Abstract

The set of metamers for a given device response can be calculated given the device’s spectral sensitivities. Knowledge of the metamer set has been useful in practical applications such as color correction and reflectance recovery. Unfortunately, the device sensitivities of a camera or scanner are not known, and they are difficult to estimate reliably outside the laboratory. We show how metamer sets can be calculated when a device’s spectral sensitivities are not known. The result is built on two observations: first, the set of all reflectance spectra consists of convex combinations of certain basic colors that tend to be very bright (or dark) and have high chroma; second, the convex combinations that describe reflectance spectra result in convex combinations of red–green–blue (RGB) values. Thus, given an RGB value, it is possible to find the set of convex combinations of the RGB values of the basic colors that generate the same RGB value. The corresponding set of convex combinations of the basic spectra is the metamer set.

© 2007 Optical Society of America

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References

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  1. CIE, Commission Internationale de l'Eclairage Proceedings, 1931 (Cambridge U. Press, 1932).
  2. F. Imai, S. Quan, M. Rosen, and R. Berns, "Digital camera filter design for colorimetric and spectral accuracy," in Proceedings of the Third International Conference on Multispectral Color Science (Society for Imaging Science and Technology, 2001), pp. 13-16.
  3. G. Sharma and H. J. Trussell, "Figures of merit for color scanners," IEEE Trans. Image Process. 6, 990-1001 (1997).
    [CrossRef] [PubMed]
  4. G. D. Finlayson and M. S. Drew, "The maximum ignorance assumption with positivity," in Proceedings of the IS&T/SID Fourth Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science and Technology, 1996), pp. 202-205.
  5. H. R. Kang, "Color scanner calibration," J. Imaging Sci. Technol. 36, 162-170 (1992).
  6. C. D. Andersen and J. Y. Hardeberg, "Colorimetric characterization of digital cameras preserving hue planes," in Proceedings of the IS&T/SID Thirteenth Color Imaging Conference (Society for Imaging Science and Technology, 2005), pp. 141-146.
  7. G. D. Finlayson and P. Morovic, "Metamer constrained colour correction," J. Imaging Sci. Technol. 44, 295-300 (2000).
  8. P. Morovic, "Metamer sets," Ph.D. thesis (University of East Anglia, 2002).
  9. G. D. Finlayson and P. Morovic, "Metamer sets," J. Opt. Soc. Am. A 22, 810-819 (2005).
    [CrossRef]
  10. P. Morovic and G. D. Finlayson, "Metamer-set-based approach to estimating surface reflectance from camera RGB," J. Opt. Soc. Am. A 23, 1814-1822 (2006).
    [CrossRef]
  11. M. S. Drew and B. Funt, "Natural metamers," Comput. Vision Graph. Image Process. Image Understanding 56, 139-151 (1992).
  12. G. Finlayson, S. Hordley, and P. Hubel, "Recovering device sensitivities with quadratic programming," in Proceedings of IS&T/SID Sixth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, 1998), pp. 90-95.
  13. G. Sharma and H. Trussell, "Characterization of scanner sensitivity," in Proceedings of IS&T/SID Color Imaging Conference (Society for Imaging Science and Technology, 1993), pp. 103-107.
  14. K. Barnard and B. Funt, "Camera characterization for color research," Color Res. Appl. 27, 153-164 (2002).
    [CrossRef]
  15. B. Dyas, "Robust sensor response characterization," in Proceedings of IS&T/SID Eighth Color Imagining Conference (Society for Imaging Science and Technology, 2000), pp. 144-148.
  16. A. Alsam and G. D. Finlayson, "Recovering spectral sensitivities with uncertainty," in Proceedings of the First European Conference on Color in Graphics, Imaging and Vision (Society for Imaging Science and Technology, 2002), pp. 22-26.
  17. A. Alsam and R. Lenz, "Calibrating color cameras using metameric blacks," J. Opt. Soc. Am. A 24, 11-17 (2007).
    [CrossRef]
  18. G. D. Finlayson and P. Morovic, "Intensity constrained error-less colour correction," in Proceedings of IS&T/SID Twelfth Color Imaging Conference (Society for Imaging Science and Technology, 2004), pp. 106-110.
  19. B. K. P. Horn, "Exact reproduction of colored images," Comput. Vis. Graph. Image Process. 26, 135-167 (1984).
    [CrossRef]
  20. B. Smith, C. Spiekermann, and R. Sember, "Numerical methods for colorimetric calculations: sampling density requirements," Color Res. Appl. 17, 394-401 (1992).
    [CrossRef]
  21. J. B. Cohen, "Color and color mixture: scalar and vector fundamentals," Color Res. Appl. 13, 5-39 (1988).
    [CrossRef]
  22. A. L. Peressini, The Mathematics of Nonlinear Programming, 2nd ed. (Springer-Verlag, 1988).
    [CrossRef]
  23. T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing (Prentice Hall, 2000).
  24. H. E. J. Neugebauer, "Die theoretischen grundlagen des mehrfarbendrucks," Z. Wiss. Photogr. 36, 36-73 (1937).
  25. Qhull. See http://www.qhull.org.
  26. G. D. Finlayson and P. Morovic, "Metamer constrained colour correction," in Proceedings of IS&T/SID Seventh Color Imaging Conference (Society for Imaging Science and Technology, 1999), pp. 26-31.

