Abstract

We propose a spectral prediction model for predicting the reflectance and transmittance of recto–verso halftone prints. A recto–verso halftone print is modeled as a diffusing substrate surrounded by two inked interfaces in contact with air (or with another medium). The interaction of light with the print comprises three components: (a) the attenuation of the incident light penetrating the print across the inked interface, (b) the internal reflectance and internal transmittance that accounts for the substrate’s intrinsic reflectance and transmittance and for the multiple Fresnel internal reflections at the inked interfaces, and (c) the attenuation of light exiting the print across the inked interfaces. Both the classical Williams–Clapper and Clapper–Yule spectral prediction models are special cases of the proposed recto–verso reflectance and transmittance model. We also extend the Kubelka–Munk model to predict the reflectance and transmittance of recto–verso halftone prints. The extended Kubelka–Munk model is compatible with the proposed recto–verso reflectance and transmittance model. In the case of a homogeneous substrate, the recto–verso model’s internal reflectance and transmittance can be expressed as a function Kubelka–Munk’s scattering and absorption parameters, or the Kubelka–Munk’s scattering and absorption parameters can be inferred from the recto–verso model’s internal reflectance and transmittance, deduced from spectral measurements. The proposed model offers new perspectives both for spectral transmission and reflection predictions and for characterizing the properties of printed diffuse substrates.

© 2006 Optical Society of America

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  1. P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).
  2. H. E. J. Neugebauer, "Die theoretischen Grundlagen des Mehrfarbendrucks," Z. Wiss. Photog. 36, 73-89 (1937); reprinted in Neubegauer Memorial Seminar on Color Reproduction, K.Sayanasi, ed., Proc. SPIE 1184, 194-202 (1989).
  3. J. A. C. Yule and W. J. Nielsen, "The penetration of light into paper and its effect on halftone reproductions," in Proceedings of the Technical Association of the Graphic Arts (TAGA, 1951), Vol. 3, pp. 65-76.
  4. J. A. S. Viggiano, "Modeling the color of multi-colored halftones," in Proceedings of the Technical Association of the Graphic Arts (TAGA, 1990), pp. 44-62.
  5. F. R. Clapper and J. A. C. Yule, "Effect of multiple internal reflections on the densities of halftone prints on paper," J. Opt. Soc. Am. 43, 600-603 (1953).
    [CrossRef]
  6. F. C. Williams and F. R. Clapper, "Multiple internal reflections in photographic color prints," J. Opt. Soc. Am. 43, 595-599 (1953).
    [CrossRef] [PubMed]
  7. P. Kubelka, "New contributions to the optics of intensely light-scattering material. Part I," J. Opt. Soc. Am. 38, 448-457 (1948).
    [CrossRef] [PubMed]
  8. P. Kubelka, "New contributions to the optics of intensely light-scattering materials. Part II: Nonhomogeneous layers," J. Opt. Soc. Am. 44, 330-335 (1954).
    [CrossRef]
  9. J. L. Saunderson, "Calculation of the color pigmented plastics," J. Opt. Soc. Am. 32, 727-736 (1942).
    [CrossRef]
  10. J. S. Arney, "A probability description of the Yule-Nielsen effect: I," J. Imaging Sci. Technol. 41, 633-636 (1997).
  11. G. Rogers, "Effect of light scatter on halftone color," J. Opt. Soc. Am. A 15, 1813-1821 (1998).
    [CrossRef]
  12. 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]
  13. G. Rogers, "A generalized Clapper-Yule model of halftone reflectance," Color Res. Appl. 25, 402-407 (2000).
    [CrossRef]
  14. R. D. Hersch, P. Emmel, F. Collaud, and F. Crété, "Spectral reflection and dot surface prediction models for color halftone prints," J. Electron. Imaging 14, 33001-12 (2005).
    [CrossRef]
  15. M. Hébert and R. D. Hersch, "Extending the Clapper-Yule model to rough printing supports," J. Opt. Soc. Am. A 22, 1952-1967 (2005).
    [CrossRef]
  16. P. Emmel and R. D. Hersch, "A unified model for color prediction of halftoned prints," J. Imaging Sci. Technol. 44, 351-359 (2000).
  17. L. Yang and B. Kruse, "Revised Kubelka-Munk theory. I. Theory and application," J. Opt. Soc. Am. A 21, 1933-1941 (2004).
    [CrossRef]
  18. L. Yang and B. Kruse, "Revised Kubelka-Munk theory. II. Unified framework for homogeneous and inhomogeneous optical media," J. Opt. Soc. Am. A 21, 1942-1952 (2004).
    [CrossRef]
  19. L. Yang and B. Kruse, "Revised Kubelka-Munk theory. III. A general theory of light propagation in scattering and absorptive media," J. Opt. Soc. Am. A 22, 1866-1873 (2005).
    [CrossRef]
  20. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  21. W. E. Vargas and G. A. Niklasson, "Applicability conditions of the Kubelka-Munk theory," Appl. Opt. 36, 5580-5586 (1997).
    [CrossRef] [PubMed]
  22. M. Born and E. Wolf, Principle of Optics, 7th expanded ed. (Pergamon, 1999), p. 47.
  23. W. R. McCluney, Introduction to Radiometry and Photometry (Artech, 1994), pp. 7-13.
  24. D. B. Judd, "Fresnel reflection of diffusely incident light," J. Res. Natl. Bur. Stand. 29, 329-332 (1942).
  25. H.-H. Perkampus, Encyclopedia of Spectroscopy (VCH, 1995).
  26. J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).
  27. H. Hébert and R. D. Hersch, "Classical print reflection models: a radiometric approach," J. Imaging Sci. Technol. 48, 363-374 (2004).
  28. M. E. Demichel, Procédé, 26, 17-21 (1924), see also Ref. .
  29. D. R. Wyble and R. S. Berns, "A critical review of spectral models applied to binary color printing," Color Res. Appl. 25, 4-19 (2000).
    [CrossRef]
  30. G. Sharma, "Color fundamentals for digital imaging," in Digital Color Imaging Handbook, G.Sharma, ed. (CRC Press, 2003), pp. 1-114.
  31. J. W. Harris and H. Stocker, Handbook of Mathematics and Computational Science (Springer-Verlag, 1998), pp. 736-758.

