Abstract

We calculate the fast Fourier transforms of the color-matching functions for the CIE 1931 standard observer with reference to the XYZ primaries. From an analysis of the Fourier-transform moduli thus obtained, we were then able to study the sampling theorem to get mathematical formulas that lead to the reconstruction of the color-matching functions at limiting frequencies of 0.02 and 0.05 cycle/nm. This reconstruction proves to be highly reliable at a sampling interval of 10 nm and perfectly acceptable at 25 nm and even wider intervals.

© 1992 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).
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  3. L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A 3, 1673–1683 (1986).
    [CrossRef] [PubMed]
  4. H. B. Barlow, “What causes trichromacy? A theoretical analysis using comb-filtered spectra,” Vision Res. 22, 635–643 (1982).
    [CrossRef] [PubMed]
  5. T. Benzschawel, M. H. Brill, T. E. Cohn, “Analysis of human mechanisms using sinusoidal spectral power distributions,” J. Opt. Soc. Am. A 3, 1713–1725 (1986).
    [CrossRef] [PubMed]
  6. J. W. Goodman, Introduction a l’Optique de Fourier et a l’Holographie (Masson, Paris, 1972), pp. 20–24.
  7. E. I. Stearns, “The determination of weights for use in calculating tristimulus values,” Color Res. Appl. 6, 210–212 (1981).
    [CrossRef]
  8. W. Erb, M. Krystek, “Truncation error in colorimetric computations,” Color Res. Appl. 8, 17–22 (1983).
    [CrossRef]
  9. G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1982).

1986 (2)

1984 (1)

1983 (1)

W. Erb, M. Krystek, “Truncation error in colorimetric computations,” Color Res. Appl. 8, 17–22 (1983).
[CrossRef]

1982 (1)

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

1981 (1)

E. I. Stearns, “The determination of weights for use in calculating tristimulus values,” Color Res. Appl. 6, 210–212 (1981).
[CrossRef]

1977 (1)

Barlow, H. B.

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

Benzschawel, T.

Brill, M. H.

Buchsbaum, G.

Cohn, T. E.

Erb, W.

W. Erb, M. Krystek, “Truncation error in colorimetric computations,” Color Res. Appl. 8, 17–22 (1983).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction a l’Optique de Fourier et a l’Holographie (Masson, Paris, 1972), pp. 20–24.

Gottschalk, A.

Krystek, M.

W. Erb, M. Krystek, “Truncation error in colorimetric computations,” Color Res. Appl. 8, 17–22 (1983).
[CrossRef]

Maloney, L. T.

Ohta, N.

Stearns, E. I.

E. I. Stearns, “The determination of weights for use in calculating tristimulus values,” Color Res. Appl. 6, 210–212 (1981).
[CrossRef]

Stiles, W. S.

Wyszecki, G.

Color Res. Appl. (2)

E. I. Stearns, “The determination of weights for use in calculating tristimulus values,” Color Res. Appl. 6, 210–212 (1981).
[CrossRef]

W. Erb, M. Krystek, “Truncation error in colorimetric computations,” Color Res. Appl. 8, 17–22 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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 (2)

J. W. Goodman, Introduction a l’Optique de Fourier et a l’Holographie (Masson, Paris, 1972), pp. 20–24.

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

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Figures (4)

Fig. 1
Fig. 1

FFT modulus for different CIE 1931 standard-observer color-matching functions. Ordinates are in arbitrary units.

Fig. 2
Fig. 2

Original x ¯ λ (dotted curve) and its reconstruction sampled at intervals of 25 nm (solid curve).

Fig. 3
Fig. 3

Originaly y ¯ λ (dotted curve) and its reconstruction sampled at intervals of 50 nm (solid curve).

Fig. 4
Fig. 4

Original z ¯ λ (dotted curve) and its reconstruction sampled at intervals of 25 nm (solid curve).

Tables (4)

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Table 1 Original x ¯ λ and Its Reconstruction Sampled at Two Intervals

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Table 2 Original y ¯ λ and Its Reconstruction Sampled at Two Intervals

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Table 3 Original z ¯ λ and Its Reconstruction Sampled at Two Intervals

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Table 4 Mean Absolute Error 〈d〉 and Relative Error 〈r〉 for Two Reconstructions of the Color-Matching Functions Analyzeda

Equations (5)

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M ( λ ) = n = - + M ( n 2 f 1 ) sinc [ 2 f 1 ( λ - n 2 f 1 ) ] ,
M ( λ ) = n = 15 30 M ( n 2 f 1 ) sinc [ 2 f 1 ( λ - n 2 f 1 ) ]
M ( λ ) = n = 36 75 M ( n 2 f 1 ) sinc [ 2 f 1 ( λ - n 2 f 1 ) ] ,
d = 1 N vis M ( λ ) - M ( λ ) ,
r = d / 1 N vis M ( λ ) ,

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