Abstract

Recovery of spectral reflectances of objects being imaged through the use of sensor responses is important to reproduce color images under various illuminations. Although the Wiener estimation is usually used for the recovery, the recovery performance of the estimation depends on the autocorrelation matrix of the spectral reflectances and the noise present in an image acquisition system. The purpose of the present paper is to show that the Wiener estimation with the noise variance estimated by the previous proposal [IEEE Trans. Image Process. 16, 1848 (2006) ] and with the autocorrelation matrix that uses the features of the spectral reflectances recovered by the previous method is very effective in greatly improving the performance.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. D. Fairchild, Color Appearance Models (Addison-Wesley, 1997).
  2. Y. Nayatani, K. Takahama, and H. Sobagaki, “Prediction of color appearance under various adapting conditions,” Color Res. Appl. 11, 62-72 (1986).
    [CrossRef]
  3. R. W. G. Hunt, “Revised colour-appearance model for related and unrelated colours,” Color Res. Appl. 16, 146-163 (1991).
    [CrossRef]
  4. H. C. Lee, E. J. Breneman, and C. P. Schulte, “Modeling light reflection for computer color vision,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 79-86 (1990).
    [CrossRef]
  5. G. Healey and D. Slater, “Global color constancy: recognition of objects by use of illumination-invariant properties of color distributions,” J. Opt. Soc. Am. A 11, 3003-3010 (1994).
    [CrossRef]
  6. G. J. Klinker, S. A. Shafer, and T. Kanade, “A physical approach to color image understanding,” Int. J. Comput. Vis. 4, 7-38 (1990).
    [CrossRef]
  7. 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]
  8. J. Ho, B. V. Funt, and M. S. Drew, “Separating a color signal into illumination and surface reflectance components: Theory and applications,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 966-977 (1990).
    [CrossRef]
  9. G. Iverson and M. D'Zumura, “Criteria for color constancy in trichromatic linear models,” J. Opt. Soc. Am. A 11, 1970-1975 (1994).
    [CrossRef]
  10. A. Rosenfeld and A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, 1982).
  11. G. Sharma and H. J. Trussell, “Figures of merit for color scanners,” IEEE Trans. Image Process. 6, 990-1001 (1997).
    [CrossRef] [PubMed]
  12. H. Haneishi, T. Hasegawa, N. A. Hosoi, Y. Yokoyama, N. Tsumura, and Y. Miyake, “System design for accurately estimating the spectral reflectance of art paintings,” Appl. Opt. 39, 6621-6632 (2000).
    [CrossRef]
  13. J. Cohen, “Dependency of spectral reflectance curves of the Munsell color chips,” Psychonomic Sci. 1, 369-370 (1964).
  14. 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]
  15. J. P. S. Parkkinen, J. Hallikainen, and T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318-322 (1989).
    [CrossRef]
  16. M. Shi and G. Healey, “Using reflectance models for color scanner calibration,” J. Opt. Soc. Am. A 19, 645-656 (2002).
    [CrossRef]
  17. Y. Zhao, L. A. Taplin, M. Nezamabadi, and R. S. Berns, “Using the matrix R method for spectral image archives,” in Proceedings of The 10th Congress of the International Colour Association(AIC'5) (AIC, 2005), pp. 469-472.
  18. H. L. Shen and H. H. Xin, “Spectral characterization of a color scanner based on optimized adaptive estimation,” J. Opt. Soc. Am. A 23, 1566-1569 (2006).
    [CrossRef]
  19. D. Connah, J. Y. Hardeberg, and S. Westland, “Comparison of linear spectral reconstruction methods for multispectral imaging,” Proceedings IEEE International Conference on Image Processing (IEEE, 2004), pp. 1497-1500.
  20. F. H. Imai and R. S. Berns, “Spectral estimation using trichromatic digital cameras,” in Proceedings of International Symposium on Multispectral Imaging and Color Reproduction for Digital Archives, Chiba, Japan, 21-22 October 1999, pp. 42-49.
  21. N. Shimano, “Recovery of spectral reflectances of objects being imaged without prior knowledge,” IEEE Trans. Image Process. 15, 1848-1856 (2006).
    [CrossRef] [PubMed]
  22. N. Shimano, K. Terai, and M. Hironaga, “Recovery of spectral reflectances of objects being imaged by multispectral cameras,” J. Opt. Soc. Am. A 24, 3211-3219 (2007).
    [CrossRef]
  23. N. Shimano and M. Hironaga, “A new proposal for the accurate recovery of spectral reflectances of imaged objects without prior knowledge,” in Archiving 2008, Proceedings of Society of Imaging Science and Technology, Bern (IS&T, 2008), pp. 155-158.
  24. N. Shimano, “Application of a colorimetric evaluation model to multispectral color image acquisition systems,” J. Imaging Sci. Technol. 49, 588-593 (2005).
  25. N. Shimano, “Optimization of spectral sensitivities with Gaussian distribution functions for a color image acquisition device in the presence of noise,” Opt. Eng. (Bellingham) 45, 013201-1-8 (2006).
    [CrossRef]
  26. “Standard Object Colour Spectra Database for Colour Reproduction Evaluation (SOCS),” Japanese Industrial Standards Association, TR X 0012:1998 (1998).
  27. A. A. Afifi and S. P. Azen, Statistical Analysis (Academic, 1972), Chap. 3.
  28. G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (The Johns Hopkins Univ. Press, 1996), p. 55.
  29. B. Noble and J. W. Daniel, Applied Linear Algebra, 3rd. ed. (Prentice-Hall, 1988), pp. 338-346.
  30. M. A. López-Álvarez, J. Hernández-Andrés, J. Romero, and R. L. Lee, Jr., “Designing a practical system for spectral imaging of skylight,” Appl. Opt. 44, 5688-5695 (2005).
    [CrossRef] [PubMed]

