Abstract

We applied independent component analysis (ICA) to hyperspectral images in order to learn an efficient representation of color in natural scenes. In the spectra of single pixels, the algorithm found basis functions that had broadband spectra and basis functions that were similar to natural reflectance spectra. When applied to small image patches, the algorithm found some basis functions that were achromatic and others with overall chromatic variation along lines in color space, indicating color opponency. The directions of opponency were not strictly orthogonal. Comparison with principal-component analysis on the basis of statistical measures such as average mutual information, kurtosis, and entropy, shows that the ICA transformation results in much sparser coefficients and gives higher coding efficiency. Our findings suggest that nonorthogonal opponent encoding of photoreceptor signals leads to higher coding efficiency and that ICA may be used to reveal the underlying statistical properties of color information in natural scenes.

© 2001 Optical Society of America

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2000 (2)

1999 (5)

D. R. Tailor, L. H. Finkel, G. Buchsbaum, “Spatiochromatic independent component filters of natural scenes,” Soc. Neurosci. Abstr. 25, 1935 (1999).

M. S. Lewicki, B. A. Olshausen, “A probablistic framework for the adaptation and comparison of image codes,” J. Opt. Soc. Am. A 16, 1587–1601 (1999).
[CrossRef]

T. Wachtler, T. J. Sejnowski, T. D. Albright, “Interactions between stimulus and background chromaticities in macaque primary visual cortex,” Invest. Ophthalmol. Visual Sci. Suppl. 40, S641 (1999).

T.-W. Lee, M. Girolami, T. J. Sejnowski, “Independent component analysis using an extended infomax algorithm for mixed sub-gaussian and super-gaussian sources,” Neural Comput. 11, 409–433 (1999).

J. F. Cardoso, “High-order contrasts for independent component analysis,” Neural Comput. 11, 157–192 (1999).
[CrossRef] [PubMed]

1998 (2)

1997 (5)

J-F. Cardoso, “Infomax and maximum likelihood for blind source separation,” IEEE Signal Process. Lett. 4(4), 112–114 (1997).
[CrossRef]

J. Romero, A. Garcia-Beltrán, J. Hernández-Andrés, “Linear basis for representation of natural and artificial illuminants,” J. Opt. Soc. Am. A 14, 1007–1014 (1997).
[CrossRef]

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

A. J. Bell, T. J. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
[CrossRef]

A. Hyvärinen, E. Oja, “A fast fixed-point algorithm for independent component analysis,” Neural Comput. 9, 1483–1492 (1997).
[CrossRef]

1996 (2)

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

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 (1)

A. J. Bell, T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
[CrossRef] [PubMed]

1994 (3)

P. Comon, “Independent component analysis—a new concept?” Signal Process. 36, 287–314 (1994).
[CrossRef]

D. J. Field, “What is the goal of sensory coding?” Neural Comput. 6, 559–601 (1994).
[CrossRef]

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

1991 (1)

C. Jutten, J. Herault, “Blind separation of sources, part I: an adaptive algorithm based on neuromimetic architecture,” Signal Process. 24, 1–10 (1991).
[CrossRef]

1990 (1)

P. Lennie, J. Krauskopf, G. Sclar, “Chromatic mechanisms in striate cortex of macaque,” J. Neurosci. 10, 649–669 (1990).
[PubMed]

1989 (2)

J. N. Lythgoe, J. C. Partridge, “Visual pigments and the acquisition of visual information,” J. Exp. Biol. 146, 1–20 (1989).
[PubMed]

J. D. Mollon, “‘Tho’ she kneel’d in that place where they grew …’: the uses and origins of primate colour vision,” J. Exp. Biol. 146, 21–38 (1989).
[PubMed]

1986 (1)

1984 (1)

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

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]

1979 (1)

Albright, T. D.

T. Wachtler, T. J. Sejnowski, T. D. Albright, “Interactions between stimulus and background chromaticities in macaque primary visual cortex,” Invest. Ophthalmol. Visual Sci. Suppl. 40, S641 (1999).

