Abstract

When a transparency printed with a first halftone color is deposited on top of a paper printed with a second halftone color, we obtain a third color that we are able to predict in both reflectance and transmittance modes, thanks to a spectral prediction model. The model accounts for the multiple reflections of light between the printed paper and the printed transparency, which are themselves described by specific reflectance and transmittance models, each one being calibrated using a small number of printed colors. The model can account for light scattering by the inks. The measuring geometry and the orientations of light in the transparency are taken into account on the basis of radiometric rules and geometrical optical laws. Experimental testing carried out from several inkjet-printed CMY halftones shows fairly good agreement between predictions and measurements.

© 2012 Optical Society of America

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References

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  1. R. D. Hersch and F. Crété, “Improving the Yule–Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions,” Proc. SPIE 5667, 434–447 (2005).
    [CrossRef]
  2. 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.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. M. Hébert and R. D. Hersch, “Yule–Nielsen based recto-verso color halftone transmittance prediction model,” Appl. Opt. 50, 519–525 (2011).
    [CrossRef]
  8. M. Hébert and R. D. Hersch, “Deducing ink-transmittance spectra from reflectance and transmittance measurements of prints,” Proc. SPIE 6493, 649314 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. P. Emmel, I. Amidror, V. Ostromoukhov, and R. D. Hersch, “Predicting the spectral behaviour of colour printers for transparent inks on transparent support,” in Proceedings of IS&T/SID 96, Color Imaging Conference (Society for Imaging Science, 1996), pp. 86–91.
  13. J. McElvain, J. Miller, and E. Jin, “Spectral printer modeling for transparency media: toward high dynamic range scene reproduction,” Proc. SPIE 7241, 72410U (2009).
    [CrossRef]
  14. I. Amidror, The Theory of the Moiré Phenomenon: Periodic Layers, 2nd ed. (Springer, 2009).
  15. V. Ostromoukhov and R. D. Hersch, “Stochastic clustered-dot dithering,” J. Electron. Imag. 8, 439–445 (1999).
  16. M. Qi, C. Yang, C. Tu, X. Meng, and Y. Sun, “A GPU-based algorithm for building stochastic clustered-dot screens,” Adv. Vis. Comput. 4841, 98–105 (2007).
    [CrossRef]
  17. J. Stover, Optical Scattering: Measurement and Analysis (SPIE Press, 1995).
  18. F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (U.S. Dept. of Commerce, 1977).
  19. M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
  20. CIE, Colorimetry CIE Technical Report, 3rd ed. (CIE, 1998).
  21. H.-H. Perkampus, Encyclopedia of Spectroscopy (VCH, 1995).
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    [CrossRef]
  23. I. Amidror and R. D. Hersch, “Neugebauer and Demichel: dependence and independence in n-screen superpositions for colour printing,” Color Res. Appl. 25, 267–277 (2000).
    [CrossRef]
  24. J. A. C. Yule and W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” in Proceedings of the Technical Association of the Graphic Arts, Vol. 3 (TAGA, 1951), pp. 65–76.
  25. T. Bugnon and R. D. Hersch, “Constrained acquisition of ink spreading curves from printed color images,” IEEE Trans. Image Process. 20, 513–522 (2011).
    [CrossRef]
  26. A. Glassner, Principles of Digital Image Synthesis, Vol. 2 (Kaufmann, 1995).
  27. G. Sharma, Digital Color Imaging Handbook (CRC Press, 2003), pp. 30–36.

2011 (2)

T. Bugnon and R. D. Hersch, “Constrained acquisition of ink spreading curves from printed color images,” IEEE Trans. Image Process. 20, 513–522 (2011).
[CrossRef]

M. Hébert and R. D. Hersch, “Yule–Nielsen based recto-verso color halftone transmittance prediction model,” Appl. Opt. 50, 519–525 (2011).
[CrossRef]

2009 (2)

M. Hébert and R. D. Hersch, “Reflectance and transmittance model for recto-verso halftone prints: spectral predictions with multi-ink halftones,” J. Opt. Soc. Am. A 26, 356–364 (2009).
[CrossRef]