2007 (1)

2006 (1)

2005 (1)

2002 (1)

K. Barnard and B. Funt, "Camera characterization for color research," Color Res. Appl. 27, 153-164 (2002).
[CrossRef]

2000 (1)

G. D. Finlayson and P. Morovic, "Metamer constrained colour correction," J. Imaging Sci. Technol. 44, 295-300 (2000).

1997 (1)

G. Sharma and H. J. Trussell, "Figures of merit for color scanners," IEEE Trans. Image Process. 6, 990-1001 (1997).
[CrossRef] [PubMed]

1992 (3)

H. R. Kang, "Color scanner calibration," J. Imaging Sci. Technol. 36, 162-170 (1992).

M. S. Drew and B. Funt, "Natural metamers," Comput. Vision Graph. Image Process. Image Understanding 56, 139-151 (1992).

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

1988 (1)

J. B. Cohen, "Color and color mixture: scalar and vector fundamentals," Color Res. Appl. 13, 5-39 (1988).
[CrossRef]

1984 (1)

B. K. P. Horn, "Exact reproduction of colored images," Comput. Vis. Graph. Image Process. 26, 135-167 (1984).
[CrossRef]

1937 (1)

H. E. J. Neugebauer, "Die theoretischen grundlagen des mehrfarbendrucks," Z. Wiss. Photogr. 36, 36-73 (1937).

Alsam, A.

A. Alsam and R. Lenz, "Calibrating color cameras using metameric blacks," J. Opt. Soc. Am. A 24, 11-17 (2007).
[CrossRef]

A. Alsam and G. D. Finlayson, "Recovering spectral sensitivities with uncertainty," in Proceedings of the First European Conference on Color in Graphics, Imaging and Vision (Society for Imaging Science and Technology, 2002), pp. 22-26.

Andersen, C. D.

C. D. Andersen and J. Y. Hardeberg, "Colorimetric characterization of digital cameras preserving hue planes," in Proceedings of the IS&T/SID Thirteenth Color Imaging Conference (Society for Imaging Science and Technology, 2005), pp. 141-146.

Barnard, K.

K. Barnard and B. Funt, "Camera characterization for color research," Color Res. Appl. 27, 153-164 (2002).
[CrossRef]

Berns, R.

F. Imai, S. Quan, M. Rosen, and R. Berns, "Digital camera filter design for colorimetric and spectral accuracy," in Proceedings of the Third International Conference on Multispectral Color Science (Society for Imaging Science and Technology, 2001), pp. 13-16.

Cohen, J. B.

J. B. Cohen, "Color and color mixture: scalar and vector fundamentals," Color Res. Appl. 13, 5-39 (1988).
[CrossRef]

Drew, M. S.

M. S. Drew and B. Funt, "Natural metamers," Comput. Vision Graph. Image Process. Image Understanding 56, 139-151 (1992).

G. D. Finlayson and M. S. Drew, "The maximum ignorance assumption with positivity," in Proceedings of the IS&T/SID Fourth Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science and Technology, 1996), pp. 202-205.

Dyas, B.

B. Dyas, "Robust sensor response characterization," in Proceedings of IS&T/SID Eighth Color Imagining Conference (Society for Imaging Science and Technology, 2000), pp. 144-148.

Finlayson, G.

G. Finlayson, S. Hordley, and P. Hubel, "Recovering device sensitivities with quadratic programming," in Proceedings of IS&T/SID Sixth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, 1998), pp. 90-95.

Finlayson, G. D.

