Abstract

A spectral model incorporating three major physical phenomena, lateral light scattering, substrate fluorescence, and interface reflections governing light-print interaction is presented. In the model, light scattering inside a paper substrate is described by probabilities applicable for any degree of light diffusion. The concept of fluorescence enhanced transmittance has been proven very useful for extending the probability approach to prints on fluorescent substrates. The contribution of multiple internal reflections is accounted for by series expansion of fast convergence. Previously developed models like the Neugebauer equation and the Clapper–Yule model can easily be obtained as special cases of the present model.

© 2010 Optical Society of America

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References

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  1. F. Grum, “Colorimetry of fluorescent materials,” in Optical Radiation Measurements, Color Measurements, Vol. 2, F.Grum and C.J.Bartleson, eds. (Academic, 1980), pp. 236–287.
  2. N. Pauler, Paper Optics (AB Lorentzen & Wettre, Corporate, P.O. Box 4, 164 93 Kista, Sweden, 2002).
  3. R. D. Hersch and M. Hébert, “Interaction between light, paper and color halftones: Challenges and modelization approaches,” in Proceedings of IS&T 3rd European Conference on Color in Graphics, Imaging and Vision (CGIV) (IS&T, Springfield, VA, 2006), pp.1–7.
  4. F. R. Clapper and J. A. C. Yule, “The effect of multiple internal reflections on the densities of halftone print on paper,” J. Opt. Soc. Am. 43, 600–603 (1953).
    [CrossRef]
  5. J. R. Huntsman, “A new model of dot gain and its application to a multilayer color proof,” J. Imaging Sci. Technol. 13, 136–145 (1987).
  6. J. S. Arney, “A probability description of the Yule-Nielsen effect, I.,” J. Imaging Sci. Technol. 41, 633–636 (1997).
  7. J. S. Arney and M. Katsube, “A probability description of the Yule-Nielsen effect II: The impact of halftone geometry,” J. Imaging Sci. Technol. 41, 637–642 (1997).
  8. J. S. Arney, T. Wu, and C. Blehm, “Modeling the Yule-Nielsen effect on color halftone,” J. Imaging Sci. Technol. 42, 335–340 (1998).
  9. L. Yang, R. Lenz, and B. Kruse, “Light scattering and ink penetration effects on tone reproduction,” J. Opt. Soc. Am. A 18, 360–366 (2001).
    [CrossRef]
  10. L. Yang, “Spectral model of halftone on a fluorescent substrate,” J. Imaging Sci. Technol. 49, 179–184 (2005).
  11. L. Yang, S. Gooran, and B. Kruse, “Simulation of optical dot gain in multichromatic tone production,” J. Imaging Sci. Technol. 45, 198–204 (2001).
  12. G. L. Rogers, “Effect of light scatter on halftone color,” J. Opt. Soc. Am. A 15, 1813–1821 (1998).
    [CrossRef]
  13. G. L. Rogers, “A generalized Clapper-Yule model of halftone reflectance,” J. Color Res. Appl. 25, 402–407 (2000).
    [CrossRef]
  14. G. L. Rogers, “Spectral model of a fluorescent ink halftone,” J. Opt. Soc. Am. A 17, 1975–1981 (2000).
    [CrossRef]
  15. P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).
  16. P. Emmel,”Modèles de prédiction couleur appliqués à l’impression jet d’encre,” Thesis No. 1857 (École Polytechnique Fédérale de Lausanne, Switzerland, 1998).
  17. P. Emmel and R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).
  18. P. Emmel, “Physical model for color prediction,” in Digital Color Imaging Handbook (CRC Press, 2003), pp.173–238.
  19. M. Hébert and R. D. Hersch, “Reflectance and transmittance model for recto-verso halftone prints,” J. Opt. Soc. Am. A 23, 2415–2432 (2006).
    [CrossRef]
  20. R. D. Hersch, “Spectral prediction model for color prints on paper with fluorescent additives,” Appl. Opt. 47, 6710–6722 (2008).
    [CrossRef] [PubMed]
  21. B. Kruse and S. Gustavson, “Rendering of color on scattering media,” Proc. SPIE 2657, 696–705 (1996).
  22. S. Gustavson, “Color gamut of halftone reproduction,” J. Imaging Sci. Technol. 41, 283–290 (1997).
  23. M. Sormaz, T. Stamm, S. Mourad, and P. Jenny, “Stochastic modeling of light scattering with fluorescence using a Monte Carlo-based multiscale approach,” J. Opt. Soc. Am. A 26, 1403–1413 (2009).
    [CrossRef]
  24. H. E. J. Neugebauer, “Die theoretischen Grundlagen des Mehrfarbenbuchdrucks,” Z. Tech. Phys. (Leipzig) 36, 75–89 (1937).
  25. F. R. Ruckdeschel and O. G. Hauser, “Yule–Nielsen effect in printing: a physical analysis,” Appl. Opt. 17, 3376–3383 (1978).
    [CrossRef] [PubMed]
  26. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  27. R. D. Hersch, P. Emmel, F. Collaud, and F. Créte, “Modeling ink spreading for color prediction,” J. Electron. Imaging 14, 33001–12 (2005).
    [CrossRef]

