Abstract

We examined the effects of probing human color mechanisms using sinusoidal spectral power distributions (SPD’s) varying in frequency (i.e., from 0.1 to 5.0 cycles/300 nm for a constant starting phase) and phase (i.e., from 0 to 360 deg for a fixed frequency of 1 cycle/300 nm) through computer simulation using several color models. Predicted modulation sensitivity functions (MSF’s) in spectral frequency and phase differ among the models and indicate that measurements of the minimum amplitudes necessary to detect sinusoidal SPD’s would be useful for distinguishing among theories of color vision. MSF’s obtained from similar analyses of dichromats’ color mechanisms reveal characteristic patterns of modulation sensitivities and suggest that such measures could serve to distinguish type and degree of color-vision defect. Some implications based on sinusoidal approximations to illuminant and reflectance spectra are discussed along with more general considerations regarding sine-wave SPD’s as a probe for mechanisms of color vision.

© 1986 Optical Society of America

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  1. H. B. Barlow, “What causes trichromacy? A theoretical analysis using comb-filtered spectra,” Vision Res. 22, 635–644 (1982).
    [CrossRef] [PubMed]
  2. G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding, and optimum colour information transmission in the retina,” Proc. R. Soc. London Ser. B 220, 89–113 (1983).
    [CrossRef]
  3. G. Buchsbaum, A. Gottschalk, “Chromaticity coordinates of frequency-limited functions,” J. Opt. Soc. Am. A 1, 885–887 (1984).
    [CrossRef] [PubMed]
  4. H. Wolter, “Physikalische Begründung eines Farbenkreises und Anzätze einer physikalischen Farbenlehre,” Ann. Phys. (Leipzig) Ser. 6 8, 11–29 (1950).
    [CrossRef]
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    [CrossRef]
  6. M. H. Brill, T. Benzschawel, “Remarks on signal-processing explanations of the trichromacy of vision,” J. Opt. Soc. Am. A 2, 1794–1796 (1985).
    [CrossRef] [PubMed]
  7. Note that, unlike previous reports using sine-wave spectra, angular measure is expressed in degrees and frequency is specified in cycles per 300 nm. In those articles, angular measure is given in radians and spectral frequency is given in units of terahertz or in cycles per nanometer.
  8. D. S. Goodman, “Some methods for producing straight achromatic white-light fringes of variable contrast and constant average radiance,” Appl. Opt. 21, 3876–3878 (1982).
    [CrossRef] [PubMed]
  9. L. M. Hurvich, D. Jameson, “An opponent-process theory of color vision,” Psych. Rev. 64, 384–404 (1957).
    [CrossRef]
  10. D. Jameson, “Theoretical issues of color vision,” in Handbook of Sensory Physiology, D. Jameson, L. M. Hurvich, eds. (Springer-Verlag, Berlin, 1972), Vol. 7/4, Chap. 14.
    [CrossRef]
  11. L. M. Hurvich, Color Vision (Sinauer, 1981).
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    [CrossRef] [PubMed]
  13. C. R. Ingling, B. H.-P. Tsou, “Orthogonal combination of three visual channels,” Vision Res. 17, 1075–1082 (1977).
    [CrossRef]
  14. Although each of the models described herein uses slightly different nomenclature for analogous receptor types and opponent-colors mechanisms, for simplicity we have chosen to use the same set of terms for each. That is, the photoreceptors are referred to as L, M, and S for the long-, short- and medium-wavelength sensitive photoreceptors, respectively. The second stage mechanisms of the opponent-colors models are designated as A, T, and D for an achromatic or luminance system, a tritanopic system (i.e., the system that tritanopes have that signals red or green), and a deuteranopic system (i.e., the system that deuteranopes and protanopes have that signals blue or yellow), respectively.
  15. R. M. Boynton, W. Schafer, M. E. Neun, “Hue-wavelength relation measured by color-naming method for three retinal locations,” Science 146, 666–668 (1964); S. M. Luria, “Color-name as a function of stimulus-intensity and duration,” Am. J. Psychol. 80, 14–27 (1967).
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    [CrossRef] [PubMed]
  18. J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
    [CrossRef] [PubMed]
  19. B. W. Tansley, R. M. Boynton, “A single line predicts the distinctness of borders by different colors,” Science 191, 954–957 (1976); B. W. Tansley, R. M. Boynton, “Chromatic border perception: the role of red- and green-sensitive cones,” Vision Res. 18, 683–697 (1978).
    [CrossRef] [PubMed]
  20. 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]
  21. A. Eisner, D. I. A. MacLeod, “Blue-sensitive cones do not contribute to luminance,” J. Opt. Soc. Am. 70, 121–123 (1980).
    [CrossRef] [PubMed]
  22. B. Drum, “Short-wavelength cones contribute to achromatic sensitivity,” Vision Res. 23, 1433–1439 (1983).
    [CrossRef] [PubMed]
  23. J. D. Mollon, P. G. Polden, “An anomaly in the response of the eye to short wavelengths,” Phil. Trans. Soc. London Ser. B 278, 207–240 (1977); J. J. Wisowaty, “An action spectrum for the production of transient tritanopia,” Vision Res. 23, 769–774 (1983).
    [CrossRef] [PubMed]
  24. R. W. Massof, S. J. Starr, “Vector magnitude operation in color vision models: derivation from signal detection theory,” J. Opt. Soc. Am. 70, 870–872 (1980).
    [CrossRef] [PubMed]
  25. W. S. Stiles, “Increment thresholds and the mechanisms of colour vision,” Doc. Ophthalmol. 3, 138–163 (1949); W. S. Stiles, “Colour vision: The approach through increment-threshold sensitivity,” Proc. Nat. Acad. Sci. USA 3, 100–114 (1959).
    [CrossRef] [PubMed]
  26. E. N. Pugh, “The nature of the π1colour mechanism of W. S. Stiles,” J. Physiol. 257, 713–747 (1976);B. A. Wandell, E. N. Pugh, “Detection of long-duration, long-wavelength incremental flashes by a chromatically coded pathway,” Vision Res. 20, 613–624 (1980); E. N. Pugh, J. Larimer, “Test of the identity of the site of blue/yellow hue cancellation and the site of chromatic antagonism in the π1pathway,” Vision Res. 20, 779–788 (1980); C. Sigel, L. Brousseau, “Pi-4: adaptation of more than one class of cone,” J. Opt. Soc. Am. 72, 237–246 (1980).
    [CrossRef]
  27. G. B. Rollman, J. Nachmias, “Simultaneous detection and recognition of chromatic flashes,” Percept. Psychophys. 12, 309–314 (1972); B. A. Wandell, J. Sanchez, B. Quinn, “Detection/discrimination in the long-wavelength pathways,” Vision Res. 22, 1061–1069 (1982).
