Abstract

The Bidirectional Reflectance Distribution Function (BRDF) is essential to characterize an object’s reflectance properties. This function depends both on the various illumination-observation geometries as well as on the wavelength. As a result, the comprehensive interpretation of the data becomes rather complex. In this work we assess the use of the multivariable analysis technique of Principal Components Analysis (PCA) applied to the experimental BRDF data of a ceramic colour standard. It will be shown that the result may be linked to the various reflection processes occurring on the surface, assuming that the incoming spectral distribution is affected by each one of these processes in a specific manner. Moreover, this procedure facilitates the task of interpolating a series of BRDF measurements obtained for a particular sample.

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References

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  1. F. E. Nicodemus, J. C. Richmond, and J. J. Hsia, “Geometrical considerations and nomenclature for reflectance,” National Bureau of Standards Monograph (National Bureau of Standards, 1977), Vol. 160.
  2. C. Bordier, C. Andraud, and J. Lafait, “Model of light scattering that includes polarization effects by multilayered media,” J. Opt. Soc. Am. A 25, 1406–1419 (2008).
    [CrossRef]
  3. L. Simonot, “Photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets,” Appl. Opt. 48, 5793–5801 (2009).
    [CrossRef] [PubMed]
  4. R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” Technical report, Computer Graphics (ACM, 1981), Vol.  15, No. 3.
  5. B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18(6), 311–317 (1975).
    [CrossRef]
  6. G. J. Ward, “Measuring and modelling anisotropic reflection,” Comput. Graphics 26, 265–272 (1992).
    [CrossRef]
  7. X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Technical report, Computer Graphics (1991), Vol.  25, No. 4.
  8. J. F. Blinn, “Models of light reflection for computer synthesized pictures,” Comput. Graphics 11, 192–198 (1977).
    [CrossRef]
  9. E. P. F. Lafortune, S. C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” Technical report (Cornell University, 1997).
  10. S. H. Westin, H. Li, and K. E. Torrance, “A comparison of four BRDF models,” Technical report PCG-04-02, Program of Computer Graphics (Cornell University, April2004).
  11. A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” Eurographics Symposium on Rendering, K. Bala and P. Dutre, eds. (2005).
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    [CrossRef] [PubMed]
  13. J. L. Simonds, “Application of characteristic vector analysis to photographic and optical response data,” J. Opt. Soc. Am. 53, 968–971 (1963).
    [CrossRef]
  14. J. M. López-Alonso, J. Alda, and E. Bernabéu, “Principal-component characterization of noise for infrared images,” Appl. Opt.41, 320–331 (2002).
    [CrossRef] [PubMed]
  15. A. Ferrero, J. Alda, J. Campos, J. M. López-Alonso, and A. Pons, “Principal components analysis of the photoresponse nonuniformity of a matrix detector,” Appl. Opt. 46, 9–17 (2007).
    [CrossRef]
  16. A. M. Rabal, A. Ferrero, J. L. Fontecha, A. Pons, J. Campos, A. Corróns, and A. M. Rubio, “Gonio-spectrophotometer for low-uncertainty measurements of bidirectional scattering distribution function (BSDF),” Proceedings of CIE Expert Symposium on “Spectral and Imaging Methods for Photometry and Radiometry,” Publication CIE x036:2010 (CIE, Vienna, Austria, 2010), pp. 79–84.
  17. T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
    [CrossRef]
  18. D. Hünerhoff, U. Grusemann, and A Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection.” Metrologia 43, S11–S16 (2006).
    [CrossRef]
  19. F. B. Leloup, S. Forment, P. Dutré, M. R. Pointer, and P. Hanselaer, “Design of an instrument for measuring the spectral bidirectional scatter distribution function,” Appl. Opt. 47(29), 5454–5467 (2008).
    [PubMed]
  20. R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. 63, 561–564 (1995).
    [CrossRef]
  21. T. J. Papetti, W. E. Walker, C. E. Keffer, and B. E. Johnson, “Coherent backscatter: measurement of the retroreflective BRDF peak exhibited by several surfaces relevant to ladar applications,” Proc. SPIE6682, 66820E (2007).
    [CrossRef]

