Abstract

Fluorescence tomography (FT) is depth-resolved three-dimensional (3D) localization and quantification of fluorescence distribution in biological tissue and entails a highly ill-conditioned problem as depth information must be extracted from boundary measurements. Conventionally, L2 regularization schemes that penalize the Euclidean norm of the solution and possess smoothing effects are used for FT reconstruction. Oversmooth, continuous reconstructions lack high-frequency edge-type features of the original distribution and yield poor resolution. We propose an alternative regularization method for FT that penalizes the total variation (TV) norm of the solution to preserve sharp transitions in the reconstructed fluorescence map while overcoming ill-posedness. We have developed two iterative methods for fast 3D reconstruction in FT based on TV regularization inspired by Rudin–Osher–Fatemi and split Bregman algorithms. The performance of the proposed method is studied in a phantom-based experiment using a noncontact constant-wave trans-illumination FT system. It is observed that the proposed method performs better in resolving fluorescence inclusions at different depths.

© 2012 Optical Society of America

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    [CrossRef]
  31. P. C. Hansen, “The L-curve and its use in the numerical treatment of inverse problems,” in Computational Inverse Problems in Electrocardiology, P. D. Johnston, ed. (WIT Press, 2001), pp. 119–142.
  32. X. Liu and L. Huang, “Split Bregman iteration algorithm for total bounded variation regularization based image deblurring,” J. Math. Anal. Appl. 372, 486–495 (2010).
    [CrossRef]
  33. C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations (SIAM, 1995).
  34. A. Michelson, Studies in Optics (University of Chicago, 1927).
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    [CrossRef]
  36. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42, 1971–1979 (1997).
    [CrossRef]
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    [CrossRef]
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  39. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
    [CrossRef]

2012

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

2011

J. C. Baritaux, K. Hassler, M. Bucher, S. Sanyal, and M. Unser, “Sparsity-driven reconstruction for FDOT with anatomical priors,” IEEE Trans. Med. Imag. 30, 1143–1153 (2011).
[CrossRef]

2010

2009

2007

2006

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8, 1–33 (2006).
[CrossRef]

2005

2004

R. Schultz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imag. 23, 492–500 (2004).

2003

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
[CrossRef]

2002

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
[CrossRef]

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1, 82–88 (2002).
[CrossRef]

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. SevickMuraca, “Three-dimensional, Bayesian image reconstruction from sparse and noisy data sets: near-infrared fluorescence tomography,” Proc. Natl. Acad. Sci. USA 99, 9619–9624 (2002).
[CrossRef]

2001

2000

S. Chang, B. Yu, and M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Proces. 9, 1532–1546 (2000).
[CrossRef]

1998

1997

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42, 1971–1979 (1997).
[CrossRef]

1996

J. A. Fessler and W. L. Rogers, “Spatial resolution properties of penalized-likelihood image reconstruction: Spatial-invariant tomographs,” IEEE Trans. Image Process. 9, 1346–1358 (1996).
[CrossRef]

1995

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

1994

R. Acar and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Probl. 10, 1217–1229 (1994).
[CrossRef]

1992

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D. 60, 259–268 (1992).
[CrossRef]

1970

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef]

1928

R. Courant, K. Friedrichs, and H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik,” Math. Ann. 100, 32–74 (1928).
[CrossRef]

Acar, R.

R. Acar and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Probl. 10, 1217–1229 (1994).
[CrossRef]

Adibi, A.

Ahn, S.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

Arridge, S. R.

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, and S. R. Arridge, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15, 6696–6716 (2007).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Bai, J.

Baritaux, J. C.

J. C. Baritaux, K. Hassler, M. Bucher, S. Sanyal, and M. Unser, “Sparsity-driven reconstruction for FDOT with anatomical priors,” IEEE Trans. Med. Imag. 30, 1143–1153 (2011).
[CrossRef]

J. C. Baritaux, K. Hassler, and M. Unser, “An efficient numerical method for general Lp regularization in fluorescence molecular tomography,” IEEE Trans. Med. Imag. 29, 1075–1087 (2010).
[CrossRef]

Bender, R.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef]

Bremer, C.

