Abstract

We compute the gamut of coordinates in the Commission Internationale de l’Eclairage chromaticity diagram that correspond to positive frequency-limited functions having only three degrees of freedom (a trichromatic system). The results suggest that practical colors are either suitably frequency limited to be represented by three samples or have suitably frequency-limited metamers.

© 1984 Optical Society of America

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References

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  1. 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]
  2. R. G. Gallager, Information Theory and Reliable Communication (Wiley, New York, 1968).
  3. 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]
  4. H. Wolter, “Physikalische Begründung eines Farbenkreises und Ansätze einer physikalischen Farbenlehre,” Ann. Phys. (Leipzig), Ser. 6, 8, 11–29 (1950); M. H. Brill, T. Benzschawel, “Spectral-phase modulation transfer function as a test for color vision,” J. Opt. Soc. Am. 72, 1741 (A) (1982).
  5. G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).
  6. H. B. Barlow, “What causes trichromacy? A theoretical analysis using comb-filtered spectra,” Vision Res. 22, 635–644 (1982).
    [CrossRef] [PubMed]
  7. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
  8. D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 53, 1031–1040 (1964).
    [CrossRef]

1983 (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]

1982 (1)

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

1977 (1)

1964 (2)

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

D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 53, 1031–1040 (1964).
[CrossRef]

1950 (1)

H. Wolter, “Physikalische Begründung eines Farbenkreises und Ansätze einer physikalischen Farbenlehre,” Ann. Phys. (Leipzig), Ser. 6, 8, 11–29 (1950); M. H. Brill, T. Benzschawel, “Spectral-phase modulation transfer function as a test for color vision,” J. Opt. Soc. Am. 72, 1741 (A) (1982).

Barlow, H. B.

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

Buchsbaum, G.

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]

Cohen, J.

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

Gallager, R. G.

R. G. Gallager, Information Theory and Reliable Communication (Wiley, New York, 1968).

Gottschalk, A.

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]

Judd, D. B.

D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 53, 1031–1040 (1964).
[CrossRef]

MacAdam, D. L.

D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 53, 1031–1040 (1964).
[CrossRef]

Ohta, N.

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).

Wolter, H.

H. Wolter, “Physikalische Begründung eines Farbenkreises und Ansätze einer physikalischen Farbenlehre,” Ann. Phys. (Leipzig), Ser. 6, 8, 11–29 (1950); M. H. Brill, T. Benzschawel, “Spectral-phase modulation transfer function as a test for color vision,” J. Opt. Soc. Am. 72, 1741 (A) (1982).

Wyszecki, G.

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]

D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 53, 1031–1040 (1964).
[CrossRef]

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

Ann. Phys. (Leipzig) (1)

H. Wolter, “Physikalische Begründung eines Farbenkreises und Ansätze einer physikalischen Farbenlehre,” Ann. Phys. (Leipzig), Ser. 6, 8, 11–29 (1950); M. H. Brill, T. Benzschawel, “Spectral-phase modulation transfer function as a test for color vision,” J. Opt. Soc. Am. 72, 1741 (A) (1982).

J. Opt. Soc. Am. (2)

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]

D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 53, 1031–1040 (1964).
[CrossRef]

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]

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, 635–644 (1982).
[CrossRef] [PubMed]

Other (2)

R. G. Gallager, Information Theory and Reliable Communication (Wiley, New York, 1968).

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

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Figures (3)

Fig. 1
Fig. 1

Shown on the CIE chromaticity diagram are the span of visual color spectra (functions of wavelength) whose Fourier transforms are frequency limited by 0.005 cycle/nm (outer solid curve), the NTSC color-television standard primaries (triangle vertices), and the gamut of available Munsell samples (dashed curve). The Munsell samples shown are, starting at the arrow, 10RP3/10, 5RP3/10, 10P3/10, 5P3/10, 10PB3/10, 5PB3/12, 10B3/8, 5B4/8, 5BG5/8, 10G5/8, 5G5/8, 10GY6/10, 5GY7/10, 10Y7/8, 5Y8/12, 10YR6/10, 5YR6/12, 10R5/10, and 5R4/14.

Fig. 2
Fig. 2

Shown on the CIE chromaticity diagram is the span of color coordinates that are metameric to a representation by function of a given Fourier frequency in Eq. (3). Frequencies are given in cycles per nanometer. The triangles are the NTSC (American) and PAL (European) standards for color television.

Fig. 3
Fig. 3

Fourier-frequency spectrum (magnitude) of the color-matching function. The transform was performed digitally using 1-nm samples from 380 to 700 nm. The ordinate is linearly scaled.

Equations (4)

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n = ( 2 B T ) + 1 ,
( 2 B T ) + 1 = 3.
S ( λ ) = 1 + m sin ( 2 π f λ + ϕ ) ,
0 m 1 , 0 ϕ 2 π , 0 f 0.005 cycle / nm .

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