Abstract

We develop sensor transformations, collectively called spectral sharpening, that convert a given set of sensor sensitivity functions into a new set that will improve the performance of any color-constancy algorithm that is based on an independent adjustment of the sensor response channels. Independent adjustment of multiplicative coefficients corresponds to the application of a diagonal-matrix transform (DMT) to the sensor response vector and is a common feature of many theories of color constancy, Land’s retinex and von Kries adaptation in particular. We set forth three techniques for spectral sharpening. Sensor-based sharpening focuses on the production of new sensors as linear combinations of the given ones such that each new sensor has its spectral sensitivity concentrated as much as possible within a narrow band of wavelengths. Data-based sharpening, on the other hand, extracts new sensors by optimizing the ability of a DMT to account for a given illumination change by examining the sensor response vectors obtained from a set of surfaces under two different illuminants. Finally in perfect sharpening we demonstrate that, if illumination and surface reflectance are described by two- and three-parameter finite-dimensional models, there exists a unique optimal sharpening transform. All three sharpening methods yield similar results. When sharpened cone sensitivities are used as sensors, a DMT models illumination change extremely well. We present simulation results suggesting that in general nondiagonal transforms can do only marginally better. Our sharpening results correlate well with the psychophysical evidence of spectral sharpening in the human visual system.

© 1994 Optical Society of America

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References

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  1. J. Beck, Surface Color Perception (Cornell U. Press, Ithaca, N.Y., 1972).
  2. G. West, M. H. Brill, “Necessary and sufficient conditions for von Kries chromatic adaption to give colour constancy,” J. Math. Biol. 15, 249–258 (1982).
    [CrossRef]
  3. E. H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).
    [CrossRef] [PubMed]
  4. B. K. P. Horn, “Determining lightness from an image,” Comput. Vision Graphics Image Process. 3, 277–299 (1974).
    [CrossRef]
  5. A. Blake, “Boundary conditions for lightness computation in Mondrian world,” Comput. Vision Graphics Image Process. 32, 314–327 (1985).
    [CrossRef]
  6. D. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vision 5, 5–36 (1990).
    [CrossRef]
  7. M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1672 (1986).
    [CrossRef]
  8. B. V. Funt, M. S. Drew, “Color constancy computation in near-Mondrian scenes using a finite dimensional linear model,” in Computer Vision and Pattern Recognition Proceedings (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 544–549.
  9. M. H. Brill, “Computer-simulated object-color recognizer,” Tech. Rep. 122 (MIT Research Laboratory of Electronics, Cambridge, Mass., 1980).
  10. G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).
  11. E. H. Land, “The retinex theory of color vision,” Sci. Am. 237, 108–129 (1977).
    [CrossRef] [PubMed]
  12. D. H. Foster, “Changes in field spectral sensitivities of red-, green- and blue-sensitive colour mechanisms obtained on small background fields,” Vision Res. 21, 1433–1455 (1981).
    [CrossRef] [PubMed]
  13. H. G. Sperling, R. S. Harwerth, “Red–green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
    [CrossRef] [PubMed]
  14. M. Kalloniatis, R. S. Harwerth, “Spectral sensitivity and adaptation characteristics of cone mechanisms under white-light adaptation,” J. Opt. Soc. Am. A 7, 1912–1928 (1990).
    [CrossRef] [PubMed]
  15. D. C. Hood, M. A. Finkelstein, “A case for the revision of textbook models of colour vision,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 385–398.
  16. A. B. Poirson, B. A. Wandell, “Task-dependent color discrimination,” J. Opt. Soc. Am. A 7, 776–782 (1990).
    [CrossRef] [PubMed]
  17. J. H. Wilkinson, Algebraic Eigenvalue Problem, Monographs on Numerical Analysis (Oxford U. Press, Oxford, 1965).
  18. J. K. Bowmaker, H. J. A. Dartnell, “Visual pigments of rods and cones in the human retina,” J. Physiol. 298, 501–511 (1980).
  19. D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031–1040 (1964).
    [CrossRef]
  20. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
  21. L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A 3, 1673–1683 (1986).
    [CrossRef] [PubMed]
  22. C. L. Novak, S. A. Shafer, “Supervised color constancy using a color chart,” Tech. Rep. CMU-CS-90-140 (Carnegie–Mellon University School of Computer Science, Pittsburgh, Pa., 1990).
  23. G. D. Finlayson, M. S. Drew, B. V. Funt, “Enhancing von Kries adaptation via sensor transformations,” in Human Vision, Visual Processing, and Digital Display TV, J. P. Allebach, Bernice E. Roqowitz, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1913, 473–484 (1993).
  24. D. H. Marimont, B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913 (1992).
    [CrossRef] [PubMed]
  25. M. D’Zmura, “Color constancy: surface color from changing illumination,” J. Opt. Soc. Am. A 9, 490–493 (1992).
    [CrossRef]
  26. D. H. Foster, “Colour vision,” Contemp. Phys. 25, 477–497 (1984).
    [CrossRef]
  27. D. H. Foster, R. S. Snelgar, “Test and field spectral sensitivities of colour mechanisms obtained on small white backgrounds: action of unitary opponent-colour processes?” Vision Res. 23, 787–797 (1983).
    [CrossRef] [PubMed]
  28. D. H. Foster, R. S. Snelgar, “Initial analysis of opponent-colour interactions revealed in sharpened field sensitivities,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 303–312.
  29. W. Jaeger, H. Krastel, S. Braun, “An incrementthreshold evaluation of mechanisms underlying colour constancy,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 545–552.
  30. Kodak Filters: for Scientific and Technical Uses, 2nd ed. (Eastman Kodak, Rochester, N.Y., 1981).

