Abstract

Natural illuminant and reflectance spectra can be roughly approximated by a linear model with as few as three basis functions, and this has suggested that the visual system might construct a linear representation of the spectra by estimating the weights of these functions. However, such models do not accommodate nonlinearities in color appearance, such as the Abney effect. Previously, we found that these nonlinearities are qualitatively consistent with a perceptual inference that stimulus spectra are instead roughly Gaussian, with the hue tied to the inferred centroid of the spectrum [J. Vision 6(9), 12 (2006)]. Here, we examined to what extent a Gaussian inference provides a sufficient approximation of natural color signals. Reflectance and illuminant spectra from a wide set of databases were analyzed to test how well the curves could be fit by either a simple Gaussian with three parameters (amplitude, peak wavelength, and standard deviation) versus the first three principal component analysis components of standard linear models. The resulting Gaussian fits were comparable to linear models with the same degrees of freedom, suggesting that the Gaussian model could provide a plausible perceptual assumption about stimulus spectra for a trichromatic visual system.

© 2012 Optical Society of America

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2011 (1)

D. H. Foster, “Color constancy,” Vision Res. 51, 674-700 (2011).
[CrossRef]

2009 (1)

A. D. Logvinenko, “An object-color space,” J. Vision 9(11), 5 (2009).
[CrossRef]

2008 (1)

S. K. Shevell and F. A. Kingdom, “Color in complex scenes,” Ann. Rev. Psychol. 59, 143-166 (2008).
[CrossRef]

2007 (2)

R. D. Pridmore, “Effect of purity on hue (Abney effect) in various conditions,” Color Res. Appl. 32, 25-39 (2007).
[CrossRef]

M. A. Webster, Y. Mizokami, and S. M. Webster, “Seasonal variations in the color statistics of natural images,” Network 18, 213-233 (2007).
[CrossRef] [PubMed]

2006 (2)

O. Kohonen, J. Parkkinen, and T. Jääskeläinen, “Databases for spectral color science,” Color Res. Appl. 31, 381-390 (2006).
[CrossRef]

Y. Mizokami, J. S. Werner, M. A. Crognale, and M. A. Webster, “Nonlinearities in color coding: compensating color appearance for the eye's spectral sensitivity,” J. Vision 6(9), 12(2006).
[CrossRef]

2005 (2)

S. M. Nascimento, D. H. Foster, and K. Amano, “Psychophysical estimates of the number of spectral-reflectance basis functions needed to reproduce natural scenes,” J. Opt. Soc. Am. A 22, 1017-1022 (2005).
[CrossRef]

H. E. Smithson, “Sensory, computational and cognitive components of human colour constancy,” Philos. Trans. R. Soc. London B 360, 1329-1346 (2005).
[CrossRef]

2004 (1)

D. H. Foster, S. M. Nascimento, and K. Amano, “Information limits on neural identification of colored surfaces in natural scenes,” Visual Neurosci. 21, 331-336 (2004).
[CrossRef]

2003 (2)

D. I. A. MacLeod and J. Golz, “A computational analysis of colour constancy,” in Colour Perception: Mind and the Physical World, R.Mausfeld and D.Heyer, eds. (Oxford University, 2003), pp. 205-242.

L. T. Maloney, “The importance of realistic models of surface and light in the study of human color vision,” in Colour Perception: Mind and the Physical World, R.Mausfeld and D.Heyer, eds. (Oxford University, 2003), pp. 243-246.

2002 (2)

2000 (1)

1999 (1)

L. T. Maloney, “Physics-based approaches to modeling surface color perception,” in Color Vision: From Genes to Perception, K.R.Gegenfurtner and L.T.Sharpe, eds. (Cambridge Univ., 1999), pp. 387-416.

1998 (1)

A. Hurlbert, “Computational models of color constancy,” in Perceptual Constancies, V.Walsh and J.J.Kulikowski, eds. (Cambridge Univ., 1998), pp. 283-322.

