Abstract

The problem of calculation of the light field eikonal function providing focusing into a prescribed region is formulated as a variational problem and as a Monge–Kantorovich mass transportation problem. It is obtained that the cost function in the Monge–Kantorovich problem corresponds to the distance between a point of the source region (in which the eikonal function is defined) and a point of the target region. This result demonstrates that the sought-for eikonal function corresponds to a mapping, for which the total distance between the points of the original plane and the target region is minimized. The formalism proposed in the present work makes it possible to reduce the calculation of the eikonal function to a linear programming problem. Besides, the calculation of the “ray mapping” corresponding to the eikonal function is reduced to the solution of a linear assignment problem. The proposed approach is illustrated by examples of calculation of optical elements for focusing a circular beam into a rectangle and a beam of square section into a ring.

© 2017 Optical Society of America

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References

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  1. R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the Monge–Ampère equation,” Opt. Lett. 38(2), 229–231 (2013).
    [Crossref] [PubMed]
  2. R. Wu, P. Benítez, Y. Zhang, and J. C. Miñano, “Influence of the characteristics of a light source and target on the Monge–Ampère equation method in freeform optics design,” Opt. Lett. 39(3), 634–637 (2014).
    [Crossref] [PubMed]
  3. Y. Ma, H. Zhang, Z. Su, Y. He, L. Xu, X. Lui, and H. Li, “Hybrid method of free-form lens design for arbitrary illumination target,” Appl. Opt. 54(14), 4503–4508 (2015).
    [Crossref] [PubMed]
  4. 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]
  5. V. I. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” in Trends in Nonlinear Analysis, M. Kirkilionis, S. Krömker, R. Rannacher, and F. Tomi, eds. (Springer, 2003).
    [Crossref]
  6. S. A. Kochengin and V. I. Oliker, “Computational algorithms for constructing reflectors,” Comput. Vis. Sci. 6(1), 15–21 (2003).
    [Crossref]
  7. V. I. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method (SQM),” Opt. Express 25(4), A58–A72 (2017).
    [Crossref] [PubMed]
  8. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express 18(5), 5295–5304 (2010).
    [Crossref] [PubMed]
  9. L. L. Doskolovich, K. V. Borisova, M. A. Moiseev, and N. L. Kazanskiy, “Design of mirrors for generating prescribed continuous illuminance distributions on the basis of the supporting quadric method,” Appl. Opt. 55(4), 687–695 (2016).
    [Crossref] [PubMed]
  10. D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. 36(6), 918–920 (2011).
    [Crossref] [PubMed]
  11. L. L. Doskolovich, M. A. Moiseev, E. A. Bezus, and V. Oliker, “On the use of the supporting quadric method in the problem of the light field eikonal calculation,” Opt. Express 23(15), 19605–19617 (2015).
    [Crossref] [PubMed]
  12. V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd., 1997), p. 245.
  13. L. L. Doskolovich, A. Y. Dmitriev, and S. I. Kharitonov, “Analytic design of optical elements generating a line focus,” Opt. Eng. 52(9), 091707 (2013).
    [Crossref]
  14. 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]
  15. 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]
  16. X. Mao, S. Xu, X. Hu, and Y. Xie, “Design of a smooth freeform illumination system for a point light source based on polar-type optimal transport mapping,” Appl. Opt. 56(22), 6324–6331 (2017).
    [Crossref]
  17. R. Wu, Y. Zhang, M. M. Sulman, Z. Zheng, P. Benítez, and J. C. Miñano, “Initial design with L2 Monge–Kantorovich theory for the Monge–Ampère equation method in freeform surface illumination design,” Opt. Express 22(13), 16161–16177 (2014).
    [Crossref] [PubMed]
  18. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20(13), 14477–14485 (2012).
    [Crossref] [PubMed]
  19. 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]
  20. 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]
  21. X.-J. Wang, “On the design of a reflector antenna II,” Calc. Var. 20(3), 329–341 (2004).
    [Crossref]
  22. J. Munkres, “Algorithms for the assignment and transportation problems,” SIAM J. Appl. Math. 5(1), 32–38 (1957).
    [Crossref]
  23. É. Tardos, “A strongly polynomial algorithm to solve combinatorial linear programs,” Oper. Res. 34(2), 250–256 (1986).
    [Crossref]
  24. D. P. Bertsekas, “The auction algorithm: A distributed relaxation method for the assignment problem,” Ann. Oper. Res. 14(1), 105–123 (1988).
    [Crossref]
  25. R. Jonker and A. Volgenant, “A shortest augmenting path algorithm for dense and sparse linear assignment problems,” Computing 38(4), 325–340 (1987).
    [Crossref]
  26. Opto-mechanical software TracePro. https://www.lambdares.com/tracepro/
  27. Implementation of Bertsekas’ auction algorithm. http://www.mathworks.com/matlabcentral/fileexchange/48448
  28. L. L. Doskolovich, S. N. Khonina, V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, and G. V. Uspleniev, “Focusators into a ring,” Opt. Quant. Electron. 25(11), 801–814 (1993).
    [Crossref]
  29. C. Canavesi, W. J. Cassarly, and J. P. Rolland, “Direct calculation algorithm for two-dimensional reflector design,” Opt. Lett. 37(18), 3852–3854 (2012).
    [Crossref] [PubMed]
  30. C. Canavesi, W. J. Cassarly, and J. P. Rolland, “Observations on the linear programming formulation of the single reflector design problem,” Opt. Express 20(4), 4050–4055 (2012).
    [Crossref] [PubMed]

