Abstract

We reconstruct images of the absorption and the scattering coefficients for diffuse optical tomography using five different models for light propagation in tissues: (1) the radiative transport equation, (2) the delta-Eddington approximation, (3) the Fokker-Planck approximation, (4) the Fokker-Planck-Eddington approximation and (5) the generalized Fokker-Planck-Eddington approximation. The last four models listed are approximations of the radiative transport equation that take into account forward-peaked scattering analytically. Using simulated data from the numerical solution of radiative transport equation, we solve the inverse problem for the absorption and scattering coefficients using the transport-backtransport method. Through comparison of the numerical results, we show that all of these light scattering models produce good image reconstructions. In addition, we show that these approximations afford considerable computational savings over solving the radiative transport equation. However, all of the models exhibit significant “crosstalk” between absorption and scattering coefficient images. Among the approximations, we have found that the generalized Fokker-Planck-Eddington equation produced the best image reconstructions in comparison with the image reconstructions produced by the radiative transport equation.

© 2009 Optical Society of America

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References

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  1. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  2. A. P. Gibson, J. C. Hebden, and S. R. Arridge “Recent advances in diffuse optical imaging,” Phys. Med. Bio. 50, R1–R43 (2005).
    [CrossRef]
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997).
  4. P. Gonzàlez-Rodríguez and A. D. Kim,, “Light propagation in tissues with forward-peaked and large-angle scattering,” Appl. Opt. 47, 2599–2609 (2008).
    [CrossRef] [PubMed]
  5. O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
    [CrossRef]
  6. O. Dorn, “Scattering and absorption transport sensitivity functions for optical tomography,” Opt. Express 13, 492–506 (2000).
    [CrossRef]
  7. F. Natterer and F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
    [CrossRef]
  8. M. Vogeler, “Reconstruction of the three-dimensional refractive index in electromagnetic scattering by using a propagation-backpropagation method,” Inverse Probl. 19, 739–753 (2003).
    [CrossRef]
  9. O. Dorn, H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, “A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields,” Inverse Probl. 15, 1523–1558 (1999).
    [CrossRef]
  10. O. Dorn, E. L. Miller, and C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adoint fields and level sets,” Inverse Probl. 16, 1119–1156 (2000).
    [CrossRef]
  11. P. Gonzàlez-Rodríguez, M. Kindelan, M. Moscoso, and O. Dorn, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
    [CrossRef]
  12. E. Haber, U. Ascher, and D. Oldenburg, “On optimization techniques for solving nonlinear inverse problems,” Inverse Probl. 16, 1263–1280 (2000).
    [CrossRef]
  13. F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001).
    [CrossRef]
  14. C. R. Vogel, Computational Methods for Inverse Problems (SIAM, Philadelphia, PA, 2002).
    [CrossRef]
  15. F. Natterer, H. Sielschott, O. Dorn, T. Dierkes, and V. Palamodov, “Fréchet derivatives for some bilinear inverse problems,” SIAM J. Appl. Math. 62, 2092–2113 (2002).
    [CrossRef]
  16. J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
    [CrossRef]
  17. A. D. Fokker, “Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld,” Ann. Phys. 43, 810–820 (1914).
    [CrossRef]
  18. M. Planck, “Ueber einen Satz der statistichen Dynamik und eine Erweiterung in der Quantumtheorie,” Sitzung-ber. Preuss. Akad. Wiss. 5, 324–341 (1917).
  19. A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92–98 (2003).
    [CrossRef]
  20. J. Heino, S. Arridge, J. Sikora, and E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908 (2003).
    [CrossRef]
  21. E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, Inc., La Grange Park, IL, 1993).
  22. L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge University Press, Cambridge, 1985).
    [CrossRef]

2008 (1)

2005 (2)

A. P. Gibson, J. C. Hebden, and S. R. Arridge “Recent advances in diffuse optical imaging,” Phys. Med. Bio. 50, R1–R43 (2005).
[CrossRef]

