Abstract

We have constructed a wine-glass-type five-layer neural network and generated an identity mapping of the surface spectral-reflectance data of 1280 Munsell color chips, using a backpropagation learning algorithm. To achieve an identity mapping, the same data set is used for the input and for the teacher. After the learning was completed, we analyzed the responses to individual chips of the three hidden units in the middle layer in order to obtain the internal representation of the color information. We found that each of the three hidden units corresponds to a psychological color attribute, that is, the Munsell value (luminance), red–green, and yellow–blue. We also examined the relationship between the internal representation and the number of hidden units and found that the network with three hidden units acquires optimum color representation. The five-layer neural network is shown to be an efficient method for reproducing the transformation of color information (or color coding) in the visual system.

© 1992 Optical Society of America

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References

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  1. S. Usui, S. Saruyama, G. Mitarai, M. Sakakibara, T. Yagi, “Image-sensing mechanisms in the vertebrate retina,” Biomechanism (U. Tokyo Press, Tokyo, 1984), Vol. 7, pp. 41–49.
  2. T. N. Wiesel, D. H. Hubel, “Spatial and chromatic interactions in the lateral geniculate nucleus of the rhesus monkey,” J. Neurophysiol. 29, 1115–1156 (1966).
    [PubMed]
  3. R. L. De Valois, I. Abramov, G. H. Jacobs, “Analysis of response patterns of LGN cells,”J. Opt. Soc. Am. 56, 966–977 (1966).
    [Crossref] [PubMed]
  4. A. M. Derrington, J. Krauskopf, P. Lennie, “Chromatic mechanisms in lateral geniculate nucleus of macaque,” J. Physiol. 357, 241–265 (1984).
    [PubMed]
  5. E. Oja, Subspace Methods of Pattern Recognition (Research Studies, Letchworth, England, 1983).
  6. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonomic Sci. 1, 369–370 (1964).
  7. 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]
  8. H. Sobagaki, “New approach to the colorimetric standardization for object colors,” Bull. Electrotechnical Lab. Jpn. 48, 785–792 (1984).
  9. J. P. S. Parkkinen, J. Hallikaineln, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [Crossref]
  10. J. P. S. Parkkinen, T. Jaaskelainen, “Color vision: machine and human,” in Visual Communications and Image Processing IV, W. A. Pearlman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1199, 1184–1192 (1989).
    [Crossref]
  11. R. A. Young, “Principal-component analysis of macaque lateral geniculate nucleus chromatic data,” J. Opt. Soc. Am. A 3, 1735–1742 (1986).
    [Crossref] [PubMed]
  12. G. W. Cottrell, P. Munro, “Principal component analysis of image via back propagation,” in Visual Communications and Image Processing ’88: Third in a Series, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1001, 1070–1076 (1988).
    [Crossref]
  13. H. Bourlard, Y. Kamp, “Auto-association by multilayer perceptrons and singular value decomposition,” Biol. Cybern. 59, 291–294 (1988).
    [Crossref] [PubMed]
  14. D. E. Rumelhart, J. L. McClelland, Parallel Distributed Processing (MIT Press, Cambridge, Mass., 1986).
  15. K. Funahashi, “On the approximate realization of identity mappings by three-layer neural networks,” Inst. Electron. Inform. Commun. Eng. Trans. J73-A, 139–145 (1990).
  16. B. Irie, S. Miyake, “Capabilities of three-layered perceptrons,”IEEE Int. Conf. Neural Networks 1, 641–648 (1988).
    [Crossref]

1990 (1)

K. Funahashi, “On the approximate realization of identity mappings by three-layer neural networks,” Inst. Electron. Inform. Commun. Eng. Trans. J73-A, 139–145 (1990).

1989 (1)

1988 (2)

B. Irie, S. Miyake, “Capabilities of three-layered perceptrons,”IEEE Int. Conf. Neural Networks 1, 641–648 (1988).
[Crossref]

H. Bourlard, Y. Kamp, “Auto-association by multilayer perceptrons and singular value decomposition,” Biol. Cybern. 59, 291–294 (1988).
[Crossref] [PubMed]

1986 (2)

1984 (2)

H. Sobagaki, “New approach to the colorimetric standardization for object colors,” Bull. Electrotechnical Lab. Jpn. 48, 785–792 (1984).

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

1966 (2)

T. N. Wiesel, D. H. Hubel, “Spatial and chromatic interactions in the lateral geniculate nucleus of the rhesus monkey,” J. Neurophysiol. 29, 1115–1156 (1966).
[PubMed]

R. L. De Valois, I. Abramov, G. H. Jacobs, “Analysis of response patterns of LGN cells,”J. Opt. Soc. Am. 56, 966–977 (1966).
[Crossref] [PubMed]

1964 (1)

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

Abramov, I.

