Abstract

Fluorescence molecular tomography (FMT) has become an important method for in-vivo imaging of small animals. It has been widely used for tumor genesis, cancer detection, metastasis, drug discovery, and gene therapy. In this study, an algorithm for FMT is proposed to obtain accurate and fast reconstruction by combining an adaptive mesh refinement technique and an analytical solution of diffusion equation. Numerical studies have been performed on a parallel plate FMT system with matching fluid. The reconstructions obtained show that the algorithm is efficient in computation time, and they also maintain image quality.

© 2007 Optical Society of America

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References

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2006 (2)

2005 (2)

A. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express 13, 9847–9857(2005).
[Crossref] [PubMed]

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[Crossref] [PubMed]

2004 (5)

E. E. Graves, R. Weissleder, and V. Ntziachristos, “Fluorescence molecular imaging of small animal tumor models,” Curr. Mol. Med. 4, 419–430 (2004).
[Crossref] [PubMed]

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE 5535, 679–686 (2004).
[Crossref]

R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging 23, 492–500 (2004).
[Crossref] [PubMed]

E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21, 231–241 (2004).
[Crossref]

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12, 5402–5417 (2004).
[Crossref] [PubMed]

2003 (3)

J. Ripoll and V. Ntziachristos, “Iterative boundary method for diffuse optical tomography,” J. Opt. Soc. Am. A 20, 1103–1110 (2003).
[Crossref]

J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys Rev Lett. 91, 103901 (2003).
[Crossref] [PubMed]

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911 (2003).
[Crossref] [PubMed]

2002 (4)

E. E. Graves, A. Petrovsky, R. Weissleder, and V. Ntziachristos, “In vivo time-resolved optical spectroscopy of mice,” presented at the Optical Society of America Biomedical Optical Spectroscopy and Diagnostics Meeting, Miami, Fla., April 7–10, 2002.
[PubMed]

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. Bio. 47, N1–N10 (2002).
[Crossref]

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,” Proc. Nat. Acad. Sci. 99, 9619–9624 (2002).
[Crossref] [PubMed]

J. Ripoll, M. Nieto-Vesperinas, R. Weissleder, and V. Ntziachristos, “Fast analytical approximation for arbitrary geometries in diffuse optical tomography,” Opt. Lett. 27, 527–529 (2002).
[Crossref]

2001 (2)

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

J. Ripoll, V. Ntziachrisos, R. Carminati, and M. Nieto-Vesperinas, “The Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917 (2001).
[Crossref]

1999 (2)

R. H. Byrd, M. E. Hribar, and J. Nocedal, “An interior point algorithm for large scale nonlinear programming,” SIAM J. Optimization 9, 877–900 (1999).
[Crossref]

R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express 4, 353–371 (1999).
[Crossref] [PubMed]

1998 (1)

1997 (1)

1996 (1)

1995 (1)

Arridge, S. R.

Bangerth, W.

Boas, D. A.

Byrd, R. H.

R. H. Byrd, M. E. Hribar, and J. Nocedal, “An interior point algorithm for large scale nonlinear programming,” SIAM J. Optimization 9, 877–900 (1999).
[Crossref]

Carminati, R.

J. Ripoll, V. Ntziachrisos, R. Carminati, and M. Nieto-Vesperinas, “The Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917 (2001).
[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. Bio. 47, N1–N10 (2002).
[Crossref]

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21, 158–160 (1996).
[Crossref] [PubMed]

Chen, A. U.

Cong, A.

A. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express 13, 9847–9857(2005).
[Crossref] [PubMed]

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE 5535, 679–686 (2004).
[Crossref]

Cong, W.

Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, J. Shi, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express 14, 8211–8223 (2006).
[Crossref] [PubMed]

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE 5535, 679–686 (2004).
[Crossref]

Culver, J. P.

E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21, 231–241 (2004).
[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. Bio. 47, N1–N10 (2002).
[Crossref]

Dougherty, D. E.

Eppstein, M. J.

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,” Proc. Nat. Acad. Sci. 99, 9619–9624 (2002).
[Crossref] [PubMed]

M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevick-Muraca, “Biomedical optical tomography using dynamic parameterization and Bayesian conditioning on photon migration measurements,” Appl. Opt. 38, 2138–2150 (1998).
[Crossref]

Godavarty, A.

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,” Proc. Nat. Acad. Sci. 99, 9619–9624 (2002).
[Crossref] [PubMed]

Graves, E. E.

E. E. Graves, R. Weissleder, and V. Ntziachristos, “Fluorescence molecular imaging of small animal tumor models,” Curr. Mol. Med. 4, 419–430 (2004).
[Crossref] [PubMed]

E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21, 231–241 (2004).
[Crossref]

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911 (2003).
[Crossref] [PubMed]

E. E. Graves, A. Petrovsky, R. Weissleder, and V. Ntziachristos, “In vivo time-resolved optical spectroscopy of mice,” presented at the Optical Society of America Biomedical Optical Spectroscopy and Diagnostics Meeting, Miami, Fla., April 7–10, 2002.
[PubMed]

Hawrysz, D. J.

