Abstract

We consider the problem of calculating a refractive surface generating a prescribed irradiance distribution in the far field in the case of a point light source. We show that this problem can be formulated as a mass transportation problem with a non-quadratic cost function. A method for calculating the refractive surface is proposed, which is based on reducing the problem of calculating an integrable ray mapping to finding a solution to a linear assignment problem. We discuss the application of the developed method for the design of optical elements with two “working” refractive surfaces. The method was applied to the calculation of refractive optical elements generating uniform irradiance distributions in a square and in a region in the form of the letters “AB” on a zero background. The results of the simulations of the designed optical elements demonstrate high performance of the proposed method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, and J. C. Miñano, “Design of freeform illumination optics,” Laser Photon. Rev. 12(7), 1700310 (2018).
    [Crossref]
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    [Crossref] [PubMed]
  3. R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Xiu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge–Ampère equation,” Opt. Lett. 38(2), 229–231 (2013).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  7. C. R. Prins, J. H. M. ten Thije Boonkkamp, J. van Roosmalen, W. L. IJzerman, and T. W. Tukker, “A Monge–Ampèresolver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
    [Crossref]
  8. C. Bösel and H. Gross, “Ray mapping approach for the efficient design of continuous freeform surfaces,” Opt. Express 24(13), 14271–14282 (2016).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  11. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of the Monge– Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
    [Crossref]
  12. K. Brix, Y. Hafizogullari, and A. Platen, “Designing illumination lenses and mirrors by the numerical solution of Monge–Ampère equations,” J. Opt. Soc. Am. A 32(11), 2227–2236 (2015).
    [Crossref]
  13. K. Brix, Y. Hafizogullari, and A. Platen, “Solving the Monge–Ampère equations for the inverse reflector problem,” Math. Model. Methods Appl. Sci. 25(6), 803–837 (2015).
    [Crossref]
  14. T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
    [Crossref]
  15. X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20(3), 329–341 (2004).
    [Crossref]
  16. C. E. Gutiérrez, “Refraction problems in geometric optics,” Fully Nonlinear PDEs in Real and Complex Geometry and Optics, Vol. 2087 of the Series Lecture Notes in Mathematics (Springer, 2014), pp. 95–150.
    [Crossref]
  17. C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
    [Crossref]
  18. L. L. Doskolovich, A. A. Mingazov, D. A. Bykov, E. S. Andreev, and E. A. Bezus, “Variational approach to calculation of light field eikonal function for illuminating a prescribed region,” Opt. Express 25(22), 26378–26392 (2017).
    [Crossref] [PubMed]
  19. D. A. Bykov, L. L. Doskolovich, A. A. Mingazov, E. A. Bezus, and N. L. Kazanskiy, “Linear assignment problem in the design of freeform refractive optical elements generating prescribed irradiance distributions,” Opt. Express 26(21), 27812–27825 (2018).
    [Crossref] [PubMed]
  20. J. Munkres, “Algorithms for the assignment and transportation problems,” J. Soc. for Ind. Appl. Math. 5(1), 32–38 (1957).
    [Crossref]
  21. D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
    [Crossref]
  22. M. A. Moiseev, S. V. Kravchenko, and L. L. Doskolovich, “Design of efficient LED optics with two free-form surfaces,” Opt. Express 22(S7), A1926–A1935 (2014).
    [Crossref]
  23. L. L. Doskolovich, D. A. Bykov, E. S. Andreev, E. A. Bezus, and V. Oliker, “Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems,” Opt. Express 26(19), 24602–24613 (2018).
    [Crossref] [PubMed]
  24. L. L. Doskolovich, E. S. Andreev, S. I. Kharitonov, and N. L. Kazansky, “Reconstruction of an optical surface from a given source-target map,” J. Opt. Soc. Am. A 33(8), 1504–1508 (2016).
    [Crossref]
  25. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).
  26. L. C. Evans, “Partial differential equations and Monge–Kantorovich mass transfer,” in Current Developments in Mathematics, R. Bott, A. Jaffe, D. Jerison, G. Lusztig, I. Singer, and S.-T. Yau, eds. (International Press of Boston, 1999).
  27. V. Oliker, L. L. Doskolovich, and D. A. Bykov, “Beam shaping with a plano-freeform lens pair,” Opt. Express 26(15), 19406–19419 (2018).
    [Crossref] [PubMed]
  28. X. Mao, H. Li, Y. Han, and Y. Luo, “Polar-grids based source-target mapping construction method for designing freeform illumination system for a lighting target with arbitrary shape,” Opt. Express 23(4), 4313–4328 (2015).
    [Crossref] [PubMed]
  29. Fast linear assignment problem using auction algorithm (mex). http://www.mathworks.com/matlabcentral/fileexchange/48448

