Abstract

Source optimization (SO) has emerged as a key technique for improving lithographic imaging over a range of process variations. Current SO approaches are pixel-based, where the source pattern is designed by solving a quadratic optimization problem using gradient-based algorithms or solving a linear programming problem. Most of these methods, however, are either computational intensive or result in a process window (PW) that may be further extended. This paper applies the rich theory of compressive sensing (CS) to develop an efficient and robust SO method. In order to accelerate the SO design, the source optimization is formulated as an underdetermined linear problem, where the number of equations can be much less than the source variables. Assuming the source pattern is a sparse pattern on a certain basis, the SO problem is transformed into a l1-norm image reconstruction problem based on CS theory. The linearized Bregman algorithm is applied to synthesize the sparse optimal source pattern on a representation basis, which effectively improves the source manufacturability. It is shown that the proposed linear SO formulation is more effective for improving the contrast of the aerial image than the traditional quadratic formulation. The proposed SO method shows that sparse-regularization in inverse lithography can indeed extend the PW of lithography systems. A set of simulations and analysis demonstrate the superiority of the proposed SO method over the traditional approaches.

© 2014 Optical Society of America

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2013

2012

2011

J. L. Paredes and G. R. Arce, “Compressive sampling signal reconstruction by weighted median regression estimates,” IEEE Trans. Signal Proc. 59(6), 2585–2601 (2011).
[CrossRef]

J. Yu and P. Yu, “Gradient-based fast source maskoptimization (SMO),” Proc. SPIE,vol.  7973, p. 797320 (2011).
[CrossRef]

Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE Trans. Image Proc. 20, 2856–2864 (2011).
[CrossRef]

N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19,384–19,398 (2011).
[CrossRef]

K. Lai and et al., “Design specific joint optimization of masks and sources on a very large scale,” Proc. SPIE, vol.  7973, p. 797308 (2011).
[CrossRef]

2010

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE, vol.  7640, p. 764024 (2010).
[CrossRef]

J. Carriere, J. Stack, J. Childers, K. Welch, and M. D. Himel, “Advances in DOE modeling and optical performance for SMO applications,” Proc. SPIE, vol.  7640, p. 764025 (2010).
[CrossRef]

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

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Proc. 19(1), 264–270 (2010).
[CrossRef]

2009

J. F. Cai, S. Osher, and Z. Shen, “Linearized bregman iterations for compressed sensing,” Mathematics of Computation 78(267), 1515–1536 (2009).
[CrossRef]

K. Lai and et al., “Experimental result and simulation analysis for the use of pixelated illumination from source mask optimization for 22nm logic lithography process,” Proc. SPIE, vol.  7274, p. 72740A (2009).
[CrossRef]

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of globalsource optimization in optical lithography,”Proc. SPIE, vol.  7274, p.72740C (2009).
[CrossRef]

X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Optics Express 17(7), 5783–5793 (2009).
[CrossRef] [PubMed]

2008

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE, vol.  7140, p. 714010 (2008).
[CrossRef]

2006

A. E. Rosenbluth and N. Seong, “Global optimization of the illumination distribution to maximize integrated process window,” Proc. SPIE, vol.  6154, p. 61540H (2006).
[CrossRef]

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

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

2005

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[CrossRef]

2004

A. Erdmann, T. Fühner, T. Schnattinger, and B. Tollkühn, “Towards automatic mask and sourceoptimization for optical lithography,” Proc.SPIE, vol.  5377, pp.646–657 (2004).
[CrossRef]

Y. Granik, “Source optimization for image fidelity and throughput,” J. Microlith. Microfab. Microsyst. 3, 509–522 (2004).

