Abstract

The Shannon entropy [Bell Syst. Tech J. 27, 379 (1948)] of spectral distributions is applied to the problem of color rendering. With this novel approach, calculations for visual white entropy, spectral entropy, and color rendering are proposed, indices that are unreliant on the subjectivity inherent in reference spectra and color samples. The indices are tested against real lamp spectra, showing a simple and robust system for color rendering assessment. The discussion considers potential roles for white entropy in several areas of color theory and psychophysics and nonextensive entropy generalizations of the entropy indices in mathematical color spaces.

© 2012 Optical Society of America

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References

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  1. Illuminating Engineering Society (IES), “100 significant papers” (2004), http://www.ies.org/edoppts/100papers.cfm .
  2. D. Nickerson, “Measurement and specification of color rendition properties of light sources,” in Proceedings of National Technical Conference of the Illuminating Engineering Society (1957), pp. 77–90.
  3. W. Davis, “Recent developments of the color rendering index,” presented at 27th Session of the CIE (Sun City, South Africa, 2011).
  4. D. L. MacAdam, ed., Selected Papers on Colorimetry: Fundamentals, Vol. MS 77 of Milestone Series (SPIE, 1993).
  5. Commission Internationale de l’Eclairage, Method of Measuring and Specifying Colour Rendering Properties of Light Sources, CIE Publication 13.3 (CIE, 1995).
  6. C. Li, R. Luo, M. Pointer, X. Li, C. Li, and W. Ji, “A new method for quantifying colour rendering,” presented at 26th Session of the CIE (Beijing, 2007).
  7. Commission Internationale de l’Eclairage, ILV: International Lighting Vocabulary, CIE Publication S 017/E (CIE, 2011).
  8. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
  9. C. Tsallis, “Nonextensive statistical mechanics: construction and physical interpretation,” in Nonextensive Entropy: Interdisciplinary Applications, M. Gell-Mann and C. Tsallis, eds., (Oxford University, 2004) pp. 1–53.
  10. A. Hurlbert, “Colour vision: is colour constancy real?” Curr. Biol. 9, R558–R561 (1999).
    [CrossRef]
  11. Commission Internationale de l’Eclairage, Colorimetry, CIE Publication 15 (CIE, 2004).
  12. A. Plastino, M. T. Martin, and O. Rosso, “Generalised information measures and the analysis of brain electrical signals,” in Nonextensive Entropy: Interdisciplinary Applications, M. Gell-Mann and C. Tsallis, eds. (Oxford University, 2004), pp. 261–293.
  13. G. N. Lewis, “The entropy of radiation,” Proc. Natl. Acad. Sci. USA 13, 307–313 (1927).
    [CrossRef]
  14. P. Rosen, “Entropy of radiation,” Phys. Rev. 96, 555–556 (1954).
    [CrossRef]
  15. A. Ore, “Entropy of radiation,” Phys. Rev. 98, 887–888(1955).
    [CrossRef]
  16. J. J. Clark and S. Skaff, “A spectral theory of color perception,” J. Opt. Soc. Am. A 26, 2488–2502 (2009).
    [CrossRef]
  17. S. Skaff and J. J. Clark, “Spectral color constancy using a maximum entropy approach,” J. Opt. Soc. Am. A 28, 2385–2399 (2011).
    [CrossRef]
  18. The unit name n-mits is intended to be short for a generalized multibase information unit. Most nongeneralized information units are “X”its where the letter replacing X describes the number used for n (e.g., bits, where b stands for binary, n=2). It is not known where or indeed whether such a general unit has previously been proposed, but it adds convenience here as it ensures 1 is always the maximum value of CESp whatever the number of wavelength intervals used.
  19. D. B. Judd, “A flattery index for artificial illuminants,” Illum. Eng. 62, 593–598 (1967).
  20. W. Davis and Y. Ohno, “Toward an improved color rendering metric,” in Proceedings of Fifth International Conference on Solid State Lighting (SPIE, 2005), p. 59411G.
  21. W. Davis and Y. Ohno, “The color quality scale,” Opt. Eng. 49, 033602 (2010).
    [CrossRef]
  22. K. Baczynska and L. L. A. Price, “Efficacy and ocular safety of bright light therapy lamps,” Light. Res. Technol. (online first, 12April2012).
    [CrossRef]
  23. K. H. Norwich, “On the information received by sensory receptors,” Bull. Math. Biol. 39, 153–161 (1977).
    [CrossRef]
  24. M. S. Rea, M. G. Figueiro, A. Bierman, and J. D. Bullough, “Circadian light,” J. Circadian Rhythms 8, 2 (2010).
    [CrossRef]

