Abstract

The paper adopts the philosophical stance that colors are real and can be identified with spectral models based on the photoreceptor signals. A statistical setting represents spectral profiles as probability density functions. This permits the use of analytic tools from the field of information geometry to determine a new kind of color space and structure deriving therefrom. In particular, the metric of the color space is shown to be the Fisher information matrix. A maximum entropy technique for spectral modeling is proposed that takes into account measurement noise. Theoretical predictions provided by our approach are compared with empirical colorfulness and color similarity data.

© 2009 Optical Society of America

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  1. B. Maund, “Color,” in The Stanford Encyclopedia of Philosophy, Winter 2008 ed., Edward N. Zalta, ed., http://plato.stanford.edu/archives/win2008/entries/color/.
  2. A. Byrne and D. Hilbert, “Color realism and color science,” Behav. Brain Sci. 26, 3-64 (2003).
    [PubMed]
  3. M. Tye, “The puzzle of true blue,” Analysis 66, 173-178 (2006).
    [CrossRef]
  4. C. L. Hardin, “A spectral reflectance doth not a color make,” J. Philos. 100, 191-202 (2003).
  5. J. K. O'Regan and A. Noë, “A sensorimotor account of vision and visual consciousness,” Behav. Brain Sci. 24, 939-1011 (2001).
    [CrossRef]
  6. D. H. Brainard and W. T. Freeman, “Bayesian color constancy,” J. Opt. Soc. Am. A 14, 1393-1411 (1997).
    [CrossRef]
  7. L. T. Maloney and B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29-33 (1986).
    [CrossRef] [PubMed]
  8. R. T. Cox, The Algebra of Probable Inference (Johns Hopkins Univ. Press, 1961).
  9. 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]
  10. D. H. Krantz, “Color measurement and color theory: I. Representation theorem for Grassmann structures,” J. Meteorol. Soc. Jpn. 12, 283-303 (1975).
  11. C. H. Graham and Y. Hsia, “Studies of color blindness: a unilaterally dichromatic subject,” Proc. Natl. Acad. Sci. U.S.A. 45, 96-99 (1959).
    [CrossRef]
  12. D. I. A. MacLeod and P. Lennie, “Red-green blindness confined to one eye,” Vision Res. 16, 691-702 (1976).
    [CrossRef] [PubMed]
  13. M. A. Webster, E. Miyahara, G. Malkoc, and V. E. Raker, “Variations in normal color vision. II. Unique hues,” J. Opt. Soc. Am. A 17, 1545-1555 (2000).
    [CrossRef]
  14. J. J. Clark and S. Skaff, “Maximum entropy models of surface reflectance spectra,” in Proceedings of the IEEE Instrumentation and Measurement Technology Conference, IMTC 2005 (IEEE, 2005), Vol. 2, pp. 1557-1560.
    [CrossRef]
  15. E. T. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. SSC-4, 227-241 (1968).
    [CrossRef]
  16. E. T. Jaynes, “On the rationale of maximum-entropy methods,” Proc. IEEE 70, 939-952 (1982).
    [CrossRef]
  17. E. T. Jaynes, “Concentration of distributions at entropy maxima,” in E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics, R.D.Rosenkrantz, ed. (D. Reidel, 1979), pp. 315-334.
  18. S.-I. Amari and H. Nagaoka, Methods of Information Geometry, Vol. 191 of Translations of Mathematical Monographs (American Mathematical Society, 2000).
  19. R. A. Fisher, “Theory of statistical estimation,” Proc. Cambridge Philos. Soc. 22, 700-725 (1925).
    [CrossRef]