2005 (3)

2004 (3)

2001 (2)

J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (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]

2000 (3)

G. Rogers, "A generalized Clapper-Yule model of halftone reflectance," Color Res. Appl. 25, 402-407 (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).

D. R. Wyble and R. S. Berns, "A critical review of spectral models applied to binary color printing," Color Res. Appl. 25, 4-19 (2000).
[CrossRef]

1998 (1)

1997 (2)

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

W. E. Vargas and G. A. Niklasson, "Applicability conditions of the Kubelka-Munk theory," Appl. Opt. 36, 5580-5586 (1997).
[CrossRef] [PubMed]

1954 (1)

1953 (2)

1948 (1)

1942 (2)

J. L. Saunderson, "Calculation of the color pigmented plastics," J. Opt. Soc. Am. 32, 727-736 (1942).
[CrossRef]

D. B. Judd, "Fresnel reflection of diffusely incident light," J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

1937 (1)

H. E. J. Neugebauer, "Die theoretischen Grundlagen des Mehrfarbendrucks," Z. Wiss. Photog. 36, 73-89 (1937); reprinted in Neubegauer Memorial Seminar on Color Reproduction, K.Sayanasi, ed., Proc. SPIE 1184, 194-202 (1989).

1931 (1)

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, "A probability description of the Yule-Nielsen effect: I," J. Imaging Sci. Technol. 41, 633-636 (1997).

Berns, R. S.

D. R. Wyble and R. S. Berns, "A critical review of spectral models applied to binary color printing," Color Res. Appl. 25, 4-19 (2000).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principle of Optics, 7th expanded ed. (Pergamon, 1999), p. 47.

Chandrasekhar, S.

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

Clapper, F. R.

Collaud, F.

R. D. Hersch, P. Emmel, F. Collaud, and F. Crété, "Spectral reflection and dot surface prediction models for color halftone prints," J. Electron. Imaging 14, 33001-12 (2005).
[CrossRef]

Crété, F.

