de Vries-de Mol, L. N. Went, “The colour vision characteristics of an observer with unilateral defective colour vision: results and analysis,” Vision Res. 12, 2035–2057 (1972).
[CrossRef] [PubMed]

E. C. de Vries-de Mol, L. N. Went, “Unilateral colour vision disturbance: a family study,” Clin. Genet. 2, 15–27 (1971).

Wollach, L.

Yager, D.

M. Romeskie, D. Yager, “Psychopysical measures and theoretical analysis of dichromatic opponent response functions,” Mod. Probl. Ophthalmol. 19, 212–217 (1978).

Clin. Genet. (1)

E. C. de Vries-de Mol, L. N. Went, “Unilateral colour vision disturbance: a family study,” Clin. Genet. 2, 15–27 (1971).

Color Res. Appl. (2)

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

T. Seim, A. Valberg, “Towards a uniform color space. A better formula to describe the Munsell and OSA color scales,” Color Res. Appl. 11, 11–24 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

J. Physiol. (London) (2)

M. Alpern, K. Kitahara, D. H. Krantz, “Perception of colour in unilateral tritanopia,” J. Physiol. (London) 335, 683–697 (1983).

M. Hendricks, K. H. Ruddock, W. A. Waterfield, “Spectral sensitivity functions of post-receptoral responses in human vision,” J. Physiol. (London) 331, 17–33 (1982).

J. Res. Natl. Bur. Stand. (1)

D. B. Judd, “Color perceptions of deuteranopic and protanopic observers,” J. Res. Natl. Bur. Stand. 41, 247–271 (1948).
[CrossRef]

Klin. Mbl. Augenheilk. (1)

W. Trendelenburg, “Ueber Vererbung bei einen Fall von anomalen Farbensinn des einen und normalen Farbensinn des anderen Auges beim Mann,” Klin. Mbl. Augenheilk. 107, 280–293 (1941).

Mod. Probl. Ophthalmol. (1)

M. Romeskie, D. Yager, “Psychopysical measures and theoretical analysis of dichromatic opponent response functions,” Mod. Probl. Ophthalmol. 19, 212–217 (1978).

Nature (London) (1)

F. Viénot, H. Brettel, L. Ott, A. Ben M’Barek, J. D. Mollon, “What do color blind people see?” Nature (London) 376, 127–128 (1995).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

C. H. Graham, Y. Hsia, “The spectral luminosity curves for a dichromatic eye and a normal eye in the same person,” Proc. Natl. Acad. Sci. USA 44, 46–49 (1958).
[CrossRef] [PubMed]

Science (1)

C. H. Graham, Y. Hsia, “Color defect and color theory: studies in normal and blind persons, including a subject color-blind in one eye but not in the other,” Science 127, 675–682 (1958).
[CrossRef] [PubMed]

Vision Res. (8)

C. R. Ingling, B. H. Tsou, “Orthogonal combinations of three visual channels,” Vision Res. 17, 1075–1082 (1977).
[CrossRef]

R. L. De Valois, K. K. De Valois, “A multi-stage color model,” Vision Res. 33, 1053–1065 (1993).
[CrossRef] [PubMed]

R. L. De Valois, K. K. De Valois, E. Switkes, M. Luke, “Hue scaling of isoluminant and cone-specific lights,” Vision Res. 37, 885–897 (1997).
[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]

D. I. A. MacLeod, P. Lennie, “Red–green blindness confined to one eye,” Vision Res. 16, 691–702 (1976).
[CrossRef]

B. G. Bender, K. H. Ruddock, E. C. de Vries-de Mol, L. N. Went, “The colour vision characteristics of an observer with unilateral defective colour vision: results and analysis,” Vision Res. 12, 2035–2057 (1972).
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[CrossRef]

K. H. Ruddock, “Psychophysics of inherited colour vision deficiencies,” in Inherited and Acquired Colour Vision Deficiencies, D. H. Foster, ed., Vol 7 of Vision and Visual Dysfunction (Macmillan, New York, 1991), pp. 4–37.

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K. H. Ruddock, “Psychophysics of inherited colour vision deficiencies,” in Inherited and Acquired Colour Vision Deficiencies, D. H. Foster, ed., Vol 7 of Vision and Visual Dysfunction (Macmillan, New York, 1991), pp. 4–37.

L. T. Sharpe, A. Stockman, H. Jägle, J. Nathans, “Opsin genes, cone photopigments, color vision, and color blindness,” in K. R. Gegenfurtner, L. T. Sharpe, eds., Color Vision: From Genes to Perception (Cambridge U. Press, 1999), Part 1, Chap. 1, pp. 3–52.

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Figures (8)

Fig. 1
Fig. 1

Corresponding-pair procedure. The stimulus S that produces the same sensation in a normal observer as stimulus T in a dichromat is the corresponding pair of T for the normal observer. S is obtained by inverting the dichromatic perceptual descriptors of T, BhS (brightness, hue, and saturation, or whatever color descriptors are given by the model), with the normal model. If the perceptual descriptors are computed from the output of more than one stage of the model, as happens with ATD95, we must invert the model from the perceptual stage. If the perceptual descriptors of the model are computed from its last opponent stage, it suffices to invert the model from that stage. This is the case for linear ATD models, for instance.

