Abstract

Optical proximity correction (OPC) is an extensively used resolution enhancement technique (RET) in optical lithography. To date, the computational efficiency has become a big issue for pixelated OPC techniques due to the increasing complexity of lithographic masks in modern integrated circuits. This paper is the first to apply nonlinear compressive sensing (CS) theory to break through the computational efficiency of gradient-based pixelated OPC methods. The proposed method reduces the dimensionality of the OPC problem by downsampling the layout pattern. Then, a nonlinear cost function is established to guarantee the lithography imaging performance on the downsampled layout. Under the sparsity assumption of the mask, the OPC problem is formulated as an inverse nonlinear CS reconstruction problem. The iterative hard thresholding (IHT) algorithm is then used to solve for the OPC problem. The proposed method proves to improve the computational efficiency of traditional gradient-based OPC methods, while improving the process windows of the lithography systems. Benefiting from the sparse property of the mask patterns, the mask manufacturability can also be improved compared to traditional methods.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  33. X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30(1), 112–123 (2013).
    [Crossref]
  34. D. Peng, P. Hu, V. Tolani, and T. Dam, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
    [Crossref]
  35. X. Ma, Z. Song, Y. Li, and G. R. Arce, “Block-based mask optimization for optical lithography,” Appl. Opt. 52(14), 3351–3363 (2013).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]

2017 (1)

2014 (2)

Z. Song, X. Ma, J. Gao, J. Wang, Y. Li, and G. R. Arce, “Inverse lithography source optimization via compressive sensing,” Opt. Express 22(12), 14180–14198 (2014).
[Crossref] [PubMed]

W. Lv, E. Y. Lam, H. Wei, and S. Liu, “Cascadic multigrid algorithm for robust inverse mask synthesis in optical lithography,” J. Micro/Nanolith. MEMS MOEMS 13(2), 023003 (2014).
[Crossref]

2013 (6)

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

X. Ma, Z. Song, Y. Li, and G. R. Arce, “Block-based mask optimization for optical lithography,” Appl. Opt. 52(14), 3351–3363 (2013).
[Crossref] [PubMed]

X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30(1), 112–123 (2013).
[Crossref]

A. Beck and Y. C. Eldar, “Sparsity constrained nonlinear optimization: Optimality conditions and algorithms,” SIAM J. Optimiz. 23(3), 1480–1509, (2013).
[Crossref]

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “Nonlinear basis pursuit,” 47th Asilomar Conference on Signals, Systems and Computers 118(1), 115–119 (2013).

T. Blumensath, “Compressed sensing with nonlinear observations and related nonlinear optimization problems,” IEEE Trans. Inf. Theory 59(6), 3466–3474 (2013).
[Crossref]

2012 (1)

2011 (6)

X. Ma and G. R. Arce, “Pixel-based OPC optimization based on conjugate gradients,” Opt. Express 19(3), 2165–2180 (2011).
[Crossref] [PubMed]

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust levelset-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
[Crossref] [PubMed]

X. Ma, G. R. Arce, and Y. Li, “Optimal 3D phase-shifting masks in partially coherent illumination,” Appl. Opt. 50(28), 5567–5576 (2011).
[Crossref] [PubMed]

T. Blumensath, “Sampling and reconstructing signals from a union of linear subspaces,” IEEE Trans. Inf. Theory 57(7), 4660–4671 (2011).
[Crossref]

X. Ma and Y. Li, “Resolution enhancement optimization methods in optical lithography with improved manufacturability,” J. Micro/Nanolith. MEMS MOEMS 10(2), 023009 (2011).
[Crossref]

S. Jiang, X. Ma, and A. Zakhor, “A recursive cost-based approach to fracturing,” Proc. SPIE 7973, 79732P (2011).
[Crossref]

2010 (4)

D. Peng, P. Hu, V. Tolani, and T. Dam, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

T. Blumensath, “Compressed sensing with non-linear observations,” IEEE Trans. Inf. Theory 59(6), 3466–3474 (2010).
[Crossref]

J. Yu and P. Yu, “Impacts of cost functions on inverse lithography patterning,” Opt. Express 18(22), 23331–23342 (2010).
[Crossref] [PubMed]

N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 45601–45609 (2010).
[Crossref]

2009 (3)

2008 (1)

P. Gao, A. Gu, and A. Zakhor, “Optical proximity correction with principal component regression,” Proc. SPIE 6924, 69243N (2008).
[Crossref]

2007 (1)

A. Poonawala and P. Milanfar, “Mask design for optical microlithography-an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
[Crossref] [PubMed]

2006 (3)

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. 5(4), 043002 (2006).

