Abstract

A novel emission spectral tomography algorithm based on multiobjective optimization is proposed. Its reconstruction results for asymmetrical emission coefficient fields are studied with computer simulation. The results show that this algorithm provides a significant improvement in reconstruction precision and convergence over traditional algorithms and is suitable for real-time reconstruction of an emission-coefficient field with incomplete data. In an experiment of the argon-arc plasma diagnosis, we adopted this algorithm and the spectrum relative-intensity method to obtain the three-dimensional distributions of temperature, ionization coefficient, and electron (ion) and atom densities.

© 2004 Optical Society of America

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    [CrossRef]
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2003 (2)

X. Wan, Y. Gao, Y. Wang, “3-D flame temperature field reconstruction with multiobjective neural network,” Chin. Opt. Lett. 1, 78–81 (2003).

X. Wan, Y. Gao, Q. Wang, S. Le, S. Yu, “Limited-angle optical computed tomography algorithms,” Opt. Eng. (Bellingham) 42, 2659–2669 (2003).
[CrossRef]

2001 (1)

Y. Gao, “Reconstruction of arc temperature fields by projection space iteration reconstruction-reprojection,” Acta Photon. Sin. 30, 196–200 (2001).

1998 (1)

Y. Gao, “3-D arc temperature diagnosis with more computed tomography,” Acta Opt. Sin. 18, 376–380 (1998).

1997 (1)

Y. Wang, F. M. Wahl, “Multiobjective neural network for image reconstruction,” IEE Proc. Vision Image Signal Process. 144, 233–236 (1997).
[CrossRef]

1995 (1)

Y. Takakura, S. Ogawa, Y. Wada, “Transonic wind-tunnel flow about a fully configured aircraft,” AIAA J. 33, 557–559 (1995).
[CrossRef]

1994 (1)

1993 (1)

1991 (2)

1989 (1)

N. Iwama, H. Yoshida, H. Takimoto, Y. Shen, S. Takamura, T. Tsukishima, “Phillips–Tikhonov regularization of plasma image reconstruction with the generalized cross validation,” Appl. Phys. Lett. 54, 502–504 (1989).
[CrossRef]

1988 (3)

1987 (3)

1986 (1)

1985 (4)

K. M. Hanson, G. W. Wecksung, “Local basis-function approach to computed tomography,” Appl. Opt. 24, 4028–4039 (1985).
[CrossRef] [PubMed]

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imaging MI-4, 65–71 (1985).
[CrossRef]

H. Uchiyama, M. Nakajima, S. Yuta, “Measurement of flame temperature distribution by IR emission computed tomography,” Appl. Opt. 24, 4111–4116 (1985).
[CrossRef] [PubMed]

H. Uchiyama, K. Kazahaya, M. Nakajima, M. Mizomoto, S. Yuta, “Attenuation correction for infrared rays emission CT,” Trans. Inst. Electron. Commun. Eng. Jpn., Part D 68, 2073–2081 (1985).

1984 (3)

H. Uchiyama, T. Takezawa, K. Abe, S. Haku, M. Nakajima, S. Yuta, “Proposal on infrared rays emission CT,” Trans. Inst. Electron. Commun. Eng. Jpn., Part D 67, 1021–1026 (1984).

R. Synder, L. Hesselink, “Optical tomography for flow visualization of the density field around a revolving helicopter rotor blade,” Appl. Opt. 23, 3650–3656 (1984).
[CrossRef]

T. S. Melnikova, V. V. Pickalov, “Tomographic measurements of temperature fields in non-stationary arc plasma,” Beitr. Plasmaphys. 24, 431–445 (1984).
[CrossRef]

1983 (1)

1982 (1)

1980 (1)

N. Sebald, “Measurement of the temperature and flow fields of the magnetically stabilized cross-flow N2 arc,” Appl. Phys. 21, 221–236 (1980).
[CrossRef]

1978 (1)

B. R. Myers, M. A. Levine, “Two-dimensional spectral line emission reconstruction as a plasma diagnostic,” Rev. Sci. Instrum. 49, 610–616 (1978).
[CrossRef] [PubMed]

1974 (1)

D. W. Sweeney, C. M. Vest, “Measurement of three-dimensional temperature fields above heated surfaces by holographic interferometry,” Int. J. Heat Mass Transfer 17, 1443–1454 (1974).
[CrossRef]

1973 (1)

1972 (1)

1968 (2)

E. T. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. SSC-4, 227–241 (1968).
[CrossRef]

H. N. Olsen, C. D. Maldonado, G. D. Duckworth, “A numerical method for obtaining internal emission coefficients from externally measured spectral intensities of asymmetrical plasmas,” J. Quant. Spectrosc. Radiat. Transf. 8, 1419–1430 (1968).
[CrossRef]

Abbiss, J. B.