2018 (6)

2017 (3)

2016 (2)

2015 (4)

2014 (2)

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. van Roosmalen, W. L. IJzerman, and T. W. Tukker, “A Monge–Ampèresolver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

M. A. Moiseev, S. V. Kravchenko, and L. L. Doskolovich, “Design of efficient LED optics with two free-form surfaces,” Opt. Express 22(S7), A1926–A1935 (2014).
[Crossref]

2013 (2)

2011 (1)

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of the Monge– Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

2009 (1)

C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[Crossref]

2004 (1)

X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20(3), 329–341 (2004).
[Crossref]

2003 (1)

T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
[Crossref]

1988 (1)

D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
[Crossref]

1957 (1)

J. Munkres, “Algorithms for the assignment and transportation problems,” J. Soc. for Ind. Appl. Math. 5(1), 32–38 (1957).
[Crossref]

Andreev, E. S.

Benítez, P.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, and J. C. Miñano, “Design of freeform illumination optics,” Laser Photon. Rev. 12(7), 1700310 (2018).
[Crossref]

Bertsekas, D. P.

D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
[Crossref]

Bezus, E. A.

Bösel, C.

Brix, K.

K. Brix, Y. Hafizogullari, and A. Platen, “Designing illumination lenses and mirrors by the numerical solution of Monge–Ampère equations,” J. Opt. Soc. Am. A 32(11), 2227–2236 (2015).
[Crossref]

K. Brix, Y. Hafizogullari, and A. Platen, “Solving the Monge–Ampère equations for the inverse reflector problem,” Math. Model. Methods Appl. Sci. 25(6), 803–837 (2015).
[Crossref]

Bykov, D. A.

Chang, S.

Doskolovich, L. L.

Evans, L. C.

L. C. Evans, “Partial differential equations and Monge–Kantorovich mass transfer,” in Current Developments in Mathematics, R. Bott, A. Jaffe, D. Jerison, G. Lusztig, I. Singer, and S.-T. Yau, eds. (International Press of Boston, 1999).

Feng, Z.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, and J. C. Miñano, “Design of freeform illumination optics,” Laser Photon. Rev. 12(7), 1700310 (2018).
[Crossref]

Glimm, T.

T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
[Crossref]

Gross, H.

Gutiérrez, C. E.

C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[Crossref]

C. E. Gutiérrez, “Refraction problems in geometric optics,” Fully Nonlinear PDEs in Real and Complex Geometry and Optics, Vol. 2087 of the Series Lecture Notes in Mathematics (Springer, 2014), pp. 95–150.
[Crossref]

Hafizogullari, Y.

K. Brix, Y. Hafizogullari, and A. Platen, “Solving the Monge–Ampère equations for the inverse reflector problem,” Math. Model. Methods Appl. Sci. 25(6), 803–837 (2015).
[Crossref]

K. Brix, Y. Hafizogullari, and A. Platen, “Designing illumination lenses and mirrors by the numerical solution of Monge–Ampère equations,” J. Opt. Soc. Am. A 32(11), 2227–2236 (2015).
[Crossref]

Han, Y.

He, Y.

Hu, X.

Huang, Q.

C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[Crossref]

IJzerman, W. L.