D. Donoho, “Compressive sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

E. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

1995 (2)

S. Sherif, B. Saleh, and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Trans. Image Process. 4(9), 1252–1257 (1995).
[Crossref] [PubMed]

N. Cobb and A. Zakhor, “Fast, low-complexity mask design,” Proc. SPIE 2440, 313–327 (1995).
[Crossref]

1994 (1)

1992 (1)

Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Trans. Semicond. Manuf. 5(2), 138–152 (1992).
[Crossref]

Arce, G. R.

Beck, A.

A. Beck and Y. C. Eldar, “Sparsity constrained nonlinear optimization: Optimality conditions and algorithms,” SIAM J. Optimiz. 23(3), 1480–1509, (2013).
[Crossref]

Blumensath, T.

T. Blumensath, “Compressed sensing with nonlinear observations and related nonlinear optimization problems,” IEEE Trans. Inf. Theory 59(6), 3466–3474 (2013).
[Crossref]

T. Blumensath, “Sampling and reconstructing signals from a union of linear subspaces,” IEEE Trans. Inf. Theory 57(7), 4660–4671 (2011).
[Crossref]

T. Blumensath, “Compressed sensing with non-linear observations,” IEEE Trans. Inf. Theory 59(6), 3466–3474 (2010).
[Crossref]

T. Blumensath and M. E. Davies, “Iterative hard thresholding for compressed sensing,” Appl. Comput. Harmon. A 27(3), 265–274 (2009).
[Crossref]

T. Blumensath and M. Davies, “Gradient pursuit for non-linear sparse signal modelling,” 16th European Signal Processing Conference, 1–5 (2008).

Candés, E.

E. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Cobb, N.

N. Cobb and A. Zakhor, “Fast, low-complexity mask design,” Proc. SPIE 2440, 313–327 (1995).
[Crossref]

Dam, T.

D. Peng, P. Hu, V. Tolani, and T. Dam, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Davies, M.

T. Blumensath and M. Davies, “Gradient pursuit for non-linear sparse signal modelling,” 16th European Signal Processing Conference, 1–5 (2008).

Davies, M. E.

T. Blumensath and M. E. Davies, “Iterative hard thresholding for compressed sensing,” Appl. Comput. Harmon. A 27(3), 265–274 (2009).
[Crossref]

Dong, L.

Dong, R.

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “Nonlinear basis pursuit,” 47th Asilomar Conference on Signals, Systems and Computers 118(1), 115–119 (2013).

H. Ohlsson, A. Y. Yang, R. Dong, M. Verhaegen, and S. Sastry, “Quadratic basis pursuit,” arXiv:1301.7002v2, (2013).

Donoho, D.

D. Donoho, “Compressive sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

Eldar, Y. C.

A. Beck and Y. C. Eldar, “Sparsity constrained nonlinear optimization: Optimality conditions and algorithms,” SIAM J. Optimiz. 23(3), 1480–1509, (2013).
[Crossref]

Gao, J.

Gao, P.

P. Gao, A. Gu, and A. Zakhor, “Optical proximity correction with principal component regression,” Proc. SPIE 6924, 69243N (2008).
[Crossref]

Granik, Y.

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. 5(4), 043002 (2006).

Gu, A.

P. Gao, A. Gu, and A. Zakhor, “Optical proximity correction with principal component regression,” Proc. SPIE 6924, 69243N (2008).
[Crossref]

Han, C.

Hu, P.

D. Peng, P. Hu, V. Tolani, and T. Dam, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Jafarpour, B.

L. Li and B. Jafarpour, “An iteratively reweighted algorithm for sparse reconstruction of subsurface flow properties from nonlinear dynamic data,” arXiv:0911.2270, (2009).

Jia, N.

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust levelset-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
[Crossref] [PubMed]

N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 45601–45609 (2010).
[Crossref]

Jiang, S.

S. Jiang, X. Ma, and A. Zakhor, “A recursive cost-based approach to fracturing,” Proc. SPIE 7973, 79732P (2011).
[Crossref]

Kailath, T.

Lam, E. Y.

W. Lv, E. Y. Lam, H. Wei, and S. Liu, “Cascadic multigrid algorithm for robust inverse mask synthesis in optical lithography,” J. Micro/Nanolith. MEMS MOEMS 13(2), 023003 (2014).
[Crossref]

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust levelset-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
[Crossref] [PubMed]

N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 45601–45609 (2010).
[Crossref]

E. Y. Lam and A. K. Wong, “Computation lithography: virtual reality and virtual virtuality,” Opt. Express 17(5), 12259–12268 (2009).
[Crossref] [PubMed]

Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
[Crossref]

Leone, R.

