Abstract

This study proposes an effective and accurate mechanism for spectral reflectivity recovery based on a hybrid technique. Adaptive non-negative matrix transformation, three-dimensional interpolation, and two-dimensional interpolation were reconstructed to an integrative hybrid recovery method. The existing spectral reflectivity data of 1269 Munsell matte color chips were used as reference data. Under the standard condition of a D65 illuminant and a 10° observer of 1964 CIE, the spectral reflectivity of the 1269 Munsell colors was reconstructed successfully using the optimized hybrid recovery method. The root mean square error and goodness of fitting were used to determine the quality of the presented method. Using the hybrid method, the strategy for fast and reliable spectral reflectivity recovery of given images were also presented and demonstrated.

© 2012 Optical Society of America

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References

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  1. R. S. Berns, Billmeyer and Saltzman’s Principles of Color Technology, 3rd ed. (Wiley, 2000).
  2. J. Schanda, Colorimetry: Understanding the CIE System, 1st ed. (Wiley, 2007).
  3. N. Ohta and A. Robertson, Colorimetry: Fundamentals and Applications, 1st ed. (Wiley, 2006).
  4. S. G. Kandi and M. A. Tehran, “Applying metamer sets to investigate data dependency of principal component analysis method in recovery of spectral data,” Color Res. Appl. 36, 349–354 (2011).
    [CrossRef]
  5. D. A. Landgrebe, “Information extraction principles and methods for multispectral and hyperspectral remote sensing,” in Information Processing for Remote Sensing, C. H. Chen, ed. (World Scientific, 1999).
  6. L. W. MacDonald and R. Luo, Colour Imaging: Vision and Technology (Wiley, 1999).
  7. S. Westland and C. Ripamonti, Computational Color Science Using Matlab (Wiley, 2004).
  8. G. Camps-Valls and L. Bruzzone, “Kernel-based methods for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens. 43, 1351–1362 (2005).
    [CrossRef]
  9. H. S. Fairman and M. H. Brill, “The principal components of reflectance,” Color Res. Appl. 29, 104–110 (2004).
    [CrossRef]
  10. V. Babaei, S. H. Amirshahi, and F. Agahian, “Using weighted pseudo-inverse method for reconstruction of reflectance spectra and analyzing the dataset in terms of normality,” Color Res. Appl. 36, 295–305 (2011).
    [CrossRef]
  11. F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
    [CrossRef]
  12. 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]
  13. S. Zuffi, S. Santini, and R. Schettini, “From color sensor space to feasible reflectance spectra,” IEEE Trans. Signal Process. 56, 518–531 (2008).
    [CrossRef]
  14. S. Bianco, “Reflectance spectra recovery from tristimulus values by adaptive estimation with metameric shape correction,” J. Opt. Soc. Am. A 27, 1868–1877 (2010).
    [CrossRef]
  15. S. Farajikhah, F. Madanchi, and S. H. Amirshahi, “Nonlinear principal component analysis for compression of spectral data,” in Proceedings of Conference on Data Mining and Data Warehouses (2011), http://ailab.ijs.si/dunja/SiKDD2011/ .
  16. D. D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788–791 (1999).
    [CrossRef]
  17. D. D. Lee and H. S. Seung, “Algorithms for non-negative matrix factorization,” in Proceedings of Neural Information Processing Systems (MIT, 2000), Vol. 13, pp. 556–562.
  18. J. Kim and H. Park, “Fast nonnegative matrix factorization: an active-set-like method and comparisons,” SIAM J. Sci. Comput. 33, 3261–3281 (2011).
    [CrossRef]
  19. S. H. Amirshahi and S. A. Amirhahi, “Adaptive non-negative bases for reconstruction of spectral data from colorimetric information,” Opt. Rev. 17, 562–569 (2010).
    [CrossRef]
  20. F. M. Abed, S. H. Amirshahi, and M. R. M. Abed, “Reconstruction of reflectance data using an interpolation technique,” J. Opt. Soc. Am. A 26, 613–624 (2009).
    [CrossRef]
  21. W. L. Martinez, A. R. Martinez, and J. L. Solka, Exploratory Data Analysis with Matlab, 2nd ed. (Chapman & Hall, 2011).
  22. University of Joensuu Color Group, “Spectral Database,” http://spectral.joensuu.fi .
  23. The Bebel Color Company, “RGB coordinates of the Macbeth Color Checker,” http://www.babelcolor.com/main_level/ColorChecker.htm .
  24. The IRI Color Design, “Hue and Tone 120 system,” http://www.iricolor.com .
  25. Nippon Color and Design Research Institute Inc., “The color chart, marketing, and merchandising color chart,” http://www.ncd-ri.co.jp .
  26. E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E. Herold, and G. C. Walsh, An Engineer’s Guide to Matlab, 3rd ed. (Pearson, 2011).
  27. M. de Berg, M. van Krefeld, M. Overmars, and O. Schwarzkopf, Computational Geometry: Algorithms and Applications, 2nd ed. (Springer, 2000).
  28. I. Amidror, “Scattered data interpolation methods for electronic imaging systems: A survey,” J. Electron. Imaging 11, 157–176 (2002).
    [CrossRef]
  29. P. Green and L. MacDonald, Color Engineering Achieving Device Independent Colour (Addison-Wesley, 2002).
  30. J. Kasson, W. Plouffe, and S. Nin, “A tetrahedral interpolation technique for color space conversion,” Proc. SPIE 1909, 127–138 (1993).
    [CrossRef]
  31. W. L. Martinez and A. R. Martinez, Computational Statistics Handbook with Matlab, 2nd ed. (Chapman & Hall, 2002).
  32. We have prepared the supplementary materials for the detailed comparison of our technique with that of adaptive PCA and the method of [14]. It can be downloaded from our website: http://home.pusan.ac.kr/~boggikim/hybrid/supforjosa1.docx .