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. van Roosmalen, W. L. IJzerman, and T. W. Tukker, “A Monge–Ampèresolver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

Kazanskiy, N. L.

Kazansky, N. L.

Kharitonov, S. I.

Kravchenko, S. V.

Li, H.

Liang, R.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, and J. C. Miñano, “Design of freeform illumination optics,” Laser Photon. Rev. 12(7), 1700310 (2018).
[Crossref]

Liu, P.

Liu, X.

Lui, X.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

Luo, Y.

Ma, Y.

Mao, X.

Miñano, J. C.

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, and J. C. Miñano, “Design of freeform illumination optics,” Laser Photon. Rev. 12(7), 1700310 (2018).
[Crossref]

Mingazov, A. A.

Moiseev, M. A.

Munkres, J.

J. Munkres, “Algorithms for the assignment and transportation problems,” J. Soc. for Ind. Appl. Math. 5(1), 32–38 (1957).
[Crossref]

Oliker, V.

Platen, A.

K. Brix, Y. Hafizogullari, and A. Platen, “Designing illumination lenses and mirrors by the numerical solution of Monge–Ampère equations,” J. Opt. Soc. Am. A 32(11), 2227–2236 (2015).
[Crossref]

K. Brix, Y. Hafizogullari, and A. Platen, “Solving the Monge–Ampère equations for the inverse reflector problem,” Math. Model. Methods Appl. Sci. 25(6), 803–837 (2015).
[Crossref]

Prins, C. R.

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. van Roosmalen, W. L. IJzerman, and T. W. Tukker, “A Monge–Ampèresolver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

Russell, R. D.

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of the Monge– Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

Su, Z.

Sulman, M. M.

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of the Monge– Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

ten Thije Boonkkamp, J. H. M.

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. van Roosmalen, W. L. IJzerman, and T. W. Tukker, “A Monge–Ampèresolver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

Tukker, T. W.

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. van Roosmalen, W. L. IJzerman, and T. W. Tukker, “A Monge–Ampèresolver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

van Roosmalen, J.

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. van Roosmalen, W. L. IJzerman, and T. W. Tukker, “A Monge–Ampèresolver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

Wang, X.-J.

X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20(3), 329–341 (2004).
[Crossref]

Williams, J. F.

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of the Monge– Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

Wu, R.

Xie, Y.

Xiu, X.

Xu, L.

Xu, S.

Zhang, H.

Zhang, Y.

Zhao, L.

Zheng, Z.

Ann. Oper. Res. (1)

D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
[Crossref]

Appl. Numer. Math. (1)

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of the Monge– Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

Appl. Opt. (2)

Arch. Ration. Mech. Anal. (1)

C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[Crossref]

Calc. Var. (1)

X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20(3), 329–341 (2004).
[Crossref]

J. Math. Sci. (1)

T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
[Crossref]

J. Opt. Soc. Am. A (4)

J. Soc. for Ind. Appl. Math. (1)

J. Munkres, “Algorithms for the assignment and transportation problems,” J. Soc. for Ind. Appl. Math. 5(1), 32–38 (1957).
[Crossref]

Laser Photon. Rev. (1)

R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, and J. C. Miñano, “Design of freeform illumination optics,” Laser Photon. Rev. 12(7), 1700310 (2018).
[Crossref]

Math. Model. Methods Appl. Sci. (1)

K. Brix, Y. Hafizogullari, and A. Platen, “Solving the Monge–Ampère equations for the inverse reflector problem,” Math. Model. Methods Appl. Sci. 25(6), 803–837 (2015).
[Crossref]

Opt. Express (8)

C. Bösel and H. Gross, “Ray mapping approach for the efficient design of continuous freeform surfaces,” Opt. Express 24(13), 14271–14282 (2016).
[Crossref] [PubMed]

L. L. Doskolovich, A. A. Mingazov, D. A. Bykov, E. S. Andreev, and E. A. Bezus, “Variational approach to calculation of light field eikonal function for illuminating a prescribed region,” Opt. Express 25(22), 26378–26392 (2017).
[Crossref] [PubMed]