S. Sherif, B. Saleh, and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Trans. Image Process. 4(9), 1252–1257 (1995).
[Crossref] [PubMed]

Li, L.

L. Li and B. Jafarpour, “An iteratively reweighted algorithm for sparse reconstruction of subsurface flow properties from nonlinear dynamic data,” arXiv:0911.2270, (2009).

Li, Y.

Liu, S.

W. Lv, E. Y. Lam, H. Wei, and S. Liu, “Cascadic multigrid algorithm for robust inverse mask synthesis in optical lithography,” J. Micro/Nanolith. MEMS MOEMS 13(2), 023003 (2014).
[Crossref]

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Liu, Y.

Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Trans. Semicond. Manuf. 5(2), 138–152 (1992).
[Crossref]

Lv, W.

W. Lv, E. Y. Lam, H. Wei, and S. Liu, “Cascadic multigrid algorithm for robust inverse mask synthesis in optical lithography,” J. Micro/Nanolith. MEMS MOEMS 13(2), 023003 (2014).
[Crossref]

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Ma, X.

X. Ma, D. Shi, Z. Wang, Y. Li, and G. R. Arce, “Lithographic source optimization based on adaptive projection compressive sensing,” Opt. Express 25(6), 7131–7149 (2017).
[Crossref] [PubMed]

Z. Song, X. Ma, J. Gao, J. Wang, Y. Li, and G. R. Arce, “Inverse lithography source optimization via compressive sensing,” Opt. Express 22(12), 14180–14198 (2014).
[Crossref] [PubMed]

X. Ma, Z. Song, Y. Li, and G. R. Arce, “Block-based mask optimization for optical lithography,” Appl. Opt. 52(14), 3351–3363 (2013).
[Crossref] [PubMed]

X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30(1), 112–123 (2013).
[Crossref]

X. Ma, Y. Li, and L. Dong, “Mask optimization approaches in optical lithography based on a vector imaging model,” J. Opt. Soc. Am. A 29(7), 1300–1312 (2012).
[Crossref]

X. Ma and G. R. Arce, “Pixel-based OPC optimization based on conjugate gradients,” Opt. Express 19(3), 2165–2180 (2011).
[Crossref] [PubMed]

X. Ma, G. R. Arce, and Y. Li, “Optimal 3D phase-shifting masks in partially coherent illumination,” Appl. Opt. 50(28), 5567–5576 (2011).
[Crossref] [PubMed]

S. Jiang, X. Ma, and A. Zakhor, “A recursive cost-based approach to fracturing,” Proc. SPIE 7973, 79732P (2011).
[Crossref]

X. Ma and Y. Li, “Resolution enhancement optimization methods in optical lithography with improved manufacturability,” J. Micro/Nanolith. MEMS MOEMS 10(2), 023009 (2011).
[Crossref]

X. Ma and G. R. Arce, Computational Lithography, Wiley Series in Pure and Applied Optics, 1st ed. (John Wiley and Sons, 2010).
[Crossref]

Milanfar, P.

A. Poonawala and P. Milanfar, “Mask design for optical microlithography-an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
[Crossref] [PubMed]

Ohlsson, H.

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “Nonlinear basis pursuit,” 47th Asilomar Conference on Signals, Systems and Computers 118(1), 115–119 (2013).

H. Ohlsson, A. Y. Yang, R. Dong, M. Verhaegen, and S. Sastry, “Quadratic basis pursuit,” arXiv:1301.7002v2, (2013).

Pati, Y. C.

Peng, D.

D. Peng, P. Hu, V. Tolani, and T. Dam, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Poonawala, A.

A. Poonawala and P. Milanfar, “Mask design for optical microlithography-an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
[Crossref] [PubMed]

Romberg, J.

E. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Saleh, B.

S. Sherif, B. Saleh, and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Trans. Image Process. 4(9), 1252–1257 (1995).
[Crossref] [PubMed]

Sastry, S.

H. Ohlsson, A. Y. Yang, R. Dong, M. Verhaegen, and S. Sastry, “Quadratic basis pursuit,” arXiv:1301.7002v2, (2013).

Sastry, S. S.

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “Nonlinear basis pursuit,” 47th Asilomar Conference on Signals, Systems and Computers 118(1), 115–119 (2013).

Shen, Y.

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust levelset-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011).
[Crossref] [PubMed]

Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
[Crossref]

Sherif, S.

