Abstract

The almost-illuminant-independent achromatic variable ξ [J. Opt. Soc. Am A 11, 1003 (1994)] is supplemented by two achromatic variables, both almost illuminant-independent. The results are numerically verified and obtained by means of so-called (human) visual system-response functions, defined as those linear combinations of the color-matching functions that constitute the best possible approximations to δ functions. It is argued that they and not the cone sensitivities are basic to understanding the visual system as a color-constant signal detection system. The three variables solve the color-constancy problem and define a chromatic adaptation transform for blackbody radiators of temperature T. The generalization of the results to arbitrary well-behaved illuminants is discussed.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1982).
  2. C. van Trigt, “Smoothest reflectance functions. I. Definition and main results,” J. Opt. Soc. Am. A 7, 1891–1904 (1990).
    [CrossRef]
  3. C. van Trigt, “Smoothest reflectance functions. II. Complete results,” J. Opt. Soc. Am. A 7, 2208–2222 (1990).
    [CrossRef]
  4. C. van Trigt, “Metameric blacks and estimating reflectance,” J. Opt. Soc. Am. A 11, 1003–1024 (1994).
    [CrossRef]
  5. G. Wyszecki, “Valenzmetrische Untersuchung des Zusammenhanges zwischen normaler und anomaler Trichromasie,” Die Farbe 2, 39–52 (1953).
  6. P. Moon, D. E. Spencer, “Polynomial representations of reflectance curves,” J. Opt. Soc. Am. 35, 597–600 (1945).
    [CrossRef]
  7. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Physchon. Sci. 1, 369–370 (1964).
  8. J. P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [CrossRef]
  9. D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vision 5, 5–36 (1990).
    [CrossRef]
  10. M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1683 (1986).
    [CrossRef] [PubMed]
  11. 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]
  12. 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]
  13. D. H. Marimont, B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913 (1992).
    [CrossRef] [PubMed]
  14. M. D’Zmura, G. Iverson, “Color constancy. I. Basic theory of two-stage linear recovery of spectral descriptions for lights and surfaces,” J. Opt. Soc. Am. A 10, 2148–2165 (1993).
    [CrossRef]
  15. J. M. Troost, C. M. M. de Weert, “Techniques for simulating object color under changing illuminant conditions on electronic displays,” Color Res. Appl. 17, 316–327 (1992).
    [CrossRef]
  16. R. S. Berns, Color Constant Extensions of the Munsell Book of Color (Rensselaer Polytechnic Institute, Troy, N.Y., 1983).
  17. J. L. Dannemiller, “Spectral reflectance of natural objects: how many basis functions are necessary?” J. Opt. Soc. Am. A 9, 507–515 (1992).
    [CrossRef]
  18. G. West, M. H. Brill, “Necessary and sufficient conditions for von Kries chromatic adaptation to give colour constancy,” J. Math. Biol. 15, 249–258 (1982).
    [CrossRef]
  19. N. Y. Vilenkin, Functional Analysis (Wolters-Noordhof, Groningen, The Netherlands, 1972).
  20. W. A. Thornton, “Matching lights, metamers and human visual response,” J. Color Appearance 2, 23–29 (1973).
  21. W. A. Thornton, “Evidence for the three spectral responses of the normal human visual system,” Color Res. Appl. 11, 160–163 (1986).
    [CrossRef]
  22. N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977).
    [CrossRef]
  23. N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987).
    [CrossRef]
  24. R. G. Kuehni, “Peaks of operation of the human color vision system and metamers,” Color Res. Appl. 19, 390–392 (1994).
    [CrossRef]
  25. J. J. Vos, O. Estevez, P. L. Walraven, “Improved color fundamentals offer a new view on photometric additivity,” Vis. Res. 30, 937–943 (1990).
    [CrossRef] [PubMed]
  26. V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
    [CrossRef] [PubMed]
  27. J. A. C. Yule, Principles of Color Reproduction (Wiley, New York, 1967).
  28. G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
    [CrossRef]
  29. C. van Trigt, “Color video system with illuminant-independent properties,” international patent applicationPCT/NL94/00049 (February28, 1994).
  30. W. N. Sproson, Colour Science in Television and Display Systems (Hilger, Bristol, UK, 1983).
  31. C. van Trigt, “Estimating reflectance: how to discount the illuminant,” Die Farbe 40, 9–24 (1994).
  32. D. B. Judd, D. L. MacAdam, G. W. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031–1040 (1964).
    [CrossRef]
  33. J. L. Dannemiller, “Computational approaches to color constancy: adaptive and ontogenetic considerations,” Psychol. Rev. 96, 255–266 (1989).
    [CrossRef] [PubMed]
  34. J. Troost, Ch. de Weert, “Surface reflectance and human color constancy: comment on Dannemiller (1989),” Psychol. Rev. 98, 143–145 (1991).
    [CrossRef]
  35. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1962).
  36. G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
    [CrossRef]
  37. J. J. McCann, “The role of simple nonlinear operations in modeling human lightness and color sensations,” in Human Vision, Visual Processing, and Digital Display, B. E. Rogowitz, ed., Proc. SPIE1077, 355–363 (1989).
    [CrossRef]
  38. E. Beckenbach, R. Bellman, Inequalities (Springer-Verlag, Berlin, 1961).

