Abstract

This paper proposes a solution to the spectral color constancy problem. The method is based on a statistical model for the surface reflectance spectrum and applies a maximum entropy constraint. Unlike prior methods based on linear models, the solution process does not require a set of basis functions to be defined, nor does it require a database of spectra to be specified in advance. Experiments on simulated and real data show that spectral estimation using the maximum entropy approach is feasible and performs similarly to existing spectral methods in spite of the lower level of a priori information required.

© 2011 Optical Society of America

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2010 (1)

B. Funt and L. Shi, “The Rehabilitation of MaxRGB,” in Proceedings of the Eighteenth Color Imaging Conference (The Society for Imaging Science and Technology, 2010), pp. 256–259.

2009 (2)

J. J. Clark and S. Skaff, “A spectral theory of color perception,” J. Opt. Soc. Am. A 26, 2488–2502 (2009).
[CrossRef]

S. Skaff, T. Arbel, and J. J. Clark, “A sequential Bayesian approach to color constancy using multiple sensors,” Computer Vision and Image Understanding 113, 993–1004 (2009).
[CrossRef]

2008 (1)

P. V. Gehler, C. Rother, A. Blake, T. Sharp, and T. Minka, “Bayesian color constancy revisited,” in Proceedings of the IEEE International Conference on Computer Vision and Patter Recognition (IEEE, 2008), pp. 1–8.

2007 (1)

S. Skaff and J. J. Clark, “Maximum entropy spectral models for color constancy,” in Proceedings of the Color Imaging Conference (The Society for Imaging Science and Technology, 2007), pp. 100–105.

2006 (5)

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]

C. Lu, “Removing shadows from color images,” Ph.D. thesis (School of Computing, Simon Fraser University, 2006).

I. Marin-Franch and D. H. Foster, “Estimating the information available from coloured surfaces in natural scenes,” in Proceedings of the European Conference on Color in Graphics, Imaging and Vision (The Society for Imaging Science and Technology, 2006), pp. 44–47.

M. Mosny and B. Funt, “Multispectral color constancy,” in Proceedings of the Color Imaging Conference (The Society for Imaging Science and Technology, 2006), pp. 309–313.

S. D. Hordley and G. D. Finlayson, “Reevaluation of color constancy algorithm performance,” J. Opt. Soc. Am. A 23, 1008–1020 (2006).
[CrossRef]

2005 (1)

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

2004 (1)

G. D. Finlayson, M. S. Drew, and C. Lu, “Intrinsic images by entropy minimization,” in Proceedings of the European Conference on Computer Vision (Springer, 2004), pp. 582–595.

2003 (2)

B. Jedynak, H. Zheng, M. Daoudi, and D. Barret, “Maximum entropy models for skin detection,” in Proceedings of the IEEE International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (IEEE, 2003), pp. 180–193.
[CrossRef]

M. Toews and T. Arbel, “Entropy-of-likelihood feature selection for image correspondence,” in Proceedings of the IEEE Conference on Computer Vision (IEEE, 2003), pp. 1041–1047.
[CrossRef]

2002 (4)

K. Barnard, V. Cardei, and B. Funt, “A comparison of computational color constancy algorithms. I.: methodology and experiments with synthesized data,” IEEE Trans. Image Process. 11, 972–984 (2002).
[CrossRef]

S. Skaff, T. Arbel, and J. J. Clark, “Active Bayesian color constancy with non-uniform sensors,” in Proceedings of the IEEE International Conference on Pattern Recognition (IEEE, 2002), pp. 681–684.

S. Skaff, “Active Bayesian color constancy with non-uniform sensors,” master’s thesis (McGill University, 2002).

K. Devlin, A. Chalmers, A. Wilkie, and W. Purgathofer, “Tone reproduction and physically based spectral rendering,” in Computer Graphics Forum (Eurographics, 2002), pp. 101–123.

2001 (3)

G. D. Finlayson, S. D. Hordley, and P. M. Hubel, “Color by correlation: a simple unifying framework for color constancy,” IEEE Trans. Pattern Anal. Machine Intell. 23, 1209–1221 (2001).
[CrossRef]

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification and Scene Analysis (Wiley, 2001).

M. R. Gupta and R. M. Gray, “Color conversions using maximum entropy estimation,” in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2001), pp. 118–121.

2000 (1)

D. J. Sheskin, Handbook of Parametric and Nonparametric Statistical Procedures (Chapman and Hall, 2000).

1999 (3)

G. Finlayson and S. Hordley, “Selection for gamut mapping colour constancy,” Image Vis. Comp. 17, 597–604 (1999).
[CrossRef]

K. Barnard, “Practical colour constancy,” Ph.D. thesis (School of Computing, Simon Fraser University, 1999).

R. Hall, “Comparing spectral color computation methods,” IEEE Comp. Grap. Appl. 19, 36–45 (1999).
[CrossRef]

1997 (2)

D. H. Brainard and W. T. Freeman, “Bayesian color constancy,” J. Opt. Soc. Am. A 14, 1393–1411 (1997).
[CrossRef]

G. D. Finlayson, P. M. Hubel, and S. Hordley, “Color by correlation,” in Proceedings of the Fifth Color Imaging Conference (The Society for Imaging Science and Technology, 1997), pp. 6–11.

1996 (1)

G. D. Finlayson, “Color in perspective,” IEEE Trans. Pattern Anal. Machine Intell. 18, 1034–1038 (1996).
[CrossRef]

1995 (2)

K. Barnard, “Computational colour constancy: taking theory into practice,” Master’s thesis (School of Computing, Simon Fraser University, 1995).