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1, 82–88 (2002).
[CrossRef]

Bucher, M.

J. C. Baritaux, K. Hassler, M. Bucher, S. Sanyal, and M. Unser, “Sparsity-driven reconstruction for FDOT with anatomical priors,” IEEE Trans. Med. Imag. 30, 1143–1153 (2011).
[CrossRef]

Cai, J. F.

J. F. Cai, S. Osher, and Z. Shen, “Split Bregman methods and frame based image restoration,” SIAM J. Multisc. Model. Simul.8, 337–369 (2009).

Chan, T.

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
[CrossRef]

Chance, B.

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
[CrossRef]

Chang, S.

S. Chang, B. Yu, and M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Proces. 9, 1532–1546 (2000).
[CrossRef]

Chen, N.

Cherry, S. R.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

Choe, R.

Clason, C.

Cong, A. X.

Corlu, A.

Courant, R.

R. Courant, K. Friedrichs, and H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik,” Math. Ann. 100, 32–74 (1928).
[CrossRef]

Cubeddu, R.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42, 1971–1979 (1997).
[CrossRef]

Culver, J. P.

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
[CrossRef]

Davis, S. C.

Dehghani, H.

Delpy, D. T.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Durduran, T.

Dutta, J.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

Eftekhar, A. A.

Eppstein, M. J.

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. SevickMuraca, “Three-dimensional, Bayesian image reconstruction from sparse and noisy data sets: near-infrared fluorescence tomography,” Proc. Natl. Acad. Sci. USA 99, 9619–9624 (2002).
[CrossRef]

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D. 60, 259–268 (1992).
[CrossRef]

Feng, J.

Fessler, J. A.

J. A. Fessler and W. L. Rogers, “Spatial resolution properties of penalized-likelihood image reconstruction: Spatial-invariant tomographs,” IEEE Trans. Image Process. 9, 1346–1358 (1996).
[CrossRef]

Freiberger, M.

Friedrichs, K.

R. Courant, K. Friedrichs, and H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik,” Math. Ann. 100, 32–74 (1928).
[CrossRef]

Gao, H.

Godavarty, A.

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. SevickMuraca, “Three-dimensional, Bayesian image reconstruction from sparse and noisy data sets: near-infrared fluorescence tomography,” Proc. Natl. Acad. Sci. USA 99, 9619–9624 (2002).
[CrossRef]

Goldstein, T.

T. Goldstein and S. Osher, “The Split Bregman method for L1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
[CrossRef]

Gordon, R.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef]

Graves, E. E.

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1, 82–88 (2002).
[CrossRef]

Gulsen, G.

Han, D.

Hansen, P. C.

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1997).

P. C. Hansen, “The L-curve and its use in the numerical treatment of inverse problems,” in Computational Inverse Problems in Electrocardiology, P. D. Johnston, ed. (WIT Press, 2001), pp. 119–142.

Hanson, R J.

C. L. Lawson and R J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).

Hassler, K.

J. C. Baritaux, K. Hassler, M. Bucher, S. Sanyal, and M. Unser, “Sparsity-driven reconstruction for FDOT with anatomical priors,” IEEE Trans. Med. Imag. 30, 1143–1153 (2011).
[CrossRef]

J. C. Baritaux, K. Hassler, and M. Unser, “An efficient numerical method for general Lp regularization in fluorescence molecular tomography,” IEEE Trans. Med. Imag. 29, 1075–1087 (2010).
[CrossRef]

Hawrysz, D. J.

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. SevickMuraca, “Three-dimensional, Bayesian image reconstruction from sparse and noisy data sets: near-infrared fluorescence tomography,” Proc. Natl. Acad. Sci. USA 99, 9619–9624 (2002).
[CrossRef]

Herman, G. T.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef]

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Huang, J.

Huang, L.