2003

T. E. Brist and G. E. Bailey, “Effective multicutline QUASAR illumination optimization for SRAM and logic,” Proc. SPIE, vol.  5042, pp. 153–159 (2003).
[CrossRef]

2002

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

2001

G. M. Gallatin, “High-numerical-aperture scalar imaging,” Appl. Opt. 40(28), 4958–4964 (2001).
[CrossRef]

D. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inform. Theory 47(7), 2845–2862 (2001).
[CrossRef]

2000

T. S. Gau, R. G. Liu, C. K. Chen, C. M. Lai, F. J. Liang, and C. C. Hsia, “The customized illumination aperture filter for low k1 photolithography process,” Proc. SPIE, vol.  4000, pp. 271–282 (2000).
[CrossRef]

1998

M. Burkhardt, A. Yen, C. Progler, and G. Wells, “Illumination design for printing of regular contact patterns,” Microelectron. Eng. 41, 91–96 (1998).
[CrossRef]

1996

R. R. Vallishayee, S. A. Orszag, and E. Barouch, “Optimization of stepper parameters and their influence on OPC,” Proc. SPIE, vol.  2726, pp. 660–669 (1996).
[CrossRef]

Adrichem, P. V.

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2x node logic,” Proc. SPIE, vol. 8166 (2011).
[CrossRef]

Arce, G. R.

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, C. Han, Y. Li, B. Wu, Z. Song, L. Dong, and G. R. Arce, “Hybrid source mask optimization for robust immersion lithography,” Appl. Opt. 52(18), 4200–4211 (2013).
[CrossRef] [PubMed]

J. L. Paredes and G. R. Arce, “Compressive sampling signal reconstruction by weighted median regression estimates,” IEEE Trans. Signal Proc. 59(6), 2585–2601 (2011).
[CrossRef]

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Proc. 19(1), 264–270 (2010).
[CrossRef]

X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Optics Express 17(7), 5783–5793 (2009).
[CrossRef] [PubMed]

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

Aschke, L.

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE, vol.  7640, p. 764024 (2010).
[CrossRef]

Bagheri, S.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of globalsource optimization in optical lithography,”Proc. SPIE, vol.  7274, p.72740C (2009).
[CrossRef]

Bailey, G. E.

T. E. Brist and G. E. Bailey, “Effective multicutline QUASAR illumination optimization for SRAM and logic,” Proc. SPIE, vol.  5042, pp. 153–159 (2003).
[CrossRef]

Barouch, E.

R. R. Vallishayee, S. A. Orszag, and E. Barouch, “Optimization of stepper parameters and their influence on OPC,” Proc. SPIE, vol.  2726, pp. 660–669 (1996).
[CrossRef]

Bisschop, P. D.

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2x node logic,” Proc. SPIE, vol. 8166 (2011).
[CrossRef]

Bizjak, T.

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE, vol.  7640, p. 764024 (2010).
[CrossRef]

Brist, T. E.

T. E. Brist and G. E. Bailey, “Effective multicutline QUASAR illumination optimization for SRAM and logic,” Proc. SPIE, vol.  5042, pp. 153–159 (2003).
[CrossRef]

Bu, Y.

S. Li, X. Wang, and Y. Bu, “Robust pixel-based source and mask optimization for inverse lithography,” Opt. Laser Technol. 45, 285–293 (2013).
[CrossRef]

Bukofsky, S.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Burger, M.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[CrossRef]

Burkhardt, M.

M. Burkhardt, A. Yen, C. Progler, and G. Wells, “Illumination design for printing of regular contact patterns,” Microelectron. Eng. 41, 91–96 (1998).
[CrossRef]

Cai, J. F.

J. F. Cai, S. Osher, and Z. Shen, “Linearized bregman iterations for compressed sensing,” Mathematics of Computation 78(267), 1515–1536 (2009).
[CrossRef]

Callan, N.

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE, vol.  7140, p. 714010 (2008).
[CrossRef]

Candés, E.

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

Carriere, J.

J. Carriere, J. Stack, J. Childers, K. Welch, and M. D. Himel, “Advances in DOE modeling and optical performance for SMO applications,” Proc. SPIE, vol.  7640, p. 764025 (2010).
[CrossRef]

Chao, H. Y.

