Abstract

The bioluminescence tomography is a novel molecular imaging technology for small animal studies. Known reconstruction methods require the completely measured data on the external surface, although only partially measured data is available in practice. In this work, we formulate a mathematical model for BLT from partial data and generalize our previous results on the solution uniqueness to the partial data case. Then we extend two of our reconstruction methods for BLT to this case. The first method is a variant of the well-known EM algorithm. The second one is based on the Landweber scheme. Both methods allow the incorporation of knowledgebased constraints. Two practical constraints, the source non-negativity and support constraints, are introduced to regularize the BLT problem and produce stability. The initial choice of both methods and its influence on the regularization and stability are also discussed. The proposed algorithms are evaluated and validated with intensive numerical simulation and a physical phantom experiment. Quantitative results including the location and source power accuracy are reported. Various algorithmic issues are investigated, especially how to avoid the inverse crime in numerical simulations.

© 2007 Optical Society of America

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  1. C. Contag and M. H. Bachmann, “Advances in bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4, 235–260 (2002).
    [Crossref] [PubMed]
  2. V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotech. 23, 313–320 (2005).
    [Crossref]
  3. B. W. Rice, M. D. Cable, and M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6, 432–440 (2001).
    [Crossref] [PubMed]
  4. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
    [Crossref] [PubMed]
  5. Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, “Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden,” Molecular Imaging 3, 117–124 (2004).
    [Crossref] [PubMed]
  6. A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia 2, 491–495 (2002).
    [Crossref]
  7. A. McCaffrey, M. A. Kay, and C. H. Contag, “Advancing molecular therapies through in vivo bioluminescent imaging,” Molecuar Imaging 2, 75–86 (2003).
    [Crossref]
  8. A. Soling and N. G. Rainov, “Bioluminescence imaging in vivo-application to cancer research,” Expert Opinion on Biological Therapy 3, 1163–1172 (2003).
    [PubMed]
  9. J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003).
    [Crossref] [PubMed]
  10. C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. 16, 378–387 (2002).
    [Crossref]
  11. G. Wang, E. A. Hoffman, and G. McLennan, “Bioluminescent CT method and apparatus,” (2003). US provisional patent application.
  12. G. Wang et al, “Development of the first bioluminescent tomography system,” Radiology Suppl. (Proceedings of the RSNA) 229(P) (2003).
  13. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004).
    [Crossref] [PubMed]
  14. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of SPIE: Developments in X-Ray Tomography IV,”, vol. 5535 (2004), vol. 5535, pp. 335–351. Invited talk.
  15. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2004).
  16. H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Academic Radiology 11, 1029–1038 (2004).
    [Crossref] [PubMed]
  17. X. J. Gu, Q. H. Zhang, L. Larcom, and H. B. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12, 3996–4000 (2004).
    [Crossref] [PubMed]
  18. M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).
  19. W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005).
    [Crossref] [PubMed]
  20. A. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express 13, 9847–9857 (2005).
    [Crossref] [PubMed]
  21. C. Kuo, O. Coquoz, T. Troy, N. Zhang, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” in “SMI Fourth Conference,” (Cologne, Germany, 2005).
  22. A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005).
    [Crossref] [PubMed]
  23. N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33, 61–68 (2006).
    [Crossref] [PubMed]
  24. H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters 31, 365–367 (2005).
    [Crossref]
  25. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999).
    [Crossref]
  26. F. Natterer and F. Wuübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001).
    [Crossref]
  27. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
    [Crossref] [PubMed]
  28. A. Cong and G. Wang, “Multispectral bioluminescence tomography: Methodology and simulation,” International Journal of Biomedical Imaging 2006 (2006). Article ID 57614. doi:10.1155/IJBI/2006/57614.
  29. C. Q. Li and H. B. Jiang, “Imaging of particle size and concentration in heterogeneous turbid media with multispectral diffuse optical tomography,” Opt. Express 12, 6313–6318 (2004).
    [Crossref] [PubMed]
  30. A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1987).
  31. F. Natterer, The Mathematics of Computerized Tomography (SIAM, Philadelphia, PA, 2001).
    [Crossref]
  32. A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximal likelihood form incomplete data via the EM algorithm,” Journal of the Royal Statistical Society. Series B. 39, 1–38 (1977).
  33. L. A. Shepp and Y. Vardi, “Maximum likelihood restoration for emission tomography,” IEEE Transactions on Medical Imaging 1, 113–122 (1982).
    [Crossref] [PubMed]
  34. D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Transactions on Signal Processing 40, 1143–1150 (1992).
    [Crossref]
  35. M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Transactions on Medical Imaging 22, 569–579 (2003).
    [Crossref] [PubMed]
  36. M. Jiang and G. Wang, “Development of iterative algorithms for image reconstruction,” J. X-Ray Sci. Technol. 10, 77–86 (2002). Invited Review.
  37. M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Problems 13, 441–463 (1997).
    [Crossref]
  38. A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative leat-squares and regularization,” IEEE Transactions on Signal Processing 46, 2345–2352 (1998).
    [Crossref]
  39. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997).
  40. A. D. Klose and A. H. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Problems 19, 387–409 (2003).
    [Crossref]
  41. D. S. Anikonov, A. E. Kovtanyuk, and I. V. Prokhorov, Transport equation and tomography, Inverse and Ill-posed Problems Series (VSP, Utrecht, 2002).
  42. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin-Heideberg-New York, 1983).
  43. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. I (Springer-Verlag, Berlin, 1990).
  44. V. Isakov, Inverse Problems for Partial Differential Equations, vol. 127 of Applied Mathematical Series (Springer, New York-Berlin-Heideberg, 1998).
  45. W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1991), 2nd ed.
  46. M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Prentice-Hall, Englewood Cliffs, N. J., 1967).
  47. B. Eicke, “Konvex-resringierte schlechtgestellte Problems und ihr Regularisierung durch Iterationsverfahren,” Thesis, Technischen Universität Berlin (1991).
  48. B. Eicke, “Iteration methods for convexly constrained ill-posed problems in Hilbert space,” Numerical Functional Analysis and Optimization 13, 413–429 (1992).
    [Crossref]
  49. S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Texts in applied mathematics; 15 (Springer-Verlag, New York, NY, 2002), 2nd ed.
  50. D. L. Colton and R. Kress, Inverse acoustic and elctromagnetic scattering theory (Springer, Berlin; New York, 1998), 2nd ed.
  51. A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescent sources,” J. Opt. Soc. Am., A 24, 1601–1608 (2007).
    [Crossref]
  52. E. A. Marengo, A. J. Devaney, and R. W. Ziolkowski, “Inverse source problem and mimnimum-energy sources,” J. Opt. Soc. Am., A 17, 34–45 (2000).
    [Crossref]
  53. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems (W. H. Winston, Washington, D. C., 1977).
  54. M. Bertero and P. Boccacci, Inverse Problems in Imaging (Institute of Physical Publishing, Bristol and Philadelphia, 1998).
    [Crossref]
  55. R. J. Santos, “Equivalence of regularization and truncated iteration for general ill-posed problems,” Linear Algebra and Its applications 236, 25–33 (1996).
    [Crossref]
  56. R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Transactions on Medical Imaging 23, 492–500 (2004).
    [Crossref] [PubMed]
  57. M. D. Buhmann, Radial basis functions: theory and implementations, vol. 12 of Cambridge Monographs on Applied and Computational Mathematics (Cambridge University Press, Cambridge, 2003).
    [Crossref]

