Abstract

A reflectance function occurring in practice is usually smooth. The square of its derivative, integrated over the entire visual range, is defined as a measure of its smoothness. The problem (to determine for given tristimulus values under a given illuminant the smoothest reflectance function in this sense) has 16 types of solution, each valid in a certain domain of the object color solid. The four simplest solutions, which are also those of most practical importance, are studied in this paper, together with their domains of validity. A simple algorithm for their generation is described.

© 1990 Optical Society of America

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References

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  1. G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1982).
  2. Commission International de l’Eclairage, Method of Measuring and Specifying Colour Rendering Properties of Light Sources (Paris, 1974).
  3. J. von Kries, Handbuch der Physiologie des Menschen (Brunswick, Vieweg, 1905).
  4. Y. Nayatani, K. Takahama, H. Sobagaki, “Formulation of a nonlinear model on chromatic adaptation,” Color Res. Appl. 6, 161–171 (1981).
    [Crossref]
  5. Scandanavian Color Institute, Natural Colour System (Stockholm, 1982). Reflectance functions have been measured by A. Stierum, to whom the author is indebted.
  6. J. T. C. van Kemenade, R. A. J. Keyser, B. F. Bolt, “New possibilities for HPS lamps in indoor lighting,” J. Illum. Eng. Soc. 16, 150–161 (1987).
  7. J. T. C. van Kemenade, P. J. M. v. d. Burgt, “Light sources and colour rendering, additional information to the Raindex,” presented at the National Lighting Conference, Cambridge, UK, 1988.
  8. Munsell Color Company, Munsell Book of Color (Hoffman, Baltimore, Md., 1929).
  9. Uniform Color Scales, Optical Society of America, Washington, D.C., 1977.
  10. DIN, Beuth-Vertrieb GmbH, Berlin, 1953.
  11. P. Moon, D. E. Spencer, “Polynomial representations of reflectance curves,” J. Opt. Soc. Am. 35, 597–600 (1945).
    [Crossref]
  12. W. L. Brewer, F. R. Holley, “Synthesis of spectral distribution curves,” J. Opt. Soc. Am. 38, 858–874 (1948).
    [Crossref] [PubMed]
  13. W. S. Stiles, G. Wyszecki, N. Ohta, “Counting metameric object-color stimuli using frequency-limited spectral distribution functions,” J. Opt. Soc. Am. 67, 779–784 (1977).
    [Crossref]
  14. K. Takahama, Y. Nayatani, “New method for generating metameric stimuli of object colors,” J. Opt. Soc. Am. 62, 1516–1520 (1972).
    [Crossref]
  15. K. Takahama, Y. Nayatani, “Derivation of spectral-reflectance functions of object colors with good color constancy,” Acta Chromat. 2, 187–191 (1973).
  16. N. Ohta, “A simplified method for formulating pseudo-object colors,” Color Res. Appl. 7, 78–81 (1982).
    [Crossref]
  17. R. S. Berns, Color Constant Extensions of the ‘Munsell Book of Color’ (Rensselaer Polytechnic Institute, Troy, New York, 1983).
  18. J. B. Cohen, “Color and color mixture, scalar and vector fundamentals,” Color Res. Appl. 13, 5–39 (1987).
    [Crossref]
  19. E. Schrödinger, “Theorie der pigmente von grosster Leuchtkraft,” Ann. Phys. (Paris) 62, 603–622 (1920).
  20. W. Ostwald, “Neue Forschungen zur Farbenlehre,” Phys. Z. 17, 322–332 (1916).
  21. D. MacAdam, Color Measurement: Theme and Variations (Springer-Verlag, New York, 1981).
  22. G. West, M. H. Brill, “Conditions under which Schrödinger object colors are optimal,” J. Opt. Soc. Am. 73, 1223–1225 (1983).
    [Crossref]
  23. C. van Trigt, “Smoothest reflectance functions. II. Complete results,” J. Opt. Soc. Am. A. (to be published, November1990).
  24. E. C. Titchmarsh, The Theory of Functions (Oxford U. Press, London, 1960), p. 393.
  25. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
  26. J. P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [Crossref]
  27. G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London Ser. B 220, 89–113 (1983).
    [Crossref]
  28. G. Buchsbaum, “Color signal coding: color vision and color television,” Color Res. Appl. 12, 266–269 (1987).
    [Crossref]
  29. L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small number of parameters,” J. Opt. Soc. Am. A 3, 1673–1683 (1986).
    [Crossref] [PubMed]
  30. L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
    [Crossref] [PubMed]
  31. M. H. Brill, G. West, “Chromatic adaptation and color constancy: a possible dichotomy,” Color Res. Appl. 11, 196–204 (1986).
    [Crossref]
  32. R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985).
    [Crossref]
  33. C. Fox, An Introduction to the Calculus of Variations (Oxford U. Press, London, 1954).
  34. G. Polya, G. Szego, Problems and Theorems in Analysis (Springer-Verlag, Berlin, 1972).
  35. A. C. Aitken, Determinants and Matrices (Oliver and Boyd, Edinburgh, 1959).
  36. P. J. Bouma, Physical Aspects of Colour (Macmillan, London, 1971).
  37. D. B. Judd, G. Wyszecki, Color in Business, Science and Industry (Wiley, New York, 1975).