W. T. Freeman and D. H. Brainard, “Bayesian decision theory, the maximum local mass estimate, and color constancy,” in Proceedings of the IEEE International Conference on Computer Vision (IEEE, 1995), pp. 210–217.
[CrossRef]

1994 (2)

R. Schettini, “Deriving spectral reflectance functions of computer-simulated object colors,” Comput. Graph. Forum 13, 211–217 (1994).
[CrossRef]

M. D. D’Zmura and G. Iverson, “Color Constancy III. General linear recovery of spectral descriptions for lights and surfaces,” J. Opt. Soc. Am. A 11, 2389–2400 (1994).
[CrossRef]

1993 (2)

1992 (1)

1991 (2)

M. Hardy, “Entropies of likelihood functions,” in Maximum Entropy and Bayesian Methods, C.R.Smith, G.J.Erickson, and P. O. Neudorfer, eds. (Academic, 1991), pp. 127–130.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

1990 (2)

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–36 (1990).
[CrossRef]

J. Ho, B. V. Funt, and M. S. Drew, “Separating a color signal into illumination and surface reflectance components: theory and applications,” IEEE Trans. Pattern Anal. Machine Intell. 12, 966–977 (1990).
[CrossRef]

1989 (1)

1987 (3)

R. Gershon, A. D. Jepson, and J. K. Tsotsos, “From [R, G, B] to surface reflectance: computing color constant descriptors in images,” in Proceedings of the International Joint Conference on Artificial Intelligence (AAAI/MIT, 1987), pp. 755–758.

B. A. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Machine Intell. 9, 2–13 (1987).
[CrossRef]

A. Yuille, “A method for computing spectral reflectance,” Biol. Cybern. 56, 195–201 (1987).
[CrossRef] [PubMed]

1986 (4)

1985 (2)

A. Blake, “Boundary conditions for lightness computation in Mondrian World,” Lect. Notes Comput. Sci. 32, 314–327(1985).

J. A. Worthey, “Limitations of color constancy,” J. Opt. Soc. Am. A 2, 1014–1026 (1985).
[CrossRef]

1984 (1)

L. T. Maloney, “Computational approaches to color constancy,” Ph.D. thesis (Stanford University, 1984).

1982 (1)

G. West and M. H. Brill, “Necessary and sufficient conditions for Von Kries chromatic adaptation to give color constancy,” J. Math. Biol. 15, 249–258 (1982).
[CrossRef] [PubMed]

1981 (2)

M. H. Brill and G. West, “Contributions to the theory of invariance of colour under the condition of varying illumination,” J. Math. Biol. 11, 337–350 (1981).
[CrossRef]

J. E. Kaufman, IES Lighting Handbook (Illuminating Engineering Society of North America, 1981).

1980 (1)

G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

1979 (3)

M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,” J. Theor. Biol. 78, 305–308 (1979).
[CrossRef] [PubMed]

G. West, “Color perception and the limits of color constancy,” J. Math. Biol. 8, 47–53 (1979).
[CrossRef] [PubMed]

E. L. Hall, “TV and color coding,” in Computer Image Processing and Recognition, W.Rheinboldt, ed. (Academic, 1979), pp. 214–239.

1978 (1)

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,” J. Theor. Biol. 71, 473–478 (1978).
[CrossRef] [PubMed]

1977 (1)

E. H. Land, “The Retinex theory of color vision,” Sci. Am. 237, 108–128 (1977).
[CrossRef] [PubMed]

1976 (1)

Munsell Color Corporation, Munsell Book of Color-Matte Finish Collection (Munsell Color, 1976).

1974 (1)

B. K. P. Horn, “Determining lightness from an image,” Comput. Vision Graph. 3, 277–299 (1974).

1971 (1)

1968 (1)

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

1967 (1)

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas (Wiley, 1967).

1964 (2)

D. B. Judd, D. L. MacAdam, and G. Wyszecky, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031–1040 (1964).
[CrossRef]

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

1962 (1)

1945 (1)

1940 (1)

1878 (1)

J. von Kries, “Beitrag zur Physiologie der Gesichtsempfinding,” Arch. Anat. Physiol. 2, 5050–5524 (1878).

Alapati, N.

N. H. Younan, K. Ponnala, and N. Alapati, “Edge detection in multispectral imagery via maximum entropy,” presented at the International Symposium on Remote Sensing of Environment, San José, Costa Rica (25–29 June 2007).

Arbel, T.

S. Skaff, T. Arbel, and J. J. Clark, “A sequential Bayesian approach to color constancy using multiple sensors,” Computer Vision and Image Understanding 113, 993–1004 (2009).
[CrossRef]

M. Toews and T. Arbel, “Entropy-of-likelihood feature selection for image correspondence,” in Proceedings of the IEEE Conference on Computer Vision (IEEE, 2003), pp. 1041–1047.
[CrossRef]

S. Skaff, T. Arbel, and J. J. Clark, “Active Bayesian color constancy with non-uniform sensors,” in Proceedings of the IEEE International Conference on Pattern Recognition (IEEE, 2002), pp. 681–684.

Barnard, K.

K. Barnard, V. Cardei, and B. Funt, “A comparison of computational color constancy algorithms. I.: methodology and experiments with synthesized data,” IEEE Trans. Image Process. 11, 972–984 (2002).
[CrossRef]

K. Barnard, “Practical colour constancy,” Ph.D. thesis (School of Computing, Simon Fraser University, 1999).

K. Barnard, “Computational colour constancy: taking theory into practice,” Master’s thesis (School of Computing, Simon Fraser University, 1995).

Barret, D.

B. Jedynak, H. Zheng, M. Daoudi, and D. Barret, “Maximum entropy models for skin detection,” in Proceedings of the IEEE International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (IEEE, 2003), pp. 180–193.
[CrossRef]

Blake, A.

P. V. Gehler, C. Rother, A. Blake, T. Sharp, and T. Minka, “Bayesian color constancy revisited,” in Proceedings of the IEEE International Conference on Computer Vision and Patter Recognition (IEEE, 2008), pp. 1–8.