  20. S. J. Maybank, “The Fisher-Rao metric,” Math. Today 44(6), 255-257 (2008).
  21. N. N. Cencov, Statistical Decisions Rules and Optimal Inference, Vol. 53 of Transactions on Mathematical Monographs (American Mathematical Society, 1982).
  22. C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81-91 (1945).
  23. S. Kullback and R. A. Leibler, “On information and sufficiency,” Ann. Math. Stat. 22, 79-86 (1951).
    [CrossRef]
  24. N. Silver, D. S. Sivia, and J. E. Gubernatis, “Maximum-entropy method for analytic continuation of quantum Monte Carlo data,” Phys. Rev. B 41, 2380-2389 (1990).
    [CrossRef]
  25. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems (Winston, 1977).
  26. A. Stockman, L. T. Sharpe, and C. C. Fach, “The spectral sensitivity of the human short-wavelength cones,” Vision Res. 39, 2901-2927 (1999).
    [CrossRef] [PubMed]
  27. D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry, 3rd ed. (Wiley, 1975).
  28. N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. J. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in 10th IS&T/SID Color Imaging Conference (2002) pp. 23-27.
  29. R. W. G. Hunt, “Saturation, superfluous or superior?” 9th IS&T/SID Color Imaging Conference (2001), pp. 1-5.
  30. M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivner, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166-180 (1991).
    [CrossRef]
  31. M. H. Kim, T. Weyrich, and J. Kautz, “Modeling human color perception under extended luminance levels,” in International Conference on Computer Graphics and Interactive Techniniques (ACM Digital Library, 2009), article 27.
  32. A. H. Munsell, Munsell Book of Color: Matte Finish Collection (Munsell Color, 1979).
  33. J. P. S. Parkkinen, J. Hallikainen, and T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318-322 (1989).
    [CrossRef]
  34. K. Uchikawa, H. Uchikawa and P. K. Kaiser, “Luminance and saturation of equality bright colors,” Color Res. Appl. 9, 5-14 (1984).
    [CrossRef]
  35. T. D. Kulp and K. Fuld, “The prediction of hue and saturation for non-spectral lights,” Vision Res. 35, 2967-2983 (1995).
    [CrossRef] [PubMed]
  36. J. J. Vos, “Line elements and physiological models of color vision,” Color Res. Appl. 4, 208-216 (1979).
    [CrossRef]
  37. J. J. Vos, “From lower to higher colour metrics: a historical account,” Clin. Exp. Optom. 86, 348-360 (2006).
    [CrossRef]
  38. H. Helmholtz, Handbuch der physiologischen Optik, 2nd ed. (Voss, 1896).
  39. E. Schrödinger, “Grundlinien einer Theorie der Farbenmetrik im Tagessehen,” Ann. Phys. 368, 481-584 (1920).
    [CrossRef]
  40. M. A. Bouman and P. L. Walraven, “Quantum theory of colour discrimination of dichromats,” Vision Res. 2, 177-187 (1962).
    [CrossRef]
  41. W. S. Stiles, “A modified Helmholtz line element in brightness-colour space,” Proc. Phys. Soc. London 58, 41-65 (1946).
    [CrossRef]
  42. J. J. Vos and P. L. Walraven, “An analytical description of the line element in the zone fluctuation model of color vision, I and II,” Vision Res. 12, 1327-1365 (1972).
    [CrossRef] [PubMed]
  43. K. J. McCree, “Small field tritanopia and the effects of voluntary fixation,” J. Mod. Opt. 7, 4, 317-323 (1960).
    [CrossRef]
  44. T. W. Cronin and J. Marshall, “Parallel processing and image analysis in the eyes of mantis shrimps,” Biol. Bull. 200, 177-183 (2001).
    [CrossRef] [PubMed]