P. Morovic and G. D. Finlayson, "Metamer-set-based approach to estimating surface reflectance from camera RGB," J. Opt. Soc. Am. A 23, 1814-1822 (2006).
[CrossRef]

G. D. Finlayson and P. Morovic, "Metamer sets," J. Opt. Soc. Am. A 22, 810-819 (2005).
[CrossRef]

G. D. Finlayson and P. Morovic, "Metamer constrained colour correction," J. Imaging Sci. Technol. 44, 295-300 (2000).

G. D. Finlayson and M. S. Drew, "The maximum ignorance assumption with positivity," in Proceedings of the IS&T/SID Fourth Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science and Technology, 1996), pp. 202-205.

A. Alsam and G. D. Finlayson, "Recovering spectral sensitivities with uncertainty," in Proceedings of the First European Conference on Color in Graphics, Imaging and Vision (Society for Imaging Science and Technology, 2002), pp. 22-26.

G. D. Finlayson and P. Morovic, "Intensity constrained error-less colour correction," in Proceedings of IS&T/SID Twelfth Color Imaging Conference (Society for Imaging Science and Technology, 2004), pp. 106-110.

G. D. Finlayson and P. Morovic, "Metamer constrained colour correction," in Proceedings of IS&T/SID Seventh Color Imaging Conference (Society for Imaging Science and Technology, 1999), pp. 26-31.

Funt, B.

K. Barnard and B. Funt, "Camera characterization for color research," Color Res. Appl. 27, 153-164 (2002).
[CrossRef]

M. S. Drew and B. Funt, "Natural metamers," Comput. Vision Graph. Image Process. Image Understanding 56, 139-151 (1992).

Hardeberg, J. Y.

C. D. Andersen and J. Y. Hardeberg, "Colorimetric characterization of digital cameras preserving hue planes," in Proceedings of the IS&T/SID Thirteenth Color Imaging Conference (Society for Imaging Science and Technology, 2005), pp. 141-146.

Hordley, S.

G. Finlayson, S. Hordley, and P. Hubel, "Recovering device sensitivities with quadratic programming," in Proceedings of IS&T/SID Sixth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, 1998), pp. 90-95.

Horn, B. K. P.

B. K. P. Horn, "Exact reproduction of colored images," Comput. Vis. Graph. Image Process. 26, 135-167 (1984).
[CrossRef]

Hubel, P.

G. Finlayson, S. Hordley, and P. Hubel, "Recovering device sensitivities with quadratic programming," in Proceedings of IS&T/SID Sixth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, 1998), pp. 90-95.

Imai, F.

F. Imai, S. Quan, M. Rosen, and R. Berns, "Digital camera filter design for colorimetric and spectral accuracy," in Proceedings of the Third International Conference on Multispectral Color Science (Society for Imaging Science and Technology, 2001), pp. 13-16.

Kang, H. R.

H. R. Kang, "Color scanner calibration," J. Imaging Sci. Technol. 36, 162-170 (1992).

Lenz, R.

Moon, T. K.

T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing (Prentice Hall, 2000).

Morovic, P.

P. Morovic and G. D. Finlayson, "Metamer-set-based approach to estimating surface reflectance from camera RGB," J. Opt. Soc. Am. A 23, 1814-1822 (2006).
[CrossRef]

G. D. Finlayson and P. Morovic, "Metamer sets," J. Opt. Soc. Am. A 22, 810-819 (2005).
[CrossRef]

G. D. Finlayson and P. Morovic, "Metamer constrained colour correction," J. Imaging Sci. Technol. 44, 295-300 (2000).

P. Morovic, "Metamer sets," Ph.D. thesis (University of East Anglia, 2002).

G. D. Finlayson and P. Morovic, "Intensity constrained error-less colour correction," in Proceedings of IS&T/SID Twelfth Color Imaging Conference (Society for Imaging Science and Technology, 2004), pp. 106-110.

G. D. Finlayson and P. Morovic, "Metamer constrained colour correction," in Proceedings of IS&T/SID Seventh Color Imaging Conference (Society for Imaging Science and Technology, 1999), pp. 26-31.

Neugebauer, H. E. J.

H. E. J. Neugebauer, "Die theoretischen grundlagen des mehrfarbendrucks," Z. Wiss. Photogr. 36, 36-73 (1937).

Peressini, A. L.

A. L. Peressini, The Mathematics of Nonlinear Programming, 2nd ed. (Springer-Verlag, 1988).
[CrossRef]

Quan, S.

F. Imai, S. Quan, M. Rosen, and R. Berns, "Digital camera filter design for colorimetric and spectral accuracy," in Proceedings of the Third International Conference on Multispectral Color Science (Society for Imaging Science and Technology, 2001), pp. 13-16.