2009

2008

2006

M. Hébert and R. D. Hersch, “Reflectance and transmittance model for recto-verso halftone prints,” J. Opt. Soc. Am. A 23, 2415–2432 (2006).
[CrossRef]

R. D. Hersch and M. Hébert, “Interaction between light, paper and color halftones: Challenges and modelization approaches,” in Proceedings of IS&T 3rd European Conference on Color in Graphics, Imaging and Vision (CGIV) (IS&T, Springfield, VA, 2006), pp.1–7.

2005

L. Yang, “Spectral model of halftone on a fluorescent substrate,” J. Imaging Sci. Technol. 49, 179–184 (2005).

R. D. Hersch, P. Emmel, F. Collaud, and F. Créte, “Modeling ink spreading for color prediction,” J. Electron. Imaging 14, 33001–12 (2005).
[CrossRef]

2003

P. Emmel, “Physical model for color prediction,” in Digital Color Imaging Handbook (CRC Press, 2003), pp.173–238.

2002

N. Pauler, Paper Optics (AB Lorentzen & Wettre, Corporate, P.O. Box 4, 164 93 Kista, Sweden, 2002).

2001

L. Yang, R. Lenz, and B. Kruse, “Light scattering and ink penetration effects on tone reproduction,” J. Opt. Soc. Am. A 18, 360–366 (2001).
[CrossRef]

L. Yang, S. Gooran, and B. Kruse, “Simulation of optical dot gain in multichromatic tone production,” J. Imaging Sci. Technol. 45, 198–204 (2001).

2000

G. L. Rogers, “A generalized Clapper-Yule model of halftone reflectance,” J. Color Res. Appl. 25, 402–407 (2000).
[CrossRef]

G. L. Rogers, “Spectral model of a fluorescent ink halftone,” J. Opt. Soc. Am. A 17, 1975–1981 (2000).
[CrossRef]

P. Emmel and R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

1998

P. Emmel,”Modèles de prédiction couleur appliqués à l’impression jet d’encre,” Thesis No. 1857 (École Polytechnique Fédérale de Lausanne, Switzerland, 1998).

G. L. Rogers, “Effect of light scatter on halftone color,” J. Opt. Soc. Am. A 15, 1813–1821 (1998).
[CrossRef]

J. S. Arney, T. Wu, and C. Blehm, “Modeling the Yule-Nielsen effect on color halftone,” J. Imaging Sci. Technol. 42, 335–340 (1998).

1997

J. S. Arney, “A probability description of the Yule-Nielsen effect, I.,” J. Imaging Sci. Technol. 41, 633–636 (1997).

J. S. Arney and M. Katsube, “A probability description of the Yule-Nielsen effect II: The impact of halftone geometry,” J. Imaging Sci. Technol. 41, 637–642 (1997).

