Abstract

The smoothest reflectance function [ J. Opt. Soc. Am. A 7, 1891 ( 1990)] is considered an estimate of the actual reflectance function with the same tristimulus values under the given illuminant. The estimate differs from the actual function by a so-called metameric black. The metameric black depends on a number of parameters that are inaccessible to the visual system and describes the uncertainty with which the visual system has to cope when illuminant-independent properties of reflectance are being predicted. Illuminant-independent properties of reflectance are determined that can be predicted with little uncertainty and that therefore can be calculated in good approximation from the estimate of the actual reflectance function. The result is an estimate of the property that is, by construction, almost independent of the illuminant and thus in principle is able to explain color constancy. Such a property, a weighted mean of reflectance yielding an achromatic variable, is constructed; the predictions are verified numerically; and the result is compared with experiment.

© 1994 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. W D. Wright, The Measurement of Colour (Hilger, London, 1969).
  7. W A. Thornton, “Matching lights, metamers and human visual response,” J. Color Appear. 2, 23–29 (1973).
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    [CrossRef]
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    [CrossRef]
  10. M. H. Brill, “Statistical confirmation of Thornton’s zero crossing conjecture,” Color Res. Appl. 12, 51–53 (1987).
    [CrossRef]
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    [CrossRef]
  12. R. S. Berns, R. G. Kuehni, “What determines crossover wavelength of metameric pairs with three crossovers,” Color Res. Appl. 15, 23–28 (1990).
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    [CrossRef]
  22. Commission Internationale de l’Eclairage, Method of Measuring and Specifying Colour Rendering Properties of Light Sources (CIE, Paris, 1974).
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    [CrossRef]
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    [CrossRef]
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  34. J. J. Vos, O. Estévez, P. L. Walraven, “Improved color fundamentals offer a new view on photometric additivity,” Vision Res. 30, 937–943 (1990).
    [CrossRef] [PubMed]
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    [CrossRef]
  38. J. M. Troost, C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Col. Res. Appl. 17, 316–327 (1992).
    [CrossRef]
  39. N. Y. Vilenkin, Functional Analysis (Wolters-Noordhof, Groningen, The Netherlands, 1972).
  40. The chromaticity coordinates of test color 4 should have been 0.2871,0.4141; the values of the correct reflectance function differ slightly from those in the table.
  41. Commission Internationale de l’Eclairage, Colorimetry (CIE, Paris, 1970).
  42. G. West, M. H. Brill, “Necessary and sufficient conditions for von Kries adaptation to give color constancy,” J. Math. Biol. 15, 249–258 (1982).
    [CrossRef]
  43. D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–36 (1990).
    [CrossRef]
  44. J. Vos, “Colorimetric and photometric properties of a 2° fundamental observer,” Col. Res. Appl. 3, 125–128 (1978).
    [CrossRef]
  45. J. E. M. Janssen, Philips Research Laboratories, Eindhoven, The Netherlands (personal communication, December1991).

1992 (2)

J. L. Dannemiller, “Spectral reflectance of natural objects: how many basis functions are necessary?” J. Opt. Soc. Am. A 9, 507–515 (1992).
[CrossRef]

J. M. Troost, C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Col. Res. Appl. 17, 316–327 (1992).
[CrossRef]

1991 (1)

R. G. Kuehni, “On the evolution of the color vision system,” Color Res. Appl. 16, 279–281 (1991).
[CrossRef]

1990 (5)

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

J. J. Vos, O. Estévez, P. L. Walraven, “Improved color fundamentals offer a new view on photometric additivity,” Vision Res. 30, 937–943 (1990).
[CrossRef] [PubMed]

R. S. Berns, R. G. Kuehni, “What determines crossover wavelength of metameric pairs with three crossovers,” Color Res. Appl. 15, 23–28 (1990).
[CrossRef]

C. van Trigt, “Smoothest reflectance functions. I. Definition and main results,” J. Opt. Soc. Am. A 7, 1891–1904 (1990).
[CrossRef]

C. van Trigt, “Smoothest reflectance functions. II. Complete results,” J. Opt. Soc. Am. A 7, 2208–2222 (1990).
[CrossRef]

1989 (1)

1987 (2)

M. H. Brill, “Statistical confirmation of Thornton’s zero crossing conjecture,” Color Res. Appl. 12, 51–53 (1987).
[CrossRef]

N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987).
[CrossRef]

1983 (1)

1982 (1)

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

1981 (1)

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

1978 (3)

C. J. Bartleson, “Comparison of chromatic-adaptation transforms,” Color Res. Appl. 3, 129–136 (1978).
[CrossRef]

W A. Thornton, “Reply to Ohta–Wyszecki on location nodes of metameric color stimuli,” Color Res. Appl. 3, 202–204 (1978).
[CrossRef]

J. Vos, “Colorimetric and photometric properties of a 2° fundamental observer,” Col. Res. Appl. 3, 125–128 (1978).
[CrossRef]

1977 (1)

N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
[CrossRef]

1975 (1)

1973 (1)

W A. Thornton, “Matching lights, metamers and human visual response,” J. Color Appear. 2, 23–29 (1973).