2007

2006

H. L. Shen and H. H. Xin, “Spectral characterization of a color scanner based on optimized adaptive estimation,” J. Opt. Soc. Am. A 23, 1566-1569 (2006).
[CrossRef]

N. Shimano, “Recovery of spectral reflectances of objects being imaged without prior knowledge,” IEEE Trans. Image Process. 15, 1848-1856 (2006).
[CrossRef] [PubMed]

N. Shimano, “Optimization of spectral sensitivities with Gaussian distribution functions for a color image acquisition device in the presence of noise,” Opt. Eng. (Bellingham) 45, 013201-1-8 (2006).
[CrossRef]

2005

M. A. López-Álvarez, J. Hernández-Andrés, J. Romero, and R. L. Lee, Jr., “Designing a practical system for spectral imaging of skylight,” Appl. Opt. 44, 5688-5695 (2005).
[CrossRef] [PubMed]

N. Shimano, “Application of a colorimetric evaluation model to multispectral color image acquisition systems,” J. Imaging Sci. Technol. 49, 588-593 (2005).

2002

2000

1997

G. Sharma and H. J. Trussell, “Figures of merit for color scanners,” IEEE Trans. Image Process. 6, 990-1001 (1997).
[CrossRef] [PubMed]

1994

1991

R. W. G. Hunt, “Revised colour-appearance model for related and unrelated colours,” Color Res. Appl. 16, 146-163 (1991).
[CrossRef]

1990

H. C. Lee, E. J. Breneman, and C. P. Schulte, “Modeling light reflection for computer color vision,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 79-86 (1990).
[CrossRef]

G. J. Klinker, S. A. Shafer, and T. Kanade, “A physical approach to color image understanding,” Int. J. Comput. Vis. 4, 7-38 (1990).
[CrossRef]

J. Ho, B. V. Funt, and M. S. Drew, “Separating a color signal into illumination and surface reflectance components: Theory and applications,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 966-977 (1990).
[CrossRef]

1989

1986

1964

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

Afifi, A. A.