2009 (1)

2008 (3)

2007 (1)

2006 (1)

D. Hünerhoff, U. Grusemann, and A Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection.” Metrologia 43, S11–S16 (2006).
[CrossRef]

1999 (1)

T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

1995 (1)

R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. 63, 561–564 (1995).
[CrossRef]

1992 (1)

G. J. Ward, “Measuring and modelling anisotropic reflection,” Comput. Graphics 26, 265–272 (1992).
[CrossRef]

1991 (1)

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Technical report, Computer Graphics (1991), Vol.  25, No. 4.

1981 (1)

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” Technical report, Computer Graphics (ACM, 1981), Vol.  15, No. 3.

1977 (1)

J. F. Blinn, “Models of light reflection for computer synthesized pictures,” Comput. Graphics 11, 192–198 (1977).
[CrossRef]

1975 (1)

B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18(6), 311–317 (1975).
[CrossRef]

1963 (1)

Alda, J.

A. Ferrero, J. Alda, J. Campos, J. M. López-Alonso, and A. Pons, “Principal components analysis of the photoresponse nonuniformity of a matrix detector,” Appl. Opt. 46, 9–17 (2007).
[CrossRef]

J. M. López-Alonso, J. Alda, and E. Bernabéu, “Principal-component characterization of noise for infrared images,” Appl. Opt.41, 320–331 (2002).
[CrossRef] [PubMed]

Andraud, C.

Asmail, C. C.

T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

Bernabéu, E.

J. M. López-Alonso, J. Alda, and E. Bernabéu, “Principal-component characterization of noise for infrared images,” Appl. Opt.41, 320–331 (2002).
[CrossRef] [PubMed]

Blinn, J. F.

J. F. Blinn, “Models of light reflection for computer synthesized pictures,” Comput. Graphics 11, 192–198 (1977).
[CrossRef]

Bordier, C.

Boreman, G. D.

Campos, J.

A. Ferrero, J. Alda, J. Campos, J. M. López-Alonso, and A. Pons, “Principal components analysis of the photoresponse nonuniformity of a matrix detector,” Appl. Opt. 46, 9–17 (2007).
[CrossRef]

A. M. Rabal, A. Ferrero, J. L. Fontecha, A. Pons, J. Campos, A. Corróns, and A. M. Rubio, “Gonio-spectrophotometer for low-uncertainty measurements of bidirectional scattering distribution function (BSDF),” Proceedings of CIE Expert Symposium on “Spectral and Imaging Methods for Photometry and Radiometry,” Publication CIE x036:2010 (CIE, Vienna, Austria, 2010), pp. 79–84.

Cook, R. L.

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” Technical report, Computer Graphics (ACM, 1981), Vol.  15, No. 3.

Corey, R.

R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. 63, 561–564 (1995).
[CrossRef]

Corróns, A.

A. M. Rabal, A. Ferrero, J. L. Fontecha, A. Pons, J. Campos, A. Corróns, and A. M. Rubio, “Gonio-spectrophotometer for low-uncertainty measurements of bidirectional scattering distribution function (BSDF),” Proceedings of CIE Expert Symposium on “Spectral and Imaging Methods for Photometry and Radiometry,” Publication CIE x036:2010 (CIE, Vienna, Austria, 2010), pp. 79–84.

Durand, F.

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” Eurographics Symposium on Rendering, K. Bala and P. Dutre, eds. (2005).

Dutré, P.

Ferrero, A.

A. Ferrero, J. Alda, J. Campos, J. M. López-Alonso, and A. Pons, “Principal components analysis of the photoresponse nonuniformity of a matrix detector,” Appl. Opt. 46, 9–17 (2007).
[CrossRef]

A. M. Rabal, A. Ferrero, J. L. Fontecha, A. Pons, J. Campos, A. Corróns, and A. M. Rubio, “Gonio-spectrophotometer for low-uncertainty measurements of bidirectional scattering distribution function (BSDF),” Proceedings of CIE Expert Symposium on “Spectral and Imaging Methods for Photometry and Radiometry,” Publication CIE x036:2010 (CIE, Vienna, Austria, 2010), pp. 79–84.