2017 (2)

2016 (2)

2015 (3)

2014 (2)

2013 (3)

2012 (3)

2011 (2)

D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. 36(6), 918–920 (2011).
[Crossref] [PubMed]

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]

2010 (1)

2004 (1)

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

2003 (2)

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]

S. A. Kochengin and V. I. Oliker, “Computational algorithms for constructing reflectors,” Comput. Vis. Sci. 6(1), 15–21 (2003).
[Crossref]

1993 (1)

L. L. Doskolovich, S. N. Khonina, V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, and G. V. Uspleniev, “Focusators into a ring,” Opt. Quant. Electron. 25(11), 801–814 (1993).
[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]

1987 (1)

R. Jonker and A. Volgenant, “A shortest augmenting path algorithm for dense and sparse linear assignment problems,” Computing 38(4), 325–340 (1987).
[Crossref]

1986 (1)

É. Tardos, “A strongly polynomial algorithm to solve combinatorial linear programs,” Oper. Res. 34(2), 250–256 (1986).
[Crossref]

1957 (1)

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

Bäuerle, A.

Benítez, P.

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.

Borisova, K. V.

Bösel, C.

Bräuer, A.

Bruneton, A.

Canavesi, C.

Cassarly, W. J.

Dmitriev, A. Y.

L. L. Doskolovich, A. Y. Dmitriev, and S. I. Kharitonov, “Analytic design of optical elements generating a line focus,” Opt. Eng. 52(9), 091707 (2013).
[Crossref]

Doskolovich, L. L.

L. L. Doskolovich, K. V. Borisova, M. A. Moiseev, and N. L. Kazanskiy, “Design of mirrors for generating prescribed continuous illuminance distributions on the basis of the supporting quadric method,” Appl. Opt. 55(4), 687–695 (2016).
[Crossref] [PubMed]

L. L. Doskolovich, M. A. Moiseev, E. A. Bezus, and V. Oliker, “On the use of the supporting quadric method in the problem of the light field eikonal calculation,” Opt. Express 23(15), 19605–19617 (2015).
[Crossref] [PubMed]

L. L. Doskolovich, A. Y. Dmitriev, and S. I. Kharitonov, “Analytic design of optical elements generating a line focus,” Opt. Eng. 52(9), 091707 (2013).
[Crossref]

L. L. Doskolovich, S. N. Khonina, V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, and G. V. Uspleniev, “Focusators into a ring,” Opt. Quant. Electron. 25(11), 801–814 (1993).
[Crossref]

V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd., 1997), p. 245.

Fournier, F. R.

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.

Han, Y.

He, Y.

Hu, X.