P. Gonzàlez-Rodríguez, M. Kindelan, M. Moscoso, and O. Dorn, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
[CrossRef]

2003 (3)

M. Vogeler, “Reconstruction of the three-dimensional refractive index in electromagnetic scattering by using a propagation-backpropagation method,” Inverse Probl. 19, 739–753 (2003).
[CrossRef]

J. Heino, S. Arridge, J. Sikora, and E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908 (2003).
[CrossRef]

A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92–98 (2003).
[CrossRef]

2002 (1)

F. Natterer, H. Sielschott, O. Dorn, T. Dierkes, and V. Palamodov, “Fréchet derivatives for some bilinear inverse problems,” SIAM J. Appl. Math. 62, 2092–2113 (2002).
[CrossRef]

2000 (3)

O. Dorn, “Scattering and absorption transport sensitivity functions for optical tomography,” Opt. Express 13, 492–506 (2000).
[CrossRef]

E. Haber, U. Ascher, and D. Oldenburg, “On optimization techniques for solving nonlinear inverse problems,” Inverse Probl. 16, 1263–1280 (2000).
[CrossRef]

O. Dorn, E. L. Miller, and C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adoint fields and level sets,” Inverse Probl. 16, 1119–1156 (2000).
[CrossRef]

1999 (2)

O. Dorn, H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, “A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields,” Inverse Probl. 15, 1523–1558 (1999).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

1998 (1)

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
[CrossRef]

1995 (1)

F. Natterer and F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
[CrossRef]

1976 (1)

J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

1917 (1)

M. Planck, “Ueber einen Satz der statistichen Dynamik und eine Erweiterung in der Quantumtheorie,” Sitzung-ber. Preuss. Akad. Wiss. 5, 324–341 (1917).

1914 (1)

A. D. Fokker, “Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld,” Ann. Phys. 43, 810–820 (1914).
[CrossRef]

Arridge, S.

J. Heino, S. Arridge, J. Sikora, and E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908 (2003).
[CrossRef]

Arridge, S. R.

A. P. Gibson, J. C. Hebden, and S. R. Arridge “Recent advances in diffuse optical imaging,” Phys. Med. Bio. 50, R1–R43 (2005).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

Ascher, U.

E. Haber, U. Ascher, and D. Oldenburg, “On optimization techniques for solving nonlinear inverse problems,” Inverse Probl. 16, 1263–1280 (2000).
[CrossRef]

Berryman, J. G.

O. Dorn, H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, “A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields,” Inverse Probl. 15, 1523–1558 (1999).
[CrossRef]

Bertete-Aguirre, H.

O. Dorn, H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, “A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields,” Inverse Probl. 15, 1523–1558 (1999).
[CrossRef]

Delves, L. M.

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge University Press, Cambridge, 1985).
[CrossRef]

Dierkes, T.

F. Natterer, H. Sielschott, O. Dorn, T. Dierkes, and V. Palamodov, “Fréchet derivatives for some bilinear inverse problems,” SIAM J. Appl. Math. 62, 2092–2113 (2002).
[CrossRef]

Dorn, O.

P. Gonzàlez-Rodríguez, M. Kindelan, M. Moscoso, and O. Dorn, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
[CrossRef]

F. Natterer, H. Sielschott, O. Dorn, T. Dierkes, and V. Palamodov, “Fréchet derivatives for some bilinear inverse problems,” SIAM J. Appl. Math. 62, 2092–2113 (2002).
[CrossRef]

O. Dorn, “Scattering and absorption transport sensitivity functions for optical tomography,” Opt. Express 13, 492–506 (2000).
[CrossRef]

O. Dorn, E. L. Miller, and C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adoint fields and level sets,” Inverse Probl. 16, 1119–1156 (2000).
[CrossRef]

O. Dorn, H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, “A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields,” Inverse Probl. 15, 1523–1558 (1999).
[CrossRef]

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
[CrossRef]

Fokker, A. D.