J. McElvain, J. Miller, and E. Jin, “Spectral printer modeling for transparency media: toward high dynamic range scene reproduction,” Proc. SPIE 7241, 72410U (2009).
[CrossRef]

2008 (1)

2007 (3)

M. Hébert, R. D. Hersch, and J. Becker, “Compositional reflectance and transmittance model for multilayer specimens,” J. Opt. Soc. Am. A 24, 2628–2644 (2007).
[CrossRef]

M. Qi, C. Yang, C. Tu, X. Meng, and Y. Sun, “A GPU-based algorithm for building stochastic clustered-dot screens,” Adv. Vis. Comput. 4841, 98–105 (2007).
[CrossRef]

M. Hébert and R. D. Hersch, “Deducing ink-transmittance spectra from reflectance and transmittance measurements of prints,” Proc. SPIE 6493, 649314 (2007).
[CrossRef]

2006 (2)

2005 (1)

R. D. Hersch and F. Crété, “Improving the Yule–Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions,” Proc. SPIE 5667, 434–447 (2005).
[CrossRef]

2000 (1)

I. Amidror and R. D. Hersch, “Neugebauer and Demichel: dependence and independence in n-screen superpositions for colour printing,” Color Res. Appl. 25, 267–277 (2000).
[CrossRef]

1999 (1)

V. Ostromoukhov and R. D. Hersch, “Stochastic clustered-dot dithering,” J. Electron. Imag. 8, 439–445 (1999).

1978 (1)

1953 (2)

Amidror, I.

I. Amidror and R. D. Hersch, “Neugebauer and Demichel: dependence and independence in n-screen superpositions for colour printing,” Color Res. Appl. 25, 267–277 (2000).
[CrossRef]

P. Emmel, I. Amidror, V. Ostromoukhov, and R. D. Hersch, “Predicting the spectral behaviour of colour printers for transparent inks on transparent support,” in Proceedings of IS&T/SID 96, Color Imaging Conference (Society for Imaging Science, 1996), pp. 86–91.

I. Amidror, The Theory of the Moiré Phenomenon: Periodic Layers, 2nd ed. (Springer, 2009).

Becker, J.

Bhatia, A.

M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Born, M.

M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Bugnon, T.

T. Bugnon and R. D. Hersch, “Constrained acquisition of ink spreading curves from printed color images,” IEEE Trans. Image Process. 20, 513–522 (2011).
[CrossRef]

Clapper, F.

Clapper, F. R.

Crété, F.

R. D. Hersch and F. Crété, “Improving the Yule–Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions,” Proc. SPIE 5667, 434–447 (2005).
[CrossRef]

Emmel, P.

P. Emmel, I. Amidror, V. Ostromoukhov, and R. D. Hersch, “Predicting the spectral behaviour of colour printers for transparent inks on transparent support,” in Proceedings of IS&T/SID 96, Color Imaging Conference (Society for Imaging Science, 1996), pp. 86–91.

Ginsberg, I.

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (U.S. Dept. of Commerce, 1977).

Glassner, A.

A. Glassner, Principles of Digital Image Synthesis, Vol. 2 (Kaufmann, 1995).

Hauser, O. G.

Hébert, M.

Hersch, R. D.

M. Hébert and R. D. Hersch, “Yule–Nielsen based recto-verso color halftone transmittance prediction model,” Appl. Opt. 50, 519–525 (2011).
[CrossRef]

T. Bugnon and R. D. Hersch, “Constrained acquisition of ink spreading curves from printed color images,” IEEE Trans. Image Process. 20, 513–522 (2011).
[CrossRef]

M. Hébert and R. D. Hersch, “Reflectance and transmittance model for recto-verso halftone prints: spectral predictions with multi-ink halftones,” J. Opt. Soc. Am. A 26, 356–364 (2009).
[CrossRef]

M. Hébert, R. D. Hersch, and L. Simonot, “Spectral prediction model for piles of nonscattering sheets,” J. Opt. Soc. Am. A 25, 2066–2077 (2008).
[CrossRef]

