Abstract

Principal component analysis (PCA) and weighted PCA were applied to spectra of optimal colors belonging to the outer surface of the object-color solid or to so-called MacAdam limits. The correlation matrix formed from this data is a circulant matrix whose biggest eigenvalue is simple and the corresponding eigenvector is constant. All other eigenvalues are double, and the eigenvectors can be expressed with trigonometric functions. Found trigonometric functions can be used as a general basis to reconstruct all possible smooth reflectance spectra. When the spectral data are weighted with an appropriate weight function, the essential part of the color information is compressed to the first three components and the shapes of the first three eigenvectors correspond to one achromatic response function and to two chromatic response functions, the latter corresponding approximately to Munsell opponent-hue directions 9YR-9B and 2BG-2R.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Flinkman, H. Laamanen, P. Vahimaa, and M. Hauta-Kasari, “Number of colors generated by smooth nonfluorescent reflectance spectra,” J. Opt. Soc. Am. A 29, 2566–2575 (2012).
    [CrossRef]
  2. R. Ramanath, R. G. Kuehni, W. E. Snyder, and D. Hinks, “Spectral spaces and color spaces,” Color Res. Appl. 29, 29–37 (2004).
    [CrossRef]
  3. D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
    [CrossRef]
  4. B. G. Kim, J. Han, and S. Parkm, “Spectral reflectivity recovery from the tristimulus values using a hybrid method,” J. Opt. Soc. Am. A 29, 2612–2621 (2012).
    [CrossRef]
  5. 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]
  6. H. Laamanen, T. Jetsu, T. Jaaskelainen, and J. Parkkinen, “Weighted compression of spectral color information,” J. Opt. Soc. Am. A 25, 1383–1388 (2008).
    [CrossRef]
  7. 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]
  8. H. Laamanen, T. Jaaskelainen, and J. P. S. Parkkinen, “Comparison of PCA and ICA in color recognition,” Proc. SPIE 4197, 367–377 (2000).
    [CrossRef]
  9. T. Jaaskelainen, J. Parkkinen, and S. Toyooka, “Vector-subspace model for color representation,” J. Opt. Soc. Am. A 7, 725–730 (1990).
    [CrossRef]
  10. Y. Mizokami and M. A. Webster, “Are Gaussian spectra a viable perceptual assumption in color appearance?” J. Opt. Soc. Am. A 29, A10–A18 (2012).
    [CrossRef]
  11. G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 2000), pp. 179–183.
  12. R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2, 155–239 (2005).
    [CrossRef]
  13. P. J. Davis, Circulant Matrices (Wiley, 1979).
  14. Spectral Database, University of Eastern Finland Color Group, http://www.uef.fi/spectral/spectral-database .
  15. Colorimetry, CIE 15:2004 (CIE, 2004).
  16. C. S. McCamy, H. Marcus, and J. G. Davidson, “A color-rendition chart,” J. App. Photo. Eng. 2, 95–99 (1978).
  17. Spectral Image Database, University of Eastern Finland Color Group, http://www.uef.fi/spectral/spectral-image-database .
  18. H. Laamanen, T. Jääskeläinen, and J. Parkkinen, “Conversion between the reflectance spectra and the Munsell notations,” Color Res. Appl. 31, 57–66 (2006).
    [CrossRef]

2012 (3)

2008 (2)

H. Laamanen, T. Jetsu, T. Jaaskelainen, and J. Parkkinen, “Weighted compression of spectral color information,” J. Opt. Soc. Am. A 25, 1383–1388 (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]

2006 (1)

H. Laamanen, T. Jääskeläinen, and J. Parkkinen, “Conversion between the reflectance spectra and the Munsell notations,” Color Res. Appl. 31, 57–66 (2006).
[CrossRef]

2005 (2)

D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2, 155–239 (2005).
[CrossRef]

2004 (1)

R. Ramanath, R. G. Kuehni, W. E. Snyder, and D. Hinks, “Spectral spaces and color spaces,” Color Res. Appl. 29, 29–37 (2004).
[CrossRef]

2000 (1)

H. Laamanen, T. Jaaskelainen, and J. P. S. Parkkinen, “Comparison of PCA and ICA in color recognition,” Proc. SPIE 4197, 367–377 (2000).
[CrossRef]

1990 (1)

1986 (1)

1978 (1)

C. S. McCamy, H. Marcus, and J. G. Davidson, “A color-rendition chart,” J. App. Photo. Eng. 2, 95–99 (1978).

Agahian, F.

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. 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.

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]

Berns, R. S.

D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

Davidson, J. G.

C. S. McCamy, H. Marcus, and J. G. Davidson, “A color-rendition chart,” J. App. Photo. Eng. 2, 95–99 (1978).

Davis, P. J.