1989 (1)

1987 (3)

J. T. C. van Kemenade, R. A. J. Keyser, B. F. Bolt, “New possibilities for HPS lamps in indoor lighting,” J. Illum. Eng. Soc. 16, 150–161 (1987).

J. B. Cohen, “Color and color mixture, scalar and vector fundamentals,” Color Res. Appl. 13, 5–39 (1987).
[Crossref]

G. Buchsbaum, “Color signal coding: color vision and color television,” Color Res. Appl. 12, 266–269 (1987).
[Crossref]

1986 (3)

1985 (1)

R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985).
[Crossref]

1983 (2)

G. West, M. H. Brill, “Conditions under which Schrödinger object colors are optimal,” J. Opt. Soc. Am. 73, 1223–1225 (1983).
[Crossref]

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London Ser. B 220, 89–113 (1983).
[Crossref]

1982 (1)

N. Ohta, “A simplified method for formulating pseudo-object colors,” Color Res. Appl. 7, 78–81 (1982).
[Crossref]

1981 (1)

Y. Nayatani, K. Takahama, H. Sobagaki, “Formulation of a nonlinear model on chromatic adaptation,” Color Res. Appl. 6, 161–171 (1981).
[Crossref]

1977 (1)

1973 (1)

K. Takahama, Y. Nayatani, “Derivation of spectral-reflectance functions of object colors with good color constancy,” Acta Chromat. 2, 187–191 (1973).

1972 (1)

1964 (1)

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

1948 (1)

1945 (1)

1920 (1)

E. Schrödinger, “Theorie der pigmente von grosster Leuchtkraft,” Ann. Phys. (Paris) 62, 603–622 (1920).

1916 (1)

W. Ostwald, “Neue Forschungen zur Farbenlehre,” Phys. Z. 17, 322–332 (1916).

Aitken, A. C.

A. C. Aitken, Determinants and Matrices (Oliver and Boyd, Edinburgh, 1959).

Berns, R. S.

R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985).
[Crossref]

R. S. Berns, Color Constant Extensions of the ‘Munsell Book of Color’ (Rensselaer Polytechnic Institute, Troy, New York, 1983).

Billmeyer, F. W.

R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985).
[Crossref]

Bolt, B. F.

J. T. C. van Kemenade, R. A. J. Keyser, B. F. Bolt, “New possibilities for HPS lamps in indoor lighting,” J. Illum. Eng. Soc. 16, 150–161 (1987).

Bouma, P. J.

P. J. Bouma, Physical Aspects of Colour (Macmillan, London, 1971).

Brewer, W. L.

Brill, M. H.

M. H. Brill, G. West, “Chromatic adaptation and color constancy: a possible dichotomy,” Color Res. Appl. 11, 196–204 (1986).
[Crossref]

G. West, M. H. Brill, “Conditions under which Schrödinger object colors are optimal,” J. Opt. Soc. Am. 73, 1223–1225 (1983).
[Crossref]

Buchsbaum, G.

G. Buchsbaum, “Color signal coding: color vision and color television,” Color Res. Appl. 12, 266–269 (1987).
[Crossref]

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London Ser. B 220, 89–113 (1983).
[Crossref]

Burgt, P. J. M. v. d.