D. A. Bykov, L. L. Doskolovich, A. A. Mingazov, E. A. Bezus, and N. L. Kazanskiy, “Linear assignment problem in the design of freeform refractive optical elements generating prescribed irradiance distributions,” Opt. Express 26(21), 27812–27825 (2018).
[Crossref] [PubMed]

R. Wu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “A mathematical model of the single freeform surface design for collimated beam shaping,” Opt. Express 21(18), 20974–20989 (2013).
[Crossref] [PubMed]

V. Oliker, L. L. Doskolovich, and D. A. Bykov, “Beam shaping with a plano-freeform lens pair,” Opt. Express 26(15), 19406–19419 (2018).
[Crossref] [PubMed]

X. Mao, H. Li, Y. Han, and Y. Luo, “Polar-grids based source-target mapping construction method for designing freeform illumination system for a lighting target with arbitrary shape,” Opt. Express 23(4), 4313–4328 (2015).
[Crossref] [PubMed]

M. A. Moiseev, S. V. Kravchenko, and L. L. Doskolovich, “Design of efficient LED optics with two free-form surfaces,” Opt. Express 22(S7), A1926–A1935 (2014).
[Crossref]

L. L. Doskolovich, D. A. Bykov, E. S. Andreev, E. A. Bezus, and V. Oliker, “Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems,” Opt. Express 26(19), 24602–24613 (2018).
[Crossref] [PubMed]

Opt. Lett. (2)

SIAM J. Sci. Comput. (1)

C. R. Prins, J. H. M. ten Thije Boonkkamp, J. van Roosmalen, W. L. IJzerman, and T. W. Tukker, “A Monge–Ampèresolver for free-form reflector design,” SIAM J. Sci. Comput. 36(3), B640–B660 (2014).
[Crossref]

Other (4)

C. E. Gutiérrez, “Refraction problems in geometric optics,” Fully Nonlinear PDEs in Real and Complex Geometry and Optics, Vol. 2087 of the Series Lecture Notes in Mathematics (Springer, 2014), pp. 95–150.
[Crossref]

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

L. C. Evans, “Partial differential equations and Monge–Kantorovich mass transfer,” in Current Developments in Mathematics, R. Bott, A. Jaffe, D. Jerison, G. Lusztig, I. Singer, and S.-T. Yau, eds. (International Press of Boston, 1999).

Fast linear assignment problem using auction algorithm (mex). http://www.mathworks.com/matlabcentral/fileexchange/48448

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Figures (6)

Fig. 1
Fig. 1 Geometry of the problem.
Fig. 2
Fig. 2 Equal-flux-cell approximations of the circular domain Ω corresponding to the Lambertian source O (a), of the domain Ω corresponding to the virtual source O′ (b), and of the square domain D (c). Dots correspond to the centers of the cells. (d)–(f) Ray mapping of a square mesh from the source O (d) to the virtual source O′ (e) and to the target plane (f). The presented mappings correspond to the optical element designed in Subsection 5.2.
Fig. 3
Fig. 3 Refractive surface generating a uniform irradiance distribution in a square region (a) and the simulated irradiance distribution (b).
Fig. 4
Fig. 4 (a) Profile of an optical element with two working surfaces. The lower surface forms a virtual image O′ of the source O. (b) Geometry of refraction of a ray at the lower surface.
Fig. 5
Fig. 5 Refractive optical element with two surfaces generating a uniform irradiance distribution in a square region (a) and the simulated irradiance distribution (b).
Fig. 6
Fig. 6 Refractive optical element with two working surfaces generating uniformly illuminated letters “AB” (a) and the simulated irradiance distribution (b). The inset shows the sharp bends on the surface of the optical element.