1972 (1)

1964 (1)

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

1963 (1)

1958 (1)

1957 (1)

1953 (1)

G. Wyszecki, “Valenzmetrische Untersuchung des Zusam menhanges zwischen normaler und anomaler Trichromasie,” Farbe 2, 39–52 (1953).

1940 (1)

1935 (1)

V A. Kohlrausch, “Zur Photometrie farbiger Lichter,” Licht 5, 259–275 (1935).

1920 (1)

E. Schrodinger, “Theorie der Pigmente von grosster Leuchtkraft,” Ann. Physik 62, 603–622 (1920).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1968).

Bartleson, C. J.

C. J. Bartleson, “Comparison of chromatic-adaptation transforms,” Color Res. Appl. 3, 129–136 (1978).
[CrossRef]

Berns, R. S.

R. S. Berns, R. G. Kuehni, “What determines crossover wavelength of metameric pairs with three crossovers,” Color Res. Appl. 15, 23–28 (1990).
[CrossRef]

Boas, R. P.

R. P. Boas, R. C. Buck, Polynomial Expansions of Analytic Functions (Springer-Verlag, Berlin, 1958).
[CrossRef]

Brill, M. H.

M. H. Brill, “Statistical confirmation of Thornton’s zero crossing conjecture,” Color Res. Appl. 12, 51–53 (1987).
[CrossRef]

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

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

Buck, R. C.

R. P. Boas, R. C. Buck, Polynomial Expansions of Analytic Functions (Springer-Verlag, Berlin, 1958).
[CrossRef]

Cohen, J.

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

Dannemiller, J. L.

de Weert, C. M. M.

J. M. Troost, C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Col. Res. Appl. 17, 316–327 (1992).
[CrossRef]

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2.

Estévez, O.

J. J. Vos, O. Estévez, P. L. Walraven, “Improved color fundamentals offer a new view on photometric additivity,” Vision Res. 30, 937–943 (1990).
[CrossRef] [PubMed]

Forsyth, D. A.

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

Froberg, C. E.

C. E. Froberg, Introduction to Numerical Analysis (Addison-Wesley, Reading, Mass., 1965).

Hallikainen, J.

Hering, E.

E. Hering, Outlines of a Theory of the Light Sense, L. Hurvich, D. Jameson, transl. (Harvard U. Press, Cambridge, Mass., 1964).

Hurvich, L. M.

L. M. Hurvich, Color Vision (Sinauer, Sunderland, Mass., 1981).

Jaaskelainen, T.

Janssen, J. E. M.

J. E. M. Janssen, Philips Research Laboratories, Eindhoven, The Netherlands (personal communication, December1991).

Judd, D. B.

Knuth, D. E.

D. E. Knuth, The Art of Computer Programming (Addison-Wesley, Reading, Mass., 1975), Vol. 1.

Kohlrausch, V A.

V A. Kohlrausch, “Zur Photometrie farbiger Lichter,” Licht 5, 259–275 (1935).

Kuehni, R. G.

R. G. Kuehni, “On the evolution of the color vision system,” Color Res. Appl. 16, 279–281 (1991).
[CrossRef]

R. S. Berns, R. G. Kuehni, “What determines crossover wavelength of metameric pairs with three crossovers,” Color Res. Appl. 15, 23–28 (1990).
[CrossRef]

MacAdam, D. L.

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2.

Nayatani, Y.

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

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

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2.

Ohta, N.

N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987).
[CrossRef]

N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
[CrossRef]

N. Ohta, “Generating metameric object colors,” J. Opt. Soc. Am. 65, 1081–1082 (1975).
[CrossRef]

Parkkinen, J. P. S.

Sanders, C. L.

Schrodinger, E.

E. Schrodinger, “Theorie der Pigmente von grosster Leuchtkraft,” Ann. Physik 62, 603–622 (1920).
[CrossRef]

Sobagaki, H.

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1968).

Stiles, W. S.

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

Takahama, K.

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

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

Thornton, W A.

W A. Thornton, “Reply to Ohta–Wyszecki on location nodes of metameric color stimuli,” Color Res. Appl. 3, 202–204 (1978).
[CrossRef]

W A. Thornton, “Matching lights, metamers and human visual response,” J. Color Appear. 2, 23–29 (1973).

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2.

Troost, J. M.

J. M. Troost, C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Col. Res. Appl. 17, 316–327 (1992).
[CrossRef]

van Trigt, C.

Vilenkin, N. Y.

N. Y. Vilenkin, Functional Analysis (Wolters-Noordhof, Groningen, The Netherlands, 1972).

von Helmholtz, H.