Chen, C. C.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of globalsource optimization in optical lithography,”Proc. SPIE, vol.  7274, p.72740C (2009).
[CrossRef]

Chen, C. K.

T. S. Gau, R. G. Liu, C. K. Chen, C. M. Lai, F. J. Liang, and C. C. Hsia, “The customized illumination aperture filter for low k1 photolithography process,” Proc. SPIE, vol.  4000, pp. 271–282 (2000).
[CrossRef]

Chen, L.

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE, vol.  7140, p. 714010 (2008).
[CrossRef]

Childers, J.

J. Carriere, J. Stack, J. Childers, K. Welch, and M. D. Himel, “Advances in DOE modeling and optical performance for SMO applications,” Proc. SPIE, vol.  7640, p. 764025 (2010).
[CrossRef]

Dam, T.

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

David, L.

S. Robert, X. Shi, and L. David, “Simultaneous source mask optimization(SMO),” Proc. SPIE, vol.5853, pp. 180–193(Yokohama, Japan, 2005).
[CrossRef]

Dong, L.

Donoho, D.

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

D. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inform. Theory 47(7), 2845–2862 (2001).
[CrossRef]

Erdmann, A.

A. Erdmann, T. Fühner, T. Schnattinger, and B. Tollkühn, “Towards automatic mask and sourceoptimization for optical lithography,” Proc.SPIE, vol.  5377, pp.646–657 (2004).
[CrossRef]

Fonseca, C.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Fühner, T.

A. Erdmann, T. Fühner, T. Schnattinger, and B. Tollkühn, “Towards automatic mask and sourceoptimization for optical lithography,” Proc.SPIE, vol.  5377, pp.646–657 (2004).
[CrossRef]

Gallatin, G. M.

Gau, T. S.

T. S. Gau, R. G. Liu, C. K. Chen, C. M. Lai, F. J. Liang, and C. C. Hsia, “The customized illumination aperture filter for low k1 photolithography process,” Proc. SPIE, vol.  4000, pp. 271–282 (2000).
[CrossRef]

Gil, D.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of globalsource optimization in optical lithography,”Proc. SPIE, vol.  7274, p.72740C (2009).
[CrossRef]

Goldfarb, D.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[CrossRef]

Granik, Y.

Y. Granik, “Source optimization for image fidelity and throughput,” J. Microlith. Microfab. Microsyst. 3, 509–522 (2004).

Gronlund, K.

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE, vol.  7140, p. 714010 (2008).
[CrossRef]

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2x node logic,” Proc. SPIE, vol. 8166 (2011).
[CrossRef]

Han, C.

Hansen, S.

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE, vol.  7140, p. 714010 (2008).
[CrossRef]

Hibbs, M.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Himel, M. D.

J. Carriere, J. Stack, J. Childers, K. Welch, and M. D. Himel, “Advances in DOE modeling and optical performance for SMO applications,” Proc. SPIE, vol.  7640, p. 764025 (2010).
[CrossRef]

Hsia, C. C.

T. S. Gau, R. G. Liu, C. K. Chen, C. M. Lai, F. J. Liang, and C. C. Hsia, “The customized illumination aperture filter for low k1 photolithography process,” Proc. SPIE, vol.  4000, pp. 271–282 (2000).
[CrossRef]

Hsu, S.

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE, vol.  7140, p. 714010 (2008).
[CrossRef]

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2x node logic,” Proc. SPIE, vol. 8166 (2011).
[CrossRef]

Hu, P.

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

Huo, X.

D. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inform. Theory 47(7), 2845–2862 (2001).
[CrossRef]

Imgrunt, W.

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE, vol.  7640, p. 764024 (2010).
[CrossRef]

Iwase, K.

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2x node logic,” Proc. SPIE, vol. 8166 (2011).
[CrossRef]

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1988).

Jia, N.

N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19,384–19,398 (2011).
[CrossRef]

Kachalov, D. G.

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE, vol.  7640, p. 764024 (2010).
[CrossRef]

Krasnoperova, A.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of globalsource optimization in optical lithography,”Proc. SPIE, vol.  7274, p.72740C (2009).
[CrossRef]

Laenens, B.

