Abstract

We gathered hyperspectral images of natural, foliage-dominated scenes and converted them to human cone quantal catches to characterize the second-order redundancy present within the retinal photoreceptor array under natural conditions. The data are expressed most simply in a logarithmic response space, wherein an orthogonal decorrelation robustly produces three principal axes, one corresponding to simple changes in radiance and two that are reminiscent of the blue–yellow and red–green chromatic-opponent mechanisms found in the primate visual system. Further inclusion of spatial stimulus dimensions demonstrates a complete spatial decorrelation of these three cone-space axes in natural cone responses.

© 1998 Optical Society of America

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  1. F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
    [CrossRef] [PubMed]
  2. H. B. Barlow, “Possible principles underlying the transformation of sensory messages,” in Sensory Communication, W. A. Rosenblith, ed. (MIT Press, Cambridge, Mass., 1961).
  3. M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London, Ser. B 216, 427–459 (1982).
    [CrossRef]
  4. J. J. Atick, N. Redlich, “Towards a theory of early visual processing,” Neural Comput. 2, 308–320 (1990).
    [CrossRef]
  5. B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature (London) 381, 607–609 (1996).
    [CrossRef]
  6. G. J. Burton, I. R. Moorhead, “Color and spatial structure in natural scenes,” Appl. Opt. 26, 157–170 (1987).
    [CrossRef] [PubMed]
  7. D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
    [CrossRef] [PubMed]
  8. J. H. van Hateren, “Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation,” J. Comp. Physiol. A 171, 157–170 (1992).
  9. D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
    [CrossRef] [PubMed]
  10. D. L. Ruderman, “The statistics of natural images,” Network 5, 517–548 (1994).
    [CrossRef]
  11. D. W. Dong, J. J. Atick, “Statistics of natural time-varying images,” Network 6, 345–358 (1995).
    [CrossRef]
  12. N. G. Nagle, D. Osorio, “The tuning of human photopigments may minimize red–green chromatic signals in natural conditions,” Proc. R. Soc. London, Ser. B 252, 209–213 (1993).
    [CrossRef]
  13. G. Brelstaff, A. Párraga, T. Troscianko, D. Carr, “Hyperspectral camera system: acquisition and analysis,” in Geographic Information Systems, Photogrammetry, and Geological/Geophysical Remote Sensing, J. B. Lurie, J. Pearson, E. Zilioli, eds., Proc. SPIE2587, 150–159 (1995).
    [CrossRef]
  14. M. A. Webster, J. D. Mollon, “Adaptation and the color statistics of natural images,” Vision Res. 37, 3283–3298 (1997).
    [CrossRef]
  15. C. A. Párraga, G. Brelstaff, T. Troscianko, “Color and luminance information in natural scenes,” J. Opt. Soc. Am. A 15, 563–569 (1998).
    [CrossRef]
  16. G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colour coding and optimum colour information transmission in the retina,” Proc. R. Soc. London, Ser. B 220, 89–113 (1983).
    [CrossRef]
  17. D. L. Ruderman, “Designing receptive fields for highest fidelity,” Network 5, 147–155 (1994).
    [CrossRef]
  18. A. Stockman, D. I. A. MacLeod, N. E. Johnson, “Spectral sensitivities of the human cones,” J. Opt. Soc. Am. A 10, 2491–2521 (1993).
    [CrossRef]
  19. G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).
  20. D. R. J. Laming, Sensory Analysis (Academic, London, 1986).
  21. D. L. Ruderman, “Origins of scaling in natural images,” Vision Res. 37, 3385–3398 (1997).
    [CrossRef]
  22. I. T. Jolliffe, Principal Component Analysis (Springer-Verlag, New York, 1986).
  23. P. Flanagan, P. Cavanagh, O. E. Favreau, “Independent orientation-selective mechanisms for the cardinal directions of colour space,” Vision Res. 30, 769–778 (1990).
    [CrossRef] [PubMed]