A. Blake, “Boundary conditions for lightness computation in Mondrian World,” Lect. Notes Comput. Sci. 32, 314–327(1985).

Brainard, D. H.

D. H. Brainard and W. T. Freeman, “Bayesian color constancy,” J. Opt. Soc. Am. A 14, 1393–1411 (1997).
[CrossRef]

W. T. Freeman and D. H. Brainard, “Bayesian decision theory, the maximum local mass estimate, and color constancy,” in Proceedings of the IEEE International Conference on Computer Vision (IEEE, 1995), pp. 210–217.
[CrossRef]

D. H. Brainard and B. A. Wandell, “Analysis of the Retinex theory of color vision,” J. Opt. Soc. Am. A 3, 1651–1661 (1986).
[CrossRef] [PubMed]

Brill, M. H.

G. West and M. H. Brill, “Necessary and sufficient conditions for Von Kries chromatic adaptation to give color constancy,” J. Math. Biol. 15, 249–258 (1982).
[CrossRef] [PubMed]

M. H. Brill and G. West, “Contributions to the theory of invariance of colour under the condition of varying illumination,” J. Math. Biol. 11, 337–350 (1981).
[CrossRef]

M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,” J. Theor. Biol. 78, 305–308 (1979).
[CrossRef] [PubMed]

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,” J. Theor. Biol. 71, 473–478 (1978).
[CrossRef] [PubMed]

Buchsbaum, G.

G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

Cardei, V.

K. Barnard, V. Cardei, and B. Funt, “A comparison of computational color constancy algorithms. I.: methodology and experiments with synthesized data,” IEEE Trans. Image Process. 11, 972–984 (2002).
[CrossRef]

Chalmers, A.

K. Devlin, A. Chalmers, A. Wilkie, and W. Purgathofer, “Tone reproduction and physically based spectral rendering,” in Computer Graphics Forum (Eurographics, 2002), pp. 101–123.

Clark, J. J.

S. Skaff, T. Arbel, and J. J. Clark, “A sequential Bayesian approach to color constancy using multiple sensors,” Computer Vision and Image Understanding 113, 993–1004 (2009).
[CrossRef]

J. J. Clark and S. Skaff, “A spectral theory of color perception,” J. Opt. Soc. Am. A 26, 2488–2502 (2009).
[CrossRef]

S. Skaff and J. J. Clark, “Maximum entropy spectral models for color constancy,” in Proceedings of the Color Imaging Conference (The Society for Imaging Science and Technology, 2007), pp. 100–105.

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

S. Skaff, T. Arbel, and J. J. Clark, “Active Bayesian color constancy with non-uniform sensors,” in Proceedings of the IEEE International Conference on Pattern Recognition (IEEE, 2002), pp. 681–684.

Cohen, J.

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

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

D’Zmura, M.

D’Zmura, M. D.

Daoudi, M.

B. Jedynak, H. Zheng, M. Daoudi, and D. Barret, “Maximum entropy models for skin detection,” in Proceedings of the IEEE International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (IEEE, 2003), pp. 180–193.
[CrossRef]

Devlin, K.

K. Devlin, A. Chalmers, A. Wilkie, and W. Purgathofer, “Tone reproduction and physically based spectral rendering,” in Computer Graphics Forum (Eurographics, 2002), pp. 101–123.

Drew, M. S.

G. D. Finlayson, M. S. Drew, and C. Lu, “Intrinsic images by entropy minimization,” in Proceedings of the European Conference on Computer Vision (Springer, 2004), pp. 582–595.

J. Ho, B. V. Funt, and M. S. Drew, “Separating a color signal into illumination and surface reflectance components: theory and applications,” IEEE Trans. Pattern Anal. Machine Intell. 12, 966–977 (1990).
[CrossRef]

Duda, R. O.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification and Scene Analysis (Wiley, 2001).

Finlayson, G.

G. Finlayson and S. Hordley, “Selection for gamut mapping colour constancy,” Image Vis. Comp. 17, 597–604 (1999).
[CrossRef]

Finlayson, G. D.

S. D. Hordley and G. D. Finlayson, “Reevaluation of color constancy algorithm performance,” J. Opt. Soc. Am. A 23, 1008–1020 (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]

G. D. Finlayson, M. S. Drew, and C. Lu, “Intrinsic images by entropy minimization,” in Proceedings of the European Conference on Computer Vision (Springer, 2004), pp. 582–595.

G. D. Finlayson, S. D. Hordley, and P. M. Hubel, “Color by correlation: a simple unifying framework for color constancy,” IEEE Trans. Pattern Anal. Machine Intell. 23, 1209–1221 (2001).
[CrossRef]

G. D. Finlayson, P. M. Hubel, and S. Hordley, “Color by correlation,” in Proceedings of the Fifth Color Imaging Conference (The Society for Imaging Science and Technology, 1997), pp. 6–11.

G. D. Finlayson, “Color in perspective,” IEEE Trans. Pattern Anal. Machine Intell. 18, 1034–1038 (1996).
[CrossRef]

Forsyth, D. A.

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–36 (1990).
[CrossRef]

Foster, D. H.

I. Marin-Franch and D. H. Foster, “Estimating the information available from coloured surfaces in natural scenes,” in Proceedings of the European Conference on Color in Graphics, Imaging and Vision (The Society for Imaging Science and Technology, 2006), pp. 44–47.

Freeman, W. T.

D. H. Brainard and W. T. Freeman, “Bayesian color constancy,” J. Opt. Soc. Am. A 14, 1393–1411 (1997).
[CrossRef]

W. T. Freeman and D. H. Brainard, “Bayesian decision theory, the maximum local mass estimate, and color constancy,” in Proceedings of the IEEE International Conference on Computer Vision (IEEE, 1995), pp. 210–217.
[CrossRef]

Funt, B.

B. Funt and L. Shi, “The Rehabilitation of MaxRGB,” in Proceedings of the Eighteenth Color Imaging Conference (The Society for Imaging Science and Technology, 2010), pp. 256–259.