Abe, A.

M. Watanabe, A. Abe, R. T. Casey, K. Takayama, “Holographic interferometric observation of shock wave phenomena,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 418–426 (1992).
[CrossRef]

Abe, K.

H. Uchiyama, T. Takezawa, K. Abe, S. Haku, M. Nakajima, S. Yuta, “Proposal on infrared rays emission CT,” Trans. Inst. Electron. Commun. Eng. Jpn., Part D 67, 1021–1026 (1984).

Aono, T.

Bahl, S.

Bryanston-Cross, P. J.

D. P. Towers, C. E. Towers, P. J. Bryanston-Cross, K. Fry, A. E. Harris, “Visualization and analysis of three-dimensional transonic flows by holographic interferometry,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 388–403 (1992).
[CrossRef]

Byer, R. L.

Casey, R. T.

M. Watanabe, A. Abe, R. T. Casey, K. Takayama, “Holographic interferometric observation of shock wave phenomena,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 418–426 (1992).
[CrossRef]

De Mol, C.

Defrise, M.

Dhadwal, H. S.

Duckworth, G. D.

H. N. Olsen, C. D. Maldonado, G. D. Duckworth, “A numerical method for obtaining internal emission coefficients from externally measured spectral intensities of asymmetrical plasmas,” J. Quant. Spectrosc. Radiat. Transf. 8, 1419–1430 (1968).
[CrossRef]

Faris, G. W.

Frieden, B. R.

Fry, K.

D. P. Towers, C. E. Towers, P. J. Bryanston-Cross, K. Fry, A. E. Harris, “Visualization and analysis of three-dimensional transonic flows by holographic interferometry,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 388–403 (1992).
[CrossRef]

Gao, Y.

X. Wan, Y. Gao, Y. Wang, “3-D flame temperature field reconstruction with multiobjective neural network,” Chin. Opt. Lett. 1, 78–81 (2003).

X. Wan, Y. Gao, Q. Wang, S. Le, S. Yu, “Limited-angle optical computed tomography algorithms,” Opt. Eng. (Bellingham) 42, 2659–2669 (2003).
[CrossRef]

Y. Gao, “Reconstruction of arc temperature fields by projection space iteration reconstruction-reprojection,” Acta Photon. Sin. 30, 196–200 (2001).

Y. Gao, “3-D arc temperature diagnosis with more computed tomography,” Acta Opt. Sin. 18, 376–380 (1998).

X. Wan, Y. Gao, S. Yu, “Study of limited-view tomography alogrithms for plasma diagnostics,” in Optical Design and Testing, Z. Weng, J. M. Sasian, Y. Wang, eds., Proc. SPIE4927, 625–632 (2002).
[CrossRef]

Gull, S. F.

Haku, S.

H. Uchiyama, T. Takezawa, K. Abe, S. Haku, M. Nakajima, S. Yuta, “Proposal on infrared rays emission CT,” Trans. Inst. Electron. Commun. Eng. Jpn., Part D 67, 1021–1026 (1984).

Hanson, K. M.

Harris, A. E.

D. P. Towers, C. E. Towers, P. J. Bryanston-Cross, K. Fry, A. E. Harris, “Visualization and analysis of three-dimensional transonic flows by holographic interferometry,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 388–403 (1992).
[CrossRef]

Hesselink, L.

Hino, M.

Iwama, N.

N. Iwama, H. Yoshida, H. Takimoto, Y. Shen, S. Takamura, T. Tsukishima, “Phillips–Tikhonov regularization of plasma image reconstruction with the generalized cross validation,” Appl. Phys. Lett. 54, 502–504 (1989).
[CrossRef]

Jaynes, E. T.

E. T. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. SSC-4, 227–241 (1968).
[CrossRef]

Kawata, S.

Kazahaya, K.