  24. F. M. DeMonasterio, P. Gouras, “Functional properties of ganglion cells of the rhesus monkey retina,” J. Physiol. (London) 251, 167–195 (1975).
  25. F. M. DeMonasterio, P. Gouras, D. J. Tolhurst, “Trichromatic colour opponency in ganglion cells of the rhesus monkey retina,” J. Physiol. (London) 251, 197–216 (1975).
  26. A. M. Derrington, J. Krauskopf, P. Lennie, “Chromatic mechanisms in lateral geniculate nucleus of macaque,” J. Physiol. (London) 357, 241–265 (1984).
  27. R. C. Reid, R. M. Shapley, “Spatial structure of cone inputs to receptive fields in primate lateral geniculate nucleus,” Nature (London) 356, 716–718 (1992).
    [CrossRef]
  28. E. Hering, Outlines of a Theory of the Light Sense (Harvard U. Press, Cambridge, Mass., 1964).
  29. D. Jameson, L. M. Hurvich, “Some quantitative aspects of an opponent-colors theory. I. Chromatic responses and spectral saturation,” J. Opt. Soc. Am. 45, 546–552 (1955).
    [CrossRef]
  30. J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
    [CrossRef] [PubMed]
  31. M. A. Webster, “Human colour perception and its adaptation,” Network 7, 587–634 (1996).
    [CrossRef]
  32. P. Lennie, M. D’Zmura, “Mechanisms of color vision,” CRC Crit. Rev. Clin. Neurobiol. 3, 333–400 (1988).
  33. J. H. van Hateren, “Spatial, temporal and spectral pre-processing for colour vision,” Proc. R. Soc. London, Ser. B 251, 61–68 (1993).
    [CrossRef]
  34. Translation invariance implies that correlations depend only on pixel separations. From scale invariance the correlation matrix in the cone subspace is not a function of the pixel separation (up to an overall multiplicative factor). Thus the correlation matrix is of the form F(x)Cab, where x is the pixel separation vector and a and b are index cone-space directions. Such a correlation matrix can be diagonalized through a Fourier transform in the variable x and a separate diagonalization of the matrix C (yielding the l,α, and β directions). The associated eigenvalues will be a product of the F (spatial) eigenvalues and the C (cone-space) eigenvalues, as in Eq. (6). This is analogous to two decorrelated multiplicative processes, one in real space and the other in cone space.
  35. This invariance of the spectral shape is consistent with the hypothesis that spatial statistics in natural images are dominated by the size distribution of objects within the scenes.21
  36. J. J. Atick, Z. Li, A. N. Redlich, “Understanding retinal color coding from first principles,” Neural Comput. 4, 559–572 (1992).
    [CrossRef]
  37. J. B. Derrico, G. Buchsbaum, “A computational model of spatiochromatic image coding in early vision,” J. Visual Commun. Image Represent. 2, 31–38 (1991).
    [CrossRef]
  38. I. R. Moorhead, “Human color vision and natural images,” in Colour in Information Technology and Information Displays (Institution of Electronic and Radio Engineers, London, 1985).
  39. J. J. Vos, P. L. Walraven, “On the derivation of the foveal receptor primaries,” Vision Res. 11, 799–818 (1971).
    [CrossRef] [PubMed]
  40. W. S. Stiles, “Color vision: the approach through increment threshold sensitivity,” Proc. Natl. Acad. Sci. USA 45, 100–114 (1959).
    [CrossRef]
  41. D. R. Williams, D. I. A. MacLeod, M. M. Hayhoe, “Foveal tritanopia,” Vision Res. 21, 1341–1356 (1981).
    [CrossRef] [PubMed]
  42. D. Osorio, M. Vorobyev, “Colour vision as an adaptation to frugivory in primates,” Proc. R. Soc. London, Ser. B 263, 593–599 (1996).
    [CrossRef]

1998

1997

D. L. Ruderman, “Origins of scaling in natural images,” Vision Res. 37, 3385–3398 (1997).
[CrossRef]

M. A. Webster, J. D. Mollon, “Adaptation and the color statistics of natural images,” Vision Res. 37, 3283–3298 (1997).
[CrossRef]

1996

M. A. Webster, “Human colour perception and its adaptation,” Network 7, 587–634 (1996).
[CrossRef]

B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature (London) 381, 607–609 (1996).
[CrossRef]