M. Mosny and B. Funt, “Multispectral color constancy,” in Proceedings of the Color Imaging Conference (The Society for Imaging Science and Technology, 2006), pp. 309–313.

K. Barnard, V. Cardei, and B. Funt, “A comparison of computational color constancy algorithms. I.: methodology and experiments with synthesized data,” IEEE Trans. Image Process. 11, 972–984 (2002).
[CrossRef]

Funt, B. V.

J. Ho, B. V. Funt, and M. S. Drew, “Separating a color signal into illumination and surface reflectance components: theory and applications,” IEEE Trans. Pattern Anal. Machine Intell. 12, 966–977 (1990).
[CrossRef]

Gehler, P. V.

P. V. Gehler, C. Rother, A. Blake, T. Sharp, and T. Minka, “Bayesian color constancy revisited,” in Proceedings of the IEEE International Conference on Computer Vision and Patter Recognition (IEEE, 2008), pp. 1–8.

Gershon, R.

R. Gershon, A. D. Jepson, and J. K. Tsotsos, “From [R, G, B] to surface reflectance: computing color constant descriptors in images,” in Proceedings of the International Joint Conference on Artificial Intelligence (AAAI/MIT, 1987), pp. 755–758.

Gray, R. M.

M. R. Gupta and R. M. Gray, “Color conversions using maximum entropy estimation,” in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2001), pp. 118–121.

Gupta, M. R.

M. R. Gupta and R. M. Gray, “Color conversions using maximum entropy estimation,” in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2001), pp. 118–121.

Hall, E. L.

E. L. Hall, “TV and color coding,” in Computer Image Processing and Recognition, W.Rheinboldt, ed. (Academic, 1979), pp. 214–239.

Hall, R.

R. Hall, “Comparing spectral color computation methods,” IEEE Comp. Grap. Appl. 19, 36–45 (1999).
[CrossRef]

Hallikainen, J.

Hardy, M.

M. Hardy, “Entropies of likelihood functions,” in Maximum Entropy and Bayesian Methods, C.R.Smith, G.J.Erickson, and P. O. Neudorfer, eds. (Academic, 1991), pp. 127–130.

Hart, P. E.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification and Scene Analysis (Wiley, 2001).

Ho, J.

J. Ho, B. V. Funt, and M. S. Drew, “Separating a color signal into illumination and surface reflectance components: theory and applications,” IEEE Trans. Pattern Anal. Machine Intell. 12, 966–977 (1990).
[CrossRef]

Hordley, S.

G. Finlayson and S. Hordley, “Selection for gamut mapping colour constancy,” Image Vis. Comp. 17, 597–604 (1999).
[CrossRef]

G. D. Finlayson, P. M. Hubel, and S. Hordley, “Color by correlation,” in Proceedings of the Fifth Color Imaging Conference (The Society for Imaging Science and Technology, 1997), pp. 6–11.

Hordley, S. D.

S. D. Hordley and G. D. Finlayson, “Reevaluation of color constancy algorithm performance,” J. Opt. Soc. Am. A 23, 1008–1020 (2006).
[CrossRef]

G. D. Finlayson, S. D. Hordley, and P. M. Hubel, “Color by correlation: a simple unifying framework for color constancy,” IEEE Trans. Pattern Anal. Machine Intell. 23, 1209–1221 (2001).
[CrossRef]

Horn, B. K. P.

B. K. P. Horn, “Determining lightness from an image,” Comput. Vision Graph. 3, 277–299 (1974).

Hubel, P. M.

G. D. Finlayson, S. D. Hordley, and P. M. Hubel, “Color by correlation: a simple unifying framework for color constancy,” IEEE Trans. Pattern Anal. Machine Intell. 23, 1209–1221 (2001).
[CrossRef]

G. D. Finlayson, P. M. Hubel, and S. Hordley, “Color by correlation,” in Proceedings of the Fifth Color Imaging Conference (The Society for Imaging Science and Technology, 1997), pp. 6–11.

Hurlbert, A.

Iverson, G.

Jaaskelainen, T.

Jaynes, E. T.

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

Jedynak, B.

B. Jedynak, H. Zheng, M. Daoudi, and D. Barret, “Maximum entropy models for skin detection,” in Proceedings of the IEEE International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (IEEE, 2003), pp. 180–193.
[CrossRef]

Jepson, A. D.

R. Gershon, A. D. Jepson, and J. K. Tsotsos, “From [R, G, B] to surface reflectance: computing color constant descriptors in images,” in Proceedings of the International Joint Conference on Artificial Intelligence (AAAI/MIT, 1987), pp. 755–758.

Judd, D. B.

Kaufman, J. E.

J. E. Kaufman, IES Lighting Handbook (Illuminating Engineering Society of North America, 1981).

Kries, J. von

J. von Kries, “Beitrag zur Physiologie der Gesichtsempfinding,” Arch. Anat. Physiol. 2, 5050–5524 (1878).

Land, E. H.

Lennie, P.

Lu, C.

C. Lu, “Removing shadows from color images,” Ph.D. thesis (School of Computing, Simon Fraser University, 2006).

G. D. Finlayson, M. S. Drew, and C. Lu, “Intrinsic images by entropy minimization,” in Proceedings of the European Conference on Computer Vision (Springer, 2004), pp. 582–595.

MacAdam, D. L.

Maloney, L. T.

Marin-Franch, I.

I. Marin-Franch and D. H. Foster, “Estimating the information available from coloured surfaces in natural scenes,” in Proceedings of the European Conference on Color in Graphics, Imaging and Vision (The Society for Imaging Science and Technology, 2006), pp. 44–47.

McCann, J. J.

Minka, T.

P. V. Gehler, C. Rother, A. Blake, T. Sharp, and T. Minka, “Bayesian color constancy revisited,” in Proceedings of the IEEE International Conference on Computer Vision and Patter Recognition (IEEE, 2008), pp. 1–8.