    [CrossRef] [PubMed]
  28. R. A. Normann, F. S. Werblin, “Control of retinal sensitivity: I. Light dark adaptation of vertebrate rods and cones,” J. Gen. Physiol. 63, 37–61 (1974);J. Larimer, D. H. Krantz, C. M. Cicerone, “Opponent-process additivity-II. Yellow/blue equilibria and nonlinear models,” Vision Res. 18, 1521–1532 (1975); J. Raaijmakers, C. de Weert, “Linear and non-linear opponent color coding,” Percept. Psychophys. 18, 474–480 (1975).
    [CrossRef]
  29. R. W. Massof, J. F. Bird, “A general zone theory of color and brightness vision,” J. Opt. Soc. Am. 68, 1465–1471 (1978); R. W. Massof, “Color-vision theory and linear models of color vision,” Color Res. Appl. 10, 133–146 (1985). According to this view, the derivatives of the intensity-response functions of the color mechanisms define, for given static and uniform conditions, a transformation matrix of receptor outputs like those in Eq. (5). Thus for small perturbations about a given adaptive state, nonlinear models that embody similar receptor transformations to those of Eqs. (8)–(10) are virtually indistinguishable from their linear counterparts.
    [CrossRef] [PubMed]
  30. V. C. Smith, J. Pokorny, “Spectral sensitivity of colorblind observers and the cone photopigments,” Vision Res. 12, 2059–2071 (1972).
    [CrossRef] [PubMed]
  31. D. B. Judd, “Colorimetry and Artificial Daylight,” in Technical Committee No. 7 Report of Secretariat United States Commission, International Commission on Illumination, Twelfth Session, Stockholm (Bureau Central de la CIE, Paris1951), pp. 1–60.
  32. D. Jameson, L. M. Hurvich, “Opponent-response functions related to measured cone photopigments,” J. Opt. Soc. Am. 58, 429–430 (1968).
    [CrossRef]
  33. Both the models of Guth et al. and that of Ingling specify different mechanism outputs as the adaptation level (determined by the prevailing luminance level) is changed. For example, at high intensities the spectrum appears more bluish and yellowish relative to red and green. For the Guth model, this simply requires adjustment of the relative outputs of T and D by constant amounts. Hence, we applied the coefficient 12.0 to the output of D as prescribed by the model. In Ingling’s scheme, the transformation equations change with adaptation level. That is, for suprathreshold levels, there is input from S into T and A, and the relative balance of inputs from (L + M) and S into D are altered. Because we are generating predictions for the light-adapted observer, we chose the suprathreshold equations (3a)–(3c) of Ref. 13 for the present calculations. Note that later versions of the Ingling model incorporate additional intensity-dependent nonlinearities, including inhibition between L and M receptors and the rod system as well as summation among rods and S receptor outputs.34
  34. C. R. Ingling, “The spectral sensitivity of the opponent-color channels,” Vision Res. 17, 1083–1089 (1977).
    [CrossRef] [PubMed]
  35. E. N. Pugh, C. Sigel, “Evaluation of the candidacy of the π-mechanisms of Stiles for color-matching functions,” Vision Res. 18, 317–330 (1978).
    [CrossRef]
  36. G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1982), p. 553.
  37. H. M. O. Scheibner, R. M. Boynton, “Residual red-green discrimination in dichromats,” J. Opt. Soc. Am. 58, 1151–1158 (1968); M. Alpern, E. N. Pugh, “Variation in the action spectrum of erythrolabe among deuteranopes,” J. Physiol. 266, 613–646 (1977);V. C. Smith, J. Pokorny, “Large field trichromacy in protanopes and deuteranopes,” J. Opt. Soc. Am. 67, 213–220 (1977); A. L. Nagy, R. M. Boynton, “Large-field color naming of dichromats with rods bleached,” J. Opt. Soc. Am. 69, 1259–1265 (1979); M. E. Breton, W. B. Cowan, “Deuteranomalous color matching in the deuteranopic eye,” J. Opt. Soc. Am. 71, 1220–1223 (1981); F. S. Frome, T. P. Piantanida, D. H. Kelly, “Psychophysical evidence for more than two kinds of cone in dichromatic color blindness,” Science 215, 417–419 (1982); J. Nathans, T. P. Piantanida, R. L. Eddy, T. B. Shows, D. S. Hogness, “Molecular genetics of inherited variation in human color vision,” Science 232, 203–210 (1986).
    [CrossRef] [PubMed]
  38. W. S. Stiles, G. Wyszecki, N. Ohta, “Counting metameric object-color stimuli using frequency-limited spectral reflectance functions,” J. Opt. Soc. Am. 67, 779–784 (1977).
    [CrossRef]
  39. M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,” J. Theor. Biol. 71, 473–478 (1978); M. H. Brill, “Further features of the illuminant invariant trichromatic photosensor,” J. Theor. Biol. 78, 305–308 (1979); L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
    [CrossRef] [PubMed]
  40. B. A. Wandell, “The synthesis and analysis of color images,” NASA Tech. Memo. 86844 (1985), pp. 1–35.
  41. S. A. Burns, V. C. Smith, J. Pokorny, A. E. Elsner, “Brightness of equal-luminance lights,” J. Opt. Soc. Am. 72, 1225–1231 (1982).
    [CrossRef] [PubMed]
  42. Equation (21) would describe a general conic section, but the inequality t≥ r > 0 ensures ellipticity for all but the degenerative case t= r. This inequality can be proved by noting that a and c are the square magnitudes of two 3-vectors, noting that b is the dot product of these vectors and invoking the Schwarz inequality. The condition t≥ r> 0 also ensures that the right-hand side of Eq. (21) is nonnegative.
  43. H. Yaguchi, M. Ikeda, “Contribution of opponent-colour channels to brightness,” in Colour Vision, J. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 353–360; J. Thornton, E. N. Pugh, “Relationship of opponent-colours cancellation measures to cone-antagonistic signals deduced from increment threshold data,” ibid., pp. 361–373.
  44. R. M. Boynton, M. Ikeda, W. S. Stiles, “Interactions among chromatic mechanisms as inferred from positive and negative increment thresholds,” Vision Res. 4, 87–117 (1964).
    [CrossRef] [PubMed]
  45. R. L. DeValois, I. Abramov, G. H. Jacobs, “Analysis of response patterns of LGN cells,” J. Opt. Soc. Am. 56, 966–977 (1966); F. M. de Monasterio, P. Gouras, “Functional properties of ganglion cells of rhesus monkey retina,” J. Physiol. 185, 587–599 (1975); A. Derrington, P. Lennie, J. Krauskopf, “Chromatic response properties of parvocellular neurons in the macaque LGN,” in Colour Vision, J. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 245–251.
    [CrossRef]