1992 (2)

1990 (3)

1986 (2)

1985 (1)

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

1984 (1)

D. H. Foster, “Colour vision,” Contemp. Phys. 25, 477–497 (1984).
[CrossRef]

1983 (1)

D. H. Foster, R. S. Snelgar, “Test and field spectral sensitivities of colour mechanisms obtained on small white backgrounds: action of unitary opponent-colour processes?” Vision Res. 23, 787–797 (1983).
[CrossRef] [PubMed]

1982 (1)

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

1981 (1)

D. H. Foster, “Changes in field spectral sensitivities of red-, green- and blue-sensitive colour mechanisms obtained on small background fields,” Vision Res. 21, 1433–1455 (1981).
[CrossRef] [PubMed]

1980 (1)

J. K. Bowmaker, H. J. A. Dartnell, “Visual pigments of rods and cones in the human retina,” J. Physiol. 298, 501–511 (1980).

1977 (1)

E. H. Land, “The retinex theory of color vision,” Sci. Am. 237, 108–129 (1977).
[CrossRef] [PubMed]

1974 (1)

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

1971 (2)

H. G. Sperling, R. S. Harwerth, “Red–green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
[CrossRef] [PubMed]

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

1964 (2)

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

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

Beck, J.

J. Beck, Surface Color Perception (Cornell U. Press, Ithaca, N.Y., 1972).

Blake, A.

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

Bowmaker, J. K.

J. K. Bowmaker, H. J. A. Dartnell, “Visual pigments of rods and cones in the human retina,” J. Physiol. 298, 501–511 (1980).

Braun, S.

W. Jaeger, H. Krastel, S. Braun, “An incrementthreshold evaluation of mechanisms underlying colour constancy,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 545–552.

Brill, M. H.

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

M. H. Brill, “Computer-simulated object-color recognizer,” Tech. Rep. 122 (MIT Research Laboratory of Electronics, Cambridge, Mass., 1980).

Cohen, J.

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

D’Zmura, M.

Dartnell, H. J. A.

J. K. Bowmaker, H. J. A. Dartnell, “Visual pigments of rods and cones in the human retina,” J. Physiol. 298, 501–511 (1980).

Drew, M. S.

G. D. Finlayson, M. S. Drew, B. V. Funt, “Enhancing von Kries adaptation via sensor transformations,” in Human Vision, Visual Processing, and Digital Display TV, J. P. Allebach, Bernice E. Roqowitz, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1913, 473–484 (1993).

