Abstract

Ideal-observer analysis was used to determine the number of basis functions necessary to represent the spectral reflectances of natural objects. The ideal observer was placed at the level of photon catch in the foveal photoreceptors of a typical human eye. Daylight illumination was used. Discrimination of an original reflectance spectrum from its approximation typically reached an asymptote after three or four basis functions were used. A consideration of various factors involved in real rather than ideal performance led to the conclusion that three basis functions are necessary and probably sufficient for representing the spectral reflectance functions of natural objects.

© 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).
    [Crossref]
  2. L. 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]
  3. See also samples of rock in G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1967), Fig. 1.56.
  4. M. Brill, “A device performing illuminant-invariant assessment of chromatic relations,”J. Theor. Biol. 71, 473–478 (1978).
    [Crossref] [PubMed]
  5. G. Buchsbaum, “A spatial processor model for object color perception,” J. Franklin Inst. 310, 1–26 (1980).
    [Crossref]
  6. L. Maloney, B. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
    [Crossref] [PubMed]
  7. E. L. Krinov, “Spectral’naye otrazhatel’naya sposobnost’prirodnykh obrazovanii,” Izd. Akad. Nauk USSR (Proc. Acad. Sci. USSR) (1947); translation by G. Belkov, “Spectral reflectance properties of natural formations,” Technical Translation TT-439 (National Research Council of Canada, Ottawa, Canada, 1953).
  8. W. Geisler, “Sequential ideal-observer analysis of visual discriminations,” Psychol. Rev. 96, 267–314 (1989).
    [Crossref] [PubMed]
  9. J. P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [Crossref]
  10. D. L. MacAdam, “Visual sensitivities to color differences in daylight,”J. Opt. Soc. Am. 32, 247–274 (1942).
    [Crossref]
  11. H. Barlow, “What causes trichromacy? A theoretical analysis using comb-filtered spectra,” Vision Res. 22, 635–643 (1982).
    [Crossref] [PubMed]
  12. The ideal observer was assumed to have perfect memory over the duration of the trial.
  13. P. L. Walraven, “A closer look at the tritanopic convergence point,” Vision Res. 14, 1339–1343 (1974).
    [Crossref] [PubMed]
  14. G. Osterberg, “Topography of the layer of rods and cones in the human retina,” Acta Ophthalmol. Kobenhaven Suppl. 6, 1–102 (1935).
  15. W. Miller, G. Bernard, “Averaging over the foveal receptor aperture curtails aliasing,” Vision Res. 23, 1365–1369 (1983).
    [Crossref] [PubMed]
  16. G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1967).
  17. V. C. Smith, J. Pokorny, “Spectral sensitivities of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
    [Crossref] [PubMed]
  18. The luminances of the following objects were recorded at approximately noon on May 24, 1990, in Madison, Wisconsin, under a slightly hazy, blue sky. Luminance values are in candelas per square meter. All the measurements, with the exception of the leaves and the grass, were taken with the photometer perpendicular to the reflecting object, which itself was attached to the wall of a building: light brown brick, 3260; black felt, 450; white paper, 12,500; red matte paper, 1700; blue matte paper, 1150; green tree leaves, 2400; and green grass, 2000.
  19. D. Judd, D. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,”J. Opt. Soc. Am. 54, 1031–1040 (1964).
    [Crossref]
  20. J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision. I. Basic concepts,” Vision Res. 12, 1327–1344 (1972).
    [Crossref] [PubMed]
  21. W. Geisler, University of Texas at Austin, Austin, Texas 78712 (personal communication, May19, 1991).
  22. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
  23. T. Jaaskelainen, J. Parkkinen, S. Toyooka, “Vector-subspace model for color representation,” J. Opt. Soc. Am. A 7, 725–730 (1990).
    [Crossref]
  24. M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1672 (1986).
    [Crossref]
  25. A. Valberg, B. Lange-Malecki, “‘Colour constancy’ in mondrian patterns: a partial cancellation of physical chromaticity shifts by simultaneous contrast,” Vision Res. 30, 371–380 (1990).
    [Crossref]

1990 (2)

T. Jaaskelainen, J. Parkkinen, S. Toyooka, “Vector-subspace model for color representation,” J. Opt. Soc. Am. A 7, 725–730 (1990).
[Crossref]