2011 (3)

V. Babaei, S. H. Amirshahi, and F. Agahian, “Using weighted pseudo-inverse method for reconstruction of reflectance spectra and analyzing the dataset in terms of normality,” Color Res. Appl. 36, 295–305 (2011).
[CrossRef]

S. G. Kandi and M. A. Tehran, “Applying metamer sets to investigate data dependency of principal component analysis method in recovery of spectral data,” Color Res. Appl. 36, 349–354 (2011).
[CrossRef]

J. Kim and H. Park, “Fast nonnegative matrix factorization: an active-set-like method and comparisons,” SIAM J. Sci. Comput. 33, 3261–3281 (2011).
[CrossRef]

2010 (2)

S. H. Amirshahi and S. A. Amirhahi, “Adaptive non-negative bases for reconstruction of spectral data from colorimetric information,” Opt. Rev. 17, 562–569 (2010).
[CrossRef]

S. Bianco, “Reflectance spectra recovery from tristimulus values by adaptive estimation with metameric shape correction,” J. Opt. Soc. Am. A 27, 1868–1877 (2010).
[CrossRef]

2009 (1)

2008 (3)

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (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]

2005 (1)

G. Camps-Valls and L. Bruzzone, “Kernel-based methods for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens. 43, 1351–1362 (2005).
[CrossRef]

2004 (1)

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

2002 (1)

I. Amidror, “Scattered data interpolation methods for electronic imaging systems: A survey,” J. Electron. Imaging 11, 157–176 (2002).
[CrossRef]

1999 (1)

D. D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788–791 (1999).
[CrossRef]

1993 (1)

J. Kasson, W. Plouffe, and S. Nin, “A tetrahedral interpolation technique for color space conversion,” Proc. SPIE 1909, 127–138 (1993).
[CrossRef]

Abed, F. M.

Abed, M. R. M.

Agahian, F.

V. Babaei, S. H. Amirshahi, and F. Agahian, “Using weighted pseudo-inverse method for reconstruction of reflectance spectra and analyzing the dataset in terms of normality,” Color Res. Appl. 36, 295–305 (2011).
[CrossRef]

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (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]

Amidror, I.

I. Amidror, “Scattered data interpolation methods for electronic imaging systems: A survey,” J. Electron. Imaging 11, 157–176 (2002).
[CrossRef]

Amirhahi, S. A.