R. D. Hersch, P. Emmel, F. Collaud, and F. Crété, "Spectral reflection and dot surface prediction models for color halftone prints," J. Electron. Imaging 14, 33001-12 (2005).
[CrossRef]

Demichel, M. E.

M. E. Demichel, Procédé, 26, 17-21 (1924), see also Ref. .

Emmel, P.

R. D. Hersch, P. Emmel, F. Collaud, and F. Crété, "Spectral reflection and dot surface prediction models for color halftone prints," 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).

Harris, J. W.

J. W. Harris and H. Stocker, Handbook of Mathematics and Computational Science (Springer-Verlag, 1998), pp. 736-758.

Hébert, H.

H. Hébert and R. D. Hersch, "Classical print reflection models: a radiometric approach," J. Imaging Sci. Technol. 48, 363-374 (2004).

Hébert, M.

Hersch, R. D.

M. Hébert and R. D. Hersch, "Extending the Clapper-Yule model to rough printing supports," J. Opt. Soc. Am. A 22, 1952-1967 (2005).
[CrossRef]

R. D. Hersch, P. Emmel, F. Collaud, and F. Crété, "Spectral reflection and dot surface prediction models for color halftone prints," J. Electron. Imaging 14, 33001-12 (2005).
[CrossRef]

H. Hébert and R. D. Hersch, "Classical print reflection models: a radiometric approach," J. Imaging Sci. Technol. 48, 363-374 (2004).

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

Judd, D. B.

D. B. Judd, "Fresnel reflection of diffusely incident light," J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

Kruse, B.

Kubelka, P.

Lenz, R.

McCluney, W. R.

W. R. McCluney, Introduction to Radiometry and Photometry (Artech, 1994), pp. 7-13.

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 Mehrfarbendrucks," Z. Wiss. Photog. 36, 73-89 (1937); reprinted in Neubegauer Memorial Seminar on Color Reproduction, K.Sayanasi, ed., Proc. SPIE 1184, 194-202 (1989).

Nielsen, W. J.

J. A. C. Yule and W. J. Nielsen, "The penetration of light into paper and its effect on halftone reproductions," in Proceedings of the Technical Association of the Graphic Arts (TAGA, 1951), Vol. 3, pp. 65-76.

Niklasson, G. A.

Perkampus, H.-H.

H.-H. Perkampus, Encyclopedia of Spectroscopy (VCH, 1995).

Rogers, G.

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

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

Saunderson, J. L.

Sharma, G.

G. Sharma, "Color fundamentals for digital imaging," in Digital Color Imaging Handbook, G.Sharma, ed. (CRC Press, 2003), pp. 1-114.

Shore, J. D.

J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).

Spoonhower, J. P.

J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).

Stocker, H.

J. W. Harris and H. Stocker, Handbook of Mathematics and Computational Science (Springer-Verlag, 1998), pp. 736-758.

Vargas, W. E.

Viggiano, J. A. S.

J. A. S. Viggiano, "Modeling the color of multi-colored halftones," in Proceedings of the Technical Association of the Graphic Arts (TAGA, 1990), pp. 44-62.

Williams, F. C.

Wolf, E.

M. Born and E. Wolf, Principle of Optics, 7th expanded ed. (Pergamon, 1999), p. 47.

Wyble, D. R.

D. R. Wyble and R. S. Berns, "A critical review of spectral models applied to binary color printing," Color Res. Appl. 25, 4-19 (2000).
[CrossRef]

Yang, L.

Yule, J. A. C.

F. R. Clapper and J. A. C. Yule, "Effect of multiple internal reflections on the densities of halftone prints on paper," J. Opt. Soc. Am. 43, 600-603 (1953).
[CrossRef]

J. A. C. Yule and W. J. Nielsen, "The penetration of light into paper and its effect on halftone reproductions," in Proceedings of the Technical Association of the Graphic Arts (TAGA, 1951), Vol. 3, pp. 65-76.