A. A. Afifi and S. P. Azen, Statistical Analysis (Academic, 1972), Chap. 3.

Azen, S. P.

A. A. Afifi and S. P. Azen, Statistical Analysis (Academic, 1972), Chap. 3.

Berns, R. S.

Y. Zhao, L. A. Taplin, M. Nezamabadi, and R. S. Berns, “Using the matrix R method for spectral image archives,” in Proceedings of The 10th Congress of the International Colour Association(AIC'5) (AIC, 2005), pp. 469-472.

F. H. Imai and R. S. Berns, “Spectral estimation using trichromatic digital cameras,” in Proceedings of International Symposium on Multispectral Imaging and Color Reproduction for Digital Archives, Chiba, Japan, 21-22 October 1999, pp. 42-49.

Breneman, E. J.

H. C. Lee, E. J. Breneman, and C. P. Schulte, “Modeling light reflection for computer color vision,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 79-86 (1990).
[CrossRef]

Cohen, J.

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

Connah, D.

D. Connah, J. Y. Hardeberg, and S. Westland, “Comparison of linear spectral reconstruction methods for multispectral imaging,” Proceedings IEEE International Conference on Image Processing (IEEE, 2004), pp. 1497-1500.

Daniel, J. W.

B. Noble and J. W. Daniel, Applied Linear Algebra, 3rd. ed. (Prentice-Hall, 1988), pp. 338-346.

Drew, M. S.

J. Ho, B. V. Funt, and M. S. Drew, “Separating a color signal into illumination and surface reflectance components: Theory and applications,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 966-977 (1990).
[CrossRef]

D'Zumura, M.

Fairchild, M. D.

M. D. Fairchild, Color Appearance Models (Addison-Wesley, 1997).

Funt, B. V.

J. Ho, B. V. Funt, and M. S. Drew, “Separating a color signal into illumination and surface reflectance components: Theory and applications,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 966-977 (1990).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (The Johns Hopkins Univ. Press, 1996), p. 55.

Hallikainen, J.

Haneishi, H.

Hardeberg, J. Y.

D. Connah, J. Y. Hardeberg, and S. Westland, “Comparison of linear spectral reconstruction methods for multispectral imaging,” Proceedings IEEE International Conference on Image Processing (IEEE, 2004), pp. 1497-1500.

Hasegawa, T.

Healey, G.

Hernández-Andrés, J.

Hironaga, M.

N. Shimano, K. Terai, and M. Hironaga, “Recovery of spectral reflectances of objects being imaged by multispectral cameras,” J. Opt. Soc. Am. A 24, 3211-3219 (2007).
[CrossRef]

N. Shimano and M. Hironaga, “A new proposal for the accurate recovery of spectral reflectances of imaged objects without prior knowledge,” in Archiving 2008, Proceedings of Society of Imaging Science and Technology, Bern (IS&T, 2008), pp. 155-158.

Ho, J.

J. Ho, B. V. Funt, and M. S. Drew, “Separating a color signal into illumination and surface reflectance components: Theory and applications,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 966-977 (1990).
[CrossRef]

Hosoi, N. A.

Hunt, R. W. G.

R. W. G. Hunt, “Revised colour-appearance model for related and unrelated colours,” Color Res. Appl. 16, 146-163 (1991).
[CrossRef]

Imai, F. H.

F. H. Imai and R. S. Berns, “Spectral estimation using trichromatic digital cameras,” in Proceedings of International Symposium on Multispectral Imaging and Color Reproduction for Digital Archives, Chiba, Japan, 21-22 October 1999, pp. 42-49.

Iverson, G.

Jaaskelainen, T.

Kak, A. C.

A. Rosenfeld and A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, 1982).

Kanade, T.

G. J. Klinker, S. A. Shafer, and T. Kanade, “A physical approach to color image understanding,” Int. J. Comput. Vis. 4, 7-38 (1990).
[CrossRef]

Klinker, G. J.