1994 (4)

C. van Trigt, “Metameric blacks and estimating reflectance,” J. Opt. Soc. Am. A 11, 1003–1024 (1994).
[CrossRef]

R. G. Kuehni, “Peaks of operation of the human color vision system and metamers,” Color Res. Appl. 19, 390–392 (1994).
[CrossRef]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
[CrossRef]

C. van Trigt, “Estimating reflectance: how to discount the illuminant,” Die Farbe 40, 9–24 (1994).

1993 (1)

1992 (3)

1991 (1)

J. Troost, Ch. de Weert, “Surface reflectance and human color constancy: comment on Dannemiller (1989),” Psychol. Rev. 98, 143–145 (1991).
[CrossRef]

1990 (4)

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

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]

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

1989 (2)

J. P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
[CrossRef]

J. L. Dannemiller, “Computational approaches to color constancy: adaptive and ontogenetic considerations,” Psychol. Rev. 96, 255–266 (1989).
[CrossRef] [PubMed]

1987 (1)

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

1986 (4)

1982 (1)

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

1980 (1)

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

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

1973 (1)

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

1964 (2)

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

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

1953 (1)

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

1945 (1)

Beckenbach, E.

E. Beckenbach, R. Bellman, Inequalities (Springer-Verlag, Berlin, 1961).

Bellman, R.

E. Beckenbach, R. Bellman, Inequalities (Springer-Verlag, Berlin, 1961).

Berns, R. S.

R. S. Berns, Color Constant Extensions of the Munsell Book of Color (Rensselaer Polytechnic Institute, Troy, N.Y., 1983).

Brill, M. H.

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

Buchsbaum, G.

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

Cohen, J.

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

D’Zmura, M.

Dannemiller, J. L.

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. L. Dannemiller, “Computational approaches to color constancy: adaptive and ontogenetic considerations,” Psychol. Rev. 96, 255–266 (1989).
[CrossRef] [PubMed]

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,” Color Res. Appl. 17, 316–327 (1992).
[CrossRef]

de Weert, Ch.

J. Troost, Ch. de Weert, “Surface reflectance and human color constancy: comment on Dannemiller (1989),” Psychol. Rev. 98, 143–145 (1991).
[CrossRef]

Drew, M. S.

Estevez, O.

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

Finlayson, G. D.

Forsyth, D. A.

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

Funt, B. V.

Hallikainen, J.

Iverson, G.

Jaaskelainen, T.

Judd, D. B.

Kuehni, R. G.

R. G. Kuehni, “Peaks of operation of the human color vision system and metamers,” Color Res. Appl. 19, 390–392 (1994).
[CrossRef]

Lennie, P.

MacAdam, D. L.

Maloney, L. T.

