Abstract

Nonlinear estimation method of spectral reflectance from camera responses is proposed. The proposed method minimizes the mean square error of spectral reflectance when the reflectance can be regarded as a random sequence of Gaussian mixture distribution. In computer simulations, 168 samples of spectral reflectance from a color chart are estimated from their image signals obtained by three- and six-band cameras. It is confirmed that the proposed method improves the accuracy in comparison with the conventional Wiener estimation method.

© 2002 Optical Society of America

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References

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  1. Y. Ohya, T. Obi, M. Yamaguchi, N. Ohyama, Y. Komiya, “Natural color reproduction of human skin for telemedicine,” in Medical Imaging 1998: Image Display, Y. Kim, S. K. Mun, eds. Proc. SPIE3335, 263–270 (1998).
    [CrossRef]
  2. H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, Y. Miyake, “System design for accurately estimating spectral reflectance of art paintings,” Appl. Opt. 39, 6621–6632 (2000).
    [CrossRef]
  3. Th. Keusen, “Multispectral color system with an encoding format compatible with the conventional tristimulus model,” J. Imaging Sci. Technol. 40, 510–515 (1996).
  4. F. Koenig, “Reconstruction of natural spectra from color sensor using nonlinear estimation methods,” in Proceedings of IS&T’s 50th Annual Conference: A Celebration of All Imaging, (The Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 454–458.
  5. N. Tsumura, H. Sato, T. Hasegawa, H. Haneishi, Y. Miyake, “Limitation of color samples for spectral estimation from sensor responses in fine art painting,” Opt. Rev. 6, 57–61 (1999).
    [CrossRef]
  6. F. H. Imai, R. S. Berns, Di-Y. Tzeng, “A comparative analysis of spectral reflecntace estimated in various spaces using a trichromatic camera system,” J. Imaging Sci. Technol. 44, 280–287 (2000).
  7. W. Menke, “Geophysical Data Analysis: Discrete Inverse Theory,” (Academic Press, Inc., San Diego, Calif., 1989), pp. 92–99.
  8. J. MacQueen, “Some methods for classification and analysis of multivariate observations,” in Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, (University of California Press, Berkeley, 1967) Vol. 1, pp. 281–297.
  9. H. Akaike, “Factor analysis and AIC,” Psychometrika, 52, 317–332 (1987).
    [CrossRef]

2000

H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, Y. Miyake, “System design for accurately estimating spectral reflectance of art paintings,” Appl. Opt. 39, 6621–6632 (2000).
[CrossRef]

F. H. Imai, R. S. Berns, Di-Y. Tzeng, “A comparative analysis of spectral reflecntace estimated in various spaces using a trichromatic camera system,” J. Imaging Sci. Technol. 44, 280–287 (2000).

1999

N. Tsumura, H. Sato, T. Hasegawa, H. Haneishi, Y. Miyake, “Limitation of color samples for spectral estimation from sensor responses in fine art painting,” Opt. Rev. 6, 57–61 (1999).
[CrossRef]

1996

Th. Keusen, “Multispectral color system with an encoding format compatible with the conventional tristimulus model,” J. Imaging Sci. Technol. 40, 510–515 (1996).

1987

H. Akaike, “Factor analysis and AIC,” Psychometrika, 52, 317–332 (1987).
[CrossRef]

Akaike, H.

H. Akaike, “Factor analysis and AIC,” Psychometrika, 52, 317–332 (1987).
[CrossRef]

Berns, R. S.

F. H. Imai, R. S. Berns, Di-Y. Tzeng, “A comparative analysis of spectral reflecntace estimated in various spaces using a trichromatic camera system,” J. Imaging Sci. Technol. 44, 280–287 (2000).

Haneishi, H.

H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, Y. Miyake, “System design for accurately estimating spectral reflectance of art paintings,” Appl. Opt. 39, 6621–6632 (2000).
[CrossRef]

N. Tsumura, H. Sato, T. Hasegawa, H. Haneishi, Y. Miyake, “Limitation of color samples for spectral estimation from sensor responses in fine art painting,” Opt. Rev. 6, 57–61 (1999).
[CrossRef]

Hasegawa, T.