Jonker, R.

R. Jonker and A. Volgenant, “A shortest augmenting path algorithm for dense and sparse linear assignment problems,” Computing 38(4), 325–340 (1987).
[Crossref]

Kazanskiy, N. L.

Kharitonov, S. I.

L. L. Doskolovich, A. Y. Dmitriev, and S. I. Kharitonov, “Analytic design of optical elements generating a line focus,” Opt. Eng. 52(9), 091707 (2013).
[Crossref]

Khonina, S. N.

L. L. Doskolovich, S. N. Khonina, V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, and G. V. Uspleniev, “Focusators into a ring,” Opt. Quant. Electron. 25(11), 801–814 (1993).
[Crossref]

Kochengin, S. A.

S. A. Kochengin and V. I. Oliker, “Computational algorithms for constructing reflectors,” Comput. Vis. Sci. 6(1), 15–21 (2003).
[Crossref]

Kotlyar, V. V.

L. L. Doskolovich, S. N. Khonina, V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, and G. V. Uspleniev, “Focusators into a ring,” Opt. Quant. Electron. 25(11), 801–814 (1993).
[Crossref]

V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd., 1997), p. 245.

Li, H.

Liu, P.

Liu, X.

Loosen, P.

Lui, X.

Luo, Y.

Ma, Y.

Mao, X.

Michaelis, D.

Miñano, J. C.

Moiseev, M. A.

Munkres, J.

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

Nikolsky, I. V.

L. L. Doskolovich, S. N. Khonina, V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, and G. V. Uspleniev, “Focusators into a ring,” Opt. Quant. Electron. 25(11), 801–814 (1993).
[Crossref]

Oliker, V.

L. L. Doskolovich, M. A. Moiseev, E. A. Bezus, and V. Oliker, “On the use of the supporting quadric method in the problem of the light field eikonal calculation,” Opt. Express 23(15), 19605–19617 (2015).
[Crossref] [PubMed]

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]

Oliker, V. I.

V. I. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method (SQM),” Opt. Express 25(4), A58–A72 (2017).
[Crossref] [PubMed]

S. A. Kochengin and V. I. Oliker, “Computational algorithms for constructing reflectors,” Comput. Vis. Sci. 6(1), 15–21 (2003).
[Crossref]

V. I. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” in Trends in Nonlinear Analysis, M. Kirkilionis, S. Krömker, R. Rannacher, and F. Tomi, eds. (Springer, 2003).
[Crossref]

Rolland, J. P.

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]

Schreiber, P.

Soifer, V. A.

L. L. Doskolovich, S. N. Khonina, V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, and G. V. Uspleniev, “Focusators into a ring,” Opt. Quant. Electron. 25(11), 801–814 (1993).
[Crossref]

V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd., 1997), p. 245.

Stollenwerk, J.

Su, Z.

Sulman, M. M.

Tardos, É.

É. Tardos, “A strongly polynomial algorithm to solve combinatorial linear programs,” Oper. Res. 34(2), 250–256 (1986).
[Crossref]

Uspleniev, G. V.

L. L. Doskolovich, S. N. Khonina, V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, and G. V. Uspleniev, “Focusators into a ring,” Opt. Quant. Electron. 25(11), 801–814 (1993).
[Crossref]

Volgenant, A.

R. Jonker and A. Volgenant, “A shortest augmenting path algorithm for dense and sparse linear assignment problems,” Computing 38(4), 325–340 (1987).
[Crossref]

Wang, X.-J.

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

Wester, R.

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.

Xu, L.

Xu, S.

Zhang, H.

Zhang, Y.