S. Gustavson, “Color gamut of halftone reproduction,” J. Imaging Sci. Technol. 41, 283–290 (1997).

1996

B. Kruse and S. Gustavson, “Rendering of color on scattering media,” Proc. SPIE 2657, 696–705 (1996).

1987

J. R. Huntsman, “A new model of dot gain and its application to a multilayer color proof,” J. Imaging Sci. Technol. 13, 136–145 (1987).

1980

F. Grum, “Colorimetry of fluorescent materials,” in Optical Radiation Measurements, Color Measurements, Vol. 2, F.Grum and C.J.Bartleson, eds. (Academic, 1980), pp. 236–287.

1978

1960

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

1953

1937

H. E. J. Neugebauer, “Die theoretischen Grundlagen des Mehrfarbenbuchdrucks,” Z. Tech. Phys. (Leipzig) 36, 75–89 (1937).

1931

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Arney, J. S.

J. S. Arney, T. Wu, and C. Blehm, “Modeling the Yule-Nielsen effect on color halftone,” J. Imaging Sci. Technol. 42, 335–340 (1998).

J. S. Arney, “A probability description of the Yule-Nielsen effect, I.,” J. Imaging Sci. Technol. 41, 633–636 (1997).

J. S. Arney and M. Katsube, “A probability description of the Yule-Nielsen effect II: The impact of halftone geometry,” J. Imaging Sci. Technol. 41, 637–642 (1997).

Blehm, C.

J. S. Arney, T. Wu, and C. Blehm, “Modeling the Yule-Nielsen effect on color halftone,” J. Imaging Sci. Technol. 42, 335–340 (1998).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Clapper, F. R.

Collaud, F.

R. D. Hersch, P. Emmel, F. Collaud, and F. Créte, “Modeling ink spreading for color prediction,” J. Electron. Imaging 14, 33001–12 (2005).
[CrossRef]

Créte, F.

R. D. Hersch, P. Emmel, F. Collaud, and F. Créte, “Modeling ink spreading for color prediction,” J. Electron. Imaging 14, 33001–12 (2005).
[CrossRef]

Emmel, P.

R. D. Hersch, P. Emmel, F. Collaud, and F. Créte, “Modeling ink spreading for color prediction,” J. Electron. Imaging 14, 33001–12 (2005).
[CrossRef]

P. Emmel, “Physical model for color prediction,” in Digital Color Imaging Handbook (CRC Press, 2003), pp.173–238.

P. Emmel and R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

P. Emmel,”Modèles de prédiction couleur appliqués à l’impression jet d’encre,” Thesis No. 1857 (École Polytechnique Fédérale de Lausanne, Switzerland, 1998).

Gooran, S.

L. Yang, S. Gooran, and B. Kruse, “Simulation of optical dot gain in multichromatic tone production,” J. Imaging Sci. Technol. 45, 198–204 (2001).

Grum, F.

F. Grum, “Colorimetry of fluorescent materials,” in Optical Radiation Measurements, Color Measurements, Vol. 2, F.Grum and C.J.Bartleson, eds. (Academic, 1980), pp. 236–287.

Gustavson, S.

S. Gustavson, “Color gamut of halftone reproduction,” J. Imaging Sci. Technol. 41, 283–290 (1997).

B. Kruse and S. Gustavson, “Rendering of color on scattering media,” Proc. SPIE 2657, 696–705 (1996).

Hauser, O. G.

Hébert, M.

M. Hébert and R. D. Hersch, “Reflectance and transmittance model for recto-verso halftone prints,” J. Opt. Soc. Am. A 23, 2415–2432 (2006).
[CrossRef]

R. D. Hersch and M. Hébert, “Interaction between light, paper and color halftones: Challenges and modelization approaches,” in Proceedings of IS&T 3rd European Conference on Color in Graphics, Imaging and Vision (CGIV) (IS&T, Springfield, VA, 2006), pp.1–7.

Hersch, R. D.