H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, Y. Miyake, “System design for accurately estimating spectral reflectance of art paintings,” Appl. Opt. 39, 6621–6632 (2000).
[CrossRef]

N. Tsumura, H. Sato, T. Hasegawa, H. Haneishi, Y. Miyake, “Limitation of color samples for spectral estimation from sensor responses in fine art painting,” Opt. Rev. 6, 57–61 (1999).
[CrossRef]

Hosoi, A.

H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, Y. Miyake, “System design for accurately estimating spectral reflectance of art paintings,” Appl. Opt. 39, 6621–6632 (2000).
[CrossRef]

Imai, F. H.

F. H. Imai, R. S. Berns, Di-Y. Tzeng, “A comparative analysis of spectral reflecntace estimated in various spaces using a trichromatic camera system,” J. Imaging Sci. Technol. 44, 280–287 (2000).

Keusen, Th.

Th. Keusen, “Multispectral color system with an encoding format compatible with the conventional tristimulus model,” J. Imaging Sci. Technol. 40, 510–515 (1996).

Koenig, F.

F. Koenig, “Reconstruction of natural spectra from color sensor using nonlinear estimation methods,” in Proceedings of IS&T’s 50th Annual Conference: A Celebration of All Imaging, (The Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 454–458.

Komiya, Y.

Y. Ohya, T. Obi, M. Yamaguchi, N. Ohyama, Y. Komiya, “Natural color reproduction of human skin for telemedicine,” in Medical Imaging 1998: Image Display, Y. Kim, S. K. Mun, eds. Proc. SPIE3335, 263–270 (1998).
[CrossRef]

MacQueen, J.

J. MacQueen, “Some methods for classification and analysis of multivariate observations,” in Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, (University of California Press, Berkeley, 1967) Vol. 1, pp. 281–297.

Menke, W.

W. Menke, “Geophysical Data Analysis: Discrete Inverse Theory,” (Academic Press, Inc., San Diego, Calif., 1989), pp. 92–99.

Miyake, Y.

H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, Y. Miyake, “System design for accurately estimating spectral reflectance of art paintings,” Appl. Opt. 39, 6621–6632 (2000).
[CrossRef]

N. Tsumura, H. Sato, T. Hasegawa, H. Haneishi, Y. Miyake, “Limitation of color samples for spectral estimation from sensor responses in fine art painting,” Opt. Rev. 6, 57–61 (1999).
[CrossRef]

Obi, T.

Y. Ohya, T. Obi, M. Yamaguchi, N. Ohyama, Y. Komiya, “Natural color reproduction of human skin for telemedicine,” in Medical Imaging 1998: Image Display, Y. Kim, S. K. Mun, eds. Proc. SPIE3335, 263–270 (1998).
[CrossRef]

Ohya, Y.

Y. Ohya, T. Obi, M. Yamaguchi, N. Ohyama, Y. Komiya, “Natural color reproduction of human skin for telemedicine,” in Medical Imaging 1998: Image Display, Y. Kim, S. K. Mun, eds. Proc. SPIE3335, 263–270 (1998).
[CrossRef]

Ohyama, N.

Y. Ohya, T. Obi, M. Yamaguchi, N. Ohyama, Y. Komiya, “Natural color reproduction of human skin for telemedicine,” in Medical Imaging 1998: Image Display, Y. Kim, S. K. Mun, eds. Proc. SPIE3335, 263–270 (1998).
[CrossRef]

Sato, H.

N. Tsumura, H. Sato, T. Hasegawa, H. Haneishi, Y. Miyake, “Limitation of color samples for spectral estimation from sensor responses in fine art painting,” Opt. Rev. 6, 57–61 (1999).
[CrossRef]

Tsumura, N.

H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, Y. Miyake, “System design for accurately estimating spectral reflectance of art paintings,” Appl. Opt. 39, 6621–6632 (2000).
[CrossRef]

N. Tsumura, H. Sato, T. Hasegawa, H. Haneishi, Y. Miyake, “Limitation of color samples for spectral estimation from sensor responses in fine art painting,” Opt. Rev. 6, 57–61 (1999).
[CrossRef]

Tzeng, Di-Y.

F. H. Imai, R. S. Berns, Di-Y. Tzeng, “A comparative analysis of spectral reflecntace estimated in various spaces using a trichromatic camera system,” J. Imaging Sci. Technol. 44, 280–287 (2000).