1985 (1)

1984 (1)

1983 (2)

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding, and optimum colour information transmission in the retina,” Proc. R. Soc. London Ser. B 220, 89–113 (1983).
[CrossRef]

B. Drum, “Short-wavelength cones contribute to achromatic sensitivity,” Vision Res. 23, 1433–1439 (1983).
[CrossRef] [PubMed]

1982 (4)

1980 (3)

1978 (3)

R. W. Massof, J. F. Bird, “A general zone theory of color and brightness vision,” J. Opt. Soc. Am. 68, 1465–1471 (1978); R. W. Massof, “Color-vision theory and linear models of color vision,” Color Res. Appl. 10, 133–146 (1985). According to this view, the derivatives of the intensity-response functions of the color mechanisms define, for given static and uniform conditions, a transformation matrix of receptor outputs like those in Eq. (5). Thus for small perturbations about a given adaptive state, nonlinear models that embody similar receptor transformations to those of Eqs. (8)–(10) are virtually indistinguishable from their linear counterparts.
[CrossRef] [PubMed]

E. N. Pugh, C. Sigel, “Evaluation of the candidacy of the π-mechanisms of Stiles for color-matching functions,” Vision Res. 18, 317–330 (1978).
[CrossRef]

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,” J. Theor. Biol. 71, 473–478 (1978); M. H. Brill, “Further features of the illuminant invariant trichromatic photosensor,” J. Theor. Biol. 78, 305–308 (1979); L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
[CrossRef] [PubMed]

1977 (4)

C. R. Ingling, “The spectral sensitivity of the opponent-color channels,” Vision Res. 17, 1083–1089 (1977).
[CrossRef] [PubMed]

W. S. Stiles, G. Wyszecki, N. Ohta, “Counting metameric object-color stimuli using frequency-limited spectral reflectance functions,” J. Opt. Soc. Am. 67, 779–784 (1977).
[CrossRef]

J. D. Mollon, P. G. Polden, “An anomaly in the response of the eye to short wavelengths,” Phil. Trans. Soc. London Ser. B 278, 207–240 (1977); J. J. Wisowaty, “An action spectrum for the production of transient tritanopia,” Vision Res. 23, 769–774 (1983).
[CrossRef] [PubMed]

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

1976 (2)

E. N. Pugh, “The nature of the π1colour mechanism of W. S. Stiles,” J. Physiol. 257, 713–747 (1976);B. A. Wandell, E. N. Pugh, “Detection of long-duration, long-wavelength incremental flashes by a chromatically coded pathway,” Vision Res. 20, 613–624 (1980); E. N. Pugh, J. Larimer, “Test of the identity of the site of blue/yellow hue cancellation and the site of chromatic antagonism in the π1pathway,” Vision Res. 20, 779–788 (1980); C. Sigel, L. Brousseau, “Pi-4: adaptation of more than one class of cone,” J. Opt. Soc. Am. 72, 237–246 (1980).
[CrossRef]

B. W. Tansley, R. M. Boynton, “A single line predicts the distinctness of borders by different colors,” Science 191, 954–957 (1976); B. W. Tansley, R. M. Boynton, “Chromatic border perception: the role of red- and green-sensitive cones,” Vision Res. 18, 683–697 (1978).
[CrossRef] [PubMed]

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]

1974 (1)

R. A. Normann, F. S. Werblin, “Control of retinal sensitivity: I. Light dark adaptation of vertebrate rods and cones,” J. Gen. Physiol. 63, 37–61 (1974);J. Larimer, D. H. Krantz, C. M. Cicerone, “Opponent-process additivity-II. Yellow/blue equilibria and nonlinear models,” Vision Res. 18, 1521–1532 (1975); J. Raaijmakers, C. de Weert, “Linear and non-linear opponent color coding,” Percept. Psychophys. 18, 474–480 (1975).
[CrossRef]

1972 (2)

V. C. Smith, J. Pokorny, “Spectral sensitivity of colorblind observers and the cone photopigments,” Vision Res. 12, 2059–2071 (1972).
[CrossRef] [PubMed]

G. B. Rollman, J. Nachmias, “Simultaneous detection and recognition of chromatic flashes,” Percept. Psychophys. 12, 309–314 (1972); B. A. Wandell, J. Sanchez, B. Quinn, “Detection/discrimination in the long-wavelength pathways,” Vision Res. 22, 1061–1069 (1982).
[CrossRef] [PubMed]

1969 (1)

S. L. Guth, N. J. Donley, R. T. Marrocco, “On luminance additivity and related topics,” Vision Res. 9, 537–575 (1969); K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–745 (1979).
[CrossRef] [PubMed]

1968 (2)

1966 (1)

1964 (2)

R. M. Boynton, M. Ikeda, W. S. Stiles, “Interactions among chromatic mechanisms as inferred from positive and negative increment thresholds,” Vision Res. 4, 87–117 (1964).
[CrossRef] [PubMed]

R. M. Boynton, W. Schafer, M. E. Neun, “Hue-wavelength relation measured by color-naming method for three retinal locations,” Science 146, 666–668 (1964); S. M. Luria, “Color-name as a function of stimulus-intensity and duration,” Am. J. Psychol. 80, 14–27 (1967).
[CrossRef] [PubMed]

1957 (1)

L. M. Hurvich, D. Jameson, “An opponent-process theory of color vision,” Psych. Rev. 64, 384–404 (1957).
[CrossRef]

1955 (1)

1950 (1)

H. Wolter, “Physikalische Begründung eines Farbenkreises und Anzätze einer physikalischen Farbenlehre,” Ann. Phys. (Leipzig) Ser. 6 8, 11–29 (1950).
[CrossRef]

1949 (1)

W. S. Stiles, “Increment thresholds and the mechanisms of colour vision,” Doc. Ophthalmol. 3, 138–163 (1949); W. S. Stiles, “Colour vision: The approach through increment-threshold sensitivity,” Proc. Nat. Acad. Sci. USA 3, 100–114 (1959).
[CrossRef] [PubMed]

Abramov, I.

Barlow, H. B.

H. B. Barlow, “What causes trichromacy? A theoretical analysis using comb-filtered spectra,” Vision Res. 22, 635–644 (1982).
[CrossRef] [PubMed]

Benzschawel, T.

