Abstract

In frequency-domain optical diffusion imaging, the magnitude and the phase of modulated light propagated through a highly scattering medium are used to reconstruct an image of the scattering and absorption coefficients in the medium. Although current reconstruction algorithms have been applied with some success, there are opportunities for improving both the accuracy of the reconstructions and the speed of convergence. In particular, conventional integral equation approaches such as the Born iterative method and the distorted Born iterative method can suffer from slow convergence, especially for large spatial variations in the constitutive parameters. We show that slow convergence of conventional algorithms is due to the linearized integral equations’ not being the correct Fréchet derivative with respect to the absorption and scattering coefficients. The correct Fréchet derivative operator is derived here. However, the Fréchet derivative suffers from numerical instability because it involves gradients of both the Green’s function and the optical flux near singularities, a result of the use of near-field imaging data. To ameliorate these effects we derive an approximation to the Fréchet derivative and implement it in an inversion algorithm. Simulation results show that this inversion algorithm outperforms conventional iterative methods.

© 1999 Optical Society of America

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References

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1998 (2)

1997 (3)

1996 (3)

1995 (2)

1994 (2)

Q. H. Liu, “Nonlinear inversion of electrode-type resistivity measurements,” IEEE Trans. Geosci. Remote Sens. 32, 499–507 (1994).
[CrossRef]

O. Arikan, “Regularized inversion of a two-dimensional integral equation with applications in borehole induction measurement,” Radio Sci. 29, 519–538 (1994).
[CrossRef]

1992 (1)

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

1991 (1)

N. Joachimowicz, C. Pichot, J. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

1990 (1)

T. J. Connolly, D. J. Wall, “On Fréchet differentiability of some nonlinear operators occurring in inverse problems: an implicit function theorem approach,” Inverse Probl. 6, 949–966 (1990).
[CrossRef]

1989 (2)

J. C. Adams, “mudpack: multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113–146 (1989).
[CrossRef]

M. S. Patterson, B. Chance, B. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

1988 (1)

T. J. Connolly, D. J. Wall, “On an inverse problem, with boundary measurements, for the steady state diffusion equation,” Inverse Probl. 4, 995–1012 (1988).
[CrossRef]

Adams, J. C.

J. C. Adams, “mudpack: multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113–146 (1989).
[CrossRef]

Adams, R. A.

R. A. Adams, Sobolev Spaces, Vol. 65 of Pure and Applied Mathematics (Academic, New York, 1975).

Arikan, O.

O. Arikan, “Regularized inversion of a two-dimensional integral equation with applications in borehole induction measurement,” Radio Sci. 29, 519–538 (1994).
[CrossRef]

Arridge, S. R.

S. R. Arridge, M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, “Photon-measurement density functions. 2. Finite-element calculations,” Appl. Opt. 34, 8026–8037 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scattering imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Arsenin, V.

A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Winston, New York, 1977).

Barbour, R. L.

Boas, D.

Bouman, C. A.

C. A. Bouman, K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process. 5, 480–492 (1996).
[CrossRef] [PubMed]

Chance, B.

Chew, W.

W. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

Connolly, T. J.

T. J. Connolly, D. J. Wall, “On Fréchet differentiability of some nonlinear operators occurring in inverse problems: an implicit function theorem approach,” Inverse Probl. 6, 949–966 (1990).
[CrossRef]

T. J. Connolly, D. J. Wall, “On an inverse problem, with boundary measurements, for the steady state diffusion equation,” Inverse Probl. 4, 995–1012 (1988).
[CrossRef]

Cunningham, G. S.

S. S. Saquib, K. M. Hanson, G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging 1997: Image Processing, K. M. Hanson, ed., Proc. SPIE3034, 369–380 (1997).
[CrossRef]

Delpy, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scattering imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Dennis, J. E.

J. E. Dennis, Numerical Methods for Unconstrained Optimization and Nonlinear Equation (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Downar, T. J.

J. C. Ye, R. P. Millane, K. J. Webb, T. J. Downar, “Importance of the ∇D term in frequency-resolved optical diffusion imaging,” Opt. Lett. 23, 1423–1425 (1998).
[CrossRef]

J. C. Ye, K. J. Webb, T. J. Downar, R. P. Millane, “Weighted cost function reconstruction in optical diffusion imaging,” in Computational, Experimental and Numerical Methods for Solving Ill-Posed Inverse Imaging Problems: Medical and Nonmedical Application, R. L. Barber, M. J. Carvin, M. A. Fiddy, eds., Proc. SPIE3171, 118–127 (1997).
[CrossRef]

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Farrell, T. J.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Fletcher, R.