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2x node logic,” Proc. SPIE, vol. 8166 (2011).
[CrossRef]

Lai, C. M.

T. S. Gau, R. G. Liu, C. K. Chen, C. M. Lai, F. J. Liang, and C. C. Hsia, “The customized illumination aperture filter for low k1 photolithography process,” Proc. SPIE, vol.  4000, pp. 271–282 (2000).
[CrossRef]

Lai, K.

K. Lai and et al., “Design specific joint optimization of masks and sources on a very large scale,” Proc. SPIE, vol.  7973, p. 797308 (2011).
[CrossRef]

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of globalsource optimization in optical lithography,”Proc. SPIE, vol.  7274, p.72740C (2009).
[CrossRef]

K. Lai and et al., “Experimental result and simulation analysis for the use of pixelated illumination from source mask optimization for 22nm logic lithography process,” Proc. SPIE, vol.  7274, p. 72740A (2009).
[CrossRef]

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Lam, E.

Lam, E. Y.

N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19,384–19,398 (2011).
[CrossRef]

Li, J.

Li, S.

S. Li, X. Wang, and Y. Bu, “Robust pixel-based source and mask optimization for inverse lithography,” Opt. Laser Technol. 45, 285–293 (2013).
[CrossRef]

Li, X.

X. Li, “An improved Bregman iterative algorithm,” Master’s thesis, Beijing Jiaotong University (2010).

Li, Y.

Li, Z.

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE, vol.  7140, p. 714010 (2008).
[CrossRef]

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2x node logic,” Proc. SPIE, vol. 8166 (2011).
[CrossRef]

Liang, F. J.

T. S. Gau, R. G. Liu, C. K. Chen, C. M. Lai, F. J. Liang, and C. C. Hsia, “The customized illumination aperture filter for low k1 photolithography process,” Proc. SPIE, vol.  4000, pp. 271–282 (2000).
[CrossRef]

Lissotschenko, V. N.

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE, vol.  7640, p. 764024 (2010).
[CrossRef]

Liu, H.

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE, vol.  7140, p. 714010 (2008).
[CrossRef]

Liu, R. G.

T. S. Gau, R. G. Liu, C. K. Chen, C. M. Lai, F. J. Liang, and C. C. Hsia, “The customized illumination aperture filter for low k1 photolithography process,” Proc. SPIE, vol.  4000, pp. 271–282 (2000).
[CrossRef]

Liu, S.

Ma, X.

X. Ma, C. Han, Y. Li, B. Wu, Z. Song, L. Dong, and G. R. Arce, “Hybrid source mask optimization for robust immersion lithography,” Appl. Opt. 52(18), 4200–4211 (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 and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Optics Express 17(7), 5783–5793 (2009).
[CrossRef] [PubMed]

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

Y. Li, L. Dong, and X. Ma, “Aerial image calculation method based on the Abbe imaging vector model,” Chinese Patent, ZL 201110268282.X (Authorized in 2013).

Melville, D.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of globalsource optimization in optical lithography,”Proc. SPIE, vol.  7274, p.72740C (2009).
[CrossRef]

Miklyaev, Y. V.

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE, vol.  7640, p. 764024 (2010).
[CrossRef]

Molless, A.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Morgenfeld, B.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of globalsource optimization in optical lithography,”Proc. SPIE, vol.  7274, p.72740C (2009).
[CrossRef]

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, New York, 2006).

Orszag, S. A.

R. R. Vallishayee, S. A. Orszag, and E. Barouch, “Optimization of stepper parameters and their influence on OPC,” Proc. SPIE, vol.  2726, pp. 660–669 (1996).
[CrossRef]

Osher, S.

J. F. Cai, S. Osher, and Z. Shen, “Linearized bregman iterations for compressed sensing,” Mathematics of Computation 78(267), 1515–1536 (2009).
[CrossRef]

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[CrossRef]

Paredes, J. L.