A. D. Fokker, “Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld,” Ann. Phys. 43, 810–820 (1914).
[CrossRef]

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge “Recent advances in diffuse optical imaging,” Phys. Med. Bio. 50, R1–R43 (2005).
[CrossRef]

Gonzàlez-Rodríguez, P.

P. Gonzàlez-Rodríguez and A. D. Kim,, “Light propagation in tissues with forward-peaked and large-angle scattering,” Appl. Opt. 47, 2599–2609 (2008).
[CrossRef] [PubMed]

P. Gonzàlez-Rodríguez, M. Kindelan, M. Moscoso, and O. Dorn, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
[CrossRef]

Haber, E.

E. Haber, U. Ascher, and D. Oldenburg, “On optimization techniques for solving nonlinear inverse problems,” Inverse Probl. 16, 1263–1280 (2000).
[CrossRef]

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge “Recent advances in diffuse optical imaging,” Phys. Med. Bio. 50, R1–R43 (2005).
[CrossRef]

Heino, J.

J. Heino, S. Arridge, J. Sikora, and E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908 (2003).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997).

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Keller, J. B.

Kim, A. D.

Kim,, A. D.

Kindelan, M.

P. Gonzàlez-Rodríguez, M. Kindelan, M. Moscoso, and O. Dorn, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
[CrossRef]

Lewis, E. E.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, Inc., La Grange Park, IL, 1993).

Miller, E. L.

O. Dorn, E. L. Miller, and C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adoint fields and level sets,” Inverse Probl. 16, 1119–1156 (2000).
[CrossRef]

Miller, W. F.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, Inc., La Grange Park, IL, 1993).

Mohamed, J. L.

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge University Press, Cambridge, 1985).
[CrossRef]

Moscoso, M.

P. Gonzàlez-Rodríguez, M. Kindelan, M. Moscoso, and O. Dorn, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
[CrossRef]

Natterer, F.

F. Natterer, H. Sielschott, O. Dorn, T. Dierkes, and V. Palamodov, “Fréchet derivatives for some bilinear inverse problems,” SIAM J. Appl. Math. 62, 2092–2113 (2002).
[CrossRef]

F. Natterer and F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
[CrossRef]

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001).
[CrossRef]

Oldenburg, D.

E. Haber, U. Ascher, and D. Oldenburg, “On optimization techniques for solving nonlinear inverse problems,” Inverse Probl. 16, 1263–1280 (2000).
[CrossRef]

Palamodov, V.

F. Natterer, H. Sielschott, O. Dorn, T. Dierkes, and V. Palamodov, “Fréchet derivatives for some bilinear inverse problems,” SIAM J. Appl. Math. 62, 2092–2113 (2002).
[CrossRef]

Papanicolaou, G. C.

O. Dorn, H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, “A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields,” Inverse Probl. 15, 1523–1558 (1999).
[CrossRef]

Planck, M.

M. Planck, “Ueber einen Satz der statistichen Dynamik und eine Erweiterung in der Quantumtheorie,” Sitzung-ber. Preuss. Akad. Wiss. 5, 324–341 (1917).

Rappaport, C.

O. Dorn, E. L. Miller, and C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adoint fields and level sets,” Inverse Probl. 16, 1119–1156 (2000).
[CrossRef]

Sielschott, H.

F. Natterer, H. Sielschott, O. Dorn, T. Dierkes, and V. Palamodov, “Fréchet derivatives for some bilinear inverse problems,” SIAM J. Appl. Math. 62, 2092–2113 (2002).
[CrossRef]

Sikora, J.

J. Heino, S. Arridge, J. Sikora, and E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908 (2003).
[CrossRef]

Somersalo, E.

J. Heino, S. Arridge, J. Sikora, and E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908 (2003).
[CrossRef]

Vogel, C. R.

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, Philadelphia, PA, 2002).
[CrossRef]

Vogeler, M.