A. Valberg, B. Lange-Malecki, “‘Colour constancy’ in mondrian patterns: a partial cancellation of physical chromaticity shifts by simultaneous contrast,” Vision Res. 30, 371–380 (1990).
[Crossref]

1989 (2)

W. Geisler, “Sequential ideal-observer analysis of visual discriminations,” Psychol. Rev. 96, 267–314 (1989).
[Crossref] [PubMed]

J. P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
[Crossref]

1986 (3)

1983 (1)

W. Miller, G. Bernard, “Averaging over the foveal receptor aperture curtails aliasing,” Vision Res. 23, 1365–1369 (1983).
[Crossref] [PubMed]

1982 (1)

H. 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 color perception,” J. Franklin Inst. 310, 1–26 (1980).
[Crossref]

1978 (1)

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

1977 (1)

1975 (1)

V. C. Smith, J. Pokorny, “Spectral sensitivities of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[Crossref] [PubMed]

1974 (1)

P. L. Walraven, “A closer look at the tritanopic convergence point,” Vision Res. 14, 1339–1343 (1974).
[Crossref] [PubMed]

1972 (1)

J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision. I. Basic concepts,” Vision Res. 12, 1327–1344 (1972).
[Crossref] [PubMed]

1964 (2)

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

D. Judd, D. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,”J. Opt. Soc. Am. 54, 1031–1040 (1964).
[Crossref]

1947 (1)

E. L. Krinov, “Spectral’naye otrazhatel’naya sposobnost’prirodnykh obrazovanii,” Izd. Akad. Nauk USSR (Proc. Acad. Sci. USSR) (1947); translation by G. Belkov, “Spectral reflectance properties of natural formations,” Technical Translation TT-439 (National Research Council of Canada, Ottawa, Canada, 1953).

1942 (1)

1935 (1)

G. Osterberg, “Topography of the layer of rods and cones in the human retina,” Acta Ophthalmol. Kobenhaven Suppl. 6, 1–102 (1935).

Barlow, H.

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

Bernard, G.

W. Miller, G. Bernard, “Averaging over the foveal receptor aperture curtails aliasing,” Vision Res. 23, 1365–1369 (1983).
[Crossref] [PubMed]

Brill, M.

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

Buchsbaum, G.

G. Buchsbaum, “A spatial processor model for object color 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).

D’Zmura, M.

Geisler, W.

W. Geisler, “Sequential ideal-observer analysis of visual discriminations,” Psychol. Rev. 96, 267–314 (1989).
[Crossref] [PubMed]

W. Geisler, University of Texas at Austin, Austin, Texas 78712 (personal communication, May19, 1991).

Hallikainen, J.

Jaaskelainen, T.

Judd, D.

Krinov, E. L.

E. L. Krinov, “Spectral’naye otrazhatel’naya sposobnost’prirodnykh obrazovanii,” Izd. Akad. Nauk USSR (Proc. Acad. Sci. USSR) (1947); translation by G. Belkov, “Spectral reflectance properties of natural formations,” Technical Translation TT-439 (National Research Council of Canada, Ottawa, Canada, 1953).

Lange-Malecki, B.

A. Valberg, B. Lange-Malecki, “‘Colour constancy’ in mondrian patterns: a partial cancellation of physical chromaticity shifts by simultaneous contrast,” Vision Res. 30, 371–380 (1990).
[Crossref]

Lennie, P.

MacAdam, D.

MacAdam, D. L.

Maloney, L.

Miller, W.

W. Miller, G. Bernard, “Averaging over the foveal receptor aperture curtails aliasing,” Vision Res. 23, 1365–1369 (1983).
[Crossref] [PubMed]

Ohta, N.

Osterberg, G.

G. Osterberg, “Topography of the layer of rods and cones in the human retina,” Acta Ophthalmol. Kobenhaven Suppl. 6, 1–102 (1935).

Parkkinen, J.

Parkkinen, J. P. S.

Pokorny, J.

V. C. Smith, J. Pokorny, “Spectral sensitivities of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[Crossref] [PubMed]

Smith, V. C.

V. C. Smith, J. Pokorny, “Spectral sensitivities of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[Crossref] [PubMed]

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 (Wiley, New York, 1967).

See also samples of rock in G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1967), Fig. 1.56.