H. Uchiyama, K. Kazahaya, M. Nakajima, M. Mizomoto, S. Yuta, “Attenuation correction for infrared rays emission CT,” Trans. Inst. Electron. Commun. Eng. Jpn., Part D 68, 2073–2081 (1985).

Le, S.

X. Wan, Y. Gao, Q. Wang, S. Le, S. Yu, “Limited-angle optical computed tomography algorithms,” Opt. Eng. (Bellingham) 42, 2659–2669 (2003).
[CrossRef]

Levine, M. A.

B. R. Myers, M. A. Levine, “Two-dimensional spectral line emission reconstruction as a plasma diagnostic,” Rev. Sci. Instrum. 49, 610–616 (1978).
[CrossRef] [PubMed]

Liburdy, J. A.

Lu, W.

Maldonado, C. D.

H. N. Olsen, C. D. Maldonado, G. D. Duckworth, “A numerical method for obtaining internal emission coefficients from externally measured spectral intensities of asymmetrical plasmas,” J. Quant. Spectrosc. Radiat. Transf. 8, 1419–1430 (1968).
[CrossRef]

Meier, G. E. A.

Meinel, E. S.

Melnikova, T. S.

T. S. Melnikova, V. V. Pickalov, “Tomographic measurements of temperature fields in non-stationary arc plasma,” Beitr. Plasmaphys. 24, 431–445 (1984).
[CrossRef]

Middendorf, P.

Minami, S.

Mizomoto, M.

H. Uchiyama, K. Kazahaya, M. Nakajima, M. Mizomoto, S. Yuta, “Attenuation correction for infrared rays emission CT,” Trans. Inst. Electron. Commun. Eng. Jpn., Part D 68, 2073–2081 (1985).

Montgomery, G. P.

Myers, B. R.

B. R. Myers, M. A. Levine, “Two-dimensional spectral line emission reconstruction as a plasma diagnostic,” Rev. Sci. Instrum. 49, 610–616 (1978).
[CrossRef] [PubMed]

Nakajima, M.

M. Hino, T. Aono, M. Nakajima, S. Yuta, “Light emission computed tomography system for plasma diagnostics,” Appl. Opt. 26, 4742–4746 (1987).
[CrossRef] [PubMed]

H. Uchiyama, K. Kazahaya, M. Nakajima, M. Mizomoto, S. Yuta, “Attenuation correction for infrared rays emission CT,” Trans. Inst. Electron. Commun. Eng. Jpn., Part D 68, 2073–2081 (1985).

H. Uchiyama, M. Nakajima, S. Yuta, “Measurement of flame temperature distribution by IR emission computed tomography,” Appl. Opt. 24, 4111–4116 (1985).
[CrossRef] [PubMed]

H. Uchiyama, T. Takezawa, K. Abe, S. Haku, M. Nakajima, S. Yuta, “Proposal on infrared rays emission CT,” Trans. Inst. Electron. Commun. Eng. Jpn., Part D 67, 1021–1026 (1984).

Nakamura, O.

Nalcioglu, O.

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imaging MI-4, 65–71 (1985).
[CrossRef]

Newton, T. J.

Obermeier, F.

Ogawa, S.

Y. Takakura, S. Ogawa, Y. Wada, “Transonic wind-tunnel flow about a fully configured aircraft,” AIAA J. 33, 557–559 (1995).
[CrossRef]

Olsen, H. N.

H. N. Olsen, C. D. Maldonado, G. D. Duckworth, “A numerical method for obtaining internal emission coefficients from externally measured spectral intensities of asymmetrical plasmas,” J. Quant. Spectrosc. Radiat. Transf. 8, 1419–1430 (1968).
[CrossRef]

Pickalov, V. V.

T. S. Melnikova, V. V. Pickalov, “Tomographic measurements of temperature fields in non-stationary arc plasma,” Beitr. Plasmaphys. 24, 431–445 (1984).
[CrossRef]

Poplevina, L. I.

L. I. Poplevina, I. M. Tokmulin, G. N. Vishnyakov, “Emission spectral tomography of multijet plasma flow,” in Inverse Optics III, M. A. Fiddy, ed., Proc. SPIE2241, 90–98 (1994).
[CrossRef]

Reuss, D. L.

Romine, G. L.

A. J. Senol, G. L. Romine, “Three dimensional refraction-diffraction of EM waves through rocket exhaust plumes,” in Proceedings of the AIAA 17th Fluid Dynamics, Plasma Dynamics and Lasers Conference (Institute of Aeronautics and Astronautics, Reston, Va., 1984), paper 84-1597.