M. Hébert, R. D. Hersch, and J. Becker, “Compositional reflectance and transmittance model for multilayer specimens,” J. Opt. Soc. Am. A 24, 2628–2644 (2007).
[CrossRef]

M. Hébert and R. D. Hersch, “Deducing ink-transmittance spectra from reflectance and transmittance measurements of prints,” Proc. SPIE 6493, 649314 (2007).
[CrossRef]

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]

L. Simonot, M. Hébert, and R. D. Hersch, “Extension of the Williams–Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432–1441 (2006).
[CrossRef]

R. D. Hersch and F. Crété, “Improving the Yule–Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions,” Proc. SPIE 5667, 434–447 (2005).
[CrossRef]

I. Amidror and R. D. Hersch, “Neugebauer and Demichel: dependence and independence in n-screen superpositions for colour printing,” Color Res. Appl. 25, 267–277 (2000).
[CrossRef]

V. Ostromoukhov and R. D. Hersch, “Stochastic clustered-dot dithering,” J. Electron. Imag. 8, 439–445 (1999).

P. Emmel, I. Amidror, V. Ostromoukhov, and R. D. Hersch, “Predicting the spectral behaviour of colour printers for transparent inks on transparent support,” in Proceedings of IS&T/SID 96, Color Imaging Conference (Society for Imaging Science, 1996), pp. 86–91.

Hsia, J.

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (U.S. Dept. of Commerce, 1977).

Jin, E.

J. McElvain, J. Miller, and E. Jin, “Spectral printer modeling for transparency media: toward high dynamic range scene reproduction,” Proc. SPIE 7241, 72410U (2009).
[CrossRef]

Limperis, T.

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (U.S. Dept. of Commerce, 1977).

McElvain, J.

J. McElvain, J. Miller, and E. Jin, “Spectral printer modeling for transparency media: toward high dynamic range scene reproduction,” Proc. SPIE 7241, 72410U (2009).
[CrossRef]

Meng, X.

M. Qi, C. Yang, C. Tu, X. Meng, and Y. Sun, “A GPU-based algorithm for building stochastic clustered-dot screens,” Adv. Vis. Comput. 4841, 98–105 (2007).
[CrossRef]

Miller, J.

J. McElvain, J. Miller, and E. Jin, “Spectral printer modeling for transparency media: toward high dynamic range scene reproduction,” Proc. SPIE 7241, 72410U (2009).
[CrossRef]

Nicodemus, F.

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (U.S. Dept. of Commerce, 1977).

Nielsen, W. J.

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

Ostromoukhov, V.

V. Ostromoukhov and R. D. Hersch, “Stochastic clustered-dot dithering,” J. Electron. Imag. 8, 439–445 (1999).

P. Emmel, I. Amidror, V. Ostromoukhov, and R. D. Hersch, “Predicting the spectral behaviour of colour printers for transparent inks on transparent support,” in Proceedings of IS&T/SID 96, Color Imaging Conference (Society for Imaging Science, 1996), pp. 86–91.

Perkampus, H.-H.

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

Qi, M.

M. Qi, C. Yang, C. Tu, X. Meng, and Y. Sun, “A GPU-based algorithm for building stochastic clustered-dot screens,” Adv. Vis. Comput. 4841, 98–105 (2007).
[CrossRef]

Richmond, J.

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (U.S. Dept. of Commerce, 1977).

Ruckdeschel, F. R.

Sharma, G.

G. Sharma, Digital Color Imaging Handbook (CRC Press, 2003), pp. 30–36.

Simonot, L.

Stover, J.

J. Stover, Optical Scattering: Measurement and Analysis (SPIE Press, 1995).

Sun, Y.

M. Qi, C. Yang, C. Tu, X. Meng, and Y. Sun, “A GPU-based algorithm for building stochastic clustered-dot screens,” Adv. Vis. Comput. 4841, 98–105 (2007).
[CrossRef]

Tu, C.

M. Qi, C. Yang, C. Tu, X. Meng, and Y. Sun, “A GPU-based algorithm for building stochastic clustered-dot screens,” Adv. Vis. Comput. 4841, 98–105 (2007).
[CrossRef]

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, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Yang, C.