G. J. Klinker, S. A. Shafer, and T. Kanade, “A physical approach to color image understanding,” Int. J. Comput. Vis. 4, 7-38 (1990).
[CrossRef]

Lee, H. C.

H. C. Lee, E. J. Breneman, and C. P. Schulte, “Modeling light reflection for computer color vision,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 79-86 (1990).
[CrossRef]

Lee, R. L.

Loan, C. F. V.

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (The Johns Hopkins Univ. Press, 1996), p. 55.

López-Álvarez, M. A.

Maloney, L. T.

Miyake, Y.

Nayatani, Y.

Y. Nayatani, K. Takahama, and H. Sobagaki, “Prediction of color appearance under various adapting conditions,” Color Res. Appl. 11, 62-72 (1986).
[CrossRef]

Nezamabadi, M.

Y. Zhao, L. A. Taplin, M. Nezamabadi, and R. S. Berns, “Using the matrix R method for spectral image archives,” in Proceedings of The 10th Congress of the International Colour Association(AIC'5) (AIC, 2005), pp. 469-472.

Noble, B.

B. Noble and J. W. Daniel, Applied Linear Algebra, 3rd. ed. (Prentice-Hall, 1988), pp. 338-346.

Parkkinen, J. P. S.

Romero, J.

Rosenfeld, A.

A. Rosenfeld and A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, 1982).

Schulte, C. P.

H. C. Lee, E. J. Breneman, and C. P. Schulte, “Modeling light reflection for computer color vision,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 79-86 (1990).
[CrossRef]

Shafer, S. A.

G. J. Klinker, S. A. Shafer, and T. Kanade, “A physical approach to color image understanding,” Int. J. Comput. Vis. 4, 7-38 (1990).
[CrossRef]

Sharma, G.

G. Sharma and H. J. Trussell, “Figures of merit for color scanners,” IEEE Trans. Image Process. 6, 990-1001 (1997).
[CrossRef] [PubMed]

Shen, H. L.

Shi, M.

Shimano, N.

N. Shimano, K. Terai, and M. Hironaga, “Recovery of spectral reflectances of objects being imaged by multispectral cameras,” J. Opt. Soc. Am. A 24, 3211-3219 (2007).
[CrossRef]

N. Shimano, “Recovery of spectral reflectances of objects being imaged without prior knowledge,” IEEE Trans. Image Process. 15, 1848-1856 (2006).
[CrossRef] [PubMed]

N. Shimano, “Optimization of spectral sensitivities with Gaussian distribution functions for a color image acquisition device in the presence of noise,” Opt. Eng. (Bellingham) 45, 013201-1-8 (2006).
[CrossRef]

N. Shimano, “Application of a colorimetric evaluation model to multispectral color image acquisition systems,” J. Imaging Sci. Technol. 49, 588-593 (2005).

N. Shimano and M. Hironaga, “A new proposal for the accurate recovery of spectral reflectances of imaged objects without prior knowledge,” in Archiving 2008, Proceedings of Society of Imaging Science and Technology, Bern (IS&T, 2008), pp. 155-158.

Slater, D.

Sobagaki, H.

Y. Nayatani, K. Takahama, and H. Sobagaki, “Prediction of color appearance under various adapting conditions,” Color Res. Appl. 11, 62-72 (1986).
[CrossRef]

Takahama, K.

Y. Nayatani, K. Takahama, and H. Sobagaki, “Prediction of color appearance under various adapting conditions,” Color Res. Appl. 11, 62-72 (1986).
[CrossRef]

Taplin, L. A.

Y. Zhao, L. A. Taplin, M. Nezamabadi, and R. S. Berns, “Using the matrix R method for spectral image archives,” in Proceedings of The 10th Congress of the International Colour Association(AIC'5) (AIC, 2005), pp. 469-472.

Terai, K.

Trussell, H. J.

G. Sharma and H. J. Trussell, “Figures of merit for color scanners,” IEEE Trans. Image Process. 6, 990-1001 (1997).
[CrossRef] [PubMed]

Tsumura, N.

