Abstract

We present an approach to estimating the reflectance of a surface given its camera response. In this approach we first solve the general form of this problem: We calculate the set of all possible surface reflectances, called the metamer set, and then choose a member from this set. Three possibilities in choosing a single reflectance are described here. First, we assume that all reflectances are equally likely and minimize worst-case error. Second, we adopt the assumption that reflectances follow a normal probability distribution and maximize this probability. Finally, we assume that reflectances are smooth and maximize this property. The results of our experiments show that there is significant benefit from the proposed approach in terms of the accuracy of the estimation compared with that of standard estimation methods. Moreover, the present approach introduces a notion of robustness of estimates in the form of error bars.

© 2006 Optical Society of America

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References

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  1. L. T. Maloney and B. A. Wandell, "Color constancy: a method for recovering surface spectral reflectance," J. Opt. Soc. Am. A 3, 29-33 (1986).
    [CrossRef] [PubMed]
  2. S. M. Newhall, D. Nickerson, and D. B. Judd, "Final Report of the OSA sub-committee on the spacing of the Munsell colors," J. Opt. Soc. Am. 33, 385-418 (1943).
    [CrossRef]
  3. G. Wyszecki, "Evaluation of metameric colors," J. Opt. Soc. Am. 48, 451-454 (1958).
    [CrossRef]
  4. C. van Trigt, "Smoothest reflectance functions. I. Definitions and main results," J. Opt. Soc. Am. A 7, 1891-1904 (1990).
    [CrossRef]
  5. J. A. Worthey, "Spectrally smooth reflectances that match," Color Res. Appl. 19, 395-396 (1994).
    [CrossRef]
  6. E. H. Land, "The retinex theory of color vision," Sci. Am. 237, 108-128 (1977).
    [CrossRef] [PubMed]
  7. G. D. Finlayson and P. Morovic, "Metamer sets," J. Opt. Soc. Am. A 22, 810-819 (2005).
    [CrossRef]
  8. 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]
  9. X. Gonzalez, Image Processing Handbook (Addison Wesley, 1990).
  10. B. Smith, C. Spiekermann, and R. Sember, "Numerical methods for colorimetric calculations: sampling density requirements," Color Res. Appl. 17, 394-401 (1992).
    [CrossRef]
  11. J. Parkkinen, T. Jaaskelainen, and M. Kuttinen, "Spectral representation of color images," in IEEE 9th International Conference on Pattern Recognition (1988), Vol. 2, pp. 933-935.
  12. R. Gershon, A. D. Jepson, and J. K. Tsotsos, "From [R,G,B] to surface reflectance: computing color constant descriptors in images," Percept. Psychophys. 17, 755-758 (1988).
  13. J. J. Dannemiller, "Spectral reflectance of natural objects: how many basis functions are necessary?" J. Opt. Soc. Am. A 9, 507-515 (1992).
    [CrossRef]
  14. D. H. Marimont and B. A. Wandell, "Linear models of surface and illuminant spectra," J. Opt. Soc. Am. A 11, 1905-1913 (1992).
    [CrossRef]
  15. H. Cramer, Mathematical Methods of Statistics (Princeton U. Press, 1999).
  16. J. C. Maxwell, "Experiments on colour, as perceived by the eye, with remarks on colour-blindness," Trans. R. Soc. Edinbrgh 21, 275-298 (1855).
  17. T. Katrinak, M. Galvanec, E. Gedeonova, and J. Smital, Algebra a Teoreticka Aritmetika (Algebra and Theoretical Arithmetic) (Alfa Bratislava, Czechoslovakia, 1985).
  18. P. Morovic, "Metamer sets," Ph.D. thesis (University of East Anglia, School of Information Systems, 2002).
  19. R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis (Wiley, 1973).
  20. W. T. Freeman and D. H. Brainard, "Bayesian color constancy," J. Opt. Soc. Am. A 14, 1393-1411 (1997).
    [CrossRef]
  21. A. L. Peressini, F. E. Sullivan, and J. J. J. Uhl, The Mathematics of Nonlinear Programming (Springer-Verlag, 1991).
  22. K. Takahama and Y. Nayatani, "New method for generating metameric stimuli of object colors," J. Opt. Soc. Am. 62, 1516-1520 (1972).
    [CrossRef]
  23. N. Ohta, "Generating metameric object colors," J. Opt. Soc. Am. 65, 1081-1082 (1975), Letter to the Editor.
    [CrossRef]
  24. M. S. Drew and B. V. Funt, "Natural metamers," CVGIP: Image Understand. 56, 139-151 (1992).
    [CrossRef]
  25. C. van Trigt, "Smoothest reflectance functions. II. Complete results," J. Opt. Soc. Am. A 7, 1891-1904 (1990).
    [CrossRef]
  26. W. L. Brewer and J. F. R. Holly, "Syntheses of spectral-distribution curves," J. Opt. Soc. Am. 38, 858-874 (1948).
    [CrossRef] [PubMed]
  27. N. Shimano, "Illuminant invariant recovery of surface reflectances and estimation of colorimetric values from image signals," Opt. Rev. 4, 358-361 (1997).
    [CrossRef]
  28. N. Shimano, "Recovery of surface reflectances using video camera image signals under unknown illuminations," Opt. Rev. 4, 707-712 (1997).
    [CrossRef]
  29. R. Schettini, "Deriving spectral reflectance functions of computer-simulated object colours," Comput. Graph. Forum 13, 211-217 (1994).
    [CrossRef]
  30. R. Schettini and B. Banolo, "Estimating reflectance functions from tristimulus values," Applied Signal Processing 3, 104-115 (1996).
  31. G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed. (The Johns Hopkins U. Press, 1996).
  32. C. Li and M. R. Luo, "The estimation of spectral reflectances using the smoothest constraint condition," in Proceedings of the IS&T/SID 9th Color Imaging Conference (Society for Imaging Science and Technology, 2001), pp. 62-67.
  33. J. P. S. Parkkinen, J. Hallikanen, and T. Jaaskelainen, "Characteristic spectra of Munsell colors," J. Opt. Soc. Am. A 6, 318-322 (1989).
    [CrossRef]
  34. M. J. Vrhel, R. Gershon, and L. S. Iwan, "Measurement and analysis of object reflectance spectra," Color Res. Appl. 19, 4-9 (1994).
  35. C. S. McCamy, H. Marcus, and J. G. Davidson, "A color-rendition chart," J. Appl. Photogr. Eng. 2, 95-99 (1976).