D. Osorio, M. Vorobyev, “Colour vision as an adaptation to frugivory in primates,” Proc. R. Soc. London, Ser. B 263, 593–599 (1996).
[CrossRef]

1995

D. W. Dong, J. J. Atick, “Statistics of natural time-varying images,” Network 6, 345–358 (1995).
[CrossRef]

1994

D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

D. L. Ruderman, “The statistics of natural images,” Network 5, 517–548 (1994).
[CrossRef]

D. L. Ruderman, “Designing receptive fields for highest fidelity,” Network 5, 147–155 (1994).
[CrossRef]

1993

A. Stockman, D. I. A. MacLeod, N. E. Johnson, “Spectral sensitivities of the human cones,” J. Opt. Soc. Am. A 10, 2491–2521 (1993).
[CrossRef]

N. G. Nagle, D. Osorio, “The tuning of human photopigments may minimize red–green chromatic signals in natural conditions,” Proc. R. Soc. London, Ser. B 252, 209–213 (1993).
[CrossRef]

J. H. van Hateren, “Spatial, temporal and spectral pre-processing for colour vision,” Proc. R. Soc. London, Ser. B 251, 61–68 (1993).
[CrossRef]

1992

J. J. Atick, Z. Li, A. N. Redlich, “Understanding retinal color coding from first principles,” Neural Comput. 4, 559–572 (1992).
[CrossRef]

J. H. van Hateren, “Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation,” J. Comp. Physiol. A 171, 157–170 (1992).

R. C. Reid, R. M. Shapley, “Spatial structure of cone inputs to receptive fields in primate lateral geniculate nucleus,” Nature (London) 356, 716–718 (1992).
[CrossRef]

1991

J. B. Derrico, G. Buchsbaum, “A computational model of spatiochromatic image coding in early vision,” J. Visual Commun. Image Represent. 2, 31–38 (1991).
[CrossRef]

1990

J. J. Atick, N. Redlich, “Towards a theory of early visual processing,” Neural Comput. 2, 308–320 (1990).
[CrossRef]

P. Flanagan, P. Cavanagh, O. E. Favreau, “Independent orientation-selective mechanisms for the cardinal directions of colour space,” Vision Res. 30, 769–778 (1990).
[CrossRef] [PubMed]

1988

P. Lennie, M. D’Zmura, “Mechanisms of color vision,” CRC Crit. Rev. Clin. Neurobiol. 3, 333–400 (1988).

1987

1984

A. M. Derrington, J. Krauskopf, P. Lennie, “Chromatic mechanisms in lateral geniculate nucleus of macaque,” J. Physiol. (London) 357, 241–265 (1984).

1983

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colour coding and optimum colour information transmission in the retina,” Proc. R. Soc. London, Ser. B 220, 89–113 (1983).
[CrossRef]

1982

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London, Ser. B 216, 427–459 (1982).
[CrossRef]

1981

D. R. Williams, D. I. A. MacLeod, M. M. Hayhoe, “Foveal tritanopia,” Vision Res. 21, 1341–1356 (1981).
[CrossRef] [PubMed]

1975

F. M. DeMonasterio, P. Gouras, “Functional properties of ganglion cells of the rhesus monkey retina,” J. Physiol. (London) 251, 167–195 (1975).

F. M. DeMonasterio, P. Gouras, D. J. Tolhurst, “Trichromatic colour opponency in ganglion cells of the rhesus monkey retina,” J. Physiol. (London) 251, 197–216 (1975).

1971

J. J. Vos, P. L. Walraven, “On the derivation of the foveal receptor primaries,” Vision Res. 11, 799–818 (1971).
[CrossRef] [PubMed]

1959

W. S. Stiles, “Color vision: the approach through increment threshold sensitivity,” Proc. Natl. Acad. Sci. USA 45, 100–114 (1959).
[CrossRef]

1955

1954

F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

Atick, J. J.