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987).

Friedman, A.

A. Friedman, Partial Differential Equations (Rinehart and Winston, New York, 1969).

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Hanson, K. M.

S. S. Saquib, K. M. Hanson, G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging 1997: Image Processing, K. M. Hanson, ed., Proc. SPIE3034, 369–380 (1997).
[CrossRef]

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scattering imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Hugonin, J.

N. Joachimowicz, C. Pichot, J. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

Jiang, H.

Joachimowicz, N.

N. Joachimowicz, C. Pichot, J. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

Liu, Q. H.

Q. H. Liu, “Nonlinear inversion of electrode-type resistivity measurements,” IEEE Trans. Geosci. Remote Sens. 32, 499–507 (1994).
[CrossRef]

Millane, R. P.

J. C. Ye, R. P. Millane, K. J. Webb, T. J. Downar, “Importance of the ∇D term in frequency-resolved optical diffusion imaging,” Opt. Lett. 23, 1423–1425 (1998).
[CrossRef]

J. C. Ye, K. J. Webb, T. J. Downar, R. P. Millane, “Weighted cost function reconstruction in optical diffusion imaging,” in Computational, Experimental and Numerical Methods for Solving Ill-Posed Inverse Imaging Problems: Medical and Nonmedical Application, R. L. Barber, M. J. Carvin, M. A. Fiddy, eds., Proc. SPIE3171, 118–127 (1997).
[CrossRef]

Morozov, V. A.

V. A. Morozov, Methods for Solving Incorrectly Posed Problems (Springer-Verlag, New York, 1984).

Naylor, A. W.

A. W. Naylor, G. R. Sell, Linear Operator Theory in Engineering and Science, 2nd ed., Vol. 40 of Applied Mathematical Science (Springer-Verlag, New York, 1982).
[CrossRef]

O’Leary, M.

Osterberg, U. L.

Patterson, M. S.

Paulsen, K. D.

Pei, Y.

Pichot, C.

N. Joachimowicz, C. Pichot, J. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

Pogue, B. W.

Poor, H. V.

H. V. Poor, An Introduction of Signal Detection and Estimation, 2nd ed. (Springer-Verlag, New York, 1994).

Przadka, A.

Reynolds, J. S.

Saquib, S. S.

S. S. Saquib, K. M. Hanson, G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging 1997: Image Processing, K. M. Hanson, ed., Proc. SPIE3034, 369–380 (1997).
[CrossRef]

Sauer, K.

C. A. Bouman, K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process. 5, 480–492 (1996).
[CrossRef] [PubMed]

Schweiger, M.

S. R. Arridge, M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, “Photon-measurement density functions. 2. Finite-element calculations,” Appl. Opt. 34, 8026–8037 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scattering imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Sell, G. R.

A. W. Naylor, G. R. Sell, Linear Operator Theory in Engineering and Science, 2nd ed., Vol. 40 of Applied Mathematical Science (Springer-Verlag, New York, 1982).
[CrossRef]

Tautenhahn, U.

U. Tautenhahn, “On a general regularization scheme for nonlinear ill-posed problems,” Inverse Probl. 13, 1427–1437 (1997).
[CrossRef]

Thompson, C. A.

Tikhonov, A.

A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Winston, New York, 1977).

Wall, D. J.

T. J. Connolly, D. J. Wall, “On Fréchet differentiability of some nonlinear operators occurring in inverse problems: an implicit function theorem approach,” Inverse Probl. 6, 949–966 (1990).
[CrossRef]

T. J. Connolly, D. J. Wall, “On an inverse problem, with boundary measurements, for the steady state diffusion equation,” Inverse Probl. 4, 995–1012 (1988).
[CrossRef]

Wang, Y.

Webb, K. J.

Weiner, A. M.

Wilson, B.

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, B. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

Yao, Y.

Ye, J. C.

J. C. Ye, R. P. Millane, K. J. Webb, T. J. Downar, “Importance of the ∇D term in frequency-resolved optical diffusion imaging,” Opt. Lett. 23, 1423–1425 (1998).
[CrossRef]

J. C. Ye, K. J. Webb, T. J. Downar, R. P. Millane, “Weighted cost function reconstruction in optical diffusion imaging,” in Computational, Experimental and Numerical Methods for Solving Ill-Posed Inverse Imaging Problems: Medical and Nonmedical Application, R. L. Barber, M. J. Carvin, M. A. Fiddy, eds., Proc. SPIE3171, 118–127 (1997).
[CrossRef]

Yeung, S.