Fontecha, J. L.

A. M. Rabal, A. Ferrero, J. L. Fontecha, A. Pons, J. Campos, A. Corróns, and A. M. Rubio, “Gonio-spectrophotometer for low-uncertainty measurements of bidirectional scattering distribution function (BSDF),” Proceedings of CIE Expert Symposium on “Spectral and Imaging Methods for Photometry and Radiometry,” Publication CIE x036:2010 (CIE, Vienna, Austria, 2010), pp. 79–84.

Foo, S. C.

E. P. F. Lafortune, S. C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” Technical report (Cornell University, 1997).

Forment, S.

Germer, T. A.

T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

Greenberg, D. P.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Technical report, Computer Graphics (1991), Vol.  25, No. 4.

E. P. F. Lafortune, S. C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” Technical report (Cornell University, 1997).

Grusemann, U.

D. Hünerhoff, U. Grusemann, and A Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection.” Metrologia 43, S11–S16 (2006).
[CrossRef]

Hanselaer, P.

He, X. D.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Technical report, Computer Graphics (1991), Vol.  25, No. 4.

Höpe, A

D. Hünerhoff, U. Grusemann, and A Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection.” Metrologia 43, S11–S16 (2006).
[CrossRef]

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, and J. J. Hsia, “Geometrical considerations and nomenclature for reflectance,” National Bureau of Standards Monograph (National Bureau of Standards, 1977), Vol. 160.

Hünerhoff, D.

D. Hünerhoff, U. Grusemann, and A Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection.” Metrologia 43, S11–S16 (2006).
[CrossRef]

Johnson, B. E.

T. J. Papetti, W. E. Walker, C. E. Keffer, and B. E. Johnson, “Coherent backscatter: measurement of the retroreflective BRDF peak exhibited by several surfaces relevant to ladar applications,” Proc. SPIE6682, 66820E (2007).
[CrossRef]

Keffer, C. E.

T. J. Papetti, W. E. Walker, C. E. Keffer, and B. E. Johnson, “Coherent backscatter: measurement of the retroreflective BRDF peak exhibited by several surfaces relevant to ladar applications,” Proc. SPIE6682, 66820E (2007).
[CrossRef]

Kissner, M.

R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. 63, 561–564 (1995).
[CrossRef]

Lafait, J.

Lafortune, E. P. F.

E. P. F. Lafortune, S. C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” Technical report (Cornell University, 1997).

Leloup, F. B.

Li, H.

S. H. Westin, H. Li, and K. E. Torrance, “A comparison of four BRDF models,” Technical report PCG-04-02, Program of Computer Graphics (Cornell University, April2004).

López-Alonso, J. M.

A. Ferrero, J. Alda, J. Campos, J. M. López-Alonso, and A. Pons, “Principal components analysis of the photoresponse nonuniformity of a matrix detector,” Appl. Opt. 46, 9–17 (2007).
[CrossRef]

J. M. López-Alonso, J. Alda, and E. Bernabéu, “Principal-component characterization of noise for infrared images,” Appl. Opt.41, 320–331 (2002).
[CrossRef] [PubMed]

Matusik, W.

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” Eurographics Symposium on Rendering, K. Bala and P. Dutre, eds. (2005).

Ngan, A.

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” Eurographics Symposium on Rendering, K. Bala and P. Dutre, eds. (2005).

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, and J. J. Hsia, “Geometrical considerations and nomenclature for reflectance,” National Bureau of Standards Monograph (National Bureau of Standards, 1977), Vol. 160.

Papetti, T. J.

T. J. Papetti, W. E. Walker, C. E. Keffer, and B. E. Johnson, “Coherent backscatter: measurement of the retroreflective BRDF peak exhibited by several surfaces relevant to ladar applications,” Proc. SPIE6682, 66820E (2007).
[CrossRef]

Phong, B. T.

B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18(6), 311–317 (1975).
[CrossRef]

Pointer, M. R.

Pons, A.