Moon, P.

Morovic, P.

Mosny, M.

M. Mosny and B. Funt, “Multispectral color constancy,” in Proceedings of the Color Imaging Conference (The Society for Imaging Science and Technology, 2006), pp. 309–313.

Neudorfer, P. O.

M. Hardy, “Entropies of likelihood functions,” in Maximum Entropy and Bayesian Methods, C.R.Smith, G.J.Erickson, and P. O. Neudorfer, eds. (Academic, 1991), pp. 127–130.

Parkkinen, J. P. S.

Ponnala, K.

N. H. Younan, K. Ponnala, and N. Alapati, “Edge detection in multispectral imagery via maximum entropy,” presented at the International Symposium on Remote Sensing of Environment, San José, Costa Rica (25–29 June 2007).

Purgathofer, W.

K. Devlin, A. Chalmers, A. Wilkie, and W. Purgathofer, “Tone reproduction and physically based spectral rendering,” in Computer Graphics Forum (Eurographics, 2002), pp. 101–123.

Rother, C.

P. V. Gehler, C. Rother, A. Blake, T. Sharp, and T. Minka, “Bayesian color constancy revisited,” in Proceedings of the IEEE International Conference on Computer Vision and Patter Recognition (IEEE, 2008), pp. 1–8.

Schettini, R.

R. Schettini, “Deriving spectral reflectance functions of computer-simulated object colors,” Comput. Graph. Forum 13, 211–217 (1994).
[CrossRef]

Sharp, T.

P. V. Gehler, C. Rother, A. Blake, T. Sharp, and T. Minka, “Bayesian color constancy revisited,” in Proceedings of the IEEE International Conference on Computer Vision and Patter Recognition (IEEE, 2008), pp. 1–8.

Sheskin, D. J.

D. J. Sheskin, Handbook of Parametric and Nonparametric Statistical Procedures (Chapman and Hall, 2000).

Shi, L.

B. Funt and L. Shi, “The Rehabilitation of MaxRGB,” in Proceedings of the Eighteenth Color Imaging Conference (The Society for Imaging Science and Technology, 2010), pp. 256–259.

Skaff, S.

S. Skaff, T. Arbel, and J. J. Clark, “A sequential Bayesian approach to color constancy using multiple sensors,” Computer Vision and Image Understanding 113, 993–1004 (2009).
[CrossRef]

J. J. Clark and S. Skaff, “A spectral theory of color perception,” J. Opt. Soc. Am. A 26, 2488–2502 (2009).
[CrossRef]

S. Skaff and J. J. Clark, “Maximum entropy spectral models for color constancy,” in Proceedings of the Color Imaging Conference (The Society for Imaging Science and Technology, 2007), pp. 100–105.

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

S. Skaff, T. Arbel, and J. J. Clark, “Active Bayesian color constancy with non-uniform sensors,” in Proceedings of the IEEE International Conference on Pattern Recognition (IEEE, 2002), pp. 681–684.

S. Skaff, “Active Bayesian color constancy with non-uniform sensors,” master’s thesis (McGill University, 2002).

Spencer, D. E.

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas (Wiley, 1967).

W. S. Stiles and G. Wyszecki, “Counting metameric object colors,” J. Opt. Soc. Am. 52, 313–322 (1962).
[CrossRef]

Stork, D. G.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification and Scene Analysis (Wiley, 2001).

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Toews, M.

M. Toews and T. Arbel, “Entropy-of-likelihood feature selection for image correspondence,” in Proceedings of the IEEE Conference on Computer Vision (IEEE, 2003), pp. 1041–1047.
[CrossRef]

Tsotsos, J. K.

R. Gershon, A. D. Jepson, and J. K. Tsotsos, “From [R, G, B] to surface reflectance: computing color constant descriptors in images,” in Proceedings of the International Joint Conference on Artificial Intelligence (AAAI/MIT, 1987), pp. 755–758.

Wandell, B. A.

West, G.

G. West and M. H. Brill, “Necessary and sufficient conditions for Von Kries chromatic adaptation to give color constancy,” J. Math. Biol. 15, 249–258 (1982).
[CrossRef] [PubMed]

M. H. Brill and G. West, “Contributions to the theory of invariance of colour under the condition of varying illumination,” J. Math. Biol. 11, 337–350 (1981).
[CrossRef]

G. West, “Color perception and the limits of color constancy,” J. Math. Biol. 8, 47–53 (1979).
[CrossRef] [PubMed]

Wilkie, A.

K. Devlin, A. Chalmers, A. Wilkie, and W. Purgathofer, “Tone reproduction and physically based spectral rendering,” in Computer Graphics Forum (Eurographics, 2002), pp. 101–123.

Worthey, J. A.

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas (Wiley, 1967).

W. S. Stiles and G. Wyszecki, “Counting metameric object colors,” J. Opt. Soc. Am. 52, 313–322 (1962).
[CrossRef]

Wyszecky, G.

Younan, N. H.

N. H. Younan, K. Ponnala, and N. Alapati, “Edge detection in multispectral imagery via maximum entropy,” presented at the International Symposium on Remote Sensing of Environment, San José, Costa Rica (25–29 June 2007).

Yuille, A.

A. Yuille, “A method for computing spectral reflectance,” Biol. Cybern. 56, 195–201 (1987).
[CrossRef] [PubMed]

Zheng, H.