H. von Helmholtz, Helmholt’s Treatise on Physiological Optics, J. P. Southall, transl. (Optical Society of America, Washington, D.C., 1924).

von Kries, J.

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

Vos, J.

J. Vos, “Colorimetric and photometric properties of a 2° fundamental observer,” Col. Res. Appl. 3, 125–128 (1978).
[CrossRef]

Vos, J. J.

J. J. Vos, O. Estévez, P. L. Walraven, “Improved color fundamentals offer a new view on photometric additivity,” Vision Res. 30, 937–943 (1990).
[CrossRef] [PubMed]

Walraven, P. L.

J. J. Vos, O. Estévez, P. L. Walraven, “Improved color fundamentals offer a new view on photometric additivity,” Vision Res. 30, 937–943 (1990).
[CrossRef] [PubMed]

West, G.

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

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

Wright, W D.

W D. Wright, The Measurement of Colour (Hilger, London, 1969).

Wyszecki, G.

N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
[CrossRef]

G. Wyszecki, “Evaluation of metameric colors,” J. Opt. Soc. Am. 48, 451–454 (1958).
[CrossRef]

C. L. Sanders, G. Wyszecki, “Correlate of lightness in terms of CIE tristimulus values. Part 1,” J. Opt. Soc. Am. 47, 398–404 (1957).
[CrossRef]

G. Wyszecki, “Valenzmetrische Untersuchung des Zusam menhanges zwischen normaler und anomaler Trichromasie,” Farbe 2, 39–52 (1953).

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

Ann. Physik (1)

E. Schrodinger, “Theorie der Pigmente von grosster Leuchtkraft,” Ann. Physik 62, 603–622 (1920).
[CrossRef]

Col. Res. Appl. (2)

J. M. Troost, C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Col. Res. Appl. 17, 316–327 (1992).
[CrossRef]

J. Vos, “Colorimetric and photometric properties of a 2° fundamental observer,” Col. Res. Appl. 3, 125–128 (1978).
[CrossRef]

Color Res. Appl. (8)

R. G. Kuehni, “On the evolution of the color vision system,” Color Res. Appl. 16, 279–281 (1991).
[CrossRef]

N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
[CrossRef]

W A. Thornton, “Reply to Ohta–Wyszecki on location nodes of metameric color stimuli,” Color Res. Appl. 3, 202–204 (1978).
[CrossRef]

M. H. Brill, “Statistical confirmation of Thornton’s zero crossing conjecture,” Color Res. Appl. 12, 51–53 (1987).
[CrossRef]

N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987).
[CrossRef]

R. S. Berns, R. G. Kuehni, “What determines crossover wavelength of metameric pairs with three crossovers,” Color Res. Appl. 15, 23–28 (1990).
[CrossRef]

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

C. J. Bartleson, “Comparison of chromatic-adaptation transforms,” Color Res. Appl. 3, 129–136 (1978).
[CrossRef]

Farbe (1)

G. Wyszecki, “Valenzmetrische Untersuchung des Zusam menhanges zwischen normaler und anomaler Trichromasie,” Farbe 2, 39–52 (1953).

Int. J. Comput. Vis. (1)

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

J. Color Appear. (1)

W A. Thornton, “Matching lights, metamers and human visual response,” J. Color Appear. 2, 23–29 (1973).

J. Math. Biol. (1)

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

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (4)

Licht (1)

V A. Kohlrausch, “Zur Photometrie farbiger Lichter,” Licht 5, 259–275 (1935).

Psychon. Sci. (1)

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

Vision Res. (1)

J. J. Vos, O. Estévez, P. L. Walraven, “Improved color fundamentals offer a new view on photometric additivity,” Vision Res. 30, 937–943 (1990).
[CrossRef] [PubMed]

Other (16)

C. E. Froberg, Introduction to Numerical Analysis (Addison-Wesley, Reading, Mass., 1965).

D. E. Knuth, The Art of Computer Programming (Addison-Wesley, Reading, Mass., 1975), Vol. 1.

R. P. Boas, R. C. Buck, Polynomial Expansions of Analytic Functions (Springer-Verlag, Berlin, 1958).
[CrossRef]

N. Y. Vilenkin, Functional Analysis (Wolters-Noordhof, Groningen, The Netherlands, 1972).

The chromaticity coordinates of test color 4 should have been 0.2871,0.4141; the values of the correct reflectance function differ slightly from those in the table.

Commission Internationale de l’Eclairage, Colorimetry (CIE, Paris, 1970).

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

H. von Helmholtz, Helmholt’s Treatise on Physiological Optics, J. P. Southall, transl. (Optical Society of America, Washington, D.C., 1924).

E. Hering, Outlines of a Theory of the Light Sense, L. Hurvich, D. Jameson, transl. (Harvard U. Press, Cambridge, Mass., 1964).

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

W D. Wright, The Measurement of Colour (Hilger, London, 1969).