Bourlard, H.

H. Bourlard, Y. Kamp, “Auto-association by multilayer perceptrons and singular value decomposition,” Biol. Cybern. 59, 291–294 (1988).
[Crossref] [PubMed]

Cohen, J.

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

Cottrell, G. W.

G. W. Cottrell, P. Munro, “Principal component analysis of image via back propagation,” in Visual Communications and Image Processing ’88: Third in a Series, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1001, 1070–1076 (1988).
[Crossref]

De Valois, R. L.

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]

Funahashi, K.

K. Funahashi, “On the approximate realization of identity mappings by three-layer neural networks,” Inst. Electron. Inform. Commun. Eng. Trans. J73-A, 139–145 (1990).

Hallikaineln, J.

Hubel, D. H.

T. N. Wiesel, D. H. Hubel, “Spatial and chromatic interactions in the lateral geniculate nucleus of the rhesus monkey,” J. Neurophysiol. 29, 1115–1156 (1966).
[PubMed]

Irie, B.

B. Irie, S. Miyake, “Capabilities of three-layered perceptrons,”IEEE Int. Conf. Neural Networks 1, 641–648 (1988).
[Crossref]

Jaaskelainen, T.

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

J. P. S. Parkkinen, T. Jaaskelainen, “Color vision: machine and human,” in Visual Communications and Image Processing IV, W. A. Pearlman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1199, 1184–1192 (1989).
[Crossref]

Jacobs, G. H.

Kamp, Y.

H. Bourlard, Y. Kamp, “Auto-association by multilayer perceptrons and singular value decomposition,” Biol. Cybern. 59, 291–294 (1988).
[Crossref] [PubMed]

Krauskopf, J.

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

Lennie, P.

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

Maloney, L. T.

McClelland, J. L.

D. E. Rumelhart, J. L. McClelland, Parallel Distributed Processing (MIT Press, Cambridge, Mass., 1986).

Mitarai, G.

S. Usui, S. Saruyama, G. Mitarai, M. Sakakibara, T. Yagi, “Image-sensing mechanisms in the vertebrate retina,” Biomechanism (U. Tokyo Press, Tokyo, 1984), Vol. 7, pp. 41–49.

Miyake, S.

B. Irie, S. Miyake, “Capabilities of three-layered perceptrons,”IEEE Int. Conf. Neural Networks 1, 641–648 (1988).
[Crossref]

Munro, P.

G. W. Cottrell, P. Munro, “Principal component analysis of image via back propagation,” in Visual Communications and Image Processing ’88: Third in a Series, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1001, 1070–1076 (1988).
[Crossref]

Oja, E.

E. Oja, Subspace Methods of Pattern Recognition (Research Studies, Letchworth, England, 1983).

Parkkinen, J. P. S.

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

J. P. S. Parkkinen, T. Jaaskelainen, “Color vision: machine and human,” in Visual Communications and Image Processing IV, W. A. Pearlman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1199, 1184–1192 (1989).
[Crossref]

Rumelhart, D. E.

D. E. Rumelhart, J. L. McClelland, Parallel Distributed Processing (MIT Press, Cambridge, Mass., 1986).

Sakakibara, M.

S. Usui, S. Saruyama, G. Mitarai, M. Sakakibara, T. Yagi, “Image-sensing mechanisms in the vertebrate retina,” Biomechanism (U. Tokyo Press, Tokyo, 1984), Vol. 7, pp. 41–49.

Saruyama, S.

S. Usui, S. Saruyama, G. Mitarai, M. Sakakibara, T. Yagi, “Image-sensing mechanisms in the vertebrate retina,” Biomechanism (U. Tokyo Press, Tokyo, 1984), Vol. 7, pp. 41–49.

Sobagaki, H.

H. Sobagaki, “New approach to the colorimetric standardization for object colors,” Bull. Electrotechnical Lab. Jpn. 48, 785–792 (1984).

Usui, S.

S. Usui, S. Saruyama, G. Mitarai, M. Sakakibara, T. Yagi, “Image-sensing mechanisms in the vertebrate retina,” Biomechanism (U. Tokyo Press, Tokyo, 1984), Vol. 7, pp. 41–49.

Wiesel, T. N.

T. N. Wiesel, D. H. Hubel, “Spatial and chromatic interactions in the lateral geniculate nucleus of the rhesus monkey,” J. Neurophysiol. 29, 1115–1156 (1966).
[PubMed]

Yagi, T.

S. Usui, S. Saruyama, G. Mitarai, M. Sakakibara, T. Yagi, “Image-sensing mechanisms in the vertebrate retina,” Biomechanism (U. Tokyo Press, Tokyo, 1984), Vol. 7, pp. 41–49.

Young, R. A.

