Abstract

Recent computational models of color vision demonstrate that it is possible to achieve exact color constancy over a limited range of lights and surfaces described by linear models. The success of these computational models hinges on whether any sizable range of surface spectral reflectances can be described by a linear model with about three parameters. In the first part of this paper, I analyze two large sets of empirical surface spectral reflectances and examine three conjectures concerning constraints on surface reflectance: (1) that empirical surface reflectances fall within a linear model with a small number of parameters, (2) that empirical surface reflectances fall within a linear model composed of band-limited functions with a small number of parameters, and (3) that the shape of the spectral-sensitivity curves of human vision enhance the fit between empirical surface reflectances and a linear model. I conclude that the first and second conjectures hold for the two sets of spectral reflectances analyzed but that the number of parameters required to model the spectral reflectances is five to seven, not three. A reanalysis of the empirical data that takes human visual sensitivity into account gives more promising results. The linear models derived provide excellent fits to the data with as few as three or four parameters, confirming the third conjecture. The results suggest that constraints on possible surface-reflectance functions and the “filtering” properties of the shapes of the spectral-sensitivity curves of photoreceptors can both contribute to color constancy. In the last part of the paper I derive the relation between the number of photoreceptor classes present in vision and the “filtering” properties of each class. The results of this analysis reverse a conclusion reached by Barlow: the “filtering” properties of human photoreceptors are consistent with a trichromatic visual system that is color constant.