1997 (1)

S. M. Nascimento and D. H. Foster, “Detecting natural changes of cone-excitation ratios in simple and complex coloured images,” Proc. Biol. Sci. 264, 1395-1402 (1997).
[CrossRef] [PubMed]

1995 (3)

M. A. Webster and J. D. Mollon, “Colour constancy influenced by contrast adaptation,” Nature 373, 694-698 (1995).
[CrossRef] [PubMed]

E. J. Chichilnisky and B. A. Wandell, “Photoreceptor sensitivity changes explain color appearance shifts induced by large uniform backgrounds in dichoptic matching,” Vision Res. 35, 239-254 (1995).
[CrossRef] [PubMed]

T. D. Kulp and K. Fuld, “The prediction of hue and saturation for non-spectral lights,” Vision Res. 35, 2967-2983 (1995).
[CrossRef] [PubMed]

1992 (3)

1989 (1)

1987 (1)

1986 (2)

1984 (2)

S. A. Burns, A. E. Elsner, J. Pokorny, and V. C. Smith, “The Abney effect: chromaticity coordinates of unique and other constant hues,” Vision Res. 24, 479-489 (1984).
[CrossRef] [PubMed]

W. Kurtenbach, C. E. Sternheim, and L. Spillmann, “Change in hue of spectral colors by dilution with white light (Abney effect),” J. Opt. Soc. Am. 1, 365-372 (1984).
[CrossRef]

1977 (1)

1964 (2)

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

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

1909 (1)

W. de W. Abney, “On the change in hue of spectrum colors by dilution with white light,” Proc. R. Soc. London Ser. A 83, 120-127 (1909).
[CrossRef]

Abney, W. de W.

W. de W. Abney, “On the change in hue of spectrum colors by dilution with white light,” Proc. R. Soc. London Ser. A 83, 120-127 (1909).
[CrossRef]

Amano, K.

S. M. Nascimento, D. H. Foster, and K. Amano, “Psychophysical estimates of the number of spectral-reflectance basis functions needed to reproduce natural scenes,” J. Opt. Soc. Am. A 22, 1017-1022 (2005).
[CrossRef]

D. H. Foster, S. M. Nascimento, and K. Amano, “Information limits on neural identification of colored surfaces in natural scenes,” Visual Neurosci. 21, 331-336 (2004).
[CrossRef]

Ayama, M.

Bonnardel, V.

Brainard, D. H.

Burns, S. A.

S. A. Burns, A. E. Elsner, J. Pokorny, and V. C. Smith, “The Abney effect: chromaticity coordinates of unique and other constant hues,” Vision Res. 24, 479-489 (1984).
[CrossRef] [PubMed]

Chichilnisky, E. J.

E. J. Chichilnisky and B. A. Wandell, “Photoreceptor sensitivity changes explain color appearance shifts induced by large uniform backgrounds in dichoptic matching,” Vision Res. 35, 239-254 (1995).
[CrossRef] [PubMed]

Cohen, J.

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

Crognale, M. A.

Y. Mizokami, J. S. Werner, M. A. Crognale, and M. A. Webster, “Nonlinearities in color coding: compensating color appearance for the eye's spectral sensitivity,” J. Vision 6(9), 12(2006).
[CrossRef]

Dannemiller, J. L.

D'Zmura, M.

Elsner, A. E.

S. A. Burns, A. E. Elsner, J. Pokorny, and V. C. Smith, “The Abney effect: chromaticity coordinates of unique and other constant hues,” Vision Res. 24, 479-489 (1984).
[CrossRef] [PubMed]

Ferreira, F. P.

Foster, D. H.