J. L. Paredes and G. R. Arce, “Compressive sampling signal reconstruction by weighted median regression estimates,” IEEE Trans. Signal Proc. 59(6), 2585–2601 (2011).
[CrossRef]

Park, S.

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE, vol.  7140, p. 714010 (2008).
[CrossRef]

Pavelyev, V. S.

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE, vol.  7640, p. 764024 (2010).
[CrossRef]

Peng, D.

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

Peng, Y.

Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE Trans. Image Proc. 20, 2856–2864 (2011).
[CrossRef]

Progler, C.

M. Burkhardt, A. Yen, C. Progler, and G. Wells, “Illumination design for printing of regular contact patterns,” Microelectron. Eng. 41, 91–96 (1998).
[CrossRef]

Robert, S.

S. Robert, X. Shi, and L. David, “Simultaneous source mask optimization(SMO),” Proc. SPIE, vol.5853, pp. 180–193(Yokohama, Japan, 2005).
[CrossRef]

Romberg, J.

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

Rosenbluth, A. E.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of globalsource optimization in optical lithography,”Proc. SPIE, vol.  7274, p.72740C (2009).
[CrossRef]

A. E. Rosenbluth and N. Seong, “Global optimization of the illumination distribution to maximize integrated process window,” Proc. SPIE, vol.  6154, p. 61540H (2006).
[CrossRef]

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Schnattinger, T.

A. Erdmann, T. Fühner, T. Schnattinger, and B. Tollkühn, “Towards automatic mask and sourceoptimization for optical lithography,” Proc.SPIE, vol.  5377, pp.646–657 (2004).
[CrossRef]

Seong, N.

A. E. Rosenbluth and N. Seong, “Global optimization of the illumination distribution to maximize integrated process window,” Proc. SPIE, vol.  6154, p. 61540H (2006).
[CrossRef]

Shen, Y.

Shen, Z.

J. F. Cai, S. Osher, and Z. Shen, “Linearized bregman iterations for compressed sensing,” Mathematics of Computation 78(267), 1515–1536 (2009).
[CrossRef]

Shi, X.

S. Robert, X. Shi, and L. David, “Simultaneous source mask optimization(SMO),” Proc. SPIE, vol.5853, pp. 180–193(Yokohama, Japan, 2005).
[CrossRef]

Singh, R. N.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

Socha, R.

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE, vol.  7140, p. 714010 (2008).
[CrossRef]

Song, Z.

Stack, J.

J. Carriere, J. Stack, J. Childers, K. Welch, and M. D. Himel, “Advances in DOE modeling and optical performance for SMO applications,” Proc. SPIE, vol.  7640, p. 764025 (2010).
[CrossRef]

Tao, T.

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

Tian, K.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of globalsource optimization in optical lithography,”Proc. SPIE, vol.  7274, p.72740C (2009).
[CrossRef]

Tirapu-Azpiroz, J.

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of globalsource optimization in optical lithography,”Proc. SPIE, vol.  7274, p.72740C (2009).
[CrossRef]

Tolani, V.

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

Tollkühn, B.

A. Erdmann, T. Fühner, T. Schnattinger, and B. Tollkühn, “Towards automatic mask and sourceoptimization for optical lithography,” Proc.SPIE, vol.  5377, pp.646–657 (2004).
[CrossRef]

Vallishayee, R. R.

R. R. Vallishayee, S. A. Orszag, and E. Barouch, “Optimization of stepper parameters and their influence on OPC,” Proc. SPIE, vol.  2726, pp. 660–669 (1996).
[CrossRef]

Wang, X.

S. Li, X. Wang, and Y. Bu, “Robust pixel-based source and mask optimization for inverse lithography,” Opt. Laser Technol. 45, 285–293 (2013).
[CrossRef]

Wang, Y.

Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE Trans. Image Proc. 20, 2856–2864 (2011).
[CrossRef]

Wang, Z.

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Proc. 19(1), 264–270 (2010).
[CrossRef]

Welch, K.