2007 (1)

A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescent sources,” J. Opt. Soc. Am., A 24, 1601–1608 (2007).
[Crossref]

2006 (1)

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33, 61–68 (2006).
[Crossref] [PubMed]

2005 (7)

H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters 31, 365–367 (2005).
[Crossref]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005).
[Crossref] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[Crossref] [PubMed]

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005).
[Crossref] [PubMed]

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

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
[Crossref] [PubMed]

2004 (6)

Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, “Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden,” Molecular Imaging 3, 117–124 (2004).
[Crossref] [PubMed]

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004).
[Crossref] [PubMed]

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Academic Radiology 11, 1029–1038 (2004).
[Crossref] [PubMed]

X. J. Gu, Q. H. Zhang, L. Larcom, and H. B. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12, 3996–4000 (2004).
[Crossref] [PubMed]

C. Q. Li and H. B. Jiang, “Imaging of particle size and concentration in heterogeneous turbid media with multispectral diffuse optical tomography,” Opt. Express 12, 6313–6318 (2004).
[Crossref] [PubMed]

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

2003 (5)

A. D. Klose and A. H. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Problems 19, 387–409 (2003).
[Crossref]

M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Transactions on Medical Imaging 22, 569–579 (2003).
[Crossref] [PubMed]

A. McCaffrey, M. A. Kay, and C. H. Contag, “Advancing molecular therapies through in vivo bioluminescent imaging,” Molecuar Imaging 2, 75–86 (2003).
[Crossref]

A. Soling and N. G. Rainov, “Bioluminescence imaging in vivo-application to cancer research,” Expert Opinion on Biological Therapy 3, 1163–1172 (2003).
[PubMed]

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003).
[Crossref] [PubMed]

2002 (4)

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. 16, 378–387 (2002).
[Crossref]

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia 2, 491–495 (2002).
[Crossref]

M. Jiang and G. Wang, “Development of iterative algorithms for image reconstruction,” J. X-Ray Sci. Technol. 10, 77–86 (2002). Invited Review.