J. T. C. van Kemenade, P. J. M. v. d. Burgt, “Light sources and colour rendering, additional information to the Raindex,” presented at the National Lighting Conference, Cambridge, UK, 1988.

Cohen, J.

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

Cohen, J. B.

J. B. Cohen, “Color and color mixture, scalar and vector fundamentals,” Color Res. Appl. 13, 5–39 (1987).
[Crossref]

Fox, C.

C. Fox, An Introduction to the Calculus of Variations (Oxford U. Press, London, 1954).

Gottschalk, A.

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London Ser. B 220, 89–113 (1983).
[Crossref]

Hallikainen, J.

Holley, F. R.

Jaaskelainen, T.

Judd, D. B.

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

Keyser, R. A. J.

J. T. C. van Kemenade, R. A. J. Keyser, B. F. Bolt, “New possibilities for HPS lamps in indoor lighting,” J. Illum. Eng. Soc. 16, 150–161 (1987).

MacAdam, D.

D. MacAdam, Color Measurement: Theme and Variations (Springer-Verlag, New York, 1981).

Maloney, L. T.

Moon, P.

Nayatani, Y.

Y. Nayatani, K. Takahama, H. Sobagaki, “Formulation of a nonlinear model on chromatic adaptation,” Color Res. Appl. 6, 161–171 (1981).
[Crossref]

K. Takahama, Y. Nayatani, “Derivation of spectral-reflectance functions of object colors with good color constancy,” Acta Chromat. 2, 187–191 (1973).

K. Takahama, Y. Nayatani, “New method for generating metameric stimuli of object colors,” J. Opt. Soc. Am. 62, 1516–1520 (1972).
[Crossref]

Ohta, N.

Ostwald, W.

W. Ostwald, “Neue Forschungen zur Farbenlehre,” Phys. Z. 17, 322–332 (1916).

Parkkinen, J. P. S.

Polya, G.

G. Polya, G. Szego, Problems and Theorems in Analysis (Springer-Verlag, Berlin, 1972).

Sacher, R. S.

R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985).
[Crossref]

Schrödinger, E.

E. Schrödinger, “Theorie der pigmente von grosster Leuchtkraft,” Ann. Phys. (Paris) 62, 603–622 (1920).

Sobagaki, H.

Y. Nayatani, K. Takahama, H. Sobagaki, “Formulation of a nonlinear model on chromatic adaptation,” Color Res. Appl. 6, 161–171 (1981).
[Crossref]

Spencer, D. E.

Stiles, W. S.

Szego, G.

G. Polya, G. Szego, Problems and Theorems in Analysis (Springer-Verlag, Berlin, 1972).

Takahama, K.

Y. Nayatani, K. Takahama, H. Sobagaki, “Formulation of a nonlinear model on chromatic adaptation,” Color Res. Appl. 6, 161–171 (1981).
[Crossref]

K. Takahama, Y. Nayatani, “Derivation of spectral-reflectance functions of object colors with good color constancy,” Acta Chromat. 2, 187–191 (1973).

K. Takahama, Y. Nayatani, “New method for generating metameric stimuli of object colors,” J. Opt. Soc. Am. 62, 1516–1520 (1972).
[Crossref]

Titchmarsh, E. C.

E. C. Titchmarsh, The Theory of Functions (Oxford U. Press, London, 1960), p. 393.

van Kemenade, J. T. C.

J. T. C. van Kemenade, R. A. J. Keyser, B. F. Bolt, “New possibilities for HPS lamps in indoor lighting,” J. Illum. Eng. Soc. 16, 150–161 (1987).

J. T. C. van Kemenade, P. J. M. v. d. Burgt, “Light sources and colour rendering, additional information to the Raindex,” presented at the National Lighting Conference, Cambridge, UK, 1988.

van Trigt, C.

C. van Trigt, “Smoothest reflectance functions. II. Complete results,” J. Opt. Soc. Am. A. (to be published, November1990).

von Kries, J.

J. von Kries, Handbuch der Physiologie des Menschen (Brunswick, Vieweg, 1905).

Wandell, B. A.

West, G.