Schultz, P. H.

Sebald, N.

N. Sebald, “Measurement of the temperature and flow fields of the magnetically stabilized cross-flow N2 arc,” Appl. Phys. 21, 221–236 (1980).
[CrossRef]

Senol, A. J.

A. J. Senol, G. L. Romine, “Three dimensional refraction-diffraction of EM waves through rocket exhaust plumes,” in Proceedings of the AIAA 17th Fluid Dynamics, Plasma Dynamics and Lasers Conference (Institute of Aeronautics and Astronautics, Reston, Va., 1984), paper 84-1597.

Shen, Y.

N. Iwama, H. Yoshida, H. Takimoto, Y. Shen, S. Takamura, T. Tsukishima, “Phillips–Tikhonov regularization of plasma image reconstruction with the generalized cross validation,” Appl. Phys. Lett. 54, 502–504 (1989).
[CrossRef]

Soller, C.

Sweeney, D. W.

D. W. Sweeney, C. M. Vest, “Measurement of three-dimensional temperature fields above heated surfaces by holographic interferometry,” Int. J. Heat Mass Transfer 17, 1443–1454 (1974).
[CrossRef]

D. W. Sweeney, C. M. Vest, “Reconstruction of three-dimensional refractive index field from multidirectional interferometric data,” Appl. Opt. 12, 2649–2664 (1973).
[CrossRef] [PubMed]

Synder, R.

Takakura, Y.

Y. Takakura, S. Ogawa, Y. Wada, “Transonic wind-tunnel flow about a fully configured aircraft,” AIAA J. 33, 557–559 (1995).
[CrossRef]

Takamura, S.

N. Iwama, H. Yoshida, H. Takimoto, Y. Shen, S. Takamura, T. Tsukishima, “Phillips–Tikhonov regularization of plasma image reconstruction with the generalized cross validation,” Appl. Phys. Lett. 54, 502–504 (1989).
[CrossRef]

Takayama, K.

M. Watanabe, A. Abe, R. T. Casey, K. Takayama, “Holographic interferometric observation of shock wave phenomena,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 418–426 (1992).
[CrossRef]

Takezawa, T.

H. Uchiyama, T. Takezawa, K. Abe, S. Haku, M. Nakajima, S. Yuta, “Proposal on infrared rays emission CT,” Trans. Inst. Electron. Commun. Eng. Jpn., Part D 67, 1021–1026 (1984).

Takimoto, H.

N. Iwama, H. Yoshida, H. Takimoto, Y. Shen, S. Takamura, T. Tsukishima, “Phillips–Tikhonov regularization of plasma image reconstruction with the generalized cross validation,” Appl. Phys. Lett. 54, 502–504 (1989).
[CrossRef]

Tokmulin, I. M.

L. I. Poplevina, I. M. Tokmulin, G. N. Vishnyakov, “Emission spectral tomography of multijet plasma flow,” in Inverse Optics III, M. A. Fiddy, ed., Proc. SPIE2241, 90–98 (1994).
[CrossRef]

Towers, C. E.

D. P. Towers, C. E. Towers, P. J. Bryanston-Cross, K. Fry, A. E. Harris, “Visualization and analysis of three-dimensional transonic flows by holographic interferometry,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 388–403 (1992).
[CrossRef]

Towers, D. P.

D. P. Towers, C. E. Towers, P. J. Bryanston-Cross, K. Fry, A. E. Harris, “Visualization and analysis of three-dimensional transonic flows by holographic interferometry,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 388–403 (1992).
[CrossRef]

Tsukishima, T.

N. Iwama, H. Yoshida, H. Takimoto, Y. Shen, S. Takamura, T. Tsukishima, “Phillips–Tikhonov regularization of plasma image reconstruction with the generalized cross validation,” Appl. Phys. Lett. 54, 502–504 (1989).
[CrossRef]

Uchiyama, H.

H. Uchiyama, M. Nakajima, S. Yuta, “Measurement of flame temperature distribution by IR emission computed tomography,” Appl. Opt. 24, 4111–4116 (1985).
[CrossRef] [PubMed]

H. Uchiyama, K. Kazahaya, M. Nakajima, M. Mizomoto, S. Yuta, “Attenuation correction for infrared rays emission CT,” Trans. Inst. Electron. Commun. Eng. Jpn., Part D 68, 2073–2081 (1985).