B. V. Funt, M. S. Drew, “Color constancy computation in near-Mondrian scenes using a finite dimensional linear model,” in Computer Vision and Pattern Recognition Proceedings (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 544–549.

Finkelstein, M. A.

D. C. Hood, M. A. Finkelstein, “A case for the revision of textbook models of colour vision,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 385–398.

Finlayson, G. D.

G. D. Finlayson, M. S. Drew, B. V. Funt, “Enhancing von Kries adaptation via sensor transformations,” in Human Vision, Visual Processing, and Digital Display TV, J. P. Allebach, Bernice E. Roqowitz, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1913, 473–484 (1993).

Forsyth, D.

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

Foster, D. H.

D. H. Foster, “Colour vision,” Contemp. Phys. 25, 477–497 (1984).
[CrossRef]

D. H. Foster, R. S. Snelgar, “Test and field spectral sensitivities of colour mechanisms obtained on small white backgrounds: action of unitary opponent-colour processes?” Vision Res. 23, 787–797 (1983).
[CrossRef] [PubMed]

D. H. Foster, “Changes in field spectral sensitivities of red-, green- and blue-sensitive colour mechanisms obtained on small background fields,” Vision Res. 21, 1433–1455 (1981).
[CrossRef] [PubMed]

D. H. Foster, R. S. Snelgar, “Initial analysis of opponent-colour interactions revealed in sharpened field sensitivities,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 303–312.

Funt, B. V.

B. V. Funt, M. S. Drew, “Color constancy computation in near-Mondrian scenes using a finite dimensional linear model,” in Computer Vision and Pattern Recognition Proceedings (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 544–549.

G. D. Finlayson, M. S. Drew, B. V. Funt, “Enhancing von Kries adaptation via sensor transformations,” in Human Vision, Visual Processing, and Digital Display TV, J. P. Allebach, Bernice E. Roqowitz, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1913, 473–484 (1993).

Harwerth, R. S.

M. Kalloniatis, R. S. Harwerth, “Spectral sensitivity and adaptation characteristics of cone mechanisms under white-light adaptation,” J. Opt. Soc. Am. A 7, 1912–1928 (1990).
[CrossRef] [PubMed]

H. G. Sperling, R. S. Harwerth, “Red–green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
[CrossRef] [PubMed]

Hood, D. C.

D. C. Hood, M. A. Finkelstein, “A case for the revision of textbook models of colour vision,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 385–398.

Horn, B. K. P.

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

Jaeger, W.

W. Jaeger, H. Krastel, S. Braun, “An incrementthreshold evaluation of mechanisms underlying colour constancy,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 545–552.

Judd, D. B.

Kalloniatis, M.

Krastel, H.

W. Jaeger, H. Krastel, S. Braun, “An incrementthreshold evaluation of mechanisms underlying colour constancy,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 545–552.

Land, E. H.

Lennie, P.

MacAdam, D. L.

Maloney, L. T.

Marimont, D. H.

McCann, J. J.

Novak, C. L.

C. L. Novak, S. A. Shafer, “Supervised color constancy using a color chart,” Tech. Rep. CMU-CS-90-140 (Carnegie–Mellon University School of Computer Science, Pittsburgh, Pa., 1990).

Poirson, A. B.

Shafer, S. A.

C. L. Novak, S. A. Shafer, “Supervised color constancy using a color chart,” Tech. Rep. CMU-CS-90-140 (Carnegie–Mellon University School of Computer Science, Pittsburgh, Pa., 1990).

Snelgar, R. S.

D. H. Foster, R. S. Snelgar, “Test and field spectral sensitivities of colour mechanisms obtained on small white backgrounds: action of unitary opponent-colour processes?” Vision Res. 23, 787–797 (1983).
[CrossRef] [PubMed]

D. H. Foster, R. S. Snelgar, “Initial analysis of opponent-colour interactions revealed in sharpened field sensitivities,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 303–312.