2011 (1)

2010 (2)

M. S. Rea, M. G. Figueiro, A. Bierman, and J. D. Bullough, “Circadian light,” J. Circadian Rhythms 8, 2 (2010).
[CrossRef]

W. Davis and Y. Ohno, “The color quality scale,” Opt. Eng. 49, 033602 (2010).
[CrossRef]

2009 (1)

1999 (1)

A. Hurlbert, “Colour vision: is colour constancy real?” Curr. Biol. 9, R558–R561 (1999).
[CrossRef]

1977 (1)

K. H. Norwich, “On the information received by sensory receptors,” Bull. Math. Biol. 39, 153–161 (1977).
[CrossRef]

1967 (1)

D. B. Judd, “A flattery index for artificial illuminants,” Illum. Eng. 62, 593–598 (1967).

1955 (1)

A. Ore, “Entropy of radiation,” Phys. Rev. 98, 887–888(1955).
[CrossRef]

1954 (1)

P. Rosen, “Entropy of radiation,” Phys. Rev. 96, 555–556 (1954).
[CrossRef]

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).

1927 (1)

G. N. Lewis, “The entropy of radiation,” Proc. Natl. Acad. Sci. USA 13, 307–313 (1927).
[CrossRef]

Baczynska, K.

K. Baczynska and L. L. A. Price, “Efficacy and ocular safety of bright light therapy lamps,” Light. Res. Technol. (online first, 12April2012).
[CrossRef]

Bierman, A.

M. S. Rea, M. G. Figueiro, A. Bierman, and J. D. Bullough, “Circadian light,” J. Circadian Rhythms 8, 2 (2010).
[CrossRef]

Bullough, J. D.

M. S. Rea, M. G. Figueiro, A. Bierman, and J. D. Bullough, “Circadian light,” J. Circadian Rhythms 8, 2 (2010).
[CrossRef]

Clark, J. J.

Davis, W.

W. Davis and Y. Ohno, “The color quality scale,” Opt. Eng. 49, 033602 (2010).
[CrossRef]

W. Davis and Y. Ohno, “Toward an improved color rendering metric,” in Proceedings of Fifth International Conference on Solid State Lighting (SPIE, 2005), p. 59411G.

W. Davis, “Recent developments of the color rendering index,” presented at 27th Session of the CIE (Sun City, South Africa, 2011).

Figueiro, M. G.

M. S. Rea, M. G. Figueiro, A. Bierman, and J. D. Bullough, “Circadian light,” J. Circadian Rhythms 8, 2 (2010).
[CrossRef]

Hurlbert, A.

A. Hurlbert, “Colour vision: is colour constancy real?” Curr. Biol. 9, R558–R561 (1999).
[CrossRef]

Ji, W.

C. Li, R. Luo, M. Pointer, X. Li, C. Li, and W. Ji, “A new method for quantifying colour rendering,” presented at 26th Session of the CIE (Beijing, 2007).

Judd, D. B.

D. B. Judd, “A flattery index for artificial illuminants,” Illum. Eng. 62, 593–598 (1967).

Lewis, G. N.

G. N. Lewis, “The entropy of radiation,” Proc. Natl. Acad. Sci. USA 13, 307–313 (1927).
[CrossRef]

Li, C.

C. Li, R. Luo, M. Pointer, X. Li, C. Li, and W. Ji, “A new method for quantifying colour rendering,” presented at 26th Session of the CIE (Beijing, 2007).

C. Li, R. Luo, M. Pointer, X. Li, C. Li, and W. Ji, “A new method for quantifying colour rendering,” presented at 26th Session of the CIE (Beijing, 2007).

Li, X.

C. Li, R. Luo, M. Pointer, X. Li, C. Li, and W. Ji, “A new method for quantifying colour rendering,” presented at 26th Session of the CIE (Beijing, 2007).

Luo, R.

C. Li, R. Luo, M. Pointer, X. Li, C. Li, and W. Ji, “A new method for quantifying colour rendering,” presented at 26th Session of the CIE (Beijing, 2007).