2008 (1)

S. J. Maybank, “The Fisher-Rao metric,” Math. Today 44(6), 255-257 (2008).

2006 (3)

M. Tye, “The puzzle of true blue,” Analysis 66, 173-178 (2006).
[CrossRef]

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]

J. J. Vos, “From lower to higher colour metrics: a historical account,” Clin. Exp. Optom. 86, 348-360 (2006).
[CrossRef]

2003 (2)

A. Byrne and D. Hilbert, “Color realism and color science,” Behav. Brain Sci. 26, 3-64 (2003).
[PubMed]

C. L. Hardin, “A spectral reflectance doth not a color make,” J. Philos. 100, 191-202 (2003).

2001 (2)

J. K. O'Regan and A. Noë, “A sensorimotor account of vision and visual consciousness,” Behav. Brain Sci. 24, 939-1011 (2001).
[CrossRef]

T. W. Cronin and J. Marshall, “Parallel processing and image analysis in the eyes of mantis shrimps,” Biol. Bull. 200, 177-183 (2001).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

A. Stockman, L. T. Sharpe, and C. C. Fach, “The spectral sensitivity of the human short-wavelength cones,” Vision Res. 39, 2901-2927 (1999).
[CrossRef] [PubMed]

1997 (1)

1995 (1)

T. D. Kulp and K. Fuld, “The prediction of hue and saturation for non-spectral lights,” Vision Res. 35, 2967-2983 (1995).
[CrossRef] [PubMed]

1991 (1)

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivner, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166-180 (1991).
[CrossRef]

1990 (1)

N. Silver, D. S. Sivia, and J. E. Gubernatis, “Maximum-entropy method for analytic continuation of quantum Monte Carlo data,” Phys. Rev. B 41, 2380-2389 (1990).
[CrossRef]

1989 (1)

1986 (1)

1984 (1)

K. Uchikawa, H. Uchikawa and P. K. Kaiser, “Luminance and saturation of equality bright colors,” Color Res. Appl. 9, 5-14 (1984).
[CrossRef]

1982 (1)

E. T. Jaynes, “On the rationale of maximum-entropy methods,” Proc. IEEE 70, 939-952 (1982).
[CrossRef]

1979 (1)

J. J. Vos, “Line elements and physiological models of color vision,” Color Res. Appl. 4, 208-216 (1979).
[CrossRef]

1976 (1)

D. I. A. MacLeod and P. Lennie, “Red-green blindness confined to one eye,” Vision Res. 16, 691-702 (1976).
[CrossRef] [PubMed]

1975 (1)

D. H. Krantz, “Color measurement and color theory: I. Representation theorem for Grassmann structures,” J. Meteorol. Soc. Jpn. 12, 283-303 (1975).

1972 (1)

J. J. Vos and P. L. Walraven, “An analytical description of the line element in the zone fluctuation model of color vision, I and II,” Vision Res. 12, 1327-1365 (1972).
[CrossRef] [PubMed]

1968 (1)

E. T. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. SSC-4, 227-241 (1968).
[CrossRef]

1962 (1)

M. A. Bouman and P. L. Walraven, “Quantum theory of colour discrimination of dichromats,” Vision Res. 2, 177-187 (1962).
[CrossRef]

1960 (1)

K. J. McCree, “Small field tritanopia and the effects of voluntary fixation,” J. Mod. Opt. 7, 4, 317-323 (1960).
[CrossRef]

1959 (1)

C. H. Graham and Y. Hsia, “Studies of color blindness: a unilaterally dichromatic subject,” Proc. Natl. Acad. Sci. U.S.A. 45, 96-99 (1959).
[CrossRef]

1951 (1)

S. Kullback and R. A. Leibler, “On information and sufficiency,” Ann. Math. Stat. 22, 79-86 (1951).
[CrossRef]

1946 (1)

W. S. Stiles, “A modified Helmholtz line element in brightness-colour space,” Proc. Phys. Soc. London 58, 41-65 (1946).
[CrossRef]

1945 (1)

C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81-91 (1945).

1925 (1)

R. A. Fisher, “Theory of statistical estimation,” Proc. Cambridge Philos. Soc. 22, 700-725 (1925).
[CrossRef]

1920 (1)

E. Schrödinger, “Grundlinien einer Theorie der Farbenmetrik im Tagessehen,” Ann. Phys. 368, 481-584 (1920).
[CrossRef]

Amari, S.-I.

S.-I. Amari and H. Nagaoka, Methods of Information Geometry, Vol. 191 of Translations of Mathematical Monographs (American Mathematical Society, 2000).

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems (Winston, 1977).

Bouman, M. A.

M. A. Bouman and P. L. Walraven, “Quantum theory of colour discrimination of dichromats,” Vision Res. 2, 177-187 (1962).
[CrossRef]

Brainard, D. H.

Byrne, A.

A. Byrne and D. Hilbert, “Color realism and color science,” Behav. Brain Sci. 26, 3-64 (2003).
[PubMed]

Cencov, N. N.

N. N. Cencov, Statistical Decisions Rules and Optimal Inference, Vol. 53 of Transactions on Mathematical Monographs (American Mathematical Society, 1982).

Clark, J. J.

J. J. Clark and S. Skaff, “Maximum entropy models of surface reflectance spectra,” in Proceedings of the IEEE Instrumentation and Measurement Technology Conference, IMTC 2005 (IEEE, 2005), Vol. 2, pp. 1557-1560.
[CrossRef]

Clarke, A. A.