© 1986 Optical Society of America

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References

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  1. H. Helson, “Fundamental problems in color vision. I. The principle governing changes in hue saturation and lightness of non-selective samples in chromatic illumination,” J. Exp. Psychol. 23, 439–476 (1938).
    [CrossRef]
  2. D. B. Judd, “Hue saturation and lightness of surface colors with chromatic illumination,” J. Opt. Soc. Am. 30, 2–32 (1940).
    [CrossRef]
  3. H. Yilmaz, “Color vision and a new approach to color perception,” in Biological Prototypes and Synthetic Systems (Plenum, New York, 1962), pp. 126–141.
    [CrossRef]
  4. J. M. Tenenbaum, Ph.D. dissertation (Stanford University, Stanford, Calif., 1971).
  5. P. Sallstrom, “Colour and physics: some remarks concerning the physical aspects of human colour vision,” Inst. Phys. Rep. no. 73-09 (Institute of Physics, University of Stockholm, 1973).
  6. M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,” J. Theor. Biol. 71, 473–478 (1978).
    [CrossRef] [PubMed]
  7. M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,” J. Theor. Biol. 78, 305–308 (1979).
    [CrossRef] [PubMed]
  8. G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
    [CrossRef]
  9. L. T. Maloney, B. A. Wandell, “Color constancy: A method for recovering surface spectral reflectance,” J. Opt. Soc. Am. 3, 29–33 (1986).
    [CrossRef]
  10. L. T. Maloney, “Computational approaches to color constancy,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1984).
  11. G. Buchsbaum, A. Gottschalk, “Chromaticity coordinates of frequency-limited functions,” J. Opt Soc. Am 1, 885–887 (1984).
    [CrossRef]
  12. H. B. Barlow, “What causes trichromacy? A theoretical analysis using comb-filtered spectra,” Vision Res. 22, 635–643 (1982).
    [CrossRef] [PubMed]
  13. D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031–1040 (1964); S. R. Das, V. D. P. Sastri, “Spectral distribution and color of tropical daylight,” J. Opt. Soc. Am. 55, 319–323 (1965); V. D. P. Sastri, S. R. Das, “Spectral distribution of color of north sky at Delhi,” J. Opt. Soc. Am. 56, 829–830 (1966); “Typical spectra distributions and color for tropical daylight,” J. Opt. Soc. Am. 58, 391–398 (1968); E. R. Dixon, “Spectral distribution of Australian daylight,” J. Opt. Soc. Am. 68, 437–450 (1978).
    [CrossRef]
  14. D. L. MacAdam, Color Measurement; Theme and Variations (Springer-Verlag, Berlin, 1981), p. 128.
  15. J. N. Lythgoe, The Ecology of Vision (Clarendon, Oxford, 1979), p. 31.
  16. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
  17. K. L. Kelley, K. S. Gibson, D. Nickerson, “Tristimulus specification of the Munsell Book of Color from spectrophotometric measurements,” J. Opt. Soc. Am. 33, 355–376 (1943); The original spectrophotometric data for Kelley et al. were obtained from the Macbeth Corporation, Little Britain Road, P.O. Box 950, Newburgh, N.Y. 12550.
    [CrossRef]
  18. E. L. Krinov, Spectral Reflectance Properties of Natural Formations (National Research Council of Canada, Ottawa, technical translation: TT-439, 1947). Krinov reports 370 surface spectral reflectances. The 337 analyzed were those that had complete and legible data available from 400 to 650 nm.
  19. The analyses reported below were repeated for the full range of Munsell data, 380 to 770 nm, inclusive. The analyses were also repeated after each surface spectral reflectance was transformed from equal-spaced wavelength measurements to equal-spaced wave-number measurements by 4-point Lagrangean interpolation as described by Barlow. The results for both these analyses are qualitatively similar to the results reported here and are not reported.
  20. Regression on the first three characteristic vectors derived from the Munsell data directly produced similar results.
  21. 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]
  22. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
    [CrossRef]
  23. H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961); “Prolate spheroidal wave functions, Fourier analysis and uncertainty. III: The dimensions of the space of essentially time and band-limited signals.” Bell Syst. Tech. J. 41, 1295–1336 (1962). The consequences of these results for vision was first set out by Buchsbaum and Gottschalk.11
    [CrossRef]
  24. K. Nassau, The Physics and Chemistry of Color: The Fifteen Causes of Color (Wiley, New York, 1983).
  25. H. Suzuki, Electronic Absorption Spectra and Geometry of Organic Molecules; An Application of Molecular Orbital Theory (Academic, New York, 1967), p. 79.
  26. W. Kauzmann, Quantum Chemistry (Academic, New York, 1957), pp. 582 and 669.
  27. W. Kauzmann, ibid., pp. 669–670.
  28. P. Pringsheim, Fluorescence and Phosphorescence (Interscience, New York, 1949).
  29. T. H. Wonnacott, R. J. Wonnacott, Regression: A Second Course in Statistics (Wiley, New York, 1981).
  30. G. Wyszecki, W. S. Stiles, Color Science; Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982), p. 256.
  31. Barlow performs his analysis with color signals plotted against wave number, not wavelength. To avoid confusion, I continue to use wavelength as the independent variable. A change to wave number does not substantially affect the following argument.
  32. The spectrum of the light is convolved with the spectrum of the surface reflectance. See R. N. Bracewell, The Fourier Transform and Its Applications., 2nd ed. (McGraw-Hill, New York, 1978), Chap. 3. It is easily shown that the highest frequency present in a convolution is the sum of the two highest frequencies present in the functions being convolved.
  33. Barlow uses terahertz rather than inverse centimeters. The conversion factor is 33.33. Barlow assumes that the width of the visible spectrum is W= 228 THz. Then, using the result of Slepian et al., N= 2WF becomes N= 0.456F, where N is the number of samples required to capture the essentially frequency limited functions with cutoff F in cycles/1000 THz. Note that Barlow’s computations use a different formula based on the Shannon–Whittaker Theorem that gives N= 2WF+ 1.
  34. See D. A. Belsey, E. Kuh, R. W. Welsch, Regression Diagnostics; Identifying Influential Data and Sources of Collinearity (Wiley, New York, 1980).
    [CrossRef]
  35. The analyses reported in the text were also carried out on the unnormalized data. Comparison of the fits obtained on normalized and unnormalized data showed that normalized data produced worse fits in all cases. In effect, the penalty associated with failing to fit surface reflectances with small entries is reduced by neglecting to normalize.
  36. Also known as principal-components analysis or the Karhunen–Loeve decomposition. See K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, London, 1979). All characteristic vector analyses and multiple regressions were done in AT&T Bell Laboratories’ statistical language S. See R. A. Becker, J. M. Chambers, S: An Interactive Environment for Data Analysis and Graphics (Wadsworth, Belmont, Calif., 1984).
  37. See R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra from the Point of View of Communications Engineering (Dover, New York; 1959); P. Bloomfield, Fourier Analysis of Time Series: An Introduction (Wiley, New York, 1976).