Toyooka, S.

Valberg, A.

A. Valberg, B. Lange-Malecki, “‘Colour constancy’ in mondrian patterns: a partial cancellation of physical chromaticity shifts by simultaneous contrast,” Vision Res. 30, 371–380 (1990).
[Crossref]

Vos, J. J.

J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision. I. Basic concepts,” Vision Res. 12, 1327–1344 (1972).
[Crossref] [PubMed]

Walraven, P. L.

P. L. Walraven, “A closer look at the tritanopic convergence point,” Vision Res. 14, 1339–1343 (1974).
[Crossref] [PubMed]

J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision. I. Basic concepts,” Vision Res. 12, 1327–1344 (1972).
[Crossref] [PubMed]

Wandell, B.

Wyszecki, G.

Acta Ophthalmol. Kobenhaven Suppl. (1)

G. Osterberg, “Topography of the layer of rods and cones in the human retina,” Acta Ophthalmol. Kobenhaven Suppl. 6, 1–102 (1935).

Izd. Akad. Nauk USSR (Proc. Acad. Sci. USSR) (1)

E. L. Krinov, “Spectral’naye otrazhatel’naya sposobnost’prirodnykh obrazovanii,” Izd. Akad. Nauk USSR (Proc. Acad. Sci. USSR) (1947); translation by G. Belkov, “Spectral reflectance properties of natural formations,” Technical Translation TT-439 (National Research Council of Canada, Ottawa, Canada, 1953).

J. Franklin Inst. (1)

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

J. Theor. Biol. (1)

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

Psychol. Rev. (1)

W. Geisler, “Sequential ideal-observer analysis of visual discriminations,” Psychol. Rev. 96, 267–314 (1989).
[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. (6)

P. L. Walraven, “A closer look at the tritanopic convergence point,” Vision Res. 14, 1339–1343 (1974).
[Crossref] [PubMed]

A. Valberg, B. Lange-Malecki, “‘Colour constancy’ in mondrian patterns: a partial cancellation of physical chromaticity shifts by simultaneous contrast,” Vision Res. 30, 371–380 (1990).
[Crossref]

J. J. Vos, P. L. Walraven, “An analytical description of the line element in the zone-fluctuation model of colour vision. I. Basic concepts,” Vision Res. 12, 1327–1344 (1972).
[Crossref] [PubMed]

V. C. Smith, J. Pokorny, “Spectral sensitivities of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[Crossref] [PubMed]

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

W. Miller, G. Bernard, “Averaging over the foveal receptor aperture curtails aliasing,” Vision Res. 23, 1365–1369 (1983).
[Crossref] [PubMed]

Other (5)

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1967).

The ideal observer was assumed to have perfect memory over the duration of the trial.

The luminances of the following objects were recorded at approximately noon on May 24, 1990, in Madison, Wisconsin, under a slightly hazy, blue sky. Luminance values are in candelas per square meter. All the measurements, with the exception of the leaves and the grass, were taken with the photometer perpendicular to the reflecting object, which itself was attached to the wall of a building: light brown brick, 3260; black felt, 450; white paper, 12,500; red matte paper, 1700; blue matte paper, 1150; green tree leaves, 2400; and green grass, 2000.

W. Geisler, University of Texas at Austin, Austin, Texas 78712 (personal communication, May19, 1991).

See also samples of rock in G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1967), Fig. 1.56.

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Figures (6)

Fig. 1
Fig. 1

Spectral power distribution for a 6500-K daylight illuminant.

Fig. 2
Fig. 2

Original reflectance spectrum (solid curve, #3 from Krinov’s collection7) and its least-squares approximation (dashed curve) based on the number of basis functions shown in each panel.

Fig. 3
Fig. 3

Average value of d′ plotted as a function of the number of basis functions used to represent Krinov’s original reflectance spectra (n = 337).

Fig. 4
Fig. 4

Frequency distribution showing the number of basis functions required in an approximation to produce a d′ of 1.00 or lower for the ideal observer’s discrimination of the approximation from the original spectrum.