Martin, M. T.

A. Plastino, M. T. Martin, and O. Rosso, “Generalised information measures and the analysis of brain electrical signals,” in Nonextensive Entropy: Interdisciplinary Applications, M. Gell-Mann and C. Tsallis, eds. (Oxford University, 2004), pp. 261–293.

Nickerson, D.

D. Nickerson, “Measurement and specification of color rendition properties of light sources,” in Proceedings of National Technical Conference of the Illuminating Engineering Society (1957), pp. 77–90.

Norwich, K. H.

K. H. Norwich, “On the information received by sensory receptors,” Bull. Math. Biol. 39, 153–161 (1977).
[CrossRef]

Ohno, Y.

W. Davis and Y. Ohno, “The color quality scale,” Opt. Eng. 49, 033602 (2010).
[CrossRef]

W. Davis and Y. Ohno, “Toward an improved color rendering metric,” in Proceedings of Fifth International Conference on Solid State Lighting (SPIE, 2005), p. 59411G.

Ore, A.

A. Ore, “Entropy of radiation,” Phys. Rev. 98, 887–888(1955).
[CrossRef]

Plastino, A.

A. Plastino, M. T. Martin, and O. Rosso, “Generalised information measures and the analysis of brain electrical signals,” in Nonextensive Entropy: Interdisciplinary Applications, M. Gell-Mann and C. Tsallis, eds. (Oxford University, 2004), pp. 261–293.

Pointer, M.

C. Li, R. Luo, M. Pointer, X. Li, C. Li, and W. Ji, “A new method for quantifying colour rendering,” presented at 26th Session of the CIE (Beijing, 2007).

Price, L. L. A.

K. Baczynska and L. L. A. Price, “Efficacy and ocular safety of bright light therapy lamps,” Light. Res. Technol. (online first, 12April2012).
[CrossRef]

Rea, M. S.

M. S. Rea, M. G. Figueiro, A. Bierman, and J. D. Bullough, “Circadian light,” J. Circadian Rhythms 8, 2 (2010).
[CrossRef]

Rosen, P.

P. Rosen, “Entropy of radiation,” Phys. Rev. 96, 555–556 (1954).
[CrossRef]

Rosso, O.

A. Plastino, M. T. Martin, and O. Rosso, “Generalised information measures and the analysis of brain electrical signals,” in Nonextensive Entropy: Interdisciplinary Applications, M. Gell-Mann and C. Tsallis, eds. (Oxford University, 2004), pp. 261–293.

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).

Skaff, S.

Tsallis, C.

C. Tsallis, “Nonextensive statistical mechanics: construction and physical interpretation,” in Nonextensive Entropy: Interdisciplinary Applications, M. Gell-Mann and C. Tsallis, eds., (Oxford University, 2004) pp. 1–53.

Bell Syst. Tech. J. (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).

Bull. Math. Biol. (1)

K. H. Norwich, “On the information received by sensory receptors,” Bull. Math. Biol. 39, 153–161 (1977).
[CrossRef]

Curr. Biol. (1)

A. Hurlbert, “Colour vision: is colour constancy real?” Curr. Biol. 9, R558–R561 (1999).
[CrossRef]

Illum. Eng. (1)

D. B. Judd, “A flattery index for artificial illuminants,” Illum. Eng. 62, 593–598 (1967).

J. Circadian Rhythms (1)

M. S. Rea, M. G. Figueiro, A. Bierman, and J. D. Bullough, “Circadian light,” J. Circadian Rhythms 8, 2 (2010).
[CrossRef]

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

Opt. Eng. (1)

W. Davis and Y. Ohno, “The color quality scale,” Opt. Eng. 49, 033602 (2010).
[CrossRef]

Phys. Rev. (2)

P. Rosen, “Entropy of radiation,” Phys. Rev. 96, 555–556 (1954).
[CrossRef]

A. Ore, “Entropy of radiation,” Phys. Rev. 98, 887–888(1955).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

G. N. Lewis, “The entropy of radiation,” Proc. Natl. Acad. Sci. USA 13, 307–313 (1927).
[CrossRef]

Other (13)

K. Baczynska and L. L. A. Price, “Efficacy and ocular safety of bright light therapy lamps,” Light. Res. Technol. (online first, 12April2012).
[CrossRef]

The unit name n-mits is intended to be short for a generalized multibase information unit. Most nongeneralized information units are “X”its where the letter replacing X describes the number used for n (e.g., bits, where b stands for binary, n=2). It is not known where or indeed whether such a general unit has previously been proposed, but it adds convenience here as it ensures 1 is always the maximum value of CESp whatever the number of wavelength intervals used.