Appl. Opt. (1)

Color Res. Appl. (2)

D. R. Wyble and R. S. Berns, "A critical review of spectral models applied to binary color printing," Color Res. Appl. 25, 4-19 (2000).
[CrossRef]

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

J. Electron. Imaging (1)

R. D. Hersch, P. Emmel, F. Collaud, and F. Crété, "Spectral reflection and dot surface prediction models for color halftone prints," J. Electron. Imaging 14, 33001-12 (2005).
[CrossRef]

J. Imaging Sci. Technol. (4)

J. S. Arney, "A probability description of the Yule-Nielsen effect: I," J. Imaging Sci. Technol. 41, 633-636 (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. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).

H. Hébert and R. D. Hersch, "Classical print reflection models: a radiometric approach," J. Imaging Sci. Technol. 48, 363-374 (2004).

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (6)

J. Res. Natl. Bur. Stand. (1)

D. B. Judd, "Fresnel reflection of diffusely incident light," J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

Z. Tech. Phys. (Leipzig) (1)

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

Z. Wiss. Photog. (1)

H. E. J. Neugebauer, "Die theoretischen Grundlagen des Mehrfarbendrucks," Z. Wiss. Photog. 36, 73-89 (1937); reprinted in Neubegauer Memorial Seminar on Color Reproduction, K.Sayanasi, ed., Proc. SPIE 1184, 194-202 (1989).

Other (9)

J. A. C. Yule and W. J. Nielsen, "The penetration of light into paper and its effect on halftone reproductions," in Proceedings of the Technical Association of the Graphic Arts (TAGA, 1951), Vol. 3, pp. 65-76.

J. A. S. Viggiano, "Modeling the color of multi-colored halftones," in Proceedings of the Technical Association of the Graphic Arts (TAGA, 1990), pp. 44-62.

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

H.-H. Perkampus, Encyclopedia of Spectroscopy (VCH, 1995).

M. Born and E. Wolf, Principle of Optics, 7th expanded ed. (Pergamon, 1999), p. 47.

W. R. McCluney, Introduction to Radiometry and Photometry (Artech, 1994), pp. 7-13.

M. E. Demichel, Procédé, 26, 17-21 (1924), see also Ref. .

G. Sharma, "Color fundamentals for digital imaging," in Digital Color Imaging Handbook, G.Sharma, ed. (CRC Press, 2003), pp. 1-114.

J. W. Harris and H. Stocker, Handbook of Mathematics and Computational Science (Springer-Verlag, 1998), pp. 736-758.

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Figures (4)

Fig. 1
Fig. 1

Interaction of light with the recto–verso print comprising the attenuation of the penetrating light ( T in ) , the attenuation of the emerging light ( T ex ) , the print’s internal reflectance ( R m ) , and internal transmittance ( T m ) . The substrate is characterized by its intrinsic reflectance ρ 1 at its recto side, ρ 2 at its verso side, and by its intrinsic transmittance τ. The reflectances of the colored interfaces are r 1 at the recto and r 2 at the verso.

Fig. 2
Fig. 2

Representation of the paths followed by the diffuse light within the print, accounting for the reflections by the substrate (reflectance ρ 1 at the recto side and ρ 2 at the verso side), the transmissions by the substrate (transmittance τ), and the reflections at the colored interfaces (reflectance r 1 at the recto and r 2 at the verso).

Fig. 3
Fig. 3

Upward and downward irradiances crossing a sublayer of thickness d x at a depth x in the substrate layer.

Fig. 4
Fig. 4

(a) Grounded substrate reflectance ρ B , as defined in the Williams–Clapper model, is the ratio between the irradiance W r reflected by the substrate bounded on its verso and the incident irradiance W i . (b) The reflected irradiance W r can be expressed as a function of the substrate’s intrinsic reflectances ( ρ 1 at the recto side, ρ 2 at the verso side) and its intrinsic transmittance τ by taking into account the multiple internal reflections at the verso interface (reflectance r 2 ).