Bird, J. F.

Boynton, R. M.

B. W. Tansley, R. M. Boynton, “A single line predicts the distinctness of borders by different colors,” Science 191, 954–957 (1976); B. W. Tansley, R. M. Boynton, “Chromatic border perception: the role of red- and green-sensitive cones,” Vision Res. 18, 683–697 (1978).
[CrossRef] [PubMed]

H. M. O. Scheibner, R. M. Boynton, “Residual red-green discrimination in dichromats,” J. Opt. Soc. Am. 58, 1151–1158 (1968); M. Alpern, E. N. Pugh, “Variation in the action spectrum of erythrolabe among deuteranopes,” J. Physiol. 266, 613–646 (1977);V. C. Smith, J. Pokorny, “Large field trichromacy in protanopes and deuteranopes,” J. Opt. Soc. Am. 67, 213–220 (1977); A. L. Nagy, R. M. Boynton, “Large-field color naming of dichromats with rods bleached,” J. Opt. Soc. Am. 69, 1259–1265 (1979); M. E. Breton, W. B. Cowan, “Deuteranomalous color matching in the deuteranopic eye,” J. Opt. Soc. Am. 71, 1220–1223 (1981); F. S. Frome, T. P. Piantanida, D. H. Kelly, “Psychophysical evidence for more than two kinds of cone in dichromatic color blindness,” Science 215, 417–419 (1982); J. Nathans, T. P. Piantanida, R. L. Eddy, T. B. Shows, D. S. Hogness, “Molecular genetics of inherited variation in human color vision,” Science 232, 203–210 (1986).
[CrossRef] [PubMed]

R. M. Boynton, M. Ikeda, W. S. Stiles, “Interactions among chromatic mechanisms as inferred from positive and negative increment thresholds,” Vision Res. 4, 87–117 (1964).
[CrossRef] [PubMed]

R. M. Boynton, W. Schafer, M. E. Neun, “Hue-wavelength relation measured by color-naming method for three retinal locations,” Science 146, 666–668 (1964); S. M. Luria, “Color-name as a function of stimulus-intensity and duration,” Am. J. Psychol. 80, 14–27 (1967).
[CrossRef] [PubMed]

Brill, M. H.

M. H. Brill, T. Benzschawel, “Remarks on signal-processing explanations of the trichromacy of vision,” J. Opt. Soc. Am. A 2, 1794–1796 (1985).
[CrossRef] [PubMed]

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,” J. Theor. Biol. 71, 473–478 (1978); M. H. Brill, “Further features of the illuminant invariant trichromatic photosensor,” J. Theor. Biol. 78, 305–308 (1979); L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
[CrossRef] [PubMed]

Buchsbaum, G.

G. Buchsbaum, A. Gottschalk, “Chromaticity coordinates of frequency-limited functions,” J. Opt. Soc. Am. A 1, 885–887 (1984).
[CrossRef] [PubMed]

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding, and optimum colour information transmission in the retina,” Proc. R. Soc. London Ser. B 220, 89–113 (1983).
[CrossRef]

Burns, S. A.

DeValois, R. L.

Donley, N. J.

S. L. Guth, N. J. Donley, R. T. Marrocco, “On luminance additivity and related topics,” Vision Res. 9, 537–575 (1969); K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–745 (1979).
[CrossRef] [PubMed]

Drum, B.

B. Drum, “Short-wavelength cones contribute to achromatic sensitivity,” Vision Res. 23, 1433–1439 (1983).
[CrossRef] [PubMed]

Eisner, A.

Elsner, A. E.

Goodman, D. S.

Gottschalk, A.

G. Buchsbaum, A. Gottschalk, “Chromaticity coordinates of frequency-limited functions,” J. Opt. Soc. Am. A 1, 885–887 (1984).
[CrossRef] [PubMed]

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding, and optimum colour information transmission in the retina,” Proc. R. Soc. London Ser. B 220, 89–113 (1983).
[CrossRef]

Guth, S. L.

S. L. Guth, R. W. Massof, T. Benzschawel, “Vector model for normal and dichromatic vision,” J. Opt. Soc. Am. 70, 197–212 (1980).
[CrossRef] [PubMed]

S. L. Guth, N. J. Donley, R. T. Marrocco, “On luminance additivity and related topics,” Vision Res. 9, 537–575 (1969); K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–745 (1979).
[CrossRef] [PubMed]

Heeley, D. W.

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

Hurvich, L. M.

Ikeda, M.

R. M. Boynton, M. Ikeda, W. S. Stiles, “Interactions among chromatic mechanisms as inferred from positive and negative increment thresholds,” Vision Res. 4, 87–117 (1964).
[CrossRef] [PubMed]

H. Yaguchi, M. Ikeda, “Contribution of opponent-colour channels to brightness,” in Colour Vision, J. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 353–360; J. Thornton, E. N. Pugh, “Relationship of opponent-colours cancellation measures to cone-antagonistic signals deduced from increment threshold data,” ibid., pp. 361–373.

Ingling, C. R.

C. R. Ingling, “The spectral sensitivity of the opponent-color channels,” Vision Res. 17, 1083–1089 (1977).
[CrossRef] [PubMed]

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

Jacobs, G. H.

Jameson, D.

Judd, D. B.

D. B. Judd, “Colorimetry and Artificial Daylight,” in Technical Committee No. 7 Report of Secretariat United States Commission, International Commission on Illumination, Twelfth Session, Stockholm (Bureau Central de la CIE, Paris1951), pp. 1–60.

Krauskopf, J.

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

MacLeod, D. I. A.

Marrocco, R. T.

S. L. Guth, N. J. Donley, R. T. Marrocco, “On luminance additivity and related topics,” Vision Res. 9, 537–575 (1969); K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–745 (1979).
[CrossRef] [PubMed]

Massof, R. W.

Mollon, J. D.

J. D. Mollon, P. G. Polden, “An anomaly in the response of the eye to short wavelengths,” Phil. Trans. Soc. London Ser. B 278, 207–240 (1977); J. J. Wisowaty, “An action spectrum for the production of transient tritanopia,” Vision Res. 23, 769–774 (1983).
[CrossRef] [PubMed]

Nachmias, J.

G. B. Rollman, J. Nachmias, “Simultaneous detection and recognition of chromatic flashes,” Percept. Psychophys. 12, 309–314 (1972); B. A. Wandell, J. Sanchez, B. Quinn, “Detection/discrimination in the long-wavelength pathways,” Vision Res. 22, 1061–1069 (1982).
[CrossRef] [PubMed]

Neun, M. E.

