Abstract

If color appearance is to be a useful feature in identifying an object, then color appearance must remain roughly constant when the object is viewed in different contexts. People maintain approximate color constancy despite variation in the color of nearby objects and despite variation in the spectral power distribution of the ambient light. Land’s retinex algorithm is a model of human color constancy. We analyze the retinex algorithm and discuss its general properties. We show that the algorithm is too sensitive to changes in the color of nearby objects to serve as an adequate model of human color constancy.

© 1986 Optical Society of America

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References

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  1. H. von Helmholtz, Handbuch der Physiologischen Optik, 2nd ed. (Voss, Hamburg, 1896).
  2. H. Helson, “Fundamental problems in color vision. I. The principle governing changes in hue, saturation and lightness of nonselective samples in chromatic illumination,” J. Exp. Psychol. 23, 439–476 (1938).
    [CrossRef]
  3. D. B. Judd, “Hue saturation and lightness of surface colors with chromatic illumination,” J. Opt. Soc. Am. 30, 2–32 (1940).
    [CrossRef]
  4. E. H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).
    [CrossRef] [PubMed]
  5. M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,” J. Theor. Biol. 71, 473–478 (1978).
    [CrossRef] [PubMed]
  6. G. Buchsbaum, “A spatial processor model for object color perception,” J. Franklin Inst. 310, 1–26 (1980).
    [CrossRef]
  7. L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
    [CrossRef] [PubMed]
  8. E. H. Land, “Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image,” Proc. Nat. Acad. Sci. USA 80, 5163–5169 (1983).
  9. E. H. Land, D. H. Hubel, M. S. Livingston, S. H. Hollis, M. M. Burns, “Colour-generating interactions across the corpus callosum,” Nature 303, 616–618 (1983).
    [CrossRef] [PubMed]
  10. J. Frankle, J. J. McCann, “Method and apparatus for lightness imaging,” United States Patent No.4,384,336, 1983.
  11. J. J. McCann, K. Houston, “Calculating color sensations from arrays of physical stimuli,” IEEE Trans. Systems Man Cybern. SMC-13, 1000–1007 (1983).
    [CrossRef]
  12. M. Livingston, D. Hubel, “Anatomy and physiology of a color system in the primate visual cortex,” J. Neurosci. 4, 309–356 (1984).
  13. D. Ingle, “The goldfish as a retinex animal,” Science 227, 651–654 (1985).
    [CrossRef] [PubMed]
  14. S. Grossberg, E. Mingolla, “Neural dynamics of form perception: boundary completions, illusory figures, and neon color spreading,” Psychol. Rev. 92, 173–207 (1985).
    [CrossRef] [PubMed]
  15. A. Blake, “Boundary conditions for lightness computation in Mondrian world,” Comput. Vis. Graphics Image Process. 32, 314–327 (1985).
    [CrossRef]
  16. D. Terzopoulos, “Image analysis using multigrid relaxation methods,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 129–139 (1986).
    [CrossRef]
  17. E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–22 (1986).
    [CrossRef] [PubMed]
  18. M. M. Woolfson, “Some new aspects of color perception,” IBM J. Res. Dev. 3, 313 (1959).
  19. G. Walls, “Land! Land!” Psychol. Bull. 57, 29–48 (1960).
    [CrossRef] [PubMed]
  20. D. B. Judd, “Appraisal of Land’s work on two-primary color projections,” J. Opt. Soc. Am. 50, 254–268 (1960).
    [CrossRef] [PubMed]
  21. M. Brill, G. West, “Contributions to the theory of color,” J. Math. Biol. 11, 337–350 (1981).
    [CrossRef]
  22. J. Worthey, “Limitations of color constancy,” J. Opt. Soc. Am. A 2, 1014–1026 (1985).
    [CrossRef]
  23. R. M. Shapley, “The importance of contrast for the activity of single neurons, the VEP and perception,” Vision Res. 26, 45–62 (1986).
    [CrossRef] [PubMed]
  24. B. K. P. Horn, “Determining lightness from an image,” Comput. Graphics Image Process. 3, 277–299 (1974).
    [CrossRef]
  25. J. J. McCann, S. P. McKee, T. H. Taylor, “Quantitative studies in retinex theory: a comparison between theoretical predictions and observer responses to the color Mondrian experiments,” Vision Res. 16, 445–448 (1976).
    [CrossRef]
  26. Partial descriptions of the path-generation process can be found in McCann et al. (Ref. 25, p. 454), where they write: “The origin of the path and its direction in the 480-point array were determined by a random-number generator. The paths traveled straight ahead until they reached the perimeter of the target where they either reflected back across the target or traveled along the perimeter. The direction of the reflection from the perimeter was also chosen by the random-number generator.” Land (Ref. 8, p. 5165) writes “The signal will proceed … with freedom to branch within the rule of implied directionality inherent in the concept of a signal radiating from a single source,” whereas in Land (Ref. 17, p. 12) the paths are generated by an “optical mouse system [which] is swept along the random pathways” of the image.
  27. McCann et al.25 emphasize that the details of the path-generation process are not critical when they write (Ref. 25, p. 453): “The model described in this paper is one of many embodiments using the ratio-multiplication process.”
  28. L. Maloney, “Computational approaches to color vision,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1984).
  29. B. A. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Vis. (to be published).
  30. R. M. Boynton, Human Color Vision (Holt, New York, 1979).
  31. D. Nickerson, “Spectrophometric data for a collection of Munsell samples,” Report, Cotton Division, U.S. Department of Agriculture, Washington, D.C., 1957.
  32. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonanal. Sci. 1, 369–370 (1964).
  33. D. Judd, G. Wyszecki, Color in Business, Science, and Industry (Wiley, New York, 1975).

