Abstract

Optical proximity correction (OPC) and phase shifting masks (PSM) are resolution enhancement techniques (RET) used extensively in the semiconductor industry to improve the resolution and pattern fidelity of optical lithography. In this paper, we develop generalized gradient-based RET optimization methods to solve for the inverse lithography problem, where the search space is not constrained to a finite phase tessellation but where arbitrary search trajectories in the complex space are allowed. Subsequent mask quantization leads to efficient design of PSMs having an arbitrary number of discrete phases. In order to influence the solution patterns to have more desirable manufacturability properties, a wavelet regularization framework is introduced offering more localized flexibility than total-variation regularization methods traditionally employed in inverse problems. The proposed algorithms provide highly effective four-phase PSMs capable of generating mask patterns with arbitrary Manhattan geometries. Furthermore, a double-exposure optimization method for general inverse lithography is developed where each exposure uses an optimized two-phase mask.

© 2007 Optical Society of America

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References

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  1. S. V. G. J. Schneider, J. Murakowski and D. W. Prather, "Combination lithography for photoniccrystal circuits,"J.of Vacuum Science and Technology B 22(1), 146-151 (2004).
    [CrossRef]
  2. J. M. M. J. M. P. Yao, G. J. Schneider and D.W. Prather, "Micro/nano lithography realized by chemical printing," in Proceedings of SPIE - The International Society for Optical Engineering, vol. 6151I of Emerging Lithographic Technologies X, p. 61511N (2006).
  3. D. W. P. E. D. W. P. Yao, G. J. Schneider and D. J. O’Brien, "Fabrication of three-dimensional photonic crystals with multilayer photolithography," Optics Express 13, 2370-2376 (2005).
    [CrossRef] [PubMed]
  4. B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider and D. J. O’Brien, "Multilayer three-dimensional photolithography with traditional planar method," Applied Physics Letters 85, 2920-2922 (2004).
    [CrossRef]
  5. A. K. Wong, Resolution enhancement techniques, vol. 1 (SPIE Press, 2001).
    [CrossRef]
  6. S. A. Campbell, The science and engineering of microelectronic fabrication, 2nd ed. (Publishing House of Electronics Industry, 2003).
  7. F. Schellenberg, "Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography," Proc. SPIE 5377, 1-20 (2004).
    [CrossRef]
  8. F. Schellenberg, Resolution enhancement techniques in optical lithography (SPIE Press, 2004).
  9. A. W. M. L. W. L. L. Liebmann, S. Mansfield and T. Dunham, "TCAD development for lithography resolution enhancement," IBM Journal of Research and Development pp. 651-665 (2001).
    [CrossRef]
  10. N. S. V. M. D. Levenson and R. A. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982).
    [CrossRef]
  11. B. S. S. Sherif and R. Leone, "Binary image synthesis using mixed integer programming," IEEE Transactions on Image Processing 4(9), 1252-1257 (1995).
    [CrossRef] [PubMed]
  12. Y. Liu and A. Zakhor, "Binary and phase shifting mask design for optical lithography," IEEE Transactions on Semiconductor Manufacturing 5(2) (1992).
    [CrossRef]
  13. Y. C. Pati and T. Kailath, "Phase-shifting masks for microlithography: Automated design and mask requirements," Optical Society of America 11 (1994).
  14. T. F. B. T. A. Erdmann, R. Farkas and G. Kokai, "Towards automatic mask and source optimization for optical lithography," Optical Microlithography, Proc. SPIE 5377, 646-657 (2004).
  15. Y. L. L. Pang and D. Abrams, "Inverse lithography technology (ILT): What is the impact to the photomask industry?" Proc. SPIE (2006).
    [CrossRef]
  16. Y. Granik, "Illuminator optimization methods in microlithography," Optical Microlithography Proc. SPIE 5754, 217-229 (2005).
  17. A. Poonawala and P. Milanfar, "OPC and PSM design using inverse lithography: A non-linear optimization approach," in Proceedings of the SPIE, vol. 6154, pp. 1159-1172 (San Jose, CA, 2006).
  18. C. Vogel, Computational methods for inverse problems (SIAM Press, 2002).
    [CrossRef]
  19. X. Ma and G. R. Arce, "Generalized inverse lithography methods for phase-shifting mask design," in Proceedings of SPIE, vol. 65200U (2007).
  20. A. Poonawala and P. Milanfar, "Double Exposure Mask Synthesis using Inverse Lithography," submitted toJournal of Microlithography, Microfabrication, and Microsystems.
  21. M. Born and E. Wolfe, Principles of optics (Cambridge University Press, 1999).
  22. N. Cobb and A. Zakhor, "Fast sparse aerial image calculation for OPC," BACUS Symposium on Photomask Technology, Proc. SPIE 2440, 313-327 (1995).