R. M. Boynton, W. Schafer, M. E. Neun, “Hue-wavelength relation measured by color-naming method for three retinal locations,” Science 146, 666–668 (1964); S. M. Luria, “Color-name as a function of stimulus-intensity and duration,” Am. J. Psychol. 80, 14–27 (1967).
[CrossRef] [PubMed]

Normann, R. A.

R. A. Normann, F. S. Werblin, “Control of retinal sensitivity: I. Light dark adaptation of vertebrate rods and cones,” J. Gen. Physiol. 63, 37–61 (1974);J. Larimer, D. H. Krantz, C. M. Cicerone, “Opponent-process additivity-II. Yellow/blue equilibria and nonlinear models,” Vision Res. 18, 1521–1532 (1975); J. Raaijmakers, C. de Weert, “Linear and non-linear opponent color coding,” Percept. Psychophys. 18, 474–480 (1975).
[CrossRef]

Ohta, N.

Pokorny, J.

S. A. Burns, V. C. Smith, J. Pokorny, A. E. Elsner, “Brightness of equal-luminance lights,” J. Opt. Soc. Am. 72, 1225–1231 (1982).
[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]

V. C. Smith, J. Pokorny, “Spectral sensitivity of colorblind observers and the cone photopigments,” Vision Res. 12, 2059–2071 (1972).
[CrossRef] [PubMed]

Polden, P. G.

J. D. Mollon, P. G. Polden, “An anomaly in the response of the eye to short wavelengths,” Phil. Trans. Soc. London Ser. B 278, 207–240 (1977); J. J. Wisowaty, “An action spectrum for the production of transient tritanopia,” Vision Res. 23, 769–774 (1983).
[CrossRef] [PubMed]

Pugh, E. N.

E. N. Pugh, C. Sigel, “Evaluation of the candidacy of the π-mechanisms of Stiles for color-matching functions,” Vision Res. 18, 317–330 (1978).
[CrossRef]

E. N. Pugh, “The nature of the π1colour mechanism of W. S. Stiles,” J. Physiol. 257, 713–747 (1976);B. A. Wandell, E. N. Pugh, “Detection of long-duration, long-wavelength incremental flashes by a chromatically coded pathway,” Vision Res. 20, 613–624 (1980); E. N. Pugh, J. Larimer, “Test of the identity of the site of blue/yellow hue cancellation and the site of chromatic antagonism in the π1pathway,” Vision Res. 20, 779–788 (1980); C. Sigel, L. Brousseau, “Pi-4: adaptation of more than one class of cone,” J. Opt. Soc. Am. 72, 237–246 (1980).
[CrossRef]

Rollman, G. B.

G. B. Rollman, J. Nachmias, “Simultaneous detection and recognition of chromatic flashes,” Percept. Psychophys. 12, 309–314 (1972); B. A. Wandell, J. Sanchez, B. Quinn, “Detection/discrimination in the long-wavelength pathways,” Vision Res. 22, 1061–1069 (1982).
[CrossRef] [PubMed]

Schafer, W.

R. M. Boynton, W. Schafer, M. E. Neun, “Hue-wavelength relation measured by color-naming method for three retinal locations,” Science 146, 666–668 (1964); S. M. Luria, “Color-name as a function of stimulus-intensity and duration,” Am. J. Psychol. 80, 14–27 (1967).
[CrossRef] [PubMed]

Scheibner, H. M. O.

Sigel, C.

E. N. Pugh, C. Sigel, “Evaluation of the candidacy of the π-mechanisms of Stiles for color-matching functions,” Vision Res. 18, 317–330 (1978).
[CrossRef]

Smith, V. C.

S. A. Burns, V. C. Smith, J. Pokorny, A. E. Elsner, “Brightness of equal-luminance lights,” J. Opt. Soc. Am. 72, 1225–1231 (1982).
[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]

V. C. Smith, J. Pokorny, “Spectral sensitivity of colorblind observers and the cone photopigments,” Vision Res. 12, 2059–2071 (1972).
[CrossRef] [PubMed]

Starr, S. J.

Stiles, W. S.

W. S. Stiles, G. Wyszecki, N. Ohta, “Counting metameric object-color stimuli using frequency-limited spectral reflectance functions,” J. Opt. Soc. Am. 67, 779–784 (1977).
[CrossRef]

R. M. Boynton, M. Ikeda, W. S. Stiles, “Interactions among chromatic mechanisms as inferred from positive and negative increment thresholds,” Vision Res. 4, 87–117 (1964).
[CrossRef] [PubMed]

W. S. Stiles, “Increment thresholds and the mechanisms of colour vision,” Doc. Ophthalmol. 3, 138–163 (1949); W. S. Stiles, “Colour vision: The approach through increment-threshold sensitivity,” Proc. Nat. Acad. Sci. USA 3, 100–114 (1959).
[CrossRef] [PubMed]

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1982), p. 553.

Tansley, B. W.

B. W. Tansley, R. M. Boynton, “A single line predicts the distinctness of borders by different colors,” Science 191, 954–957 (1976); B. W. Tansley, R. M. Boynton, “Chromatic border perception: the role of red- and green-sensitive cones,” Vision Res. 18, 683–697 (1978).
[CrossRef] [PubMed]

Tsou, B. H.-P.

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

Wandell, B. A.

B. A. Wandell, “The synthesis and analysis of color images,” NASA Tech. Memo. 86844 (1985), pp. 1–35.

Werblin, F. S.

R. A. Normann, F. S. Werblin, “Control of retinal sensitivity: I. Light dark adaptation of vertebrate rods and cones,” J. Gen. Physiol. 63, 37–61 (1974);J. Larimer, D. H. Krantz, C. M. Cicerone, “Opponent-process additivity-II. Yellow/blue equilibria and nonlinear models,” Vision Res. 18, 1521–1532 (1975); J. Raaijmakers, C. de Weert, “Linear and non-linear opponent color coding,” Percept. Psychophys. 18, 474–480 (1975).
[CrossRef]

Williams, D. R.

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

Wolter, H.

H. Wolter, “Physikalische Begründung eines Farbenkreises und Anzätze einer physikalischen Farbenlehre,” Ann. Phys. (Leipzig) Ser. 6 8, 11–29 (1950).
[CrossRef]

Wyszecki, G.

Yaguchi, H.

H. Yaguchi, M. Ikeda, “Contribution of opponent-colour channels to brightness,” in Colour Vision, J. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 353–360; J. Thornton, E. N. Pugh, “Relationship of opponent-colours cancellation measures to cone-antagonistic signals deduced from increment threshold data,” ibid., pp. 361–373.