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivner, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166-180 (1991).
[CrossRef]

Cox, R. T.

R. T. Cox, The Algebra of Probable Inference (Johns Hopkins Univ. Press, 1961).

Cronin, T. W.

T. W. Cronin and J. Marshall, “Parallel processing and image analysis in the eyes of mantis shrimps,” Biol. Bull. 200, 177-183 (2001).
[CrossRef] [PubMed]

Fach, C. C.

A. Stockman, L. T. Sharpe, and C. C. Fach, “The spectral sensitivity of the human short-wavelength cones,” Vision Res. 39, 2901-2927 (1999).
[CrossRef] [PubMed]

Fairchild, M. D.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. J. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in 10th IS&T/SID Color Imaging Conference (2002) pp. 23-27.

Finlayson, G. D.

Fisher, R. A.

R. A. Fisher, “Theory of statistical estimation,” Proc. Cambridge Philos. Soc. 22, 700-725 (1925).
[CrossRef]

Freeman, W. T.

Fuld, K.

T. D. Kulp and K. Fuld, “The prediction of hue and saturation for non-spectral lights,” Vision Res. 35, 2967-2983 (1995).
[CrossRef] [PubMed]

Graham, C. H.

C. H. Graham and Y. Hsia, “Studies of color blindness: a unilaterally dichromatic subject,” Proc. Natl. Acad. Sci. U.S.A. 45, 96-99 (1959).
[CrossRef]

Gubernatis, J. E.

N. Silver, D. S. Sivia, and J. E. Gubernatis, “Maximum-entropy method for analytic continuation of quantum Monte Carlo data,” Phys. Rev. B 41, 2380-2389 (1990).
[CrossRef]

Hallikainen, J.

Hardin, C. L.

C. L. Hardin, “A spectral reflectance doth not a color make,” J. Philos. 100, 191-202 (2003).

Helmholtz, H.

H. Helmholtz, Handbuch der physiologischen Optik, 2nd ed. (Voss, 1896).

Hilbert, D.

A. Byrne and D. Hilbert, “Color realism and color science,” Behav. Brain Sci. 26, 3-64 (2003).
[PubMed]

Hsia, Y.

C. H. Graham and Y. Hsia, “Studies of color blindness: a unilaterally dichromatic subject,” Proc. Natl. Acad. Sci. U.S.A. 45, 96-99 (1959).
[CrossRef]

Hunt, R. W. G.

R. W. G. Hunt, “Saturation, superfluous or superior?” 9th IS&T/SID Color Imaging Conference (2001), pp. 1-5.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. J. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in 10th IS&T/SID Color Imaging Conference (2002) pp. 23-27.

Jaaskelainen, T.

Jaynes, E. T.

E. T. Jaynes, “On the rationale of maximum-entropy methods,” Proc. IEEE 70, 939-952 (1982).
[CrossRef]

E. T. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. SSC-4, 227-241 (1968).
[CrossRef]

E. T. Jaynes, “Concentration of distributions at entropy maxima,” in E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics, R.D.Rosenkrantz, ed. (D. Reidel, 1979), pp. 315-334.

Judd, D. B.

D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry, 3rd ed. (Wiley, 1975).

Kaiser, P. K.

K. Uchikawa, H. Uchikawa and P. K. Kaiser, “Luminance and saturation of equality bright colors,” Color Res. Appl. 9, 5-14 (1984).
[CrossRef]

Kautz, J.

M. H. Kim, T. Weyrich, and J. Kautz, “Modeling human color perception under extended luminance levels,” in International Conference on Computer Graphics and Interactive Techniniques (ACM Digital Library, 2009), article 27.

Kim, M. H.

M. H. Kim, T. Weyrich, and J. Kautz, “Modeling human color perception under extended luminance levels,” in International Conference on Computer Graphics and Interactive Techniniques (ACM Digital Library, 2009), article 27.

Krantz, D. H.

D. H. Krantz, “Color measurement and color theory: I. Representation theorem for Grassmann structures,” J. Meteorol. Soc. Jpn. 12, 283-303 (1975).

Kullback, S.

S. Kullback and R. A. Leibler, “On information and sufficiency,” Ann. Math. Stat. 22, 79-86 (1951).
[CrossRef]

Kulp, T. D.