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

L. M. Hurvich, Color Vision (Sinauer, Sunderland, Mass., 1981).

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1968).

J. E. M. Janssen, Philips Research Laboratories, Eindhoven, The Netherlands (personal communication, December1991).

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Figures (8)

Fig. 1
Fig. 1

CIE test colors 2, 4, 6, and 8 (dashed curves) compared with the corresponding (metameric) smoothest reflectance functions (solid curves); illuminant A.

Fig. 2
Fig. 2

Metameric blacks of orders k = 1 and k = 2; illuminant D65.

Fig. 3
Fig. 3

Metameric blacks of orders k = 1 and k = 2; illuminant A.

Fig. 4
Fig. 4

Metameric blacks of orders k = 3 and k = 5 (m = 1,2); illuminant D65.

Fig. 5
Fig. 5

Metameric blacks of order k = 3 and k = 5 (m = 1,2); illuminant A.

Fig. 6
Fig. 6

Functions Δi; (λ), i = 1, 2, 3, and 4, constructed in Appendix B, normalized to unity at their maximum values; with peaks (breadths at half-maximum): 445 (55), 545 (80), 555 (100), and 580 nm (110 nm). Δ 3 ( λ ) , Δ 1 ( λ ) z ¯ ( λ ) , y ¯ ( λ ) .

Fig. 7
Fig. 7

CIE test colors 2, 4, 6, and 8 (dashed curves) compared with their approximations Eq. (23) (solid curves); illuminant D65.

Fig. 8
Fig. 8

Achromatic response functions A(λ) for D65 and A, normalized to unity at their maximum values. The first function almost coincides with Δ2(λ) of Fig. 6.

Tables (3)

Tables Icon

Table 1 Smoothest Reflectance Functions under A

Tables Icon

Table 2 Values of the Achromatic Variables

Tables Icon

Table 3 Values of the Achromatic Variables

Equations (92)