B. Jedynak, H. Zheng, M. Daoudi, and D. Barret, “Maximum entropy models for skin detection,” in Proceedings of the IEEE International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (IEEE, 2003), pp. 180–193.
[CrossRef]

Arch. Anat. Physiol. (1)

J. von Kries, “Beitrag zur Physiologie der Gesichtsempfinding,” Arch. Anat. Physiol. 2, 5050–5524 (1878).

Biol. Cybern. (1)

A. Yuille, “A method for computing spectral reflectance,” Biol. Cybern. 56, 195–201 (1987).
[CrossRef] [PubMed]

Comput. Graph. Forum (1)

R. Schettini, “Deriving spectral reflectance functions of computer-simulated object colors,” Comput. Graph. Forum 13, 211–217 (1994).
[CrossRef]

Comput. Vision Graph. (1)

B. K. P. Horn, “Determining lightness from an image,” Comput. Vision Graph. 3, 277–299 (1974).

Computer Vision and Image Understanding (1)

S. Skaff, T. Arbel, and J. J. Clark, “A sequential Bayesian approach to color constancy using multiple sensors,” Computer Vision and Image Understanding 113, 993–1004 (2009).
[CrossRef]

IEEE Comp. Grap. Appl. (1)

R. Hall, “Comparing spectral color computation methods,” IEEE Comp. Grap. Appl. 19, 36–45 (1999).
[CrossRef]

IEEE Trans. Image Process. (1)

K. Barnard, V. Cardei, and B. Funt, “A comparison of computational color constancy algorithms. I.: methodology and experiments with synthesized data,” IEEE Trans. Image Process. 11, 972–984 (2002).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (4)

J. Ho, B. V. Funt, and M. S. Drew, “Separating a color signal into illumination and surface reflectance components: theory and applications,” IEEE Trans. Pattern Anal. Machine Intell. 12, 966–977 (1990).
[CrossRef]

B. A. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Machine Intell. 9, 2–13 (1987).
[CrossRef]

G. D. Finlayson, S. D. Hordley, and P. M. Hubel, “Color by correlation: a simple unifying framework for color constancy,” IEEE Trans. Pattern Anal. Machine Intell. 23, 1209–1221 (2001).
[CrossRef]

G. D. Finlayson, “Color in perspective,” IEEE Trans. Pattern Anal. Machine Intell. 18, 1034–1038 (1996).
[CrossRef]

IEEE Trans. Syst. Sci. Cyber. (1)

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

Image Vis. Comp. (1)

G. Finlayson and S. Hordley, “Selection for gamut mapping colour constancy,” Image Vis. Comp. 17, 597–604 (1999).
[CrossRef]

Int. J. Comput. Vis. (1)

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–36 (1990).
[CrossRef]

J. Franklin Inst. (1)

G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

J. Math. Biol. (3)

G. West, “Color perception and the limits of color constancy,” J. Math. Biol. 8, 47–53 (1979).
[CrossRef] [PubMed]

M. H. Brill and G. West, “Contributions to the theory of invariance of colour under the condition of varying illumination,” J. Math. Biol. 11, 337–350 (1981).
[CrossRef]

G. West and M. H. Brill, “Necessary and sufficient conditions for Von Kries chromatic adaptation to give color constancy,” J. Math. Biol. 15, 249–258 (1982).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (14)

M. D’Zmura and P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1672 (1986).
[CrossRef] [PubMed]

M. D. D’Zmura, “Color constancy: surface color from changing illumination,” J. Opt. Soc. Am. A 9, 490–493 (1992).
[CrossRef]

M. D. D’Zmura and G. Iverson, “Color Constancy II. Results for two-stage linear recovery of spectral descriptions for lights and surfaces,” J. Opt. Soc. Am. A 10, 2166–2180 (1993).
[CrossRef]

M. D. D’Zmura and G. Iverson, “Color Constancy I. Basic theory of two-stage linear recovery of spectral descriptions for lights and surfaces,” J. Opt. Soc. Am. A 10, 2148–2165 (1993).
[CrossRef]

M. D. D’Zmura and G. Iverson, “Color Constancy III. General linear recovery of spectral descriptions for lights and surfaces,” J. Opt. Soc. Am. A 11, 2389–2400 (1994).
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J. P. S. Parkkinen, J. Hallikainen, and T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322(1989).
[CrossRef]

S. D. Hordley and G. D. Finlayson, “Reevaluation of color constancy algorithm performance,” J. Opt. Soc. Am. A 23, 1008–1020 (2006).
[CrossRef]

D. H. Brainard and B. A. Wandell, “Analysis of the Retinex theory of color vision,” J. Opt. Soc. Am. A 3, 1651–1661 (1986).
[CrossRef] [PubMed]

A. Hurlbert, “Formal connections between lightness algorithms,” J. Opt. Soc. Am. A 3, 1684–1693 (1986).
[CrossRef] [PubMed]

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. A. Worthey, “Limitations of color constancy,” J. Opt. Soc. Am. A 2, 1014–1026 (1985).
[CrossRef]

J. J. Clark and S. Skaff, “A spectral theory of color perception,” J. Opt. Soc. Am. A 26, 2488–2502 (2009).
[CrossRef]

D. H. Brainard and W. T. Freeman, “Bayesian color constancy,” J. Opt. Soc. Am. A 14, 1393–1411 (1997).
[CrossRef]

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]

J. Theor. Biol. (2)

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,” J. Theor. Biol. 71, 473–478 (1978).
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M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,” J. Theor. Biol. 78, 305–308 (1979).
[CrossRef] [PubMed]

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A. Blake, “Boundary conditions for lightness computation in Mondrian World,” Lect. Notes Comput. Sci. 32, 314–327(1985).

Psychon. Sci. (1)

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

Sci. Am. (1)

E. H. Land, “The Retinex theory of color vision,” Sci. Am. 237, 108–128 (1977).
[CrossRef] [PubMed]

Other (30)

K. Barnard, “Computational colour constancy: taking theory into practice,” Master’s thesis (School of Computing, Simon Fraser University, 1995).

G. D. Finlayson, P. M. Hubel, and S. Hordley, “Color by correlation,” in Proceedings of the Fifth Color Imaging Conference (The Society for Imaging Science and Technology, 1997), pp. 6–11.