C. Contag and M. H. Bachmann, “Advances in bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4, 235–260 (2002).
[Crossref] [PubMed]

2001 (1)

B. W. Rice, M. D. Cable, and M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6, 432–440 (2001).
[Crossref] [PubMed]

2000 (1)

E. A. Marengo, A. J. Devaney, and R. W. Ziolkowski, “Inverse source problem and mimnimum-energy sources,” J. Opt. Soc. Am., A 17, 34–45 (2000).
[Crossref]

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999).
[Crossref]

1998 (1)

A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative leat-squares and regularization,” IEEE Transactions on Signal Processing 46, 2345–2352 (1998).
[Crossref]

1997 (1)

M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Problems 13, 441–463 (1997).
[Crossref]

1996 (1)

R. J. Santos, “Equivalence of regularization and truncated iteration for general ill-posed problems,” Linear Algebra and Its applications 236, 25–33 (1996).
[Crossref]

1992 (2)

B. Eicke, “Iteration methods for convexly constrained ill-posed problems in Hilbert space,” Numerical Functional Analysis and Optimization 13, 413–429 (1992).
[Crossref]

D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Transactions on Signal Processing 40, 1143–1150 (1992).
[Crossref]

1982 (1)

L. A. Shepp and Y. Vardi, “Maximum likelihood restoration for emission tomography,” IEEE Transactions on Medical Imaging 1, 113–122 (1982).
[Crossref] [PubMed]

1977 (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximal likelihood form incomplete data via the EM algorithm,” Journal of the Royal Statistical Society. Series B. 39, 1–38 (1977).

Alexandrakis, G.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
[Crossref] [PubMed]

Anikonov, D. S.

D. S. Anikonov, A. E. Kovtanyuk, and I. V. Prokhorov, Transport equation and tomography, Inverse and Ill-posed Problems Series (VSP, Utrecht, 2002).

Antich, P. P.

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33, 61–68 (2006).
[Crossref] [PubMed]

Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, “Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden,” Molecular Imaging 3, 117–124 (2004).
[Crossref] [PubMed]

Arridge, S. R.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[Crossref] [PubMed]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999).
[Crossref]

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems (W. H. Winston, Washington, D. C., 1977).

Bachmann, M. H.

C. Contag and M. H. Bachmann, “Advances in bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4, 235–260 (2002).
[Crossref] [PubMed]

Bading, J. R.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005).
[Crossref] [PubMed]

Bertero, M.

M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Problems 13, 441–463 (1997).
[Crossref]

M. Bertero and P. Boccacci, Inverse Problems in Imaging (Institute of Physical Publishing, Bristol and Philadelphia, 1998).
[Crossref]

Boccacci, P.

M. Bertero and P. Boccacci, Inverse Problems in Imaging (Institute of Physical Publishing, Bristol and Philadelphia, 1998).
[Crossref]

Bollinger, R. A.

Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, “Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden,” Molecular Imaging 3, 117–124 (2004).
[Crossref] [PubMed]

Braasch, D. A.

Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, “Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden,” Molecular Imaging 3, 117–124 (2004).
[Crossref] [PubMed]

Brenner, S. C.

S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Texts in applied mathematics; 15 (Springer-Verlag, New York, NY, 2002), 2nd ed.

Buhmann, M. D.

M. D. Buhmann, Radial basis functions: theory and implementations, vol. 12 of Cambridge Monographs on Applied and Computational Mathematics (Cambridge University Press, Cambridge, 2003).
[Crossref]

Cable, M. D.

B. W. Rice, M. D. Cable, and M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6, 432–440 (2001).
[Crossref] [PubMed]

Cardozo, S. J.

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia 2, 491–495 (2002).
[Crossref]

Chatziioannou, A. F.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
[Crossref] [PubMed]

Chaudhari, A. J.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005).
[Crossref] [PubMed]

Chen, I. Y.

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003).
[Crossref] [PubMed]

Cheng, J. T.

M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).

Cherry, S. R.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005).
[Crossref] [PubMed]

Colton, D. L.

D. L. Colton and R. Kress, Inverse acoustic and elctromagnetic scattering theory (Springer, Berlin; New York, 1998), 2nd ed.

Cong, A.

Cong, W.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005).
[Crossref] [PubMed]

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Academic Radiology 11, 1029–1038 (2004).
[Crossref] [PubMed]

M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).

Contag, C.

C. Contag and M. H. Bachmann, “Advances in bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4, 235–260 (2002).
[Crossref] [PubMed]

Contag, C. H.

A. McCaffrey, M. A. Kay, and C. H. Contag, “Advancing molecular therapies through in vivo bioluminescent imaging,” Molecuar Imaging 2, 75–86 (2003).
[Crossref]

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia 2, 491–495 (2002).
[Crossref]

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. 16, 378–387 (2002).
[Crossref]

Conti, P. S.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005).
[Crossref] [PubMed]

Coquoz, O.

C. Kuo, O. Coquoz, T. Troy, N. Zhang, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” in “SMI Fourth Conference,” (Cologne, Germany, 2005).

Corey, D. R.

Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, “Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden,” Molecular Imaging 3, 117–124 (2004).
[Crossref] [PubMed]

Darvas, F.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005).
[Crossref] [PubMed]

Dautray, R.

R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. I (Springer-Verlag, Berlin, 1990).

Davis, S.

H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters 31, 365–367 (2005).
[Crossref]

De, A.

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003).
[Crossref] [PubMed]

Dehghani, H.

H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters 31, 365–367 (2005).
[Crossref]

Dempster, A. P.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximal likelihood form incomplete data via the EM algorithm,” Journal of the Royal Statistical Society. