Abstract

Anomalous trichromatic color systems are examined within the framework of a three variable, bimodal opponent-colors theory. Specific types of anomaly are assumed to be related to specific changes in the normal photoreceptor distributions and to impairments in the normal chromatic response processes. The aberrant color equations, color sensations, and color discriminations of both protanomalous and deuteranomalous observers are shown to be meaningfully related to the postulated, systematically aberrant visual mechanisms.

© 1956 Optical Society of America

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  11. W. M. McKeon and W. D. Wright, Proc. Phys. Soc. (London) 52, 464 (1940).
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  28. F. C. Donders, Arch. Physiol. 8, 518 (1884).
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  39. C. v. Hess, Arch. Augenheilk. 91, 133 (1922).
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  41. K. Koffka, Z. Sinnesphysiol. 43, 123 (1909).
  42. H. Köllner, Arch. Augenheilk. 78, 302 (1915).
  43. A. Franceschetti, Schweiz. med. Wochschr. 58, 1273 (1928).
  44. E. Engelking, Klin. Monatsbl. Augenheilk. 77, 61 (1926).
  45. H. Köllner and E. Engelking, Handb. Norm. Path. Physiol. 12, 502 (1929), p. 515.
  46. W. Trendelenburg and I. Schmidt, Sitzber. preuss. Akad. Wiss., Physik. Math. Kl., 13–81 (1935).
  47. Dr. Scheidemann, Z. Bahn. Bahnkassenartze 9, 8 (1914).
  48. E. Francke, Arch. Augenheilk. 82, 1 (1917).
  49. G. H. M. Waaler, Z. Indukt. Abstamm. Vererbungslehre 45, 279 (1927).
  50. P. von Planta, Arch. Ophthalmol. 120, 253 (1928).
  51. E. Holzlöhner and W. H. Stein, Z. Sinnesphysiol. 61, 209 (1930/31).
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    [Crossref]
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    [Crossref]
  54. J. Stilling, Z. Sinnesphysiol. 44, 371 (1910).
  55. C. v. Hess, Ergeb. Physiol. 20, 1 (1922).
    [Crossref]
  56. G. H. Rautian, Doklady Akad. Nauk. S.S.S.R. 74, 1073 (1950).
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  58. G. L. Walls and R. W. Mathews, New Means of Studying Color Blindness and Normal Foveal Color Vision (University of California Press, Berkeley, 1952).
  59. R. W. Pickford, Brit. J. Psychol. 45, 115 (1954).
  60. Hering, Pflügers Arch. ges. Physiol. 57, 308 (1894).
    [Crossref]
  61. W. Abney and W. Watson, Proc. Roy. Soc. (London) 89A, 232 (1913).
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    [Crossref]
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  68. A. Samoljoff, Z. Sinnesphysiol. 41, 367 (1907).
  69. A. Tschermak, Handb. Norm. Path. Physiol. 12, 295 (1929).
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  71. W. D. Wright, Appendix II, in F. H. G. Pitt, (1935).
  72. H. Westphal, Z. Sinnesphysiol. 44, 182 (1910).
  73. H. Köllner, Arch. Augenheilk. 84, 177 (1919).
  74. H. Goldmann, Pflügers Arch. ges. Physiol. 210, 70 (1925).
    [Crossref]
  75. G. Schubert, Pflügers Arch. ges. Physiol. 220, 82 (1928).
    [Crossref]
  76. J. G. Hailwood and H. E. Roaf, J. Physiol. 91, 37 (1937).
  77. A. Chapanis, J. Exptl. Psychol. 34, 24 (1944).
    [Crossref]
  78. C. Rosencrantz, Z. Sinnesphysiol. 58, 5 (1927).
  79. E. Sachs, Z. Sinnesphysiol. 59, 243 (1928).
  80. P. M. Ladekarl, Acta Ophthalmol. 12 (Supplement III) (1934).
  81. H. V. Corbett, J. Physiol. 88, 176 (1937).
  82. R. A. Weale, J. Physiol. 113, 115 (1951).

1956 (2)

1955 (5)

1954 (1)

R. W. Pickford, Brit. J. Psychol. 45, 115 (1954).

1953 (2)

H. J. A. Dartnall, Brit. Med. Bull. 9, 24 (1953).

Hubbard, Gregerman, and Wald, J. Gen. Physiol. 36, 415 (1953).

1951 (2)

E. N. Yustova, Doklady Akad. Nauk. S.S.S.R. 81, 1051 (1951).

R. A. Weale, J. Physiol. 113, 115 (1951).

1950 (2)

G. H. Rautian, Doklady Akad. Nauk. S.S.S.R. 74, 1073 (1950).

L. L. Sloan, J. Opt. Soc. Am. 40, 41 (1950).
[Crossref]

1949 (1)

F. H. G. Pitt, Documenta Ophthalmol. 3, 307 (1949).
[Crossref]

1947 (1)

D. Volk and G. A. Fry, Am. J. Optom. 24, 99 (1947).
[Crossref]

1944 (1)

A. Chapanis, J. Exptl. Psychol. 34, 24 (1944).
[Crossref]

1940 (1)

W. M. McKeon and W. D. Wright, Proc. Phys. Soc. (London) 52, 464 (1940).
[Crossref]

1938 (1)

J. H. Nelson, Proc. Phys. Soc. (London) 50, 661 (1938).
[Crossref]

1937 (2)

H. V. Corbett, J. Physiol. 88, 176 (1937).

J. G. Hailwood and H. E. Roaf, J. Physiol. 91, 37 (1937).

1935 (1)

W. Trendelenburg and I. Schmidt, Sitzber. preuss. Akad. Wiss., Physik. Math. Kl., 13–81 (1935).

1934 (2)

P. M. Ladekarl, Acta Ophthalmol. 12 (Supplement III) (1934).

D. Katzin and E. Murray, Am. J. Psychol. 46, 611 (1934).
[Crossref]

1933 (1)

K. Schönewald, Z. Sinnesphysiol. 63, 285 (1933).

1931 (1)

A. Kohlrausch, Handb. Norm. Path. Physiol. 12, 1499 (1931), p. 1562.

1929 (4)

J. v. Kries, Handb. Norm. Path. Physiol. 12, 585 (1929), p. 593.

W. Trendelenburg, Klin. Monatsbl. Augenheilk. 83, 721 (1929).

H. Köllner and E. Engelking, Handb. Norm. Path. Physiol. 12, 502 (1929), p. 515.

A. Tschermak, Handb. Norm. Path. Physiol. 12, 295 (1929).

1928 (4)

G. Schubert, Pflügers Arch. ges. Physiol. 220, 82 (1928).
[Crossref]

E. Sachs, Z. Sinnesphysiol. 59, 243 (1928).

P. von Planta, Arch. Ophthalmol. 120, 253 (1928).

A. Franceschetti, Schweiz. med. Wochschr. 58, 1273 (1928).

1927 (3)

E. Engelking, Klin. Monatsbl. Augenheilk. 78 (Supp.), 209 (1927).

G. H. M. Waaler, Z. Indukt. Abstamm. Vererbungslehre 45, 279 (1927).

C. Rosencrantz, Z. Sinnesphysiol. 58, 5 (1927).

1926 (1)

E. Engelking, Klin. Monatsbl. Augenheilk. 77, 61 (1926).

1925 (2)

E. Engelking, Arch. Ophthalmol. 116, 196 (1925).

H. Goldmann, Pflügers Arch. ges. Physiol. 210, 70 (1925).
[Crossref]

1923 (1)

G. F. Göthlin, Skand. Arch. Physiol. 43, 13 (1923).
[Crossref]

1922 (2)

C. v. Hess, Ergeb. Physiol. 20, 1 (1922).
[Crossref]

C. v. Hess, Arch. Augenheilk. 91, 133 (1922).

1919 (2)

J. v. Kries, Z. Sinnesphysiol. 50, 137 (1919).

H. Köllner, Arch. Augenheilk. 84, 177 (1919).

1917 (1)

E. Francke, Arch. Augenheilk. 82, 1 (1917).

1915 (1)

H. Köllner, Arch. Augenheilk. 78, 302 (1915).

1914 (1)

Dr. Scheidemann, Z. Bahn. Bahnkassenartze 9, 8 (1914).