Fig. 5
Fig. 5

Photon catches for the three different photoreceptor types as a function of the phase of various frequency basis functions. A phase of 0.0 rad corresponds to sine phase at 400 nm. Frequencies of 1–7 cycles/250 nm were used. Each of these frequencies was added alone to a constant reflectance of 0.13 across wavelength with the amplitude shown for that frequency in Fig. 6. The phase was changed in steps of 0.125π rad. These artificial reflectance spectra thus consisted of a mean reflectance across wavelength to which was added a single frequency of sinusoidal variation in different phases. The photon catch at each phase was determined after these synthesized spectra were illuminated with 6500-K daylight to 6400 Td. The photon catches for a frequency of 1 cycle/250 nm are shown by the solid curves, with a prominent sinusoidal variation in each plot. Increasing frequencies are shown by the other curves; the amplitude of sinusoidal variation decreases monotonically with increasing frequency.

Fig. 6
Fig. 6

Average amplitude and standard deviation plotted as a function of the frequency of the basis function for the 337 reflectance functions in Krinov’s collection. For example, the average value for a frequency of 1 cycle/250 nm corresponds to the average modulus of the coefficients for the second (sine) and the third (cosine) basis functions, while the value for a frequency of 2 cycles/250 nm corresponds to the average modulus of the coefficients for the fourth and the fifth basis functions. The first basis function has a frequency of 0 cycles/250 nm.

Tables (2)

Tables Icon

Table 1 Ratios (Original Object: Approximation) of Standard Deviations (Photons Receptor−1 100 ms−1)

Tables Icon

Table 2 Terms Used To Compute Photon Catches

Equations (16)

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d = ( n r Δ R 2 / R + n g Δ G 2 / G + n b Δ B 2 / B ) 1 / 2 .
L = 680 E ( λ ) S ( λ ) V ( λ ) Δ λ ,
I = 680 π ( a / 2 ) 2 E ( λ ) S ( λ ) V ( λ ) Δ λ .
I 0 = 680 k 1 π ( a / 2 ) 2 E ( λ ) S 1 ( λ ) V ( λ ) Δ λ .
k X = I 0 [ 170 π a 2 E ( λ ) S X ( λ ) V ( λ ) Δ λ ] - 1 .
q ( λ ) = k E ( λ ) S ( λ )             ( W m - 2 sr - 1 nm - 1 ) .
q ( λ ) = π ( a / 2 ) 2 ( 10 - 3 ) 2 k E ( λ ) S ( λ )             ( W sr - 1 nm - 1 ) .
q ( λ ) = 10 - 6 π ( a / 2 ) 2 k E ( λ ) S ( λ ) 278.3 × 10 - 6             ( W m - 2 nm - 1 ) .
q r ( λ ) = π ( a / 2 ) 2 k E ( λ ) S ( λ ) t ( λ ) r ( λ ) 278.3             ( W m - 2 nm - 1 ) .
q r ( λ ) = 10 7 π ( a / 2 ) 2 k E ( λ ) S ( λ ) t ( λ ) r ( λ ) 278.3             ( ergs s - 1 m - 2 nm - 1 ) .
q r ( λ ) = 10 7 π ( a / 2 ) 2 k E ( λ ) S ( λ ) t ( λ ) r ( λ ) λ 278.3 × 6.626 × 10 - 27 × 3 × 10 6             ( photons s - 1 m - 2 nm - 1 ) .
q r ( λ ) = 10 22 π a 2 d k E ( λ ) S ( λ ) t ( λ ) r ( λ ) λ 2.21             ( photons m - 2 nm - 1 ) .
q r ( λ ) = 10 22 π a 2 d π ( e / 2 ) 2 ( 10 - 6 ) k E ( λ ) S ( λ ) t ( λ ) r ( λ ) λ 2.21             ( photons receptor - 1 nm - 1 ) .
q r ( λ ) = 1.13 ( 10 15 ) π 2 a 2 e 2 d k E ( λ ) S ( λ ) t ( λ ) r ( λ ) λ             ( photons receptor - 1 nm - 1 ) .
q r = 1.13 ( 10 15 ) π 2 a 2 e 2 d k E ( λ ) S ( λ ) t ( λ ) r ( λ ) λ Δ λ             ( photons receptor - 1 ) .
q r = 1.13 ( 10 6 ) π 2 a 2 e 2 d k E ( λ ) S ( λ ) t ( λ ) r ( λ ) λ Δ λ             ( photons receptor - 1 ) .

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