Yilmaz, H.

H. Yilmaz, “Color vision and a new approach to general perception,” in Biological Prototypes and Synthetic Systems, E. Bernard, M. Kare, eds. (Plenum, New York, 1962), pp. 126–141, Vol. I.
[CrossRef]

Ann. Phys. (Leipzig) Ser. 6 (1)

H. Wolter, “Physikalische Begründung eines Farbenkreises und Anzätze einer physikalischen Farbenlehre,” Ann. Phys. (Leipzig) Ser. 6 8, 11–29 (1950).
[CrossRef]

Appl. Opt. (1)

Doc. Ophthalmol. (1)

W. S. Stiles, “Increment thresholds and the mechanisms of colour vision,” Doc. Ophthalmol. 3, 138–163 (1949); W. S. Stiles, “Colour vision: The approach through increment-threshold sensitivity,” Proc. Nat. Acad. Sci. USA 3, 100–114 (1959).
[CrossRef] [PubMed]

J. Gen. Physiol. (1)

R. A. Normann, F. S. Werblin, “Control of retinal sensitivity: I. Light dark adaptation of vertebrate rods and cones,” J. Gen. Physiol. 63, 37–61 (1974);J. Larimer, D. H. Krantz, C. M. Cicerone, “Opponent-process additivity-II. Yellow/blue equilibria and nonlinear models,” Vision Res. 18, 1521–1532 (1975); J. Raaijmakers, C. de Weert, “Linear and non-linear opponent color coding,” Percept. Psychophys. 18, 474–480 (1975).
[CrossRef]

J. Opt. Soc. Am. (10)

R. W. Massof, J. F. Bird, “A general zone theory of color and brightness vision,” J. Opt. Soc. Am. 68, 1465–1471 (1978); R. W. Massof, “Color-vision theory and linear models of color vision,” Color Res. Appl. 10, 133–146 (1985). According to this view, the derivatives of the intensity-response functions of the color mechanisms define, for given static and uniform conditions, a transformation matrix of receptor outputs like those in Eq. (5). Thus for small perturbations about a given adaptive state, nonlinear models that embody similar receptor transformations to those of Eqs. (8)–(10) are virtually indistinguishable from their linear counterparts.
[CrossRef] [PubMed]

D. Jameson, L. M. Hurvich, “Opponent-response functions related to measured cone photopigments,” J. Opt. Soc. Am. 58, 429–430 (1968).
[CrossRef]

R. W. Massof, S. J. Starr, “Vector magnitude operation in color vision models: derivation from signal detection theory,” J. Opt. Soc. Am. 70, 870–872 (1980).
[CrossRef] [PubMed]

A. Eisner, D. I. A. MacLeod, “Blue-sensitive cones do not contribute to luminance,” J. Opt. Soc. Am. 70, 121–123 (1980).
[CrossRef] [PubMed]

S. L. Guth, R. W. Massof, T. Benzschawel, “Vector model for normal and dichromatic vision,” J. Opt. Soc. Am. 70, 197–212 (1980).
[CrossRef] [PubMed]

D. Jameson, L. M. Hurvich, “Some quantitative aspects of an opponent-colors theory. I. Chromatic responses and spectral saturation,” J. Opt. Soc. Am. 45, 546–552 (1955); C. R. Ingling, P. W. Russell, M. S. Rea, B. H.-P. Tsou, “Red-green opponent spectral sensitivity: disparity between cancellation and direct matching methods,” Science 201, 221–1223 (1978); B. R. Wooten, J. S. Werner, “Short-wave cone input to the red-green opponent channel,” Vision Res. 19, 1053–1054 (1978).
[CrossRef]

H. M. O. Scheibner, R. M. Boynton, “Residual red-green discrimination in dichromats,” J. Opt. Soc. Am. 58, 1151–1158 (1968); M. Alpern, E. N. Pugh, “Variation in the action spectrum of erythrolabe among deuteranopes,” J. Physiol. 266, 613–646 (1977);V. C. Smith, J. Pokorny, “Large field trichromacy in protanopes and deuteranopes,” J. Opt. Soc. Am. 67, 213–220 (1977); A. L. Nagy, R. M. Boynton, “Large-field color naming of dichromats with rods bleached,” J. Opt. Soc. Am. 69, 1259–1265 (1979); M. E. Breton, W. B. Cowan, “Deuteranomalous color matching in the deuteranopic eye,” J. Opt. Soc. Am. 71, 1220–1223 (1981); F. S. Frome, T. P. Piantanida, D. H. Kelly, “Psychophysical evidence for more than two kinds of cone in dichromatic color blindness,” Science 215, 417–419 (1982); J. Nathans, T. P. Piantanida, R. L. Eddy, T. B. Shows, D. S. Hogness, “Molecular genetics of inherited variation in human color vision,” Science 232, 203–210 (1986).
[CrossRef] [PubMed]

W. S. Stiles, G. Wyszecki, N. Ohta, “Counting metameric object-color stimuli using frequency-limited spectral reflectance functions,” J. Opt. Soc. Am. 67, 779–784 (1977).
[CrossRef]

S. A. Burns, V. C. Smith, J. Pokorny, A. E. Elsner, “Brightness of equal-luminance lights,” J. Opt. Soc. Am. 72, 1225–1231 (1982).
[CrossRef] [PubMed]

R. L. DeValois, I. Abramov, G. H. Jacobs, “Analysis of response patterns of LGN cells,” J. Opt. Soc. Am. 56, 966–977 (1966); F. M. de Monasterio, P. Gouras, “Functional properties of ganglion cells of rhesus monkey retina,” J. Physiol. 185, 587–599 (1975); A. Derrington, P. Lennie, J. Krauskopf, “Chromatic response properties of parvocellular neurons in the macaque LGN,” in Colour Vision, J. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 245–251.
[CrossRef]

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

J. Physiol. (1)

E. N. Pugh, “The nature of the π1colour mechanism of W. S. Stiles,” J. Physiol. 257, 713–747 (1976);B. A. Wandell, E. N. Pugh, “Detection of long-duration, long-wavelength incremental flashes by a chromatically coded pathway,” Vision Res. 20, 613–624 (1980); E. N. Pugh, J. Larimer, “Test of the identity of the site of blue/yellow hue cancellation and the site of chromatic antagonism in the π1pathway,” Vision Res. 20, 779–788 (1980); C. Sigel, L. Brousseau, “Pi-4: adaptation of more than one class of cone,” J. Opt. Soc. Am. 72, 237–246 (1980).
[CrossRef]