Yodh, A.

Zeidler, E.

E. Zeidler, Applied Functional Analysis, Vol. 108 of Applied Mathematical Sciences (Springer-Verlag, New York, 1995).

Zhu, W.

Appl. Math. Comput. (1)

J. C. Adams, “mudpack: multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113–146 (1989).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (1)

N. Joachimowicz, C. Pichot, J. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

Q. H. Liu, “Nonlinear inversion of electrode-type resistivity measurements,” IEEE Trans. Geosci. Remote Sens. 32, 499–507 (1994).
[CrossRef]

IEEE Trans. Image Process. (1)

C. A. Bouman, K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process. 5, 480–492 (1996).
[CrossRef] [PubMed]

Inverse Probl. (3)

T. J. Connolly, D. J. Wall, “On Fréchet differentiability of some nonlinear operators occurring in inverse problems: an implicit function theorem approach,” Inverse Probl. 6, 949–966 (1990).
[CrossRef]

T. J. Connolly, D. J. Wall, “On an inverse problem, with boundary measurements, for the steady state diffusion equation,” Inverse Probl. 4, 995–1012 (1988).
[CrossRef]

U. Tautenhahn, “On a general regularization scheme for nonlinear ill-posed problems,” Inverse Probl. 13, 1427–1437 (1997).
[CrossRef]

J. Opt. Soc. Am. A (2)

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (2)

Radio Sci. (1)

O. Arikan, “Regularized inversion of a two-dimensional integral equation with applications in borehole induction measurement,” Radio Sci. 29, 519–538 (1994).
[CrossRef]

Other (14)

J. C. Ye, K. J. Webb, T. J. Downar, R. P. Millane, “Weighted cost function reconstruction in optical diffusion imaging,” in Computational, Experimental and Numerical Methods for Solving Ill-Posed Inverse Imaging Problems: Medical and Nonmedical Application, R. L. Barber, M. J. Carvin, M. A. Fiddy, eds., Proc. SPIE3171, 118–127 (1997).
[CrossRef]

E. Zeidler, Applied Functional Analysis, Vol. 108 of Applied Mathematical Sciences (Springer-Verlag, New York, 1995).

H. V. Poor, An Introduction of Signal Detection and Estimation, 2nd ed. (Springer-Verlag, New York, 1994).

S. S. Saquib, K. M. Hanson, G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging 1997: Image Processing, K. M. Hanson, ed., Proc. SPIE3034, 369–380 (1997).
[CrossRef]

A. W. Naylor, G. R. Sell, Linear Operator Theory in Engineering and Science, 2nd ed., Vol. 40 of Applied Mathematical Science (Springer-Verlag, New York, 1982).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scattering imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Winston, New York, 1977).

J. E. Dennis, Numerical Methods for Unconstrained Optimization and Nonlinear Equation (Prentice-Hall, Englewood Cliffs, N.J., 1983).

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987).

W. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

V. A. Morozov, Methods for Solving Incorrectly Posed Problems (Springer-Verlag, New York, 1984).

R. A. Adams, Sobolev Spaces, Vol. 65 of Pure and Applied Mathematics (Academic, New York, 1975).

A. Friedman, Partial Differential Equations (Rinehart and Winston, New York, 1969).

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

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Figures (9)

Fig. 1
Fig. 1

Zero-input photon current or absorbing boundary condition for the diffusion equation, where all incident light from within the scattering boundary is lost to free space. Setting ϕ=0 on an extrapolated boundary at 0.7104(3D), where 3D is the mean free path, is equivalent to the zero-input current condition on the physical boundary Ωd. z is distance perpendicular to the interface, and az is the unit vector.

Fig. 2
Fig. 2

Simulation geometry showing the locations of uniformly spaced sources and detectors.

Fig. 3
Fig. 3

True images (a) μa and (b) μs. (c) μa and (d) μs as reconstructed by the conventional DBIM. (e) μa and (f) μs as reconstructed by the new algorithm.

Fig. 4
Fig. 4

NRMSE versus iteration for the new algorithm and for the conventional DBIM for (a) μa and (b) μs.

Fig. 5
Fig. 5

True images (a) μa and (b) μs. (c) μa and (d) μs as reconstructed by the conventional DBIM. (e) μa and (f) μs as reconstructed by the new algorithm.

Fig. 6
Fig. 6

NRMSE versus iteration for the new algorithm and for the conventional DBIM for (a) μa and (b) μs.