T. D. Kulp and K. Fuld, “The prediction of hue and saturation for non-spectral lights,” Vision Res. 35, 2967-2983 (1995).
[CrossRef] [PubMed]

Leibler, R. A.

S. Kullback and R. A. Leibler, “On information and sufficiency,” Ann. Math. Stat. 22, 79-86 (1951).
[CrossRef]

Lennie, P.

D. I. A. MacLeod and P. Lennie, “Red-green blindness confined to one eye,” Vision Res. 16, 691-702 (1976).
[CrossRef] [PubMed]

Li, C. J.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. J. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in 10th IS&T/SID Color Imaging Conference (2002) pp. 23-27.

Luo, M. R.

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivner, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166-180 (1991).
[CrossRef]

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. J. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in 10th IS&T/SID Color Imaging Conference (2002) pp. 23-27.

MacLeod, D. I. A.

D. I. A. MacLeod and P. Lennie, “Red-green blindness confined to one eye,” Vision Res. 16, 691-702 (1976).
[CrossRef] [PubMed]

Malkoc, G.

Maloney, L. T.

Marshall, J.

T. W. Cronin and J. Marshall, “Parallel processing and image analysis in the eyes of mantis shrimps,” Biol. Bull. 200, 177-183 (2001).
[CrossRef] [PubMed]

Maund, B.

B. Maund, “Color,” in The Stanford Encyclopedia of Philosophy, Winter 2008 ed., Edward N. Zalta, ed., http://plato.stanford.edu/archives/win2008/entries/color/.

Maybank, S. J.

S. J. Maybank, “The Fisher-Rao metric,” Math. Today 44(6), 255-257 (2008).

McCree, K. J.

K. J. McCree, “Small field tritanopia and the effects of voluntary fixation,” J. Mod. Opt. 7, 4, 317-323 (1960).
[CrossRef]

Miyahara, E.

Moroney, N.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. J. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in 10th IS&T/SID Color Imaging Conference (2002) pp. 23-27.

Morovic, P.

Munsell, A. H.

A. H. Munsell, Munsell Book of Color: Matte Finish Collection (Munsell Color, 1979).

Nagaoka, H.

S.-I. Amari and H. Nagaoka, Methods of Information Geometry, Vol. 191 of Translations of Mathematical Monographs (American Mathematical Society, 2000).

Newman, T.

N. Moroney, M. D. Fairchild, R. W. G. Hunt, C. J. Li, M. R. Luo, and T. Newman, “The CIECAM02 color appearance model,” in 10th IS&T/SID Color Imaging Conference (2002) pp. 23-27.

Noë, A.

J. K. O'Regan and A. Noë, “A sensorimotor account of vision and visual consciousness,” Behav. Brain Sci. 24, 939-1011 (2001).
[CrossRef]

O'Regan, J. K.

J. K. O'Regan and A. Noë, “A sensorimotor account of vision and visual consciousness,” Behav. Brain Sci. 24, 939-1011 (2001).
[CrossRef]

Parkkinen, J. P. S.

Raker, V. E.

Rao, C. R.

C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81-91 (1945).

Rhodes, P. A.

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivner, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166-180 (1991).
[CrossRef]

Schappo, A.

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivner, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166-180 (1991).
[CrossRef]

Schrödinger, E.

E. Schrödinger, “Grundlinien einer Theorie der Farbenmetrik im Tagessehen,” Ann. Phys. 368, 481-584 (1920).
[CrossRef]

Scrivner, S. A. R.

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivner, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166-180 (1991).
[CrossRef]

Sharpe, L. T.

A. Stockman, L. T. Sharpe, and C. C. Fach, “The spectral sensitivity of the human short-wavelength cones,” Vision Res. 39, 2901-2927 (1999).
[CrossRef] [PubMed]

Silver, N.

N. Silver, D. S. Sivia, and J. E. Gubernatis, “Maximum-entropy method for analytic continuation of quantum Monte Carlo data,” Phys. Rev. B 41, 2380-2389 (1990).
[CrossRef]

Sivia, D. S.

N. Silver, D. S. Sivia, and J. E. Gubernatis, “Maximum-entropy method for analytic continuation of quantum Monte Carlo data,” Phys. Rev. B 41, 2380-2389 (1990).
[CrossRef]

Skaff, S.