2005 (1)

1997 (3)

W. T. Freeman and D. H. Brainard, "Bayesian color constancy," J. Opt. Soc. Am. A 14, 1393-1411 (1997).
[CrossRef]

N. Shimano, "Illuminant invariant recovery of surface reflectances and estimation of colorimetric values from image signals," Opt. Rev. 4, 358-361 (1997).
[CrossRef]

N. Shimano, "Recovery of surface reflectances using video camera image signals under unknown illuminations," Opt. Rev. 4, 707-712 (1997).
[CrossRef]

1996 (1)

R. Schettini and B. Banolo, "Estimating reflectance functions from tristimulus values," Applied Signal Processing 3, 104-115 (1996).

1994 (3)

R. Schettini, "Deriving spectral reflectance functions of computer-simulated object colours," Comput. Graph. Forum 13, 211-217 (1994).
[CrossRef]

M. J. Vrhel, R. Gershon, and L. S. Iwan, "Measurement and analysis of object reflectance spectra," Color Res. Appl. 19, 4-9 (1994).

J. A. Worthey, "Spectrally smooth reflectances that match," Color Res. Appl. 19, 395-396 (1994).
[CrossRef]

1992 (4)

M. S. Drew and B. V. Funt, "Natural metamers," CVGIP: Image Understand. 56, 139-151 (1992).
[CrossRef]

B. Smith, C. Spiekermann, and R. Sember, "Numerical methods for colorimetric calculations: sampling density requirements," Color Res. Appl. 17, 394-401 (1992).
[CrossRef]

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

D. H. Marimont and B. A. Wandell, "Linear models of surface and illuminant spectra," J. Opt. Soc. Am. A 11, 1905-1913 (1992).
[CrossRef]

1990 (2)

1989 (1)

1988 (1)

R. Gershon, A. D. Jepson, and J. K. Tsotsos, "From [R,G,B] to surface reflectance: computing color constant descriptors in images," Percept. Psychophys. 17, 755-758 (1988).