2005

D. W. P. E. D. W. P. Yao, G. J. Schneider and D. J. O’Brien, "Fabrication of three-dimensional photonic crystals with multilayer photolithography," Optics Express 13, 2370-2376 (2005).
[CrossRef] [PubMed]

Y. Granik, "Illuminator optimization methods in microlithography," Optical Microlithography Proc. SPIE 5754, 217-229 (2005).

2004

B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider and D. J. O’Brien, "Multilayer three-dimensional photolithography with traditional planar method," Applied Physics Letters 85, 2920-2922 (2004).
[CrossRef]

F. Schellenberg, "Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography," Proc. SPIE 5377, 1-20 (2004).
[CrossRef]

T. F. B. T. A. Erdmann, R. Farkas and G. Kokai, "Towards automatic mask and source optimization for optical lithography," Optical Microlithography, Proc. SPIE 5377, 646-657 (2004).

1995

N. Cobb and A. Zakhor, "Fast sparse aerial image calculation for OPC," BACUS Symposium on Photomask Technology, Proc. SPIE 2440, 313-327 (1995).

B. S. S. Sherif and R. Leone, "Binary image synthesis using mixed integer programming," IEEE Transactions on Image Processing 4(9), 1252-1257 (1995).
[CrossRef] [PubMed]

1992

Y. Liu and A. Zakhor, "Binary and phase shifting mask design for optical lithography," IEEE Transactions on Semiconductor Manufacturing 5(2) (1992).
[CrossRef]

1982

N. S. V. M. D. Levenson and R. A. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982).
[CrossRef]

Cobb, N.

N. Cobb and A. Zakhor, "Fast sparse aerial image calculation for OPC," BACUS Symposium on Photomask Technology, Proc. SPIE 2440, 313-327 (1995).

Erdmann, T.

T. F. B. T. A. Erdmann, R. Farkas and G. Kokai, "Towards automatic mask and source optimization for optical lithography," Optical Microlithography, Proc. SPIE 5377, 646-657 (2004).

Farkas, R.

T. F. B. T. A. Erdmann, R. Farkas and G. Kokai, "Towards automatic mask and source optimization for optical lithography," Optical Microlithography, Proc. SPIE 5377, 646-657 (2004).

Granik, Y.

Y. Granik, "Illuminator optimization methods in microlithography," Optical Microlithography Proc. SPIE 5754, 217-229 (2005).

Kokai, G.

T. F. B. T. A. Erdmann, R. Farkas and G. Kokai, "Towards automatic mask and source optimization for optical lithography," Optical Microlithography, Proc. SPIE 5377, 646-657 (2004).

Leone, R.

B. S. S. Sherif and R. Leone, "Binary image synthesis using mixed integer programming," IEEE Transactions on Image Processing 4(9), 1252-1257 (1995).
[CrossRef] [PubMed]

Levenson, N.

N. S. V. M. D. Levenson and R. A. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982).
[CrossRef]

Liu, Y.