1913 (4)

R. Hilbert, Z. Bahn. Bahnkässenartze 8, 62 (1913).

F. W. Edridge-Green, Proc. Roy. Soc. (London) 86B, 164 (1913).

W. Watson, Proc. Roy. Soc. (London) 88A, 404 (1913).

W. Abney and W. Watson, Proc. Roy. Soc. (London) 89A, 232 (1913).

1910 (2)

J. Stilling, Z. Sinnesphysiol. 44, 371 (1910).

H. Westphal, Z. Sinnesphysiol. 44, 182 (1910).

1909 (1)

K. Koffka, Z. Sinnesphysiol. 43, 123 (1909).

1908 (2)

A. Guttmann, Z. Sinnesphysiol. 42, 24, 250 (1908); Z. Sinnesphysiol. 43, 146, 199, 255 (1909).

W. Nagel, Z. Sinnesphysiol. 42, 65 (1908).

1907 (1)

A. Samoljoff, Z. Sinnesphysiol. 41, 367 (1907).

1904 (1)

W. Nagel, Klin. Monatsbl. Augenheilk. 42, 366 (1904).

1899 (1)

J. v. Kries, Z. Psychol. 19, 63 (1899).

1894 (1)

Hering, Pflügers Arch. ges. Physiol. 57, 308 (1894).
[Crossref]

1893 (1)

A. König and C. Dieterici, Z. Psychol. 4, 241 (1893).

1890 (1)

E. Hering, Arch. Ophthalmol. 36, 1 (1890).

1885 (1)

E. Hering and Lotos, Jahrb. Naturwiss. N. F. 6, 142 (1885).

1884 (1)

F. C. Donders, Arch. Physiol. 8, 518 (1884).

1881 (1)

J. W. Strutt ( Rayleigh), Nature 25, 64 (1881).
[Crossref]

J. W. Strutt ( Rayleigh), Nature 25, 64 (1881).
[Crossref]

Abney, W.

W. Abney and W. Watson, Proc. Roy. Soc. (London) 89A, 232 (1913).

W. Abney, Researches in Colour Vision and the Trichromatic Theory (Longmans Green and Company, London, 1913), Chap. XX.

Chapanis, A.

A. Chapanis, J. Exptl. Psychol. 34, 24 (1944).
[Crossref]

Corbett, H. V.

H. V. Corbett, J. Physiol. 88, 176 (1937).

Dartnall, H. J. A.

H. J. A. Dartnall, Brit. Med. Bull. 9, 24 (1953).

Dieterici, C.

A. König and C. Dieterici, Z. Psychol. 4, 241 (1893).

Donders, F. C.

F. C. Donders, Arch. Physiol. 8, 518 (1884).

Edridge-Green, F. W.

F. W. Edridge-Green, Proc. Roy. Soc. (London) 86B, 164 (1913).

Engelking, E.

H. Köllner and E. Engelking, Handb. Norm. Path. Physiol. 12, 502 (1929), p. 515.

E. Engelking, Klin. Monatsbl. Augenheilk. 78 (Supp.), 209 (1927).

E. Engelking, Klin. Monatsbl. Augenheilk. 77, 61 (1926).

E. Engelking, Arch. Ophthalmol. 116, 196 (1925).

Farnsworth, D.

M. P. Willis and D. Farnsworth, “Comparative evaluation of anomaloscopes,” , XI, No. 7 (August18, 1952).

Franceschetti, A.

A. Franceschetti, Schweiz. med. Wochschr. 58, 1273 (1928).

Francke, E.

E. Francke, Arch. Augenheilk. 82, 1 (1917).

Fry, G. A.

D. Volk and G. A. Fry, Am. J. Optom. 24, 99 (1947).
[Crossref]

Goldmann, H.

H. Goldmann, Pflügers Arch. ges. Physiol. 210, 70 (1925).
[Crossref]

Göthlin, G. F.

G. F. Göthlin, Skand. Arch. Physiol. 43, 13 (1923).
[Crossref]

Gregerman,

Hubbard, Gregerman, and Wald, J. Gen. Physiol. 36, 415 (1953).

Guttmann, A.

A. Guttmann, Z. Sinnesphysiol. 42, 24, 250 (1908); Z. Sinnesphysiol. 43, 146, 199, 255 (1909).

Hailwood, J. G.

J. G. Hailwood and H. E. Roaf, J. Physiol. 91, 37 (1937).

Helmholtz, H. v.

H. v. Helmholtz, Handbuch der Physiologischen Optik (L. Voss, Hamburg and Leipzig, 1911), Vol. II, p. 346. See English edition (Optical Society of America, 1924), Vol. II, p. 409.