D. W. Dong, J. J. Atick, “Statistics of natural time-varying images,” Network 6, 345–358 (1995).
[CrossRef]

J. J. Atick, Z. Li, A. N. Redlich, “Understanding retinal color coding from first principles,” Neural Comput. 4, 559–572 (1992).
[CrossRef]

J. J. Atick, N. Redlich, “Towards a theory of early visual processing,” Neural Comput. 2, 308–320 (1990).
[CrossRef]

Attneave, F.

F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

Barlow, H. B.

H. B. Barlow, “Possible principles underlying the transformation of sensory messages,” in Sensory Communication, W. A. Rosenblith, ed. (MIT Press, Cambridge, Mass., 1961).

Bialek, W.

D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

Brelstaff, G.

C. A. Párraga, G. Brelstaff, T. Troscianko, “Color and luminance information in natural scenes,” J. Opt. Soc. Am. A 15, 563–569 (1998).
[CrossRef]

G. Brelstaff, A. Párraga, T. Troscianko, D. Carr, “Hyperspectral camera system: acquisition and analysis,” in Geographic Information Systems, Photogrammetry, and Geological/Geophysical Remote Sensing, J. B. Lurie, J. Pearson, E. Zilioli, eds., Proc. SPIE2587, 150–159 (1995).
[CrossRef]

Buchsbaum, G.

J. B. Derrico, G. Buchsbaum, “A computational model of spatiochromatic image coding in early vision,” J. Visual Commun. Image Represent. 2, 31–38 (1991).
[CrossRef]

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colour coding and optimum colour information transmission in the retina,” Proc. R. Soc. London, Ser. B 220, 89–113 (1983).
[CrossRef]

Burton, G. J.

Carr, D.

G. Brelstaff, A. Párraga, T. Troscianko, D. Carr, “Hyperspectral camera system: acquisition and analysis,” in Geographic Information Systems, Photogrammetry, and Geological/Geophysical Remote Sensing, J. B. Lurie, J. Pearson, E. Zilioli, eds., Proc. SPIE2587, 150–159 (1995).
[CrossRef]

Cavanagh, P.

P. Flanagan, P. Cavanagh, O. E. Favreau, “Independent orientation-selective mechanisms for the cardinal directions of colour space,” Vision Res. 30, 769–778 (1990).
[CrossRef] [PubMed]

D’Zmura, M.

P. Lennie, M. D’Zmura, “Mechanisms of color vision,” CRC Crit. Rev. Clin. Neurobiol. 3, 333–400 (1988).

DeMonasterio, F. M.

F. M. DeMonasterio, P. Gouras, D. J. Tolhurst, “Trichromatic colour opponency in ganglion cells of the rhesus monkey retina,” J. Physiol. (London) 251, 197–216 (1975).

F. M. DeMonasterio, P. Gouras, “Functional properties of ganglion cells of the rhesus monkey retina,” J. Physiol. (London) 251, 167–195 (1975).

Derrico, J. B.

J. B. Derrico, G. Buchsbaum, “A computational model of spatiochromatic image coding in early vision,” J. Visual Commun. Image Represent. 2, 31–38 (1991).
[CrossRef]

Derrington, A. M.

A. M. Derrington, J. Krauskopf, P. Lennie, “Chromatic mechanisms in lateral geniculate nucleus of macaque,” J. Physiol. (London) 357, 241–265 (1984).

Dong, D. W.

D. W. Dong, J. J. Atick, “Statistics of natural time-varying images,” Network 6, 345–358 (1995).
[CrossRef]

Dubs, A.

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London, Ser. B 216, 427–459 (1982).
[CrossRef]

Favreau, O. E.

P. Flanagan, P. Cavanagh, O. E. Favreau, “Independent orientation-selective mechanisms for the cardinal directions of colour space,” Vision Res. 30, 769–778 (1990).
[CrossRef] [PubMed]

Field, D. J.

B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature (London) 381, 607–609 (1996).
[CrossRef]

D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
[CrossRef] [PubMed]

Flanagan, P.

P. Flanagan, P. Cavanagh, O. E. Favreau, “Independent orientation-selective mechanisms for the cardinal directions of colour space,” Vision Res. 30, 769–778 (1990).
[CrossRef] [PubMed]

Gottschalk, A.