M. H. Brill, G. West, “Chromatic adaptation and color constancy: a possible dichotomy,” Color Res. Appl. 11, 196–204 (1986).
[Crossref]

G. West, M. H. Brill, “Conditions under which Schrödinger object colors are optimal,” J. Opt. Soc. Am. 73, 1223–1225 (1983).
[Crossref]

Wyszecki, G.

W. S. Stiles, G. Wyszecki, N. Ohta, “Counting metameric object-color stimuli using frequency-limited spectral distribution functions,” J. Opt. Soc. Am. 67, 779–784 (1977).
[Crossref]

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1982).

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

Acta Chromat. (1)

K. Takahama, Y. Nayatani, “Derivation of spectral-reflectance functions of object colors with good color constancy,” Acta Chromat. 2, 187–191 (1973).

Ann. Phys. (Paris) (1)

E. Schrödinger, “Theorie der pigmente von grosster Leuchtkraft,” Ann. Phys. (Paris) 62, 603–622 (1920).

Color Res. Appl. (6)

N. Ohta, “A simplified method for formulating pseudo-object colors,” Color Res. Appl. 7, 78–81 (1982).
[Crossref]

Y. Nayatani, K. Takahama, H. Sobagaki, “Formulation of a nonlinear model on chromatic adaptation,” Color Res. Appl. 6, 161–171 (1981).
[Crossref]

J. B. Cohen, “Color and color mixture, scalar and vector fundamentals,” Color Res. Appl. 13, 5–39 (1987).
[Crossref]

G. Buchsbaum, “Color signal coding: color vision and color television,” Color Res. Appl. 12, 266–269 (1987).
[Crossref]

M. H. Brill, G. West, “Chromatic adaptation and color constancy: a possible dichotomy,” Color Res. Appl. 11, 196–204 (1986).
[Crossref]

R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985).
[Crossref]

J. Illum. Eng. Soc. (1)

J. T. C. van Kemenade, R. A. J. Keyser, B. F. Bolt, “New possibilities for HPS lamps in indoor lighting,” J. Illum. Eng. Soc. 16, 150–161 (1987).

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

Phys. Z. (1)

W. Ostwald, “Neue Forschungen zur Farbenlehre,” Phys. Z. 17, 322–332 (1916).

Proc. R. Soc. London Ser. B (1)

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London Ser. B 220, 89–113 (1983).
[Crossref]

Psychon. Sci. (1)

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

Other (17)

C. van Trigt, “Smoothest reflectance functions. II. Complete results,” J. Opt. Soc. Am. A. (to be published, November1990).

E. C. Titchmarsh, The Theory of Functions (Oxford U. Press, London, 1960), p. 393.

C. Fox, An Introduction to the Calculus of Variations (Oxford U. Press, London, 1954).

G. Polya, G. Szego, Problems and Theorems in Analysis (Springer-Verlag, Berlin, 1972).

A. C. Aitken, Determinants and Matrices (Oliver and Boyd, Edinburgh, 1959).

P. J. Bouma, Physical Aspects of Colour (Macmillan, London, 1971).

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

D. MacAdam, Color Measurement: Theme and Variations (Springer-Verlag, New York, 1981).

R. S. Berns, Color Constant Extensions of the ‘Munsell Book of Color’ (Rensselaer Polytechnic Institute, Troy, New York, 1983).

J. T. C. van Kemenade, P. J. M. v. d. Burgt, “Light sources and colour rendering, additional information to the Raindex,” presented at the National Lighting Conference, Cambridge, UK, 1988.

Munsell Color Company, Munsell Book of Color (Hoffman, Baltimore, Md., 1929).

Uniform Color Scales, Optical Society of America, Washington, D.C., 1977.

DIN, Beuth-Vertrieb GmbH, Berlin, 1953.

Scandanavian Color Institute, Natural Colour System (Stockholm, 1982). Reflectance functions have been measured by A. Stierum, to whom the author is indebted.

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1982).

Commission International de l’Eclairage, Method of Measuring and Specifying Colour Rendering Properties of Light Sources (Paris, 1974).

J. von Kries, Handbuch der Physiologie des Menschen (Brunswick, Vieweg, 1905).

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Figures (14)

Fig. 1
Fig. 1

Color shifts in a*b* diagram undergone by a number of test colors in the transition D65 → light source A. The dashed curves denote three subsets of test colors. Figure by courtesy of J. J. Opstelten.