H. Uchiyama, T. Takezawa, K. Abe, S. Haku, M. Nakajima, S. Yuta, “Proposal on infrared rays emission CT,” Trans. Inst. Electron. Commun. Eng. Jpn., Part D 67, 1021–1026 (1984).

Verhoeven, D.

Vest, C. M.

D. W. Sweeney, C. M. Vest, “Measurement of three-dimensional temperature fields above heated surfaces by holographic interferometry,” Int. J. Heat Mass Transfer 17, 1443–1454 (1974).
[CrossRef]

D. W. Sweeney, C. M. Vest, “Reconstruction of three-dimensional refractive index field from multidirectional interferometric data,” Appl. Opt. 12, 2649–2664 (1973).
[CrossRef] [PubMed]

Vishnyakov, G. N.

L. I. Poplevina, I. M. Tokmulin, G. N. Vishnyakov, “Emission spectral tomography of multijet plasma flow,” in Inverse Optics III, M. A. Fiddy, ed., Proc. SPIE2241, 90–98 (1994).
[CrossRef]

Wada, Y.

Y. Takakura, S. Ogawa, Y. Wada, “Transonic wind-tunnel flow about a fully configured aircraft,” AIAA J. 33, 557–559 (1995).
[CrossRef]

Wahl, F. M.

Y. Wang, F. M. Wahl, “Multiobjective neural network for image reconstruction,” IEE Proc. Vision Image Signal Process. 144, 233–236 (1997).
[CrossRef]

Wan, X.

X. Wan, Y. Gao, Y. Wang, “3-D flame temperature field reconstruction with multiobjective neural network,” Chin. Opt. Lett. 1, 78–81 (2003).

X. Wan, Y. Gao, Q. Wang, S. Le, S. Yu, “Limited-angle optical computed tomography algorithms,” Opt. Eng. (Bellingham) 42, 2659–2669 (2003).
[CrossRef]

X. Wan, Y. Gao, S. Yu, “Study of limited-view tomography alogrithms for plasma diagnostics,” in Optical Design and Testing, Z. Weng, J. M. Sasian, Y. Wang, eds., Proc. SPIE4927, 625–632 (2002).
[CrossRef]

Wang, Q.

X. Wan, Y. Gao, Q. Wang, S. Le, S. Yu, “Limited-angle optical computed tomography algorithms,” Opt. Eng. (Bellingham) 42, 2659–2669 (2003).
[CrossRef]

Wang, Y.

Watanabe, M.

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M. Watanabe, A. Abe, R. T. Casey, K. Takayama, “Holographic interferometric observation of shock wave phenomena,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 418–426 (1992).
[CrossRef]

X. Wan, Y. Gao, S. Yu, “Study of limited-view tomography alogrithms for plasma diagnostics,” in Optical Design and Testing, Z. Weng, J. M. Sasian, Y. Wang, eds., Proc. SPIE4927, 625–632 (2002).
[CrossRef]

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[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Relationship between emission coefficient εν with spectral intensity Iν.

Fig. 2
Fig. 2

Grid divisions of the series-expansion-based algorithms.

Fig. 3
Fig. 3

Single-peak cosine simulative emission-coefficient phantom.

Fig. 4
Fig. 4

Four-peak Gaussian simulative emission-coefficient phantom.

Fig. 5
Fig. 5

Reconstruction of a single-peak cosine phantom with the MCIR algorithm: (a) no noise (geometry A), (b) little noise (geometry A), (c) strong noise (geometry A).

Fig. 6
Fig. 6

Reconstruction of a four-peak Gaussian phantom with the MCIR algorithm: (a) no noise (geometry C), (b) little noise (geometry C), (c) strong noise (geometry B).

Fig. 7
Fig. 7

Convergence of four algorithms for the single-peak cosine phantom reconstructions (geometry A): (a) no noise, (b) little noise, (c) strong noise.

Fig. 8
Fig. 8

Convergence of four algorithms for the four-peak Gaussian phantom reconstructions (geometry B): (a) no noise, (b) little noise, (c) strong noise.

Fig. 9
Fig. 9

Experimental device of argon-arc plasma field reconstruction.