M. Vogeler, “Reconstruction of the three-dimensional refractive index in electromagnetic scattering by using a propagation-backpropagation method,” Inverse Probl. 19, 739–753 (2003).
[CrossRef]

Weinman, J. A.

J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Wiscombe, W. J.

J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Wübbeling, F.

F. Natterer and F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
[CrossRef]

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001).
[CrossRef]

Ann. Phys. (1)

A. D. Fokker, “Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld,” Ann. Phys. 43, 810–820 (1914).
[CrossRef]

Appl. Opt. (1)

Inverse Probl. (8)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
[CrossRef]

F. Natterer and F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
[CrossRef]

M. Vogeler, “Reconstruction of the three-dimensional refractive index in electromagnetic scattering by using a propagation-backpropagation method,” Inverse Probl. 19, 739–753 (2003).
[CrossRef]

O. Dorn, H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, “A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields,” Inverse Probl. 15, 1523–1558 (1999).
[CrossRef]

O. Dorn, E. L. Miller, and C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adoint fields and level sets,” Inverse Probl. 16, 1119–1156 (2000).
[CrossRef]

P. Gonzàlez-Rodríguez, M. Kindelan, M. Moscoso, and O. Dorn, “History matching problem in reservoir engineering using the propagation-backpropagation method,” Inverse Probl. 21, 565–590 (2005).
[CrossRef]

E. Haber, U. Ascher, and D. Oldenburg, “On optimization techniques for solving nonlinear inverse problems,” Inverse Probl. 16, 1263–1280 (2000).
[CrossRef]

J. Atmos. Sci. (1)

J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Express (1)

O. Dorn, “Scattering and absorption transport sensitivity functions for optical tomography,” Opt. Express 13, 492–506 (2000).
[CrossRef]

Phys. Med. Bio. (1)

A. P. Gibson, J. C. Hebden, and S. R. Arridge “Recent advances in diffuse optical imaging,” Phys. Med. Bio. 50, R1–R43 (2005).
[CrossRef]

Phys. Rev. E (1)

J. Heino, S. Arridge, J. Sikora, and E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E 68, 031908 (2003).
[CrossRef]

SIAM J. Appl. Math. (1)

F. Natterer, H. Sielschott, O. Dorn, T. Dierkes, and V. Palamodov, “Fréchet derivatives for some bilinear inverse problems,” SIAM J. Appl. Math. 62, 2092–2113 (2002).
[CrossRef]

Sitzung-ber. Preuss. Akad. Wiss. (1)

M. Planck, “Ueber einen Satz der statistichen Dynamik und eine Erweiterung in der Quantumtheorie,” Sitzung-ber. Preuss. Akad. Wiss. 5, 324–341 (1917).

Other (5)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997).

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001).
[CrossRef]

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, Philadelphia, PA, 2002).
[CrossRef]

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, Inc., La Grange Park, IL, 1993).

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge University Press, Cambridge, 1985).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

The actual absorption coefficient (a) and reconstructions of the absorption coefficient produced by the five light scattering models (b)–(f).

Fig. 2.
Fig. 2.

The actual scattering coefficient (a) and reconstructions of the scattering coefficient produced by the five light scattering models (b)–(f).

Fig. 3.
Fig. 3.

Comparison of the values of the cost functional as a function of successful iterations for the radiative transport equation (RTE), the generalized Fokker-Planck-Eddington approximation (gFPE), the Fokker-Planck-Eddington approximation (FPE), the Fokker-Planck approximation (FP) and delta-Eddington approximation (DE).

Tables (1)

Tables Icon

Table 1. Time required to compute the numerical solution of a single direct problem in seconds.