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

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 ,
h ( λ ) = | x ¯ ( λ ) y ¯ ( λ ) z ¯ ( λ ) x ¯ ( λ 1 ) y ¯ ( λ 1 ) z ¯ ( λ 1 ) x ¯ ( λ 2 ) y ¯ ( λ 2 ) z ¯ ( λ 2 ) |
| x s ( λ ) x s ( λ ) 1 x s ( λ 1 ) x s ( λ 1 ) 1 x s ( λ 2 ) x s ( λ 2 ) 1 |
r ( λ ) S ( λ ) h ( λ ) d λ = 0 .
r ( λ ) = i = 1 4 a i p i ( λ ) , a 4 = 1 .
ρ ( λ ) = ρ 0 ( λ ) + k = 1 c k r k ( λ ) ,
P ( λ ) ρ ( λ ) d λ = P ( λ ) ρ 0 ( λ ) d λ + k = 1 c k P ( λ ) r k ( λ ) d λ .
f 1 ( m + 1 ) ( λ ) = 1 m ! λ b λ ( λ λ λ e λ b ) m S ( λ ) x ¯ ( λ ) X 0 d λ , f 2 ( m + 1 ) ( λ ) = 1 m ! λ b λ ( λ λ λ e λ b ) m S ( λ ) y ¯ ( λ ) Y 0 d λ , f 3 ( m + 1 ) ( λ ) = 1 m ! λ b λ ( λ λ λ e λ b ) m S ( λ ) z ¯ ( λ ) Z 0 d λ .
f 1 ( 0 ) ( λ ) = ( λ e λ b ) S ( λ ) x ¯ ( λ ) X 0 ,
d n f i ( m + 1 ) d λ n = f i ( m n + 1 ) ( λ ) ( λ e λ b ) n .
f i ( m + 1 ) ( λ ) = 1 n ! λ b λ ( λ λ λ e λ b ) n f i ( m n ) ( λ ) d λ λ e λ b
g 1 ( m + 1 ) ( λ ) = 1 m ! λ λ e ( λ λ λ e λ b ) m S ( λ ) x ¯ ( λ ) X 0 d λ , g 2 ( m + 1 ) ( λ ) = 1 m ! λ λ e ( λ λ λ e λ b ) m S ( λ ) y ¯ ( λ ) Y 0 d λ , g 3 ( m + 1 ) ( λ ) = 1 m ! λ λ e ( λ λ λ e λ b ) m S ( λ ) z ¯ ( λ ) Z 0 d λ .
g i ( m + 1 ) ( λ ) = 1 n ! λ λ e ( λ λ λ e λ b ) n g i m n ( λ ) d λ λ e λ b .
f i ( m ) ( λ ) f i ( m ) ( λ e ) = ( λ λ b λ e λ b ) m + α i [ 1 + ( α i + 1 ) A i m + α i + 1 λ λ e λ e λ b + . . . ] , g i ( m ) ( λ ) g i ( m ) ( λ b ) = ( λ e λ λ e λ b ) m + β i [ 1 + ( β i + 1 ) B i m + β i + 1 λ λ b λ e λ b + . . . ] .
A 3 λ e λ b 2 ( 2 S ( λ b ) d S ( λ b ) d λ + 1 z ¯ ( λ b ) d z ¯ ( λ b ) d λ ) .
r k ( λ ) = 1 m ! ( λ λ e λ e λ b ) m j = 1 3 π j k m ! λ λ e ( λ λ λ e λ b ) m f j ( m + 1 ) ( λ ) d λ
d n r k ( λ e ) d λ n = δ m , n ( λ e λ b ) n , d m + 1 r k ( λ ) d λ m + 1 = 1 ( λ e λ b ) m j = 1 3 π j k f j ( m + 1 ) ( λ ) .
r k ( λ ) = 1 m ! ( λ λ b λ e λ b ) m j = 1 3 π j k m ! λ b λ ( λ λ λ e λ b ) m g j ( m + 1 ) ( λ ) d λ
d n r k ( λ b ) d λ n = δ m , n ( λ e λ b ) n , d m + 1 r k ( λ ) d λ m + 1 = 1 ( λ e λ b ) m j = 1 3 π j k g j ( m + 1 ) ( λ )
n = 0 N 1 ( 1 ) n ( λ e λ b ) n f i n + 1 ( λ ) d n r k ( λ ) d λ n λ e | λ b + ( 1 ) N λ ( λ e λ b ) N 1 λ b λ e f i ( N ) ( λ ) d N r k ( λ ) d λ N d λ = 0.
f i ( m + 1 ) ( λ e ) j = 1 3 π j k f i ( m + 1 ) ( λ ) f j ( m + 1 ) ( λ ) d λ = 0 .
A ( m + 1 ) [ π 1 k π 2 k π 3 k ] = [ f 1 ( m + 1 ) ( λ e ) f 2 ( m + 1 ) ( λ e ) f 3 ( m + 1 ) ( λ e ) ] ,
a i , j ( m + 1 ) = f i ( m + 1 ) ( λ ) f j ( m + 1 ) ( λ ) d λ .
( λ e λ b ) j = 1 3 â i , j ( m + 1 ) π j k f j ( m + 1 ) ( λ e ) = 1 â i , j ( m ) = [ f i ( m ) ( λ e ) f j ( m ) ( λ e ) ] 1 f i ( m ) ( λ ) f j ( m ) ( λ ) d y λ e λ b .