Sperling, H. G.

H. G. Sperling, R. S. Harwerth, “Red–green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
[CrossRef] [PubMed]

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).

Wandell, B. A.

West, G.

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

Wilkinson, J. H.

J. H. Wilkinson, Algebraic Eigenvalue Problem, Monographs on Numerical Analysis (Oxford U. Press, Oxford, 1965).

Wyszecki, G.

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

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).

Comput. Vision Graphics Image Process. (2)

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

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

Contemp. Phys. (1)

D. H. Foster, “Colour vision,” Contemp. Phys. 25, 477–497 (1984).
[CrossRef]

Int. J. Comput. Vision (1)

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

J. Math. Biol. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

J. Physiol. (1)

J. K. Bowmaker, H. J. A. Dartnell, “Visual pigments of rods and cones in the human retina,” J. Physiol. 298, 501–511 (1980).

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, 108–129 (1977).
[CrossRef] [PubMed]

Science (1)

H. G. Sperling, R. S. Harwerth, “Red–green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
[CrossRef] [PubMed]

Vision Res. (2)

D. H. Foster, “Changes in field spectral sensitivities of red-, green- and blue-sensitive colour mechanisms obtained on small background fields,” Vision Res. 21, 1433–1455 (1981).
[CrossRef] [PubMed]

D. H. Foster, R. S. Snelgar, “Test and field spectral sensitivities of colour mechanisms obtained on small white backgrounds: action of unitary opponent-colour processes?” Vision Res. 23, 787–797 (1983).
[CrossRef] [PubMed]

Other (11)

D. H. Foster, R. S. Snelgar, “Initial analysis of opponent-colour interactions revealed in sharpened field sensitivities,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 303–312.

W. Jaeger, H. Krastel, S. Braun, “An incrementthreshold evaluation of mechanisms underlying colour constancy,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 545–552.

Kodak Filters: for Scientific and Technical Uses, 2nd ed. (Eastman Kodak, Rochester, N.Y., 1981).

C. L. Novak, S. A. Shafer, “Supervised color constancy using a color chart,” Tech. Rep. CMU-CS-90-140 (Carnegie–Mellon University School of Computer Science, Pittsburgh, Pa., 1990).

G. D. Finlayson, M. S. Drew, B. V. Funt, “Enhancing von Kries adaptation via sensor transformations,” in Human Vision, Visual Processing, and Digital Display TV, J. P. Allebach, Bernice E. Roqowitz, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1913, 473–484 (1993).

D. C. Hood, M. A. Finkelstein, “A case for the revision of textbook models of colour vision,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 385–398.

J. H. Wilkinson, Algebraic Eigenvalue Problem, Monographs on Numerical Analysis (Oxford U. Press, Oxford, 1965).

J. Beck, Surface Color Perception (Cornell U. Press, Ithaca, N.Y., 1972).

B. V. Funt, M. S. Drew, “Color constancy computation in near-Mondrian scenes using a finite dimensional linear model,” in Computer Vision and Pattern Recognition Proceedings (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 544–549.

M. H. Brill, “Computer-simulated object-color recognizer,” Tech. Rep. 122 (MIT Research Laboratory of Electronics, Cambridge, Mass., 1980).

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).

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Figures (7)

Fig. 1
Fig. 1

VW fundamentals (solid curves) are contrasted with the sharpened sensitivities derived through sensor-based (dotted curves) and data-based (dashed curves) sharpening. Top, long-wavelength mechanism; middle, medium-wavelength mechanism; bottom, short-wavelength mechanism.

Fig. 2
Fig. 2

Data-based sharpening generates different sharpened sensors for each illuminant. The range of sharpened curves over all the test illuminants (CIE A, D48, D65, D75, and D100) mapped to D55 is shown for the VW cone mechanisms. Top, long-wavelength mechanism; middle, medium-wavelength mechanism; bottom, short-wavelength mechanism.