Equations (30)

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e ( m ) = ( m 1 , m 2 , 1 m 1 2 m 2 2 ) ,
R ( m ) = ρ ( m ) e ( m ) ,
p ( x ) = ( x 1 , x 2 , f ) / x 1 2 + x 2 2 + f 2 .
R el ( m ; x ) = e ( m ) ρ el ( m ; x ) = e ( m ) τ ( x ) 1 ν e ( m ) p ( x ) ,
ρ el ( m ; x ) x i = 0 , i = 1 , 2 .
log τ ( x ) x i = e ( m ) p ( x ) x i ν 1 e ( m ) p ( x ) , i = 1 , 2 .
log τ ( x ) x i = m i p i ( x ) e ( m ) p ( x ) ν 1 e ( m ) p ( x ) p 3 ( x ) f , i = 1 , 2 .
ρ ( m ) = max x D τ ( x ) 1 ν e ( m ) p ( x ) ,
ρ ( m ) = min x D τ ( x ) 1 ν e ( m ) p ( x ) .
γ : m arg max x D τ ( x ) 1 ν e ( m ) p ( x ) ,
γ 1 ( S ) I ( m ) d S Ω = S E ( x ) d x ,
I pr ( m ) = E ( γ ( m ) ) | J γ ( m ) | ,
𝒦 ( m , x ) = log { 1 ν e ( m ) p ( x ) } .
( γ ) = Ω 𝒦 ( m , γ ( m ) ) I pr ( m ) d m .
( γ , μ ) = Ω { 𝒦 ( m , γ ( m ) ) I pr ( m ) + μ ( m ) [ I pr ( m ) E ( γ ( m ) ) | J γ ( m ) | ] } d m ,
δ μ ( γ , μ ) = ε Ω η ( m ) [ I pr ( m ) E ( γ ( m ) ) | J γ ( m ) | ] d m ,
( γ , λ ) = Ω [ 𝒦 ( m , γ ( m ) ) + log λ ( γ ( m ) ) ] I pr ( m ) d m Ω log λ ( γ ( m ) ) E ( γ ( m ) ) | J γ ( m ) | d m .
( γ , λ ) = Ω [ 𝒦 ( m , γ ( m ) ) + log λ ( γ ( m ) ) ] I pr ( m ) d m D log λ ( x ) E ( x ) d x .
δ γ e ( m ) p ( γ ( m ) ) = δ γ j = 1 3 e j ( m ) p j ( γ ( m ) ) = j = 1 3 e j ( m ) δ γ p j ( γ ( m ) ) = ε j = 1 3 e j ( m ) [ p j x 1 ( γ ( m ) ) ω 1 ( m ) + p j x 2 ( γ ( m ) ) ω 2 ( m ) ] = ε i = 1 2 e ( m ) p x i ( γ ( m ) ) ω i ( m ) .
δ γ 𝒦 ( m , γ ( m ) ) = ε ν 1 e ( m ) p ( γ ( m ) ) i = 1 2 e ( m ) p x i ( γ ( m ) ) ω i ( m ) .
δ γ ( γ , λ ) = ε i = 1 2 Ω [ e ( m ) p x i ( γ ( m ) ) ν 1 e ( m ) p ( γ ( m ) ) + log λ x i ( γ ( m ) ) ] ω i ( m ) I pr ( m ) d m .
( C ) i , j = 𝒦 ( m i , x j ) , i , j = 1 , , N ,
d ( j 1 , , j N ) = i = 1 N 𝒦 ( m i , x j i ) ,
d ( j 1 , , j N ) max ,
d ( j 1 , , j N ) min ,
log τ ( x ) = m , n p m n B m ( x 1 ) P n ( x 2 ) ,
x 2 + z 2 n 1 x 2 + ( z + Δ ) 2 = c 0 ,
r ( θ ) = c 0 n 1 2 Δ cos θ + n 1 ( c 0 + Δ cos θ ) 2 ( n 1 2 1 ) Δ 2 sin 2 θ n 1 2 1 , θ [ 0 , π 2 ] .
θ v ( θ ) = arcsin ( sin θ r ( θ ) / t ( θ ) ) ,
m ^ i = m i r ( θ i ) / t ( θ i ) , i = 1 , , N .

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