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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  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. Gonzalez-Rodriguez 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. Wubbeling, "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. Gonzalez-Rodri;guez, 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. W¨ubbeling, 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, "Fr’echet 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," Sitzungber. 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, W. F. MillerJr., 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

2005

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. Gonzalez-Rodri;guez, 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

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

F. Natterer, H. Sielschott, O. Dorn, T. Dierkes and V. Palamodov, "Fr’echet derivatives for some bilinear inverse problems," SIAM J. Appl. Math. 62, 2092-2113 (2002).
[CrossRef]

2000

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

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

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

1995

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

1976

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

M. Planck, "Ueber einen Satz der statistichen Dynamik und eine Erweiterung in der Quantumtheorie," Sitzungber. Preuss. Akad. Wiss. 5, 324-341 (1917).

1914

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]

Dierkes, T.

F. Natterer, H. Sielschott, O. Dorn, T. Dierkes and V. Palamodov, "Fr’echet derivatives for some bilinear inverse problems," SIAM J. Appl. Math. 62, 2092-2113 (2002).
[CrossRef]

Dorn, O.

P. Gonzalez-Rodri;guez, 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, "Fr’echet 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]

Gonz’alez-Rodr’iguez, P.

P. Gonzalez-Rodriguez and A. D. Kim, "Light propagation in tissues with forward-peaked and large-angle scattering," Appl. Opt. 47, 2599-2609 (2008).
[CrossRef] [PubMed]

P. Gonzalez-Rodri;guez, 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]

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.

Kindelan, M.

P. Gonzalez-Rodri;guez, 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]

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]

Moscoso, M.

P. Gonzalez-Rodri;guez, 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, "Fr’echet derivatives for some bilinear inverse problems," SIAM J. Appl. Math. 62, 2092-2113 (2002).
[CrossRef]

F. Natterer and F. Wubbeling, "A propagation-backpropagation method for ultrasound tomography," Inverse Probl. 11, 1225-1232 (1995).
[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, "Fr’echet 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," Sitzungber. 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, "Fr’echet 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]

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]

Wubbeling, F.

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

Ann. Phys.

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

Appl. Opt.

Inverse Probl.

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. Wubbeling, "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. Gonzalez-Rodri;guez, 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.

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

Opt. Express

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

Phys. Med. Bio.

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

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.

F. Natterer, H. Sielschott, O. Dorn, T. Dierkes and V. Palamodov, "Fr’echet derivatives for some bilinear inverse problems," SIAM J. Appl. Math. 62, 2092-2113 (2002).
[CrossRef]

Sitzungber. Preuss. Akad. Wiss.

M. Planck, "Ueber einen Satz der statistichen Dynamik und eine Erweiterung in der Quantumtheorie," Sitzungber. Preuss. Akad. Wiss. 5, 324-341 (1917).

Other

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

F. Natterer and F. W¨ubbeling, 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, W. F. MillerJr., 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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