S. H. Amirshahi and S. A. Amirhahi, “Adaptive non-negative bases for reconstruction of spectral data from colorimetric information,” Opt. Rev. 17, 562–569 (2010).
[CrossRef]

Amirshahi, S. A.

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

Amirshahi, S. H.

V. Babaei, S. H. Amirshahi, and F. Agahian, “Using weighted pseudo-inverse method for reconstruction of reflectance spectra and analyzing the dataset in terms of normality,” Color Res. Appl. 36, 295–305 (2011).
[CrossRef]

S. H. Amirshahi and S. A. Amirhahi, “Adaptive non-negative bases for reconstruction of spectral data from colorimetric information,” Opt. Rev. 17, 562–569 (2010).
[CrossRef]

F. M. Abed, S. H. Amirshahi, and M. R. M. Abed, “Reconstruction of reflectance data using an interpolation technique,” J. Opt. Soc. Am. A 26, 613–624 (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]

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

Azarm, S.

E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E. Herold, and G. C. Walsh, An Engineer’s Guide to Matlab, 3rd ed. (Pearson, 2011).

Babaei, V.

V. Babaei, S. H. Amirshahi, and F. Agahian, “Using weighted pseudo-inverse method for reconstruction of reflectance spectra and analyzing the dataset in terms of normality,” Color Res. Appl. 36, 295–305 (2011).
[CrossRef]

Balachandran, B.

E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E. Herold, and G. C. Walsh, An Engineer’s Guide to Matlab, 3rd ed. (Pearson, 2011).

Berns, R. S.

R. S. Berns, Billmeyer and Saltzman’s Principles of Color Technology, 3rd ed. (Wiley, 2000).

Bianco, S.

Brill, M. H.

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

Bruzzone, L.

G. Camps-Valls and L. Bruzzone, “Kernel-based methods for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens. 43, 1351–1362 (2005).
[CrossRef]

Camps-Valls, G.

G. Camps-Valls and L. Bruzzone, “Kernel-based methods for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens. 43, 1351–1362 (2005).
[CrossRef]

de Berg, M.

M. de Berg, M. van Krefeld, M. Overmars, and O. Schwarzkopf, Computational Geometry: Algorithms and Applications, 2nd ed. (Springer, 2000).

Duncan, J. H.

E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E. Herold, and G. C. Walsh, An Engineer’s Guide to Matlab, 3rd ed. (Pearson, 2011).

Fairman, H. S.

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

Green, P.

P. Green and L. MacDonald, Color Engineering Achieving Device Independent Colour (Addison-Wesley, 2002).

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]

Herold, K. E.

E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E. Herold, and G. C. Walsh, An Engineer’s Guide to Matlab, 3rd ed. (Pearson, 2011).

Kandi, S. G.

S. G. Kandi and M. A. Tehran, “Applying metamer sets to investigate data dependency of principal component analysis method in recovery of spectral data,” Color Res. Appl. 36, 349–354 (2011).
[CrossRef]

Kasson, J.

J. Kasson, W. Plouffe, and S. Nin, “A tetrahedral interpolation technique for color space conversion,” Proc. SPIE 1909, 127–138 (1993).
[CrossRef]

Kim, J.

J. Kim and H. Park, “Fast nonnegative matrix factorization: an active-set-like method and comparisons,” SIAM J. Sci. Comput. 33, 3261–3281 (2011).
[CrossRef]

Landgrebe, D. A.

D. A. Landgrebe, “Information extraction principles and methods for multispectral and hyperspectral remote sensing,” in Information Processing for Remote Sensing, C. H. Chen, ed. (World Scientific, 1999).

Lee, D. D.

D. D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788–791 (1999).
[CrossRef]

D. D. Lee and H. S. Seung, “Algorithms for non-negative matrix factorization,” in Proceedings of Neural Information Processing Systems (MIT, 2000), Vol. 13, pp. 556–562.

Luo, R.

L. W. MacDonald and R. Luo, Colour Imaging: Vision and Technology (Wiley, 1999).

MacDonald, L.

P. Green and L. MacDonald, Color Engineering Achieving Device Independent Colour (Addison-Wesley, 2002).

MacDonald, L. W.