Fig. 7
Fig. 7

Effect of the starting image on the reconstructions. (a), (b) Reconstructed μa and μs, respectively, from starting images μab=0.02 cm-1 and μsb=10.0 cm-1. (c), (d) Reconstructed μa and μs, respectively, from starting images μab=0.022 cm-1 and μsb=10.0 cm-1. (e), (f) Reconstructed μa and μs, respectively, from starting images μab=0.02 cm-1 and μsb=11.0 cm-1.

Fig. 8
Fig. 8

True images (a) μa and (b) μs. (c) μa and (d) μs as reconstructed by the conventional DBIM. (e) μa and (f) μs as reconstructed by the new algorithm.

Fig. 9
Fig. 9

True images (a) μa and (b) μs. (c) μa and (d) μs as reconstructed by the conventional DBIM. (e) μa and (f) μs as reconstructed by the new algorithm.

Equations (72)

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

y=F(x),y-yδ<δ,
Jα(x)=y-F(xk)-F(xk)(x-xk)2+αx-xk2
xk+1=xk+[FH(xk)F(xk)+αI]-1×FH(xk)[yδ-F(xk)],
(Dϕ)+(-μa+jω/c)ϕ=-s,
2ϕ+k2ϕ=-sˆ,
k2=3(μa+μs)(-μa+jω/c),sˆ=3(μa+μs)s.
ϕ(r; k2)=ϕ(r; kb2)+Ωdrg(r, r; kb2)ϕ(r; k2)Δk2(r),
F(k2)=ϕ(r; kb2)+Ωdrg(r, r; kb2)ϕ(r; k2)Δk2(r),
y=ϕ(r; k2).
F(kb2)Δk2=Ωdrg(r, r; kb2)ϕ(r; kb2)Δk2(r).
Δμa=-μabω/c Im(Δk2)+Re(Δk2)Im(Δk2)ω/c+3(μab+μsb),
Δμs=Im(Δk2)3ω/c-Δμa,
Δμa(r)=μa(r)-μab(r),Δμs(r)=μs(r)-μsb(r).
(Dbϕ)+(-μab+jω/c)ϕ
=-s+Δμaϕ-[(D-Db)ϕ]=-s+Δμaϕ+Δμa+Δμsμab+μsb Dϕ,
ϕ(r; μa, μs)=ϕ(r; μab, μsb)
-Ωdrg(r, r; μab, μsb)ϕ(r; μa, μs)Δμa(r)
-Ωdrg(r, r; μab, μsb)
×Δμa(r)+Δμs(r)μab(r)+μsb(r) D(r)ϕ(r; μa, μs),
(sA)=s(A)+s(A)
ϕ(r; μa, μs)=ϕ(r; μab, μsb)-Ωdrg(r, r; μab, μsb)ϕ(r; μa, μs)Δμa
-Ωdrg(r, r; μab, μsb)[D(r)ϕ(r; μa, μs)]×Δμa(r)+Δμs(r)μab(r)+μsb(r).
ϕ(r; μa, μs)ϕ(r; μab, μsb)
-Ωdrg(r, r; μab, μsb)ϕ(r; μab, μsb)Δμa(r)-Ωdrg(r, r; μab, μsb)[Db(r)ϕ(r; μab, μsb)] Δμa(r)μab(r)+μsb(r)-Ωdrg(r, r; μab, μsb)[Db(r)ϕ(r; μab, μsb)] Δμs(r)μab(r)+μsb(r).
F(μab, μsb)ΔμaΔμs=Fμa(μab, μsb)Δμa+Fμs(μab, μsb)Δμs,
Fμa(μab, μsb)Δμa
=-Ωdrg(r, r; μab, μsb)ϕ(r; μab, μsb)Δμa(r)
-Ωdr{3[Db(r)]2g(r, r; μab, μsb)
ϕ(r; μab, μsb)}Δμa(r),
Fμs(μab, μsb)Δμs
=-Ωdr{3[Db(r)]2g(r, r; μab, μsb)ϕ(r; μab, μsb)}Δμs(r).
ϕ(r)=14π|r-rs| exp( jk|r-rs|),