W. Davis and Y. Ohno, “Toward an improved color rendering metric,” in Proceedings of Fifth International Conference on Solid State Lighting (SPIE, 2005), p. 59411G.

Commission Internationale de l’Eclairage, Colorimetry, CIE Publication 15 (CIE, 2004).

A. Plastino, M. T. Martin, and O. Rosso, “Generalised information measures and the analysis of brain electrical signals,” in Nonextensive Entropy: Interdisciplinary Applications, M. Gell-Mann and C. Tsallis, eds. (Oxford University, 2004), pp. 261–293.

C. Tsallis, “Nonextensive statistical mechanics: construction and physical interpretation,” in Nonextensive Entropy: Interdisciplinary Applications, M. Gell-Mann and C. Tsallis, eds., (Oxford University, 2004) pp. 1–53.

Illuminating Engineering Society (IES), “100 significant papers” (2004), http://www.ies.org/edoppts/100papers.cfm .

D. Nickerson, “Measurement and specification of color rendition properties of light sources,” in Proceedings of National Technical Conference of the Illuminating Engineering Society (1957), pp. 77–90.

W. Davis, “Recent developments of the color rendering index,” presented at 27th Session of the CIE (Sun City, South Africa, 2011).

D. L. MacAdam, ed., Selected Papers on Colorimetry: Fundamentals, Vol. MS 77 of Milestone Series (SPIE, 1993).

Commission Internationale de l’Eclairage, Method of Measuring and Specifying Colour Rendering Properties of Light Sources, CIE Publication 13.3 (CIE, 1995).

C. Li, R. Luo, M. Pointer, X. Li, C. Li, and W. Ji, “A new method for quantifying colour rendering,” presented at 26th Session of the CIE (Beijing, 2007).

Commission Internationale de l’Eclairage, ILV: International Lighting Vocabulary, CIE Publication S 017/E (CIE, 2011).

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Figures (4)

Fig. 1.
Fig. 1.

The color entropy for whiteness in a general color space with white point (,). The 10 isoentropic closed curves correspond to a geometric progression in CEWh from 1, represented by a point at (,), to 0.91. The following Table 2 illuminants are superimposed, based on their xy coordinates: indirect daylight (0.28, 0.29), tungsten halogen (TH) lamp (0.46, 0.41), and standard illuminant D65 (0.31, 0.33), and the thick line shows spectral locus. The axis labels, illuminant and spectral locus positions are correct only for the specific standard xy color space diagram. The diagonal line shows p3=0 generally, e.g., z=0.

Fig. 2.
Fig. 2.

Color entropies of blackbody spectra white entropy CEWh, spectral entropy CESp, and color entropy CE [see Eqs. (2), (3), and (4a) above] as a function of temperature T in degrees Kelvin.

Fig. 3.
Fig. 3.

Color entropies of sample 1 illuminant spectra (see Table 1). The sample includes important reference spectra and a selection of domestic illuminants. For the first spectrum only, the spectral and color entropies are 0. From top to bottom, the three bars for each illuminant show white entropy, spectral entropy, and color entropy.

Fig. 4.
Fig. 4.

Color entropies of sample 2 illuminant spectra (see Table 2), from the CQS model. The reference numbers are from that model. Again the illuminants are ranked in order of increasing CE values. Qa and Ra agree closely, except for 9: HPS. From top to bottom, the five bars for each illuminant show white entropy, spectral entropy, color entropy, CQS Qa and CRI Ra (n=81, but see comments below about illuminants 29 and 30).

Tables (2)

Tables Icon

Table 1. Color Entropies and xy-Coordinates of Sample 1 Illuminant Spectra

Tables Icon

Table 2. Color Rendering Scores and CCT of Sample 2 Illuminant Spectra, Real and Artificial (n=81, but see comments about Illuminants 29 and 30)

Equations (5)

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

H=Kpilogpi,
CEWh=pilogpilog3,
CESp=p(λi)logp(λi)logn,
CE=|CEWhCESp|.
CE=|CEWhCESp|,or{CEWh,CESp}.

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