Abstract

In this work a local optimization-based method that is able to recover the reflectance spectra with the desired tristimulus values, choosing the metamer with the most similar shape to the reflectances available in the training set, is proposed. Four different datasets of reflectance spectra and three different error metrics have been used in this study. According to all the error metrics considered, the proposed algorithm was able to recover the spectral reflectances with a higher accuracy than all the state of the art methods considered.

© 2010 Optical Society of America

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References

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  1. D. Dupont, “Study of the reconstruction of reflectance curves based on tristimulus values: comparison of methods of optimization,” Color Res. Appl. 27, 88–99 (2002).
    [CrossRef]
  2. X. Zhang and H. Xu, “Reconstructing spectral reflectance by dividing spectral space and extending the principal components in principal component analysis,” J. Opt. Soc. Am. A 25, 371–378 (2008).
    [CrossRef]
  3. A. Mansouri, T. Sliwa, J. Y. Hardeberg, and Y. Voisin, “An adaptive-pca algorithm for reflectance estimation from color images,” in Proceedings of the 19th IEEE International Conference on Pattern Recognition (IEEE, 2008), pp. 1–4.
    [CrossRef]
  4. R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406–413 (1955).
    [CrossRef]
  5. H. S. Fairman and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
    [CrossRef]
  6. T. Harifi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
    [CrossRef]
  7. J. Hernández-Andrés and J. Romero, “Colorimetric and spectroradiometric characteristics of narrow-field-of-view clear skylight in Granada, Spain,” J. Opt. Soc. Am. A 18, 412–420 (2001).
    [CrossRef]
  8. C. J. Hawkyard, “Synthetic reflectance curves by additive mixing,” J. Soc. Dyers Colour. 109, 323–329 (1993).
    [CrossRef]
  9. S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
    [CrossRef]
  10. P. Comon, “Independent component analysis, a new concept?” Signal Process. 36, 287–314 (1994).
    [CrossRef]
  11. A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
    [CrossRef]
  12. R. M. Lewis and V. Torczon, “Pattern search methods for linearly constrained minimization,” SIAM J. Optim. 10, 917–941 (2000).
    [CrossRef]
  13. M. J. Vhrel, R. Gershon, and L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).
  14. J. Lehtonen, J. Parkkinen, T. Jaaskelainen, and A. Kamshilin, “Principal component and sampling analysis of color spectra,” Opt. Rev. 16, 81–90 (2009).
    [CrossRef]
  15. F. Wilcoxon, “Individual comparisons by ranking methods,” Biometrics 1, 80–83 (1945).
    [CrossRef]

2009 (1)

J. Lehtonen, J. Parkkinen, T. Jaaskelainen, and A. Kamshilin, “Principal component and sampling analysis of color spectra,” Opt. Rev. 16, 81–90 (2009).
[CrossRef]

2008 (3)

X. Zhang and H. Xu, “Reconstructing spectral reflectance by dividing spectral space and extending the principal components in principal component analysis,” J. Opt. Soc. Am. A 25, 371–378 (2008).
[CrossRef]

T. Harifi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
[CrossRef]

2004 (1)

H. S. Fairman and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

2002 (1)

D. Dupont, “Study of the reconstruction of reflectance curves based on tristimulus values: comparison of methods of optimization,” Color Res. Appl. 27, 88–99 (2002).
[CrossRef]

2001 (1)

2000 (1)

R. M. Lewis and V. Torczon, “Pattern search methods for linearly constrained minimization,” SIAM J. Optim. 10, 917–941 (2000).
[CrossRef]

1994 (2)

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

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

1993 (1)

C. J. Hawkyard, “Synthetic reflectance curves by additive mixing,” J. Soc. Dyers Colour. 109, 323–329 (1993).
[CrossRef]

1955 (1)

R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406–413 (1955).
[CrossRef]

1945 (1)

F. Wilcoxon, “Individual comparisons by ranking methods,” Biometrics 1, 80–83 (1945).
[CrossRef]

Agahian, F.

T. Harifi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

Amirshahi, S. H.

T. Harifi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

Brill, M. H.

H. S. Fairman and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

Comon, P.

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

Dupont, D.

D. Dupont, “Study of the reconstruction of reflectance curves based on tristimulus values: comparison of methods of optimization,” Color Res. Appl. 27, 88–99 (2002).
[CrossRef]

Fairman, H. S.

H. S. Fairman and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

Gershon, R.

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

Hardeberg, J. Y.

A. Mansouri, T. Sliwa, J. Y. Hardeberg, and Y. Voisin, “An adaptive-pca algorithm for reflectance estimation from color images,” in Proceedings of the 19th IEEE International Conference on Pattern Recognition (IEEE, 2008), pp. 1–4.
[CrossRef]

Harifi, T.