1986 (1)

L. T. Maloney, B. A. Wandell, “Color constancy: A method for recovering surface spectral reflectance,” J. Opt. Soc. Am. 3, 29–33 (1986).
[CrossRef]

1984 (1)

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

1982 (1)

H. B. Barlow, “What causes trichromacy? A theoretical analysis using comb-filtered spectra,” Vision Res. 22, 635–643 (1982).
[CrossRef] [PubMed]

1980 (1)

G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

1979 (1)

M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,” J. Theor. Biol. 78, 305–308 (1979).
[CrossRef] [PubMed]

1978 (1)

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,” J. Theor. Biol. 71, 473–478 (1978).
[CrossRef] [PubMed]

1977 (1)

1964 (2)

1961 (2)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961); “Prolate spheroidal wave functions, Fourier analysis and uncertainty. III: The dimensions of the space of essentially time and band-limited signals.” Bell Syst. Tech. J. 41, 1295–1336 (1962). The consequences of these results for vision was first set out by Buchsbaum and Gottschalk.11
[CrossRef]

1943 (1)

1940 (1)

1938 (1)

H. Helson, “Fundamental problems in color vision. I. The principle governing changes in hue saturation and lightness of non-selective samples in chromatic illumination,” J. Exp. Psychol. 23, 439–476 (1938).
[CrossRef]

Barlow, H. B.

H. B. Barlow, “What causes trichromacy? A theoretical analysis using comb-filtered spectra,” Vision Res. 22, 635–643 (1982).
[CrossRef] [PubMed]

Belsey, D. A.

See D. A. Belsey, E. Kuh, R. W. Welsch, Regression Diagnostics; Identifying Influential Data and Sources of Collinearity (Wiley, New York, 1980).
[CrossRef]

Bibby, J. M.

Also known as principal-components analysis or the Karhunen–Loeve decomposition. See K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, London, 1979). All characteristic vector analyses and multiple regressions were done in AT&T Bell Laboratories’ statistical language S. See R. A. Becker, J. M. Chambers, S: An Interactive Environment for Data Analysis and Graphics (Wadsworth, Belmont, Calif., 1984).

Blackman, R. B.

See R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra from the Point of View of Communications Engineering (Dover, New York; 1959); P. Bloomfield, Fourier Analysis of Time Series: An Introduction (Wiley, New York, 1976).

Bracewell, R. N.

The spectrum of the light is convolved with the spectrum of the surface reflectance. See R. N. Bracewell, The Fourier Transform and Its Applications., 2nd ed. (McGraw-Hill, New York, 1978), Chap. 3. It is easily shown that the highest frequency present in a convolution is the sum of the two highest frequencies present in the functions being convolved.

Brill, M. H.

M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,” J. Theor. Biol. 78, 305–308 (1979).
[CrossRef] [PubMed]

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,” J. Theor. Biol. 71, 473–478 (1978).
[CrossRef] [PubMed]

Buchsbaum, G.

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

G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

Cohen, J.

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).

Gibson, K. S.

Gottschalk, A.

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

Helson, H.

H. Helson, “Fundamental problems in color vision. I. The principle governing changes in hue saturation and lightness of non-selective samples in chromatic illumination,” J. Exp. Psychol. 23, 439–476 (1938).
[CrossRef]

Judd, D. B.

Kauzmann, W.

W. Kauzmann, ibid., pp. 669–670.

W. Kauzmann, Quantum Chemistry (Academic, New York, 1957), pp. 582 and 669.

Kelley, K. L.

Kent, J. T.

Also known as principal-components analysis or the Karhunen–Loeve decomposition. See K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, London, 1979). All characteristic vector analyses and multiple regressions were done in AT&T Bell Laboratories’ statistical language S. See R. A. Becker, J. M. Chambers, S: An Interactive Environment for Data Analysis and Graphics (Wadsworth, Belmont, Calif., 1984).

Krinov, E. L.

E. L. Krinov, Spectral Reflectance Properties of Natural Formations (National Research Council of Canada, Ottawa, technical translation: TT-439, 1947). Krinov reports 370 surface spectral reflectances. The 337 analyzed were those that had complete and legible data available from 400 to 650 nm.