Y. Liu and A. Zakhor, "Binary and phase shifting mask design for optical lithography," IEEE Transactions on Semiconductor Manufacturing 5(2) (1992).
[CrossRef]

Milanfar, P.

A. Poonawala and P. Milanfar, "Double Exposure Mask Synthesis using Inverse Lithography," submitted toJournal of Microlithography, Microfabrication, and Microsystems.

O’Brien, D. J.

D. W. P. E. D. W. P. Yao, G. J. Schneider and D. J. O’Brien, "Fabrication of three-dimensional photonic crystals with multilayer photolithography," Optics Express 13, 2370-2376 (2005).
[CrossRef] [PubMed]

B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider and D. J. O’Brien, "Multilayer three-dimensional photolithography with traditional planar method," Applied Physics Letters 85, 2920-2922 (2004).
[CrossRef]

Poonawala, A.

A. Poonawala and P. Milanfar, "Double Exposure Mask Synthesis using Inverse Lithography," submitted toJournal of Microlithography, Microfabrication, and Microsystems.

Schellenberg, F.

F. Schellenberg, "Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography," Proc. SPIE 5377, 1-20 (2004).
[CrossRef]

Schneider, G. J.

D. W. P. E. D. W. P. Yao, G. J. Schneider and D. J. O’Brien, "Fabrication of three-dimensional photonic crystals with multilayer photolithography," Optics Express 13, 2370-2376 (2005).
[CrossRef] [PubMed]

B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider and D. J. O’Brien, "Multilayer three-dimensional photolithography with traditional planar method," Applied Physics Letters 85, 2920-2922 (2004).
[CrossRef]

Sherif, B.

B. S. S. Sherif and R. Leone, "Binary image synthesis using mixed integer programming," IEEE Transactions on Image Processing 4(9), 1252-1257 (1995).
[CrossRef] [PubMed]

Simpson, R. A.

N. S. V. M. D. Levenson and R. A. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982).
[CrossRef]

Yao, B.

B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider and D. J. O’Brien, "Multilayer three-dimensional photolithography with traditional planar method," Applied Physics Letters 85, 2920-2922 (2004).
[CrossRef]

Yao, D.

D. W. P. E. D. W. P. Yao, G. J. Schneider and D. J. O’Brien, "Fabrication of three-dimensional photonic crystals with multilayer photolithography," Optics Express 13, 2370-2376 (2005).
[CrossRef] [PubMed]

Zakhor, A.

N. Cobb and A. Zakhor, "Fast sparse aerial image calculation for OPC," BACUS Symposium on Photomask Technology, Proc. SPIE 2440, 313-327 (1995).

Y. Liu and A. Zakhor, "Binary and phase shifting mask design for optical lithography," IEEE Transactions on Semiconductor Manufacturing 5(2) (1992).
[CrossRef]

Applied Physics Letters

B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider and D. J. O’Brien, "Multilayer three-dimensional photolithography with traditional planar method," Applied Physics Letters 85, 2920-2922 (2004).
[CrossRef]

IEEE Trans. Electron Devices

N. S. V. M. D. Levenson and R. A. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982).
[CrossRef]

IEEE Transactions on Image Processing

B. S. S. Sherif and R. Leone, "Binary image synthesis using mixed integer programming," IEEE Transactions on Image Processing 4(9), 1252-1257 (1995).
[CrossRef] [PubMed]

IEEE Transactions on Semiconductor Manufacturing

Y. Liu and A. Zakhor, "Binary and phase shifting mask design for optical lithography," IEEE Transactions on Semiconductor Manufacturing 5(2) (1992).
[CrossRef]

Journal of Microlithography, Microfabrication, and Microsystems

A. Poonawala and P. Milanfar, "Double Exposure Mask Synthesis using Inverse Lithography," submitted toJournal of Microlithography, Microfabrication, and Microsystems.