B ( m + 1 ) [ π 1 k π 2 k π 3 k ] = [ g 1 ( m + 1 ) ( λ b ) g 2 ( m + 1 ) ( λ b ) g 3 ( m + 1 ) ( λ b ) ] ,
b i , j ( m + 1 ) = g i ( m + 1 ) ( λ ) g j ( m + 1 ) ( λ ) d λ .
( λ λ b λ e λ b ) α r k ( λ ) d λ λ e λ b = ( 1 ) m Γ ( α + 1 ) Γ ( m + α + 2 ) × [ 1 j = 1 3 π j k ( λ λ b λ e λ b ) m + α + 1 f j ( m + 1 ) d ( λ ) ] .
( λ e λ λ e λ b ) β r k ( λ ) d λ λ e λ b = Γ ( β + 1 ) Γ ( m + β + 2 ) × [ 1 j = 1 3 π j k ( λ e λ λ e λ b ) m + β + 1 g j ( m + 1 ) d ( λ ) ] .
[ x ( λ ) y ( λ ) z ( λ ) ] = T [ x ¯ ( λ ) y ¯ ( λ ) z ¯ ( λ ) ] .
[ X 0 f 1 ( m + 1 ) ( λ e ) Y 0 f 2 ( m + 1 ) ( λ e ) Z 0 f 3 ( m + 1 ) ( λ e ) ] = Λ A ( m + 1 ) Λ [ π 1 k / X 0 π 2 k / Y 0 π 3 k / Z 0 ] .
T f ( λ e ) = T Λ A ( m + 1 ) Λ T + ( T + ) 1 π ,
T f ( λ e ) = f ( λ e ) , T Λ A ( m + 1 ) Λ T + = ( Λ A ( m + 1 ) Λ ) ,
π = ( T 1 ) + π .
π i k f i ( m + 1 ) ( λ ) = [ π , f ( λ ) ] = [ ( T 1 ) + π , T f ( λ ) ] = [ π , f ( λ ) ] ,
d r 1 d λ = j = 1 3 π i 1 f i ( 1 ) ( λ ) .
[ f 1 ( λ 1 ) f 2 ( λ 1 ) f 3 ( λ 1 ) f 1 ( λ 2 ) f 2 ( λ 2 ) f 3 ( λ 2 ) f 1 ( λ 3 ) f 2 ( λ 3 ) f 3 ( λ 3 ) ] [ π 1 π 2 π 3 ] = [ 0 0 0 ]
d m + 1 r k ( λ ) d λ m + 1 = 1 ( λ e λ b ) m i = 1 3 π j k f i ( m + 1 ) ( λ ) .
f i ( m + 1 ) ( λ ) = 0 λ < λ i = 1 m ! ( λ λ i λ e λ b ) m λ > λ i .
r k ( λ i ) = 1 m ! ( λ i λ e λ e λ b ) m j = 1 3 π j k m ! λ m λ e ( λ i λ λ e λ b ) m × ( λ λ j λ e λ b ) m d λ m ! .
a i , j ( m + 1 ) = ( λ e λ b ) 2 m m ! m ! λ m λ e ( λ λ i ) m ( λ λ j ) m d λ .
r 1 ( λ ) = 0 λ < λ 3 = λ λ 3 λ e λ 3 λ > λ 3 ,
 ( m + 1 ) [ σ 1 k σ 2 k σ 3 k ] = [ 1 1 1 ] , σ i k = ( λ e λ b ) π i k f i ( m + 1 ) ( λ e ) .
â i , j = 1 2 m + 3 + α i + α j ( 1 + ɛ i + ɛ j 2 m + 4 + α i + α j + . . . ) , ɛ i = ( α i + 1 ) A i m + α i + 2 .
c i , j 1 = c i c j 2 m + α i + α j + 3 , c i = Π p = 1 3 ( 2 m + 3 + α i + α p ) Π q i 3 , 1 α i + α q .
Det ( C ) = 1 / i = 1 3 c i , j = 1 3 c i , j c j = 1 , i = 1 , 2 , 3 .
[ Â ( m + 1 ) ] 1 = ( C + δ C ) 1 = C 1 C 1 δ C C 1 + . . .
σ 1 k Det [ Â ( m + 1 ) ] = ( â 2 , 2 â 2 , 1 ) ( â 3 , 3 â 3 , 1 ) ( â 2 , 3 â 2 , 1 ) × ( â 2 , 3 â 3 , 1 ) , σ 2 k Det [ Â ( m + 1 ) ] = ( â 1 , 1 â 1 , 2 ) ( â 3 , 3 â 3 , 2 ) ( â 1 , 3 â 1 , 2 ) × ( â 1 , 3 â 3 , 2 ) , σ 3 k Det [ Â ( m + 1 ) ] = ( â 1 , 1 â 1 , 3 ) ( â 2 , 2 â 2 , 3 ) ( â 1 , 2 â 1 , 3 ) × ( â 1 , 2 â 2 , 3 ) .
â p , i â p , j = α j α i ( ɛ j ɛ i ) [ 1 M 1 + . . . ] + ( α j α i ) ( ɛ i + ɛ j + 2 ɛ p ) M 1 [ 1 . . . ] ( 2 m + 3 + α p + α i ) ( 2 m + 3 + α p + α j ) ,
σ i k Det ( Â ( m + 1 ) ) Det ( C ) = c i ( 1 + p i 3 ɛ i ɛ p α i α p p = 1 3 q p 3 ɛ p ɛ q α p α q p = 1 3 ɛ p Π q p 3 1 α p α q + . . . ) .
f i ( m + 1 ) ( λ ) = 1 m ! λ b λ ( λ λ λ e λ b ) m S ( λ ) Δ i ( λ ) d λ / S ( λ ) Δ i ( λ ) d λ .