X. Liu and L. Huang, “Split Bregman iteration algorithm for total bounded variation regularization based image deblurring,” J. Math. Anal. Appl. 372, 486–495 (2010).
[CrossRef]

Intes, X.

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Jiang, H.

Jiang, S.

Kelley, C. T.

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations (SIAM, 1995).

Kisilev, P.

P. Kisilev, M. Zibulevsky, and Y. Zeevi, “Wavelet representation and total variation regularization in emission tomography,” in 2001 International Conference on Image Processing (IEEE, 2001), Vol. 1, pp. 702–705.
[CrossRef]

Lawson, C. L.

C. L. Lawson and R J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).

Leahy, R. M.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

Lewy, H.

R. Courant, K. Friedrichs, and H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik,” Math. Ann. 100, 32–74 (1928).
[CrossRef]

Li, C.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

Lin, Y.

Liu, K.

Liu, X.

X. Liu and L. Huang, “Split Bregman iteration algorithm for total bounded variation regularization based image deblurring,” J. Math. Anal. Appl. 372, 486–495 (2010).
[CrossRef]

Ma, X.

Michelson, A.

A. Michelson, Studies in Optics (University of Chicago, 1927).

Mohajerani, P.

Nalcioglu, O.

Ntziachristos, V.

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8, 1–33 (2006).
[CrossRef]

R. Schultz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imag. 23, 492–500 (2004).

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1, 82–88 (2002).
[CrossRef]

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47, N1–N10 (2002).
[CrossRef]

V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media using a normalized born approximation,” Opt. Lett. 26, 893–895 (2001).
[CrossRef]

Osher, S.

T. Goldstein and S. Osher, “The Split Bregman method for L1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
[CrossRef]

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D. 60, 259–268 (1992).
[CrossRef]

J. F. Cai, S. Osher, and Z. Shen, “Split Bregman methods and frame based image restoration,” SIAM J. Multisc. Model. Simul.8, 337–369 (2009).

Paulsen, K. D.

Pifferi, A.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42, 1971–1979 (1997).
[CrossRef]

Pogue, B. W.

Qin, C.

Ripoll, J.

R. Schultz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imag. 23, 492–500 (2004).

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1, 82–88 (2002).
[CrossRef]

Rogers, W. L.

J. A. Fessler and W. L. Rogers, “Spatial resolution properties of penalized-likelihood image reconstruction: Spatial-invariant tomographs,” IEEE Trans. Image Process. 9, 1346–1358 (1996).
[CrossRef]

Rosen, M. A.

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D. 60, 259–268 (1992).
[CrossRef]

Sanyal, S.

J. C. Baritaux, K. Hassler, M. Bucher, S. Sanyal, and M. Unser, “Sparsity-driven reconstruction for FDOT with anatomical priors,” IEEE Trans. Med. Imag. 30, 1143–1153 (2011).
[CrossRef]

Scharfetter, H.

Schultz, R.

R. Schultz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imag. 23, 492–500 (2004).

Schweiger, M.

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, and S. R. Arridge, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15, 6696–6716 (2007).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

SevickMuraca, E. M.

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. SevickMuraca, “Three-dimensional, Bayesian image reconstruction from sparse and noisy data sets: near-infrared fluorescence tomography,” Proc. Natl. Acad. Sci. USA 99, 9619–9624 (2002).
[CrossRef]

Shen, Z.

J. F. Cai, S. Osher, and Z. Shen, “Split Bregman methods and frame based image restoration,” SIAM J. Multisc. Model. Simul.8, 337–369 (2009).

Song, X.

Strong, D.

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
[CrossRef]

Taroni, P.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42, 1971–1979 (1997).
[CrossRef]

Tian, J.

Torricelli, A.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42, 1971–1979 (1997).
[CrossRef]

Unser, M.

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Figures (8)

Fig. 1.
Fig. 1.

Comparison of PSF of L2 regularization versus TV regularization for (a) on-edge and (b) off-edge voxels: (i) original fluorescence distribution, (ii) PSF for L2 regularization of a moderate level, and (iii) PSF for TV regularization of a moderate level.