1986 (2)

1977 (1)

E. H. Land, "The retinex theory of color vision," Sci. Am. 237, 108-128 (1977).
[CrossRef] [PubMed]

1976 (1)

C. S. McCamy, H. Marcus, and J. G. Davidson, "A color-rendition chart," J. Appl. Photogr. Eng. 2, 95-99 (1976).

1975 (1)

1972 (1)

1958 (1)

1948 (1)

1943 (1)

1855 (1)

J. C. Maxwell, "Experiments on colour, as perceived by the eye, with remarks on colour-blindness," Trans. R. Soc. Edinbrgh 21, 275-298 (1855).

Banolo, B.

R. Schettini and B. Banolo, "Estimating reflectance functions from tristimulus values," Applied Signal Processing 3, 104-115 (1996).

Brainard, D. H.

Brewer, W. L.

Cramer, H.

H. Cramer, Mathematical Methods of Statistics (Princeton U. Press, 1999).

Dannemiller, J. J.

Davidson, J. G.

C. S. McCamy, H. Marcus, and J. G. Davidson, "A color-rendition chart," J. Appl. Photogr. Eng. 2, 95-99 (1976).

Drew, M. S.

M. S. Drew and B. V. Funt, "Natural metamers," CVGIP: Image Understand. 56, 139-151 (1992).
[CrossRef]

Duda, R. O.

R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis (Wiley, 1973).

Finlayson, G. D.

Freeman, W. T.

Funt, B. V.

M. S. Drew and B. V. Funt, "Natural metamers," CVGIP: Image Understand. 56, 139-151 (1992).
[CrossRef]

Galvanec, M.

T. Katrinak, M. Galvanec, E. Gedeonova, and J. Smital, Algebra a Teoreticka Aritmetika (Algebra and Theoretical Arithmetic) (Alfa Bratislava, Czechoslovakia, 1985).

Gedeonova, E.

T. Katrinak, M. Galvanec, E. Gedeonova, and J. Smital, Algebra a Teoreticka Aritmetika (Algebra and Theoretical Arithmetic) (Alfa Bratislava, Czechoslovakia, 1985).

Gershon, R.

M. J. Vrhel, R. Gershon, and L. S. Iwan, "Measurement and analysis of object reflectance spectra," Color Res. Appl. 19, 4-9 (1994).

R. Gershon, A. D. Jepson, and J. K. Tsotsos, "From [R,G,B] to surface reflectance: computing color constant descriptors in images," Percept. Psychophys. 17, 755-758 (1988).

Golub, G. H.

G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed. (The Johns Hopkins U. Press, 1996).

Gonzalez, X.

X. Gonzalez, Image Processing Handbook (Addison Wesley, 1990).

Hallikanen, J.

Hart, P. E.

R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis (Wiley, 1973).

Holly, J. F. R.

Iwan, L. S.

M. J. Vrhel, R. Gershon, and L. S. Iwan, "Measurement and analysis of object reflectance spectra," Color Res. Appl. 19, 4-9 (1994).

Jaaskelainen, T.

J. P. S. Parkkinen, J. Hallikanen, and T. Jaaskelainen, "Characteristic spectra of Munsell colors," J. Opt. Soc. Am. A 6, 318-322 (1989).
[CrossRef]

J. Parkkinen, T. Jaaskelainen, and M. Kuttinen, "Spectral representation of color images," in IEEE 9th International Conference on Pattern Recognition (1988), Vol. 2, pp. 933-935.

Jepson, A. D.

R. Gershon, A. D. Jepson, and J. K. Tsotsos, "From [R,G,B] to surface reflectance: computing color constant descriptors in images," Percept. Psychophys. 17, 755-758 (1988).

Judd, D. B.

Katrinak, T.

T. Katrinak, M. Galvanec, E. Gedeonova, and J. Smital, Algebra a Teoreticka Aritmetika (Algebra and Theoretical Arithmetic) (Alfa Bratislava, Czechoslovakia, 1985).

Kuttinen, M.

J. Parkkinen, T. Jaaskelainen, and M. Kuttinen, "Spectral representation of color images," in IEEE 9th International Conference on Pattern Recognition (1988), Vol. 2, pp. 933-935.