D. H. Foster, “Color constancy,” Vision Res. 51, 674-700 (2011).
[CrossRef]

S. M. Nascimento, D. H. Foster, and K. Amano, “Psychophysical estimates of the number of spectral-reflectance basis functions needed to reproduce natural scenes,” J. Opt. Soc. Am. A 22, 1017-1022 (2005).
[CrossRef]

D. H. Foster, S. M. Nascimento, and K. Amano, “Information limits on neural identification of colored surfaces in natural scenes,” Visual Neurosci. 21, 331-336 (2004).
[CrossRef]

S. M. Nascimento, F. P. Ferreira, and D. H. Foster, “Statistics of spatial cone-excitation ratios in natural scenes,” J. Opt. Soc. Am. A 19, 1484-1490 (2002).
[CrossRef]

S. M. Nascimento and D. H. Foster, “Detecting natural changes of cone-excitation ratios in simple and complex coloured images,” Proc. Biol. Sci. 264, 1395-1402 (1997).
[CrossRef] [PubMed]

Fuld, K.

T. D. Kulp and K. Fuld, “The prediction of hue and saturation for non-spectral lights,” Vision Res. 35, 2967-2983 (1995).
[CrossRef] [PubMed]

Golz, J.

D. I. A. MacLeod and J. Golz, “A computational analysis of colour constancy,” in Colour Perception: Mind and the Physical World, R.Mausfeld and D.Heyer, eds. (Oxford University, 2003), pp. 205-242.

J. Golz and D. I. MacLeod, “Influence of scene statistics on colour constancy,” Nature 415, 637-640 (2002).
[CrossRef] [PubMed]

Hallikainen, J.

Hurlbert, A.

A. Hurlbert, “Computational models of color constancy,” in Perceptual Constancies, V.Walsh and J.J.Kulikowski, eds. (Cambridge Univ., 1998), pp. 283-322.

Jaaskelainen, T.

Jääskeläinen, T.

O. Kohonen, J. Parkkinen, and T. Jääskeläinen, “Databases for spectral color science,” Color Res. Appl. 31, 381-390 (2006).
[CrossRef]

Judd, D. B.

Kaiser, P. K.

Kingdom, F. A.

S. K. Shevell and F. A. Kingdom, “Color in complex scenes,” Ann. Rev. Psychol. 59, 143-166 (2008).
[CrossRef]

Kohonen, O.

O. Kohonen, J. Parkkinen, and T. Jääskeläinen, “Databases for spectral color science,” Color Res. Appl. 31, 381-390 (2006).
[CrossRef]

Kulp, T. D.

T. D. Kulp and K. Fuld, “The prediction of hue and saturation for non-spectral lights,” Vision Res. 35, 2967-2983 (1995).
[CrossRef] [PubMed]

Kurtenbach, W.

W. Kurtenbach, C. E. Sternheim, and L. Spillmann, “Change in hue of spectral colors by dilution with white light (Abney effect),” J. Opt. Soc. Am. 1, 365-372 (1984).
[CrossRef]

Lennie, P.

Logvinenko, A. D.

A. D. Logvinenko, “An object-color space,” J. Vision 9(11), 5 (2009).
[CrossRef]

MacAdam, D. L.

MacLeod, D. I.

J. Golz and D. I. MacLeod, “Influence of scene statistics on colour constancy,” Nature 415, 637-640 (2002).
[CrossRef] [PubMed]

MacLeod, D. I. A.

D. I. A. MacLeod and J. Golz, “A computational analysis of colour constancy,” in Colour Perception: Mind and the Physical World, R.Mausfeld and D.Heyer, eds. (Oxford University, 2003), pp. 205-242.

Maloney, L. T.

L. T. Maloney, “The importance of realistic models of surface and light in the study of human color vision,” in Colour Perception: Mind and the Physical World, R.Mausfeld and D.Heyer, eds. (Oxford University, 2003), pp. 243-246.

V. Bonnardel and L. T. Maloney, “Daylight, biochrome surfaces, and human chromatic response in the Fourier domain,” J. Opt. Soc. Am. A 17, 677-686 (2000).
[CrossRef]

L. T. Maloney, “Physics-based approaches to modeling surface color perception,” in Color Vision: From Genes to Perception, K.R.Gegenfurtner and L.T.Sharpe, eds. (Cambridge Univ., 1999), pp. 387-416.