Equations (88)

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

c1tI+ŝ·I+μaIμsI=QonSn1×Ω×[0,T].
I=I+Sn1f(ŝ·ŝ)I(ŝ,r,t)dŝ.
I=0onΓin={(ŝ,rb,t)Sn1×Ω×[0,T],ν(rb)·ŝ<0},
It=0 =0onSn1×Ω.
Mμa,μs (rb,t) = Soutn1ŝ.ν(rd)I(ŝ,rd,t)dŝonΩ×[0,T].
R:P×PD,R(μa,μs)=Mμa,μsF.
K(μa,μs)=12R(μa,μs)L22.
R (μa,μs)R(μa̅,μ̅s)+R'(μa̅,μ̅s)(δμa,δμs),
(δμa*,δμa*)=R̃(μa̅,μ̅s)[R(μa̅,μ̅s)R̃(μa̅,μ̅s)]1R(μa̅,μ̅s).
(δμa*,δμa*)βR̃(μa̅,μ̅s)R(μa̅,μ̅s).
c1tĨ+ŝ·Ĩ+μaĨμsĨ=0inSn1×Ω×[0,T],
Ĩ=ronΓout={(ŝ,rb,t)Sn1×Ω×[0,T],ν(rb)·ŝ>0},
Ĩt=T =0onSn1×Ω.
δμa*=β0TSn1Ĩ(ŝ,r,t)I(ŝ,r,t)dŝdt,
δμs*=β0TSn1Ĩ(ŝ,r,t)I(ŝ,r,t)dŝdt.
fHG(ŝ·ŝ)=14π1g2(1+g22gŝ·ŝ)3/2=n=02n+14πgnPn(ŝ·ŝ),
δEI=(1g2)I+1g24πs2[P0(ŝ·ŝ)+3g1+gP1(ŝ·ŝ)]I(ŝ,r)dŝ.
FPI=12(1g)ΔŝI,
FPE I =(1a0)I+a1ΔŝI+14πS2[b0P0(ŝ·ŝ)+3b1P1(ŝ·ŝ)]I(ŝr,t)dŝ.
a0=g2(2g),
a1=g26(1g) ,
b0=12g2+g3,
b1=g3(35g+2g2).
gFPE I=(1a0)I+a1Δŝ(𝓘a2Δŝ)1I
+14π S2[b0P0(ŝ·ŝ)+3b1P1(ŝ·ŝ)]I(s'̂,r,t)dŝ',
a0=g3 2g55g2
a1=7g36 g2125g220g+4
a2=3g460g24
b0=2g45g3+5g25g2
b1=g8g327g2+27g827g8 .
fHG=fHG(θθ)=12π1g21+g22gcos(θθ)=12π+1πn=1gncos[n(θθ)].
HGI=I+02πfHG(θθ)I(θ',x,y,t)dθ.
δEI=(1g2)I+1g22π02π[1+2g1+gcos(θθ)]I(θ',x,y,t)dθ.
FPI=(1g)θ2I.
FPEI=(1a0)I+a1θ2I+12π02π[b0+b1cos(θθ)]I(θ',x,y,t)dθ,
a0=g25 (94g) ,
a1=g25(1g),
b0=15 (59g2+4g3) ,
b1=g5(58g+3g2).
gFPEI=(1a0)I+a1θ2(Ia2θ2)1I
+12π 02π[b0+b1cos(θθ)] I (θ,x,y,t) d θ ,
a0=g3207g20g7,
a1=105g31g21600g21120g+196,
a2=75g80g28,
b0=7g420g3+20g720g7,
b1=g98g4389g3+501g2−259g+49340g2259g+49.
c1tI+cosθxI+sinθyI+μaIμsI=Q,
c1tIl+cosθxIl+sinθyIl+μaIl=Jl1
IlIl1<ε ,
J0=Ql=0,
Jl=μsIl+Q,l=1,2,.
02πf(θnθ)I(θ,x,y,t)dθk=1NfnkIk(x,y,t),n=1,,N.