R. D. Hersch, “Spectral prediction model for color prints on paper with fluorescent additives,” Appl. Opt. 47, 6710–6722 (2008).
[CrossRef] [PubMed]

M. Hébert and R. D. Hersch, “Reflectance and transmittance model for recto-verso halftone prints,” J. Opt. Soc. Am. A 23, 2415–2432 (2006).
[CrossRef]

R. D. Hersch and M. Hébert, “Interaction between light, paper and color halftones: Challenges and modelization approaches,” in Proceedings of IS&T 3rd European Conference on Color in Graphics, Imaging and Vision (CGIV) (IS&T, Springfield, VA, 2006), pp.1–7.

R. D. Hersch, P. Emmel, F. Collaud, and F. Créte, “Modeling ink spreading for color prediction,” J. Electron. Imaging 14, 33001–12 (2005).
[CrossRef]

P. Emmel and R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

Huntsman, J. R.

J. R. Huntsman, “A new model of dot gain and its application to a multilayer color proof,” J. Imaging Sci. Technol. 13, 136–145 (1987).

Jenny, P.

Katsube, M.

J. S. Arney and M. Katsube, “A probability description of the Yule-Nielsen effect II: The impact of halftone geometry,” J. Imaging Sci. Technol. 41, 637–642 (1997).

Kruse, B.

L. Yang, R. Lenz, and B. Kruse, “Light scattering and ink penetration effects on tone reproduction,” J. Opt. Soc. Am. A 18, 360–366 (2001).
[CrossRef]

L. Yang, S. Gooran, and B. Kruse, “Simulation of optical dot gain in multichromatic tone production,” J. Imaging Sci. Technol. 45, 198–204 (2001).

B. Kruse and S. Gustavson, “Rendering of color on scattering media,” Proc. SPIE 2657, 696–705 (1996).

Kubelka, P.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Lenz, R.

Mourad, S.

Munk, F.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Neugebauer, H. E. J.

H. E. J. Neugebauer, “Die theoretischen Grundlagen des Mehrfarbenbuchdrucks,” Z. Tech. Phys. (Leipzig) 36, 75–89 (1937).

Pauler, N.

N. Pauler, Paper Optics (AB Lorentzen & Wettre, Corporate, P.O. Box 4, 164 93 Kista, Sweden, 2002).

Rogers, G. L.

Ruckdeschel, F. R.

Sormaz, M.

Stamm, T.

Wu, T.

J. S. Arney, T. Wu, and C. Blehm, “Modeling the Yule-Nielsen effect on color halftone,” J. Imaging Sci. Technol. 42, 335–340 (1998).

Yang, L.

L. Yang, “Spectral model of halftone on a fluorescent substrate,” J. Imaging Sci. Technol. 49, 179–184 (2005).

L. Yang, S. Gooran, and B. Kruse, “Simulation of optical dot gain in multichromatic tone production,” J. Imaging Sci. Technol. 45, 198–204 (2001).

L. Yang, R. Lenz, and B. Kruse, “Light scattering and ink penetration effects on tone reproduction,” J. Opt. Soc. Am. A 18, 360–366 (2001).
[CrossRef]

Yule, J. A. C.

Appl. Opt.

J. Color Res. Appl.

G. L. Rogers, “A generalized Clapper-Yule model of halftone reflectance,” J. Color Res. Appl. 25, 402–407 (2000).
[CrossRef]

J. Electron. Imaging

R. D. Hersch, P. Emmel, F. Collaud, and F. Créte, “Modeling ink spreading for color prediction,” J. Electron. Imaging 14, 33001–12 (2005).
[CrossRef]

J. Imaging Sci. Technol.

S. Gustavson, “Color gamut of halftone reproduction,” J. Imaging Sci. Technol. 41, 283–290 (1997).

P. Emmel and R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

J. R. Huntsman, “A new model of dot gain and its application to a multilayer color proof,” J. Imaging Sci. Technol. 13, 136–145 (1987).

J. S. Arney, “A probability description of the Yule-Nielsen effect, I.,” J. Imaging Sci. Technol. 41, 633–636 (1997).

J. S. Arney and M. Katsube, “A probability description of the Yule-Nielsen effect II: The impact of halftone geometry,” J. Imaging Sci. Technol. 41, 637–642 (1997).

J. S. Arney, T. Wu, and C. Blehm, “Modeling the Yule-Nielsen effect on color halftone,” J. Imaging Sci. Technol. 42, 335–340 (1998).