M. Qi, C. Yang, C. Tu, X. Meng, and Y. Sun, “A GPU-based algorithm for building stochastic clustered-dot screens,” Adv. Vis. Comput. 4841, 98–105 (2007).
[CrossRef]

Yule, J.

Yule, J. A. C.

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

Adv. Vis. Comput. (1)

M. Qi, C. Yang, C. Tu, X. Meng, and Y. Sun, “A GPU-based algorithm for building stochastic clustered-dot screens,” Adv. Vis. Comput. 4841, 98–105 (2007).
[CrossRef]

Appl. Opt. (2)

Color Res. Appl. (1)

I. Amidror and R. D. Hersch, “Neugebauer and Demichel: dependence and independence in n-screen superpositions for colour printing,” Color Res. Appl. 25, 267–277 (2000).
[CrossRef]

IEEE Trans. Image Process. (1)

T. Bugnon and R. D. Hersch, “Constrained acquisition of ink spreading curves from printed color images,” IEEE Trans. Image Process. 20, 513–522 (2011).
[CrossRef]

J. Electron. Imag. (1)

V. Ostromoukhov and R. D. Hersch, “Stochastic clustered-dot dithering,” J. Electron. Imag. 8, 439–445 (1999).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Proc. SPIE (3)

R. D. Hersch and F. Crété, “Improving the Yule–Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions,” Proc. SPIE 5667, 434–447 (2005).
[CrossRef]

M. Hébert and R. D. Hersch, “Deducing ink-transmittance spectra from reflectance and transmittance measurements of prints,” Proc. SPIE 6493, 649314 (2007).
[CrossRef]

J. McElvain, J. Miller, and E. Jin, “Spectral printer modeling for transparency media: toward high dynamic range scene reproduction,” Proc. SPIE 7241, 72410U (2009).
[CrossRef]

Other (11)

I. Amidror, The Theory of the Moiré Phenomenon: Periodic Layers, 2nd ed. (Springer, 2009).

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

A. Glassner, Principles of Digital Image Synthesis, Vol. 2 (Kaufmann, 1995).

G. Sharma, Digital Color Imaging Handbook (CRC Press, 2003), pp. 30–36.

P. Emmel, I. Amidror, V. Ostromoukhov, and R. D. Hersch, “Predicting the spectral behaviour of colour printers for transparent inks on transparent support,” in Proceedings of IS&T/SID 96, Color Imaging Conference (Society for Imaging Science, 1996), pp. 86–91.

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.

J. Stover, Optical Scattering: Measurement and Analysis (SPIE Press, 1995).

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (U.S. Dept. of Commerce, 1977).

M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

CIE, Colorimetry CIE Technical Report, 3rd ed. (CIE, 1998).

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

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Figures (8)

Fig. 1.
Fig. 1.

Measuring geometry in reflectance mode (only E A is received) and transmittance mode (only E A is received).

Fig. 2.
Fig. 2.

Example of ink-spreading curve, giving the effective surface coverage of ink i when superposed on colorant j as a function of the nominal surface coverage a .

Fig. 3.
Fig. 3.

Reflection and transmission of directional light by a nonscattering film.

Fig. 4.
Fig. 4.

Light is multiply reflected within each halftone dot.

Fig. 5.
Fig. 5.

Spectral reflectance of solid ink layers printed on a transparency: cyan (dotted curve), magenta (dashed-dotted curve), yellow (dashed curve) and no ink (solid curve).

Fig. 6.
Fig. 6.

Reflection and transmission of light by a printed transparency superposed with a printed paper.

Fig. 7.
Fig. 7.

Measured spectral reflectance of a solid yellow printed on the transparency superposed with the unprinted MP101 matte paper for a d i : 8 ° geometry (solid curve) and predicted spectra with K = 1 (dashed curve) with K = 0 (dashed–dotted curve).

Fig. 8.
Fig. 8.

Measured (solid curve) and predicted (dashed curve) spectral reflectances at normal incidence of three superpositions of printed transparency and printed paper.