L. W. MacDonald and R. Luo, Colour Imaging: Vision and Technology (Wiley, 1999).

Magrab, E. B.

E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E. Herold, and G. C. Walsh, An Engineer’s Guide to Matlab, 3rd ed. (Pearson, 2011).

Martinez, A. R.

W. L. Martinez and A. R. Martinez, Computational Statistics Handbook with Matlab, 2nd ed. (Chapman & Hall, 2002).

W. L. Martinez, A. R. Martinez, and J. L. Solka, Exploratory Data Analysis with Matlab, 2nd ed. (Chapman & Hall, 2011).

Martinez, W. L.

W. L. Martinez, A. R. Martinez, and J. L. Solka, Exploratory Data Analysis with Matlab, 2nd ed. (Chapman & Hall, 2011).

W. L. Martinez and A. R. Martinez, Computational Statistics Handbook with Matlab, 2nd ed. (Chapman & Hall, 2002).

Nin, S.

J. Kasson, W. Plouffe, and S. Nin, “A tetrahedral interpolation technique for color space conversion,” Proc. SPIE 1909, 127–138 (1993).
[CrossRef]

Ohta, N.

N. Ohta and A. Robertson, Colorimetry: Fundamentals and Applications, 1st ed. (Wiley, 2006).

Overmars, M.

M. de Berg, M. van Krefeld, M. Overmars, and O. Schwarzkopf, Computational Geometry: Algorithms and Applications, 2nd ed. (Springer, 2000).

Park, H.

J. Kim and H. Park, “Fast nonnegative matrix factorization: an active-set-like method and comparisons,” SIAM J. Sci. Comput. 33, 3261–3281 (2011).
[CrossRef]

Plouffe, W.

J. Kasson, W. Plouffe, and S. Nin, “A tetrahedral interpolation technique for color space conversion,” Proc. SPIE 1909, 127–138 (1993).
[CrossRef]

Ripamonti, C.

S. Westland and C. Ripamonti, Computational Color Science Using Matlab (Wiley, 2004).

Robertson, A.

N. Ohta and A. Robertson, Colorimetry: Fundamentals and Applications, 1st ed. (Wiley, 2006).

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]

Schanda, J.

J. Schanda, Colorimetry: Understanding the CIE System, 1st ed. (Wiley, 2007).

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]

Schwarzkopf, O.

M. de Berg, M. van Krefeld, M. Overmars, and O. Schwarzkopf, Computational Geometry: Algorithms and Applications, 2nd ed. (Springer, 2000).

Seung, H. S.

D. D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788–791 (1999).
[CrossRef]

D. D. Lee and H. S. Seung, “Algorithms for non-negative matrix factorization,” in Proceedings of Neural Information Processing Systems (MIT, 2000), Vol. 13, pp. 556–562.

Solka, J. L.

W. L. Martinez, A. R. Martinez, and J. L. Solka, Exploratory Data Analysis with Matlab, 2nd ed. (Chapman & Hall, 2011).

Tehran, M. A.

S. G. Kandi and M. A. Tehran, “Applying metamer sets to investigate data dependency of principal component analysis method in recovery of spectral data,” Color Res. Appl. 36, 349–354 (2011).
[CrossRef]

van Krefeld, M.

M. de Berg, M. van Krefeld, M. Overmars, and O. Schwarzkopf, Computational Geometry: Algorithms and Applications, 2nd ed. (Springer, 2000).

Walsh, G. C.

E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E. Herold, and G. C. Walsh, An Engineer’s Guide to Matlab, 3rd ed. (Pearson, 2011).

Westland, S.