1986 (4)

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

D. Terzopoulos, “Image analysis using multigrid relaxation methods,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 129–139 (1986).
[CrossRef]

E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–22 (1986).
[CrossRef] [PubMed]

R. M. Shapley, “The importance of contrast for the activity of single neurons, the VEP and perception,” Vision Res. 26, 45–62 (1986).
[CrossRef] [PubMed]

1985 (4)

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

D. Ingle, “The goldfish as a retinex animal,” Science 227, 651–654 (1985).
[CrossRef] [PubMed]

S. Grossberg, E. Mingolla, “Neural dynamics of form perception: boundary completions, illusory figures, and neon color spreading,” Psychol. Rev. 92, 173–207 (1985).
[CrossRef] [PubMed]

A. Blake, “Boundary conditions for lightness computation in Mondrian world,” Comput. Vis. Graphics Image Process. 32, 314–327 (1985).
[CrossRef]

1984 (1)

M. Livingston, D. Hubel, “Anatomy and physiology of a color system in the primate visual cortex,” J. Neurosci. 4, 309–356 (1984).

1983 (3)

E. H. Land, “Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image,” Proc. Nat. Acad. Sci. USA 80, 5163–5169 (1983).

E. H. Land, D. H. Hubel, M. S. Livingston, S. H. Hollis, M. M. Burns, “Colour-generating interactions across the corpus callosum,” Nature 303, 616–618 (1983).
[CrossRef] [PubMed]

J. J. McCann, K. Houston, “Calculating color sensations from arrays of physical stimuli,” IEEE Trans. Systems Man Cybern. SMC-13, 1000–1007 (1983).
[CrossRef]

1981 (1)

M. Brill, G. West, “Contributions to the theory of color,” J. Math. Biol. 11, 337–350 (1981).
[CrossRef]

1980 (1)

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

1978 (1)

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

1976 (1)

J. J. McCann, S. P. McKee, T. H. Taylor, “Quantitative studies in retinex theory: a comparison between theoretical predictions and observer responses to the color Mondrian experiments,” Vision Res. 16, 445–448 (1976).
[CrossRef]

1974 (1)