Tables (3)

Tables Icon

Table 1. Prediction Accuracy for Printed Papers

Tables Icon

Table 2. Prediction Accuracy for Printed Transparencies

Tables Icon

Table 3. Average Color Differences Denoting the Deviation between Measured and Predicted Spectral Reflectances

Equations (58)

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

L = E / π .
R = ξ L D E A ,
T = ξ L D E A ,
R ^ = π L D E A ,
T ^ = π L D E A .
t = exp ( α h ) .
exp ( α h / cos θ ) = t 1 / cos θ .
n j sin θ j = n k sin θ k ,
R j k ( θ j ) = 1 2 [ n j ( n k 2 n j 2 sin 2 θ j ) 1 2 n k 2 cos θ j n j ( n k 2 n j 2 sin 2 θ j ) 1 2 + n k 2 cos θ j ] 2 + 1 2 [ ( n k 2 n j 2 sin 2 θ j ) 1 2 n j cos θ j ( n k 2 n j 2 sin 2 θ j ) 1 2 + n j cos θ j ] 2 .
R k j ( θ k ) = R j k ( θ j ) .
T j k ( θ j ) = 1 R j k ( θ j ) .
R j k ( 0 ) = ( n k n j n k + n j ) 2 .
r j k = θ j = 0 π / 2 R j k ( θ j ) sin 2 θ j d θ j ,
t j k = θ j = 0 π / 2 T j k ( θ j ) sin 2 θ j d θ j = 1 r j k .
t k j = ( n j n k ) 2 t j k ,
r k j = 1 ( n j n k ) 2 ( 1 r j k ) .
a w = ( 1 c ) ( 1 m ) ( 1 y ) a c = c ( 1 m ) ( 1 y ) a m = ( 1 c ) m ( 1 y ) a y = ( 1 c ) ( 1 m ) y a m + y = ( 1 c ) m y a c + y = c ( 1 m ) y a c + m = c m ( 1 y ) a c + m + y = c m y .
R ^ p ( λ ) = [ k = 1 8 a k R ^ k 1 / n ( λ ) ] n ,
T ^ p ( λ ) = [ k = 1 8 a k T ^ k 1 / n ( λ ) ] n .
R ^ p ( i / j ) ( a i / j , λ ) = [ ( 1 a i / j ) R ^ j 1 / n ( λ ) + a i / j R ^ i / j 1 / n ( λ ) ] n .
a i / j = arg min 0 < a < 1 λ [ R ^ p ( i / j ) ( a , λ ) P ^ p ( i / j ) ( λ ) ] 2 .
T ^ p ( i / j ) ( a i / j , λ ) = [ ( 1 a i / j ) T ^ j 1 / n ( λ ) + a i / j T ^ i / j 1 / n ( λ ) ] n .
a i / j = arg min 0 < a < 1 λ [ T ^ p ( i / j ) ( a , λ ) Q ^ p ( i / j ) ( λ ) ] 2 .
c = ( 1 m ) ( 1 y ) f c / w ( c 0 ) + m ( 1 y ) f c / m ( c 0 ) + ( 1 m ) y f c / y ( c 0 ) + m y f c / ( m + y ) ( c 0 ) m = ( 1 c ) ( 1 y ) f m / w ( m 0 ) + c ( 1 y ) f m / c ( m 0 ) + ( 1 c ) y f m / y ( m 0 ) + c y f m / ( c + y ) ( m 0 ) y = ( 1 m ) ( 1 c ) f y / w ( y 0 ) + m ( 1 c ) f y / m ( y 0 ) + ( 1 m ) c f y / c ( y 0 ) + m c f y / ( m + c ) ( y 0 ) .
n 0 sin θ 0 = n 1 sin θ 1 .
1 cos θ 1 = ( 1 sin 2 θ 0 n 1 2 ) 1 / 2 .