L. Yang, “Spectral model of halftone on a fluorescent substrate,” J. Imaging Sci. Technol. 49, 179–184 (2005).

L. Yang, S. Gooran, and B. Kruse, “Simulation of optical dot gain in multichromatic tone production,” J. Imaging Sci. Technol. 45, 198–204 (2001).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. SPIE

B. Kruse and S. Gustavson, “Rendering of color on scattering media,” Proc. SPIE 2657, 696–705 (1996).

Z. Tech. Phys. (Leipzig)

H. E. J. Neugebauer, “Die theoretischen Grundlagen des Mehrfarbenbuchdrucks,” Z. Tech. Phys. (Leipzig) 36, 75–89 (1937).

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Other

P. Emmel,”Modèles de prédiction couleur appliqués à l’impression jet d’encre,” Thesis No. 1857 (École Polytechnique Fédérale de Lausanne, Switzerland, 1998).

P. Emmel, “Physical model for color prediction,” in Digital Color Imaging Handbook (CRC Press, 2003), pp.173–238.

F. Grum, “Colorimetry of fluorescent materials,” in Optical Radiation Measurements, Color Measurements, Vol. 2, F.Grum and C.J.Bartleson, eds. (Academic, 1980), pp. 236–287.

N. Pauler, Paper Optics (AB Lorentzen & Wettre, Corporate, P.O. Box 4, 164 93 Kista, Sweden, 2002).

R. D. Hersch and M. Hébert, “Interaction between light, paper and color halftones: Challenges and modelization approaches,” in Proceedings of IS&T 3rd European Conference on Color in Graphics, Imaging and Vision (CGIV) (IS&T, Springfield, VA, 2006), pp.1–7.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

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Figures (4)

Fig. 1
Fig. 1

Reflectance values of office paper measured with ( R 0 ) and without ( R g ) fluorescence. The contribution of the fluorescence ( F 0 ) is defined in Eq. (18).

Fig. 2
Fig. 2

Spectral fluorescence values of the print solids on fluorescent paper substrate. Light absorption of the inks in the UV spectral region is responsible for the dramatic difference of the F 1 values.

Fig. 3
Fig. 3

Illustration of multiple internal reflections between the medium bulk and the air–medium interface and the fraction of light that emerges at each reflection cycle. For generality, the transmittance at each exiting point is denoted as arbitrary. It is assumed that the substrate and the ink have identical refraction indices.

Fig. 4
Fig. 4

Reflectance values of black–white print with complete light diffusion and R g = 0.9 , r ex = 0.1 , r in = 0.5 , T 1 = 0.2 , and F i = 0 . Left panel: series approximations employing Eqs. (56, 57, 58, 59) and the exact value computed with Eq. (60); right panel: the residuals of the series approximations (differences from the exact value).

Equations (72)