Land, E. H.

E. H. Land, "The retinex theory of color vision," Sci. Am. 237, 108-128 (1977).
[CrossRef] [PubMed]

Li, C.

C. Li and M. R. Luo, "The estimation of spectral reflectances using the smoothest constraint condition," in Proceedings of the IS&T/SID 9th Color Imaging Conference (Society for Imaging Science and Technology, 2001), pp. 62-67.

Luo, M. R.

C. Li and M. R. Luo, "The estimation of spectral reflectances using the smoothest constraint condition," in Proceedings of the IS&T/SID 9th Color Imaging Conference (Society for Imaging Science and Technology, 2001), pp. 62-67.

Maloney, L. T.

Marcus, H.

C. S. McCamy, H. Marcus, and J. G. Davidson, "A color-rendition chart," J. Appl. Photogr. Eng. 2, 95-99 (1976).

Marimont, D. H.

D. H. Marimont and B. A. Wandell, "Linear models of surface and illuminant spectra," J. Opt. Soc. Am. A 11, 1905-1913 (1992).
[CrossRef]

Maxwell, J. C.

J. C. Maxwell, "Experiments on colour, as perceived by the eye, with remarks on colour-blindness," Trans. R. Soc. Edinbrgh 21, 275-298 (1855).

McCamy, C. S.

C. S. McCamy, H. Marcus, and J. G. Davidson, "A color-rendition chart," J. Appl. Photogr. Eng. 2, 95-99 (1976).

Morovic, P.

G. D. Finlayson and P. Morovic, "Metamer sets," J. Opt. Soc. Am. A 22, 810-819 (2005).
[CrossRef]

P. Morovic, "Metamer sets," Ph.D. thesis (University of East Anglia, School of Information Systems, 2002).

Nayatani, Y.

Newhall, S. M.

Nickerson, D.

Ohta, N.

Parkkinen, J.

J. Parkkinen, T. Jaaskelainen, and M. Kuttinen, "Spectral representation of color images," in IEEE 9th International Conference on Pattern Recognition (1988), Vol. 2, pp. 933-935.

Parkkinen, J. P. S.

Peressini, A. L.

A. L. Peressini, F. E. Sullivan, and J. J. J. Uhl, The Mathematics of Nonlinear Programming (Springer-Verlag, 1991).

Schettini, R.

R. Schettini and B. Banolo, "Estimating reflectance functions from tristimulus values," Applied Signal Processing 3, 104-115 (1996).

R. Schettini, "Deriving spectral reflectance functions of computer-simulated object colours," Comput. Graph. Forum 13, 211-217 (1994).
[CrossRef]

Sember, R.

B. Smith, C. Spiekermann, and R. Sember, "Numerical methods for colorimetric calculations: sampling density requirements," Color Res. Appl. 17, 394-401 (1992).
[CrossRef]

Shimano, N.

N. Shimano, "Illuminant invariant recovery of surface reflectances and estimation of colorimetric values from image signals," Opt. Rev. 4, 358-361 (1997).
[CrossRef]

N. Shimano, "Recovery of surface reflectances using video camera image signals under unknown illuminations," Opt. Rev. 4, 707-712 (1997).
[CrossRef]

Smital, J.

T. Katrinak, M. Galvanec, E. Gedeonova, and J. Smital, Algebra a Teoreticka Aritmetika (Algebra and Theoretical Arithmetic) (Alfa Bratislava, Czechoslovakia, 1985).

Smith, B.

B. Smith, C. Spiekermann, and R. Sember, "Numerical methods for colorimetric calculations: sampling density requirements," Color Res. Appl. 17, 394-401 (1992).
[CrossRef]

Spiekermann, C.

B. Smith, C. Spiekermann, and R. Sember, "Numerical methods for colorimetric calculations: sampling density requirements," Color Res. Appl. 17, 394-401 (1992).
[CrossRef]

Sullivan, F. E.

A. L. Peressini, F. E. Sullivan, and J. J. J. Uhl, The Mathematics of Nonlinear Programming (Springer-Verlag, 1991).

Takahama, K.

Tsotsos, J. K.

R. Gershon, A. D. Jepson, and J. K. Tsotsos, "From [R,G,B] to surface reflectance: computing color constant descriptors in images," Percept. Psychophys. 17, 755-758 (1988).