Fig. 3
Fig. 3

Cumulative NFD histogram obtained with each test illuminant (CIE A, D48, D65, D75, and D100) for diagonal fitting of VW observations (solid curves) and diagonal fitting of sensor-based sharpened VW observations (dotted curves). The sixth cumulative NFD histogram shows the average fitting performance.

Fig. 4
Fig. 4

Cumulative NFD histogram obtained with each test illuminant (CIE A, D48, D65, D75, and D100) for diagonal fitting of VW observations (solid curves) and transformed diagonal fitting of sensor-based sharpened VW observations (dotted curves). The sixth cumulative NFD histogram shows the average fitting performance.

Fig. 5
Fig. 5

Cumulative NFD histogram obtained with each test illuminant (CIE A, D48, D65, D75, and D100) for optimal fitting of VW observations (solid curves) and transformed diagonal fitting of sensor-based sharpened VW observations (dotted curves). The sixth cumulative NFD histogram shows the average fitting performance.

Fig. 6
Fig. 6

Cumulative NFD histogram obtained with each test illuminant (CIE A, D48, D65, D75, and D100) for optimal fitting of VW observations (solid curves) and transformed white-patch normalization of VW observations (dotted curves). The sixth cumulative NFD histogram shows average color constancy performance.

Fig. 7
Fig. 7

Solid curves show results for the Kodak Wratten filters #66, #52, and #38; dotted curves show the results of sensor-based sharpening; dashed curves show the mean of the data-based sharpened sensors obtained for the five test illuminants (CIE A, D48, D65, D75, and D100). Top, long-wavelength mechanism; middle, medium-wavelength mechanism; bottom, short-wavelength mechanism.

Tables (1)

Tables Icon

Table 1 Percentage of Total Squared Norm in the Sharpening Intervals

Equations (33)

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

p i c D e p i e .
P = ω E ( λ ) S ( λ ) R ( λ ) d λ ,
ω E i ( λ ) S ( λ ) R ( λ ) d λ ω E i ( λ ) S r ( λ ) R ( λ ) d λ = ω E j ( λ ) S ( λ ) R ( λ ) d λ ω E j ( λ ) S r ( λ ) R ( λ ) d λ .
T p i c D e T p i e ,
I = ϕ [ R ( λ ) t c ] 2 d λ + μ { ω [ R ( λ ) t c ] 2 d λ 1 } ,
1 2 I c = ϕ R ( λ ) R ( λ ) t c d λ + μ [ ω R ( λ ) R ( λ ) t c d λ ] = 0 .
Λ ( ϕ ) c = μ Λ ( ω ) c .
[ Λ ( ω ) ] 1 Λ ( ϕ ) c = μ c .
W c D e W e .
T e W c D e T e W e .
D e = T e W c [ T e W e ] + ,
[ T e ] 1 D e T e = W c [ W e ] + .
W c G e W e .
G e = W c [ W e ] + .
NFD = 100 * p c q e p c .
D i i e = W i c [ W i e ] + = W i c [ W i e ] t / W i e [ W i e ] t ,
T 1 T W c T 1 D e T W e
p e = Λ e σ ,
Q e = Λ e A ,
[ Q e ] 1 p e = A 1 [ Λ e ] 1 Λ e σ = A 1 σ .
p c = D e p e ,
D i i e = p i c p i e .
p c = Λ c σ ,
Λ 2 = Λ c ,
= Λ 2 [ Λ c ] 1 .
p e = [ α I + β ] Λ c σ = [ α I + β ] p c ,
= T 1 D T ,
I = T 1 I T ,
T p e [ α I + β D ] T p c .
T p c [ α I + β D ] 1 T p e .
R p = 2.44 R 1.93 G + 0.110 B , R s = 2.46 R 1.97 G + 0.075 B , R d = 2.46 R 1.98 G + 0.100 B ,
G p = 1.55 G 0.63 R 0.16 B , G s = 1.58 G 0.66 R 0.12 B , G d = 1.52 G 0.58 R 0.14 B ,
B p = 1.0 B 0.13 G + 0.08 R , B s = 1.0 B 0.14 G + 0.09 R , B d = 1.0 B 0.13 G + 0.07 R ,

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