Abstract

In multispectral imaging it is advantageous to compress spectral information with a minimum loss of information in a way that permits accurate recovery of the spectrum. By use of the simple vector-subspace model that we propose, spectral information can be stored and recovered by the use of a few inner products, which are easy to measure optically. Two large data sets, the first consisting of 1257 Munsell colors and the other of 218 naturally occurring spectral reflectances, were analyzed to form two bases for the model. The Munsell basis can be used to represent the natural colors, and the basis derived from the natural data can be used to represent the Munsell data. The proposed vector-subspace model includes a simple relation between the inner products and conventional color coordinates. It also provides a way to estimate the spectrum of an object that has known chromaticity coordinates.

© 1990 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  12. J. Parkkinen, T. Jaaskelainen, M. Kuittinen, “Spectral representation of color images” presented at the 9th International Conference on Pattern Recognition, Rome, Italy, November 14–17, 1988.
  13. J. Parkkinen, “Theory of a new color analysis approach,” presented at the European Conference on Mathematics in Industry, Glasgow, Scotland, August 28–31, 1988.
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    [Crossref]
  15. E. L. Krinov, “Spectral reflectance properties of natural formations,” publication TT–439 (National Research Council of Canada, Ottawa, 1947).

1989 (1)

1988 (1)

J. Hallikainen, J. Parkkinen, T. Jaaskelainen, “An acousto-optic color spectrometer,” Rev. Sci. Instrum. 59, 81–83 (1988).
[Crossref]

1987 (2)

1986 (3)

1985 (1)

1983 (1)

1978 (1)

1964 (2)

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

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

Cohen, J.

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

Dixon, E. R.

Hallikainen, J.

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

J. Hallikainen, J. Parkkinen, T. Jaaskelainen, “An acousto-optic color spectrometer,” Rev. Sci. Instrum. 59, 81–83 (1988).
[Crossref]

Jaaskelainen, T.

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

J. Hallikainen, J. Parkkinen, T. Jaaskelainen, “An acousto-optic color spectrometer,” Rev. Sci. Instrum. 59, 81–83 (1988).
[Crossref]

J. Parkkinen, T. Jaaskelainen, “Color representation using statistical pattern recognition,” Appl. Opt. 26, 4240–4245 (1987).
[Crossref] [PubMed]

J. Parkkinen, T. Jaaskelainen, M. Kuittinen, “Spectral representation of color images” presented at the 9th International Conference on Pattern Recognition, Rome, Italy, November 14–17, 1988.

Judd, D. B.

Kawata, S.

Komeda, H.

Krinov, E. L.

E. L. Krinov, “Spectral reflectance properties of natural formations,” publication TT–439 (National Research Council of Canada, Ottawa, 1947).

Kuittinen, M.

J. Parkkinen, T. Jaaskelainen, M. Kuittinen, “Spectral representation of color images” presented at the 9th International Conference on Pattern Recognition, Rome, Italy, November 14–17, 1988.

MacAdam, D. L.

Maloney, L. T.

Minami, S.

Minami, T.

Parkkinen, J.

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

J. Hallikainen, J. Parkkinen, T. Jaaskelainen, “An acousto-optic color spectrometer,” Rev. Sci. Instrum. 59, 81–83 (1988).
[Crossref]

J. Parkkinen, T. Jaaskelainen, “Color representation using statistical pattern recognition,” Appl. Opt. 26, 4240–4245 (1987).
[Crossref] [PubMed]

J. Parkkinen, T. Jaaskelainen, M. Kuittinen, “Spectral representation of color images” presented at the 9th International Conference on Pattern Recognition, Rome, Italy, November 14–17, 1988.

J. Parkkinen, “Theory of a new color analysis approach,” presented at the European Conference on Mathematics in Industry, Glasgow, Scotland, August 28–31, 1988.

Sasaki, K.

Wandell, B. A.

Wyszecki, G.

Young, R. A.