S. Sherif, B. Saleh, and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Trans. Image Process. 4(9), 1252–1257 (1995).
[Crossref] [PubMed]

Shi, D.

Song, Z.

Tao, T.

E. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Tolani, V.

D. Peng, P. Hu, V. Tolani, and T. Dam, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010).
[Crossref]

Verhaegen, M.

H. Ohlsson, A. Y. Yang, R. Dong, M. Verhaegen, and S. Sastry, “Quadratic basis pursuit,” arXiv:1301.7002v2, (2013).

Wang, J.

Wang, Z.

Wei, H.

W. Lv, E. Y. Lam, H. Wei, and S. Liu, “Cascadic multigrid algorithm for robust inverse mask synthesis in optical lithography,” J. Micro/Nanolith. MEMS MOEMS 13(2), 023003 (2014).
[Crossref]

Wong, A. K.

Wong, N.

Wu, X.

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Xia, Q.

W. Lv, S. Liu, Q. Xia, X. Wu, Y. Shen, and E. Y. Lam, “Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step,” J. Vac. Sci. Technol. B 31(4), 041605 (2013).
[Crossref]

Yang, A. Y.

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Figures (11)

Fig. 1
Fig. 1 Sketch of the optical lithography system and the pixelated OPC method.
Fig. 2
Fig. 2 Simulations of different OPC methods based on vertical line-space pattern at 45nm technology node.
Fig. 3
Fig. 3 The overlapped PWs obtained by different OPC methods.
Fig. 4
Fig. 4 The measurement positions of the PWs.
Fig. 5
Fig. 5 The spectral amplitude of Θ in 2D-DFT domain for different iterations.
Fig. 6
Fig. 6 Simulations of the SD methods with downsampling.
Fig. 7
Fig. 7 Simulations of different OPC methods based on horizontal block pattern at 45nm technology node.
Fig. 8
Fig. 8 Simulations of different OPC methods based on a complex target pattern at 45nm technology node.
Fig. 9
Fig. 9 Simulations of different OPC methods based on vertical line-space pattern at 14nm technology node.
Fig. 10
Fig. 10 Simulations of different OPC methods based on horizontal block pattern at 14nm technology node.
Fig. 11
Fig. 11 The convergence curves of the proposed method using line-space pattern and horizontal block pattern at 14nm technology node.

Tables (3)

Tables Icon

Table 1 The runtime per iteration of different OPC methods.

Tables Icon

Table 2 The trapezoid counts and image fidelity obtained by the proposed method (K = 2) with different S, where “foc.” and “def.” represent focal plane and defocus plane, respectively.

Tables Icon

Table 3 The trapezoid counts and image fidelity obtained by the proposed method (K = 2) with different v, where “foc.” and “def.” represent focal plane and defocus plane, respectively.

Equations (32)