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colour coding and optimum colour information transmission in the retina,” Proc. R. Soc. London, Ser. B 220, 89–113 (1983).
[CrossRef]

Gouras, P.

F. M. DeMonasterio, P. Gouras, D. J. Tolhurst, “Trichromatic colour opponency in ganglion cells of the rhesus monkey retina,” J. Physiol. (London) 251, 197–216 (1975).

F. M. DeMonasterio, P. Gouras, “Functional properties of ganglion cells of the rhesus monkey retina,” J. Physiol. (London) 251, 167–195 (1975).

Hayhoe, M. M.

D. R. Williams, D. I. A. MacLeod, M. M. Hayhoe, “Foveal tritanopia,” Vision Res. 21, 1341–1356 (1981).
[CrossRef] [PubMed]

Heeley, D. W.

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

Hering, E.

E. Hering, Outlines of a Theory of the Light Sense (Harvard U. Press, Cambridge, Mass., 1964).

Hurvich, L. M.

Jameson, D.

Johnson, N. E.

Jolliffe, I. T.

I. T. Jolliffe, Principal Component Analysis (Springer-Verlag, New York, 1986).

Krauskopf, J.

A. M. Derrington, J. Krauskopf, P. Lennie, “Chromatic mechanisms in lateral geniculate nucleus of macaque,” J. Physiol. (London) 357, 241–265 (1984).

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

Laming, D. R. J.

D. R. J. Laming, Sensory Analysis (Academic, London, 1986).

Laughlin, S. B.

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London, Ser. B 216, 427–459 (1982).
[CrossRef]

Lennie, P.

P. Lennie, M. D’Zmura, “Mechanisms of color vision,” CRC Crit. Rev. Clin. Neurobiol. 3, 333–400 (1988).

A. M. Derrington, J. Krauskopf, P. Lennie, “Chromatic mechanisms in lateral geniculate nucleus of macaque,” J. Physiol. (London) 357, 241–265 (1984).

Li, Z.

J. J. Atick, Z. Li, A. N. Redlich, “Understanding retinal color coding from first principles,” Neural Comput. 4, 559–572 (1992).
[CrossRef]

MacLeod, D. I. A.

Mollon, J. D.

M. A. Webster, J. D. Mollon, “Adaptation and the color statistics of natural images,” Vision Res. 37, 3283–3298 (1997).
[CrossRef]

Moorhead, I. R.

G. J. Burton, I. R. Moorhead, “Color and spatial structure in natural scenes,” Appl. Opt. 26, 157–170 (1987).
[CrossRef] [PubMed]

I. R. Moorhead, “Human color vision and natural images,” in Colour in Information Technology and Information Displays (Institution of Electronic and Radio Engineers, London, 1985).

Nagle, N. G.

N. G. Nagle, D. Osorio, “The tuning of human photopigments may minimize red–green chromatic signals in natural conditions,” Proc. R. Soc. London, Ser. B 252, 209–213 (1993).
[CrossRef]

Olshausen, B. A.

B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature (London) 381, 607–609 (1996).
[CrossRef]

Osorio, D.

D. Osorio, M. Vorobyev, “Colour vision as an adaptation to frugivory in primates,” Proc. R. Soc. London, Ser. B 263, 593–599 (1996).
[CrossRef]

N. G. Nagle, D. Osorio, “The tuning of human photopigments may minimize red–green chromatic signals in natural conditions,” Proc. R. Soc. London, Ser. B 252, 209–213 (1993).
[CrossRef]

Párraga, A.

G. Brelstaff, A. Párraga, T. Troscianko, D. Carr, “Hyperspectral camera system: acquisition and analysis,” in Geographic Information Systems, Photogrammetry, and Geological/Geophysical Remote Sensing, J. B. Lurie, J. Pearson, E. Zilioli, eds., Proc. SPIE2587, 150–159 (1995).
[CrossRef]

Párraga, C. A.

Redlich, A. N.

J. J. Atick, Z. Li, A. N. Redlich, “Understanding retinal color coding from first principles,” Neural Comput. 4, 559–572 (1992).
[CrossRef]

Redlich, N.