Marimont, D. H.

McCann, J. J.

J. J. McCann, “The role of simple nonlinear operations in modeling human lightness and color sensations,” in Human Vision, Visual Processing, and Digital Display, B. E. Rogowitz, ed., Proc. SPIE1077, 355–363 (1989).
[CrossRef]

Moon, P.

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]

Parkkinen, J. P. S.

Pokorny, J.

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Smith, V. C.

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Spencer, D. E.

Sproson, W. N.

W. N. Sproson, Colour Science in Television and Display Systems (Hilger, Bristol, UK, 1983).

Stiles, W. S.

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

Thornton, W. A.

W. A. Thornton, “Evidence for the three spectral responses of the normal human visual system,” Color Res. Appl. 11, 160–163 (1986).
[CrossRef]

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

Troost, J.

J. Troost, Ch. de Weert, “Surface reflectance and human color constancy: comment on Dannemiller (1989),” Psychol. Rev. 98, 143–145 (1991).
[CrossRef]

Troost, J. M.

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

van Trigt, C.

Vilenkin, N. Y.

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

Vos, J. J.

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

Walraven, P. L.

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

Wandell, B. A.

Watson, G. N.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1962).

West, G.

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

Whittaker, E. T.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1962).

Wyszecki, G.

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

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

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

Wyszecki, G. W.

Yule, J. A. C.

J. A. C. Yule, Principles of Color Reproduction (Wiley, New York, 1967).

Color Res. Appl. (5)

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

W. A. Thornton, “Evidence for the three spectral responses of the normal human visual system,” Color Res. Appl. 11, 160–163 (1986).
[CrossRef]

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

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

R. G. Kuehni, “Peaks of operation of the human color vision system and metamers,” Color Res. Appl. 19, 390–392 (1994).
[CrossRef]

Die Farbe (2)

C. van Trigt, “Estimating reflectance: how to discount the illuminant,” Die Farbe 40, 9–24 (1994).

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

Int. J. Comput. Vision (1)

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

J. Color Appearance (1)

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

J. Franklin Inst. (1)

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

J. Math. Biol. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (11)

G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
[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]

C. van Trigt, “Metameric blacks and estimating reflectance,” J. Opt. Soc. Am. A 11, 1003–1024 (1994).
[CrossRef]

M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1683 (1986).
[CrossRef] [PubMed]

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]

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]

D. H. Marimont, B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913 (1992).
[CrossRef] [PubMed]

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

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. P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
[CrossRef]

Physchon. Sci. (1)

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

Psychol. Rev. (2)

J. L. Dannemiller, “Computational approaches to color constancy: adaptive and ontogenetic considerations,” Psychol. Rev. 96, 255–266 (1989).
[CrossRef] [PubMed]

J. Troost, Ch. de Weert, “Surface reflectance and human color constancy: comment on Dannemiller (1989),” Psychol. Rev. 98, 143–145 (1991).
[CrossRef]

Vis. Res. (1)

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

Vision Res. (1)

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Other (9)

J. A. C. Yule, Principles of Color Reproduction (Wiley, New York, 1967).

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1962).

C. van Trigt, “Color video system with illuminant-independent properties,” international patent applicationPCT/NL94/00049 (February28, 1994).

W. N. Sproson, Colour Science in Television and Display Systems (Hilger, Bristol, UK, 1983).

J. J. McCann, “The role of simple nonlinear operations in modeling human lightness and color sensations,” in Human Vision, Visual Processing, and Digital Display, B. E. Rogowitz, ed., Proc. SPIE1077, 355–363 (1989).
[CrossRef]

E. Beckenbach, R. Bellman, Inequalities (Springer-Verlag, Berlin, 1961).

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

R. S. Berns, Color Constant Extensions of the Munsell Book of Color (Rensselaer Polytechnic Institute, Troy, N.Y., 1983).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

McAdam limits for the values of ξ indicated (solid curves) compared with those for the values 0.8 and 0.4 of Y/YE (dashed curves). The achromatic variable ξ is zero on the purple line. The two curves joining the points on the McAdam limits with discontinuous derivatives are the left and right branches of the boundary locus. Predictions of lightness on the basis of ξ and luminance agree for yellowish colors and disagree for bluish ones (Helmholtz–Kohlrausch effect). The line is for ξ=Y/YE. The two lines tangential to the full figures at their intersections with the left (right) branch of the boundary locus intersect the spectrum locus at λb and λe. The proof is nontrivial.