Uhl, J. J. J.

A. L. Peressini, F. E. Sullivan, and J. J. J. Uhl, The Mathematics of Nonlinear Programming (Springer-Verlag, 1991).

van Loan, C. F.

G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed. (The Johns Hopkins U. Press, 1996).

van Trigt, C.

Vrhel, M. J.

M. J. Vrhel, R. Gershon, and L. S. Iwan, "Measurement and analysis of object reflectance spectra," Color Res. Appl. 19, 4-9 (1994).

Wandell, B. A.

D. H. Marimont and B. A. Wandell, "Linear models of surface and illuminant spectra," J. Opt. Soc. Am. A 11, 1905-1913 (1992).
[CrossRef]

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

Worthey, J. A.

J. A. Worthey, "Spectrally smooth reflectances that match," Color Res. Appl. 19, 395-396 (1994).
[CrossRef]

Wyszecki, G.

Applied Signal Processing (1)

R. Schettini and B. Banolo, "Estimating reflectance functions from tristimulus values," Applied Signal Processing 3, 104-115 (1996).

Color Res. Appl. (3)

M. J. Vrhel, R. Gershon, and L. S. Iwan, "Measurement and analysis of object reflectance spectra," Color Res. Appl. 19, 4-9 (1994).

J. A. Worthey, "Spectrally smooth reflectances that match," Color Res. Appl. 19, 395-396 (1994).
[CrossRef]

B. Smith, C. Spiekermann, and R. Sember, "Numerical methods for colorimetric calculations: sampling density requirements," Color Res. Appl. 17, 394-401 (1992).
[CrossRef]

Comput. Graph. Forum (1)

R. Schettini, "Deriving spectral reflectance functions of computer-simulated object colours," Comput. Graph. Forum 13, 211-217 (1994).
[CrossRef]

CVGIP: Image Understand. (1)

M. S. Drew and B. V. Funt, "Natural metamers," CVGIP: Image Understand. 56, 139-151 (1992).
[CrossRef]

J. Appl. Photogr. Eng. (1)

C. S. McCamy, H. Marcus, and J. G. Davidson, "A color-rendition chart," J. Appl. Photogr. Eng. 2, 95-99 (1976).

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (9)

Opt. Rev. (2)

N. Shimano, "Illuminant invariant recovery of surface reflectances and estimation of colorimetric values from image signals," Opt. Rev. 4, 358-361 (1997).
[CrossRef]

N. Shimano, "Recovery of surface reflectances using video camera image signals under unknown illuminations," Opt. Rev. 4, 707-712 (1997).
[CrossRef]

Percept. Psychophys. (1)

R. Gershon, A. D. Jepson, and J. K. Tsotsos, "From [R,G,B] to surface reflectance: computing color constant descriptors in images," Percept. Psychophys. 17, 755-758 (1988).

Sci. Am. (1)

E. H. Land, "The retinex theory of color vision," Sci. Am. 237, 108-128 (1977).
[CrossRef] [PubMed]

Trans. R. Soc. Edinbrgh (1)

J. C. Maxwell, "Experiments on colour, as perceived by the eye, with remarks on colour-blindness," Trans. R. Soc. Edinbrgh 21, 275-298 (1855).

Other (9)

T. Katrinak, M. Galvanec, E. Gedeonova, and J. Smital, Algebra a Teoreticka Aritmetika (Algebra and Theoretical Arithmetic) (Alfa Bratislava, Czechoslovakia, 1985).

P. Morovic, "Metamer sets," Ph.D. thesis (University of East Anglia, School of Information Systems, 2002).

R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis (Wiley, 1973).

H. Cramer, Mathematical Methods of Statistics (Princeton U. Press, 1999).

X. Gonzalez, Image Processing Handbook (Addison Wesley, 1990).

J. Parkkinen, T. Jaaskelainen, and M. Kuttinen, "Spectral representation of color images," in IEEE 9th International Conference on Pattern Recognition (1988), Vol. 2, pp. 933-935.

A. L. Peressini, F. E. Sullivan, and J. J. J. Uhl, The Mathematics of Nonlinear Programming (Springer-Verlag, 1991).

G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed. (The Johns Hopkins U. Press, 1996).