Hering,

Hering, Pflügers Arch. ges. Physiol. 57, 308 (1894).
[Crossref]

Hering, E.

E. Hering, Arch. Ophthalmol. 36, 1 (1890).

E. Hering and Lotos, Jahrb. Naturwiss. N. F. 6, 142 (1885).

Hess, C. v.

C. v. Hess, Ergeb. Physiol. 20, 1 (1922).
[Crossref]

C. v. Hess, Arch. Augenheilk. 91, 133 (1922).

Hilbert, R.

R. Hilbert, Z. Bahn. Bahnkässenartze 8, 62 (1913).

Holzlöhner, E.

E. Holzlöhner and W. H. Stein, Z. Sinnesphysiol. 61, 209 (1930/31).

Hubbard,

Hubbard, Gregerman, and Wald, J. Gen. Physiol. 36, 415 (1953).

Hurvich, L. M.

Jameson, D.

Judd, D. B.

D. B. Judd, Handbook of Experimental Psychology, edited by S. S. Stevens (John Wiley and Sons, Inc., New York, 1951), p. 827.

Katzin, D.

D. Katzin and E. Murray, Am. J. Psychol. 46, 611 (1934).
[Crossref]

Koffka, K.

K. Koffka, Z. Sinnesphysiol. 43, 123 (1909).

Kohlrausch, A.

A. Kohlrausch, Handb. Norm. Path. Physiol. 12, 1499 (1931), p. 1562.

Köllner, H.

H. Köllner and E. Engelking, Handb. Norm. Path. Physiol. 12, 502 (1929), p. 515.

H. Köllner, Arch. Augenheilk. 84, 177 (1919).

H. Köllner, Arch. Augenheilk. 78, 302 (1915).

H. Köllner, Die Störungen des Farbensinnes (S. Karger, Berlin, 1912), p. 141 ff.

König, A.

A. König and C. Dieterici, Z. Psychol. 4, 241 (1893).

Kries, J. v.

J. v. Kries, Handb. Norm. Path. Physiol. 12, 585 (1929), p. 593.

J. v. Kries, Z. Sinnesphysiol. 50, 137 (1919).

J. v. Kries, Z. Psychol. 19, 63 (1899).

J. v. Kries, Nagel’s Handbuch der Physiologie der Menschen (F. Vieweg and Son, Brunswick, 1905), Vol. 3.

Ladekarl, P. M.

P. M. Ladekarl, Acta Ophthalmol. 12 (Supplement III) (1934).

Levy, M.

M. Levy, “Uber einen zweiten Typus des anomalen trichromatischen Farbensystems,” dissertation, Freiburg (1903).

Lotos,

E. Hering and Lotos, Jahrb. Naturwiss. N. F. 6, 142 (1885).

Lotze, A.

A. Lotze, “Untersuchungen eines anomalen trichromatischen Farbensystems,” dissertation, Freiburg (1898).

Mathews, R. W.

G. L. Walls and R. W. Mathews, New Means of Studying Color Blindness and Normal Foveal Color Vision (University of California Press, Berkeley, 1952).

McKeon, W. M.

W. M. McKeon and W. D. Wright, Proc. Phys. Soc. (London) 52, 464 (1940).
[Crossref]

Müller, G. E.

G. E. Müller, Darstellung und Erklärung der verschiedenen Typen der Farbenblindheit (Vandenhoeck and Ruprecht, Göttigen, 1924), p. 180 ff.

Murray, E.

D. Katzin and E. Murray, Am. J. Psychol. 46, 611 (1934).
[Crossref]

Nagel, W.

W. Nagel, Z. Sinnesphysiol. 42, 65 (1908).

W. Nagel, Klin. Monatsbl. Augenheilk. 42, 366 (1904).

Nelson, J. H.

J. H. Nelson, Proc. Phys. Soc. (London) 50, 661 (1938).
[Crossref]

Parsons, J. H.

J. H. Parsons, An Introduction to the Study of Colour Vision (Cambridge University Press, Cambridge, 1924), p. 229.

Pickford, R. W.

R. W. Pickford, Brit. J. Psychol. 45, 115 (1954).

R. W. Pickford, Individual Differences in Colour Vision (Routledge and Kegan Paul, Ltd., London, 1951).

Pitt, F. H. G.