Biol. Cybern. (1)

H. Bourlard, Y. Kamp, “Auto-association by multilayer perceptrons and singular value decomposition,” Biol. Cybern. 59, 291–294 (1988).
[Crossref] [PubMed]

Bull. Electrotechnical Lab. Jpn. (1)

H. Sobagaki, “New approach to the colorimetric standardization for object colors,” Bull. Electrotechnical Lab. Jpn. 48, 785–792 (1984).

IEEE Int. Conf. Neural Networks (1)

B. Irie, S. Miyake, “Capabilities of three-layered perceptrons,”IEEE Int. Conf. Neural Networks 1, 641–648 (1988).
[Crossref]

Inst. Electron. Inform. Commun. Eng. Trans. (1)

K. Funahashi, “On the approximate realization of identity mappings by three-layer neural networks,” Inst. Electron. Inform. Commun. Eng. Trans. J73-A, 139–145 (1990).

J. Neurophysiol. (1)

T. N. Wiesel, D. H. Hubel, “Spatial and chromatic interactions in the lateral geniculate nucleus of the rhesus monkey,” J. Neurophysiol. 29, 1115–1156 (1966).
[PubMed]

J. Opt. Soc. Am. (1)

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

J. Physiol. (1)

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

Psychonomic Sci. (1)

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

Other (5)

S. Usui, S. Saruyama, G. Mitarai, M. Sakakibara, T. Yagi, “Image-sensing mechanisms in the vertebrate retina,” Biomechanism (U. Tokyo Press, Tokyo, 1984), Vol. 7, pp. 41–49.

J. P. S. Parkkinen, T. Jaaskelainen, “Color vision: machine and human,” in Visual Communications and Image Processing IV, W. A. Pearlman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1199, 1184–1192 (1989).
[Crossref]

E. Oja, Subspace Methods of Pattern Recognition (Research Studies, Letchworth, England, 1983).

G. W. Cottrell, P. Munro, “Principal component analysis of image via back propagation,” in Visual Communications and Image Processing ’88: Third in a Series, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1001, 1070–1076 (1988).
[Crossref]

D. E. Rumelhart, J. L. McClelland, Parallel Distributed Processing (MIT Press, Cambridge, Mass., 1986).

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Figures (10)

Fig. 1
Fig. 1

Munsell color solid for specifying colors on scales of hue, value, and chroma. The hues are arranged with equal angular spacing around the central axis. The vertical columns specify the levels of value. The chroma is represented by the distance from the central axis.

Fig. 2
Fig. 2

Wine-glass-type five-layer neural network that performs an identity mapping. The encoder and decoder each consist of a three-layer neural network. The same data are used for the input and for the teacher, to train the network.

Fig. 3
Fig. 3

Structure of the five-layer neural network. The layers consist of 81, 10, 3, 10, and 81 units, respectively. The numbers of input and output units correspond to the data points of the spectral reflectance. The number of middle-layer units is three, which corresponds to the fundamental number in color vision.

Fig. 4
Fig. 4

Change in the MSE during BP learning. The horizontal lines show errors by the KL expansion with three and four terms for comparison. Note that the error increases when the training data are doubled.

Fig. 5
Fig. 5

MSE for test chips at each learning phase. The dotted lines show errors by the KL expansion with three and four terms for comparison. The average error for test data is considerably lower than that of the KL expansion with three terms. According to this criterion, the network was fairly well trained after the presentation of a small training set, such as 40 to 80 examples.

Fig. 6
Fig. 6

Responses of the units in the middle layer on the constant-value plane. The size of a square shows the absolute value of the response. The open and the filled squares show the positive and the negative values, respectively.

Fig. 7
Fig. 7

Internal representation expressed by activation of each unit in the middle layer. The three axes of space are the responses of the units in the middle layer. Each grid shows unit 1, unit 2, and unit 3 outputs for Munsell colors of constant value. Note that unit 1 responds to the value axis and that both unit 2 and unit 3 represent the chromaticity of input surface spectral-reflectance data.

Fig. 8
Fig. 8

Responses of the units in the middle layer on the constant-value plane in a network with four hidden units. The responses of unit 2 and unit 3 have similar properties with opposite signs. The open and the filled squares show the positive and the negative values, respectively.

Fig. 9
Fig. 9

Response correlation of unit 2 and unit 3 of a network with four hidden units. The responses of these units are highly anticorrelated (r = −0.991).

Fig. 10
Fig. 10

Internal representation of a network with two hidden units. The responses of the hidden units to colors between yellow and blue overlap. The network with two hidden units thus exhibits tritanopia dichromacy.

Equations (2)

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R i ( λ ) = k = 1 n σ k i S k ( λ ) ,
MSE = 1 N i = 1 N j = 1 81 [ R ^ i ( λ j ) - R i ( λ j ) ] 2 ,

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