J. Theor. Biol. (1)

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,” J. Theor. Biol. 71, 473–478 (1978); M. H. Brill, “Further features of the illuminant invariant trichromatic photosensor,” J. Theor. Biol. 78, 305–308 (1979); L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
[CrossRef] [PubMed]

Percept. Psychophys. (1)

G. B. Rollman, J. Nachmias, “Simultaneous detection and recognition of chromatic flashes,” Percept. Psychophys. 12, 309–314 (1972); B. A. Wandell, J. Sanchez, B. Quinn, “Detection/discrimination in the long-wavelength pathways,” Vision Res. 22, 1061–1069 (1982).
[CrossRef] [PubMed]

Phil. Trans. Soc. London Ser. B (1)

J. D. Mollon, P. G. Polden, “An anomaly in the response of the eye to short wavelengths,” Phil. Trans. Soc. London Ser. B 278, 207–240 (1977); J. J. Wisowaty, “An action spectrum for the production of transient tritanopia,” Vision Res. 23, 769–774 (1983).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. B (1)

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding, and optimum colour information transmission in the retina,” Proc. R. Soc. London Ser. B 220, 89–113 (1983).
[CrossRef]

Psych. Rev. (1)

L. M. Hurvich, D. Jameson, “An opponent-process theory of color vision,” Psych. Rev. 64, 384–404 (1957).
[CrossRef]

Science (2)

R. M. Boynton, W. Schafer, M. E. Neun, “Hue-wavelength relation measured by color-naming method for three retinal locations,” Science 146, 666–668 (1964); S. M. Luria, “Color-name as a function of stimulus-intensity and duration,” Am. J. Psychol. 80, 14–27 (1967).
[CrossRef] [PubMed]

B. W. Tansley, R. M. Boynton, “A single line predicts the distinctness of borders by different colors,” Science 191, 954–957 (1976); B. W. Tansley, R. M. Boynton, “Chromatic border perception: the role of red- and green-sensitive cones,” Vision Res. 18, 683–697 (1978).
[CrossRef] [PubMed]

Vision Res. (10)

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]

B. Drum, “Short-wavelength cones contribute to achromatic sensitivity,” Vision Res. 23, 1433–1439 (1983).
[CrossRef] [PubMed]

C. R. Ingling, “The spectral sensitivity of the opponent-color channels,” Vision Res. 17, 1083–1089 (1977).
[CrossRef] [PubMed]

E. N. Pugh, C. Sigel, “Evaluation of the candidacy of the π-mechanisms of Stiles for color-matching functions,” Vision Res. 18, 317–330 (1978).
[CrossRef]

V. C. Smith, J. Pokorny, “Spectral sensitivity of colorblind observers and the cone photopigments,” Vision Res. 12, 2059–2071 (1972).
[CrossRef] [PubMed]

S. L. Guth, N. J. Donley, R. T. Marrocco, “On luminance additivity and related topics,” Vision Res. 9, 537–575 (1969); K. Kranda, P. E. King-Smith, “Detection of coloured stimuli by independent linear systems,” Vision Res. 19, 733–745 (1979).
[CrossRef] [PubMed]

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

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

H. B. Barlow, “What causes trichromacy? A theoretical analysis using comb-filtered spectra,” Vision Res. 22, 635–644 (1982).
[CrossRef] [PubMed]

R. M. Boynton, M. Ikeda, W. S. Stiles, “Interactions among chromatic mechanisms as inferred from positive and negative increment thresholds,” Vision Res. 4, 87–117 (1964).
[CrossRef] [PubMed]

Other (11)

Equation (21) would describe a general conic section, but the inequality t≥ r > 0 ensures ellipticity for all but the degenerative case t= r. This inequality can be proved by noting that a and c are the square magnitudes of two 3-vectors, noting that b is the dot product of these vectors and invoking the Schwarz inequality. The condition t≥ r> 0 also ensures that the right-hand side of Eq. (21) is nonnegative.

H. Yaguchi, M. Ikeda, “Contribution of opponent-colour channels to brightness,” in Colour Vision, J. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 353–360; J. Thornton, E. N. Pugh, “Relationship of opponent-colours cancellation measures to cone-antagonistic signals deduced from increment threshold data,” ibid., pp. 361–373.

B. A. Wandell, “The synthesis and analysis of color images,” NASA Tech. Memo. 86844 (1985), pp. 1–35.

H. Yilmaz, “Color vision and a new approach to general perception,” in Biological Prototypes and Synthetic Systems, E. Bernard, M. Kare, eds. (Plenum, New York, 1962), pp. 126–141, Vol. I.
[CrossRef]

D. Jameson, “Theoretical issues of color vision,” in Handbook of Sensory Physiology, D. Jameson, L. M. Hurvich, eds. (Springer-Verlag, Berlin, 1972), Vol. 7/4, Chap. 14.
[CrossRef]

L. M. Hurvich, Color Vision (Sinauer, 1981).

Although each of the models described herein uses slightly different nomenclature for analogous receptor types and opponent-colors mechanisms, for simplicity we have chosen to use the same set of terms for each. That is, the photoreceptors are referred to as L, M, and S for the long-, short- and medium-wavelength sensitive photoreceptors, respectively. The second stage mechanisms of the opponent-colors models are designated as A, T, and D for an achromatic or luminance system, a tritanopic system (i.e., the system that tritanopes have that signals red or green), and a deuteranopic system (i.e., the system that deuteranopes and protanopes have that signals blue or yellow), respectively.

Note that, unlike previous reports using sine-wave spectra, angular measure is expressed in degrees and frequency is specified in cycles per 300 nm. In those articles, angular measure is given in radians and spectral frequency is given in units of terahertz or in cycles per nanometer.

D. B. Judd, “Colorimetry and Artificial Daylight,” in Technical Committee No. 7 Report of Secretariat United States Commission, International Commission on Illumination, Twelfth Session, Stockholm (Bureau Central de la CIE, Paris1951), pp. 1–60.

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1982), p. 553.