fnk=θkΔθ/2θk+Δθ/2f(θnθ)dθ.
θ2 InIn12In+In+1(Δθ)2,n=1,,N,
Jn=μs𝓛NIn(x,y,t)+Q(θn,x,y,t).
c1tIn+cosθnxIn+sinθnyIn+μaIn=Jn,n=1,,N,
Inij(t)=1ΔxΔyxi1/2xi+1/2yi1/2yi+1/2In(x,y,t)dydx.
c1tInij+1Δx cos θn (In,i+1/2,jIn,i1/2,j) +1Δysinθn(In,i,j+1/2In,i,j1/2)
+μa,ijInij=Jnij.
In,i±1/2,j=1Δyyj1/2yj+1/2In(xi±1/2,y,t)dy,
In,i,j±1/2=1Δxxi1/2xi+1/2In(x,yj±1/2,t)dx.
1cΔt(Inijk+1/2Inijk1/2)+1Δxcosθn(In,i+1/2,jkIn,i1/2,jk)
+1Δysinθn(In,i,j+1/2kIn,i,j1/2k)+μa,ijInijk=Jnijk.
Inijk=1Δttk1/2tk+1/2Inij(t)dt.
Inijk=12(In,i+1/2,jk+In,i1/2,jk),
Inijk=12(In,i,j+1/2k+In,i,j1/2k),
Inijk=12(Inijk+1/2+Inijk1/2).
Inijk+1/2=InijkcΔtΔxcosθn(InijkIn,i1/2,jk)cΔtΔysinθn(InijkIn,i,j1/2k)
cΔt2μa,ijInijk+cΔt2Jnijk.
(1+cΔt2μa,ij)Inijk+1/2=InijkcΔtΔxcosθn(InijkIn,i1/2,jk)
cΔtΔysinθn(InijkIn,i,j1/2k)+cΔt2Jnijk.
K((μ̅a,μ̅s)+ω(δμa*,δμs*))=12R((μ̅a,μ̅s)+ω(δμa*,δμs*)L22.
K((μ̅a,μ̅s)+ω(δμa*,δμs*))=12R(μ̅a,μ̅s)+R(μ̅a,μ̅s)[ω(δμa*,δμs*)]
+O((δμa*,δμs*)L22)L22.
K((μ̅a,μ̅s)+ω(δμa*,δμs*))=12R(μ̅a,μ̅s),R(μ̅a,μ̅s)
+R(μ̅a,μ̅s),R(μ̅a,μ̅s)[ω(δμa*,δμs*)]+O((δμa*,δμs*)L22)
K((μ̅a,μ̅s)+ω(δμa*,δμs*))=12R(μ̅a,μ̅s),R(μ̅a,μ̅s)
+R(μ̅a,μ̅s),ωR(μ̅a,μ̅s)R̃(μ̅a,μ̅s)[R(μ̅a,μ̅s)R̃(μ̅a,μ̅s)]1R(μ̅a,μ̅s)
+O((δμa*,δμs*)L22).
K((μ̅a,μ̅s)+ω(δμa*,δμs*))=K(μ̅a,μ̅s)ωR(μ̅a,μ̅s)L22+O(δμa*,δμs*L22).
K((μ̅a,μ̅s)+ω(δμa*,δμs*))=K(μ̅a,μ̅s)ωβR'̃(μ̅a,μ̅s)R(μ̅a,μ̅s)L22
+ O(δμa*,δμs*L22),
Sn1Ĩ(ŝ)[I(ŝ)]dŝ=Sn1[Ĩ(ŝ)]I(ŝ)dŝ,
Ĩ,gFPI=Sn1Ĩ(ŝ)[a1Δŝ(a2Δŝ)1I(ŝ)]dŝ.
Ĩ,gFPI=Sn1[(a2Δŝ)J̃(ŝ)][a1ΔŝJ(ŝ)]dŝ
=Sn1[a1ΔŝJ̃(ŝ)][(a2Δŝ)J(ŝ)]dŝ
=Sn1[a1Δŝ(a2Δŝ)1Ĩ(ŝ)]I(ŝ)dŝ
=gFPI,Ĩ.

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