R 010 ( θ 0 , t t , λ ) = R 01 ( θ 0 ) + T 01 2 ( θ 0 ) R 01 ( θ 0 ) t t 2 / cos θ 1 ( λ ) 1 R 01 2 ( θ 0 ) t t 2 / cos θ 1 ( λ ) ,
T 010 ( θ 0 , t t , λ ) = T 01 2 ( θ 0 ) t t 1 / cos θ 1 ( λ ) 1 R 01 2 ( θ 0 ) t t 2 / cos θ 1 ( λ ) .
R 01 ( 0 ) = r 0 = ( n 1 n 0 n 1 + n 0 ) 2 ,
T 01 ( 0 ) = 1 r 0 .
R 010 ( 0 , t k , λ ) = r 0 + ( 1 r 0 ) 2 r 0 t k 2 ( λ ) 1 r 0 2 t k 2 ( λ ) ,
T 010 ( 0 , t k , λ ) = ( 1 r 0 ) 2 t k ( λ ) 1 r 0 2 t k 2 ( λ ) .
t k = [ ( 1 r 0 ) 4 + 4 r 0 2 ( T 010 ( k ) ) 2 ] 1 / 2 ( 1 r 0 ) 2 2 r 0 2 T 010 ( k ) .
R 010 ( θ 0 , a k , t k , λ ) = k = 1 8 a k R 010 ( θ 0 , t k , λ ) ,
T 010 ( θ 0 , a k , t k , λ ) = k = 1 8 a k T 010 ( θ 0 , t k , λ ) .
R 010 ( θ 0 , a k , t k , λ ) = [ k = 1 8 a k R 010 1 / n ( θ 0 , t k , λ ) ] n .
T 010 ( θ 0 , a k , t k , λ ) = [ k = 1 8 a k T 010 1 / n ( θ 0 , t k , λ ) ] n ,
R ^ ( λ ) = R 010 ( 0 , a k , t k , λ ) + K ρ ( λ ) .
ρ ( λ ) = k = 1 8 a k ρ k ( λ ) .
T t ( i / j ) ( a i / j , λ ) = { ( 1 a i / j ) [ ( 1 r 0 ) 2 t i ( λ ) 1 r 0 2 t i 2 ( λ ) ] 1 / n + a i / j [ ( 1 r 0 ) 2 t j ( λ ) 1 r 0 2 t j 2 ( λ ) ] 1 / n } n ,
a i / j = arg min a λ [ T t ( i / j ) ( a , λ ) Q t ( i / j ) ( λ ) ] 2 .
b i / j = arg min b λ [ ( 1 b ) ρ i ( λ ) + b ρ j ( λ ) P ^ t ( i / j ) ( λ ) ] 2 ,
R ^ ( θ 0 , a k , t k , b k , ρ k , λ , K ) = [ k = 1 8 a k R 010 1 / n ( θ 0 , t k , λ ) ] n + K k = 1 8 b k ρ k ( λ ) .
T ( θ 0 , a k , t k , λ ) = [ k = 1 8 a k T 010 1 / n ( θ 0 , t k , λ ) ] n .
r ( a k , t k , b k , ρ k , λ , K ) = θ 0 = 0 π / 2 R ^ ( θ 0 , a k , t k , b k , ρ k , λ , K ) sin 2 θ 0 d θ 0 ,
t ( a k , t k , λ ) = θ 0 = 0 π / 2 T ( θ 0 , a k , t k , λ ) sin 2 θ 0 d θ 0 .
E B = T in E A + r i E C E C = T p E A + R p E B L D = r s E A + T ex E C .
R ^ t + p = π r s + π T in T ex R p 1 r i R p ,
T ^ t + p = π T p T ex 1 r i R p .
T ex = 1 π T ( 0 , a k , t k , λ ) .
r s = 1 π [ k = 1 8 a k R 010 1 / n ( θ 0 , t k , λ ) ] n + K π k = 1 8 b k ρ k ( λ ) .
r s = K π k = 1 8 b k ρ k ( λ ) .
L D = d 2 Φ d d s d d Ω d .
d s d d Ω d = d s d ω cos θ d ,
d 2 Φ d d Φ i = L D d s d d Ω d E i d s = d s d cos θ d x 2 L D E i .
R = d 2 Φ d d Φ i = ξ L D E i ,
ξ = d s d cos θ d x 2 .
R ^ = R R ref = π L D E i .

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