Fig. 2
Fig. 2

(a), (b), (c), (d), CIE test colors 2, 4, 6, and 8, respectively (dashed curves), compared with the reflectance functions defined by expressions (1)(3) (solid curves).

Fig. 3
Fig. 3

Left and right branches of the boundary locus for light source A. Also shown are the constructions of λb and λe and λ1 used in defining ρ(λ; λ1) in Eq. (23).

Fig. 4
Fig. 4

As Fig. (3) but for light source D65.

Fig. 5
Fig. 5

Two reflectance functions ρ(λ; λ1) of Class 2b: 1, λ1 = 500 nm; x = 0.5455; y = 0.3829. 2, λ1 = 564 nm; x = 0.1657, y = 0.0957. The light source was D65.

Fig. 6
Fig. 6

Two reflectance functions ρ(λ; λ1) of Class 1b: 1, λ1 = 500 nm; x = 0.4500; y = 0.4707. 2, λ1 = 564 nm; x = 0.5242; y = 0.4478. The light source was D65.

Fig. 7
Fig. 7

Regions containing the chromaticity points for which the solution of the unconstrained problem belongs to the class indicated. The solution of the unconstrained problem is positive in the entire region. The end points of Classes 2a and 2b have been indicated. Equations of straight lines −26.59x1) + 20.60y1) = 1; 0.8586x1) + 1.229y1) = 1, intersecting at xa) = 0.3846, ya) = 0.5451. The light source is A.

Fig. 8
Fig. 8

As Fig. 7 but for light source D65. Equations of straight lines 9.574x1) − 3.178y1) = 1; 0.5065x1) + 1.640y1) = 1, intersecting at xa) = 0.2783, ya) = 0.5237. Also indicated are the chromaticity points of the CIE test colors that have positive solutions (i.e., all test colors except 9 and 12).

Fig. 9
Fig. 9

Luminance Y1)/Y0 (solid curve) of reflectance function ρ(λ; λ1) (this at once is the maximum admissable luminance such that a solution of the unconstrained problem for a point on the boundary of the region of Fig. 7 nowhere exceeds unity) and maximum attainable luminance YM1)/Y0 (dashed curve) for Classes 1a, 1b, 2a, and 2b. The light source is A.

Fig. 10
Fig. 10

As Fig. 9 but for light source D65.

Fig. 11
Fig. 11

Surface of maximum admissable luminance Y/Y0 such that the solution of the unconstrained problem nowhere exceeds unity. The light source is A.

Fig. 12
Fig. 12

As Fig. 15 bit for light source D65.

Fig. 13
Fig. 13

Color gamut occupied by the chromaticity points under light source D65 of 56 test colors taken from the Natural Colour System,5 with luminance 0.77 < Y/Y0 < 0.90 compared with the region Y/Y0 ≥ 0.8. Shown are 24 points; the points omitted are distributed between the points that are shown.

Fig. 14
Fig. 14

Color gamut occupied by the chromaticity points under light source D65 of 256 test colors taken from the Natural Colour System,5 with 0.23 < Y/Y0 < 0.3 compared with the region Y/Y0 ≥ 0.35. Shown are 75 points, 17 of which lie outside the region. The other 181 points are inside the region, predominantly in the space left empty.

Tables (1)

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Table 1 Reflectance Functions

Equations (53)