T. Harifi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

Hawkyard, C. J.

C. J. Hawkyard, “Synthetic reflectance curves by additive mixing,” J. Soc. Dyers Colour. 109, 323–329 (1993).
[CrossRef]

Hernández-Andrés, J.

Hyvärinen, A.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

Iwan, L. S.

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

Jaaskelainen, T.

J. Lehtonen, J. Parkkinen, T. Jaaskelainen, and A. Kamshilin, “Principal component and sampling analysis of color spectra,” Opt. Rev. 16, 81–90 (2009).
[CrossRef]

Kamshilin, A.

J. Lehtonen, J. Parkkinen, T. Jaaskelainen, and A. Kamshilin, “Principal component and sampling analysis of color spectra,” Opt. Rev. 16, 81–90 (2009).
[CrossRef]

Karhunen, J.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

Lehtonen, J.

J. Lehtonen, J. Parkkinen, T. Jaaskelainen, and A. Kamshilin, “Principal component and sampling analysis of color spectra,” Opt. Rev. 16, 81–90 (2009).
[CrossRef]

Lewis, R. M.

R. M. Lewis and V. Torczon, “Pattern search methods for linearly constrained minimization,” SIAM J. Optim. 10, 917–941 (2000).
[CrossRef]

Mansouri, A.

A. Mansouri, T. Sliwa, J. Y. Hardeberg, and Y. Voisin, “An adaptive-pca algorithm for reflectance estimation from color images,” in Proceedings of the 19th IEEE International Conference on Pattern Recognition (IEEE, 2008), pp. 1–4.
[CrossRef]

Oja, E.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

Parkkinen, J.

J. Lehtonen, J. Parkkinen, T. Jaaskelainen, and A. Kamshilin, “Principal component and sampling analysis of color spectra,” Opt. Rev. 16, 81–90 (2009).
[CrossRef]

Penrose, R.

R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406–413 (1955).
[CrossRef]

Romero, J.

Santini, S.

S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
[CrossRef]

Schettini, R.

S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
[CrossRef]

Sliwa, T.

A. Mansouri, T. Sliwa, J. Y. Hardeberg, and Y. Voisin, “An adaptive-pca algorithm for reflectance estimation from color images,” in Proceedings of the 19th IEEE International Conference on Pattern Recognition (IEEE, 2008), pp. 1–4.
[CrossRef]

Torczon, V.

R. M. Lewis and V. Torczon, “Pattern search methods for linearly constrained minimization,” SIAM J. Optim. 10, 917–941 (2000).
[CrossRef]

Vhrel, M. J.

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

Voisin, Y.

A. Mansouri, T. Sliwa, J. Y. Hardeberg, and Y. Voisin, “An adaptive-pca algorithm for reflectance estimation from color images,” in Proceedings of the 19th IEEE International Conference on Pattern Recognition (IEEE, 2008), pp. 1–4.
[CrossRef]

Wilcoxon, F.

F. Wilcoxon, “Individual comparisons by ranking methods,” Biometrics 1, 80–83 (1945).
[CrossRef]

Xu, H.

Zhang, X.

Zuffi, S.

S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
[CrossRef]

Biometrics (1)

F. Wilcoxon, “Individual comparisons by ranking methods,” Biometrics 1, 80–83 (1945).
[CrossRef]

Color Res. Appl. (3)

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

H. S. Fairman and M. H. Brill, “The principal components of reflectances,” Color Res. Appl. 29, 104–110 (2004).
[CrossRef]

D. Dupont, “Study of the reconstruction of reflectance curves based on tristimulus values: comparison of methods of optimization,” Color Res. Appl. 27, 88–99 (2002).
[CrossRef]

IEEE Trans. Signal Process. (1)

S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Soc. Dyers Colour. (1)

C. J. Hawkyard, “Synthetic reflectance curves by additive mixing,” J. Soc. Dyers Colour. 109, 323–329 (1993).
[CrossRef]

Opt. Rev. (2)

J. Lehtonen, J. Parkkinen, T. Jaaskelainen, and A. Kamshilin, “Principal component and sampling analysis of color spectra,” Opt. Rev. 16, 81–90 (2009).
[CrossRef]

T. Harifi, S. H. Amirshahi, and F. Agahian, “Recovery of reflectance spectra from colorimetric data using principal component analysis embedded regression technique,” Opt. Rev. 15, 302–308 (2008).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406–413 (1955).
[CrossRef]

SIAM J. Optim. (1)

R. M. Lewis and V. Torczon, “Pattern search methods for linearly constrained minimization,” SIAM J. Optim. 10, 917–941 (2000).
[CrossRef]

Signal Process. (1)

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

Other (2)

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

A. Mansouri, T. Sliwa, J. Y. Hardeberg, and Y. Voisin, “An adaptive-pca algorithm for reflectance estimation from color images,” in Proceedings of the 19th IEEE International Conference on Pattern Recognition (IEEE, 2008), pp. 1–4.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

The ICA basis V 0 obtained from the training set.

