Abstract

Partitive color mixing is the process by which the human eye integrates different neighboring colors to result in a single uniform surface. This process is convex: The perceived color is the weighted average of a small set of basis colors, and given that the weights represent the relative area of each color, they must sum to one. We present an efficient algorithm that generates a small number of new, natural bases such that a large set of spectra can be adequately expressed as a convex combination of these bases. Our results show that 9–11 bases are sufficient to represent a set of 1269 Munsell surfaces within the convex model.

© 2008 Optical Society of America

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References

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  1. W. S. Stiles, G. Wyszecki, and N. Ohta, “Counting metameric object-color stimuli using frequency-limited spectral reflectance functions,” J. Opt. Soc. Am. 67, 779-784 (1977).
    [CrossRef]
  2. G. Buchsbaum and A. Gottschalk, “Chromaticity coordinates of frequency-limited functions,” J. Opt. Soc. Am. A 1, 885-887 (1984).
    [CrossRef] [PubMed]
  3. K. Pearson, “On lines and planes of closest fit to systems of points in space,” Philos. Mag. 2, 559-572 (1901).
  4. H. Hotelling, “Analysis of a complex of statistical variables into principal components,” J. Educational Psychol. 24, 417-441 (1933).
    [CrossRef]
  5. L. T. Maloney and B. A. Wandell, “Color constancy: a method for recording surface spectral reflectance,” J. Opt. Soc. Am. 3, 29-33 (1986).
    [CrossRef]
  6. J. Parkkinen, J. Hallikainen, and T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. 6, 318-322 (1989).
    [CrossRef]
  7. G. Buchsbaum and O. Bloch, “Color categories revealed by non-negative matrix factorization of Munsell color spectra,” Vision Res. 42, 559-563 (2002).
    [CrossRef] [PubMed]
  8. D. D. Lee and H. S. Seung, “Algorithms for non-negative matrix factorization,” in Advances in Neural Information Processing Systems 13 (MIT, 2001), pp. 556-562.
  9. H. E. J. Neugebauer, “Die theoretischen grundlagen des mehrfarbendrucks,” Z. Wis. Photogr. 36, 36-73 (1937).
  10. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonomic Sci. 1, 369-370 (1964).
  11. L. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A 3, 1673-1683 (1986).
    [CrossRef] [PubMed]
  12. B. Smith, C. Spiekermann, and R. Sember, “Numerical methods for colorimetric calculations: sampling density requirements,” Color Res. Appl. 17, 394-401 (1992).
    [CrossRef]
  13. M. H. Brill and H. S. Fairman, “The principal components of reflectances,” Color Res. Appl. 29, 104-110 (2004).
    [CrossRef]
  14. R. Lenz, “Spaces of spectral distributions and their natural geometry,” in The First European Conference on Color in Graphics, Imaging and Vision (IS&T, 2002), pp. 249-254.
  15. A. Murray, “Monochrome reproduction in photo-engraving,” J. Franklin Inst. 221, 721-744 (1936).
    [CrossRef]
  16. J. O'Rourke, Computational Geometry in C, 2nd ed. (Cambridge Univ. Press, 1998).
  17. A. Alsam and J. Y. Hardeberg, “Convex reduction of calibration charts,” Proc. SPIE 5667, 38-46 (2005).
    [CrossRef]
  18. M. H. Brill, G. Finlayson, P. Hubel, and W. A. Thornton, “Prime colors and color imaging,” in Sixth SID/IS&T Color Imaging Conference (SID/IS&T, 1998), pp. 33-42.
  19. J. Kivinen and M. K. Warmuth, “Additive versus exponentiated gradient updates for linear prediction,” J. Inf. Comput. 132, 1-64 (1997).
    [CrossRef]

2005

A. Alsam and J. Y. Hardeberg, “Convex reduction of calibration charts,” Proc. SPIE 5667, 38-46 (2005).
[CrossRef]

2004

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

2002

G. Buchsbaum and O. Bloch, “Color categories revealed by non-negative matrix factorization of Munsell color spectra,” Vision Res. 42, 559-563 (2002).
[CrossRef] [PubMed]