ϕ(r)=r-rs|r-rs| exp( jk|r-rs|)4π|r-rs| jk-1|r-rs|.
ϕ(r; μa, μs)ϕ(r; μab, μsb)
-Ωg(r, r; μab, μsb)ϕ(r; μab, μsb)Δμa(r)dr-Ωg(r, r; μab, μsb)Δμa(r)μab(r)+μsb(r)Db(r)ϕ(r; μab, μsb)dr-Ωg(r, r; μab, μsb)×[Db(r)ϕ(r; μab, μsb)] Δμa(r)μab(r)+μsb(r) dr-Ωg(r, r; μab, μsb)Δμs(r)μab(r)+μsb(r)Db(r)ϕ(r; μab, μsb)dr-Ωg(r, r; μab, μsb)×[Db(r)ϕ(r; μab, μsb)] Δμs(r)μab(r)+μsb(r) dr.
FμaΔμa=F˜μaΔμa-Ωg(r, r; μab, μsb)Δμa(r)μab(r)+μsb(r)Db(r)ϕ(r; μab, μsb)dr,
FμsΔμs=F˜μsΔμs-Ωg(r, r; μab, μsb)Δμs(r)μab(r)+μsb(r)Db(r)ϕ(r; μab, μsb)dr,
F˜μaΔμa=Ωg(r, r; μab, μsb)ϕ(r; μab, μsb)×-1+-μab(r)+jω/cμab(r)+μsb(r)Δμa(r)dr,
F˜μsΔμs=Ωg(r, r; μab, μsb)ϕ(r; μab, μsb)×-μab(r)+jω/cμab(r)+μsb(r) Δμs(r)dr.
ϕ(r; μa, μs)ϕ(r; μab, μsb)+F˜μaΔμa+F˜μsΔμs=ϕ(r; μab, μsb)+Ωdrg(r, r; μab, μsb)×ϕ(r; μab, μsb)O(r),
O(r)=-1+-μab(r)+jω/cμab(r)+μsb(r)Δμa(r)+-μab(r)+jω/cμab(r)+μsb(r) Δμs(r).
Δμa=-μabω/c Im(O)-Re(O),
Δμs=μab+μsbω/c Im(O)-Δμa.
ΔμaηΔμaorg,ΔμsηΔμsorg,
η=3(μab+μsb)3μsb.
[FkH(xk)Fk(xk)+αI]-1FkH(xk)
=FkH(xk)[Fk(xk)FkH(xk)+αI]-1,
αk+1=trace[F˜(xk)HF˜(xk)]N y˜δ-F˜(xk)2,
SNR=10 log PSPN [dB],
NRMSE(k)=μk-μ2μ2,
F(x+h)-F(x)-FhY=o(hX)x+h,hX
limh0 F(x+h)-F(x)-FhYhX=0.
Dϕ(μa, μs)-Dbϕ(μab, μsb)2
M(Δμa+Δμs),
uY=Ω|u|2+|u|2dr1/2<,
FμaΔμa=-Ωdrg(r, r; μab, μsb)ϕ(r; μab, μsb)Δμa(r)-Ωdrg(r, r; μab, μsb)ϕ(r; μab, μsb)3[Db(r)]2Δμa(r),
|FμaΔμa|Ω|g(r, r; μab, μsb)ϕ(r; μab, μsb)|drΔμa+Ωdr|g(r, r; μab, μsb)ϕ(r; μab, μsb)|dr3(Db)2Δμa,
Ω|g(r, r; μab, μsb)ϕ(r; μab, μsb)|dr
g(r; μab,μsb)2ϕ(μab, μsb)2,
Ωdr|g(r, r; μab, μsb)ϕ(r; μab, μsb)|dr
g(r; μab, μsb)2ϕ(μab, μsb)2.
w(r; Δμa, Δμs)
=ϕ(r; μa, μs)-ϕ(r; μab, μsb)-FμaΔμa-FμsΔμs=w1+w2,
w1=-Ωg(r, r; μab, μsb)[ϕ(r; μa, μs)-ϕ(r; μab, μsb)]Δμa(r)dr,
w2=-Ωg(r, r; μab, μsb)[D(r)ϕ(r; μa, μs)-Db(r)ϕ(r; μab, μsb)]×Δμa(r)+Δμs(r)μab(r)+μsb(r)dr.
|w(r; Δμa, Δμs)||w1|+|w2|
|w1|g(r; μab, μsb)2ϕ(μa, μs)-ϕ(μab, μsb)2Δμa,
|w2|g(r; μab, μsb)2Dϕ(μa, μs)-Dbϕ(μab, μsb)2Δμa+Δμsμa+μs.
|w1|g(r; μab, μsb)2L(Δμa+Δμs)2,
|w2|g(r; μab, μsb)2M1μa+μs(Δμa+Δμs)2,
L=FμaY+FμsY.
w(r; Δμa, Δμs)=o(Δμa+Δμs).

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