J. J. Atick, N. Redlich, “Towards a theory of early visual processing,” Neural Comput. 2, 308–320 (1990).
[CrossRef]

Reid, R. C.

R. C. Reid, R. M. Shapley, “Spatial structure of cone inputs to receptive fields in primate lateral geniculate nucleus,” Nature (London) 356, 716–718 (1992).
[CrossRef]

Ruderman, D. L.

D. L. Ruderman, “Origins of scaling in natural images,” Vision Res. 37, 3385–3398 (1997).
[CrossRef]

D. L. Ruderman, “The statistics of natural images,” Network 5, 517–548 (1994).
[CrossRef]

D. L. Ruderman, “Designing receptive fields for highest fidelity,” Network 5, 147–155 (1994).
[CrossRef]

D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

Shapley, R. M.

R. C. Reid, R. M. Shapley, “Spatial structure of cone inputs to receptive fields in primate lateral geniculate nucleus,” Nature (London) 356, 716–718 (1992).
[CrossRef]

Srinivasan, M. V.

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London, Ser. B 216, 427–459 (1982).
[CrossRef]

Stiles, W. S.

W. S. Stiles, “Color vision: the approach through increment threshold sensitivity,” Proc. Natl. Acad. Sci. USA 45, 100–114 (1959).
[CrossRef]

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

Stockman, A.

Tolhurst, D. J.

F. M. DeMonasterio, P. Gouras, D. J. Tolhurst, “Trichromatic colour opponency in ganglion cells of the rhesus monkey retina,” J. Physiol. (London) 251, 197–216 (1975).

Troscianko, T.

C. A. Párraga, G. Brelstaff, T. Troscianko, “Color and luminance information in natural scenes,” J. Opt. Soc. Am. A 15, 563–569 (1998).
[CrossRef]

G. Brelstaff, A. Párraga, T. Troscianko, D. Carr, “Hyperspectral camera system: acquisition and analysis,” in Geographic Information Systems, Photogrammetry, and Geological/Geophysical Remote Sensing, J. B. Lurie, J. Pearson, E. Zilioli, eds., Proc. SPIE2587, 150–159 (1995).
[CrossRef]

van Hateren, J. H.

J. H. van Hateren, “Spatial, temporal and spectral pre-processing for colour vision,” Proc. R. Soc. London, Ser. B 251, 61–68 (1993).
[CrossRef]

J. H. van Hateren, “Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation,” J. Comp. Physiol. A 171, 157–170 (1992).

Vorobyev, M.

D. Osorio, M. Vorobyev, “Colour vision as an adaptation to frugivory in primates,” Proc. R. Soc. London, Ser. B 263, 593–599 (1996).
[CrossRef]

Vos, J. J.

J. J. Vos, P. L. Walraven, “On the derivation of the foveal receptor primaries,” Vision Res. 11, 799–818 (1971).
[CrossRef] [PubMed]

Walraven, P. L.

J. J. Vos, P. L. Walraven, “On the derivation of the foveal receptor primaries,” Vision Res. 11, 799–818 (1971).
[CrossRef] [PubMed]

Webster, M. A.

M. A. Webster, J. D. Mollon, “Adaptation and the color statistics of natural images,” Vision Res. 37, 3283–3298 (1997).
[CrossRef]

M. A. Webster, “Human colour perception and its adaptation,” Network 7, 587–634 (1996).
[CrossRef]

Williams, D. R.

J. Krauskopf, D. R. Williams, D. W. Heeley, “Cardinal directions of color space,” Vision Res. 22, 1123–1131 (1982).
[CrossRef] [PubMed]

D. R. Williams, D. I. A. MacLeod, M. M. Hayhoe, “Foveal tritanopia,” Vision Res. 21, 1341–1356 (1981).
[CrossRef] [PubMed]

Wyszecki, G.

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

Appl. Opt.

CRC Crit. Rev. Clin. Neurobiol.