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,” Proc. Nat. Acad. Sci. 99, 9619–9624 (2002).
[Crossref] [PubMed]

Hribar, M. E.

R. H. Byrd, M. E. Hribar, and J. Nocedal, “An interior point algorithm for large scale nonlinear programming,” SIAM J. Optimization 9, 877–900 (1999).
[Crossref]

Hwang, K.

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. Bio. 47, N1–N10 (2002).
[Crossref]

Joshi, A.

Kumar, D.

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE 5535, 679–686 (2004).
[Crossref]

Li, H.

Liu, Y.

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE 5535, 679–686 (2004).
[Crossref]

Luo, J.

Lv, Y.

Nieto-Vesperinas, M.

J. Ripoll, M. Nieto-Vesperinas, R. Weissleder, and V. Ntziachristos, “Fast analytical approximation for arbitrary geometries in diffuse optical tomography,” Opt. Lett. 27, 527–529 (2002).
[Crossref]

J. Ripoll, V. Ntziachrisos, R. Carminati, and M. Nieto-Vesperinas, “The Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917 (2001).
[Crossref]

Nocedal, J.

R. H. Byrd, M. E. Hribar, and J. Nocedal, “An interior point algorithm for large scale nonlinear programming,” SIAM J. Optimization 9, 877–900 (1999).
[Crossref]

Ntziachrisos, V.

J. Ripoll, V. Ntziachrisos, R. Carminati, and M. Nieto-Vesperinas, “The Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917 (2001).
[Crossref]

Ntziachristos, V.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[Crossref] [PubMed]

E. E. Graves, R. Weissleder, and V. Ntziachristos, “Fluorescence molecular imaging of small animal tumor models,” Curr. Mol. Med. 4, 419–430 (2004).
[Crossref] [PubMed]

R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging 23, 492–500 (2004).
[Crossref] [PubMed]

E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21, 231–241 (2004).
[Crossref]

J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys Rev Lett. 91, 103901 (2003).
[Crossref] [PubMed]

J. Ripoll and V. Ntziachristos, “Iterative boundary method for diffuse optical tomography,” J. Opt. Soc. Am. A 20, 1103–1110 (2003).
[Crossref]

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911 (2003).
[Crossref] [PubMed]

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. Bio. 47, N1–N10 (2002).
[Crossref]

J. Ripoll, M. Nieto-Vesperinas, R. Weissleder, and V. Ntziachristos, “Fast analytical approximation for arbitrary geometries in diffuse optical tomography,” Opt. Lett. 27, 527–529 (2002).
[Crossref]

E. E. Graves, A. Petrovsky, R. Weissleder, and V. Ntziachristos, “In vivo time-resolved optical spectroscopy of mice,” presented at the Optical Society of America Biomedical Optical Spectroscopy and Diagnostics Meeting, Miami, Fla., April 7–10, 2002.
[PubMed]

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

O’Leary, M. A.

Paithankar, D. Y.

Patterson, M. S.

Petrovsky, A.

E. E. Graves, A. Petrovsky, R. Weissleder, and V. Ntziachristos, “In vivo time-resolved optical spectroscopy of mice,” presented at the Optical Society of America Biomedical Optical Spectroscopy and Diagnostics Meeting, Miami, Fla., April 7–10, 2002.
[PubMed]

Pogue, B. W.

Rasmussen, J.

Ripoll, J.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[Crossref] [PubMed]

E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21, 231–241 (2004).
[Crossref]

R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging 23, 492–500 (2004).
[Crossref] [PubMed]

J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys Rev Lett. 91, 103901 (2003).
[Crossref] [PubMed]

J. Ripoll and V. Ntziachristos, “Iterative boundary method for diffuse optical tomography,” J. Opt. Soc. Am. A 20, 1103–1110 (2003).
[Crossref]

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911 (2003).
[Crossref] [PubMed]

J. Ripoll, M. Nieto-Vesperinas, R. Weissleder, and V. Ntziachristos, “Fast analytical approximation for arbitrary geometries in diffuse optical tomography,” Opt. Lett. 27, 527–529 (2002).
[Crossref]

J. Ripoll, V. Ntziachrisos, R. Carminati, and M. Nieto-Vesperinas, “The Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917 (2001).
[Crossref]

Roy, R.

Schulz, R. B.

R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging 23, 492–500 (2004).
[Crossref] [PubMed]

J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys Rev Lett. 91, 103901 (2003).
[Crossref] [PubMed]

Sevick-Muraca, E. M.

Shi, J.

Tian, J.

Troy, T. L.

Wang, G.

Wang, L. V.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[Crossref] [PubMed]

Weissleder, R.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[Crossref] [PubMed]

E. E. Graves, R. Weissleder, and V. Ntziachristos, “Fluorescence molecular imaging of small animal tumor models,” Curr. Mol. Med. 4, 419–430 (2004).
[Crossref] [PubMed]

E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21, 231–241 (2004).
[Crossref]

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911 (2003).
[Crossref] [PubMed]

E. E. Graves, A. Petrovsky, R. Weissleder, and V. Ntziachristos, “In vivo time-resolved optical spectroscopy of mice,” presented at the Optical Society of America Biomedical Optical Spectroscopy and Diagnostics Meeting, Miami, Fla., April 7–10, 2002.
[PubMed]

J. Ripoll, M. Nieto-Vesperinas, R. Weissleder, and V. Ntziachristos, “Fast analytical approximation for arbitrary geometries in diffuse optical tomography,” Opt. Lett. 27, 527–529 (2002).
[Crossref]

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

Yang, W.