J. J. Clark and S. Skaff, “Maximum entropy models of surface reflectance spectra,” in Proceedings of the IEEE Instrumentation and Measurement Technology Conference, IMTC 2005 (IEEE, 2005), Vol. 2, pp. 1557-1560.
[CrossRef]

Stiles, W. S.

W. S. Stiles, “A modified Helmholtz line element in brightness-colour space,” Proc. Phys. Soc. London 58, 41-65 (1946).
[CrossRef]

Stockman, A.

A. Stockman, L. T. Sharpe, and C. C. Fach, “The spectral sensitivity of the human short-wavelength cones,” Vision Res. 39, 2901-2927 (1999).
[CrossRef] [PubMed]

Tait, C. J.

M. R. Luo, A. A. Clarke, P. A. Rhodes, A. Schappo, S. A. R. Scrivner, and C. J. Tait, “Quantifying colour appearance. Part I. LUTCHI colour appearance data,” Color Res. Appl. 16, 166-180 (1991).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems (Winston, 1977).

Tye, M.

M. Tye, “The puzzle of true blue,” Analysis 66, 173-178 (2006).
[CrossRef]

Uchikawa, H.

K. Uchikawa, H. Uchikawa and P. K. Kaiser, “Luminance and saturation of equality bright colors,” Color Res. Appl. 9, 5-14 (1984).
[CrossRef]

Uchikawa, K.

K. Uchikawa, H. Uchikawa and P. K. Kaiser, “Luminance and saturation of equality bright colors,” Color Res. Appl. 9, 5-14 (1984).
[CrossRef]

Vos, J. J.

J. J. Vos, “From lower to higher colour metrics: a historical account,” Clin. Exp. Optom. 86, 348-360 (2006).
[CrossRef]

J. J. Vos, “Line elements and physiological models of color vision,” Color Res. Appl. 4, 208-216 (1979).
[CrossRef]

J. J. Vos and P. L. Walraven, “An analytical description of the line element in the zone fluctuation model of color vision, I and II,” Vision Res. 12, 1327-1365 (1972).
[CrossRef] [PubMed]

Walraven, P. L.

J. J. Vos and P. L. Walraven, “An analytical description of the line element in the zone fluctuation model of color vision, I and II,” Vision Res. 12, 1327-1365 (1972).
[CrossRef] [PubMed]

M. A. Bouman and P. L. Walraven, “Quantum theory of colour discrimination of dichromats,” Vision Res. 2, 177-187 (1962).
[CrossRef]

Wandell, B. A.

Webster, M. A.

Weyrich, T.

M. H. Kim, T. Weyrich, and J. Kautz, “Modeling human color perception under extended luminance levels,” in International Conference on Computer Graphics and Interactive Techniniques (ACM Digital Library, 2009), article 27.

Wyszecki, G.

D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry, 3rd ed. (Wiley, 1975).

Zalta, Edward N.

B. Maund, “Color,” in The Stanford Encyclopedia of Philosophy, Winter 2008 ed., Edward N. Zalta, ed., http://plato.stanford.edu/archives/win2008/entries/color/.

Analysis (1)

M. Tye, “The puzzle of true blue,” Analysis 66, 173-178 (2006).
[CrossRef]

Ann. Math. Stat. (1)

S. Kullback and R. A. Leibler, “On information and sufficiency,” Ann. Math. Stat. 22, 79-86 (1951).
[CrossRef]

Ann. Phys. (1)

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Figures (7)

Fig. 1
Fig. 1

Diagram illustrating the spectral modeling process. The photoreceptor signals define a metamer set, which is the set of all spectral profiles that can result in these signals. One element of the metamer set is distinguished by the spectral modeling approach and is taken as the spectral model. The space traced out by these distinguished elements of the metamer sets is called the spectral color manifold. It is the space of spectral models that are possible given the particular photoreceptor characteristics and spectral modeling approach.

Fig. 2
Fig. 2

Diagram illustrating the determination of the normalized measurement estimate given noisy measurements. Given a photoreceptor measurement vector r, a normalized measurement is obtained by dividing by the intensity β. The estimate of the noise-free normalized measurement will be located on the boundary of the ellipsoid centered on the normalized measurement with size given by the noise level scaled by the intensity. The intensity is chosen so as to maximize the entropy of the spectral model associated with the estimate of the noise-free normalized measurement.