Kuh, E.

See D. A. Belsey, E. Kuh, R. W. Welsch, Regression Diagnostics; Identifying Influential Data and Sources of Collinearity (Wiley, New York, 1980).
[CrossRef]

Landau, H. J.

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961); “Prolate spheroidal wave functions, Fourier analysis and uncertainty. III: The dimensions of the space of essentially time and band-limited signals.” Bell Syst. Tech. J. 41, 1295–1336 (1962). The consequences of these results for vision was first set out by Buchsbaum and Gottschalk.11
[CrossRef]

Lythgoe, J. N.

J. N. Lythgoe, The Ecology of Vision (Clarendon, Oxford, 1979), p. 31.

MacAdam, D. L.

Maloney, L. T.

L. T. Maloney, B. A. Wandell, “Color constancy: A method for recovering surface spectral reflectance,” J. Opt. Soc. Am. 3, 29–33 (1986).
[CrossRef]

L. T. Maloney, “Computational approaches to color constancy,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1984).

Mardia, K. V.

Also known as principal-components analysis or the Karhunen–Loeve decomposition. See K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, London, 1979). All characteristic vector analyses and multiple regressions were done in AT&T Bell Laboratories’ statistical language S. See R. A. Becker, J. M. Chambers, S: An Interactive Environment for Data Analysis and Graphics (Wadsworth, Belmont, Calif., 1984).

Nassau, K.

K. Nassau, The Physics and Chemistry of Color: The Fifteen Causes of Color (Wiley, New York, 1983).

Nickerson, D.

Ohta, N.

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961); “Prolate spheroidal wave functions, Fourier analysis and uncertainty. III: The dimensions of the space of essentially time and band-limited signals.” Bell Syst. Tech. J. 41, 1295–1336 (1962). The consequences of these results for vision was first set out by Buchsbaum and Gottschalk.11
[CrossRef]

Pringsheim, P.

P. Pringsheim, Fluorescence and Phosphorescence (Interscience, New York, 1949).

Sallstrom, P.

P. Sallstrom, “Colour and physics: some remarks concerning the physical aspects of human colour vision,” Inst. Phys. Rep. no. 73-09 (Institute of Physics, University of Stockholm, 1973).

Slepian, D.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

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]

G. Wyszecki, W. S. Stiles, Color Science; Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982), p. 256.

Suzuki, H.

H. Suzuki, Electronic Absorption Spectra and Geometry of Organic Molecules; An Application of Molecular Orbital Theory (Academic, New York, 1967), p. 79.

Tenenbaum, J. M.

J. M. Tenenbaum, Ph.D. dissertation (Stanford University, Stanford, Calif., 1971).

Tukey, J. W.

See R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra from the Point of View of Communications Engineering (Dover, New York; 1959); P. Bloomfield, Fourier Analysis of Time Series: An Introduction (Wiley, New York, 1976).

Wandell, B. A.

L. T. Maloney, B. A. Wandell, “Color constancy: A method for recovering surface spectral reflectance,” J. Opt. Soc. Am. 3, 29–33 (1986).
[CrossRef]

Welsch, R. W.

See D. A. Belsey, E. Kuh, R. W. Welsch, Regression Diagnostics; Identifying Influential Data and Sources of Collinearity (Wiley, New York, 1980).
[CrossRef]

Wonnacott, R. J.

T. H. Wonnacott, R. J. Wonnacott, Regression: A Second Course in Statistics (Wiley, New York, 1981).

Wonnacott, T. H.

T. H. Wonnacott, R. J. Wonnacott, Regression: A Second Course in Statistics (Wiley, New York, 1981).

Wyszecki, G.

Yilmaz, H.