Amari, S.

S. Amari, “Natural gradient works efficiently in learning,” Neural Comput. 10, 251–276 (1998).
[CrossRef]

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), pp. 217–234.

Bell, A. J.

A. J. Bell, T. J. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
[CrossRef]

A. J. Bell, T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
[CrossRef] [PubMed]

Boynton, R. M.

Brainard, D.

D. R. Williams, N. Sekiguchi, W. Haake, D. Brainard, O. Packer, “The cost of trichromacy for spatial vision,” in From Pigments to Perception, A. Valberg, B. B. Lee, eds., NATO ASI Series A, Life Sciences, Vol. 203 (Plenum, New York, 1991), pp. 11–22.

Buchsbaum, G.

D. R. Tailor, L. H. Finkel, G. Buchsbaum, “Spatiochromatic independent component filters of natural scenes,” Soc. Neurosci. Abstr. 25, 1935 (1999).

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]

Cardoso, J. F.

J. F. Cardoso, “High-order contrasts for independent component analysis,” Neural Comput. 11, 157–192 (1999).
[CrossRef] [PubMed]

Cardoso, J-F.

J-F. Cardoso, “Infomax and maximum likelihood for blind source separation,” IEEE Signal Process. Lett. 4(4), 112–114 (1997).
[CrossRef]

Chiao, C.-C.

Comon, P.

P. Comon, “Independent component analysis—a new concept?” Signal Process. 36, 287–314 (1994).
[CrossRef]

Cronin, T. W.

Derrington, A. M.

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

Engan, K.

K. Kreutz-Delgado, B. D. Rao, K. Engan, T.-W. Lee, T. J. Sejnowski, “Convex/schurconvex (csc) log-priors and sparse coding,” in 6th Joint Symposium on Neural Computation (Institute for Neural Computation, University of California, San Diego, San Diego, Calif., 1999), Vol. 9, pp. 65–71.

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 381, 607–609 (1996).
[CrossRef] [PubMed]

D. J. Field, “What is the goal of sensory coding?” Neural Comput. 6, 559–601 (1994).
[CrossRef]

Finkel, L. H.

D. R. Tailor, L. H. Finkel, G. Buchsbaum, “Spatiochromatic independent component filters of natural scenes,” Soc. Neurosci. Abstr. 25, 1935 (1999).

Garcia-Beltrán, A.

Gershon, R.

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

Girolami, M.

T.-W. Lee, M. Girolami, T. J. Sejnowski, “Independent component analysis using an extended infomax algorithm for mixed sub-gaussian and super-gaussian sources,” Neural Comput. 11, 409–433 (1999).

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]

Haake, W.

D. R. Williams, N. Sekiguchi, W. Haake, D. Brainard, O. Packer, “The cost of trichromacy for spatial vision,” in From Pigments to Perception, A. Valberg, B. B. Lee, eds., NATO ASI Series A, Life Sciences, Vol. 203 (Plenum, New York, 1991), pp. 11–22.

Herault, J.

C. Jutten, J. Herault, “Blind separation of sources, part I: an adaptive algorithm based on neuromimetic architecture,” Signal Process. 24, 1–10 (1991).
[CrossRef]

Hernández-Andrés, J.

Hyvärinen, A.

A. Hyvärinen, E. Oja, “A fast fixed-point algorithm for independent component analysis,” Neural Comput. 9, 1483–1492 (1997).
[CrossRef]

Iwan, L. S.

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

Jutten, C.

C. Jutten, J. Herault, “Blind separation of sources, part I: an adaptive algorithm based on neuromimetic architecture,” Signal Process. 24, 1–10 (1991).
[CrossRef]

Krauskopf, J.

P. Lennie, J. Krauskopf, G. Sclar, “Chromatic mechanisms in striate cortex of macaque,” J. Neurosci. 10, 649–669 (1990).
[PubMed]

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

Kreutz-Delgado, K.