Fig. 3
Fig. 3

Maximum entropy spectral models compared with true spectra for four different Munsell patches. Computed by using γ = 0.0005 .

Fig. 4
Fig. 4

Comparison between the CIECAM02 empirical chroma (colorfulness) measure and the canonical divergence colorfulness measure of Eq. (42). The dashed line represents the best linear fit to the data. The canonical divergence was computed by using γ = 0.0005 .

Fig. 5
Fig. 5

Comparison between the actual Munsell patch intensity and the intensity estimated by using the maximum entropy approach. The canonical divergence was computed by using γ = 0.0005 .

Fig. 6
Fig. 6

Square root of the canonical divergence distance measure D 0 for monochromatic stimuli as a function of wavelength. Shown are curves (solid) for two different γ values corresponding to two different effective signal to noise ratios (larger γ corresponds to lower intensity). The upper solid curve corresponds to γ = 0.00056 , and the lower solid curve to γ = 0.56 . For comparison are shown two curves derived from the experimental data from Table I of Uchikawa et al. (1984) [subject HU, purity 0.3 (dotted curve) and 0.7 (dashed curve)]. The data from Uchikawa et al. have been shifted by 20 nm to align with the theoretical curve.

Fig. 7
Fig. 7

Quantity d λ d s for the maximum entropy spectral model as a function of monochromatic stimulus wavelength, computed by using γ = 0.0005 and β = 1 . Also shown are two curves replotted from McCree’s [43] Fig. 4, showing empirical wavelength discrimination data for two different light levels. The scale for the theoretical curve is arbitrary, set for ease of comparison. It is evident in both the empirical and the theoretical curves that wavelength discrimination is greatest in the yellow and cyan regions of the spectrum.

Equations (57)