Fig. 10
Fig. 10

Structure of the optical fiber sensor head.

Fig. 11
Fig. 11

Spectral intensity data in the (a) X direction and (b) Y direction (26 RPV).

Fig. 12
Fig. 12

Reconstruction of the emission-coefficient field with the MCIR algorithm (420 nm): (a) surface and (b) contour.

Fig. 13
Fig. 13

Reconstruction of emission-coefficient field with the MCIR algorithm (425.9 nm): (a) surface and (b) contour.

Fig. 14
Fig. 14

Temperature distribution of the argon-arc plasma of the first profile (2 mm away, from the jet nozzle).

Fig. 15
Fig. 15

3-D reconstruction of the temperature field of the argon-arc plasma.

Tables (2)

Tables Icon

Table 1 Reconstruction Errors of MCIR Compared with Traditional Algorithms for Simulative Emission-Coefficient Phantomsa

Tables Icon

Table 2 Reconstruction Errors of MCIR for Rotated Cosine Phantoms

Equations (43)

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Iν(t, ϕ)=-εν(x, y)ds.
εν(x, y)=j=1MNfjb(x-xj, y-yj),
bxy=bxx-mlxlxbyy-nlyly,
bx(u)=by(u)=sin uπuπ.
Iν(t, ϕ)|i=j=1MNfjib(x-xj, y-yj)ds=j=1MNwijfj,
Iν=WF+E,
Φ1(Fˆ)=(Iν-WFˆ)TD(I-WFˆ),
Φ21(Fˆ)=jξfj-18kξjfk2,
Φ21(Fˆ)=F^TjξSjSjTFˆ=F^TBFˆ,
Sjk=1,k=j-18,kEj-18,kVj0,otherwise,
Φ22(Fˆ)=j=1MNfj2=Fˆ2=F^TFˆ,
Φ2(Fˆ)=F^T(B+I)Fˆ=F^TBFˆ.
Φ1(Fˆ)+λΦ22(Fˆ).
Φ3(Fˆ)=-j=1MNfjln fj=-F^TlnFˆ,
Φ3(Fˆ)=F^TlnFˆ.
Φ1(Fˆ)+λΦ3(Fˆ).
Φ(Fˆ)=l=13λlΦl(Fˆ)
dΦ(Fˆ)dFˆ=-2λ11σ2WT(I-WFˆ)+2λ2BFˆ+λ3(lnFˆ+1)=0.
F(0)=1,
F(j)(k+1)=R(j)(k)F(j)(k),j=1,2,,MN,
R(j)(k)=1+γ{2λ1(k) 1σ2 wij[Ii-WiF(k)]-2λ2(k)BijF(j)(k)-λ3(k)[lnF(j)(k)+1]},
i=k(modI)+1,
ΔΦl(k+1)=|Φl(k+1)-Φl(k)|,l=1, 2, 3.
λ1+λ2+λ3=1,
λ1ΔΦ1=λ2ΔΦ2=λ3ΔΦ3.
λl(k+1)=τlΔΦτ(k+1)l=13[τlΔΦτ(k+1),l,τ=1,2,3.
|F(k+1)-F(k)|<ε1.
εν(x, y)=0.25{1-cos[2π(x+0.5)4/5]}×{1-cos[2π(y+0.5)2/3]},|x, y|<0.5=0,otherwise.
εν(x, y)=i=14aiexp-4 ln 20.22 (x-xi)2-4 ln 20.22 (y-yi)2,
x1=0.15,y1=0.15;x2=0.15,y2=-0.15,
x3=-0.15,y3=0.15;x4=-0.15,y4=-0.15,
a1=1,a2=0.4,a3=0.6,a4=0.8.
Ii=[1+N(0, σ2)]Ii,
α=j=1MN|fj-fj|fj max×M×N.
β=|fj-f^j|maxfj max.
γ=j=1MN(fj-f^j)2/j=1MNfj21/2.
εν1εν2=ν1A1g1ν2A2g2exp-Eu1-Eu2kT,
ΔTTΔ(εν1/εν2)εν1/εν2+Δ(A1/A2)A1/A2kTEu1-Eu2.
ne2na=2Zi(T)Za(T)2πmekTh23/2exp-EIkT,
Zi(T)=4+2 exp-2060T,
Za(T)=1+60 exp-162500T.
P=(na+ne+ni)kT.
Γ=nena+ne.

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