Abstract

Frequency-domain diffusion imaging uses the magnitude and phase of modulated light propagating through a highly scattering medium to reconstruct an image of the spatially dependent scattering or absorption coefficients in the medium. An inversion algorithm is formulated in a Bayesian framework and an efficient optimization technique is presented for calculating the maximum a posteriori image. In this framework the data are modeled as a complex Gaussian random vector with shot-noise statistics, and the unknown image is modeled as a generalized Gaussian Markov random field. The shot-noise statistics provide correct weighting for the measurement, and the generalized Gaussian Markov random field prior enhances the reconstruction quality and retains edges in the reconstruction. A localized relaxation algorithm, the iterative-coordinate-descent algorithm, is employed as a computationally efficient optimization technique. Numerical results for two-dimensional images show that the Bayesian framework with the new optimization scheme outperforms conventional approaches in both speed and reconstruction quality.

© 1999 Optical Society of America

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1999 (2)

1998 (3)

1997 (2)

1996 (4)

1995 (1)

1994 (1)

1993 (3)

K. Sauer, C. A. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process. 41, 534–548 (1993).
[CrossRef]

C. A. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving map estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[CrossRef] [PubMed]

1992 (2)

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern. Anal. Mach. Intell. 14, 367–383 (1992).
[CrossRef]

M. O’Leary, D. Boas, B. Chance, A. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

1991 (1)

1989 (6)

J. Besag, “Towards Bayesian image analysis,” J. Appl. Stat. 16, 395–407 (1989).
[CrossRef]

S. Flock, M. Patterson, B. Wilson, D. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. Med. Imaging 8, 194–202 (1989).
[CrossRef] [PubMed]

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]

S. Jacques, “Time resolved propagation of ultrashort laser pulses within turbid tissues,” Appl. Opt. 28, 2223–2229 (1989).
[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]

1974 (1)

J. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. R. Stat. Soc. B 36, 192–236 (1974).

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]

Aronson, R.

Arridge, S. R.

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

S. R. Arridge, “Photon-measurement density functions. Part 1: Analytical forms,” Appl. Opt. 34, 7395–7409 (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, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems, F. John, transl. ed. (Winston, New York, 1977).

Barbour, R. L.

Berndtand, K. W.

Besag, J.

J. Besag, “Towards Bayesian image analysis,” J. Appl. Stat. 16, 395–407 (1989).
[CrossRef]

J. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. R. Stat. Soc. B 36, 192–236 (1974).

Boas, D.

M. O’Leary, D. Boas, B. Chance, A. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Bouman, C. A.

S. S. Saquib, C. A. Bouman, K. Sauer, “ML parameter estimation for Markov random fields with applications to Bayesian tomography,” IEEE Trans. Image Process. 7, 1029–1044 (1998).
[CrossRef]

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]

C. A. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving map estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

K. Sauer, C. A. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process. 41, 534–548 (1993).
[CrossRef]

Burch, C. L.

Carfantan, H.

H. Carfantan, A. Mohammad-Djafari, J. Idier, “A single site update algorithm for nonlinear diffraction tomography,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing IV (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 2837–2840.

Chance, B.

Chew, W. C.

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

Corngold, N.

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, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Dennis, J. E.

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

Downar, T. J.

Duderstadt, J. J.

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

Erickson, M. G.

Fishkin, J. B.

Fletcher, R.

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, Chichester, UK, 1987).

Flock, S.

S. Flock, M. Patterson, B. Wilson, D. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Frisoli, J. K.

Geman, D.

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern. Anal. Mach. Intell. 14, 367–383 (1992).
[CrossRef]

Golub, G. H.

G. H. Golub, C. F. V. Loan, Matrix Computations, 2nd ed. (The Johns Hopkins U. Press, Baltimore, 1989).

Gratton, E.

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]

Hebert, T.

T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. Med. Imaging 8, 194–202 (1989).
[CrossRef] [PubMed]

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, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Idier, J.

H. Carfantan, A. Mohammad-Djafari, J. Idier, “A single site update algorithm for nonlinear diffraction tomography,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing IV (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 2837–2840.

