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,” 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 to Journal 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 (2)

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 (4)

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).

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]

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

2001 (1)

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]

1995 (2)

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]

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

1994 (1)

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

1992 (1)

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

1982 (1)

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]

Abrams, D.

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

Arce, G. R.

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

Born, M.

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

Campbell, S. A.

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

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).

Dunham, T.

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]

Erdmann, T. F. B. T. A.

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).

Kailath, T.

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

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. S. V. M. D.

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]

Liebmann, A. W. M. L. W. L. L.

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]

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]

Ma, X.

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

Mansfield, S.

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]

Milanfar, P.

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).

A. Poonawala and P. Milanfar, “Double Exposure Mask Synthesis using Inverse Lithography,” submitted to Journal of Microlithography, Microfabrication, and Microsystems.

Murakowski, J.

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

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]

Pang, Y. L. L.

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

Pati, Y. C.

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

Poonawala, A.

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).

A. Poonawala and P. Milanfar, “Double Exposure Mask Synthesis using Inverse Lithography,” submitted to Journal of Microlithography, Microfabrication, and Microsystems.

Prather, D. W.

S. V. G. J. Schneider, J. Murakowski, and D. W. Prather, “Combination lithography for photoniccrystal circuits,” 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).

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]

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

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]

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).

Schneider, S. V. G. J.

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

Sherif, B. S. S.

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]

Vogel, C.

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

Wolfe, E.

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

Wong, A. K.

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

Yao, B. L. M. J. M. D. W. P. E. D. W. P.

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. W. P. E. D. W. P.

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]

Yao, J. M. M. J. M. P.

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).

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 (1)

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]

BACUS Symposium on Photomask Technology, Proc. SPIE (1)

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

IBM Journal of Research and Development (1)

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]

IEEE Trans. Electron Devices (1)

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 (1)

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 (1)

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

Optical Microlithography Proc. SPIE (1)

Y. Granik, “Illuminator optimization methods in microlithography,” Optical Microlithography Proc. SPIE 5754, 217–229 (2005).

Optical Microlithography, Proc. SPIE (1)

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).

Optical Society of America (1)

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

Optics Express (1)

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 (1)

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

Vacuum Science and Technology B (1)

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

Other (10)

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).

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).

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

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

A. Poonawala and P. Milanfar, “Double Exposure Mask Synthesis using Inverse Lithography,” submitted to Journal of Microlithography, Microfabrication, and Microsystems.

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).

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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.

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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.

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Fig. 4.
Fig. 4. Phase penalty for two-phase levels and four-phase levels

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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.

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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.

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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.

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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.

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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.

Download Full Size | PPT Slide | PDF

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.

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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.

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Equations (52)

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

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 * ) .
ẕ i = 1 1 + exp [ − a ∣ ∑ j = 1 N 2 h ij m̱ 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 × 1̲ + cos ϕ̲ 2 ⊙ sin θ̲ ⊙ { H T [ ( z ¯ * − z ¯ ) ⊙ z ¯ ⊙ ( 1̲ − z ¯ ) ⊙ H ( m ¯ R ) ⊙ T ( m ¯ ) ] }
− 2 a × 1̲ + cos ϕ̲ 2 ⊙ cos θ ¯ ⊙ { H T [ ( z ¯ * − z ¯ ) ⊙ z̲⊙ ( 1̲ − z ¯ ) ⊙ H ( m ¯ I ) ⊙ T ( m ¯ ) ] } ,
∇ F θ̲ ϕ̲ ϕ̲ = d ¯ ϕ̲ = a × sin ϕ̲ ⊙ cos θ̲ ⊙ { H T [ ( z ¯ * − z ¯ ) ⊙ z̲ ⊙ ( 1̲ − z ¯ ) ⊙ H ( m ¯ R ) ⊙ T ( m ¯ ) ] }
+ a × sin ϕ̲ ⊙ sin θ̲ ⊙ { H T [ ( z ¯ * − z ¯ ) ⊙ z̲⊙ ( 1̲ − 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 ∣ m̲ ∣ i 2 = ∣ m̲ ∣ T ∣ m̲ ∣ .
R pole ( θ̲ ) = ∑ i = 1 N 2 [ sin ( 2 θ̲ i − π 2 ) + 1 ] 2 = [ sin ( 2 θ̲ − π 2 ) + 1̲ ] T [ sin ( 2 θ̲ − π 2 ) + 1̲ ] .
∇ R pole ( θ̲ ) = 4 [ sin ( 2 θ̲ − π 2 ) + 1̲ ] 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 ) + 1̲ ] T [ sin ( 4 θ̲ i − 3 π 2 ) + 1̲ ] .
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 ) + 1̲ ] 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 ) − m̲ ( 2 ( i − 1 ) + p ) ( 2 ( j − 1 ) + q 1 ) − m̲ ( 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 ) − m̲ ( 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 ¯ ) ] ⊙ ( 1̲ + 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 × 1̲ + cos ϕ̲ p 2 ⊙ sin θ̲ p ⊙ { H T [ ( z ¯ * − z ¯ ) ⊙ sech 2 ( z ¯ 1 + z ¯ 2 − 1̲ ) ⊙ z ¯ p
⊙ ( 1̲ − z̲ p ) ⊙ H ( m ¯ p R ) ⊙ T ( m ¯ , p ) ] }
− a × 1̲ + cos ϕ̲ p 2 ⊙ cos θ̲ p ⊙ { H T [ ( z ¯ * − z ¯ ) ⊙ sech 2 ( z ¯ 1 + z ¯ 2 − 1̲ ) ⊙ z ¯ p
⊙ ( 1̲ − 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 − 1̲ ) ⊙ z ¯ p
⊙ ( 1̲ − z̲ 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 − 1̲ ) ⊙ z ¯ p
⊙ ( 1̲ − z ¯ ) p ⊙ H ( m ¯ p I ) ⊙ T ( m ¯ , p ) ] } ,

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