K. Kreutz-Delgado, B. D. Rao, K. Engan, T.-W. Lee, T. J. Sejnowski, “Convex/schurconvex (csc) log-priors and sparse coding,” in 6th Joint Symposium on Neural Computation (Institute for Neural Computation, University of California, San Diego, San Diego, Calif., 1999), Vol. 9, pp. 65–71.

Lee, T.-W.

T.-W. Lee, M. Girolami, T. J. Sejnowski, “Independent component analysis using an extended infomax algorithm for mixed sub-gaussian and super-gaussian sources,” Neural Comput. 11, 409–433 (1999).

T.-W. Lee, T. Wachtler, T. Sejnowski, “The spectral independent components of natural scenes,” (Institute for Neural Computation, University of California, San Diego, San Diego, Calif., 1999).

T.-W. Lee, Independent Component Analysis: Theory and Applications (Kluwer Academic, Boston, Mass., 1998).

K. Kreutz-Delgado, B. D. Rao, K. Engan, T.-W. Lee, T. J. Sejnowski, “Convex/schurconvex (csc) log-priors and sparse coding,” in 6th Joint Symposium on Neural Computation (Institute for Neural Computation, University of California, San Diego, San Diego, Calif., 1999), Vol. 9, pp. 65–71.

Lee, T-W.

T-W. Lee, M. S. Lewicki, T. J. Sejnowski, “Unsupervised classification with nongaussian mixture models using ica,” in Advances in Neural Information Processing Systems 11 (MIT Press, Cambridge, Mass., 1999), pp. 508–514.

Lennie, P.

P. Lennie, J. Krauskopf, G. Sclar, “Chromatic mechanisms in striate cortex of macaque,” J. Neurosci. 10, 649–669 (1990).
[PubMed]

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

Lewicki, M. S.

M. S. Lewicki, T. J. Sejnowski, “Learning overcomplete representations,” Neural Comput. 12, 337–365 (2000).
[CrossRef] [PubMed]

M. S. Lewicki, B. A. Olshausen, “A probablistic framework for the adaptation and comparison of image codes,” J. Opt. Soc. Am. A 16, 1587–1601 (1999).
[CrossRef]

M. S. Lewicki, “A flexible prior for independent component analysis,” Neural Comput. (to be published).

T-W. Lee, M. S. Lewicki, T. J. Sejnowski, “Unsupervised classification with nongaussian mixture models using ica,” in Advances in Neural Information Processing Systems 11 (MIT Press, Cambridge, Mass., 1999), pp. 508–514.

Lythgoe, J. N.

J. N. Lythgoe, J. C. Partridge, “Visual pigments and the acquisition of visual information,” J. Exp. Biol. 146, 1–20 (1989).
[PubMed]

MacKay, D.

D. MacKay, “Maximum likelihood and covariant algorithms for independent component analysis,” Report (University of Cambridge, Cavendish Lab, Cambridge, UK, August1996).

MacLeod, D. I. A.

Maloney, L. T.

Mollon, J. D.

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

J. D. Mollon, “‘Tho’ she kneel’d in that place where they grew …’: the uses and origins of primate colour vision,” J. Exp. Biol. 146, 21–38 (1989).
[PubMed]

Oja, E.

A. Hyvärinen, E. Oja, “A fast fixed-point algorithm for independent component analysis,” Neural Comput. 9, 1483–1492 (1997).
[CrossRef]

Olshausen, B. A.

M. S. Lewicki, B. A. Olshausen, “A probablistic framework for the adaptation and comparison of image codes,” J. Opt. Soc. Am. A 16, 1587–1601 (1999).
[CrossRef]

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

Osorio, D.

Packer, O.

D. R. Williams, N. Sekiguchi, W. Haake, D. Brainard, O. Packer, “The cost of trichromacy for spatial vision,” in From Pigments to Perception, A. Valberg, B. B. Lee, eds., NATO ASI Series A, Life Sciences, Vol. 203 (Plenum, New York, 1991), pp. 11–22.

Parra, L.