S. Westland and C. Ripamonti, Computational Color Science Using Matlab (Wiley, 2004).

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]

Color Res. Appl. (4)

S. G. Kandi and M. A. Tehran, “Applying metamer sets to investigate data dependency of principal component analysis method in recovery of spectral data,” Color Res. Appl. 36, 349–354 (2011).
[CrossRef]

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

V. Babaei, S. H. Amirshahi, and F. Agahian, “Using weighted pseudo-inverse method for reconstruction of reflectance spectra and analyzing the dataset in terms of normality,” Color Res. Appl. 36, 295–305 (2011).
[CrossRef]

F. Agahian, S. A. Amirshahi, and S. H. Amirshahi, “Reconstruction of reflectance spectra using weighted principal component analysis,” Color Res. Appl. 33, 360–371 (2008).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

G. Camps-Valls and L. Bruzzone, “Kernel-based methods for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens. 43, 1351–1362 (2005).
[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. Electron. Imaging (1)

I. Amidror, “Scattered data interpolation methods for electronic imaging systems: A survey,” J. Electron. Imaging 11, 157–176 (2002).
[CrossRef]

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

Nature (1)

D. D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788–791 (1999).
[CrossRef]

Opt. Rev. (2)

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. H. Amirshahi and S. A. Amirhahi, “Adaptive non-negative bases for reconstruction of spectral data from colorimetric information,” Opt. Rev. 17, 562–569 (2010).
[CrossRef]

Proc. SPIE (1)

J. Kasson, W. Plouffe, and S. Nin, “A tetrahedral interpolation technique for color space conversion,” Proc. SPIE 1909, 127–138 (1993).
[CrossRef]

SIAM J. Sci. Comput. (1)

J. Kim and H. Park, “Fast nonnegative matrix factorization: an active-set-like method and comparisons,” SIAM J. Sci. Comput. 33, 3261–3281 (2011).
[CrossRef]

Other (18)

D. D. Lee and H. S. Seung, “Algorithms for non-negative matrix factorization,” in Proceedings of Neural Information Processing Systems (MIT, 2000), Vol. 13, pp. 556–562.

S. Farajikhah, F. Madanchi, and S. H. Amirshahi, “Nonlinear principal component analysis for compression of spectral data,” in Proceedings of Conference on Data Mining and Data Warehouses (2011), http://ailab.ijs.si/dunja/SiKDD2011/ .

R. S. Berns, Billmeyer and Saltzman’s Principles of Color Technology, 3rd ed. (Wiley, 2000).

J. Schanda, Colorimetry: Understanding the CIE System, 1st ed. (Wiley, 2007).

N. Ohta and A. Robertson, Colorimetry: Fundamentals and Applications, 1st ed. (Wiley, 2006).

D. A. Landgrebe, “Information extraction principles and methods for multispectral and hyperspectral remote sensing,” in Information Processing for Remote Sensing, C. H. Chen, ed. (World Scientific, 1999).

L. W. MacDonald and R. Luo, Colour Imaging: Vision and Technology (Wiley, 1999).

S. Westland and C. Ripamonti, Computational Color Science Using Matlab (Wiley, 2004).

W. L. Martinez and A. R. Martinez, Computational Statistics Handbook with Matlab, 2nd ed. (Chapman & Hall, 2002).

We have prepared the supplementary materials for the detailed comparison of our technique with that of adaptive PCA and the method of [14]. It can be downloaded from our website: http://home.pusan.ac.kr/~boggikim/hybrid/supforjosa1.docx .

P. Green and L. MacDonald, Color Engineering Achieving Device Independent Colour (Addison-Wesley, 2002).

W. L. Martinez, A. R. Martinez, and J. L. Solka, Exploratory Data Analysis with Matlab, 2nd ed. (Chapman & Hall, 2011).

University of Joensuu Color Group, “Spectral Database,” http://spectral.joensuu.fi .

The Bebel Color Company, “RGB coordinates of the Macbeth Color Checker,” http://www.babelcolor.com/main_level/ColorChecker.htm .

The IRI Color Design, “Hue and Tone 120 system,” http://www.iricolor.com .

Nippon Color and Design Research Institute Inc., “The color chart, marketing, and merchandising color chart,” http://www.ncd-ri.co.jp .

E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E. Herold, and G. C. Walsh, An Engineer’s Guide to Matlab, 3rd ed. (Pearson, 2011).

M. de Berg, M. van Krefeld, M. Overmars, and O. Schwarzkopf, Computational Geometry: Algorithms and Applications, 2nd ed. (Springer, 2000).

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Figures (6)

Fig. 1.
Fig. 1.