δ i k = r k ( λ ) E ( λ ) Δ i ( λ ) d λ .
( 1 ) m δ i k = e i ( m + 1 ) ( λ e ) j = 1 3 π j k e i ( m + 1 ) ( λ ) f j ( m + 1 ) ( λ ) d λ .
δ i k = ( 1 ) m e i ( m + 1 ) ( λ e ) × j = 1 3 π j k [ f i ( m + 1 ) ( λ ) f i ( m + 1 ) ( λ e ) e i ( m + 1 ) ( λ ) e i ( m + 1 ) ( λ e ) ] f j ( m + 1 ) ( λ ) d λ , f i ( m ) ( λ ) f i ( m ) ( λ e ) e i ( m ) ( λ ) e i ( m ) ( λ e ) = ( α i + 1 ) A i m + α i + 1 ( λ λ b λ e λ b ) m + α i ( λ λ e λ e λ b + . . . ) .
e i ( m + 1 ) ( λ e ) = Γ ( α i + 1 ) e 0 Γ ( m + α i + 2 ) .
( λ λ b λ e λ b ) α i r k ( λ ) d λ λ e λ b = ( 1 ) m Γ ( α i + 1 ) Γ ( m + α i + 2 ) × j = 1 3 π j k [ f i ( m + 1 ) ( λ ) f i ( m + 1 ) ( λ e ) ( λ λ b λ b λ e ) m + α i + 1 ] f i ( m + 1 ) ( λ ) d λ , f i ( m ) ( λ ) f i ( m ) ( λ e ) ( λ λ b λ b λ e ) m + α i = ( α i + 1 ) A i m + α i + 1 ( λ λ b λ e λ b ) m + α i ( λ λ e λ e λ b + . . . ) .
r k ( λ ) E ( λ ) Δ i ( λ ) d λ = e 0 A i A i ( λ λ b λ e λ b ) α i r k ( λ ) d λ λ e λ b .
ρ ( λ ) = ρ 0 ( λ ) + A [ r 2 ( λ ) r 2 ( λ e ) r 1 ( λ ) 1 r 1 ( λ b ) r 2 ( λ e ) ] + B [ r 1 ( λ ) r 1 ( λ b ) r 2 ( λ ) 1 r 1 ( λ b ) r 2 ( λ e ) ]
A = ρ ( λ b ) ρ 0 ( λ b ) , B = ρ ( λ e ) ρ 0 ( λ e ) ,
R k ( λ ) = r k ( λ ) r k ( λ b ) R 1 ( λ ) , k = 2 m + 1 , R k ( λ ) = r k ( λ ) r k ( λ e ) R 2 ( λ ) k = 2 m + 2 .
ρ ( λ ) = ρ 1 ( λ ) + C [ R 4 ( λ e ) R 3 ( λ ) R 3 ( λ e ) R 4 ( λ ) R 4 ( λ e ) R 3 ( λ b ) R 3 ( λ e ) R 4 ( λ b ) ] + D [ R 3 ( λ b ) R 4 ( λ ) R 4 ( λ b ) R 3 ( λ ) R 4 ( λ e ) R 3 ( λ b ) R 3 ( λ e ) R 4 ( λ b ) ]
C = ρ ( λ b ) ρ 1 ( λ b ) , D = ρ ( λ e ) ρ 1 ( λ e ) .
P ( λ ) ρ ( λ ) d λ = P ( λ ) ρ 0 ( λ ) d λ + k = 1 c k P ( λ ) r k ( λ ) d λ
P 1 ( λ ) = ( τ 1 x ¯ ( λ ) X E + τ 2 y ¯ ( λ ) Y E + τ 3 z ¯ ( λ ) Z E ) E ( λ ) , τ i = 1 ,
P 2 ( λ ) = Γ ( α + β + 2 ) Γ ( α + 1 ) ( β + 1 ) ( λ λ b λ e λ b ) α ( λ e λ λ e λ b ) β 1 λ e λ b .
P 2 ( λ ) r k ( λ ) d λ = Γ ( α + β + 2 ) Γ ( α + 1 ) Γ ( β + 1 ) × [ ( λ λ b λ e λ b ) α r k ( λ ) d λ λ e λ b + . . . ] .
P 2 ( λ ) r k ( λ ) d λ = Γ ( α + β + 2 ) Γ ( α + 1 ) Γ ( β + 1 ) × [ ( λ e λ λ e λ b ) β r k ( λ ) d λ λ e λ b + . . . ] .
P ( λ ) = Δ 2 ( λ ) E ( λ ) .
ξ = ρ 0 ( λ ) E ( λ ) Δ 2 ( λ ) d λ ρ ( λ ) E ( λ ) Δ 2 ( λ ) d λ .
ξ = ρ ( λ ) S ( λ ) A ( λ ) d λ .
ξ = ρ 0 ( λ e ) + j = 1 3 μ j Δ 2 ( 1 ) ( λ ) f j ( λ ) d λ ,
Δ 2 ( 1 ) ( λ ) = λ b λ E ( λ ) Δ 2 ( λ ) d λ , ρ 0 ( λ e ) = ν 1 X X 0 + ν 2 Y Y 0 + ν 3 Z Z 0 , ν i = π i 1 / π i 1 .
τ j = ( Δ 2 ( 1 ) ( λ ) τ i f i ( λ ) ) f i ( λ ) d λ ,
ξ = [ τ 1 τ 2 τ 3 ] · [ X / X 0 Y / Y 0 Z / Z 0 ] + [ τ 1 τ 2 τ 3 ] A 1 [ X / X 0 ρ 0 ( λ e ) Y / Y 0 ρ 0 ( λ e ) Z / Z 0 ρ 0 ( λ e ) ] .