A. Ferrero, J. Alda, J. Campos, J. M. López-Alonso, and A. Pons, “Principal components analysis of the photoresponse nonuniformity of a matrix detector,” Appl. Opt. 46, 9–17 (2007).
[CrossRef]

A. M. Rabal, A. Ferrero, J. L. Fontecha, A. Pons, J. Campos, A. Corróns, and A. M. Rubio, “Gonio-spectrophotometer for low-uncertainty measurements of bidirectional scattering distribution function (BSDF),” Proceedings of CIE Expert Symposium on “Spectral and Imaging Methods for Photometry and Radiometry,” Publication CIE x036:2010 (CIE, Vienna, Austria, 2010), pp. 79–84.

Rabal, A. M.

A. M. Rabal, A. Ferrero, J. L. Fontecha, A. Pons, J. Campos, A. Corróns, and A. M. Rubio, “Gonio-spectrophotometer for low-uncertainty measurements of bidirectional scattering distribution function (BSDF),” Proceedings of CIE Expert Symposium on “Spectral and Imaging Methods for Photometry and Radiometry,” Publication CIE x036:2010 (CIE, Vienna, Austria, 2010), pp. 79–84.

Renhorn, I. G. E.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, and J. J. Hsia, “Geometrical considerations and nomenclature for reflectance,” National Bureau of Standards Monograph (National Bureau of Standards, 1977), Vol. 160.

Rubio, A. M.

A. M. Rabal, A. Ferrero, J. L. Fontecha, A. Pons, J. Campos, A. Corróns, and A. M. Rubio, “Gonio-spectrophotometer for low-uncertainty measurements of bidirectional scattering distribution function (BSDF),” Proceedings of CIE Expert Symposium on “Spectral and Imaging Methods for Photometry and Radiometry,” Publication CIE x036:2010 (CIE, Vienna, Austria, 2010), pp. 79–84.

Saulnier, P.

R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. 63, 561–564 (1995).
[CrossRef]

Sillion, F. X.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Technical report, Computer Graphics (1991), Vol.  25, No. 4.

Simonds, J. L.

Simonot, L.

Torrance, K. E.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Technical report, Computer Graphics (1991), Vol.  25, No. 4.

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” Technical report, Computer Graphics (ACM, 1981), Vol.  15, No. 3.

S. H. Westin, H. Li, and K. E. Torrance, “A comparison of four BRDF models,” Technical report PCG-04-02, Program of Computer Graphics (Cornell University, April2004).

E. P. F. Lafortune, S. C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” Technical report (Cornell University, 1997).

Walker, W. E.

T. J. Papetti, W. E. Walker, C. E. Keffer, and B. E. Johnson, “Coherent backscatter: measurement of the retroreflective BRDF peak exhibited by several surfaces relevant to ladar applications,” Proc. SPIE6682, 66820E (2007).
[CrossRef]

Ward, G. J.

G. J. Ward, “Measuring and modelling anisotropic reflection,” Comput. Graphics 26, 265–272 (1992).
[CrossRef]

Westin, S. H.

S. H. Westin, H. Li, and K. E. Torrance, “A comparison of four BRDF models,” Technical report PCG-04-02, Program of Computer Graphics (Cornell University, April2004).

Am. J. Phys. (1)

R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. 63, 561–564 (1995).
[CrossRef]

Appl. Opt. (3)

Commun. ACM (1)

B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18(6), 311–317 (1975).
[CrossRef]

Comput. Graphics (2)

G. J. Ward, “Measuring and modelling anisotropic reflection,” Comput. Graphics 26, 265–272 (1992).
[CrossRef]

J. F. Blinn, “Models of light reflection for computer synthesized pictures,” Comput. Graphics 11, 192–198 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Metrologia (1)

D. Hünerhoff, U. Grusemann, and A Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection.” Metrologia 43, S11–S16 (2006).
[CrossRef]

Opt. Express (1)

Rev. Sci. Instrum. (1)

T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

Technical report, Computer Graphics (2)

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” Technical report, Computer Graphics (ACM, 1981), Vol.  15, No. 3.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Technical report, Computer Graphics (1991), Vol.  25, No. 4.