Rosen, M.

F. Imai, S. Quan, M. Rosen, and R. Berns, "Digital camera filter design for colorimetric and spectral accuracy," in Proceedings of the Third International Conference on Multispectral Color Science (Society for Imaging Science and Technology, 2001), pp. 13-16.

Sember, R.

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

Sharma, G.

G. Sharma and H. J. Trussell, "Figures of merit for color scanners," IEEE Trans. Image Process. 6, 990-1001 (1997).
[CrossRef] [PubMed]

G. Sharma and H. Trussell, "Characterization of scanner sensitivity," in Proceedings of IS&T/SID Color Imaging Conference (Society for Imaging Science and Technology, 1993), pp. 103-107.

Smith, B.

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

Spiekermann, C.

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

Stirling, W. C.

T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing (Prentice Hall, 2000).

Trussell, H.

G. Sharma and H. Trussell, "Characterization of scanner sensitivity," in Proceedings of IS&T/SID Color Imaging Conference (Society for Imaging Science and Technology, 1993), pp. 103-107.

Trussell, H. J.

G. Sharma and H. J. Trussell, "Figures of merit for color scanners," IEEE Trans. Image Process. 6, 990-1001 (1997).
[CrossRef] [PubMed]

Color Res. Appl. (3)

K. Barnard and B. Funt, "Camera characterization for color research," Color Res. Appl. 27, 153-164 (2002).
[CrossRef]

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

J. B. Cohen, "Color and color mixture: scalar and vector fundamentals," Color Res. Appl. 13, 5-39 (1988).
[CrossRef]

Comput. Vis. Graph. Image Process. (1)

B. K. P. Horn, "Exact reproduction of colored images," Comput. Vis. Graph. Image Process. 26, 135-167 (1984).
[CrossRef]

Comput. Vision Graph. Image Process. Image Understanding (1)

M. S. Drew and B. Funt, "Natural metamers," Comput. Vision Graph. Image Process. Image Understanding 56, 139-151 (1992).

IEEE Trans. Image Process. (1)

G. Sharma and H. J. Trussell, "Figures of merit for color scanners," IEEE Trans. Image Process. 6, 990-1001 (1997).
[CrossRef] [PubMed]

J. Imaging Sci. Technol. (2)

H. R. Kang, "Color scanner calibration," J. Imaging Sci. Technol. 36, 162-170 (1992).

G. D. Finlayson and P. Morovic, "Metamer constrained colour correction," J. Imaging Sci. Technol. 44, 295-300 (2000).

J. Opt. Soc. Am. A (3)

Z. Wiss. Photogr. (1)

H. E. J. Neugebauer, "Die theoretischen grundlagen des mehrfarbendrucks," Z. Wiss. Photogr. 36, 36-73 (1937).

Other (14)

Qhull. See http://www.qhull.org.

G. D. Finlayson and P. Morovic, "Metamer constrained colour correction," in Proceedings of IS&T/SID Seventh Color Imaging Conference (Society for Imaging Science and Technology, 1999), pp. 26-31.

A. L. Peressini, The Mathematics of Nonlinear Programming, 2nd ed. (Springer-Verlag, 1988).
[CrossRef]

T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing (Prentice Hall, 2000).

G. D. Finlayson and P. Morovic, "Intensity constrained error-less colour correction," in Proceedings of IS&T/SID Twelfth Color Imaging Conference (Society for Imaging Science and Technology, 2004), pp. 106-110.

B. Dyas, "Robust sensor response characterization," in Proceedings of IS&T/SID Eighth Color Imagining Conference (Society for Imaging Science and Technology, 2000), pp. 144-148.

A. Alsam and G. D. Finlayson, "Recovering spectral sensitivities with uncertainty," in Proceedings of the First European Conference on Color in Graphics, Imaging and Vision (Society for Imaging Science and Technology, 2002), pp. 22-26.

G. Finlayson, S. Hordley, and P. Hubel, "Recovering device sensitivities with quadratic programming," in Proceedings of IS&T/SID Sixth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, 1998), pp. 90-95.

G. Sharma and H. Trussell, "Characterization of scanner sensitivity," in Proceedings of IS&T/SID Color Imaging Conference (Society for Imaging Science and Technology, 1993), pp. 103-107.

P. Morovic, "Metamer sets," Ph.D. thesis (University of East Anglia, 2002).

C. D. Andersen and J. Y. Hardeberg, "Colorimetric characterization of digital cameras preserving hue planes," in Proceedings of the IS&T/SID Thirteenth Color Imaging Conference (Society for Imaging Science and Technology, 2005), pp. 141-146.