J. Carriere, J. Stack, J. Childers, K. Welch, and M. D. Himel, “Advances in DOE modeling and optical performance for SMO applications,” Proc. SPIE, vol.  7640, p. 764025 (2010).
[CrossRef]

Wells, G.

M. Burkhardt, A. Yen, C. Progler, and G. Wells, “Illumination design for printing of regular contact patterns,” Microelectron. Eng. 41, 91–96 (1998).
[CrossRef]

Wong, A. K.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

A. K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE Press, 2001).
[CrossRef]

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, New York, 2006).

Wu, B.

Xu, J.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[CrossRef]

Yen, A.

M. Burkhardt, A. Yen, C. Progler, and G. Wells, “Illumination design for printing of regular contact patterns,” Microelectron. Eng. 41, 91–96 (1998).
[CrossRef]

Yin, W.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[CrossRef]

Yu, J.

J. Yu and P. Yu, “Gradient-based fast source maskoptimization (SMO),” Proc. SPIE,vol.  7973, p. 797320 (2011).
[CrossRef]

Yu, J. C.

Yu, P.

Yu, Z.

Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE Trans. Image Proc. 20, 2856–2864 (2011).
[CrossRef]

Zhang, J.

Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE Trans. Image Proc. 20, 2856–2864 (2011).
[CrossRef]

Appl. Opt.

IEEE Trans. Image Proc.

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Proc. 19(1), 264–270 (2010).
[CrossRef]

Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE Trans. Image Proc. 20, 2856–2864 (2011).
[CrossRef]

IEEE Trans. Inform. Theory

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

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

D. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inform. Theory 47(7), 2845–2862 (2001).
[CrossRef]

IEEE Trans. Signal Proc.

J. L. Paredes and G. R. Arce, “Compressive sampling signal reconstruction by weighted median regression estimates,” IEEE Trans. Signal Proc. 59(6), 2585–2601 (2011).
[CrossRef]

J. Microlith. Microfab. Microsyst.

Y. Granik, “Source optimization for image fidelity and throughput,” J. Microlith. Microfab. Microsyst. 3, 509–522 (2004).

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsyst. 1(1), 13–30 (2002).

J. Opt. Soc. Am. A

Mathematics of Computation

J. F. Cai, S. Osher, and Z. Shen, “Linearized bregman iterations for compressed sensing,” Mathematics of Computation 78(267), 1515–1536 (2009).
[CrossRef]

Microelectron. Eng.

M. Burkhardt, A. Yen, C. Progler, and G. Wells, “Illumination design for printing of regular contact patterns,” Microelectron. Eng. 41, 91–96 (1998).
[CrossRef]

Multiscale Model. Simul.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[CrossRef]

Opt. Express

Opt. Laser Technol.

S. Li, X. Wang, and Y. Bu, “Robust pixel-based source and mask optimization for inverse lithography,” Opt. Laser Technol. 45, 285–293 (2013).
[CrossRef]

Optics Express

X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Optics Express 17(7), 5783–5793 (2009).
[CrossRef] [PubMed]

Proc. SPIE

J. Yu and P. Yu, “Gradient-based fast source maskoptimization (SMO),” Proc. SPIE,vol.  7973, p. 797320 (2011).
[CrossRef]

K. Tian, A. Krasnoperova, D. Melville, A. E. Rosenbluth, D. Gil, J. Tirapu-Azpiroz, K. Lai, S. Bagheri, C. C. Chen, and B. Morgenfeld, “Benefits and trade-offs of globalsource optimization in optical lithography,”Proc. SPIE, vol.  7274, p.72740C (2009).
[CrossRef]

A. E. Rosenbluth and N. Seong, “Global optimization of the illumination distribution to maximize integrated process window,” Proc. SPIE, vol.  6154, p. 61540H (2006).
[CrossRef]

T. S. Gau, R. G. Liu, C. K. Chen, C. M. Lai, F. J. Liang, and C. C. Hsia, “The customized illumination aperture filter for low k1 photolithography process,” Proc. SPIE, vol.  4000, pp. 271–282 (2000).
[CrossRef]