P. J. Davis, Circulant Matrices (Wiley, 1979).

Flinkman, M.

Gray, R. M.

R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2, 155–239 (2005).
[CrossRef]

Han, J.

Hauta-Kasari, M.

Hinks, D.

R. Ramanath, R. G. Kuehni, W. E. Snyder, and D. Hinks, “Spectral spaces and color spaces,” Color Res. Appl. 29, 29–37 (2004).
[CrossRef]

Jaaskelainen, T.

Jääskeläinen, T.

H. Laamanen, T. Jääskeläinen, and J. Parkkinen, “Conversion between the reflectance spectra and the Munsell notations,” Color Res. Appl. 31, 57–66 (2006).
[CrossRef]

Jetsu, T.

Kim, B. G.

Kuehni, R. G.

R. Ramanath, R. G. Kuehni, W. E. Snyder, and D. Hinks, “Spectral spaces and color spaces,” Color Res. Appl. 29, 29–37 (2004).
[CrossRef]

Laamanen, H.

M. Flinkman, H. Laamanen, P. Vahimaa, and M. Hauta-Kasari, “Number of colors generated by smooth nonfluorescent reflectance spectra,” J. Opt. Soc. Am. A 29, 2566–2575 (2012).
[CrossRef]

H. Laamanen, T. Jetsu, T. Jaaskelainen, and J. Parkkinen, “Weighted compression of spectral color information,” J. Opt. Soc. Am. A 25, 1383–1388 (2008).
[CrossRef]

H. Laamanen, T. Jääskeläinen, and J. Parkkinen, “Conversion between the reflectance spectra and the Munsell notations,” Color Res. Appl. 31, 57–66 (2006).
[CrossRef]

H. Laamanen, T. Jaaskelainen, and J. P. S. Parkkinen, “Comparison of PCA and ICA in color recognition,” Proc. SPIE 4197, 367–377 (2000).
[CrossRef]

Maloney, L. T.

Marcus, H.

C. S. McCamy, H. Marcus, and J. G. Davidson, “A color-rendition chart,” J. App. Photo. Eng. 2, 95–99 (1978).

McCamy, C. S.

C. S. McCamy, H. Marcus, and J. G. Davidson, “A color-rendition chart,” J. App. Photo. Eng. 2, 95–99 (1978).

Mizokami, Y.

Parkkinen, J.

Parkkinen, J. P. S.

H. Laamanen, T. Jaaskelainen, and J. P. S. Parkkinen, “Comparison of PCA and ICA in color recognition,” Proc. SPIE 4197, 367–377 (2000).
[CrossRef]

Parkm, S.

Ramanath, R.

R. Ramanath, R. G. Kuehni, W. E. Snyder, and D. Hinks, “Spectral spaces and color spaces,” Color Res. Appl. 29, 29–37 (2004).
[CrossRef]

Snyder, W. E.

R. Ramanath, R. G. Kuehni, W. E. Snyder, and D. Hinks, “Spectral spaces and color spaces,” Color Res. Appl. 29, 29–37 (2004).
[CrossRef]

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 2000), pp. 179–183.

Toyooka, S.

Tzeng, D. Y.

D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[CrossRef]

Vahimaa, P.

Webster, M. A.

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 2000), pp. 179–183.

Color Res. Appl. (4)

R. Ramanath, R. G. Kuehni, W. E. Snyder, and D. Hinks, “Spectral spaces and color spaces,” Color Res. Appl. 29, 29–37 (2004).
[CrossRef]

D. Y. Tzeng and R. S. Berns, “A review of principal component analysis and its applications to color technology,” Color Res. Appl. 30, 84–98 (2005).
[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]

H. Laamanen, T. Jääskeläinen, and J. Parkkinen, “Conversion between the reflectance spectra and the Munsell notations,” Color Res. Appl. 31, 57–66 (2006).
[CrossRef]

Found. Trends Commun. Inf. Theory (1)

R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2, 155–239 (2005).
[CrossRef]

J. App. Photo. Eng. (1)

C. S. McCamy, H. Marcus, and J. G. Davidson, “A color-rendition chart,” J. App. Photo. Eng. 2, 95–99 (1978).

J. Opt. Soc. Am. A (6)

Proc. SPIE (1)

H. Laamanen, T. Jaaskelainen, and J. P. S. Parkkinen, “Comparison of PCA and ICA in color recognition,” Proc. SPIE 4197, 367–377 (2000).
[CrossRef]

Other (5)

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 2000), pp. 179–183.

Spectral Image Database, University of Eastern Finland Color Group, http://www.uef.fi/spectral/spectral-image-database .

P. J. Davis, Circulant Matrices (Wiley, 1979).

Spectral Database, University of Eastern Finland Color Group, http://www.uef.fi/spectral/spectral-database .