Fig. 2
Fig. 2

Min(λ0)/Min(λe) (curve 1) and d[Min(λ0)]/dλ0 (curve 2) as functions of λ0.

Fig. 3
Fig. 3

Visual system-response functions Pi(λ), i=1, 2, 3, defined by Eq. (9), normalized to unity at their maxima. Peaks (widths at half-maximum) are at 445 nm (52 nm), 538 nm (74 nm), and 603 nm (68 nm).

Tables (1)

Tables Icon

Table 1 Values of Achromatic Variables

Equations (58)

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

 ρ(λ)S(λ)x¯(λ)dλ=X,
 ρ(λ)S(λ)y¯(λ)dλ=Y,
 ρ(λ)S(λ)z¯(λ)dλ=Z.
ρ(λ)=ρ0(λ)+k=1 ckrk(λ)
ρ(λ)P(λ)dλ=ρ0(λ)P(λ)dλ+k=1ckrk(λ)P(λ)dλ.
ξ= ρ0(λ)Pa(λ)dλ ρ(λ)Pa(λ)dλ=a1XX0+a2YY0+a3ZZ0, aj=1.
Pa(λ)=τ1x¯(λ)XE+τ2y¯(λ)YE+τ3z¯(λ)ZEE(λ),τj=1.
ξ= ρ(λ)S(λ)A(λ)dλ.
ξi= ρ0(λ)Pi(λ)dλ ρ(λ)Pi(λ)dλ=a1(λi)XX0+a2(λi)YY0a3(λi)ZZ0,aj(λi)=1.
Pi(λ)=τ1(λi)x¯(λ)XE+τ2(λi)y¯(λ)YE+τ3(λi)z¯(λ)ZEE(λ),
τj(λi)=1
S(λ)=S0(λ)+E(λ) n=1 cnrn(λ).
fˆ1(λ)=λbλ E(λ)x¯(λ)XEdλ
P(λ0; λ)=τ1(λ0)x¯(λ)XE+τ2(λ0)y¯(λ)YE+τ3(λ0)z¯(λ)ZEE(λ),
 τj(λ0)=1.
P(1)(λ0; λ)=λbλ P(λ0; λ)dλ=j=13 τj(λ0)fˆj(λ),
P(1)(λ0; λe)=1.
λbλ[P(λ0; λ)-δ(λ-λ0)]dλ=P(1)(λ0; λ)-H(λ0; λ).
H(λ0; λ)=0,1,λ<λ0λ>λ0.
[P(1)(λ0; λ)-H(λ0; λ)]2 dλ=minimal.
Aτ1(λ0)τ2(λ0)τ3(λ0)=γ(λ0)111+h1(λ0)h2(λ0)h3(λ0).
ai,j= fˆi(λ)fˆj(λ)dλ,hi(λ0)=λ0λe fˆi(λ)dλ.
γ(λ0) j=13 πj1=1-j=13 πj1hj(λ0)=r1(λ0),
τ1(λ0)τ2(λ0)τ3(λo)=1 πj1π11π21π31
τ1(λ0)τ2(λ0)τ3(λ0)=1 πj2π12π22π32
Min(λ0)=γ(λ0)+λe-λ0-j=13 τj(λ0)hj(λ0).
Min(λ0)=1i=13πi1
Min(λ0)=1i=13πi2
d[Min(λ0)]dλ0=2 j=13 τj(λ0)fˆj(λ0)-1=0.