Y. V. Miklyaev, W. Imgrunt, V. S. Pavelyev, D. G. Kachalov, T. Bizjak, L. Aschke, and V. N. Lissotschenko, “Novel continuously shaped diffractive optical elements enable high efficiency beam shaping,” Proc. SPIE, vol.  7640, p. 764024 (2010).
[CrossRef]

J. Carriere, J. Stack, J. Childers, K. Welch, and M. D. Himel, “Advances in DOE modeling and optical performance for SMO applications,” Proc. SPIE, vol.  7640, p. 764025 (2010).
[CrossRef]

K. Lai and et al., “Design specific joint optimization of masks and sources on a very large scale,” Proc. SPIE, vol.  7973, p. 797308 (2011).
[CrossRef]

R. R. Vallishayee, S. A. Orszag, and E. Barouch, “Optimization of stepper parameters and their influence on OPC,” Proc. SPIE, vol.  2726, pp. 660–669 (1996).
[CrossRef]

T. E. Brist and G. E. Bailey, “Effective multicutline QUASAR illumination optimization for SRAM and logic,” Proc. SPIE, vol.  5042, pp. 153–159 (2003).
[CrossRef]

S. Hsu, L. Chen, Z. Li, S. Park, K. Gronlund, H. Liu, N. Callan, R. Socha, and S. Hansen, “An innovative source-mask co-optimization (SMO) method for extending low k1 imaging,” Proc. SPIE, vol.  7140, p. 714010 (2008).
[CrossRef]

K. Lai and et al., “Experimental result and simulation analysis for the use of pixelated illumination from source mask optimization for 22nm logic lithography process,” Proc. SPIE, vol.  7274, p. 72740A (2009).
[CrossRef]

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

Proc.SPIE

A. Erdmann, T. Fühner, T. Schnattinger, and B. Tollkühn, “Towards automatic mask and sourceoptimization for optical lithography,” Proc.SPIE, vol.  5377, pp.646–657 (2004).
[CrossRef]

Other

S. Robert, X. Shi, and L. David, “Simultaneous source mask optimization(SMO),” Proc. SPIE, vol.5853, pp. 180–193(Yokohama, Japan, 2005).
[CrossRef]

K. Iwase, P. D. Bisschop, B. Laenens, Z. Li, K. Gronlund, P. V. Adrichem, and S. Hsu, “A new source optimization approach for 2x node logic,” Proc. SPIE, vol. 8166 (2011).
[CrossRef]

A. K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE Press, 2001).
[CrossRef]

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

Y. Li, L. Dong, and X. Ma, “Aerial image calculation method based on the Abbe imaging vector model,” Chinese Patent, ZL 201110268282.X (Authorized in 2013).

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, New York, 2006).

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1988).

http://dsp.rice.edu/cs .

X. Li, “An improved Bregman iterative algorithm,” Master’s thesis, Beijing Jiaotong University (2010).

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Figures (14)

Fig. 1
Fig. 1

Formation of the ICC matrix. Gxsys is given by Eq. (7), which is a function of the coordinate (xs, ys). Icc is the ICC matrix, the ith column of which is the raster-scanned vector of Gxsys corresponding to the ith source point J⃗i.

Fig. 2
Fig. 2

Locations of the monitoring pixels for (a) line-space pattern and (b) horizontal block pattern. Inner margins, outer margins, and non-pattern regions are indicated by the green, red and blue colors, respectively.

Fig. 3
Fig. 3

Simulations using the compressive sensing method based on the line-space pattern.

Fig. 4
Fig. 4

Simulations using the conjugate gradient method based on the line-space pattern.

Fig. 5
Fig. 5

2D-DCT coefficients of the optimized source patterns.