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H ( p 1 , p 2 , p n ) = i = 1 n p i log ( p i ) .
r = β Λ f ( λ ) p ( λ ) d λ ,
η = Λ f ( λ ) p ( λ ) d λ .
p ̂ ( λ ; θ ) = exp ( θ f ( λ ) ψ ( θ ) ) ,
Λ p ̂ ( λ ; θ ) d λ = Λ exp ( θ f ( λ ) ) exp ( ψ ( θ ) ) d λ = 1 .
exp ( ψ ( θ ) ) Λ exp ( θ f ( λ ) ) d λ = 1 ,
ψ ( θ ) = log Λ exp ( θ f ( λ ) ) d λ .
η ̂ = E p ̂ { f ( λ ) } = Λ f ( λ ) p ̂ ( λ ; θ ) d λ .
ϕ ( η ) = max θ [ θ η ψ ( θ ) ] .
ψ ( θ ) = max η [ θ η ϕ ( η ) ] ,
ψ ( θ ) + ϕ ( η ) = θ η .
ψ ( θ ) θ i = η i ,
ϕ ( η ) η i = θ i .
H ( p ̂ ) = E { log ( p ̂ ( λ ; θ ) ) } = Λ p ̂ ( λ ; η ) log ( p ̂ ( λ ; η ) ) d λ .
H ( p ̂ ) = E { θ f ψ ( θ ) } = ( θ η ψ ( θ ) ) = ϕ ( η ) .
g i j ( θ ) = Λ p ̂ ( λ ; θ ) ( θ i log p ̂ ( λ ; θ ) θ j log p ̂ ( λ ; θ ) ) d λ .
g i j ( θ ) = η j θ i = Λ p ̂ ( λ ; θ ) ( f i ( λ ) η i ) ( f j ( λ ) η j ) d λ = Λ p ̂ ( λ ; θ ) f i ( λ ) f j ( λ ) d λ η i η j .
g i j ( θ ) = η i j ( θ ) η i ( θ ) η j ( θ ) .
d s 2 = d θ T G ( θ ) d θ = d θ d η
d s 2 = d η T G 1 ( η ) d η = d η d θ .
L ( P , Q ) = s ( P ) s ( Q ) d s = 0 1 ( d θ d t T G d θ d t ) 1 2 d t ,
D ( P Q ) = ψ ( θ P ) + ϕ ( η Q ) θ P η Q .
D ( P P + Δ θ ) = Δ θ T G Δ θ 2 + o ( Δ θ 2 ) d s 2 2 ,
D ( P Q ) = D KL ( Q P ) = Λ p ̂ Q ( λ ) log ( p ̂ Q ( λ ) p ̂ P ( λ ) ) d λ .
D ( P Q ) = ϕ ( Q ) ϕ ( P ) + θ P ( η P η Q ) .
D ( P Q ) = H ( P ) H ( Q ) + θ P ( η P η Q ) .
D 0 ( Q ) = D ( 0 Q ) = H ( 0 ) H ( Q ) .
r = β Λ f ( λ ) p ( λ ) d λ + ν .
η = Λ f ( λ ) p ( λ ) d λ + ν β .
η ̂ = Λ f ( λ ) p ̂ ( λ ) d λ .
ν T Q ν 1 ,
β ( η η ̂ ) = ν .
( η ̂ η ) T Q ( η ̂ η ) 1 β 2 .
η ̂ = arg max η ̂ Z ( η ) H ( θ ̂ ( η ̂ ) ) ,
η 0 = 1 Λ Λ f ( λ ) d λ .
η ̂ = arg min η ̂ { ( η ̂ η ) T A ( η ̂ η ) γ H ( θ ̂ ( η ̂ ) ) } ,
r 0 = β Λ f 0 ( λ ) p ( λ ) d λ = β Λ p ( λ ) d λ = β .
{ θ ̂ , β } = arg min θ , β { ( η ̂ ( θ ) r β ) T A ( η ̂ ( θ ) r β ) γ H ( θ ̂ ) } .
H ( t ) = ψ ( t θ 0 ) t θ 0 η ( t ) .
d H ( t ) d t = ψ ( t θ 0 ) t ( t θ 0 ) t η ( t ) ( t θ 0 ) η ( t ) t ,
d H ( t ) d t = ψ ( t θ 0 ) ( t θ 0 ) ( t θ 0 ) t ( t θ 0 ) t η ( t ) ( t θ 0 ) T η ( t ) θ ( t ) θ 0 = t θ 0 T G ( t ) θ 0 ,
C D ( θ ) = D 0 1 2 ( θ ) = [ H ( 0 ) H ( θ ) ] 1 2 .
D ( 0 | λ 0 ) = H ( 0 ) H ( θ ( λ 0 ) ) .
D ( 0 λ 0 ) = ψ ( 0 ) ψ ( θ ( λ 0 ) ) + θ ( λ 0 ) η .
D ( 0 λ 0 ) = ψ ( 0 ) ψ ( θ ( λ 0 ) ) + θ ( λ 0 ) f ( λ 0 ) .
ψ ( θ ( λ 0 ) ) = log Λ exp ( θ ( λ 0 ) f ) d λ .
ψ ( θ ( λ 0 ) ) = θ ( λ 0 ) f ( λ m ) + 1 2 log ( 2 π θ ( λ 0 ) | d 2 f d λ 2 | λ = λ m ) .
D ( 0 λ 0 ) ψ ( 0 ) 1 2 log ( 2 π θ ( λ 0 ) | d 2 f d λ 2 | λ = λ 0 ) .
D ( 0 λ 0 ) 1 2 log ( θ ( λ 0 ) | d 2 f d λ 2 | λ = λ 0 ) .
d s H 2 = ( d L JND ( L ) ) 2 + ( d M JND ( M ) ) 2 + ( d S JND ( S ) ) 2 ,
d s S 2 = 1 ( L + M + S ) [ ( d L L ) 2 + ( d M M ) 2 + ( d S S ) 2 ] .
d s 2 = d η T G 1 d η .
d s 2 = 1 β 2 ( [ g 1 ] 11 d L 2 + [ g 1 ] 22 d M 2 + [ g 1 ] 33 d S 2 + 2 [ g 1 ] 12 d L d M + 2 [ g 1 ] 23 d M d S + 2 [ g 1 ] 13 d S d L ) ,
d s = ( d η T d λ 0 G 1 ( η ) d η d λ 0 ) 1 2 d λ 0
d λ 0 d s = ( d η T d λ 0 G 1 ( η ) d η d λ 0 ) 1 2 ,
d η d λ 0 = | d f ( λ ) d λ | λ = λ 0 ,
d λ 0 d s = ( | d f T ( λ ) d λ | λ = λ 0 G 1 ( η ) | d f ( λ ) d λ | λ = λ 0 ) 1 2 .

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