B. Pearlmutter, L. Parra, “A context-sensitive generalization of ICA,” in Proceedings of the International Conference on Neural Information Processing (Springer, Singapore, 1996), pp. 151–157.

Partridge, J. C.

J. N. Lythgoe, J. C. Partridge, “Visual pigments and the acquisition of visual information,” J. Exp. Biol. 146, 1–20 (1989).
[PubMed]

Pearlmutter, B.

B. Pearlmutter, L. Parra, “A context-sensitive generalization of ICA,” in Proceedings of the International Conference on Neural Information Processing (Springer, Singapore, 1996), pp. 151–157.

Rao, B. D.

K. Kreutz-Delgado, B. D. Rao, K. Engan, T.-W. Lee, T. J. Sejnowski, “Convex/schurconvex (csc) log-priors and sparse coding,” in 6th Joint Symposium on Neural Computation (Institute for Neural Computation, University of California, San Diego, San Diego, Calif., 1999), Vol. 9, pp. 65–71.

Romero, J.

Ruderman, D. L.

Sclar, G.

P. Lennie, J. Krauskopf, G. Sclar, “Chromatic mechanisms in striate cortex of macaque,” J. Neurosci. 10, 649–669 (1990).
[PubMed]

Sejnowski, T.

T.-W. Lee, T. Wachtler, T. Sejnowski, “The spectral independent components of natural scenes,” (Institute for Neural Computation, University of California, San Diego, San Diego, Calif., 1999).

Sejnowski, T. J.

M. S. Lewicki, T. J. Sejnowski, “Learning overcomplete representations,” Neural Comput. 12, 337–365 (2000).
[CrossRef] [PubMed]

T.-W. Lee, M. Girolami, T. J. Sejnowski, “Independent component analysis using an extended infomax algorithm for mixed sub-gaussian and super-gaussian sources,” Neural Comput. 11, 409–433 (1999).

T. Wachtler, T. J. Sejnowski, T. D. Albright, “Interactions between stimulus and background chromaticities in macaque primary visual cortex,” Invest. Ophthalmol. Visual Sci. Suppl. 40, S641 (1999).

A. J. Bell, T. J. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
[CrossRef]

A. J. Bell, T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
[CrossRef] [PubMed]

K. Kreutz-Delgado, B. D. Rao, K. Engan, T.-W. Lee, T. J. Sejnowski, “Convex/schurconvex (csc) log-priors and sparse coding,” in 6th Joint Symposium on Neural Computation (Institute for Neural Computation, University of California, San Diego, San Diego, Calif., 1999), Vol. 9, pp. 65–71.

T-W. Lee, M. S. Lewicki, T. J. Sejnowski, “Unsupervised classification with nongaussian mixture models using ica,” in Advances in Neural Information Processing Systems 11 (MIT Press, Cambridge, Mass., 1999), pp. 508–514.

Sekiguchi, N.

D. R. Williams, N. Sekiguchi, W. Haake, D. Brainard, O. Packer, “The cost of trichromacy for spatial vision,” in From Pigments to Perception, A. Valberg, B. B. Lee, eds., NATO ASI Series A, Life Sciences, Vol. 203 (Plenum, New York, 1991), pp. 11–22.

Stiles, W. S.

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

Tailor, D. R.

D. R. Tailor, L. H. Finkel, G. Buchsbaum, “Spatiochromatic independent component filters of natural scenes,” Soc. Neurosci. Abstr. 25, 1935 (1999).

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]

Vrhel, M. J.

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

Wachtler, T.

T. Wachtler, T. J. Sejnowski, T. D. Albright, “Interactions between stimulus and background chromaticities in macaque primary visual cortex,” Invest. Ophthalmol. Visual Sci. Suppl. 40, S641 (1999).

T.-W. Lee, T. Wachtler, T. Sejnowski, “The spectral independent components of natural scenes,” (Institute for Neural Computation, University of California, San Diego, San Diego, Calif., 1999).

Webster, M. A.

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

Williams, D. R.