Fig. 6
Fig. 6

Distribution of the randomly selected monitoring pixels on wafer. Top row shows the distribution of randomly selected monitoring pixels for the line-space pattern with (a) M=200, (b) M=100 and (c) M=25. Bottom row shows the distribution of randomly selected monitoring pixels for the horizontal block pattern with (d) M=200, (e) M=100 and (f) M=25. White regions represent all of the monitoring pixels before down-sampling, and the black dots are the randomly selected monitoring pixels.

Fig. 7
Fig. 7

Comparison of convergence curves of pattern error and contrast based on the line-space pattern.

Fig. 8
Fig. 8

Process windows of the compressive sensing method and conjugate gradient method with different M based on the (a) line-space pattern and (b) horizontal block pattern.

Fig. 9
Fig. 9

Simulations using the compressive sensing method based on the horizontal block pattern.

Fig. 10
Fig. 10

Simulations using the conjugate gradient method based on the horizontal block pattern.

Fig. 11
Fig. 11

Comparison of convergence curves of pattern error and contrast based on the horizontal block pattern.

Fig. 12
Fig. 12

Simulations using the compressive sensing and conjugate gradient method by determinately selecting the monitoring pixels.

Fig. 13
Fig. 13

Simulations using the spatial basis for both of the line-space pattern and horizontal block pattern.

Fig. 14
Fig. 14

Simulations using lp-norm basis pursuit algorithm based on the 2D-DCT basis for the line-space pattern.

Tables (4)

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Table 1 Pattern errors, aerial image contrasts and runtimes of different simulations based on line-space pattern, where Def. and D.V. represents defocus and dose variation conditions, respectively.

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Table 2 The values of DOFs (nm) corresponding to different ELs. We choose EL=3%, 5%, 8%, 10% and 15% for the line-space pattern, and EL=1%, 2%, 3%, 4% and 5% for the horizontal block pattern.

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Table 3 Pattern errors, arial image contrasts and runtimes of different simulations based on horizontal block pattern, where Def. and D.V. represents defocus and dose variation conditions, respectively.

Tables Icon

Table 3 Pattern errors, arial image contrasts and runtimes of different simulations based on horizontal block pattern, where Def. and D.V. represents defocus and dose variation conditions, respectively.

Equations (21)

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F = 1 2 ( I c c in + I c c out ) J t r 2 2 + I c c 0 J δ 2 2 ,
θ ^ = min θ θ 1 subject to Z s = I s = I c c s Ψ θ ,
M = C × K × log N ˜ N ˜ ,
μ ( Ψ , Φ ) = max ( i , j ) { | ψ i , ϕ j | } , i = 1 , 2 , , N ˜ and j = 1 , 2 , , M .
I ( x s , y s ) = J ( x s , y s ) p = x , y , z H p ( B M ) 2 2 ,
I = 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 ) = x s y s J ( x s , y s ) × G x s y s ,
G x s y s = p = x , y , z H p x s y s ( B x s y s M ) 2 2 .
I = I c c J ,
Z s = I s = I c c s J ,
θ = Ψ T J ,
Ψ T ( N s ( i 1 ) + j , N s ( k 1 ) + p ) = p = 1 N s k = 1 N s Ψ ˜ ( i , k ) Ψ ˜ T ( p , j ) i , j = 1 , 2 , , N s .
g k + 1 = g k + ( Z s I c c s d θ ) ,
θ k + 1 = δ T μ ( I c c s d T g k + 1 ) ,
k = k + 1 ,
T μ ( x ) = [ t μ ( x 1 ) , t μ ( x 2 ) , , t μ ( x N ) ] T ,
t μ ( x i ) = { 0 | x i | < μ sgn ( x i ) ( | x i | μ ) | x i | μ ,
sgn ( x i ) = { 1 x i 0 1 x i 0 .
θ k + 1 = arg min θ N s 2 × 1 μ δ θ 1 + 1 2 θ δ v k + 1 2 ,
J ^ = Ψ θ ^ ,
PE = Print Image Target 2 2 .
θ ^ = min θ θ p p subject to Z s = I s = I c c s Ψ θ ,

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