F. H. G. Pitt, Documenta Ophthalmol. 3, 307 (1949).
[Crossref]

Rautian, G. H.

G. H. Rautian, Doklady Akad. Nauk. S.S.S.R. 74, 1073 (1950).

Rayleigh,

J. W. Strutt ( Rayleigh), Nature 25, 64 (1881).
[Crossref]

Roaf, H. E.

J. G. Hailwood and H. E. Roaf, J. Physiol. 91, 37 (1937).

Rosencrantz, C.

C. Rosencrantz, Z. Sinnesphysiol. 58, 5 (1927).

Rosmanit, J.

J. Rosmanit, Anleitung zur Feststettung der Farbentiichtigkeit (Deuticke, Leipzig, 1914).

Sachs, E.

E. Sachs, Z. Sinnesphysiol. 59, 243 (1928).

Samoljoff, A.

A. Samoljoff, Z. Sinnesphysiol. 41, 367 (1907).

Scheidemann, Dr.

Dr. Scheidemann, Z. Bahn. Bahnkassenartze 9, 8 (1914).

Schmidt, I.

I. Schmidt, J. Opt. Soc. Am. 45, 514 (1955). For additional references see reference 31, p. 517.
[Crossref] [PubMed]

W. Trendelenburg and I. Schmidt, Sitzber. preuss. Akad. Wiss., Physik. Math. Kl., 13–81 (1935).

Schönewald, K.

K. Schönewald, Z. Sinnesphysiol. 63, 285 (1933).

Schubert, G.

G. Schubert, Pflügers Arch. ges. Physiol. 220, 82 (1928).
[Crossref]

Sloan, L. L.

Stein, W. H.

E. Holzlöhner and W. H. Stein, Z. Sinnesphysiol. 61, 209 (1930/31).

Stilling, J.

J. Stilling, Z. Sinnesphysiol. 44, 371 (1910).

Strutt, J. W.

J. W. Strutt ( Rayleigh), Nature 25, 64 (1881).
[Crossref]

Trendelenburg, W.

W. Trendelenburg and I. Schmidt, Sitzber. preuss. Akad. Wiss., Physik. Math. Kl., 13–81 (1935).

W. Trendelenburg, Klin. Monatsbl. Augenheilk. 83, 721 (1929).

Tschermak, A.

A. Tschermak, Handb. Norm. Path. Physiol. 12, 295 (1929).

Volk, D.

D. Volk and G. A. Fry, Am. J. Optom. 24, 99 (1947).
[Crossref]

von Planta, P.

P. von Planta, Arch. Ophthalmol. 120, 253 (1928).

Waaler, G. H. M.

G. H. M. Waaler, Z. Indukt. Abstamm. Vererbungslehre 45, 279 (1927).

Wald,

Hubbard, Gregerman, and Wald, J. Gen. Physiol. 36, 415 (1953).

Wald, G.

G. Wald, Am. J. Ophthalmol. 40, 18 (1955).
[PubMed]

Walls, G. L.

G. L. Walls and R. W. Mathews, New Means of Studying Color Blindness and Normal Foveal Color Vision (University of California Press, Berkeley, 1952).

Watson, W.

W. Abney and W. Watson, Proc. Roy. Soc. (London) 89A, 232 (1913).

W. Watson, Proc. Roy. Soc. (London) 88A, 404 (1913).

Weale, R. A.

R. A. Weale, J. Physiol. 113, 115 (1951).

Westphal, H.

H. Westphal, Z. Sinnesphysiol. 44, 182 (1910).

Willis, M. P.

M. P. Willis and D. Farnsworth, “Comparative evaluation of anomaloscopes,” , XI, No. 7 (August18, 1952).

Wright, W. D.

W. M. McKeon and W. D. Wright, Proc. Phys. Soc. (London) 52, 464 (1940).
[Crossref]

W. D. Wright, Appendix II, in F. H. G. Pitt, (1935).

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Figures (18)

Fig. 1
Fig. 1

Schematic diagram showing relations between photosensitive materials α, β, and γ and visual response processes y-b, r-g, and w-bk.

Fig. 2
Fig. 2

Spectral distribution curves for four photoreceptor units.

Fig. 3
Fig. 3

Chromatic and achromatic response functions for equal energy spectrum for CIE standard observer.