(a) CIE xy and (b) CIELAB representation of 1269 Munsell color chip data with D65 illuminant and 1964 10° observer. The colors in the figure show the RGB colors of each data point. (c) Convex hull and 2D triangulation of the CIE xy values of the 1269 Munsell color chips. (d) Convex hull and 3D triangulation of the CIE XYZ values of the 1269 Munsell color chips.

Fig. 2.
Fig. 2.

(a) Example of a 3D interpolation. Four black diamonds are the vertices of the smallest tetrahedron surrounding the random target point diamond inside a tetrahedron. The cross is CIE XYZ result calculated from the reconstructed reflectivity. (b) Reconstructed reflectivity (curve with diamonds) from four reflectivity data of tetrahedral vertices (black curve). (c) ΔE between the target data and reconstructed data. (d) GOF for the 1269 Munsell color chips with the 3D interpolation.

Fig. 3.
Fig. 3.

(a) Example of 2D interpolation. Three Black diamonds are the vertices of the smallest triangle surrounding the random target point (diamond inside a triangle). The red cross is CIE XYZ result calculated from the reconstructed reflectivity. (b) Reconstructed reflectivity (line with diamonds) from the three reflectivity data of triangular vertices (black curve). (c) ΔE between the target data and reconstructed data. (d) GOF for the 1269 Munsell color chips with the 2D interpolation.

Fig. 4.
Fig. 4.

(a) Example of NMT technique. Three adapted non-negative bases used to reconstruct randomly chosen target data. (b) Reconstructed reflectivity using the adaptive NMT (line with diamonds). (c) ΔE between the target data and reconstructed data. (d) GOF for the 1269 Munsell color chips with the NMT technique.

Fig. 5.
Fig. 5.

(a) Proposed hybrid method for reflectivity reconstruction based on a conditional judgment on 3D, 2D interpolation and NMT technique. (b) GOF for the 1269 Munsell color chips with the hybrid technique.

Fig. 6.
Fig. 6.

(a) Macbeth color chart and (b) hue-and-tone 120 color chart used to test the hybrid method of reflectivity reconstruction. CIE xy color diagram of the colors from (c) Macbeth color chart and (d) hue-and-tone 120 color chart. The convex hull from the 1269 Munsell color chips is shown as a polygon. The reconstructed reflectivity from the hybrid method for (e) Macbeth color chart and (f) hue-and-tone 120 color chart.

Tables (2)

Tables Icon

Table 1. Pseudo-Code of the Proposed Algorithms, 3D Interpolation, 2D Interpolation, Adaptive NMT, and Hybrid Method of Reflectivity Reconstruction

Tables Icon

Table 2. Statistical Summary of Various Reflectivity Reconstruction Methods Applying 1269 Munsell Color Dataa

Equations (17)

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

Pt=aP1+bP2+cP3+dP4.
aX1+bX2+cX3+dX4=Xt,aY1+bY2+cY3+dY4=Yt,aZ1+bZ2+cZ3+dZ4=Zt,a+b+c+d=1.
Rt=aR1+bR2+cR3+dR4.
(abcd)=[X1X2X3X4Y1Y2Y3Y4Z1Z2Z3Z41111]1(XtYtZt1).
xi=Xi/(Xi+Yi+Zi),yi=Yi/(Xi+Yi+Zi),zi=Zi/(Xi+Yi+Zi).
αX1+(1α)X2=XT,αY1+(1α)Y2=YT,αZ1+(1α)Z2=ZT.
αx1+(1α)x2xT,αy1+(1α)y2yT,αz1+(1α)z2zT.
γT=XTYT=xTyT.
α=γTY2X2X1X2γT(Y1Y2).
RT=αR1+(1α)R2.
RUVi=1kviui,
T=ATR,
T=ATRATUV.
V=(ATU)1T.
Vf=(ATUf)1Tf,
GOF=j=1NΔRjΔRjr[j=1N(ΔRj)2]1/2[j=1N(ΔRjr)2]1/2,ΔRj=RjRj,ΔRjr=RjrRjr,
ΔE=(ΔL)2+(Δa)2+(Δb)2.

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