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ρ ( λ ) S ( λ ) x ¯ ( λ ) d λ = X , ρ ( λ ) S ( λ ) y ¯ ( λ ) d λ = Y , ρ ( λ ) S ( λ ) z ¯ ( λ ) d λ = Z
0 ρ ( λ ) 1 λ ( λ b , λ e ) .
ρ ( λ ) S ( λ ) x ¯ ( λ ) d λ = X , ρ ( λ ) S ( λ ) y ¯ ( λ ) d λ = Y , ρ ( λ ) S ( λ ) z ¯ ( λ ) d λ = Z .
( d ρ d λ ) 2 d λ = minimal .
d t ( λ ) d λ = 1 w ( λ )
d 2 ρ d λ 2 + [ μ 1 X 0 x ¯ ( λ ) + μ 2 Y 0 y ¯ ( λ ) + μ 3 Z 0 z ¯ ( λ ) ] S ( λ ) = 0.
d ρ d λ = 0
f 1 ( λ ) = 1 X 0 λ b λ S ( λ ) x ¯ ( λ ) d λ , f 2 ( λ ) = 1 Y 0 λ b λ S ( λ ) y ¯ ( λ ) d λ , f 3 ( λ ) = 1 Z 0 λ b λ S ( λ ) z ¯ ( λ ) d λ ,
d ρ d λ = j = 1 3 μ j f j ( λ ) ,
ρ ( λ ) = ρ ( λ e ) + j = 1 3 μ j λ λ e f j ( λ ) d λ ,
j = 1 3 μ j = 0 ,
A [ μ 1 μ 2 μ 3 ] = [ X / X 0 ρ ( λ e ) Y / Y 0 ρ ( λ e ) Z / Z 0 ρ ( λ e ) ] ,
a i , j = f i ( λ ) f j ( λ ) d λ .
ρ ( λ e ) = ν 1 X X 0 + ν 2 Y Y 0 + ν 3 Z Z 0 ,
ν j = i = 1 3 a i , j 1 / i , j = 1 3 a i , j 1 .
( d ρ d λ ) 2 d λ = μ 1 X X 0 + μ 2 Y Y 0 + μ 3 Z Z 0 .
g 1 ( λ ) = 1 X 0 λ λ e S ( λ ) x ¯ ( λ ) d λ , g 2 ( λ ) = 1 Y 0 λ λ e S ( λ ) y ¯ ( λ ) d λ , g 3 ( λ ) = 1 Z 0 λ λ e S ( λ ) z ¯ ( λ ) d λ ,
d ρ d λ = j = 1 3 μ j g j ( λ ) , ρ ( λ ) = ρ ( λ b ) + j = 1 3 μ j λ b λ g j ( λ ) d λ .
B [ μ 1 μ 2 μ 3 ] = [ X / X 0 ρ ( λ b ) Y / Y 0 ρ ( λ b ) Z / Z 0 ρ ( λ b ) ] ,
b i , j = g i ( λ ) g j ( λ ) d λ .
ρ ( λ b ) = ν 1 X X 0 + ν 2 Y Y 0 + ν 3 Z Z 0 ,
d ρ d λ = 0 , λ = λ 1 or λ = λ 2 , λ b < λ 1 λ 2 < λ e .
[ f 1 ( λ 1 ) f 2 ( λ 1 ) f 3 ( λ 1 ) f 1 ( λ 2 ) f 2 ( λ 2 ) f 3 ( λ 2 ) 1 1 1 ] [ μ 1 μ 2 μ 3 ] = [ 0 0 0 ] .
f 1 ( λ ) = X b ( λ ) X 0 , f 2 ( λ ) = Y b ( λ ) Y 0 , f 3 ( λ ) = Z b ( λ ) Z 0 .
[ x b ( λ 1 ) y b ( λ 1 ) 1 x b ( λ 2 ) y b ( λ 2 ) 1 x 0 y 0 1 ] 0.
ρ ( λ ) = + ( 1 ) ρ ( λ ; λ 1 ) .
d ρ d λ = ( 1 ) d ρ ( λ ; λ 1 ) d λ ,
Y = Y 0 + ( 1 ) Y ( λ 1 ) .
d ρ ( λ ; λ 1 ) d λ = γ | 1 1 1 f 1 ( λ 1 ) f 2 ( λ 1 ) f 3 ( λ 1 ) f 1 ( λ ) f 2 ( λ ) f 3 ( λ ) | .
d ρ ( λ ; λ 1 ) d λ = 0 , λ = λ 1 ,