Fig. 4
Fig. 4

Chromatic and achromatic response functions for equal energy spectrum. For observers with protanomalous, normal, and deuteranomalous photoreceptor systems and with normal strength visual response processes.

Fig. 5
Fig. 5

Chromatic and achromatic response functions for equal energy spectrum. For observers with protanomalous, normal, and deuteranomalous photoreceptor systems and with impaired red-green response processes.

Fig. 6
Fig. 6

Theoretical Rayleigh mixture ratios for postulated normal and anomalous visual systems. See text.

Fig. 7
Fig. 7

Rayleigh mixture ratios measured by Willis and Farnsworth for normal and anomalous observers.

Fig. 8
Fig. 8

Rayleigh mixture ratios. Heavy lines—theoretical ranges for observers with protanomalous, normal, and deuteranomalous photoreceptor systems and with normal strength visual response processes. Thin lines—comparable ranges for observers measured by Willis and Farnsworth.

Fig. 9
Fig. 9

Rayleigh mixture ratios. Heavy lines—theoretical ranges for observers with protanomalous, normal, and deuteranomalous photoreceptor systems and with red-green response processes of 0.1 normal strength. Thin lines—comparable ranges for observers measured by Willis and Farnsworth.

Fig. 10
Fig. 10

Rayleigh mixture ratios. Heavy lines—theoretical ranges for observers with protanomalous, normal, and deuteranomalous photoreceptor systems and with red-green response processes of 0.03 normal strength. Thin lines—comparable ranges for observers measured by Willis and Farnsworth.

Fig. 11
Fig. 11

Rayleigh mixture ratios. Heavy lines—theoretical ranges for dichromats with complete loss of red-green response processes. Thin lines—comparable ranges for observers measured by Willis and Farnsworth.

Fig. 12
Fig. 12

Average 670/535 mμ ratios to match series of intermediate spectral stimuli. (Extended Rayleigh equations.) Solid lines—theoretical ratios for a protanomalous, a normal, and a deuteranomalous observer. Symbols—experimental data from von Kries and Levy.

Fig. 13
Fig. 13

Spectral hue coefficients for a protanomalous, a normal, and a deuteranomalous observer. Theoretical.

Fig. 14
Fig. 14

Spectral saturation coefficients for a protanomalous, a normal, and a deuteranomalous observer, Theoretical.

Fig. 15
Fig. 15

Theoretical saturation discrimination functions for a protanomalous and a deuteranomalous observer.

Fig. 16
Fig. 16

Saturation discrimination functions (reciprocal of least-perceptible colorimetric purity) for a protanomalous and a deuteranomalous observer measured by Chapanis.

Fig. 17
Fig. 17

Theoretical wavelength discrimination functions based on threshold change in hue and saturation coefficients for a protanomalous and a deuteranomalous observer.

Fig. 18
Fig. 18

Wavelength discrimination functions for a protanomalous and a deuteranomalous observer measured by McKeon and Wright and by Nelson.

Tables (1)

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Table I Reported and predicted sensations of normal and deuteranomalous eyes. Reported appearances for unilateral deuteranomalous from v. Kries.a

Equations (8)

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y - b = k 1 ( Y - B ) , r - g = k 2 ( R - G ) , w - b k = k 3 ( 0.5 B + 0.5 G + Y + R ) - k 4 ( 0.5 B + 0.5 G + Y + R ) .
B λ = 13.0682 y ¯ λ + 0.2672 z ¯ λ , G λ = - 0.6736 x ¯ λ + 14.0018 y ¯ λ + 0.0040 z ¯ λ , Y λ = - 0.0039 x ¯ λ + 13.4680 y ¯ λ - 0.1327 z ¯ λ , R λ = 0.3329 x ¯ λ + 13.0012 y ¯ λ - 0.0011 z ¯ λ ,
h . c . = y - b y - b + r - g ,     or     r - g y - b + r - g .
s . c . = y - b + r - g y - b + r - g + w - b k .
a ( 535 m μ ) + b ( 670 m μ ) = c ( 589 m μ ) .
x ( r - g ) 535 + ( r - g ) 670 = ( r - g ) 589 .
x ( r - g ) 535 + ( r - g ) 670 = ( r - g ) 589 + Δ ( r - g )
x ( r - g ) 535 + ( r - g ) 670 = ( r - g ) 589 - Δ ( r - g ) .