Appl. Opt. (2)

Appl. Spectrosc. (1)

J. Opt. Soc. Am. (2)

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

Psychonomic Sci. (1)

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

Rev. Sci. Instrum. (1)

J. Hallikainen, J. Parkkinen, T. Jaaskelainen, “An acousto-optic color spectrometer,” Rev. Sci. Instrum. 59, 81–83 (1988).
[Crossref]

Other (3)

E. L. Krinov, “Spectral reflectance properties of natural formations,” publication TT–439 (National Research Council of Canada, Ottawa, 1947).

J. Parkkinen, T. Jaaskelainen, M. Kuittinen, “Spectral representation of color images” presented at the 9th International Conference on Pattern Recognition, Rome, Italy, November 14–17, 1988.

J. Parkkinen, “Theory of a new color analysis approach,” presented at the European Conference on Mathematics in Industry, Glasgow, Scotland, August 28–31, 1988.

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Figures (8)

Fig. 1
Fig. 1

First three basis vectors of Munsell colors. The vectors have been normalized to unity length. 1, 2, and 3 denote the first, second, and third basis vectors, respectively.

Fig. 2
Fig. 2

The first basis vectors of a set of 218 natural colors. 1, 2, and 3 denote the first, second, and third basis vectors, respectively.

Fig. 3
Fig. 3

Reconstruction of a yellow color spectrum by use of three basis vectors from the Munsell basis (dashed curve) and from the natural basis (dotted curve). The original spectrum is shown as a solid curve.

Fig. 4
Fig. 4

Average of absolute values of x errors as a function of the number of the basis vectors used in the representation. Here M and N refer to the Munsell and the natural data and basis, respectively.

Fig. 5
Fig. 5

Same as Fig. 4 but for the error in the y color coordinate.

Fig. 6
Fig. 6

Representation of the CIE tristimulus functions using the 8-dimensional subspace spanned by the Munsell basis vectors.

Fig. 7
Fig. 7

The maximum and mean difference between the original Y coordinate and the Y value calculated from the subspace model. The data set contained 1257 reflectance spectra.

Fig. 8
Fig. 8

Typical spectrum of an object recovered from known color coordinates. The original spectrum is shown as the solid curve.

Tables (5)

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Table 1 Noncumulative Error Distributions for 4–10-Dimensional Reconstructions of Natural-Color Spectra Using the Munsell Basis

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Table 2 Noncumulative Error Distributions for 4–10-Dimensional Reconstructions of Natural-Color Spectra Using the Natural Basis

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Table 3 Projections of the Vectors Corresponding to the CIE Tristimulus Functions Multiplied by the CIE C Source Spectrum to 1–8-Dimensional Subspacesa

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Table 4 Errors of the xy Color Coordinates Predicted from the Subspace Projectionsa

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Table 5 Color Distribution of the Sample Set

Equations (12)

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

τ = [ τ ( λ 1 ) , τ ( λ 2 ) , , τ ( λ n ) ] T ,
L = L ( υ 1 , υ 2 , , υ p ) = { x | x = 1 p a i υ i } ,
τ = 1 p ( τ T υ i ) υ i
τ = 1 p ( υ i υ i T ) τ = P τ ,
δ ( τ , L ) = [ ˆ τ ˆ 2 1 p ( τ T υ i ) 2 ] 1 / 2 = [ 1 n [ τ ( λ k ) τ ( λ k ) ] 2 ] 1 / 2 ,
( X Y Z ) = k 1 τ ( λ ) S ( λ ) [ x ¯ ( λ ) y ¯ ( λ ) z ¯ ( λ ) ] d λ ,
X m = k j = 1 n τ ( λ j ) Φ m ( λ j ) ,
X m = k τ T Φ m , m = 1 , 2 , 3 .
Φ m = i = 1 p ( Φ m T υ i ) υ i .
X m = k i = 1 p ( Φ m T υ i ) ( τ T υ i ) = τ T Φ m .
A = k = 1 N τ k τ k T / N
δ 1 = i | τ ( λ i ) τ ( λ i ) | / n .

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