D. R. Williams, N. Sekiguchi, W. Haake, D. Brainard, O. Packer, “The cost of trichromacy for spatial vision,” in From Pigments to Perception, A. Valberg, B. B. Lee, eds., NATO ASI Series A, Life Sciences, Vol. 203 (Plenum, New York, 1991), pp. 11–22.

Wyszecki, G.

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

Color Res. Appl. (1)

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Figures (10)

Fig. 1
Fig. 1

Schematic comparison between a Laplacian density (left) and a Gaussian density (right).

Fig. 2
Fig. 2

Eight hyperspectral color images of natural scenes. A, Schematic of decomposition of the observed spectrum into spectral basis functions. B, Schematic of decomposition of the observed image patch into spatiochromatic basis functions.

Fig. 3
Fig. 3

Analysis of pixel spectra using ICA: the learned basis functions of pixel spectra, ordered by decreasing L2 norm. The number in the upper left corner of each plot denotes the basis function number as referred to in the text. The inset shows the histogram of the L2 norms of the basis functions.

Fig. 4
Fig. 4

Analysis of pixel spectra using PCA: PCA basis functions of pixel spectra, ordered by decreasing eigenvalues. The inset shows the histogram of the eigenvalues.

Fig. 5
Fig. 5

Analysis of image patches using ICA. Top, spatiochromatic structure of the 147 learned ICA basis functions of image patches (7×7 pixels, three chromatic dimensions). The R, G, and B values of the color of each pixel correspond to the relative excitation of L, M, and S cones, respectively. The functions are in order of decreasing L2 norm from left to right and top to bottom. The additional column marked W on the right shows the filters for the rightmost column of basis functions (marked A). Bottom, chromaticities of the basis functions, plotted in cone-opponent color-space coordinates. Horizontal axes, L- versus M-cone variation. Vertical axes, S-cone variation. Each dot represents the coordinate of a pixel of the respective basis function, projected onto the isoluminant plane. Luminance can be inferred from the brightness of the dot. Note that for basis functions that vary mainly in luminance, the dots tend to lie on top of one another.

Fig. 6
Fig. 6

Relative contributions of ICA basis functions. A, Histogram of the L2 norms of the first 50 ICA basis functions in Fig. 5. B, Histograms of coefficient values for the first 25 ICA basis functions in Fig. 5. Most distributions have high kurtosis.

Fig. 7
Fig. 7

Example of contribution levels for ICA-patch basis function 5 (green image patch) in one of the images. Left, gray-scale rendering of original image. Right, contribution level. Dark regions correspond to low contributions of basis function 5.

Fig. 8
Fig. 8

Analysis of image patches using PCA. Top, spatiochromatic structure of the 147 PCA basis functions of image patches. The components are ordered by decreasing eigenvalues. Bottom, chromaticities of the basis functions in cone-opponent color-space coordinates. The data are plotted in the same way as the ICA basis functions in Fig. 5.

Fig. 9
Fig. 9

Eigenspectrum of the first 50 principal components in Fig. 8.

Fig. 10
Fig. 10

Comparison of ICA basis function 5 (dashed curve) with reflectance spectra of leaves after Vrhel et al.30 (solid curves). All spectra have been scaled such that their peaks near 550 nm have a value of 1.

Tables (1)

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Table 1 Coding Efficiencies of ICA and PCA for Pixel Spectra and Image Patches

Equations (13)

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

x=As,
ΔAA [I-sign(s)sT],
#bits-log2 P(xt|A)-N log2(σx),
p(s)=i=1Mpi(si).
x=As,
I(x)=p(x)logp(x)i=1Npi(xi)dx=Dp(x)i=1Npi(xi).
u=Wx=WAs,
p(x|A)=p(x|s, A)p(s)ds.
p(x|s, A)=δ(x-As).
p(x|A)=p(s)|det(A)|.
log p(x|A)=i=1Nlog p(si)-log|det(A)|.
ΔAAATAlog p(x|A)=-A[I-φ(s)xT],
ΔAA[I-sign(s)sT].

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