1997

J. Kivinen and M. K. Warmuth, “Additive versus exponentiated gradient updates for linear prediction,” J. Inf. Comput. 132, 1-64 (1997).
[CrossRef]

1992

B. Smith, C. Spiekermann, and R. Sember, “Numerical methods for colorimetric calculations: sampling density requirements,” Color Res. Appl. 17, 394-401 (1992).
[CrossRef]

1989

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

1986

L. T. Maloney and B. A. Wandell, “Color constancy: a method for recording surface spectral reflectance,” J. Opt. Soc. Am. 3, 29-33 (1986).
[CrossRef]

L. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A 3, 1673-1683 (1986).
[CrossRef] [PubMed]

1984

1977

1964

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

1937

H. E. J. Neugebauer, “Die theoretischen grundlagen des mehrfarbendrucks,” Z. Wis. Photogr. 36, 36-73 (1937).

1936

A. Murray, “Monochrome reproduction in photo-engraving,” J. Franklin Inst. 221, 721-744 (1936).
[CrossRef]

1933

H. Hotelling, “Analysis of a complex of statistical variables into principal components,” J. Educational Psychol. 24, 417-441 (1933).
[CrossRef]

1901

K. Pearson, “On lines and planes of closest fit to systems of points in space,” Philos. Mag. 2, 559-572 (1901).

Color Res. Appl.

B. Smith, C. Spiekermann, and R. Sember, “Numerical methods for colorimetric calculations: sampling density requirements,” Color Res. Appl. 17, 394-401 (1992).
[CrossRef]

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

J. Educational Psychol.

H. Hotelling, “Analysis of a complex of statistical variables into principal components,” J. Educational Psychol. 24, 417-441 (1933).
[CrossRef]

J. Franklin Inst.

A. Murray, “Monochrome reproduction in photo-engraving,” J. Franklin Inst. 221, 721-744 (1936).
[CrossRef]

J. Inf. Comput.

J. Kivinen and M. K. Warmuth, “Additive versus exponentiated gradient updates for linear prediction,” J. Inf. Comput. 132, 1-64 (1997).
[CrossRef]

J. Opt. Soc. Am.

L. T. Maloney and B. A. Wandell, “Color constancy: a method for recording surface spectral reflectance,” J. Opt. Soc. Am. 3, 29-33 (1986).
[CrossRef]

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

W. S. Stiles, G. Wyszecki, and N. Ohta, “Counting metameric object-color stimuli using frequency-limited spectral reflectance functions,” J. Opt. Soc. Am. 67, 779-784 (1977).
[CrossRef]

J. Opt. Soc. Am. A

Philos. Mag.

K. Pearson, “On lines and planes of closest fit to systems of points in space,” Philos. Mag. 2, 559-572 (1901).

Proc. SPIE

A. Alsam and J. Y. Hardeberg, “Convex reduction of calibration charts,” Proc. SPIE 5667, 38-46 (2005).
[CrossRef]

Psychonomic Sci.

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

Vision Res.

G. Buchsbaum and O. Bloch, “Color categories revealed by non-negative matrix factorization of Munsell color spectra,” Vision Res. 42, 559-563 (2002).
[CrossRef] [PubMed]

Z. Wis. Photogr.

H. E. J. Neugebauer, “Die theoretischen grundlagen des mehrfarbendrucks,” Z. Wis. Photogr. 36, 36-73 (1937).

Other

D. D. Lee and H. S. Seung, “Algorithms for non-negative matrix factorization,” in Advances in Neural Information Processing Systems 13 (MIT, 2001), pp. 556-562.

M. H. Brill, G. Finlayson, P. Hubel, and W. A. Thornton, “Prime colors and color imaging,” in Sixth SID/IS&T Color Imaging Conference (SID/IS&T, 1998), pp. 33-42.