P. Lennie, M. D’Zmura, “Mechanisms of color vision,” CRC Crit. Rev. Clin. Neurobiol. 3, 333–400 (1988).

J. Comp. Physiol. A

J. H. van Hateren, “Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation,” J. Comp. Physiol. A 171, 157–170 (1992).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

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Translation invariance implies that correlations depend only on pixel separations. From scale invariance the correlation matrix in the cone subspace is not a function of the pixel separation (up to an overall multiplicative factor). Thus the correlation matrix is of the form F(x)Cab, where x is the pixel separation vector and a and b are index cone-space directions. Such a correlation matrix can be diagonalized through a Fourier transform in the variable x and a separate diagonalization of the matrix C (yielding the l,α, and β directions). The associated eigenvalues will be a product of the F (spatial) eigenvalues and the C (cone-space) eigenvalues, as in Eq. (6). This is analogous to two decorrelated multiplicative processes, one in real space and the other in cone space.

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Figures (8)

Fig. 1
Fig. 1

Scatterplots of (top) L-versus-M and (bottom) L-versus-S data from 1000 pixels chosen at random from the image data set. The distributions show a high degree of correlation and asymmetry. Values are scaled so that the mean along each axis is 1.

Fig. 2
Fig. 2

Scatterplot of 1000 data points randomly selected as projected onto principal-axis pairs: (top) α versus l, (middle) β versus l, (bottom) α versus β. Note that the axes within each plot have the same scale, but the scales may change between plots.

Fig. 3
Fig. 3

1000 datapoints selected at random as decorrelated in the linear (L, M, S) space, projected along the l and α axes (roughly equal to those of the logarithmic space). The standard deviation along the α axis is approximately proportional to l (data not shown), representing a third-order correlation in the data. These data have been normalized to unit variance along both axes.

Fig. 4
Fig. 4

Marginal distributions of our data along the three principal axes: (top) l axis, (middle) α axis, (bottom) β axis. Logarithmic space data (solid curves) and linear space data (dotted curves) are shown together for comparison. The dashed curves are unit-variance Gaussian distributions (all data have been re- scaled to unit variance).

Fig. 5
Fig. 5

Principal axes of 3×3-pixel chromatic patches arranged in order of decreasing eigenvalue from left to right and top to bottom. The color values (R, G, and B) for each pixel in this image were determined from the L, M, and S coordinates of each element through a direct linear correspondence [e.g., R=128(L+1), giving a range of 0–255]. For instance, a component that extracts only L-cone responses would contain pixels that range from reddish (R=255; G,B=128) to cyan (R=0; G,B=128).

Fig. 6
Fig. 6

Histograms showing the fraction of energy within each principal component, which represents each of the three principal axes (l at left, α at center, β at right). Each bar can range from 0 to 1 in value, a value of 1 meaning that all the energy resides along that single principal axis. The histograms show that all of the 27 components essentially lie along only a single axis. The numbers below the histograms are the component number and its eigenvalue (variance) in parentheses.

Fig. 7
Fig. 7

Image from our data set (upper left) and its thresholded projection onto the three principal axes: l (upper right), α (lower left), β (lower right). The spatial patterns in each of these three projections are completely uncorrelated with one another.

Fig. 8
Fig. 8

Normalized variances along the principal axes in each of the three principal cone-space directions: l (solid curve), α (dashed curve), β (dotted curve). The order of the nine axes is as they appear in Fig. 5. The α- and β-eigenvalue spectra are nearly identical. The spectrum of l eigenvalues, which does not compare relative photon catches, differs significantly.

Tables (2)

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Table 1 Three Principal Axes for Image Pixels Expressed in the (L, M, S) Basis and the Standard Deviations of the Data As Projected onto These Three Axes

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Table 2 Three Principal Axes in Linear Cone Response Space Assuming a Spectral Distribution That Is Equal Energy on Average and Has White-Noise Fluctuations That Are Independent at Each Wavelengtha

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

L=log L-log L,
M=log M-log M,
S=log S-log S.
xixj=σi2δij,
yi=j=1NAijxj,
lˆ=13(Lˆ+Mˆ+Sˆ),
αˆ=16(Lˆ+Mˆ-2Sˆ),
βˆ=12(Lˆ-Mˆ),
pmn(a, i)=cm(a)sn(i),
σmn2=σc2(m)σs2(n),

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