Tables (2)

Tables Icon

Table 1 Difference in Δ E 94 between Predicted and Measured Transmittance Spectra a

Tables Icon

Table 2 Diffuse Reflectance of Colored Interfaces a

Equations (111)

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t ( λ ) = k = 1 8 a k t k ( λ ) .
R = T in T ex R m .
T = T in T ex T m .
L i ( θ 0 , ϕ 0 ) = Φ i d s cos θ 0 sin θ 0 d θ 0 d ϕ 0 .
n 0 sin θ 0 = n 1 sin θ 1 .
L r ( θ 0 , ϕ 0 ) = R 01 ( θ 0 ) Φ i d s cos θ 0 sin θ 0 d θ 0 d ϕ 0 = R 01 ( θ 0 ) L i ( θ 0 , ϕ 0 ) .
n 0 2 cos θ 0 sin θ 0 d θ 0 = n 1 2 cos θ 1 sin θ 1 d θ 1 .
L t ( θ 0 , ϕ 0 ) = T 01 ( θ 0 ) Φ i d s cos θ 1 sin θ 1 d θ 1 d ϕ 1 ,
L t ( θ 0 , ϕ 0 ) = ( n 1 n 0 ) 2 T 01 ( θ 0 ) L i ( θ 0 , ϕ 0 ) .
R 01 ( θ 0 ) + T 01 ( θ 0 ) = 1 .
T 01 ( θ 0 ) = T 10 ( θ 1 ) .
R 01 ( θ 0 ) = R 10 ( θ 1 ) ,
R 10 ( θ 1 ) + T 10 ( θ 1 ) = 1 .
d E i ( θ 0 , ϕ 0 ) = L i cos θ 0 d Ω 0 = E i π cos θ 0 sin θ 0 d θ 0 d ϕ 0 .
d E r ( θ 0 , ϕ 0 ) = R 01 ( θ 0 ) E i π cos θ 0 sin θ 0 d θ 0 d ϕ 0 .
E r = ϕ 0 = 0 2 π θ 0 = 0 π 2 R 01 ( θ 0 ) E i π cos θ 0 sin θ 0 d θ 0 d ϕ 0 .
E r = E i θ 0 = 0 π 2 R 01 ( θ 0 ) sin 2 θ 0 d θ 0 d ϕ 0 .
r 01 = θ 0 = 0 π 2 R 01 ( θ 0 ) sin 2 θ 0 d θ 0 .
t 01 = θ 0 = 0 π 2 T 01 ( θ 0 ) sin 2 θ 0 d θ 0 .
t 01 = 1 r 01 .
r 10 = θ 1 = 0 π 2 R 10 ( θ 1 ) sin 2 θ 1 d θ 1 ,
t 10 = θ 1 = 0 π 2 T 10 ( θ 1 ) sin 2 θ 1 d θ 1 = 1 r 10 .
t 10 = ( n 0 n 1 ) 2 t 01 .
d E r ( θ 1 , ϕ 1 ) = t 2 cos θ 1 R 10 ( θ 1 ) E i π cos θ 1 sin θ 1 d θ 1 d ϕ 1 .
r ( t ) = θ 1 = 0 π 2 t 2 cos θ 1 R 10 ( θ 1 ) sin 2 θ 1 d θ 1 .
r ( t ) = k a k r ( t k ) = k a k θ 1 = 0 π 2 t k 2 cos θ 1 R 10 ( θ 1 ) sin 2 θ 1 d θ 1 .
μ ( θ 0 ) = 1 cos θ 1 = [ 1 ( sin θ 0 n ) 2 ] 1 2
T in ( θ 0 ) = T 01 ( θ 0 ) t μ ( θ 0 ) .
T in ( θ 0 ) = T 01 ( θ 0 ) k a k t k μ ( θ 0 ) .
T in ( d ) = θ 0 = 0 π 2 T 01 ( θ 0 ) t μ ( θ 0 ) sin 2 θ 0 d θ 0 .
T in ( d ) t μ t 01 .
T in ( d ) = t 01 k a k t k μ .
L d = ( 1 n ) 2 T 10 ( θ 1 ) t μ ( θ 0 ) E p π ,