Jacques, S.

Jiang, H.

Kingston, R. H.

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, New York, 1978).

Lakowicz, J. R.

Leahy, R.

T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. Med. Imaging 8, 194–202 (1989).
[CrossRef] [PubMed]

Loan, C. F. V.

G. H. Golub, C. F. V. Loan, Matrix Computations, 2nd ed. (The Johns Hopkins U. Press, Baltimore, 1989).

Luenberger, D. G.

D. G. Luenberger, Linear and Nonlinear Programming, 2nd ed. (Addison-Wesley, Reading, Mass., 1989).

Millane, R. P.

Mohammad-Djafari, A.

H. Carfantan, A. Mohammad-Djafari, J. Idier, “A single site update algorithm for nonlinear diffraction tomography,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing IV (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 2837–2840.

Moultan, J. D.

O’Leary, M.

M. O’Leary, D. Boas, B. Chance, A. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Osterberg, U. L.

Patterson, M.

S. Flock, M. Patterson, B. Wilson, D. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Patterson, M. S.

Paulsen, K. D.

Pei, Y.

Pogue, B. W.

Przadka, A.

Reynolds, G.

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern. Anal. Mach. Intell. 14, 367–383 (1992).
[CrossRef]

Reynolds, J. S.

Saquib, S. S.

S. S. Saquib, C. A. Bouman, K. Sauer, “ML parameter estimation for Markov random fields with applications to Bayesian tomography,” IEEE Trans. Image Process. 7, 1029–1044 (1998).
[CrossRef]

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.

S. S. Saquib, C. A. Bouman, K. Sauer, “ML parameter estimation for Markov random fields with applications to Bayesian tomography,” IEEE Trans. Image Process. 7, 1029–1044 (1998).
[CrossRef]

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]

C. A. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving map estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

K. Sauer, C. A. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process. 41, 534–548 (1993).
[CrossRef]

Schnabel, R. B.

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

Schweiger, M.

S. R. Arridge, M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998), http://epubs.osa.org/opticsexpress .
[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, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Sevick, E. M.

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems, F. John, transl. ed. (Winston, New York, 1977).

Wang, Y.

Webb, K. J.

Wilson, B.

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]

S. Flock, M. Patterson, B. Wilson, D. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Wilson, B. C.

Wyman, D.

S. Flock, M. Patterson, B. Wilson, D. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Yao, Y.

Ye, J. C.

Yeung, S.

Yodh, A.

M. O’Leary, D. Boas, B. Chance, A. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

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. (7)

IEEE Trans. Biomed. Eng. (1)

S. Flock, M. Patterson, B. Wilson, D. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

IEEE Trans. Image Process. (3)

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]

S. S. Saquib, C. A. Bouman, K. Sauer, “ML parameter estimation for Markov random fields with applications to Bayesian tomography,” IEEE Trans. Image Process. 7, 1029–1044 (1998).
[CrossRef]

C. A. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving map estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging (1)

T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. Med. Imaging 8, 194–202 (1989).
[CrossRef] [PubMed]

IEEE Trans. Pattern. Anal. Mach. Intell. (1)

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern. Anal. Mach. Intell. 14, 367–383 (1992).
[CrossRef]

IEEE Trans. Signal Process. (1)

K. Sauer, C. A. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process. 41, 534–548 (1993).
[CrossRef]

J. Appl. Stat. (1)

J. Besag, “Towards Bayesian image analysis,” J. Appl. Stat. 16, 395–407 (1989).
[CrossRef]

J. Opt. Soc. Am. A (6)

J. R. Stat. Soc. B (1)

J. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. R. Stat. Soc. B 36, 192–236 (1974).

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

M. O’Leary, D. Boas, B. Chance, A. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Other (11)

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

G. H. Golub, C. F. V. Loan, Matrix Computations, 2nd ed. (The Johns Hopkins U. Press, Baltimore, 1989).

D. G. Luenberger, Linear and Nonlinear Programming, 2nd ed. (Addison-Wesley, Reading, Mass., 1989).

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems, F. John, transl. ed. (Winston, New York, 1977).