d 2 ρ ( λ ; λ 1 ) d λ 2 = γ S ( λ ) | 1 1 1 f 1 ( λ 1 ) f 1 ( λ 1 ) f 1 ( λ 1 ) x ¯ ( λ ) / X 0 y ¯ ( λ ) / Y 0 z ¯ ( λ ) / Z 0 | ,
ρ ( λ ; λ 1 ) = λ 1 λ d ρ ( λ ; λ 1 ) d λ d λ .
d ρ ( λ ; λ 1 ) d λ = γ | 1 1 1 x ¯ ( λ 1 ) / X 0 y ¯ ( λ 1 ) / Y 0 z ¯ ( λ 1 ) / Z 0 f 1 ( λ ) f 2 ( λ ) f 2 ( λ ) | .
| x 0 y 0 1 x s ( λ 1 ) y s ( λ 1 ) 1 x b ( λ ) y b ( λ ) 1 | ,
ρ ( λ ; λ 1 ) = λ b λ d ρ ( λ ; λ 1 ) d λ d λ ,
| X 0 Y 0 Z 0 x ¯ ( λ e ) y ¯ ( λ e ) z ¯ ( λ e ) x ¯ ( λ b ) y ¯ ( λ b ) z ¯ ( λ b ) | = 0
( 1 z 0 x 0 ν 1 ν 3 ) x ( λ 1 ) + ( 1 z 0 y 0 ν 2 ν 3 ) y ( λ 1 ) = 1 ;
( 1 z 0 x 0 ν 1 ν 3 ) x ( λ 1 ) + ( 1 z 0 y 0 ν 2 ν 3 ) y ( λ 1 ) = 1 ,
Y ( λ 1 ) Y 0 = z 0 y 0 y ( λ 1 ) ν 3 × [ 1 ( 1 ν 1 ν 3 z 0 x 0 ) x ( λ 1 ) ( 1 ν 2 ν 3 z 0 y 0 ) y ( λ 1 ) ] 1 .
Y ( λ 1 ) Y 0 = z 0 y 0 y ( λ 1 ) ν 3 × [ 1 ( 1 ν 1 ν 2 z 0 x 0 ) x ( λ 1 ) ( 1 ν 2 ν 3 z 0 y 0 ) y ( λ 1 ) ] 1 .
( 1 ν 1 ν 3 z 0 x 0 ) x + ( 1 ν 2 ν 3 z 0 y 0 ) y + Y 0 Y z 0 y 0 y ν 3 = 1 ,
( 1 ν 1 ν 3 z 0 x 0 ) x + ( 1 ν 2 ν 3 z 0 y 0 ) y + Y 0 Y z 0 y 0 y ν 3 = 1.
ρ ( λ ) = ( 1 ) [ 1 ρ ( λ ; λ 1 ) ] ,
d ρ B d λ d ρ d λ d λ = 0 ,
d ρ B d λ f j ( λ ) d λ = ρ B ( λ e ) .
ρ B ( 1 ) ( λ ) = 1 i = 1 3 π i λ λ e f i ( λ ) d λ ' with π i = ν i i , j = 1 3 a i , j 1 , ρ B ( 2 ) ( λ ) = 1 i = 1 3 π i λ b λ g i ( λ ) d λ with π i = ν i i , j = 1 3 b i , j 1
| x ¯ ( λ 1 ) y ¯ ( λ 1 ) z ¯ ( λ 1 ) x ¯ ( λ 2 ) y ¯ ( λ 2 ) z ¯ ( λ 2 ) x ¯ ( λ 3 ) y ¯ ( λ 3 ) z ¯ ( λ 3 ) | 0
| X b ( λ 1 ) Y b ( λ 1 ) Z b ( λ 1 ) X b ( λ 2 ) Y b ( λ 2 ) Z b ( λ 2 ) X b ( λ 3 ) Y b ( λ 3 ) Z b ( λ 3 ) | 0 ,
| x b ( λ 1 ) y b ( λ 1 ) 1 x b ( λ 2 ) y b ( λ 2 ) 1 x b ( λ 3 ) y b ( λ 3 ) 1 | 0 ,
| f 1 ( λ 1 ) f 2 ( λ 1 ) f 3 ( λ 1 ) f 1 ( λ 2 ) f 2 ( λ 2 ) f 3 ( λ 2 ) 1 1 1 | 0 ;
| X b ( λ ) Y b ( λ ) Z b ( λ ) X 0 Y 0 Z 0 x ¯ ( λ e ) y ¯ ( λ e ) z ¯ ( λ e ) | 0
y y b ( λ ) = d y b d x b [ x x b ( λ ) ] .
d y b d x b = d y b ( λ ) d λ [ d x b ( λ ) d λ ] 1 = y s ( λ ) y b ( λ ) x s ( λ ) x b ( λ ) .

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