Optical Microlithography Proc. SPIE

Y. Granik, "Illuminator optimization methods in microlithography," Optical Microlithography Proc. SPIE 5754, 217-229 (2005).

Optics Express

D. W. P. E. D. W. P. Yao, G. J. Schneider and D. J. O’Brien, "Fabrication of three-dimensional photonic crystals with multilayer photolithography," Optics Express 13, 2370-2376 (2005).
[CrossRef] [PubMed]

Proc. SPIE

N. Cobb and A. Zakhor, "Fast sparse aerial image calculation for OPC," BACUS Symposium on Photomask Technology, Proc. SPIE 2440, 313-327 (1995).

F. Schellenberg, "Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography," Proc. SPIE 5377, 1-20 (2004).
[CrossRef]

T. F. B. T. A. Erdmann, R. Farkas and G. Kokai, "Towards automatic mask and source optimization for optical lithography," Optical Microlithography, Proc. SPIE 5377, 646-657 (2004).

Other

Y. L. L. Pang and D. Abrams, "Inverse lithography technology (ILT): What is the impact to the photomask industry?" Proc. SPIE (2006).
[CrossRef]

F. Schellenberg, Resolution enhancement techniques in optical lithography (SPIE Press, 2004).

A. W. M. L. W. L. L. Liebmann, S. Mansfield and T. Dunham, "TCAD development for lithography resolution enhancement," IBM Journal of Research and Development pp. 651-665 (2001).
[CrossRef]

A. Poonawala and P. Milanfar, "OPC and PSM design using inverse lithography: A non-linear optimization approach," in Proceedings of the SPIE, vol. 6154, pp. 1159-1172 (San Jose, CA, 2006).

C. Vogel, Computational methods for inverse problems (SIAM Press, 2002).
[CrossRef]

X. Ma and G. R. Arce, "Generalized inverse lithography methods for phase-shifting mask design," in Proceedings of SPIE, vol. 65200U (2007).

Y. C. Pati and T. Kailath, "Phase-shifting masks for microlithography: Automated design and mask requirements," Optical Society of America 11 (1994).

A. K. Wong, Resolution enhancement techniques, vol. 1 (SPIE Press, 2001).
[CrossRef]

S. A. Campbell, The science and engineering of microelectronic fabrication, 2nd ed. (Publishing House of Electronics Industry, 2003).

S. V. G. J. Schneider, J. Murakowski and D. W. Prather, "Combination lithography for photoniccrystal circuits,"J.of Vacuum Science and Technology B 22(1), 146-151 (2004).
[CrossRef]

J. M. M. J. M. P. Yao, G. J. Schneider and D.W. Prather, "Micro/nano lithography realized by chemical printing," in Proceedings of SPIE - The International Society for Optical Engineering, vol. 6151I of Emerging Lithographic Technologies X, p. 61511N (2006).

M. Born and E. Wolfe, Principles of optics (Cambridge University Press, 1999).

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Figures (11)

Fig. 1.
Fig. 1.

Approximated forward process model. (Ref. [17], Fig. 1).

Fig. 2.
Fig. 2.

Top row (input masks), left to right: desired pattern, magnitude of the optimized complex-valued mask and optimized pole-constrained mask obtained using a threshold tm . The middle row indicates the phases of the optimized complex-valued mask. The bottom row indicates the corresponding binary output patterns. The parameters used in the simulation are a = 80, tr = 0.5, tm = 0.5, 11 × 11 Gaussian low pass filter with k = 14, s ϕ _ = 2 and s θ _ = 0.01. Green, red and blue represent 0, 1 and -1 respectively in the top and bottom rows; Dark blue, light blue, yellow and red represent π 4 , 3 π 4 , 5 π 4 and 7 π 4 respectively in the middle row.

Fig. 3.
Fig. 3.

Left to right: desired pattern, pole-constraied optimized mask and binary output pattern. s ϕ _ = 2, s θ _ = 0.01. Red and blue represent 0 and π phases respectively.