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

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, Chichester, UK, 1987).

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

H. Carfantan, A. Mohammad-Djafari, J. Idier, “A single site update algorithm for nonlinear diffraction tomography,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing IV (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 2837–2840.

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, New York, 1978).

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]

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, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Simulation geometry with the locations of sources and detectors for inversion of synthetic data. The sources and detectors are uniformly spaced along the edges.

Fig. 2
Fig. 2

Pseudo-code specification of the ICD–Born algorithm.

Fig. 3
Fig. 3

Reconstruction results for μa: (a) original absorption image; (b) reconstruction by the DBIM; (c) reconstruction by the new algorithm with a Gaussian prior (p=2.0, σ =1.00×10-3); (d) reconstruction by the new algorithm with a GGMRF prior (p=1.1, σ=2.31×10-4).

Fig. 4
Fig. 4

NRMSE versus CPU time for the ICD–Born method (with p=1.1 and p=2) and the DBIM.

Fig. 5
Fig. 5

Cost function (log scale) versus iteration for the ICD–Born algorithm with p=1.1 and p=2.

Fig. 6
Fig. 6

Reconstructions of μa showing the effect of σ for a GGMRF prior model and with p=1.1: (a) original absorption image, and reconstructions with (b) σ=2.85×10-5, (c) σ=2.31×10-4, (d) σ=1.52×10-2.

Fig. 7
Fig. 7

Variety of true absorption images used for simulations.

Fig. 8
Fig. 8

Reconstructions of the absorption images shown in Fig. 7 by the ICD–Born algorithm with the GGMRF prior with p=1.1 and σ=2.31×10-4. NRMSE values for the final reconstructions are (a) 5.83×10-2, (b) 5.56×10-2, (c) 1.92×10-1, (d) 1.25×10-1, (e) 7.70×10-2, (f) 2.18×10-1, (g) 8.34×10-2, (h) 1.26×10-1, (i) 2.08×10-1.

Fig. 9
Fig. 9

Illustration of the 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. In this figure z is the distance variable perpendicular to the interface and az is the unit vector.

Fig. 10
Fig. 10

Normalized (with respect to the source strength) magnitude of the complex optical signal versus modulation frequency for several absorption coefficients. The scattering coefficient is set to be μs=10.0 cm-1 and the distance is 4 cm. The plots are computed for the analytic form, Eq. (B2).

Tables (3)

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Table 1 Computational Complexity of the DBIM and the ICD–Born Method: Number of Complex Multiplications (cflops) per Iterationa

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Table 2 Comparison of the Computation Required (Complex Multiplications, cflops) for One Iteration of the DBIM and the ICD–Born Methoda

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Table 3 Average CPU Time (s) per Interation for the First Simulation (Fig. 3)

Equations (52)