Fig. 2.
Fig. 2.

Simulated 2D fluorescence tomography configuration: two fluorescent blobs in a 2D turbid slab with eight sources and eight detectors around it used for excitation and data acquisition.

Fig. 3.
Fig. 3.

Reconstructed fluorophore distributions for 2D simulated data with (a) SNR=50dB, (b) SNR=40dB, and (c) SNR=30dB, by (i) L2 regularization, (ii) algebraic reconstruction technique (ART), (iii) time marching ROF TV regularization, and (iv) iterative Bregman TV regularization.

Fig. 4.
Fig. 4.

Relative estimation errors for reconstructed fluorescent distributions corresponding to L2 regularization, ART, and the proposed ROF and iterative Bregman TV regularization for data SNR=30, 40, and 50 dB.

Fig. 5.
Fig. 5.

Michelson contrast [defined in Eq. (15)] is computed and plotted for the reconstructed fluorescent distributions corresponding to L2 regularization, ART, and the proposed ROF and iterative Bregman TV regularization for data SNR=30, 40, and 50 dB.

Fig. 6.
Fig. 6.

(a) Schematic diagram of the fluorescent tomography setup in the transillumination geometry. (b) Photograph of the interior of the imaging chamber.

Fig. 7.
Fig. 7.

Configuration for the experimental phantom based fluorescence tomography. Two fluorescent tubes are inserted in an intralipid-20% liquid phantom that is excited at 36 source positions (circles) and imaged by a CCD camera that yields 81 data points (dots).

Fig. 8.
Fig. 8.

Reconstructed fluorophore distributions from experimental data where fluorophore tubes are located at (i) 3 mm, (ii) 6 mm, and (iii) 9 mm depth using (a) L2 regularization, (b) time marching ROF-based TV regularization, and (c) iterative Bregman-based TV regularization.

Equations (33)

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·D(r)Φ(r)μa(r)Φ(r)=q(r),
y=Mx+n.
minxyMx2+λ2Lx2,
minxyMx22+λ2Lx22,
(M*M+λ2L*L)xrec=M*Mxorig.
minxyMx22x+λ2xTV,
xTV=|u|dΩ.
(M*M+λ2.|u|)xrec=M*Mxorig,
minuyMT(u)22+λ2|u|dΩ,
u(k+1)u(k)t=2T*M*(yMT(u))+λ2·(u|u|).
minu|u|dΩ+μ2MT(u)y22+α2uuL222,
uk+1=minuμ2MT(u)y22+α2uuL222+β2dkΔubk22,
(μT*M*MT+αIβ)uk+1=μT*M*y+αuL2+β*(dkbk),
ε=xx^2x2,
C=ImaxIminImax+Imin,
·D(r)Φexc(r)+μa(r)Φexc(r)=qexc(r),
·D(r)Φem(r)+μa(r)Φem(r)=ημflc(r)Φexc(r),
Φ(ξ)+2AD(ξ)n^·Φ(ξ)=0,
Φh(r)=1NΦjψj(r),
ψj(r)(·D(r)+μa(r))Φh(r)=ψj(r)q(r).
[K(D)+C(μa)]Φ=Qβ,
Kij=D(r)ψi(r)·ψj(r)dΩ,
Cij=μa(r)ψi(r)ψj(r)dΩ,
βi=ψi(r)Γ(r)d(δΩ),
Qi=ψi(r)q(r)dΩ,
Γ(ξ)=cD(ξ)n^·Φ(ξ)=cΦ(r)2A,
[K(D)+C(μa)+F(A)]Φ=Q,
Fij=c2Aψi(r)ψj(r)d(δΩ).
ZeΦe(i)=Qe(i),
ZmΦm(i)=Qm(i),
Qm(i)=ημfldiag(Φe(i))x,
y=[Z¯m1ημfldiag(Ze1Qe(1))Z¯m1ημfldiag(Ze1Qe(Ns))]x,
y=Mx+n,

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