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P 00 = 1 ( 1 σ ) Σ 0 Σ 0 p ( r 0 , r 0 ) d σ 0 d σ 0 ,
P 10 = 1 σ Σ 1 Σ 0 p ( r 1 , r 0 ) d σ 0 d σ 1 ,
P 01 = 1 ( 1 σ ) Σ 0 Σ 1 p ( r 0 , r 1 ) d σ 1 d σ 0 ,
P 11 = 1 σ Σ 1 Σ 1 p ( r 1 , r 1 ) d σ 1 d σ 1 .
P 01 ( 1 σ ) = P 10 σ .
P 00 + P 01 = R g ,
P 10 + P 11 = R g ,
J 00 = I s P 00 ( 1 σ ) ,
J 01 = I s T 1 P 01 ( 1 σ ) ,
J 10 = I s T 1 P 10 σ ,
J 11 = I s T 1 2 P 11 σ .
R = 1 I s i = 0 , 1 j = 0 , 1 J i j = i = 0 , 1 j = 0 , 1 T i T j P i j σ i .
R = R MD Δ R .
R MD = R g ( 1 σ ) + R g T 1 2 σ ,
Δ R = ( 1 T 1 ) 2 P 01 ( 1 σ ) = ( 1 T 1 ) 2 P 10 σ
I s ( λ ) = I vis ( λ ) + I u v ( λ ) ,
I 0 ( λ ) = [ I vis ( λ ) + u v I u v ( λ ) f ( λ , λ ) d λ ] R g ( λ ) ,
= I vis ( λ ) R g ( λ ) [ 1 + F 0 ( λ ) ] ,
F 0 ( λ ) = I u v ( λ ) f ( λ , λ ) d λ I vis ( λ ) .
R 0 ( λ ) = R g ( λ ) [ 1 + F 0 ( λ ) ] .
I 1 ( λ ) = [ I vis T 1 2 ( λ ) + T 1 ( λ ) u v I u v ( λ ) T 1 ( λ ) f ( λ , λ ) d λ ] R g ( λ ) ,
= I vis [ T 1 2 ( λ ) + T 1 F 1 ( λ ) ] R g ( λ ) .
R 1 ( λ ) = R g ( λ ) [ T 1 ( λ ) + F 1 ( λ ) ] T 1 ( λ ) ,
F 1 ( λ ) = I u v ( λ ) T 1 ( λ ) f ( λ , λ ) d λ I vis ( λ ) .
F 0 ( λ ) = R 0 ( λ ) R g ( λ ) R g ( λ ) ,
F 1 ( λ ) = R 1 ( λ ) R g ( λ ) T 1 2 ( λ ) R g ( λ ) T 1 ( λ ) .
J i j = I vis T i T j P i j σ i , ( i , j = 0 , 1 ) .
T 0 = T 0 + F 0 = 1 + F 0 ,
T 1 = T 1 + F 1 .
R = ( i = 0 , 1 j = 0 , 1 J i j ) I vis = i = 0 , 1 j = 0 , 1 T i T j P i j σ i = R MD Δ R ,
R MD = R g T 0 T 0 ( 1 σ ) + R g T 1 T 1 σ ,
Δ R = ( T 0 T 1 ) ( T 0 T 1 ) P 01 ( 1 σ ) = ( 1 + F 0 T 1 F 1 ) ( 1 T 1 ) P 01 ( 1 σ ) = ( 1 + F 0 T 1 F 1 ) ( 1 T 1 ) P 10 σ .
T 0 = 1 ,
T 1 = T I ,
T 2 = T I T II ,
T 3 = T II .
R = i = 0 N 1 j = 0 N 1 T i T j P i j σ i ,
P i j σ i = P j i σ j , ( i , j = 0 , , N 1 ) .
j = 0 N 1 P i j = R g , ( i , j = 0 , , N 1 ) .
R = i = 0 N 1 j = 0 N 1 T i T j P i j σ i = i = 0 N 1 T i 2 R g σ i i = 0 N 1 j i N 1 T i ( T i T j ) P i j σ i = R NG Δ R .
Δ R = i = 0 N 1 j i N 1 T i ( T i T j ) P i j σ i = i = 0 N 1 j < i N 1 T i ( T i T j ) P i j σ i + i = 0 N 1 j > i N 1 T i ( T i T j ) P i j σ i = i = 0 N 1 j < i N 1 ( T i T j ) 2 P i j σ i .
P i j = R g σ j , ( i , j = 0 , , N 1 ) .
R = R g i = 0 N 1 j = 0 N 1 T i T j σ i σ j = R g ( i = 0 N 1 T i σ i ) 2 ,
R 1 2 = i = 0 N 1 R i 1 2 σ i ,
R i = R g T i 2 .
T i = T i + F i , ( i = 0 , 1 , , N 1 ) .
R = i = 0 N 1 j = 0 N 1 T i T j P i j σ i .