T ex ( θ 0 ) = L d E p = T 01 ( θ 0 ) π n 2 t μ ( θ 0 ) ,
T ex ( θ 0 ) = T 01 ( θ 0 ) π n 2 k a k t k μ ( θ 0 ) .
d E ( θ 1 , ϕ 1 ) = E p π cos θ 1 sin θ 1 d θ 1 d ϕ 1 .
E = E p θ 1 = 0 π 2 t 1 cos θ 1 T 10 ( θ 1 ) sin 2 θ 1 d θ 1 .
T ex ( d ) = θ 1 = 0 π 2 t 1 cos θ 1 T 10 ( θ 1 ) sin 2 θ 1 d θ 1 .
T ex ( d ) = 1 n 2 θ 1 = 0 π 2 t μ ( θ 0 ) T 01 ( θ 0 ) sin 2 θ 0 d θ 0 .
T ex ( d ) t μ t 10 ,
T ex ( d ) t 10 k a k t k μ .
W r 0 = W i [ ρ 1 + r 1 ρ 1 2 + r 1 2 ρ 1 3 + ] = W i ρ 1 [ 1 + r 1 ρ 1 + r 1 2 ρ 1 2 ] .
W r 0 = W i ρ 1 1 r 1 ρ 1 .
W t 0 = W i [ τ 1 r 2 ρ 2 + r 1 ρ 1 τ 1 r 2 ρ 2 + ( r 1 ρ 1 ) 2 τ 1 r 2 ρ 2 + ] .
W t 0 = W i τ ( 1 r 1 ρ 1 ) ( 1 r 2 ρ 2 ) .
W r 1 = W t 0 r 2 τ ( 1 r 1 ρ 1 ) = W i r 2 τ 2 ( 1 r 1 ρ 1 ) 2 ( 1 r 2 ρ 2 ) .
W t 1 = W i r 1 r 2 τ 3 ( 1 r 1 ρ 1 ) 2 ( 1 r 2 ρ 2 ) 2 .
W r k = W i r 1 k 1 r 2 k τ 2 k ( 1 r 1 ρ 1 ) k + 1 ( 1 r 2 ρ 2 ) k , k = 1 , 2 ,
W t k = W i r 1 k r 2 k τ 2 k + 1 ( 1 r 1 ρ 1 ) k + 1 ( 1 r 2 ρ 2 ) k + 1 , k = 0 , 1 , 2 .
W r = W i ρ 1 1 r 1 ρ 1 + W i 1 r 1 ( 1 r 1 ρ 1 ) k = 1 [ r 1 r 2 τ 2 ( 1 r 1 ρ 1 ) ( 1 r 2 ρ 2 ) ] k = W i [ ρ 1 1 r 1 ρ 1 + 1 r 1 ( 1 r 1 ρ 1 ) r 1 r 2 τ 2 ( 1 r 1 ρ 1 ) ( 1 r 2 ρ 2 ) r 1 r 2 τ 2 ] = W i ρ 1 r 2 ( ρ 2 ρ 1 τ 2 ) ( 1 r 1 ρ 1 ) ( 1 r 2 ρ 2 ) r 1 r 2 τ 2 .
W t = W i τ ( 1 r 1 ρ 1 ) ( 1 r 2 ρ 2 ) k = 0 [ r 1 r 2 τ 2 ( 1 r 1 ρ 1 ) ( 1 r 2 ρ 2 ) ] k = W i τ ( 1 r 1 ρ 1 ) ( 1 r 2 ρ 2 ) r 1 r 2 τ 2 .
R m = ρ 1 r 2 ( ρ 1 ρ 2 τ 2 ) ( 1 r 1 ρ 1 ) ( 1 r 2 ρ 2 ) r 1 r 2 τ 2 ,
T m = τ ( 1 r 1 ρ 1 ) ( 1 r 2 ρ 2 ) r 1 r 2 τ 2 .
R m = ρ r 2 ( ρ 2 τ 2 ) ( 1 r 1 ρ ) ( 1 r 2 ρ ) r 1 r 2 τ 2 ,
T m = τ ( 1 r 1 ρ ) ( 1 r 2 ρ ) r 1 r 2 τ 2 .
R = T in ( θ 0 ) T ex ( θ 0 ) R m = T 01 ( θ 0 ) T 01 ( θ 0 ) π n 2 t 1 μ ( θ 0 ) + μ ( θ 0 ) ρ 1 r ( t 2 ) [ ρ 1 ρ 2 τ 2 ] [ 1 r ( t 1 ) ρ 1 ] [ 1 r ( t 2 ) ρ 2 ] r ( t 1 ) r ( t 2 ) τ 2 .
T = T in ( θ 0 ) T ex ( θ 0 ) T m = T 01 ( θ 0 ) T 01 ( θ 0 ) π n 2 t 1 μ ( θ 0 ) t 2 μ ( θ 0 ) τ [ 1 r ( t 1 ) ρ 1 ] [ 1 r ( t 2 ) ρ 2 ] r ( t 1 ) r ( t 2 ) τ 2 .