Yodh, A.

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. Bio. 47, N1–N10 (2002).
[Crossref]

Yodh, A. G.

Appl. Opt. (3)

Curr. Mol. Med. (1)

E. E. Graves, R. Weissleder, and V. Ntziachristos, “Fluorescence molecular imaging of small animal tumor models,” Curr. Mol. Med. 4, 419–430 (2004).
[Crossref] [PubMed]

IEEE Trans. Med. Imaging (1)

R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging 23, 492–500 (2004).
[Crossref] [PubMed]

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

Med. Phys. (1)

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911 (2003).
[Crossref] [PubMed]

Nat. Biotechnol. (1)

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (4)

Phys Rev Lett. (1)

J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys Rev Lett. 91, 103901 (2003).
[Crossref] [PubMed]

Phys. Med. Bio. (1)

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. Bio. 47, N1–N10 (2002).
[Crossref]

Phys. Rev. E (1)

J. Ripoll, V. Ntziachrisos, R. Carminati, and M. Nieto-Vesperinas, “The Kirchhoff approximation for diffusive waves,” Phys. Rev. E 64, 051917 (2001).
[Crossref]

presented at the Optical Society of America Biomedical Optical Spectroscopy and Diagnostics Meeting, Miami, Fla. (1)

E. E. Graves, A. Petrovsky, R. Weissleder, and V. Ntziachristos, “In vivo time-resolved optical spectroscopy of mice,” presented at the Optical Society of America Biomedical Optical Spectroscopy and Diagnostics Meeting, Miami, Fla., April 7–10, 2002.
[PubMed]

Proc. Nat. Acad. Sci. (1)

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three dimensional near infrared fluorescence tomography with Bayesian methodologies for image reconstruction from sparse and noisy data sets,” Proc. Nat. Acad. Sci. 99, 9619–9624 (2002).
[Crossref] [PubMed]

Proc. SPIE (1)

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE 5535, 679–686 (2004).
[Crossref]

SIAM J. Optimization (1)

R. H. Byrd, M. E. Hribar, and J. Nocedal, “An interior point algorithm for large scale nonlinear programming,” SIAM J. Optimization 9, 877–900 (1999).
[Crossref]

Other (1)

W. Bangerth, “Adaptive finite element methods for the identification of distributed parameters in partial differential equations,” Ph.D. thesis (University of Heidelberg, 2002).

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Figures (3)

Fig. 1.
Fig. 1.

Parallel plate FMT system setup. The schematic [5] of major components is shown in (a). Optical fibers’ arrangements are shown in (b) with black points, and the detection points’ arrangements are shown in (c).

Fig. 2.
Fig. 2.

Reconstruction of single fluorescent target. Mesh evolution in the reconstruction is demonstrated in (a), (b) and (c). Top 70% of the contour levels of reconstructed fluorescence distribution are shown in (d), in which the red cube represents the real target.

Fig. 3.
Fig. 3.

Reconstruction of two fluorescent targets for different edge-to-edge spacing: (a) 0.3 cm; (b) 0.15 cm; (c) 0.1 cm. The top 70% of the contour levels of reconstructed fluorescence distribution are shown, and red cubes represent the real targets.

Tables (1)

Tables Icon

Table 1. Summary of results for dual fluorescent targets reconstructions. dreal is the edge-to-edge targets’ separation, and nrecons is the reconstructed targets’ fluorescence distribution.

Equations (9)

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U m ( r s , r d ) = d 3 r · ( S 0 · U x ( r s , r , k λ x ) · n ( r ) D λ m · G ( r d r , k λ m ) ) ,
U m ( r s 1 , r d 1 ) = [ S 0 U x ( r s , r 1 , k λ x ) G ( r d r 1 , k λ m ) D λ m , , S 0 U x ( r s , r N , k λ x ) G ( r d r N , k λ m ) D λ m ] · [ n ( r 1 ) n ( r N ) ] ,
[ U m ( r s 1 , r d 1 ) U m ( r s M , r d M ) ] = [ W 11 W 1 N W M 1 W M N ] · [ n ( r 1 ) n ( r N ) ] .
[ U m , k ( r s 1 , r d 1 ) U m , k ( r s M , r d M ) ] = [ W 11 , k W 1 N , k W M 1 , k W M N , k ] · [ n ( r 1 , k ) n ( r N , k ) ] ,
min n k L n k n k n Ψ k ( n k ) = U m ( r s , r d ) W k n k 2 + λ k n k 2 ,
min n k , s Ψ k ( n k ) v i log s i
s . t . n k n k L s = 0 ,
n k U n k s = 0
n k + 1 0 = F k k + 1 ( n k r ) .

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