Fig. 4.
Fig. 4.

Phase penalty for two-phase levels and four-phase levels

Fig. 5.
Fig. 5.

Left to right: desired pattern, pole-constrained optimized mask magnitude, pole-constrained optimized mask phases (Dark blue is π 4 , light blue is 3 π 4 , yellow is 5 π 4 and red is 7 π 4 and binary output pattern. s ϕ _ = 2, s θ _ = 0.01, γpole, ϕ _ = 0.001 and γpole, θ _ = 0.0001.

Fig. 6.
Fig. 6.

Left to right: desired pattern, pole-constrained optimized mask magnitude, phase and binary output pattern. s ϕ _ = 2, s θ _ = 0.01, γpole, ϕ _ = 0.01, γpole, θ _ = 0.001. Wavelet regularization uses γwavelet,ϕ̲ = 0.2 and γwavelet, θ _ = 0.001.

Fig. 7.
Fig. 7.

Left to right: desired pattern, pole-constrained optimized mask magnitude, phase and binary output pattern. s ϕ _ = 2, s θ _ = 0.01, γpole, ϕ _ = 0.01, γpole, θ _ = 0.001, γtv, ϕ _ = 0.3 and γtv, θ _ = 0.001.

Fig. 8.
Fig. 8.

Left to right: desired pattern, pole-constrained optimized mask magnitude, phase and binary output pattern. s ϕ _ = 2, s θ _ = 0.01, γpole, ϕ _ = 0.001, γpole, θ _ = 0.0001, γwavelet, ϕ _ = 0.2 and γwavelet, θ _ = 0.001. The gap between the vertical bars has regional weight of 1.6. The other regions have regional weight of 0.7.

Fig. 9.
Fig. 9.

Left to right: desired pattern, pole-constrained optimized mask and binary output pattern. s ϕ _ = 2, s θ _ = 0.01, γpole, ϕ _ = 0.001, γpole, θ _ = 0.0001, γwavelet, ϕ _ = 0.03 and γwavelet, θ _ = 0.001.

Fig. 10.
Fig. 10.

Left to right: desired pattern, pole-constrained optimized mask and binary output pattern. s ϕ _ = 2, s θ _ = 0.01, γpole, ϕ _ = 0.001, γpole, θ _ = 0.0001, γwavelet, ϕ _ = 0.03 and γwavelet, θ _ = 0.001. The regions for the first and the fourth bars have regional weight of 0.8. The regions for the second and the third bars have regional weight of 1.1. The other regions have regional weight of 0.7.

Fig. 11.
Fig. 11.

Left to right: mask for first exposure, mask for second exposure and the output pattern. s ϕ̲1 = 4, s θ̲1 = 0.01, s ϕ̲2 = 4, s θ̲2 = 0.01, γ pole, ϕ _ 1 = 0.015, γ pole, θ _ 1 = 0.001, γ wavelet, ϕ _ 1 = 0.5, γ wavelet, θ _ 1 = 0.003, γ pole, ϕ _ 2 = 0.015, γ pole, θ _ 2 = 0.001, γ wavelet, ϕ _ 2 = 0.5 and γ wavelet, θ _ 2 = 0.003.

Equations (52)