J. O'Rourke, Computational Geometry in C, 2nd ed. (Cambridge Univ. Press, 1998).

R. Lenz, “Spaces of spectral distributions and their natural geometry,” in The First European Conference on Color in Graphics, Imaging and Vision (IS&T, 2002), pp. 249-254.

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Figures (11)

Fig. 1
Fig. 1

First four PCA basis functions derived from a set of 1269 surfaces measured from the Munsell Book of Color.

Fig. 2
Fig. 2

The 1269 surfaces from the Munsell Book of Color plotted in the 3D conical space.

Fig. 3
Fig. 3

Best four nonnegative basis functions obtained from the spectral data of the Munsell Book of Color.

Fig. 4
Fig. 4

Red and white reflectance (both plotted as solid curves) and some convex combinations of these two reflectances (plotted as dotted curves).

Fig. 5
Fig. 5

Red and white reflectance (plotted as solid curves) and linear combinations of the two reflectances generated without constraining the weights to be convex (plotted as dotted curves).

Fig. 6
Fig. 6

First five most significant surfaces arrived at by using Alsam’s algorithm.

Fig. 7
Fig. 7

The 5–8 basis vectors obtained with convex matrix factorization based on the 1269 spectral data of the Munsell book of colors. The y axis represents the percentage reflectance, and the x axis corresponds to the wavelength in nanometers.

Fig. 8
Fig. 8

The 5–8 basis vectors obtained with NMF based on the 1269 spectral data of the Munsell book of colors. The y axis represents the percentage reflectance, and the x axis corresponds to the wavelength in nanometers.

Fig. 9
Fig. 9

Values of the NMF and the new method bases, based on eight basis vectors in the 3D conical space. The conical values of Munsell data are shown as small asterisks, the convex bases as circles, and the NMF bases as stars. We note that the new basis vectors encapsulate the Munsell data more tightly than do those of the NMF.

Fig. 10
Fig. 10

Values of 12 convex bases, based in the conical space. The conical values of Munsell data are shown as small asterisks, and the convex bases as discs. We note that the location of the data is optimized to encapsulate the Munsell data set.

Fig. 11
Fig. 11

Twelve convex bases derived by the new algorithm based on the 1269 Munsell spectra. Here we clearly note that the new algorithm results in black and white as well as in saturated colors.

Tables (3)

Tables Icon

Table 1 Mean and Median Δ E a b * Achieved When Representing the Munsell Spectral Data as Convex Sums of the NMF Basis Vectors versus Those with the New Convex Bases Algorithm a

Tables Icon

Table 2 Mean and Median Δ E a b * Achieved by Representing the Munsell Spectral Data as Convex Sums of 20–100 Extreme Spectra Versus Those with the New Convex Bases Algorithm a

Tables Icon

Table 3 Mean and Median Δ E a b * Achieved by Representing the Spectral Data of the Esser Calibration Chart as Convex Sums of 8–12 Bases a

Equations (14)

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

c j = 1 k b j ς j ,
C auto = C C T ,
C auto = U D U T ,
c = γ 1 u 1 + γ 2 u 2 + + γ 31 u 31 ,
γ i = c T u i .
w c 1 = γ 1 , w c 2 = γ 2 γ 1 , w c 3 = γ 3 γ 1 .
C W H ,
min W , H C W H 2 subject to W , H 0 .
c ( λ ) = δ c 1 ( λ ) + ( 1 δ ) c 2 ( λ ) ,
c ( λ ) = α 1 c 1 ( λ ) + α 2 c 2 ( λ ) ,
c ( λ ) = i = 1 n δ i c i ( λ ) , δ i [ 0 , 1 ] , i = 1 n δ i = 1 ,
c W h .
min h c W h 2 subject to h i 0 and i = 1 i h i = 1 .
W i + 1 = W i + μ ( C ( H i ) T W i H i ( H i ) T ) ,

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