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1c t ψk(r, t)-D(r)ψk(r, t)+μa(r)ψk(r, t)
=S(t)δ(r-sk),
D(r)=13[μa(r)+μs(r)];
D(r)ϕk(r, ω)+[-μa(r)+jω/c]ϕk(r, ω)
=-βδ(r-sk).
x=[μa(r1),, μa(rN)]T.
f(x)=[f1(x), f2(x),, fP(x)]T=[ϕ1(d1, ω), ϕ1(d2, ω),, ϕ1(dM, ω),
ϕ2(d1, ω),, ϕK(dM, ω)]T.
y=[y11, y12 ,, y1M, y21 ,, yKM]T.
x^MAP=arg maxx log p(x|y)
=arg maxx[log p(y|x)+log p(x)],
p(ykm|x)=12πα|ϕk(dm, ω)| exp-|ykm-ϕk(dm, ω)|22α|ϕk(dm, ω)|,
Cii=α|ϕk(dm, ω)|α|ykm|,
wherei=M(k-1)+m.
Λ=½C-1.
p(y|x)=1πN|Λ|-1 exp[-y-f(x)Λ2],
p(xi|xj,ij)=p(xi|xi),
p(x)=1σ Nz(p) exp-1p uxσ, p,
u(x/σ, p)={i,j}Nbi-jρxi-xjσ, p,
p(x)=1σNz(p) exp-1pσp {i,j}Nbi-j|xi-xj|p,
1p2.
xi0,i=1 ,, N.
x^MAP=arg minx0y-f(x)Λ2+1pσp {i,j}Nbi-j|xi-xj|p.
x^i=arg minx˜i0y-f(x˜i)Λ2+1pσp jNibi-j|x˜i-xj|p,
y-f(x)Λ2y-f(xn)-f(xn)ΔxΛ2
f(x)=[f1(x)x1f1(x)xNfP(x)x1fP(x)xN]=[ϕ1(d1,ω)x1ϕ1(d1,ω)x2ϕ1(d1,ω)xN-1ϕ1(d1,ω)xNϕ1(d2,ω)x1ϕ1(d2,ω)x2ϕ1(d2,ω)xN-1ϕ1(d2,ω)xNϕ1(dM,ω)x1ϕ1(dM,ω)x2ϕ1(dM,ω)xN-1ϕ1(dM,ω)xNϕ2(d1,ω)x1ϕ2(d1,ω)x2ϕ2(d1,ω)xN-1ϕ2(d1,ω)xNϕK(dM,ω)x1ϕK(dM,ω)x2ϕK(dM,ω)xN-1ϕK(dM,ω)xN],
limN i=1N ϕk(dm, ω)xi Δxi
=Ωdr g(dm, r, ω)ϕk(r, ω)×-1+μan(r)+jω/cμan(r)+μs(r)Δμa(r),
ϕk(dm, ω)xi=g(dm, ri, ω)ϕk(ri, ω)×-1+μan(ri)+jω/cμan(ri)+μs(ri)A,
x^i=arg minx˜i0y-f(xn)-[f(xn)]*i(x˜i-xin)Λ2+1pσp jNibi-j|x˜i-xj|p,
x^i=arg minx˜i0θ1(x˜i-xin)+θ22 (x˜i-xin)2+1pσp jNibi-j|x˜i-xj|p,
θ1=-2 Re{[f(xn)*i]HΛe(i)},
θ2=2[f(xn)*i]HΛf(xn)*i,
e(1)=y-f(xn),
e(i+1)=e(i)-[f(xn)]*i(x^i-xin).
θ1+θ2(x˜i-xin)+1σp jNibi-j|x˜i-xj|p-1 sgn(x˜i-xj)
=0.
minxin-θ1θ2,xjNix^i
maxxin-θ1θ2,xjNi.
g(dm, ri, ω)=g(ri, dm, ω).
SNRmk=|E[i]|2σ0212α |ϕk(dm, ω)|,
Re[ymk]=Re[ϕk(dm, ω)]+[α|ϕk(dm, ω)|]1/2×N(0, 1),
Im[ymk]=Im[ϕk(dm, ω)]+[α|ϕk(dm, ω)|]1/2×N(0, 1),
NRMSEn=i=1N[μan(ri)-μa(ri)]2i=1N[μa(ri)]21/2,
P=|J+(dm)|Ae=D(dm)|ϕk(dm, ω)|Ae,
P=γ|ϕk(dm, 0)|,
p(i)=12πσ02 exp-|i-E[i]|22σ02,
σ02=2eBi0,
p(ykm|x)=12πηϕk(dm, 0) exp-|ykm-ϕk(dm, ω)|22ηϕk(dm, 0),
p(ykm|x)=12πα|ϕk(dm, ω)| exp-|ykm-ϕk(dm, ω)|22α|ϕk(dm, ω)|,
ψk(r, t)=Re14πDz exp-zμaD1/2+β4πDz×exp-zc2μa2+ω2c2D21/4×cos12 arctanωcμa×exp-jzc2μa2+ω2c2D21/4×sin12 arctanωcμa+jωt,
|ϕk(dm, ω)|=β4πDz exp-zc2μa2+ω2c2D21/4×cos12 arctanωcμa.

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