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]

Marimont, D. H.

Mizokami, Y.

M. A. Webster, Y. Mizokami, and S. M. Webster, “Seasonal variations in the color statistics of natural images,” Network 18, 213-233 (2007).
[CrossRef] [PubMed]

Y. Mizokami, J. S. Werner, M. A. Crognale, and M. A. Webster, “Nonlinearities in color coding: compensating color appearance for the eye's spectral sensitivity,” J. Vision 6(9), 12(2006).
[CrossRef]

Mollon, J. D.

M. A. Webster and J. D. Mollon, “Colour constancy influenced by contrast adaptation,” Nature 373, 694-698 (1995).
[CrossRef] [PubMed]

Nakatsue, T.

Nascimento, S. M.

S. M. Nascimento, D. H. Foster, and K. Amano, “Psychophysical estimates of the number of spectral-reflectance basis functions needed to reproduce natural scenes,” J. Opt. Soc. Am. A 22, 1017-1022 (2005).
[CrossRef]

D. H. Foster, S. M. Nascimento, and K. Amano, “Information limits on neural identification of colored surfaces in natural scenes,” Visual Neurosci. 21, 331-336 (2004).
[CrossRef]

S. M. Nascimento, F. P. Ferreira, and D. H. Foster, “Statistics of spatial cone-excitation ratios in natural scenes,” J. Opt. Soc. Am. A 19, 1484-1490 (2002).
[CrossRef]

S. M. Nascimento and D. H. Foster, “Detecting natural changes of cone-excitation ratios in simple and complex coloured images,” Proc. Biol. Sci. 264, 1395-1402 (1997).
[CrossRef] [PubMed]

Ohta, N.

Parkkinen, J.

O. Kohonen, J. Parkkinen, and T. Jääskeläinen, “Databases for spectral color science,” Color Res. Appl. 31, 381-390 (2006).
[CrossRef]

Parkkinen, J. P. S.

Pokorny, J.

S. A. Burns, A. E. Elsner, J. Pokorny, and V. C. Smith, “The Abney effect: chromaticity coordinates of unique and other constant hues,” Vision Res. 24, 479-489 (1984).
[CrossRef] [PubMed]

Pridmore, R. D.

R. D. Pridmore, “Effect of purity on hue (Abney effect) in various conditions,” Color Res. Appl. 32, 25-39 (2007).
[CrossRef]

Shevell, S. K.

S. K. Shevell and F. A. Kingdom, “Color in complex scenes,” Ann. Rev. Psychol. 59, 143-166 (2008).
[CrossRef]

Smith, V. C.

S. A. Burns, A. E. Elsner, J. Pokorny, and V. C. Smith, “The Abney effect: chromaticity coordinates of unique and other constant hues,” Vision Res. 24, 479-489 (1984).
[CrossRef] [PubMed]

Smithson, H. E.

H. E. Smithson, “Sensory, computational and cognitive components of human colour constancy,” Philos. Trans. R. Soc. London B 360, 1329-1346 (2005).
[CrossRef]

Spillmann, L.

W. Kurtenbach, C. E. Sternheim, and L. Spillmann, “Change in hue of spectral colors by dilution with white light (Abney effect),” J. Opt. Soc. Am. 1, 365-372 (1984).
[CrossRef]

Sternheim, C. E.

W. Kurtenbach, C. E. Sternheim, and L. Spillmann, “Change in hue of spectral colors by dilution with white light (Abney effect),” J. Opt. Soc. Am. 1, 365-372 (1984).
[CrossRef]

Stiles, W. S.

Wandell, B. A.

Webster, M. A.