Yamaguchi, M.

Y. Ohya, T. Obi, M. Yamaguchi, N. Ohyama, Y. Komiya, “Natural color reproduction of human skin for telemedicine,” in Medical Imaging 1998: Image Display, Y. Kim, S. K. Mun, eds. Proc. SPIE3335, 263–270 (1998).
[CrossRef]

Yokoyama, Y.

H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, Y. Miyake, “System design for accurately estimating spectral reflectance of art paintings,” Appl. Opt. 39, 6621–6632 (2000).
[CrossRef]

Appl. Opt.

H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, Y. Miyake, “System design for accurately estimating spectral reflectance of art paintings,” Appl. Opt. 39, 6621–6632 (2000).
[CrossRef]

J. Imaging Sci. Technol.

Th. Keusen, “Multispectral color system with an encoding format compatible with the conventional tristimulus model,” J. Imaging Sci. Technol. 40, 510–515 (1996).

F. H. Imai, R. S. Berns, Di-Y. Tzeng, “A comparative analysis of spectral reflecntace estimated in various spaces using a trichromatic camera system,” J. Imaging Sci. Technol. 44, 280–287 (2000).

Opt. Rev.

N. Tsumura, H. Sato, T. Hasegawa, H. Haneishi, Y. Miyake, “Limitation of color samples for spectral estimation from sensor responses in fine art painting,” Opt. Rev. 6, 57–61 (1999).
[CrossRef]

Psychometrika

H. Akaike, “Factor analysis and AIC,” Psychometrika, 52, 317–332 (1987).
[CrossRef]

Other

W. Menke, “Geophysical Data Analysis: Discrete Inverse Theory,” (Academic Press, Inc., San Diego, Calif., 1989), pp. 92–99.

J. MacQueen, “Some methods for classification and analysis of multivariate observations,” in Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, (University of California Press, Berkeley, 1967) Vol. 1, pp. 281–297.

Y. Ohya, T. Obi, M. Yamaguchi, N. Ohyama, Y. Komiya, “Natural color reproduction of human skin for telemedicine,” in Medical Imaging 1998: Image Display, Y. Kim, S. K. Mun, eds. Proc. SPIE3335, 263–270 (1998).
[CrossRef]

F. Koenig, “Reconstruction of natural spectra from color sensor using nonlinear estimation methods,” in Proceedings of IS&T’s 50th Annual Conference: A Celebration of All Imaging, (The Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 454–458.

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Figures (7)

Fig. 1
Fig. 1

Centers of the clusters in spectrum form, in the case of (a) K = 3 and (b) K = 6.

Fig. 2
Fig. 2

Clustered samples on the plane whose horizontal and vertical axes are the first and the second principal components in the case of (a) K = 3 and (b) K = 6. The designated numbers in the parentheses are the number of the samples classified into a respective class.

Fig. 3
Fig. 3

Spectral sensitivities of the cameras used in the simulations: (a) three bands and (b) six bands.

Fig. 4
Fig. 4

NRMSE of every class of samples for the case of K = 3 and the three-band camera.

Fig. 5
Fig. 5

Average E ab * of every class of samples in the case of K = 3, three-band camera, and viewing illuminant A.

Fig. 6
Fig. 6

Average and maximum E ab * of all samples by Wiener and GMD-based estimations.

Fig. 7
Fig. 7

Proportion of mixing parameters for all samples: (a) K = 3 and (b) K = 6. Samples are listed in the order of classes along the horizontal axis.

Equations (51)