H. Yilmaz, “Color vision and a new approach to color perception,” in Biological Prototypes and Synthetic Systems (Plenum, New York, 1962), pp. 126–141.
[CrossRef]

Bell Syst. Tech. J. (2)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961); “Prolate spheroidal wave functions, Fourier analysis and uncertainty. III: The dimensions of the space of essentially time and band-limited signals.” Bell Syst. Tech. J. 41, 1295–1336 (1962). The consequences of these results for vision was first set out by Buchsbaum and Gottschalk.11
[CrossRef]

J. Exp. Psychol. (1)

H. Helson, “Fundamental problems in color vision. I. The principle governing changes in hue saturation and lightness of non-selective samples in chromatic illumination,” J. Exp. Psychol. 23, 439–476 (1938).
[CrossRef]

J. Franklin Inst. (1)

G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

J. Opt Soc. Am (1)

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

J. Opt. Soc. Am. (5)

J. Theor. Biol. (2)

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,” J. Theor. Biol. 71, 473–478 (1978).
[CrossRef] [PubMed]

M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,” J. Theor. Biol. 78, 305–308 (1979).
[CrossRef] [PubMed]

Psychon. Sci. (1)

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).

Vision Res. (1)

H. B. Barlow, “What causes trichromacy? A theoretical analysis using comb-filtered spectra,” Vision Res. 22, 635–643 (1982).
[CrossRef] [PubMed]

Other (23)

D. L. MacAdam, Color Measurement; Theme and Variations (Springer-Verlag, Berlin, 1981), p. 128.

J. N. Lythgoe, The Ecology of Vision (Clarendon, Oxford, 1979), p. 31.

H. Yilmaz, “Color vision and a new approach to color perception,” in Biological Prototypes and Synthetic Systems (Plenum, New York, 1962), pp. 126–141.
[CrossRef]

J. M. Tenenbaum, Ph.D. dissertation (Stanford University, Stanford, Calif., 1971).

P. Sallstrom, “Colour and physics: some remarks concerning the physical aspects of human colour vision,” Inst. Phys. Rep. no. 73-09 (Institute of Physics, University of Stockholm, 1973).

E. L. Krinov, Spectral Reflectance Properties of Natural Formations (National Research Council of Canada, Ottawa, technical translation: TT-439, 1947). Krinov reports 370 surface spectral reflectances. The 337 analyzed were those that had complete and legible data available from 400 to 650 nm.

The analyses reported below were repeated for the full range of Munsell data, 380 to 770 nm, inclusive. The analyses were also repeated after each surface spectral reflectance was transformed from equal-spaced wavelength measurements to equal-spaced wave-number measurements by 4-point Lagrangean interpolation as described by Barlow. The results for both these analyses are qualitatively similar to the results reported here and are not reported.

Regression on the first three characteristic vectors derived from the Munsell data directly produced similar results.

K. Nassau, The Physics and Chemistry of Color: The Fifteen Causes of Color (Wiley, New York, 1983).

H. Suzuki, Electronic Absorption Spectra and Geometry of Organic Molecules; An Application of Molecular Orbital Theory (Academic, New York, 1967), p. 79.

W. Kauzmann, Quantum Chemistry (Academic, New York, 1957), pp. 582 and 669.

W. Kauzmann, ibid., pp. 669–670.

P. Pringsheim, Fluorescence and Phosphorescence (Interscience, New York, 1949).

T. H. Wonnacott, R. J. Wonnacott, Regression: A Second Course in Statistics (Wiley, New York, 1981).

G. Wyszecki, W. S. Stiles, Color Science; Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982), p. 256.

Barlow performs his analysis with color signals plotted against wave number, not wavelength. To avoid confusion, I continue to use wavelength as the independent variable. A change to wave number does not substantially affect the following argument.

The spectrum of the light is convolved with the spectrum of the surface reflectance. See R. N. Bracewell, The Fourier Transform and Its Applications., 2nd ed. (McGraw-Hill, New York, 1978), Chap. 3. It is easily shown that the highest frequency present in a convolution is the sum of the two highest frequencies present in the functions being convolved.

Barlow uses terahertz rather than inverse centimeters. The conversion factor is 33.33. Barlow assumes that the width of the visible spectrum is W= 228 THz. Then, using the result of Slepian et al., N= 2WF becomes N= 0.456F, where N is the number of samples required to capture the essentially frequency limited functions with cutoff F in cycles/1000 THz. Note that Barlow’s computations use a different formula based on the Shannon–Whittaker Theorem that gives N= 2WF+ 1.

See D. A. Belsey, E. Kuh, R. W. Welsch, Regression Diagnostics; Identifying Influential Data and Sources of Collinearity (Wiley, New York, 1980).
[CrossRef]

The analyses reported in the text were also carried out on the unnormalized data. Comparison of the fits obtained on normalized and unnormalized data showed that normalized data produced worse fits in all cases. In effect, the penalty associated with failing to fit surface reflectances with small entries is reduced by neglecting to normalize.