ξ = 0.5088 X X E + 1.0488 Y Y E + 0.100 Z Z E for source E = 0.4113 X X 0 + 1.3256 Y Y 0 + 0.0857 Z Z E for source D 65 = 0.7677 X X 0 + 1.6588 Y Y 0 + 0.1089 Z Z E for source A .
( λ e λ b ) S ( λ ) z ¯ ( λ ) Z 0 = ( λ λ b λ e λ b ) α p = 0 a p p ! ( λ λ b λ e λ b ) p ,
λ b λ ( λ λ λ e λ b ) m ( λ λ b λ e λ b ) p + α d λ λ e λ b = m ! Γ ( p + α + 1 ) Γ ( m + p + α + 2 ) × ( λ λ b λ e λ b ) m + p + α + 1 ,
f 1 ( m ) ( λ ) = ( λ λ b λ e λ b ) m + α k = 0 Γ ( p + α + 1 ) a p Γ ( m + p + α + 1 ) p ! ( λ λ b λ e λ b ) p .
f 1 ( m ) ( λ ) = ( λ λ b λ e λ b ) m + α k = 0 ( λ λ e λ e λ b ) k × p = k ( p k ) Γ ( p + α + 1 ) a p Γ ( m + p + α + 1 ) p ! .
Γ ( m + p + α + 1 ) Γ ( m + p + α + 2 ) = 1 m + p + α + 1 ,
A 3 ~ a 1 a 0 = λ e λ b 2 ( 2 S ( λ b ) d S ( λ b ) d λ + 1 z ¯ ( λ b ) d z ¯ ( λ b ) d λ ) ,
Δ 2 ( λ ) = | x ¯ ( λ ) x ¯ ( λ e ) x ¯ ( λ e ) y ¯ ( λ ) y ¯ ( λ e ) y ¯ ( λ e ) z ¯ ( λ ) z ¯ ( λ e ) z ¯ ( λ e ) | | X E x ¯ ( λ b ) x ¯ ( λ e ) Y E y ¯ ( λ b ) y ¯ ( λ e ) Z E z ¯ ( λ b ) z ¯ ( λ e ) | 1 = x ¯ ( λ ) + y ¯ ( λ ) + z ¯ ( λ ) X E + Y E + Z E | x s ( λ ) x s ( λ b ) x s ( λ e ) x s ( λ ) y s ( λ b ) y s ( λ e ) 1 1 1 | × | x E x s ( λ b ) x s ( λ e ) y E y s ( λ b ) y s ( λ e ) 1 1 1 | 1 .
Δ 1 ( λ ) = | x ¯ ( λ ) x ¯ ( λ b ) d x ¯ ( λ b ) / d ( λ ) y ¯ ( λ ) y ¯ ( λ b ) d y ¯ ( λ b ) / d ( λ ) z ¯ ( λ ) z ¯ ( λ b ) d z ¯ ( λ b ) / d ( λ ) | × | X E x ¯ ( λ b ) d x ¯ ( λ b ) / d ( λ ) Y E y ¯ ( λ b ) d y ¯ ( λ b ) / d ( λ ) Z E z ¯ ( λ b ) d z ¯ ( λ b ) / d ( λ ) | 1 = x ¯ ( λ ) + y ¯ ( λ ) + z ¯ ( λ ) X E + Y E + Z E | x s ( λ ) x s ( λ b ) d x s ( λ b ) / d ( λ ) x s ( λ ) y s ( λ b ) d y s ( λ b ) / d ( λ ) 1 1 0 | × | x E x s ( λ b ) d x s ( λ b ) / d ( λ ) y E y s ( λ b ) d y s ( λ b ) / d ( λ ) 1 1 0 | 1
Δ 3 ( λ ) = | x ¯ ( λ ) x ¯ ( λ e ) d x ¯ ( λ e ) / d ( λ ) y ¯ ( λ ) y ¯ ( λ e ) d y ¯ ( λ e ) / d ( λ ) z ¯ ( λ ) z ¯ ( λ e ) d z ¯ ( λ e ) / d ( λ ) | × | X E x ¯ ( λ e ) d x ¯ ( λ e ) / d ( λ ) Y E y ¯ ( λ e ) d y ¯ ( λ e ) / d ( λ ) Z E z ¯ ( λ e ) d z ¯ ( λ e ) / d ( λ ) | 1 .
Δ 3 ( λ ) ~ z ¯ ( λ ) / Z E .
Δ 1 ( λ ) ~ y ¯ ( λ ) / Y E .
ρ ( λ ) S ( λ ) x ( λ ) d λ = C S ( λ ) x ( λ ) d λ ,
[ ρ ( λ ) C ] x ( λ ) = 0
d r 1 d λ d r 2 d λ d λ = r 1 ( λ ) d r 2 d λ λ e | λ b r 1 ( λ ) d 2 r 2 d λ 2 d λ .
d r 1 d λ d r 2 d λ d λ = r 2 ( λ b ) d r 2 ( λ b ) d λ .
d r 1 d λ d r 2 d λ d λ = r 2 ( λ e ) d r 1 ( λ e ) d λ .
( d r 1 d λ ) 2 d λ = r 1 ( λ e ) d r 1 ( λ e ) d λ = d r 1 ( λ e ) d λ > 0 , ( d r 2 d λ ) 2 d λ = r 2 ( λ b ) d r 2 ( λ b ) d λ = d r 2 ( λ b ) d λ > 0 .
( d r 1 d λ d r 2 d λ d λ ) 2 = r 2 ( λ b ) r 2 ( λ e ) ( d r 1 d λ ) 2 d λ ( d r 2 d λ ) 2 d λ .

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