B. Funt and L. Shi, “The Rehabilitation of MaxRGB,” in Proceedings of the Eighteenth Color Imaging Conference (The Society for Imaging Science and Technology, 2010), pp. 256–259.

M. Mosny and B. Funt, “Multispectral color constancy,” in Proceedings of the Color Imaging Conference (The Society for Imaging Science and Technology, 2006), pp. 309–313.

L. T. Maloney, “Computational approaches to color constancy,” Ph.D. thesis (Stanford University, 1984).

K. Barnard, “Practical colour constancy,” Ph.D. thesis (School of Computing, Simon Fraser University, 1999).

P. V. Gehler, C. Rother, A. Blake, T. Sharp, and T. Minka, “Bayesian color constancy revisited,” in Proceedings of the IEEE International Conference on Computer Vision and Patter Recognition (IEEE, 2008), pp. 1–8.

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

S. Skaff, T. Arbel, and J. J. Clark, “Active Bayesian color constancy with non-uniform sensors,” in Proceedings of the IEEE International Conference on Pattern Recognition (IEEE, 2002), pp. 681–684.

S. Skaff, “Active Bayesian color constancy with non-uniform sensors,” master’s thesis (McGill University, 2002).

R. Gershon, A. D. Jepson, and J. K. Tsotsos, “From [R, G, B] to surface reflectance: computing color constant descriptors in images,” in Proceedings of the International Joint Conference on Artificial Intelligence (AAAI/MIT, 1987), pp. 755–758.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas (Wiley, 1967).

W. T. Freeman and D. H. Brainard, “Bayesian decision theory, the maximum local mass estimate, and color constancy,” in Proceedings of the IEEE International Conference on Computer Vision (IEEE, 1995), pp. 210–217.
[CrossRef]

K. Devlin, A. Chalmers, A. Wilkie, and W. Purgathofer, “Tone reproduction and physically based spectral rendering,” in Computer Graphics Forum (Eurographics, 2002), pp. 101–123.

N. H. Younan, K. Ponnala, and N. Alapati, “Edge detection in multispectral imagery via maximum entropy,” presented at the International Symposium on Remote Sensing of Environment, San José, Costa Rica (25–29 June 2007).

G. D. Finlayson, M. S. Drew, and C. Lu, “Intrinsic images by entropy minimization,” in Proceedings of the European Conference on Computer Vision (Springer, 2004), pp. 582–595.

C. Lu, “Removing shadows from color images,” Ph.D. thesis (School of Computing, Simon Fraser University, 2006).

M. R. Gupta and R. M. Gray, “Color conversions using maximum entropy estimation,” in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2001), pp. 118–121.

B. Jedynak, H. Zheng, M. Daoudi, and D. Barret, “Maximum entropy models for skin detection,” in Proceedings of the IEEE International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (IEEE, 2003), pp. 180–193.
[CrossRef]

I. Marin-Franch and D. H. Foster, “Estimating the information available from coloured surfaces in natural scenes,” in Proceedings of the European Conference on Color in Graphics, Imaging and Vision (The Society for Imaging Science and Technology, 2006), pp. 44–47.

E. L. Hall, “TV and color coding,” in Computer Image Processing and Recognition, W.Rheinboldt, ed. (Academic, 1979), pp. 214–239.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification and Scene Analysis (Wiley, 2001).

M. Hardy, “Entropies of likelihood functions,” in Maximum Entropy and Bayesian Methods, C.R.Smith, G.J.Erickson, and P. O. Neudorfer, eds. (Academic, 1991), pp. 127–130.

M. Toews and T. Arbel, “Entropy-of-likelihood feature selection for image correspondence,” in Proceedings of the IEEE Conference on Computer Vision (IEEE, 2003), pp. 1041–1047.
[CrossRef]

S. Skaff and J. J. Clark, “Maximum entropy spectral models for color constancy,” in Proceedings of the Color Imaging Conference (The Society for Imaging Science and Technology, 2007), pp. 100–105.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

University of Joensuu Color Group, “Spectral database,” http://spectral.joensuu.fi.

J. E. Kaufman, IES Lighting Handbook (Illuminating Engineering Society of North America, 1981).

D. J. Sheskin, Handbook of Parametric and Nonparametric Statistical Procedures (Chapman and Hall, 2000).

Munsell Color Corporation, Munsell Book of Color-Matte Finish Collection (Munsell Color, 1976).

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Figures (8)

Fig. 1
Fig. 1

Gray circle represents a photon coming from the illuminant and incident on the surface. The light blue circles represent this photon after it has been either reflected from the surface ( Q = T ), absorbed by the surface ( Q = A ), or transmitted by the surface ( Q = T ). The photon has wavelength λ.

Fig. 2
Fig. 2

Scene with n surface patches of reflectance spectra S 1 ( λ ) , S 2 ( λ ) , S 3 ( λ ) , … S n ( λ ) illuminated by one light source of spectrum E ( λ ) .

Fig. 3
Fig. 3

Model and actual spectra obtained in simulation for a Munsell patch scene with one, two, and four surface patches in the scene: (a) Munsell patch 2.5P 2.5 / 2 ( RMSE = 0.2022 , one surface; RMSE = 0.1516 , two surfaces; RMSE = 0.1304 , four surfaces) illuminated with (b) skylight 20 ( RMSE = 0.1501 , one surface; RMSE = 0.1352 , two surfaces; RMSE = 0.1150 , four surfaces).

Fig. 4
Fig. 4

Model and actual spectra obtained in simulation for a Munsell patch scene with one, two, and four surface patches in the scene: (a) Munsell patch 2.5Y 6 / 2 ( RMSE = 0.1813 , one surface; RMSE = 0.1416 , two surfaces; RMSE = 0.0976 , four surfaces) illuminated with (b) tungsten light at temperature 2900 K ( RMSE = 0.2434 , one surface; RMSE = 0.1861 , two surfaces; RMSE = 0.1903 , four surfaces).