P1(λ)P2(λ)P3(λ)= 0.2183-1.2313 1.9085-0.3107 2.1514-0.6282 1.0924 0.0799-0.2803×x¯(λ)E(λ)/XEy¯(λ)E(λ)/YEz¯(λ)E(λ)/ZE.
[P(1)(λ0; λ)-H(λ0; λ)]dλ=(λ0-λ)P(λ0; λ)dλ=r1(λ0) πj1-r2(λ0)πj2.
rk(λ)=rk(λ0)+(λ-λ0)drk(λ0)dλ+λ0λ(λ-λ)d2rkdλ2dλ.
rk(λ)P(λ0; λ)dλ=rk(λ0)+drk(λ0)dλ×(λ-λ0)P(λ0; λ)dλ+d2rkdλ2Pˆ(λ)dλ,
Pˆ(λ)=λbλ(λ-λ)P(λ0; λ)dλ;λbλ<λ0λλe(λ-λ)P(λ0; λ)dλ,λ0<λλe.
Pˆ(λ)=Min(λ0)-r2(λ0) πj2+(λ-λ0)P(1)(λ0; λ0)+,
Pˆ(λ)=Min(λ0)-r1(λ0) πj1+(λ-λ0)[P(1)(λ0; λ0)-1]+.
ξi=ρ0(λ)Pi(λ)dλρ(λ)Pi(λ)dλ=a1(λi)XX0+a2(λi)YY0+a3(λi)ZZ0,aj(λi)=1.
ξ1ξ2ξ3= 0.2329-1.1304 2.0309-0.3331 2.0565-0.6952 1.1002 0.0739-0.3357X/X0Y/Y0Z/Z0
ξ1ξ2ξ3= 0.4046-1.5075 1.5106-0.5899 2.4825-0.4441 1.1853 0.0250-0.0665X/X0Y/Y0Z/Z0
Pa(λ)=0.0873P1(λ)+0.7229P2(λ)+0.1898P3(λ).
P0(λ)=P1(λe)P2(λ)-P2(λe)P1(λ)P1(λe)-P2(λe)=0.1893P2(λ)+0.8107P1(λ),
P4(λ)=P3(λb)P2(λ)-P2(λb)P3(λ)P3(λb)-P2(λb)=0.4231P2(λ)+0.5769P3(λ),
P(1)(λ0; λ)=j=13 τj(λ0)fj(λ)=1-j=13 τj(λ0)gj(λ),
 τj(λ0)gj(λ)-Hx(λ0; λ)2 dλ=minimal,
Hx(λ0; λ)=1-H(λ0; λ)=1,0,λ<λ0λ>λ0.
Bτ1(λ0)τ2(λ0)τ3(λ0)=δ(λ0)111+h1x(λ0)h2x(λ0)h3x(λ0),
bi,j= gi(λ)gj(λ)dλ,hjx(λ0)=λbλ0 gj(λ)dλ.
δ(λ0) πj2=1- πj2hjx(λ0)=r2(λ0),
Min(λ0)=δ(λ0)+λ0-λb- τj(λ0)hjx(λ0).
[P(1)(λ0; λ)-H(λ0; λ)]dλ=[Min(λ0)-δ(λ0)]-[Min(λ0)-γ(λ0)],
Bπ12π22π32=111=Aπ11π21π31.
 πj1=i,j ai,j-19i,j ai,j-1>(λe-λb)-1.
dfidλ0τj(λ0)dλ0=δi,j,
 P(λ0; λ)P(λ;λ)dλ=P(λ0; λ).
ξi=ρ0(λe)+ μj  Pi(1)(λ)fj(λ)dλ,
ρ0(λe)=ν1XX0+ν2YY0+ν3ZZ0.
τj(λi)=Pi(1)(λ)-k=13 τk(λi)fk(λ)fj(λ)dλ,
ξi=τ1(λi)τ2(λi)τ3(λi)·X/X0Y/Y0Z/Z0+A-1τ1(λi)τ2(λi)τ3(λi)·X/X0-ρ0(λe)Y/Y0-ρ0(λe)Z/Z0-ρ0(λe).

Metrics