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min X X 1 : Y Φ ( X ) 2 2 E 2 2 ,
X ^ = arg min X 𝒜 f ( X ) ,
X n + 1 = 𝒫 𝒜 { X n step × f ( X n ) } ,
I = 1 J sum x s y s [ J ( x s , y s ) × p = x , y , z | H p x s y s ( B x s , y s M ) | 2 2 ] ,
Z = Γ ( I t r ) ,
f ( M ) = Π ( Z ˜ Z ) 2 2 = m = 1 N n = 1 N { Π ( m , n ) × [ Z ˜ ( m , n ) Z ( m , n ) ] } 2 ,
f K ( M ) = Π K ( Z ˜ K Z K ) 2 2 = m = 1 N / K n = 1 N / K { Π ( Km , Kn ) × [ Z ˜ ( Km , Kn ) Z ( Km , Kn ) ] } 2 ,
d ( M ) = α f K , nom ( M ) + ( 1 α ) f K , off ( M ) ,
f K , nom ( M ) = Π K ( Z ˜ K Z K , nom ) 2 2 , f K , off ( M ) = Π K ( Z ˜ K γ Z K , off ) 2 2 ,
M = 1 2 ( 1 + cos Θ ) ,
Θ ¯ = Ψ Θ Ψ T ,
Θ ^ = arg min Θ 𝒜 d ( Θ ) ,
d ( Θ ) = α f K , nom ( Θ ) + ( 1 α ) f K , off ( Θ ) ,
Z = sig ( I ) = 1 1 + exp [ a ( I t r ) ] ,
d ( Θ ) = α f K , nom ( Θ ) + ( 1 α ) f K , off ( Θ ) + γ q R q ( Θ ) + γ w R w ( Θ ) ,
d ( Θ ) = α f K , nom ( Θ ) + ( 1 α ) f K , off ( Θ ) + γ q R q ( Θ ) + γ w R w ( Θ ) ,
Θ n + 1 = 𝒫 𝒜 { Θ n step × d ( Θ n ) } ,
Θ = Ψ T Θ ¯ Ψ ,
Θ n + 1 = Re ( 𝒫 𝒜 { Θ n step × d ( Θ n ) } ) ,
R c = O [ 34 N 2 + 20 N 2 log ( N 2 / K 2 ) + 6 N 2 / K 2 ] O ( 40 N 2 + 20 N 2 log N 2 ) = O [ 34 + 20 log N 2 20 log K 2 + 6 / K 2 ] O ( 40 + 20 log N 2 ) O [ 40 log N 40 log K ] O ( 40 log N ) = 1 O ( log K log N ) .
NILS = CD I con × d I d x | I con ,
I ( Km , Kn ) = 1 J sum x s y s ( J ( x s , y s ) × p = x , y , z { r = 1 N s = 1 N H p x s y s ( Km r , Kn s ) × [ B x s y s ( r , s ) × M ( r , s ) ] } 2 ) .
I ( Km , Kn ) = 1 J sum x s y s ( J ( x s , y s ) × p = x , y , z u = 1 K v = 1 K { a = 0 N / K 1 b = 0 N / K 1 H p x s y s [ K ( m a ) u , K ( n b ) v ] × [ B x s y s ( K a + u , K b + v ) × M ( K a + u , K b + v ) ] } 2 ) .
I ( Km , Kn ) = 1 J sum x s y s ( J ( x s , y s ) × p = x , y , z u = 1 K v = 1 K { a = 0 N / K 1 b = 0 N / K 1 H p , u v x s y s ( m a , n b ) × [ B u v x s y s ( a , b ) × M u v ( a , b ) ] } 2 ) .
I K = 1 J sum x s y s [ J ( x s , y s ) × p = x , y , z u = 1 K v = 1 K | H p , u v x s y s ( B u v x s y s M u v ) | 2 2 ] .
Z K = 1 1 + exp [ a ( I K t r ) ] .
f K M ( r , s ) = 4 a J sum m = 1 N / K n = 1 N / K { Π ( Km , Kn ) × [ Z ˜ ( Km , Kn ) Z ( Km , Kn ) ] × Z ( Km , Kn ) × [ 1 Z ( Km , Kn ) ] } × x s y s J ( x s , y s ) × p = x , y , z Re ( [ B x s y s ( r , s ) ] * × [ H p x s y s ( Km r , Kn s ) ] * × { r s H p x s y s ( Km r , Kn s ) × [ B x s y s ( r , s ) × M ( r , s ) ] } ) = 4 a J sum x s y s J ( x s , y s ) × p = x , y , z Re ( [ B x s y s ( r , s ) ] * × m = 1 N / K n = 1 N / K [ H p x s y s ( Km r , Kn s ) ] * × { Π ( Km , Kn ) × Z ( Km , Kn ) × [ Z ˜ ( Km , Kn ) Z ( Km , Kn ) ] × [ 1 Z ( Km , Kn ) ] } × { r s H p x s y s ( Km r , Kn s ) × [ B x s y s ( r , s ) × M ( r , s ) ] } ) ,
f K M u v ( a , b ) = 4 a J sum x s y s J ( x s , y s ) × p = x , y , z Re ( [ B u v x s y s ( a , b ) ] * × m = 1 N / K n = 1 N / K [ H p , u v x s y s ( m a , n b ) ] * × { Π ( Km , Kn ) × Z ( Km , Kn ) × [ Z ˜ ( Km , Kn ) Z ( Km , Kn ) ] × [ 1 Z ( Km , Kn ) ] } × [ a = 0 N / K 1 b = 0 N / K 1 H p , u v x s y s ( m a , n b ) × B u v x s y s ( a , b ) × M u v ( a , b ) ] ) .
f K , u v = 4 a J sum x s y s J ( x s , y s ) × p = x , y , z Re ( ( B u v x s y s ) * ( H u v x s y s ) * { [ H p , u v x s y s ( B u v x s y s M u v ) ] Π K ( Z ˜ K Z K ) Z K ( 1 K Z K ) } ) ,
f K ( Ka + u , Kb + v ) = f K , u v ( a , b ) ,
f K Θ ( r , s ) = f K M ( r , s ) × M ( r , s ) Θ ( r , s ) .
f K ( Θ ) = 1 2 f K sin Θ .

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