C. Li and M. R. Luo, "The estimation of spectral reflectances using the smoothest constraint condition," in Proceedings of the IS&T/SID 9th Color Imaging Conference (Society for Imaging Science and Technology, 2001), pp. 62-67.

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Figures (2)

Fig. 1
Fig. 1

Spectral sensitivities of the camera used in the synthetic data experiments and the spectral power distribution of illuminant D65.

Fig. 2
Fig. 2

Spectral sensitivities of the Hasselblad/MegaVision camera and the three illuminants used in the raw data experiments.

Tables (3)

Tables Icon

Table 1 Reflectance Recovery Median Δ (Top Three Rows) and Δ T (Bottom Three Rows) Statistics for the Munsell Data Set a

Tables Icon

Table 2 Reflectance Recovery Median Δ (Top Three Rows) and Δ T (Bottom Three Rows) Statistics for the Natural Data Set a

Tables Icon

Table 3 Reflectance Estimating Median Δ Results (Top Three Rows) and Median Δ T Results (Bottom Three Rows) from Raw RGB under Illuminants D50, A, and CWF a

Equations (33)

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

χ i = ω X i ( λ ) E ( λ ) R ( λ ) d λ ,
χ ̱ = ω X ̱ ( λ ) E ( λ ) R ( λ ) d λ ,
λ ω , 0 R ( λ ) 1 .
r ̱ [ R ( λ 1 ) , R ( λ 2 ) , , R ( λ q ) ] .
r ̱ B σ ̱ .
w i 0 , i = 1 m w i = 1 , r ̱ = w 1 s ̱ 1 + + w m s ̱ m .
s ̱ i , A s ̱ i b ̱ .
A r ̱ b ̱
χ ̱ = X T D ( e ̱ ) B σ ̱ ,
χ ̱ = Λ σ ̱ ,
0 ̱ B σ ̱ 1 ̱ ,
A B σ ̱ b ̱ .
χ ̱ = Λ ( σ ̱ x + σ ̱ 0 ) .
χ ̱ = Λ ( σ ̱ + Ξ α ̱ ) ,
σ ̂ ̱ = min σ ̱ i M ( χ ̱ ) max σ ̱ j M ( χ ̱ ) σ ̱ i σ ̱ j .
max σ ̱ P ( σ ̱ χ ̱ ) ;
P ( σ ̱ χ ̱ ) = P ( χ ̱ σ ̱ ) P ( σ ̱ ) P ( χ ̱ ) .
P ( σ ̱ ) = ( 2 π ) k 2 Σ 1 2 exp [ 1 2 ( σ ̱ μ ̱ ) Σ 1 ( σ ̱ μ ̱ ) T ] ,
ln P ( σ ̱ ) = ln ( 2 π ) k 2 + ln Σ 1 2 1 2 ( σ ̱ μ ̱ ) Σ 1 ( σ ̱ μ ̱ ) T .
min σ ̱ ( σ ̱ μ ̱ ) Σ 1 ( σ ̱ μ ̱ ) T .
min α ̱ 2 ( σ ̱ μ ̱ ) T Σ 1 Ξ α ̱ + α ̱ T Ξ T Σ 1 Ξ α ̱ .
min α ̱ f ̱ α ̱ + α ̱ T H α ̱ .
( d r ̱ d i ) 2 .
D = [ 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 ] .
r ̱ T D T D r ̱ .
[ D B σ ̱ ] T D B σ ̱ .
σ ̱ = Λ 1 χ ̱ .
S T G 2 ,
σ ̱ = T χ ̱ .
min r ̱ r ̱ T D T D r ̱ ,
X D ( e ̱ ) r ̱ = χ ̱ 0 ̱ r ̱ 1 ̱ .
Δ = 1 q r ̂ ̱ r ̱ 2 = 1 q ( r ̂ ̱ r ̱ ) T ( r ̂ ̱ r ̱ ) ,
Δ T = 1 q max r ̱ i M ( χ ̱ ) r ̂ ̱ r ̱ i 2 = 1 q max r ̱ i M ( χ ̱ ) ( r ̂ ̱ r ̱ i ) T ( r ̂ ̱ r ̱ i ) .

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