B. K. P. Horn, “Determining lightness from an image,” Comput. Graphics Image Process. 3, 277–299 (1974).
[CrossRef]

1971 (1)

1964 (1)

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

1960 (2)

1959 (1)

M. M. Woolfson, “Some new aspects of color perception,” IBM J. Res. Dev. 3, 313 (1959).

1940 (1)

1938 (1)

H. Helson, “Fundamental problems in color vision. I. The principle governing changes in hue, saturation and lightness of nonselective samples in chromatic illumination,” J. Exp. Psychol. 23, 439–476 (1938).
[CrossRef]

Blake, A.

A. Blake, “Boundary conditions for lightness computation in Mondrian world,” Comput. Vis. Graphics Image Process. 32, 314–327 (1985).
[CrossRef]

Boynton, R. M.

R. M. Boynton, Human Color Vision (Holt, New York, 1979).

Brill, M.

M. Brill, G. West, “Contributions to the theory of color,” J. Math. Biol. 11, 337–350 (1981).
[CrossRef]

Brill, M. H.

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 color perception,” J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

Burns, M. M.

E. H. Land, D. H. Hubel, M. S. Livingston, S. H. Hollis, M. M. Burns, “Colour-generating interactions across the corpus callosum,” Nature 303, 616–618 (1983).
[CrossRef] [PubMed]

Cohen, J.

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

Frankle, J.

J. Frankle, J. J. McCann, “Method and apparatus for lightness imaging,” United States Patent No.4,384,336, 1983.

Grossberg, S.

S. Grossberg, E. Mingolla, “Neural dynamics of form perception: boundary completions, illusory figures, and neon color spreading,” Psychol. Rev. 92, 173–207 (1985).
[CrossRef] [PubMed]

Helson, H.

H. Helson, “Fundamental problems in color vision. I. The principle governing changes in hue, saturation and lightness of nonselective samples in chromatic illumination,” J. Exp. Psychol. 23, 439–476 (1938).
[CrossRef]

Hollis, S. H.

E. H. Land, D. H. Hubel, M. S. Livingston, S. H. Hollis, M. M. Burns, “Colour-generating interactions across the corpus callosum,” Nature 303, 616–618 (1983).
[CrossRef] [PubMed]

Horn, B. K. P.

B. K. P. Horn, “Determining lightness from an image,” Comput. Graphics Image Process. 3, 277–299 (1974).
[CrossRef]

Houston, K.

J. J. McCann, K. Houston, “Calculating color sensations from arrays of physical stimuli,” IEEE Trans. Systems Man Cybern. SMC-13, 1000–1007 (1983).
[CrossRef]

Hubel, D.

M. Livingston, D. Hubel, “Anatomy and physiology of a color system in the primate visual cortex,” J. Neurosci. 4, 309–356 (1984).

Hubel, D. H.

E. H. Land, D. H. Hubel, M. S. Livingston, S. H. Hollis, M. M. Burns, “Colour-generating interactions across the corpus callosum,” Nature 303, 616–618 (1983).
[CrossRef] [PubMed]

Ingle, D.

D. Ingle, “The goldfish as a retinex animal,” Science 227, 651–654 (1985).
[CrossRef] [PubMed]

Judd, D.

D. Judd, G. Wyszecki, Color in Business, Science, and Industry (Wiley, New York, 1975).

Judd, D. B.

Land, E. H.

E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–22 (1986).
[CrossRef] [PubMed]

E. H. Land, “Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image,” Proc. Nat. Acad. Sci. USA 80, 5163–5169 (1983).

E. H. Land, D. H. Hubel, M. S. Livingston, S. H. Hollis, M. M. Burns, “Colour-generating interactions across the corpus callosum,” Nature 303, 616–618 (1983).
[CrossRef] [PubMed]

E. H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).
[CrossRef] [PubMed]

Livingston, M.

M. Livingston, D. Hubel, “Anatomy and physiology of a color system in the primate visual cortex,” J. Neurosci. 4, 309–356 (1984).