i t ( x + d x ) = i t ( x ) ( K + S ) i t ( x ) d x + S i r ( x ) d x .
i r ( x d x ) = i r ( x ) ( K + S ) i r ( x ) d x + S i t ( x ) d x .
d d x i = i ( x ) i ( x d x ) d x = i ( x + d x ) i ( x ) d x ,
d d x i r ( x ) = ( K + S ) i r ( x ) S i t ( x ) ,
d d x i t ( x ) = S i r ( x ) ( K + S ) i t ( x ) .
R m = ( 1 a r 2 ) sinh + b r 2 cosh ( b S h ) ( a r 1 r 2 + a r 1 r 2 ) sinh ( b S h ) + b ( 1 r 1 r 2 ) cosh ( b S h ) ,
T m = b ( a r 1 r 2 + a r 1 r 2 ) sinh ( b S h ) + b ( 1 r 1 r 2 ) cosh ( b S h ) ,
a = K + S S , b = a 2 1 .
ρ = sinh ( b S h ) b cosh ( b S h ) + a sinh ( b S h ) ,
τ = b b cosh ( b S h ) + a sinh ( b S h ) .
R WC = T 01 ( θ 0 ) T 01 ( θ 0 ) π n 2 ρ B t 1 cos θ 0 t 1 cos θ 0 1 ρ B r ( t ) ,
W r = ρ 1 W i + τ 2 r 2 W i ( 1 + r 2 ρ 2 + r 2 2 ρ 2 2 + ) .
W r = ( ρ 1 + r 2 τ 2 1 r 2 ρ 2 ) W i ,
ρ B = W r W i = ρ 1 + r 2 τ 2 1 r 2 ρ 2 .
R WC = T 01 ( θ 0 ) T 01 ( θ 0 ) π n 2 t 1 cos θ 0 t t cos θ 0 ρ 1 r 2 ( ρ 1 ρ 2 τ 2 ) [ 1 r ( t ) ρ 1 ] ( 1 ρ 2 r 2 ) r ( t ) r 2 τ 2 .
R CY = T 01 ( θ 0 ) T 01 ( θ 0 ) π n 2 ρ B ( a k t k ) 2 1 ρ B r 10 a k t k 2 .
T in = T 01 ( θ 0 ) a k t k ,
T ex = T 01 ( θ 0 ) π n 2 a k t k ,
R m = ρ B 1 ρ B r 10 a k t k 2 .
R m = ρ 1 ( 1 ρ 2 r 2 ) + r 2 τ 2 ( 1 ρ 2 r 2 ) [ ( 1 ρ 2 r 2 ) ρ 1 + r 2 τ 2 ] = ρ 1 r 2 ( ρ 1 ρ 2 τ 2 ) ( 1 r 1 ρ 1 ) ( 1 r 2 ρ 2 ) r 1 r 2 τ 2 .
r ( t k ) r 10 t k 2 ,
θ = 0 π 2 t k t cos θ R 10 ( θ ) sin 2 θ d θ t k 2 θ = 0 π 2 R 10 ( θ ) sin 2 θ d θ .
R 1 = T 01 ( θ 0 ) T 01 ( θ 0 ) π n 2 ρ 1 r 10 ( ρ 1 ρ 2 τ 2 ) ( 1 r 10 ρ 1 ) ( 1 r 10 ρ 2 ) ( r 10 τ ) 2 ,
R 2 = T 01 ( θ 0 ) T 01 ( θ 0 ) π n 2 ρ 2 r 10 ( ρ 1 ρ 2 τ 2 ) ( 1 r 10 ρ 1 ) ( 1 r 10 ρ 2 ) ( r 10 τ ) 2 ,
T = T 01 ( θ 0 ) T 01 ( θ 0 ) π n 2 τ ( 1 r 10 ρ 1 ) ( 1 r 10 ρ 2 ) ( ρ 10 τ ) 2 .
R = T 01 ( θ 0 ) T 01 ( θ 0 ) π n 2 t μ ( θ 0 ) + μ ( θ 0 ) ρ 1 r 10 ( ρ 1 ρ 2 τ 2 ) ( 1 r ( t ) ρ 1 ) ( 1 r 10 ρ 2 ) r 10 r ( t ) τ 2 ,