Fig. 2
Fig. 2

The ICA basis B obtained from the spectral residuals.

Fig. 3
Fig. 3

Average spectral residuals between reconstructed and measured spectra on the Munsell training set.

Fig. 4
Fig. 4

Average spectral residuals between reconstructed and measured spectra on the Munsell test set.

Fig. 5
Fig. 5

Average spectral residuals between reconstructed and measured spectra on the Vhrel test set.

Fig. 6
Fig. 6

Average spectral residuals between reconstructed and measured spectra on the MacBeth Color Checker CC test set.

Fig. 7
Fig. 7

Average spectral residuals between reconstructed and measured spectra on the MacBeth Color Checker DC test set.

Tables (7)

Tables Icon

Table 1 Pseudo-Code of the Proposed Algorithm

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Table 2 Algorithm Performances on All the Datasets Considered: Average, 95% Percentile, and Standard Deviation (std) of the Δ E 94 Colorimetric Error under the CIE D65, A, and F2 Illuminants; Percentages of Reconstructed Spectra with Poor, Accurate, and Good Reconstructions Judged by PSNR; Percentages of Reconstructed Spectra with Poor, Accurate, Good, and Excellent Reconstructions Judged by GFC

Tables Icon

Table 3 Algorithm Performances on All the Datasets Considered: Average Δ E 2000 and Δ E CMC 2 : 1 Errors under D65, A, and F2 Illuminants

Tables Icon

Table 4 WST Scores, Evaluated on the Δ E 94 Error Distributions, Obtained by the Algorithms Subdivided for Each Dataset and Illuminant Considered a

Tables Icon

Table 5 Sensitivity Analysis of the Optimization Function with Respect to the Colorimetric Error Term Δ E 94 , i.e., the Weight α

Tables Icon

Table 6 Sensitivity Analysis of the Optimization Function with Respect to the Shape Feasibility Terms, i.e., the Weights β = γ

Tables Icon

Table 7 Sensitivity Analysis of the Optimization Function with Respect to the Spectral Error Term GFC, i.e., the Weight δ

Equations (25)

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

X = k r ( λ ) I ( λ ) x ¯ ( λ ) d λ ,
Y = k r ( λ ) I ( λ ) y ¯ ( λ ) d λ ,
Z = k r ( λ ) I ( λ ) z ¯ ( λ ) d λ ,
k = 100 I ( λ ) y ¯ ( λ ) d λ ,
[ X Y Z ] = M r ,
r = ( M T M ) 1 M T [ X Y Z ] = M + [ X Y Z ] .
r v ¯ + i = 1 k a i v i ,
[ X Y Z ] = M v ¯ + M [ v 1 v 2 v 3 ] [ a 1 a 2 a 3 ] = M v ¯ + M V [ a 1 a 2 a 3 ] ,
[ a 1 a 2 a 3 ] = ( M V ) 1 ( [ X Y Z ] M v ¯ ) .
[ a 1 a 2 a 3 a 4 a 5 a 6 ] = ( M V ) 1 ( [ X i l l 1 Y i l l 1 Z i l l 1 X i l l 2 Y i l l 2 Z i l l 2 ] M v ¯ ) ,
GFC = j = 1 N r j r ̂ j [ j = 1 N ( r j ) 2 ] 1 / 2 [ j = 1 N ( r j ˆ ) 2 ] 1 / 2 ,
r est = X o m 1 + Y o m 2 + Z o m 3 m 1 + m 2 + m 3 ,
Δ X = X X ,
Δ Y = Y Y ,
Δ Z = Z Z ,
X o = X o Δ X ,
Y o = Y o Δ Y ,
Z o = Z o Δ Z .
r 0 = r 0 , f + r 0 , + + r 0 , ,
r 1 = r 0 , f + r ̃ 0 , + + r ̃ 0 , .
u ( r ̂ ) = j = 1 N r ̂ j : r ̂ j < 0.
u + ( r ̂ ) = j = 1 N r ̂ j : r ̂ j > 1.
r ̂ = v ¯ 1 + V 1 [ a ̂ 1 a ̂ 2 a ̂ 3 ] + b ¯ + B [ a ̂ 4 a ̂ 5 a ̂ 6 ] ,
[ a ̂ 1 a ̂ 6 ] = min x R 6 ( α Δ E 94 ( r , r ̂ ) + β u ( r ̂ ) + γ u + ( r ̂ ) + δ ( 1 GFC max ) )
PSNR ( r , r ̂ ) = 20 log 10 1 1 n i = 1 n ( r i r ̂ i ) 2 ,

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