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M ̂ ( x , y ) = arg min M ( x , y ) C N × N d ( T { M ( x , y ) } , Z * ( x , y ) ) .
sig ( x ) = 1 1 + exp [ a ( x t r ) ] ,
M ̂ = arg min M d ( sig { H { M } } , Z * ) .
i = 1 1 + exp [ a j = 1 N 2 h ij j + a t r ] , i = 1 , . . . N 2 ,
F ( m ¯ ) = z ¯ * z ¯ 2 2 = i = 1 N 2 ( z i * z i ) 2 = i = 1 N 2 ( z i * 1 1 + exp [ a j = 1 N 2 h ij m ¯ j + a t r ] ) 2 .
m ¯ k = r ¯ k e j θ ¯ k , k = 1 , . . . , N 2 ,
r ¯ k = 1 + cos ϕ ¯ k 2 , k = 1 , . . . , N 2 ,
m ¯ k = 1 + cos ϕ ¯ k 2 e j θ ¯ k , k = 1 , . . . , N 2 .
( θ ̂ ¯ , ϕ ̂ ¯ ) = arg min ( θ̲ , ϕ̲ ) { F ( θ̲ , ϕ̲ ) } ,
F ( θ̲ , ϕ̲ ) = z ¯ * z ¯ 2 2 = i = 1 N 2 ( z i * z i ) 2
= i = 1 N 2 ( z i * 1 1 + exp [ a ( k = 1 N 2 h i k 1 + cos ϕ̲ k 2 e j θ̱ k ) 2 + a t r ] ) 2 .
F θ̲ ϕ̲ θ̲ = d ¯ θ̲ = 2 a × + cos ϕ̲ 2 sin θ̲ { H T [ ( z ¯ * z ¯ ) z ¯ ( z ¯ ) H ( m ¯ R ) T ( m ¯ ) ] }
2 a × + cos ϕ̲ 2 cos θ ¯ { H T [ ( z ¯ * z ¯ ) z̲⊙ ( z ¯ ) H ( m ¯ I ) T ( m ¯ ) ] } ,
F θ̲ ϕ̲ ϕ̲ = d ¯ ϕ̲ = a × sin ϕ̲ cos θ̲ { H T [ ( z ¯ * z ¯ ) ( z ¯ ) H ( m ¯ R ) T ( m ¯ ) ] }
+ a × sin ϕ̲ sin θ̲ { H T [ ( z ¯ * z ¯ ) z̲⊙ ( z ¯ ) H ( m ¯ I ) T ( m ¯ ) ] } ,
θ̲ k + 1 = θ̲ k s θ̲ d ¯ θ̲ k ,
ϕ̲ k + 1 = ϕ̲ k s ϕ̲ d ¯ ϕ̲ k ,
E = i = 1 N 2 z ¯ i * z ¯ b i = i = 1 N 2 z ¯ i * Γ H m ¯ p i .
m ̂ ¯ = arg min m ̂ ¯ { F ( m ¯ ) + γ R ( m ¯ ) } ,
R pole ( m ¯ ) = i = 1 N 2 i 2 = T .
R pole ( θ̲ ) = i = 1 N 2 [ sin ( 2 θ̲ i π 2 ) + 1 ] 2 = [ sin ( 2 θ̲ π 2 ) + ] T [ sin ( 2 θ̲ π 2 ) + ] .
R pole ( θ̲ ) = 4 [ sin ( 2 θ̲ π 2 ) + ] T cos ( 2 θ̲ π 2 ) .
R pole ( θ̲ ) = i = 1 N 2 [ sin ( 4 θ̲ i 3 π 2 ) + 1 ] 2 = [ sin ( 4 θ̲ i 3 π 2 ) + ] T [ sin ( 4 θ̲ i 3 π 2 ) + ] .
r ( θ̲ i ) = [ sin ( 4 θ̲ i 3 π 2 ) + 1 ] 2 , i = 1 , . . . , N 2
R pole ( θ̲ ) = 8 [ sin ( 4 θ̲ 3 π 2 ) + ] T cos ( 4 θ̲ 3 π 2 ) .
a i j = m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) + m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) + m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) + m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
h i j = m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) + m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
v i j = m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) + m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
d i j = m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) + m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