Livingston, M. S.

E. H. Land, D. H. Hubel, M. S. Livingston, S. H. Hollis, M. M. Burns, “Colour-generating interactions across the corpus callosum,” Nature 303, 616–618 (1983).
[CrossRef] [PubMed]

Maloney, L.

L. Maloney, “Computational approaches to color vision,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1984).

Maloney, L. T.

McCann, J. J.

J. J. McCann, K. Houston, “Calculating color sensations from arrays of physical stimuli,” IEEE Trans. Systems Man Cybern. SMC-13, 1000–1007 (1983).
[CrossRef]

J. J. McCann, S. P. McKee, T. H. Taylor, “Quantitative studies in retinex theory: a comparison between theoretical predictions and observer responses to the color Mondrian experiments,” Vision Res. 16, 445–448 (1976).
[CrossRef]

E. H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).
[CrossRef] [PubMed]

J. Frankle, J. J. McCann, “Method and apparatus for lightness imaging,” United States Patent No.4,384,336, 1983.

McKee, S. P.

J. J. McCann, S. P. McKee, T. H. Taylor, “Quantitative studies in retinex theory: a comparison between theoretical predictions and observer responses to the color Mondrian experiments,” Vision Res. 16, 445–448 (1976).
[CrossRef]

Mingolla, E.

S. Grossberg, E. Mingolla, “Neural dynamics of form perception: boundary completions, illusory figures, and neon color spreading,” Psychol. Rev. 92, 173–207 (1985).
[CrossRef] [PubMed]

Nickerson, D.

D. Nickerson, “Spectrophometric data for a collection of Munsell samples,” Report, Cotton Division, U.S. Department of Agriculture, Washington, D.C., 1957.

Shapley, R. M.

R. M. Shapley, “The importance of contrast for the activity of single neurons, the VEP and perception,” Vision Res. 26, 45–62 (1986).
[CrossRef] [PubMed]

Taylor, T. H.

J. J. McCann, S. P. McKee, T. H. Taylor, “Quantitative studies in retinex theory: a comparison between theoretical predictions and observer responses to the color Mondrian experiments,” Vision Res. 16, 445–448 (1976).
[CrossRef]

Terzopoulos, D.

D. Terzopoulos, “Image analysis using multigrid relaxation methods,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 129–139 (1986).
[CrossRef]

von Helmholtz, H.

H. von Helmholtz, Handbuch der Physiologischen Optik, 2nd ed. (Voss, Hamburg, 1896).

Walls, G.

G. Walls, “Land! Land!” Psychol. Bull. 57, 29–48 (1960).
[CrossRef] [PubMed]

Wandell, B. A.

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

B. A. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Vis. (to be published).

West, G.

M. Brill, G. West, “Contributions to the theory of color,” J. Math. Biol. 11, 337–350 (1981).
[CrossRef]

Woolfson, M. M.

M. M. Woolfson, “Some new aspects of color perception,” IBM J. Res. Dev. 3, 313 (1959).

Worthey, J.

Wyszecki, G.

D. Judd, G. Wyszecki, Color in Business, Science, and Industry (Wiley, New York, 1975).

Comput. Graphics Image Process. (1)

B. K. P. Horn, “Determining lightness from an image,” Comput. Graphics Image Process. 3, 277–299 (1974).
[CrossRef]

Comput. Vis. Graphics Image Process. (1)

A. Blake, “Boundary conditions for lightness computation in Mondrian world,” Comput. Vis. Graphics Image Process. 32, 314–327 (1985).
[CrossRef]

IBM J. Res. Dev. (1)

M. M. Woolfson, “Some new aspects of color perception,” IBM J. Res. Dev. 3, 313 (1959).

IEEE Trans. Pattern Anal. Mach. Intell. (1)

D. Terzopoulos, “Image analysis using multigrid relaxation methods,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 129–139 (1986).
[CrossRef]

IEEE Trans. Systems Man Cybern. (1)