Both the models of Guth et al. and that of Ingling specify different mechanism outputs as the adaptation level (determined by the prevailing luminance level) is changed. For example, at high intensities the spectrum appears more bluish and yellowish relative to red and green. For the Guth model, this simply requires adjustment of the relative outputs of T and D by constant amounts. Hence, we applied the coefficient 12.0 to the output of D as prescribed by the model. In Ingling’s scheme, the transformation equations change with adaptation level. That is, for suprathreshold levels, there is input from S into T and A, and the relative balance of inputs from (L + M) and S into D are altered. Because we are generating predictions for the light-adapted observer, we chose the suprathreshold equations (3a)–(3c) of Ref. 13 for the present calculations. Note that later versions of the Ingling model incorporate additional intensity-dependent nonlinearities, including inhibition between L and M receptors and the rod system as well as summation among rods and S receptor outputs.34

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Figures (11)

Fig. 1
Fig. 1

Examples of sinusoidal SPD’s of the form given by Eq. (1). Top: spectral profile of an equal-energy white stimulus. Middle: sinusoidal SPD’s having f = 1 and m = 1 for starting phase angles of 0 deg (solid curve) and 270 deg (dashed curve). Bottom: sinusoidal SPD’s having f = 2 and p0 = 0 for amplitudes of m = 1.0 and m = 0.3. Note: A in the figure corresponds to m in Eq. (1).

Fig. 2
Fig. 2

Schematic diagrams of two-stage opponent-colors models of color vision. Top: Hurvich–Jameson model in which receptors α, β, and, γ feed antagonistically into second-stage red-green and blue-yellow mechanisms and sum within an achromatic luminance system. Middle: wiring diagram proposed by Guth et al., in which R, G, B receptors feed antagonistically into deuteranopic (D) and tritanopic (T) color mechanisms and R and G sum within a nonopponent luminance system (A). Bottom: model proposed by Ingling and Tsou where R, G, and B feed antagonistically into r-g and y-b color mechanisms and an achromatic system, Vλ.

Fig. 3
Fig. 3

Block diagram of computational procedure for deriving model predictions of responses to sinusoidal SPD’s of the form given in Eq. (1). Note: A in the figure corresponds to m in Eq. (1).

Fig. 4
Fig. 4

MSF’s in spectral phase for f = 1 and m = 1 for various color theories. The predictions from a given model are indicated by the inset. For this plot, the logarithms of the differences in model responses between the equal-energy white and the particular sinusoidal SPD are calculated as a function of starting phase angle, p0.

Fig. 5
Fig. 5

MSF’s in spectral frequency for p0 = 0 and m = 1 for the various color models. The key relating the curves to the models is inset in the figure. For this plot, the logarithms of S(f, p0) are plotted as a function of frequency, f.

Fig. 6
Fig. 6

Logarithms of absolute values of responses of individual color mechanisms as functions of phase angle p0 for fully-modulated sinusoidal SPD’s of f = 1 as predicted from the Ingling (solid curves) and Guth (dashes) models for color vision. Top: responses of achromatic or luminance systems, where I denotes luminance increments and D denotes decrements. Middle: tritanopic systems that signal red (R) or green (G). Bottom: Deuteranopic systems that signal blue (B) or yellow (Y). For a given phase angle, predictions of perceived hue are given as the proportional combination of hue signals from each mechanism.

Fig. 7
Fig. 7

Responses of individual color mechanisms as functions of frequency, f, for fully modulated sinusoidal SPD’s having p0 = 0 as predicted by the Hurvich–Jameson (left) and Guth (right) models for color vision. Top: achromatic system that signals luminance increments (unshaded) or decrements (shaded). Middle: tritanopic system that signals red (unshaded) or green (shaded). Bottom: deuteranopic system that signals blue (shaded) or yellow (unshaded).

Fig. 8
Fig. 8

Predicted MSF’s in spectral phase for f = 1.0 (top) and frequency for p0 = 0 (bottom) for deuteranopes. The dashed function superimposed on each graph is that for trichromats.

Fig. 9
Fig. 9

MSF’s in spectral phase and frequency for protanopes.

Fig. 10
Fig. 10

MSF’s in spectral phase and frequency for tritanopes.

Fig. 11
Fig. 11

Polar spaces for graphical description of lights in terms of frequency (radial distance), amplitude (point size), and phase angle (angle) of sinusoidal SPD’s. Left: Sinusoidal SPD description of a single wavelength of about 620 nm. Right: Discrete representation of the spectrum at the left.

Equations (24)

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E ( λ ) = E 0 [ 1 + m sin ( f p ( λ ) + p 0 ) ] for 400 nm λ 700 nm , else E ( λ ) = 0 ,
V = | T | + | D | + | A | .
V = ( T 2 + D 2 + A 2 ) 1 / 2 .
V = max ( π 3 , π 4 , π 5 ) .
[ A T D ] = [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] [ L M S ] ,
[ L M S ] = [ 0.2435 0.8524 0.0516 0.3954 1.16425 0.0837 0 0 0.6220 ] [ X Y Z ] ,
[ L M S ] = [ 0 6.5341 0.1336 0.3368 7.0009 0.0020 0.3329 6.4671 0.1347 ] [ X Y Z ] .
[ A T D ] = [ 0.85 1.50 0.01 1.66 2.23 0.37 0.34 0.06 0.71 ] [ L M S ] .
[ A T D ] = [ 0.5967 0.3654 0 0.9553 1.2836 0 0.0248 0 0.0483 ] [ L M S ] .
[ A T D ] = [ 0.6 0.4 0 1.2 1.6 0.4 0.24 0.11 0.7 ] [ L M S ] .
[ A T ] = [ 0.5967 0.3654 0.9553 1.2836 ] [ L M ] ,
[ A D ] = [ 0.9621 0 0.04 0.0483 ] [ L S ] ,
[ A D ] = [ 0.9621 0 0.04 0.0483 ] [ M S ] ,
S ( f , p 0 ) = V ( f , p 0 ) V 0 .
S i ( x ) = E 0 [ 1 + m i F ( f x + ϕ ) ] ,
j = 1 3 [ | g j ( Q 1 ) g j ( Q 2 ) | ] n δ .
j = 1 3 [ | Q 1 j Q 2 j | ] n δ .
Q i j = E 0 q j + m i E 0 q j F ( f x + ϕ ) ,
j = 1 3 [ | E 0 q j F ( f x + ϕ ) | ] n Δ m n δ .
( a sin 2 ϕ + 2 b sin ϕ cos ϕ + c cos 2 ϕ ) ( Δ m ) 2 1 ,
a = E 0 2 δ j = 1 3 q j cos 2 π f x 2 , b = E 0 2 δ j = 1 3 q j cos 2 π f x q j sin 2 π f x ,
c = E 0 2 δ j = 1 3 q j sin 2 π f x 2 .
c + a 2 + ( c a 2 ) cos 2 ϕ + b sin 2 ϕ = r sin ( 2 ϕ s ) + t ,
Δ m 2 = [ r sin ( 2 ϕ s ) + t ] 1 .

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