Fig. 5
Fig. 5

Average RMSEs between the actual and model spectra obtained in simulation over 100 scenes for the (a) Munsell surface patches repeated in a set of five scenes, (b) daylight, skylight, and tungsten illuminants, and (c) the combination of both the surface patches and illuminants.

Fig. 6
Fig. 6

Model and actual spectra obtained in experiment for a Munsell patch scene with one, two, and four surface patches in the scene: (a) Munsell patch 2.5Y 6 / 2 ( RMSE = 0.2058 , one surface; RMSE = 0.1960 , two surfaces; RMSE = 0.1099 , four surfaces) illuminated with (b) tungsten light at 2900 K ( RMSE = 0.5896 , one surface; RMSE = 0.5192 , two surfaces; RMSE = 0.4361 , four surfaces).

Fig. 7
Fig. 7

Average RMSEs between the actual and model spectra obtained in experiment over 100 scenes for the (a) Munsell surface patches repeated in a set of five scenes, (b) tungsten illuminants, and (c) the combination of both the surface patches and illuminants.

Fig. 8
Fig. 8

Model and actual spectra of construction paper samples, (a) light blue ( RMSE = 0.1238 ) and (b) green ( RMSE = 0.1155 ).

Tables (3)

Tables Icon

Table 1 Average and Median of the RMSEs in Surface Patch and Illuminant Spectral Estimates in All Munsell Scenes for Simulation and Experimental Data

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Table 2 Average and Median of the RMSEs in Surface Patch and Illuminant Spectral Estimates in All Scenes for the Maximum Entropy Approach and the Bayesian Approach a

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Table 3 Median of the Euclidean Distances for the Chromaticities of the Illuminant Estimates of 100 Scenes (72 Scenes for 2D Gamut Mapping) Given by Five Main Previous Approaches with Their Corresponding Variations (When Applicable) and Our Maximum Entropy Approach a

Equations (48)

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I x ( λ ) = E ( λ ) S x ( λ ) ,
P k x = λ = 1 M S x ( λ ) E ( λ ) R k ( λ ) , k = 1 , 2 , , p ,
S x ( λ ) = j = 1 n σ j x S j ( λ ) ,
E ( λ ) = i = 1 m ϵ i E i ( λ ) .
E ( λ ) = β p E ( λ ) .
p S ( λ , Q = R ) + p S ( λ , Q = A ) + p S ( λ , Q = T ) = p E ( λ ) .
p S ( λ | Q ) = p S ( λ , Q ) p Q ( Q ) .
p S ( λ | Q ) = p S ( Q | λ ) p E ( λ ) p Q ( Q ) .
S ( λ ) = L ( λ | Q = R ) ,
L ( λ | Q = R ) = p S ( Q = R | λ )
S ( λ ) = p S ( Q = R | λ ) .
p Q ( Q = R ) = λ = 1 M p S ( Q = R , λ )
= λ = 1 M p S ( Q = R | λ ) p E ( λ )
= λ = 1 M S ( λ ) p E ( λ ) ,
α = p Q ( Q = R ) .
p R ( λ ) = p S ( λ | Q = R )
= p S ( Q = R | λ ) p E ( λ ) p Q ( Q = R )
= S ( λ ) α p E ( λ ) .
R ( λ ) = α β p R ( λ )
= α β [ S ( λ ) α p E ( λ ) ]
= S ( λ ) [ β p E ( λ ) ]
= S ( λ ) E ( λ ) .
H ( p S , p E ) = H ( p S ) + H ( p E ) = λ = 1 M p S ( λ ) log p S ( λ ) λ = 1 M p E ( λ ) log p E ( λ ) .
S ( λ ) 0 , λ = 1 , , M ;
E ( λ ) 0 , λ = 1 , , M ;
S ( λ ) 1 , λ = 1 , , M ;
E ( λ ) 1 , λ = 1 , , M ;
P R T ( S . * E ) = 0 ,
H n = λ = 1 M p S 1 ( λ ) log p S 1 ( λ ) λ = 1 M p S 2 ( λ ) log p S 2 ( λ ) λ = 1 M p S n ( λ ) log p S n ( λ ) λ = 1 M p E ( λ ) log p E ( λ ) .
P 1 R T ( S 1 . * E ) = 0 ;
P 2 R T ( S 2 . * E ) = 0 ;
P n R T ( S n . * E ) = 0 ,
P 1 = [ P 1 1 P 2 1 P 3 1 ] T ;
P 2 = [ P 1 2 P 2 2 P 3 2 ] T ;
P n = [ P 1 n P 2 n P 3 n ] T .
S 1 ( λ ) 0 , λ = 1 , , M ;
S 2 ( λ ) 0 , λ = 1 , , M ;
S n ( λ ) 0 , λ = 1 , , M ;
E ( λ ) 0 , λ = 1 , , M .
S 1 ( λ ) 1 , λ = 1 , , M ;
S 2 ( λ ) 1 , λ = 1 , , M ;
S n ( λ ) 1 , λ = 1 , , M ;
E ( λ ) 1 , λ = 1 , , M .
RMSE = 1 M λ = 1 M ( S A ( λ ) S M ( λ ) ) 2 .
RMSE = 1 M λ = 1 M ( E A ( λ ) E M ( λ ) ) 2 .
p ( a , b | RGB ) = p ( R G B | a , b ) p ( a , b ) p ( RGB ) p ( RGB | a , b ) p ( a , b ) = p ( RGB | a , b ) p ( a ) p ( b ) .
( r a , g a ) = ( R a / ( R a + G a + B a ) , G a / ( R a + G a + B a ) ) ; ( r m , g m ) = ( R m / ( R m + G m + B m ) , G m / ( R m + G m + B m ) ) ,
E = ( r m r a ) 2 + ( g m g a ) 2 .

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