Fig. 2
Fig. 2

Algorithm of Brettel et al.5 The stimulus S that produces the same sensation in a normal observer as stimulus T in a dichromat is the intersection of Gamut I and Gamut II. Gamut I changes with T and contains those colors that have the same perceptual descriptors as T, computed with the dichromatic model. Gamut II (which is independent of T) is the set of colors perceived equally by normals and dichromats. When these gamuts cannot be described analytically, an iterative algorithm may be used to obtain them. For example, for Gamut II the algorithm starts from a random vector Ci. This vector belongs to Gamut II if the vector that produces the same sensation in a normal, Ci*, is also Ci. If samples Ci are taken from a dense enough grid, a good estimate of Gamut II can be obtained.

Fig. 3
Fig. 3

Loci of colors that describe protanopic, deuteranopic, and tritanopic color appearance. The substitution hypothesis and the corresponding-pair algorithm were used with different cone-excitation spaces and with linear and nonlinear opponent-color models. The lines do not contain the achromatic point of the corresponding model. The loci obtained with the Brettel et al.5 algorithm are plotted as thin curves. Note that not all the colors are real.

Fig. 4
Fig. 4

Loci of colors that describe protanopic, deuteranopic, and tritanopic color appearance with the nulling hypothesis and the corresponding-pair algorithm, with one-opponent-stage models. From left to right, results from Boynton,23 Ingling and Tsou,21 and Guth et al.22 Only real corresponding pairs are plotted.

Fig. 5
Fig. 5

Loci of colors that describe protanopic, deuteranopic, and tritanopic color appearance with the nulling hypothesis and the corresponding-pair algorithm, with two-opponent-stage models. Only real corresponding pairs are plotted.

Fig. 6
Fig. 6

10-cd/m2 stimuli whose appearance for (a) a protanope, (b) a deuteranope, and (c) a tritanope cannot be simulated by Brettel et al.’s5 algorithm (top panels) and ATD95 with the corresponding-pair procedure (bottom panels).

Fig. 7
Fig. 7

Picasso’s “Dora Maar” as seen by protanopes, deuteranopes, and tritanopes, according to the predictions of Brettel et al.,5 Boynton23 (nulling hypothesis, with S=M for the tritanope), De Valois and De Valois24 (first- and second-stage nulling for red–green defectives and substitution, with S=M, for the tritanope), and ATD95 (second-stage nulling for red–green defectives and first-stage nulling for tritanopes).

Fig. 8
Fig. 8

Example of Gamut II computed with the De Valois and De Valois24 model and the first-opponent-stage nulling hypothesis. Since the analytical solution is a color with tristimulus values [0 0 0], we have considered thresholds in the color descriptors, assuming that if the differences among dominant wavelength, colorimetric purity, and luminance of color C and its corresponding pair C are less than 3 nm, 5%, and 1%, respectively, C belongs to Gamut II. Even with this lax definition, the algorithm of Brettel et al. cannot be used, since the intersection of Gamut II and Gamut I does not exist for all colors.

Equations (30)

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S=m-1(m(T, p), p),
C=m-1(m(C, p), p).
MLMSATD(I-MLMSATD-1MOMLMSATDMC)×MXYZLMSXTYTZT=000.
det(I-MLMSATD-1MOMLMSATDMC)=0.
C=m-1(m(C, p)+δ, p),
ATD=MLMSATDLMS=MLMSATDMXYZLMSXYZ,
AdTdDd=Md,LMSATDLMS=Md,LMSATD×Md,XYZLMSXYZ,
MCprotan=010010001,MCdeutan=100100001.
MCtritan=100010100or100010010.
MOprotan/deutan=100000001,MOtritan=100010000.
m(C, MC, MO)=MOMLMSATDMC×MXYZLMSXCYCZC,
XSYSZS=m-1(m(T, MC, MO),I, I)=MXYZLMS-1MLMSATD-1MOMLMSATD×MCMXYZLMSXTYTZT.
A2T2D2=MATD1ATD2A1T1D1=MATD1ATD2MLMSATDLMS=MATD1ATD2MLMSATDMXYZLMSXYZ.
m(C, MC, MO1, MO2)=MO2MATD1ATD2MO1MLMSATDMC×MXYZLMSXCYCZC.
XSYSZS=m-1(m(T, MC, MO1, MO2),I, I, I)=MXYZLMS-1MLMSATD1-1MATD1ATD2-1MO2×MATD1ATD2MO1MLMSATDMC×MXYZLMSXTYTZT.
XSYSZS=MXYZLMS-1MCMXYZLMSXTYTZT
XSYSZS=MXYZLMS-1k-1MCkMXYZLMSXTYTZT,
k=kL000kM000kS.
k-1MCk=MC,
MCk=kMC.
(MXYZLMS-1k-1MCkMXYZLMS-MXYZLMS-1×MCMXYZLMS)XTYTZT=000.
MXYZLMS-1k-1(MCk-kMC)MXYZLMS×XTYTZT=000
(MCk-kMC)MXYZLMSXTYTZT=(MCk-kMC)LTMTST=000.
(MCk-kMC)i,j=(kM-kL)δi,1δ2,jprotanopes(kL-kM)δi,2δ1,jdeuteranopes(kL-kS)δi,3δ1,jtritanopes,S=L(kM-kS)δi,3δ2,jtritanopes,S=M.
(kM-kL)LT=0protanopes & deuteranopes
(kL-kS)LT=0tritanopes,S=L,
(kM-kS)MT=0tritanopes,S=M.
XSYSZS=MXYZLMS-1MCLTMTST=MXYZLMS-1MC0MTST=MXYZLMS-100ST.
S=LXSYSZS=MXYZLMS-1MCLTMTST=MXYZLMS-1MC0MTST=MXYZLMS-10MT0,
S=MXSYSZS=MXYZLMS-1MCLTMTST=MXYZLMS-1MCLT0ST=MXYZLMS-1LT00.

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