M. A. Webster, Y. Mizokami, and S. M. Webster, “Seasonal variations in the color statistics of natural images,” Network 18, 213-233 (2007).
[CrossRef] [PubMed]

Y. Mizokami, J. S. Werner, M. A. Crognale, and M. A. Webster, “Nonlinearities in color coding: compensating color appearance for the eye's spectral sensitivity,” J. Vision 6(9), 12(2006).
[CrossRef]

M. A. Webster and J. D. Mollon, “Colour constancy influenced by contrast adaptation,” Nature 373, 694-698 (1995).
[CrossRef] [PubMed]

Webster, S. M.

M. A. Webster, Y. Mizokami, and S. M. Webster, “Seasonal variations in the color statistics of natural images,” Network 18, 213-233 (2007).
[CrossRef] [PubMed]

Werner, J. S.

Y. Mizokami, J. S. Werner, M. A. Crognale, and M. A. Webster, “Nonlinearities in color coding: compensating color appearance for the eye's spectral sensitivity,” J. Vision 6(9), 12(2006).
[CrossRef]

Wyszecki, G.

Ann. Rev. Psychol. (1)

S. K. Shevell and F. A. Kingdom, “Color in complex scenes,” Ann. Rev. Psychol. 59, 143-166 (2008).
[CrossRef]

Color Res. Appl. (2)

R. D. Pridmore, “Effect of purity on hue (Abney effect) in various conditions,” Color Res. Appl. 32, 25-39 (2007).
[CrossRef]

O. Kohonen, J. Parkkinen, and T. Jääskeläinen, “Databases for spectral color science,” Color Res. Appl. 31, 381-390 (2006).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (10)

V. Bonnardel and L. T. Maloney, “Daylight, biochrome surfaces, and human chromatic response in the Fourier domain,” J. Opt. Soc. Am. A 17, 677-686 (2000).
[CrossRef]

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]

D. H. Marimont and B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905-1913 (1992).
[CrossRef] [PubMed]

J. L. Dannemiller, “Spectral reflectance of natural objects: how many basis functions are necessary?” J. Opt. Soc. Am. A 9, 507-515 (1992).
[CrossRef]

S. M. Nascimento, D. H. Foster, and K. Amano, “Psychophysical estimates of the number of spectral-reflectance basis functions needed to reproduce natural scenes,” J. Opt. Soc. Am. A 22, 1017-1022 (2005).
[CrossRef]

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

M. D'Zmura and P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662-1672 (1986).
[CrossRef] [PubMed]

S. M. Nascimento, F. P. Ferreira, and D. H. Foster, “Statistics of spatial cone-excitation ratios in natural scenes,” J. Opt. Soc. Am. A 19, 1484-1490 (2002).
[CrossRef]

D. H. Brainard and B. A. Wandell, “Asymmetric color matching: how color appearance depends on the illuminant,” J. Opt. Soc. Am. A 9, 1433-1448 (1992).
[CrossRef] [PubMed]

M. Ayama, T. Nakatsue, and P. K. Kaiser, “Constant hue loci of unique and binary balanced hues at 10, 100, and 1000 Td,” J. Opt. Soc. Am. A 4, 1136-1144 (1987).
[CrossRef] [PubMed]

J. Vision (2)

Y. Mizokami, J. S. Werner, M. A. Crognale, and M. A. Webster, “Nonlinearities in color coding: compensating color appearance for the eye's spectral sensitivity,” J. Vision 6(9), 12(2006).
[CrossRef]

A. D. Logvinenko, “An object-color space,” J. Vision 9(11), 5 (2009).
[CrossRef]

Nature (2)

J. Golz and D. I. MacLeod, “Influence of scene statistics on colour constancy,” Nature 415, 637-640 (2002).
[CrossRef] [PubMed]

M. A. Webster and J. D. Mollon, “Colour constancy influenced by contrast adaptation,” Nature 373, 694-698 (1995).
[CrossRef] [PubMed]

Network (1)

M. A. Webster, Y. Mizokami, and S. M. Webster, “Seasonal variations in the color statistics of natural images,” Network 18, 213-233 (2007).
[CrossRef] [PubMed]