T = T 01 ( θ 0 ) T 01 ( θ 0 ) π n 2 t μ ( θ 0 ) τ ( 1 r ( t ) ρ 1 ) ( 1 r 10 ρ 2 ) r 10 r ( t ) τ 2 .
T in = T 01 ( θ 0 ) ( 1 a + a t μ ( θ 0 ) ) .
T ex = T 01 ( θ 0 ) π n 2 ( 1 a + a t μ ( θ 0 ) ) .
T ex = T 01 ( θ 0 ) π n 2 .
r 1 = ( 1 a ) r 10 + a r ( t ) ,
T in = θ = 0 π 2 T 01 ( θ ) t μ ( θ ) sin 2 θ d θ t μ θ = 0 π 2 T 01 ( θ ) sin 2 θ d θ ,
i = 0 m 1 ( t i μ t 01 θ = 0 π 2 T 01 ( θ ) t i μ ( θ ) sin 2 θ d θ ) 2 .
d d x i r ( x ) = ( K + S ) i r ( x ) S i t ( x ) ,
d d x i t ( x ) = S i r ( x ) ( K + S ) i t ( x ) .
F ( p ) = 0 f ( t ) e p t d t .
p I r ( p ) i r ( 0 ) = ( K + S ) I r ( p ) S I t ( p ) ,
p I t ( p ) i t ( 0 ) = S I r ( p ) ( K + S ) I t ( p ) .
I r ( p ) = i r ( 0 ) ( p + a S ) S i t ( 0 ) p 2 b 2 S 2 ,
I t ( p ) = i t ( 0 ) ( p a S ) + S i r ( 0 ) p 2 b 2 S 2 ,
a = ( K + S ) S ,
b = a 2 1 ,
i r ( x ) = i r ( 0 ) cosh ( b S x ) + 1 b ( a i r ( 0 ) i t ( 0 ) ) sinh ( b S x ) ,
i t ( x ) = i t ( 0 ) cosh ( b S x ) + 1 b ( i r ( 0 ) a i t ( 0 ) ) sinh ( b S x ) .
i t ( 0 ) = I 0 + r 0 i r ( 0 ) .
i r ( h ) = r h i t ( h ) .
r h i t ( h ) = i r ( 0 ) cosh ( b S h ) + 1 b [ a i r ( 0 ) ( I 0 + r 0 i r ( 0 ) ) ] sinh ( b S h ) .
i t ( h ) = ( I 0 + r h i r ( 0 ) ) cosh ( b S h ) + 1 b [ i r ( 0 ) a ( I 0 + r 0 i r ( 0 ) ) ] sinh ( b S h ) .
i r ( 0 ) = I 0 ( 1 a r h ) sinh ( b S h ) + b r h cosh ( b S h ) ( a r 0 r h + a r 0 r h ) sinh ( b S h ) + b ( 1 r 0 r h ) cosh ( b S h ) ,
i t ( h ) = I 0 b ( a r 0 r h + a r 0 r h ) sinh ( b S h ) + b ( 1 r 0 r h ) cosh ( b S h ) .
R b = ( 1 a r h ) sinh ( b S h ) + b r h cosh ( b S h ) ( a r 0 r h + a r 0 r h ) sinh ( b S h ) + b ( 1 r 0 r h ) cosh ( b S h ) ,
T b = b ( a r 0 r h + a r 0 r h ) sinh ( b S h ) + b ( 1 r 0 r h ) cosh ( b S h ) .
R KM = ( 1 a r g ) sinh ( b S h ) + b r g cosh ( b S h ) ( a r g ) sinh ( b S h ) + b cosh ( b S h ) .
R S = T in T ex R KM 1 r 10 R KM .

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