Also known as principal-components analysis or the Karhunen–Loeve decomposition. See K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, London, 1979). All characteristic vector analyses and multiple regressions were done in AT&T Bell Laboratories’ statistical language S. See R. A. Becker, J. M. Chambers, S: An Interactive Environment for Data Analysis and Graphics (Wadsworth, Belmont, Calif., 1984).

See R. B. Blackman, J. W. Tukey, The Measurement of Power Spectra from the Point of View of Communications Engineering (Dover, New York; 1959); P. Bloomfield, Fourier Analysis of Time Series: An Introduction (Wiley, New York, 1976).

L. T. Maloney, “Computational approaches to color constancy,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1984).

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Figures (3)

Fig. 1
Fig. 1

Naturally occurring surface reflectances. The two best and two worst fits obtained in fitting Krinov’s 337 surface spectral reflectances (circles) by Cohen’s first three characteristic vectors. The fitted function is shown as a solid line. Each x axis plots wavelength in nanometers; each y axis plots proportion of light reflected. Krinov’s identifying information for the surface and the values of R2 for the multiple-regression fit are given for each surface reflectance.

Fig. 2
Fig. 2

Light, surface reflectance, and “color signal.” (a)–(f) Show power spectra. The x axis is in arbitrary units of frequency. The y axes are in arbitrary units as well. The power spectrum of an illuminant shown in (a) lies between −3 and 3 and is therefore band limited. The power spectrum of a surface reflectance is also band limited to the range −3 to 3. The resulting “color signal,” the product of the light and surface reflectance, is shown in (c). It is also band limited, but its limits are the sums of the limits of the light and surface reflectance, −6 to 6. (d)–(f) Illustrate the same computation for a similar illuminant and surface reflectance that are the sum of two components: a band-limited component identical to (a) and (b), respectively, and additional high frequencies above the band limit. Contributions to the power spectrum of the resulting color signal that lie above the −6 to 6 band limit come only from the interaction of the non-band-limited components of light and surface reflectance.

Fig. 3
Fig. 3

The effect of windowing and sampling. If the constant function shown in (a) is sampled at 10-nm intervals from 400 to 650 nm, inclusive (the sampling procedure used by Krinov), then the computed power spectrum is that shown in (b). The power spectrum of a constant function that is neither windowed nor sampled is zero, except at the origin. The broadening of the power spectrum is a consequence of windowing and sampling.

Tables (7)

Tables Icon

Table 1 Proportion of Variance Accounted for by a Linear Model with 2–6 Parametersa

Tables Icon

Table 2 Distribution of 462 R2 Multiple Correlation Coefficients for Nickerson’s Munsell Surface Spectral-Reflectance Data Fitted to Linear Models Based on 2–6 Characteristic Vectorsa

Tables Icon

Table 3 Proportion of Variance Accounted for by a Linear Model with 2–6 Parametersa

Tables Icon

Table 4 Frequency Limits for Nickerson’s Munsell Surface-Reflectance Dataa

Tables Icon

Table 5 Frequency Limits for Krinov’s Surface Reflectancesa

Tables Icon

Table 6 Proportion of Variance Accounted for by a Linear Model with 2–6 Parametersa

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Table 7 Distribution of 462 R2 Multiple Correlation Coefficients for Nickerson’s Munsell Surface Spectral-Reflectance Data Fitted to Linear Models Based on 2–6 Characteristic Vectorsa

Equations (5)

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S ( λ ) = σ 1 S 1 ( λ ) + σ 2 S 2 ( λ ) + + σ n S n ( λ ) ,
| S S σ | 2 = l { S ( l ) [ σ 1 S 1 ( l ) + σ 2 S 2 ( l ) + + σ n S n ( l ) ] } 2 ,
i = 1 N | S i S σ i | 2 .
| S S σ | W 2 = l w l { S ( l ) [ σ 1 S 1 ( l ) + σ 2 S 2 ( l ) + + σ n S n ( l ) ] } 2 .
i = l N | S N S σ N | W 2

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