Philos. Trans. R. Soc. London B (1)

H. E. Smithson, “Sensory, computational and cognitive components of human colour constancy,” Philos. Trans. R. Soc. London B 360, 1329-1346 (2005).
[CrossRef]

Proc. Biol. Sci. (1)

S. M. Nascimento and D. H. Foster, “Detecting natural changes of cone-excitation ratios in simple and complex coloured images,” Proc. Biol. Sci. 264, 1395-1402 (1997).
[CrossRef] [PubMed]

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

W. de W. Abney, “On the change in hue of spectrum colors by dilution with white light,” Proc. R. Soc. London Ser. A 83, 120-127 (1909).
[CrossRef]

Psychonom. Sci. (1)

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

Vision Res. (4)

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[CrossRef] [PubMed]

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Other (7)

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Figures (8)

Fig. 1
Fig. 1

Gaussian or linear models for reflectance or illuminants and fits to the spectra.

Fig. 2
Fig. 2

Examples of best and worst fits to individual spectra from the Munsell set. Top, spectral reflectance of surfaces with the smallest and the largest rms between original and fitting curves. Left: Gaussian fit for reflectance data no. 972 ( rms = 0.0018 , thin line) and no. 364 ( rms = 0.12 , thick line). Right, linear fit for reflectance data no. 564 ( rms = 0.0015 , thin line) and no. 71 ( rms = 0.10 , thick line). Solid and dashed lines show original and fitting spectra, respectively. Bottom, best and worst fits by the criterion of Δ E ab . Left, Gaussian fit for reflectance data no. 292 ( Δ E ab = 0.49 , thin line) and no. 29 ( Δ E ab = 74 , thick line). Right, linear fit for reflectance data no. 650 ( Δ E ab = 0.096 , thin line) and no. 1058 ( Δ E ab = 33 , thick line).

Fig. 3
Fig. 3

Results of Gaussian and Linear fits to reflectance databases. Left panels, mean rms error; right panels, mean correlation coefficient (r). Error bars indicate standard deviation within each dataset. Significant differences between the linear and Gaussian fits are shown by the symbols above the bars [ * ( p < 0.05 ) , * * ( p < 0.01 ) , * * * ( p < 0.001 ) ].

Fig. 4
Fig. 4

Gaussian and linear fits to the spectra from individual hyperspectral images. Left column, mean rms; right column, mean correlation, r.

Fig. 5
Fig. 5

Gaussian and linear fits for illuminant spectra. Left, mean rms error; right, mean correlation.

Fig. 6
Fig. 6

Fits to the reflectance spectra with the illuminant basis functions and vice versa. Left, mean rms error; right, mean correlation.

Fig. 7
Fig. 7

Chromaticity coordinates for the actual Munsell spectra under illuminant C (left), or from the best-fitting Gaussian (middle) or linear (right) approximations. Coordinates are shown in the CIE 1931 x y chromaticity diagram (top) or the CIE 1976 a*b* plane (bottom).

Fig. 8
Fig. 8

(a) Gamut of chromaticities achievable with positive Gaussian spectra forming peaks (black) or troughs (red), sampled for peak intervals of 1 nm and standard deviation intervals of 5 nm . Gamut achievable with positive and negative Gaussian spectra (blue) is also shown. (b) Gamut for all-positive reflectance functions for the linear reflectance model (black) and the gamut achievable with positive and negative linear spectra (blue).

Tables (1)

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Table 1 Spectral Data for Fitting

Equations (2)

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S ( λ ) = { A · exp ( 0.5 ( ( λ peak ) / σ ) 2 ) for     A 0 1 + A · exp ( 0.5 ( ( λ peak ) / σ ) 2 for     A < 0 ,
S ( λ ) = M 1 S 1 ( λ ) + M 2 S 2 ( λ ) + M 3 S 3 ( λ ) ,

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