Other (7)

E. P. F. Lafortune, S. C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” Technical report (Cornell University, 1997).

S. H. Westin, H. Li, and K. E. Torrance, “A comparison of four BRDF models,” Technical report PCG-04-02, Program of Computer Graphics (Cornell University, April2004).

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” Eurographics Symposium on Rendering, K. Bala and P. Dutre, eds. (2005).

F. E. Nicodemus, J. C. Richmond, and J. J. Hsia, “Geometrical considerations and nomenclature for reflectance,” National Bureau of Standards Monograph (National Bureau of Standards, 1977), Vol. 160.

J. M. López-Alonso, J. Alda, and E. Bernabéu, “Principal-component characterization of noise for infrared images,” Appl. Opt.41, 320–331 (2002).
[CrossRef] [PubMed]

A. M. Rabal, A. Ferrero, J. L. Fontecha, A. Pons, J. Campos, A. Corróns, and A. M. Rubio, “Gonio-spectrophotometer for low-uncertainty measurements of bidirectional scattering distribution function (BSDF),” Proceedings of CIE Expert Symposium on “Spectral and Imaging Methods for Photometry and Radiometry,” Publication CIE x036:2010 (CIE, Vienna, Austria, 2010), pp. 79–84.

T. J. Papetti, W. E. Walker, C. E. Keffer, and B. E. Johnson, “Coherent backscatter: measurement of the retroreflective BRDF peak exhibited by several surfaces relevant to ladar applications,” Proc. SPIE6682, 66820E (2007).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Sample Coodinate System showing spherical coordinates for incidence (θi , ϕi ) and observation directions (θs , ϕs ).

Fig. 2
Fig. 2

Spectral BRDF components of a glossy ceramic colour standard.

Fig. 3
Fig. 3

Specular spectral BRDF measurements of a glossy ceramic colour standard at θi = θs = θ = 0°, 10°, 20° and 30°, and ϕi = 0° and ϕs = 180°.

Fig. 4
Fig. 4

Specular spectral BRDF measurements of a glossy ceramic colour standard at θi = θs = θ = 40°, 50°, 60° and 70°, and ϕi = 0° and ϕs = 180°.

Fig. 5
Fig. 5

Angular representation of the BRDF measurements at 500 nm in the incidence plane (0:180). Every plot shows BRDF values versus observation angles and corresponds to a different polar incidence angle (θi ).

Fig. 6
Fig. 6

BRDF measured at 500 nm outside the incidence plane (half-planes 30:150) versus observation angle (θs ) for different incidence angles (θi ).

Fig. 7
Fig. 7

Eigenspectra of the BRDF measurements of a glossy green ceramic colour standard: (a) PC1, (b) PC2, (c) PC3 and (c) PC4.

Fig. 8
Fig. 8

Relative contributions to the spectral variance of the first four principal components (half-plane 0:180) in a semilogarithmic plot.

Fig. 9
Fig. 9

Relative contributions to the spectral variance of the first four principal components (half-plane 30:150) in a semilogarithmic plot.

Fig. 10
Fig. 10

Angular representations of the relative average error εr in the incidence plane (0:180).

Equations (9)

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

A j = i = 1 N e j , i F ¯ i = F ¯ e j T
F ¯ i = j = 1 N e i , j A j ,
σ 2 [ F i ] = σ 2 [ F ¯ i ] = 1 M 1 λ = 1 M ( F ¯ i λ ) 2 ,
( F ¯ i λ ) 2 = ( j = 1 N e i , j A j λ ) 2 = j = 1 N k = 1 N e i , j e i , k A j λ A k λ .
σ 2 [ F i ] = j = 1 N k = 1 N e i , j e i , k 1 M 1 λ = 1 M A j λ A k λ = j = 1 N e i , j 2 γ j ,
S 2 = E 2 G ,
σ i , j = | e i , j | γ j .
F ¯ i , j = e i , j A j
ε i = 1 M λ = 1 M | F ¯ i λ F ¯ r , i λ |

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