R = i = 0 N 1 j = 0 N 1 T i T j P i j σ i = i = 0 N 1 ( T i + F i ) T i R g σ i i = 0 N 1 j i N 1 ( T i + F i ) ( T i T j ) P i j σ i = R NG Δ R
Δ R = i = 0 N 1 j i N 1 ( T i + F i ) ( T i T j ) P i j σ i = i = 0 N 1 j < i N 1 ( T i + F i T j F j ) ( T i T j ) P i j σ i .
R = k r ex + ( 1 r ex ) ( 1 r in ) [ R I + R II + R III + R IV + ] .
R I = i = 0 N 1 σ i ( j = 0 N 1 T i P i j T j ) = i = 0 N 1 j = 0 N 1 σ i T i P i j T j ,
R II = i = 0 N 1 σ i ( j = 0 N 1 T i P i j T j ) ( r in k = 0 N 1 T j P j k T k ) = r in i = 0 N 1 j = 0 N 1 k = 0 N 1 σ i T i T j 2 T k P i j P j k ,
R III = r in 2 i = 0 N 1 j = 0 N 1 k = 0 N 1 l = 0 N 1 σ i T i T j 2 T k 2 T l P i j P j k P k l ,
R IV = r in 3 i = 0 N 1 j = 0 N 1 k = 0 N 1 l = 0 N 1 m = 0 N 1 σ i T i T j 2 T k 2 T l 2 T m P i j P j k P k l P l m .
R I = i = 0 N 1 j = 0 N 1 σ i ( T i + F i ) T j P i j ,
R II = r in i = 0 N 1 j = 0 N 1 k = 0 N 1 σ i ( T i + F i ) T j 2 T k P i j P j k ,
R III = r in 2 i = 0 N 1 j = 0 N 1 k = 0 N 1 l = 0 N 1 σ i ( T i + F i ) T j 2 T k 2 T l P i j P j k P k l ,
R IV = r in 3 i = 0 N 1 j = 0 N 1 k = 0 N 1 l = 0 N 1 m = 0 N 1 σ i ( T i + F i ) T j 2 T k 2 T l 2 T m P i j P j k P k l P l m ,
R I MD = j = 0 N 1 σ j R g T j ( T j + F j ) ,
R II MD = j = 0 N 1 σ j R g T j ( T j + F j ) ( r in R g T j 2 ) ,
R III MD = j = 0 N 1 σ j R g T j ( T j + F j ) ( r in R g T j 2 ) 2 ,
R IV MD = j = 0 N 1 σ i R g T j ( T j + F j ) ( r in R g T j 2 ) 3 .
R = k r ex + ( 1 r ex ) ( 1 r in ) [ R I MD + R II MD + R III MD + R IV MD + ] = k r ex + ( 1 r ex ) ( 1 r in ) j = 0 N 1 σ i R g T j ( T j + F j ) 1 r in R g T j 2 .
R I = i = 0 N 1 σ i j = 0 N 1 ( T i + F i ) R g σ j T j = R g i = 0 N 1 ( T i + F i ) σ i ( j = 0 N 1 T j σ j ) ,
R II = R g [ i = 0 N 1 σ i ( T i + F i ) ( j = 0 N 1 σ j T j ) ] ( r in R g k = 0 N 1 σ k T k 2 ) ,
R III = R g [ i = 0 N 1 σ i ( T i + F i ) ( j = 0 N 1 T j σ j ) ] ( r in R g k = 0 N 1 T k 2 σ k ) 2 ,
R IV = R g [ i = 0 N 1 σ i ( T i + F i ) ( j = 0 N 1 T j σ j ) ] ( r in R g k = 0 N 1 T k 2 σ k ) 3 .
R = k r ex + ( 1 r ex ) ( 1 r in ) [ R I + R II + R III + R IV + ] = k r ex + ( 1 r ex ) ( 1 r in ) R g [ i = 0 N 1 σ i ( T i + F i ) ( j = 0 N 1 T j σ j ) ] 1 ( r in R g k = 0 N 1 T k 2 σ k ) .
T i = T i + F i ( i = 0 , 1 , 2 , ) .
R i = k r ex + ( 1 r ex ) ( 1 r in ) R g T i ( T i + F i ) 1 r in R g T i 2 ,
R i vis = k r ex + ( 1 r ex ) ( 1 r in ) R g T i 2 1 r in R g T i 2 .
F i = R i R i vis R i vis k r ex T i ( i = 0 , 1 , 2 , ) .

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