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

P M f = k = 1 K   w k p k f ,
  p k f d f = 1 ,
k = 1 K   w k = 1 ,   0 w k 1 ,
g = Hf + n ,
ε =   f - f ˆ 2
f ˆ =   f | g   =   P f | g f d f ,
P f | g     P G f P G n ,
P G f = C f   exp - 1 2 f T f - 1 f ,
P G n = C n   exp - 1 2 n T n - 1 n ,
P f | g = P G f | g     exp - 1 2 f - f * T * - 1 f - f * ,
f * = f H T H f H T + n - 1 g ,
* =   I - f H T H f H T + n - 1 H f .
f ˆ = f * .
P f | g = AP M f P G n
  P f | g d f = 1 .
P f | g = A   k = 1 K p ¯ k f | g ,
p ¯ k f | g w k p k f P n .
p ¯ k f | g = B k p k f | g ,
B k   p ¯ k f | g d f .
f ˆ =   P f | g f d f = A   k = 1 K B k     p k f | g f d f .
f ˆ = A   k = 1 K   B k f ˆ k .
p k f = p k G f = C k   exp - 1 2 f -   f k T k - 1 f -   f k ,
p k f | g = p k G f | g     exp -   1 2 f - f k * T k * - 1 ×   f - f k * ,
f k * =   f k + k H T H k H T + n - 1 g - H f k
k * =   I - k H T H k H T + n - 1 H k ,
p ¯ k f | g = p ¯ k G f | g   = w k C k C n   exp - 1 2   D k × exp - 1 2 f - f k * T k * - 1 f - f k * ,
D k = g - H f k T H k H T + n - 1 g - H f k .
B k =   w k C k C   exp - 1 2   D k × exp - 1 2 f - f k * T k * - 1 f - f k * d f   = w k C k C   exp - 1 2   D k ×   exp - 1 2 f - f k * T k * - 1 f - f k * d f   = w k C k C   exp - 1 2   D k   2 π L k * .
A = 1 k = 1 K B k .
f ˆ =   k = 1 K B k j = 1 K B j f k * =   k = 1 K   m k f ˆ k * ,
f ˆ = f k * = f ˆ k Wiener .
p ¯ k G f | g = w k C k   exp - 1 2 f - f k T k - 1 f - f k × C n   exp - 1 2 g - Hf T n - 1 g - Hf   = w k C k C n   exp - 1 2 f - f k T k - 1 f - f k +   g - Hf T n - 1 g - Hf .
f k T k - 1 f k + g T n - 1 g ,
f k * T k * - 1 f k * + D k ,
R 2 - R 2 M T R 1 + MR 2 M T - 1 MR 2 =   R 2 - 1 + M T R 1 - 1 M - 1 ,
R 2 M T R 1 + MR 2 M T - 1 =   R 2 - 1 + M T R 1 - 1 M - 1 M T R 1 - 1 ,
k - k H T H k H T + n - 1 H k =   k - 1 + H T n - 1 H - 1 ,
n - 1 - n - 1 H k - 1 + H T n - 1 H - 1 H T n - 1 =   H k H T + n - 1 ,
k H T H k H T + n - 1 = k - 1 + H T n H - 1 H T n - 1 .
k - k H T H k H T + n - 1 H k = k - 1 + H T n - 1 H - 1 = k * ,
n - 1 - n - 1 H k * H T n - 1 = H k H T + n - 1 ,
k H T H k H T + n - 1 = k * H T n - 1 .
f k * =   f k + k H T H k H T + n - 1 g - H f k   = I - k H T H k H T + n - 1 H f k + k H T H k H T + n - 1 g .
f k * = k * k - 1 f k + k * H T n - 1 g .
f k * T k * - 1 f k * =   f k T k - 1 k * k - 1 f k + 2   f k T k - 1 k * H T n - 1 g + g T n - 1 H k * H T n - 1 g .
D k =   g - H f k T H k H T + n - 1 g - H f k   = g T H k H T + n - 1 g - 2   f k T H T H k H T + n - 1 g +   f k T H T H k H T + n - 1 H f k .
g T H f H T + n - 1 g = g T n - 1 - n - 1 H k * H T n - 1 g .
- 2   f k T H T H k H T + n - 1 g = - 2   f k T k - 1 k * H T n - 1 g .
f k T H T H k H T + n - 1 H f k =   f k T k - 1 k H T H k H T + n - 1 k k - 1 H f k =   f k T k - 1 k H T H k H T + n - 1 k k - 1 H f k =   f k T k - 1 k - k * k - 1 f k .
D k = g T n - 1 - n - 1 H k * H T n - 1 g - 2   f k T k - 1 k * H T n - 1 g   + f k T k - 1 k - k * k - 1 f k .
f k * T k * - 1 f k * + D k =   f k T k - 1 f k + g T n - 1 g .

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