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. (3)

Calc. Var. (1)

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

Comput. Vis. Sci. (1)

S. A. Kochengin and V. I. Oliker, “Computational algorithms for constructing reflectors,” Comput. Vis. Sci. 6(1), 15–21 (2003).
[Crossref]

Computing (1)

R. Jonker and A. Volgenant, “A shortest augmenting path algorithm for dense and sparse linear assignment problems,” Computing 38(4), 325–340 (1987).
[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]

Oper. Res. (1)

É. Tardos, “A strongly polynomial algorithm to solve combinatorial linear programs,” Oper. Res. 34(2), 250–256 (1986).
[Crossref]

Opt. Eng. (1)

L. L. Doskolovich, A. Y. Dmitriev, and S. I. Kharitonov, “Analytic design of optical elements generating a line focus,” Opt. Eng. 52(9), 091707 (2013).
[Crossref]

Opt. Express (9)

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express 18(5), 5295–5304 (2010).
[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]

L. L. Doskolovich, M. A. Moiseev, E. A. Bezus, and V. Oliker, “On the use of the supporting quadric method in the problem of the light field eikonal calculation,” Opt. Express 23(15), 19605–19617 (2015).
[Crossref] [PubMed]

R. Wu, Y. Zhang, M. M. Sulman, Z. Zheng, P. Benítez, and J. C. Miñano, “Initial design with L2 Monge–Kantorovich theory for the Monge–Ampère equation method in freeform surface illumination design,” Opt. Express 22(13), 16161–16177 (2014).
[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]

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]

V. I. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method (SQM),” Opt. Express 25(4), A58–A72 (2017).
[Crossref] [PubMed]

C. Canavesi, W. J. Cassarly, and J. P. Rolland, “Observations on the linear programming formulation of the single reflector design problem,” Opt. Express 20(4), 4050–4055 (2012).
[Crossref] [PubMed]

A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20(13), 14477–14485 (2012).
[Crossref] [PubMed]

Opt. Lett. (4)

Opt. Quant. Electron. (1)

L. L. Doskolovich, S. N. Khonina, V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, and G. V. Uspleniev, “Focusators into a ring,” Opt. Quant. Electron. 25(11), 801–814 (1993).
[Crossref]

SIAM J. Appl. Math. (1)

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

Other (4)

V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd., 1997), p. 245.

V. I. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” in Trends in Nonlinear Analysis, M. Kirkilionis, S. Krömker, R. Rannacher, and F. Tomi, eds. (Springer, 2003).
[Crossref]

Opto-mechanical software TracePro. https://www.lambdares.com/tracepro/

Implementation of Bertsekas’ auction algorithm. http://www.mathworks.com/matlabcentral/fileexchange/48448

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Figures (5)

Fig. 1
Fig. 1

Geometry of the problem of eikonal function calculation.

Fig. 2
Fig. 2

The eikonal function in the first quadrant calculated using an LPP.

Fig. 3
Fig. 3

(insets) Optical elements for the focusing of a circular beam with 3 mm radius into a rectangle of the size 12 × 4 mm2 calculated using an LPP (a) and an LAP (c). Generated illuminance distributions (a) and (c) and its cross sections along the coordinate axes (b) and (d), respectively.

Fig. 4
Fig. 4

Rectangular grid on the aperture (a) and its mapping to the focusing plane (b).

Fig. 5
Fig. 5

(inset) Optical element for the focusing of a beam with a square section (1 × 1 mm2) into a ring (inner radius 1 mm, outer radius 2.5 mm) calculated by solving an LAP. Generated illuminance distribution (a) and its cross sections along the coordinate axes (b).

Equations (38)