G. D. Finlayson and M. S. Drew, "The maximum ignorance assumption with positivity," in Proceedings of the IS&T/SID Fourth Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science and Technology, 1996), pp. 202-205.

CIE, Commission Internationale de l'Eclairage Proceedings, 1931 (Cambridge U. Press, 1932).

F. Imai, S. Quan, M. Rosen, and R. Berns, "Digital camera filter design for colorimetric and spectral accuracy," in Proceedings of the Third International Conference on Multispectral Color Science (Society for Imaging Science and Technology, 2001), pp. 13-16.

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Figures (10)

Fig. 1
Fig. 1

Point f on the line a d can be defined as a convex combination of other points on the line.

Fig. 2
Fig. 2

Two-dimensional pictorial representation of the projection from spectral to response space.

Fig. 3
Fig. 3

Set of 24 RGB responses calculated based on the responses of the Sony DX camera to the surfaces of the Macbeth ColorChecker is shown as the solid black disks. Further, a point inside is shown as a black ring.

Fig. 4
Fig. 4

Four examples of metamer sets for the Nikon D70 camera based on the Macbeth ColorChecker. Although not shown in the figures, the original spectrum is always a member of the set. Note that the x axis of the figures represents the wavelength in 400 700 nm and the y axis represents the percentage reflectivity 0–1.

Fig. 5
Fig. 5

Metamer sets in the CIE x y space for the Nikon D70 camera based on the Macbeth ColorChecker. We note that the original value is always a member of the set. Note that the x axis of the figure represents x = X ( X + Y + Z ) and the y axis represents y = X ( X + Y + Z ) .

Fig. 6
Fig. 6

Three sets of estimates to the Nikon D70 camera. The methods are principal eigenvectors (dotted curve), quadratic programming (solid curve with circles), and Tikhonov regularization (dashed–dotted curve).

Fig. 7
Fig. 7

Metamer clouds in the RGB space. The gray disks are the result of integrating the metamers calculated with the quadratic programming sensor estimate with the sensors predicted with Tikhonov regularization. The black stars are the RGB values achieved by integrating the same metamers with the sensor curves estimated by principal eigenvectors.

Fig. 8
Fig. 8

Spectral sensitivities of a Sony DX camera. Note that the x axis of the figure represents the wavelength in 400 700 nm and the y axis represents the sensitivity.

Fig. 9
Fig. 9

Example of the proposed metamer set method compared with Finlayson’s algorithm. The calculations are based on the Sony DX camera and the Macbeth ColorChecker. Note that the x axis of the figure represents the wavelength in 400 700 nm and the y axis represents the percentage reflectivity 0–1.

Fig. 10
Fig. 10

Example of the proposed metamer set method (right), compared with Finlayson’s algorithm (left), with the added constraint of naturalness. The calculations are based on the Sony DX camera and the Macbeth ColorChecker. Note that the x axis of the figure represents the wavelength in 400 700 nm and the y axis represents the percentage reflectivity 0–1.

Tables (1)

Tables Icon

Table 1 Delta Lab Error between the Actual Color Signal and the Set of Metamers Predicted with Our Method a

Equations (23)

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

p i = ( E s ) T r i , i = 1 , 2 , 3 ,
c = E s ,
p = c T R ,
c T = c ra T + c nu T .
c ra T R = p ,
c nu T R = 0 .
Q ( p ) = { c ra T and c nu T R n , subject to c ra T R = p , c nu T R = 0 , c ra T ran ( R ) , c nu T null ( R ) } ,
[ x , y ] = { λ y + ( 1 λ ) x : 0 λ 1 } .
c = λ c 1 + ( 1 λ ) c 2 .
p 1 = c 1 T R .
λ p 1 = λ c 1 T R .
( 1 λ ) p 2 = ( 1 λ ) c 2 T R .
λ p 1 + ( 1 λ ) p 2 = ( λ c 1 T + ( 1 λ ) c 2 T ) R .
λ p 1 + ( 1 λ ) p 2 = c T R .
p = λ p 1 + ( 1 λ ) p 2 ,
p = λ 1 p 1 + λ 2 p 2 + + λ m p m , subject to
i = 1 m λ i = 1 ,
i λ i 0 ,
i λ i 1 .
p = P λ , subject to
1 T λ = 1 ,
I λ 0 ,
I λ 1 ,

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