Colorimetry, CIE 15:2004 (CIE, 2004).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Grayscale presentation of the correlation matrix of the optimal color spectra.

Fig. 2.
Fig. 2.

(a) Example of the spectral reconstructions. The spectra numbers 190 and 495 from the Munsell dataset (solid lines) are reconstructed with seven eigenvectors solved from the set of optimal color spectra (dashed lines). (b) Error bands for the reconstructions of Munsell data. Plotted curves represent the average spectral difference as a function of the wavelength. (Solid line) Spectra reconstructed with seven eigenvectors solved from the Munsell data; (dashed line) spectra reconstructed with seven eigenvectors solved from the set of optimal color spectra.

Fig. 3.
Fig. 3.

Mean color differences for the Munsell and Pantone datasets and two spectral images reconstructed by using different numbers of components in compression. Munsell and Pantone datasets were reconstructed with their own eigenvectors (dashed lines) and with the eigenvectors solved from optimal colors (solid lines). The spectral images were reconstructed using only the eigenvectors solved from the set of optimal colors.

Fig. 4.
Fig. 4.

True-color representations of the PCs and their combinations. (a)–(d) First, second, third, and fourth eigenvectors, respectively. (e) True-color representation of the original spectral image taken from an arrangement of fruits and flowers. GB is a gray background. (f)–(h) Colors related to each eigenvector against gray background. (i) Achromatic color information carried by the first eigenvector. (j)–(l) Colors related to each eigenvector against achromatic color information. (m)–(p) Combined color information from several eigenvectors.

Fig. 5.
Fig. 5.

True-color representations of the PCs and their combinations. (a)–(d) First, second, third, and fourth eigenvectors, respectively. (e) True-color representation of the original spectral image taken from the GretagMacBeth ColorChecker. GB is a gray background. (f)–(h) Colors related to each PC against gray background. (i) Achromatic color information carried by the first eigenvector. (j)–(l) Colors related to each eigenvector against achromatic color information. (m)–(p) Combined color information from several eigenvectors.

Fig. 6.
Fig. 6.

First three corrected eigenvectors calculated from the set of optimal color spectra (solid lines) compared with the estimated response functions of the human visual systems (dashed lines). The first eigenvector (square) is compared to a V(λ) curve, and the second and third eigenvectors (circle and diamond) are compared to chromatic response functions corresponding to the Munsell opponent-hue direction 9YR-9B and 2BG-2R.

Tables (4)

Tables Icon

Table 1. Munsell Data Reconstructed by Using the Eigenvectors Solved from the Munsell Data and Optimal Color Spectraa

Tables Icon

Table 2. Pantone Data Reconstructed by Using the Eigenvectors Solved from the Pantone Data and Optimal Color Spectraa

Tables Icon

Table 3. Munsell Data Reconstructed by Using the Eigenvectors Solved from the Munsell Data and Optimal Color Spectraa

Tables Icon

Table 4. Munsell and Pantone Data Reconstructed by Using the Weighted Eigenvectors Solved from the Optimal Color Spectraa

Equations (28)

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

N2N+2,
CV=VD,
Y=V^TX.
X^=V^Y=V^V^TX,
Cu=XuXuT=UX[UX]T=UCUT.
Yu=V^uTXu=[UV^u]TX.
X˜=WX^u=WV^uV^uTXu=WV^u[UV^u]TX.
x^ixi.
μ0=16N(2N23N+7),
μk=N2sin2(kπ/N),k1.
v0=1N[1,,1],
vk=2N[1,cos(2πk/N),,cos(2πk(N1)/N)],
vNk=2N[0,sin(2πk/N),,sin(2πk(N1)/N)],
μm=mandvm=1N[1,1,,1,1].
C=i=0N1Ci=i=0N1XiXiT.
X0=[010101]andX2=[10011100010000100011].
wk=[1,ωk,,ωkN1],
μk=a0+a1ωk++aN1ωkN1.
μk=c0+c1cos(2πk/n)++cn1cos(2πk(n1)/n).
vk=2N12(wk+wNk)=2N[1,cos(2πk/N),,cos(2πk(N1)/N)],
vNk=2N12i(wkwNk)=2N[0,sin(2πk/N),,sin(2πk(N1)/N)].
vm=1N[1,1,,1,1].
{cj0=1,0j<Ncji=max(0,ij)+max(0,i+jN),0j<N,i1.
cj=i=0N1cji=12N(N1)+j(jN)+1.
μ0=j=0N1cj=N+j=1N1j2=16N(2N1)(N1)+N.
s,k=j=0N1jωkj.
s1,k=Nωk1ands2,k=N(Nωk2ωkN)(ωk1)2,
μk=j=0N1cjωkj=s2,kNs1,k+12(N2N+2)s0,k=2Nωk(ωk1)2=N2sin2(kπ/N).

Metrics