E detail = i = 1 N 2 j = 1 N 2 ( h i j h i j * + v i j v i j * + d i j d i j * ) .
E detail ϕ̲ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) = sin ϕ̲ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) × Re [ e j θ̲ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) × ( 3 m ¯ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q )
m ¯ ( 2 ( i 1 ) + p 1 ) ( 2 j 1 ) + q ) ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q 1 ) ( 2 ( i 1 ) + p 1 ) ( 2 ( j 1 ) + q 1 ) ) ] ,
E detail θ̲ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) = ( 1 + cos ϕ̲ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) ) × R e [ ( j ) e j θ̲ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q )
× ( 3 m ¯ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) ( 2 ( i 1 ) + p 1 ) ( 2 ( j 1 ) + q ) m ¯ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q 1 )
m ¯ ( 2 ( i 1 ) + p 1 ) ( 2 ( j 1 ) + q 1 ) ) ] ,
J ( m ¯ ) = F ( m ¯ ) + γ pole R pole ( m ¯ ) + γ wavelet E detail ( m ¯ ) .
R T V ( ϕ̲ ) = [ Q x T sgn ( Q x T f ¯ ) + Q y T sgn ( Q y T f ¯ ) ] sin ϕ ¯ R e [ ( m ¯ z ¯ * ) e j θ̲ ] 1 2 f ¯ ,
R T V ( θ̲ ) = [ Q x T sgn ( Q x T f ¯ ) + Q y T sgn ( Q y T f ¯ ) ] ( + cos ϕ̲ ) R e [ ( m ¯ z ¯ * ) e j θ̲ ( j ) ] 1 2 f ¯ ,
J ( m ¯ i ) = F ( m ¯ i ) + γ pole R pole ( m ¯ i ) + ω ( i ) γ wavelet E detail ( m ¯ i ) ,
m ¯ 1 k = r ¯ 1 k e j θ̲ 1 k = 1 + cos ϕ̲ 1 k 2 e j θ̲ 1 k , k = 1 , . . . , N 2 ,
m ¯ 2 k = r ¯ 2 k e j θ̲ 2 k = 1 + cos ϕ̲ 2 k 2 e j θ̲ 2 k , k = 1 , . . . , N 2 ,
z ¯ 1 k = 1 1 + exp [ a k = 1 N 2 h i k 1 + cos ϕ̲ 1 k 2 e j θ̲ 1 k + a t r ] , k = 1 , . . . , N 2 ,
z ¯ 2 k = 1 1 + exp [ a k = 1 N 2 h i k 1 + cos ϕ̲ 2 k 2 e j θ̲ 2 k + a t r ] , k = 1 , . . . , N 2 ,
F = F ( θ̲ 1 , ϕ̲ 1 , θ̲ 2 , ϕ̲ 2 ) = z ¯ * z ¯ 2 2 = i = 1 N 2 { z i * 1 2 [ tanh ( z ¯ 1 i + z ¯ 2 i 1 ) + 1 ] } 2 .
F θ̲ p = a × + cos ϕ̲ p 2 sin θ̲ p { H T [ ( z ¯ * z ¯ ) sech 2 ( z ¯ 1 + z ¯ 2 ) z ¯ p
( p ) H ( m ¯ p R ) T ( m ¯ , p ) ] }
a × + cos ϕ̲ p 2 cos θ̲ p { H T [ ( z ¯ * z ¯ ) sech 2 ( z ¯ 1 + z ¯ 2 ) z ¯ p
( z ¯ ) p H ( m ¯ p I ) T ( m ¯ , p ) ] } ,
F ϕ̲ p = a 2 × sin ϕ̲ p cos θ̲ p { H T [ ( z ¯ * z ¯ ) sech 2 ( z ¯ 1 + z ¯ 2 ) z ¯ p
( p ) H ( m ¯ p R ) T ( m ¯ , p ) ] }
+ a 2 × sin ϕ̲ p sin θ̲ p { H T [ ( z ¯ * z ¯ ) sech 2 ( z ¯ 1 + z ¯ 2 ) z ¯ p
( z ¯ ) p H ( m ¯ p I ) T ( m ¯ , p ) ] } ,

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