Wandell, B. A.

Westland, S.

D. Connah, J. Y. Hardeberg, and S. Westland, “Comparison of linear spectral reconstruction methods for multispectral imaging,” Proceedings IEEE International Conference on Image Processing (IEEE, 2004), pp. 1497-1500.

Xin, H. H.

Yokoyama, Y.

Zhao, Y.

Y. Zhao, L. A. Taplin, M. Nezamabadi, and R. S. Berns, “Using the matrix R method for spectral image archives,” in Proceedings of The 10th Congress of the International Colour Association(AIC'5) (AIC, 2005), pp. 469-472.

Appl. Opt.

Color Res. Appl.

Y. Nayatani, K. Takahama, and H. Sobagaki, “Prediction of color appearance under various adapting conditions,” Color Res. Appl. 11, 62-72 (1986).
[CrossRef]

R. W. G. Hunt, “Revised colour-appearance model for related and unrelated colours,” Color Res. Appl. 16, 146-163 (1991).
[CrossRef]

IEEE Trans. Image Process.

G. Sharma and H. J. Trussell, “Figures of merit for color scanners,” IEEE Trans. Image Process. 6, 990-1001 (1997).
[CrossRef] [PubMed]

N. Shimano, “Recovery of spectral reflectances of objects being imaged without prior knowledge,” IEEE Trans. Image Process. 15, 1848-1856 (2006).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell.

J. Ho, B. V. Funt, and M. S. Drew, “Separating a color signal into illumination and surface reflectance components: Theory and applications,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 966-977 (1990).
[CrossRef]

H. C. Lee, E. J. Breneman, and C. P. Schulte, “Modeling light reflection for computer color vision,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 79-86 (1990).
[CrossRef]

Int. J. Comput. Vis.

G. J. Klinker, S. A. Shafer, and T. Kanade, “A physical approach to color image understanding,” Int. J. Comput. Vis. 4, 7-38 (1990).
[CrossRef]

J. Imaging Sci. Technol.

N. Shimano, “Application of a colorimetric evaluation model to multispectral color image acquisition systems,” J. Imaging Sci. Technol. 49, 588-593 (2005).

J. Opt. Soc. Am. A

Opt. Eng. (Bellingham)

N. Shimano, “Optimization of spectral sensitivities with Gaussian distribution functions for a color image acquisition device in the presence of noise,” Opt. Eng. (Bellingham) 45, 013201-1-8 (2006).
[CrossRef]

Psychonomic Sci.

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

Other

A. Rosenfeld and A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, 1982).

Y. Zhao, L. A. Taplin, M. Nezamabadi, and R. S. Berns, “Using the matrix R method for spectral image archives,” in Proceedings of The 10th Congress of the International Colour Association(AIC'5) (AIC, 2005), pp. 469-472.

N. Shimano and M. Hironaga, “A new proposal for the accurate recovery of spectral reflectances of imaged objects without prior knowledge,” in Archiving 2008, Proceedings of Society of Imaging Science and Technology, Bern (IS&T, 2008), pp. 155-158.

“Standard Object Colour Spectra Database for Colour Reproduction Evaluation (SOCS),” Japanese Industrial Standards Association, TR X 0012:1998 (1998).

A. A. Afifi and S. P. Azen, Statistical Analysis (Academic, 1972), Chap. 3.

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (The Johns Hopkins Univ. Press, 1996), p. 55.

B. Noble and J. W. Daniel, Applied Linear Algebra, 3rd. ed. (Prentice-Hall, 1988), pp. 338-346.

D. Connah, J. Y. Hardeberg, and S. Westland, “Comparison of linear spectral reconstruction methods for multispectral imaging,” Proceedings IEEE International Conference on Image Processing (IEEE, 2004), pp. 1497-1500.

F. H. Imai and R. S. Berns, “Spectral estimation using trichromatic digital cameras,” in Proceedings of International Symposium on Multispectral Imaging and Color Reproduction for Digital Archives, Chiba, Japan, 21-22 October 1999, pp. 42-49.