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

x = u + Φ f / 1 ( Φ ) 2 .
E ( x ) = E ( γ Φ ( u ) ) = E 0 ( u ) / J ( γ Φ ( u ) ) ,
ω E 0 ( u ) d u = γ Φ ( ω ) E ( x ) d x ,
G h ( γ Φ ( u ) ) E 0 ( u ) d u = D h ( x ) E ( x ) d x ,
Φ l ( u ; x ) = Ψ ( x ) ρ ( u , x )
ρ ( u , x ) = f 2 + ( u x ) 2
{ Φ l ( u ; x ) = Ψ ( x ) ρ ( u , x ) , x 1 [ Ψ ( x ) ρ ( u , x ) ] = 0 , x 2 [ Ψ ( x ) ρ ( u , x ) ] = 0 .
Φ ( u ) = max x D [ Ψ ( x ) ρ ( u , x ) ] ,
Φ ( u ) = min x D [ Ψ ( x ) ρ ( u , x ) ] .
Ψ l ( x ; u ) = Φ ( u ) + ρ ( u , x ) ,
{ Ψ l ( x ; u ) = Φ ( u ) + ρ ( u , x ) , u 1 [ Φ ( u ) + ρ ( u , x ) ] = 0 , u 2 [ Φ ( u ) + ρ ( u , x ) ] = 0 .
Ψ ( x ) = max u G [ Φ ( u ) + ρ ( u , x ) ] ,
Ψ ( x ) = min u G [ Φ ( u ) + ρ ( u , x ) ] .
Φ ( u ) = max x D [ Ψ ( x ) ρ ( u , x ) ] , Ψ ( x ) = min u G [ Φ ( u ) + ρ ( u , x ) ] .
γ Φ ( u ) = { x D | Ψ ( x ) Φ ( u ) = ρ ( u , x ) } .
α ( u ) = max x D [ β ( x ) ρ ( u , x ) ] , β ( x ) = min u G [ α ( u ) + ρ ( u , x ) ] .
γ Φ ( u ) = { x D | Ψ ( x ) Φ ( u ) = ρ ( u , x ) } .
Ω = { ( α , β ) C ( G ) × C ( D ) | u G , x D β ( x ) α ( u ) ρ ( u , x ) } .
I ( α , β ) =   D β ( x ) E ( x ) d x   G α ( u ) E 0 ( u ) d u .
I Λ , Θ ( α 1 , , α N ; β 1 , , β M ) = j = 1 M β j e j | τ j | i = 1 N α i e i 0 | τ i | max , β j α i ρ i j , i = 1 , , N j = 1 , , M ,
  ω E 0 ( u ) d u =   P ( ω ) E ( x ) d x .
C ( P ) =   G ρ ( u , P ( u ) ) E 0 ( u ) d u .
Φ = ( Φ u 1 , Φ u 2 ) = ( x u ) / ρ ( u , x ) .
Φ ( u 1 , u 2 ) = Φ ( 0 , 0 ) + 0 u 1 Φ u 1 ( t , 0 ) d t + 0 u 2 Φ u 2 ( u 1 , s ) d s .
C ( γ Φ ( u ) ) =   G [ γ G ( u ) u ] 2 E 0 ( u ) d u γ Φ ( u ) min
x ( u ) = γ Φ ( u ) = Φ add ( u ) ,
det [ 2 Φ add ( u ) ] = E 0 ( u ) / E ( Φ add ( u ) ) ,
Φ = [ x ( u ) u ] / f .
x ( u ) = Φ add ( u ) .
Φ add ( u ) = max x D [ x u Ψ add ( x ) ] ,
C ( j 1 , , j N ) = i = 1 N ρ ( u i , x j i ) min ,
z ( u ) = Φ ( u ) / ( n 1 ) ,
α ( u ) α ( u ) = β ( x ) ρ ( u , x ) α ( u ) ρ ( u , x ) ρ ( u , x ) .
ρ ( u , x ) ρ ( u , x ) = ρ ( u ˜ , x ) ( u x ) ,
| α ( u ) α ( u ) | | ρ ( u ˜ , x ) | | u u | L | u , u | ,
C ( P ) =   G ρ ( u , P ( u ) ) E 0 ( u ) d u   G Ψ ( P ( u ) ) E 0 ( u ) d u   G Φ ( u ) E 0 ( u ) d u .
  G Ψ ( P ( u ) ) E 0 ( u ) d u =   D Ψ ( x ) E ( x ) d x =   G Ψ ( γ Φ ( u ) ) E 0 ( u ) d u .
  G Ψ ( P ( u ) ) E 0 ( u ) d u   G Φ ( u ) E 0 ( u ) d u =   G Ψ ( γ   Φ ( u ) ) E 0 ( u ) d u   G Φ ( u ) E 0 ( u ) d u =   G [ Ψ ( γ   Φ ( u ) ) Φ ( u ) ] E 0 ( u ) d u =   G ρ ( u , γ Φ ( u ) ) E 0 ( u ) d u = C ( γ Φ ) .

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