J. J. McCann, K. Houston, “Calculating color sensations from arrays of physical stimuli,” IEEE Trans. Systems Man Cybern. SMC-13, 1000–1007 (1983).
[CrossRef]

J. Exp. Psychol. (1)

H. Helson, “Fundamental problems in color vision. I. The principle governing changes in hue, saturation and lightness of nonselective samples in chromatic illumination,” J. Exp. Psychol. 23, 439–476 (1938).
[CrossRef]

J. Franklin Inst. (1)

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

J. Math. Biol. (1)

M. Brill, G. West, “Contributions to the theory of color,” J. Math. Biol. 11, 337–350 (1981).
[CrossRef]

J. Neurosci. (1)

M. Livingston, D. Hubel, “Anatomy and physiology of a color system in the primate visual cortex,” J. Neurosci. 4, 309–356 (1984).

J. Opt. Soc. Am. (3)

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

J. Theor. Biol. (1)

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

Nature (1)

E. H. Land, D. H. Hubel, M. S. Livingston, S. H. Hollis, M. M. Burns, “Colour-generating interactions across the corpus callosum,” Nature 303, 616–618 (1983).
[CrossRef] [PubMed]

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

E. H. Land, “Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image,” Proc. Nat. Acad. Sci. USA 80, 5163–5169 (1983).

Psychol. Bull. (1)

G. Walls, “Land! Land!” Psychol. Bull. 57, 29–48 (1960).
[CrossRef] [PubMed]

Psychol. Rev. (1)

S. Grossberg, E. Mingolla, “Neural dynamics of form perception: boundary completions, illusory figures, and neon color spreading,” Psychol. Rev. 92, 173–207 (1985).
[CrossRef] [PubMed]

Psychonanal. Sci. (1)

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

Science (1)

D. Ingle, “The goldfish as a retinex animal,” Science 227, 651–654 (1985).
[CrossRef] [PubMed]

Vision Res. (3)

E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–22 (1986).
[CrossRef] [PubMed]

R. M. Shapley, “The importance of contrast for the activity of single neurons, the VEP and perception,” Vision Res. 26, 45–62 (1986).
[CrossRef] [PubMed]

J. J. McCann, S. P. McKee, T. H. Taylor, “Quantitative studies in retinex theory: a comparison between theoretical predictions and observer responses to the color Mondrian experiments,” Vision Res. 16, 445–448 (1976).
[CrossRef]

Other (9)

Partial descriptions of the path-generation process can be found in McCann et al. (Ref. 25, p. 454), where they write: “The origin of the path and its direction in the 480-point array were determined by a random-number generator. The paths traveled straight ahead until they reached the perimeter of the target where they either reflected back across the target or traveled along the perimeter. The direction of the reflection from the perimeter was also chosen by the random-number generator.” Land (Ref. 8, p. 5165) writes “The signal will proceed … with freedom to branch within the rule of implied directionality inherent in the concept of a signal radiating from a single source,” whereas in Land (Ref. 17, p. 12) the paths are generated by an “optical mouse system [which] is swept along the random pathways” of the image.

McCann et al.25 emphasize that the details of the path-generation process are not critical when they write (Ref. 25, p. 453): “The model described in this paper is one of many embodiments using the ratio-multiplication process.”

L. Maloney, “Computational approaches to color vision,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1984).

B. A. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Vis. (to be published).

R. M. Boynton, Human Color Vision (Holt, New York, 1979).

D. Nickerson, “Spectrophometric data for a collection of Munsell samples,” Report, Cotton Division, U.S. Department of Agriculture, Washington, D.C., 1957.

D. Judd, G. Wyszecki, Color in Business, Science, and Industry (Wiley, New York, 1975).

H. von Helmholtz, Handbuch der Physiologischen Optik, 2nd ed. (Voss, Hamburg, 1896).

J. Frankle, J. J. McCann, “Method and apparatus for lightness imaging,” United States Patent No.4,384,336, 1983.

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Figures (6)

Fig. 1
Fig. 1

A flat surface of colored materials is illuminated by a light source.