M. D. Fairchild, Color Appearance Models (Addison-Wesley, 1997).

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 (7)

Fig. 1
Fig. 1

Schematic diagrams of the proposal. In these diagrams, the notation W ( R S S , σ ̂ 2 ) is used to abbreviate Eq. (3), i.e., W ( R S S , σ ̂ 2 ) is the expression of W ( R S S , σ ̂ 2 ) = R S S S L T ( S L R S S S L T + σ ̂ 2 I ) 1 and W ( R a d , σ ̂ 2 ) is the expression of W ( R a d , σ ̂ 2 ) = R a d S L T ( S L R a d S L T + σ ̂ 2 I ) 1 . The spectral reflectance data base in step (3) contains spectral reflectances of various objects and color materials.

Fig. 2
Fig. 2

Spectral sensitivities of the seven-channel camera.

Fig. 3
Fig. 3

Spectral power distribution of the illuminant used for image acquisition.

Fig. 4
Fig. 4

MSE between the measured and recovered spectral reflectances by the proposal and regression model as a function of the MSE 0 j . The GretagMacbeth ColorChecker and the Kodak Q60 were used for the training and test samples, respectively.

Fig. 5
Fig. 5

MSE between the measured and recovered spectral reflectances by the proposal and regression model as a function of the MSE 0 j . The Kodak Q60 and the GretagMacbeth ColorChecker were used for the training and test samples, respectively.

Fig. 6
Fig. 6

Typical examples of the recovered spectral reflectances by the different models and steps. The recovered spectral reflectance was a patch number 9 of the GretagMacbeth ColorChecker using the learning samples of the Kodak Q60 R1. (a) Spectral reflectances recovered by the Wiener estimation. (b) Spectral reflectances recovered by the regression model.

Fig. 7
Fig. 7

Typical examples of the recovered spectral reflectances by the different models and steps. The recovered spectral reflectance was a patch number A6 of the Kodak Q60 R1 when the GretagMacbeth ColorChecker was used as the learning samples. (a) Spectral reflectances recovered by the Wiener estimation. (b) Spectral reflectances recovered by the regression model.

Tables (2)

Tables Icon

Table 1 Comparative Spectral Recovery Performance with Kodak Q60R1 Used as Training, GretagMacbeth ColorChecker Used as Test Samples, and SOCS Used as Database Samples

Tables Icon

Table 2 Comparative Spectral Recovery Performance with GretagMacbeth ColorChecker Used as Training, Kodak Q60R1 Used as Test Samples, and SOCS Used as Database Samples

Equations (13)

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

p = S L r + e ,
MSE = E { r r ̂ 2 } ,
r ̂ = R s s S L T ( S L R S S S L T + σ e 2 I ) 1 p ,
MSE = i = 1 N λ i i = 1 N j = 1 β λ i b i j 2 + i = 1 N j = 1 β σ e 4 + κ j v 2 σ 2 ( κ j v 2 + σ e 2 ) 2 λ i b i j 2 ,
MSE ( σ e 2 = 0 ) = i = 1 N λ i i = 1 N j = 1 β λ i b i j 2 + i = 1 N j = 1 β σ 2 κ j v 2 λ i b i j 2 .
σ ̂ 2 = MSE ( σ e 2 = 0 ) i = 1 N λ i + i = 1 N j = 1 β λ i b i j 2 i = 1 N j = 1 β λ i b i j 2 κ j v 2 .
MSE 0 j = 1 N test n = 1 N test r n , min j r ̂ 1 , n 2 .
W = R P + ,
r ̂ 1 , i = R P + p t , i ,
R ̂ 1 = R P + P T .
r ̂ 2 , i = R ̂ 1 P T + p t , i .
R ̂ 2 = R ̂ 1 P T + P T ,
R ̂ 2 = R P + P T P T + P T = R P + P T = R ̂ 1 ,

Metrics