Fig. 2
Fig. 2

The spatial composition of the blue, green, yellow, and purple Mondrians used in the simulations of the Land8 retinex algorithm.

Fig. 3
Fig. 3

Predicted color values and lightness triplets computed by the Land8 retinex algorithm for two chips in the standard, blue, green, yellow, and purple Mondrians. The lightness values were computed according to Eq. (8).

Fig. 4
Fig. 4

The effective reference surface for the standard, blue, green, yellow, and purple Mondrians under CIE illuminant D65.

Fig. 5
Fig. 5

The spatial composition of the red, blue, gray, and gray-red Mondrians used in the simulations of retinex algorithm with reset.

Fig. 6
Fig. 6

Predicted color values and lightness triplets computed by the McCann–Houston retinex algorithm for two chips in the standard, red, blue, gray, and gray-red Mondrians. The lightness values were computed according to Eq. (A12).

Equations (31)

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

ρ k x = n = 1 N C x ( λ n ) R k ( λ n ) = n = 1 N E ( λ n ) S x ( λ n ) R k ( λ n ) .
ρ x = Λ E σ x
A ( x 2 ) A ( x 2 ) + log ( ρ x 2 ) log ( ρ x 1 ) .
A ( x i ) A ( x i ) + log ( ρ x i ) log ( ρ x 1 ) ,
l k x = log ( ρ k x ρ k x 1 ) + + log ( ρ k x ρ k x N ) N = log [ ρ k x ( ρ k x 1 ρ k x N ) 1 / N ] = log [ ρ k x G ˆ k ( x ) ] ,
log G k ( x ) = 1 E ( N ) [ all pixels E ( x | x i ) log ( ρ k x i ) ] ,
l k x = log [ ρ k x G k ( x ) ] ,
E ( x | x i ) = n = 1 N pl P ( n = x | 1 = x i ) .
l k x = log ( ρ k x G k ) ,
ρ k x = n = 1 N E ( λ n ) S x ( λ n ) R k ( λ n ) .
l k x = log ( ρ k x G k ) ,
ρ x = Λ E σ x ,
σ x = Λ E 1 ρ x .
l x = Γ x ρ x ,
σ r x = Λ E 1 γ ,
l x ( n + 1 ) = { l x ( n ) f * [ ρ x ρ c n l c n ( n ) ] } 1 / 2 ,
l x = lim n { l x ( n ) f * [ ρ x ρ c n l c n ( n ) ] } 1 / 2 .
( l x ) 2 = lim n l x ( n ) f * [ ρ x ρ c n l c n ( n ) ] .
l x = lim n f * [ ρ x ρ c n l c n ( n ) ] .
l x s = lim n ρ x s ρ c n l c n ( n ) .
l x = ( ρ x ρ x s ) l x s .
l x s = l x l ( ρ x s ρ x l ) .
l x ρ x ρ x l
ρ x ρ c n l c n ( n ) ( ρ x ρ c n ) ( ρ c n ρ x l ) ρ x ρ x l .
f * [ ρ x ρ c n l c n ( n ) ] f * ( ρ x ρ x l ) = ρ x ρ x l .
l x ( n + 1 ) [ l x ( n ) f * ( ρ x ρ x l ) ] 1 / 2 = [ l x ( n ) ρ x ρ x l ] 1 / 2 .
exp ( l k x ) = ρ k x ρ k x l .
p ( x , y ) = s ( x , y ) r ( x , y )
log [ p ( x , y ) ] = log [ s ( x , y ) ] + log [ r ( x , y ) ] .
ρ ( x , y ) = n = 1 N E ( λ n ) S ( x , y , λ n ) R k ( λ n ) .
ρ ( x , y ) = E 2 ( x , y ) n = 1 N E 1 ( λ ) S ( x , y , λ n ) R k ( λ n ) .

Metrics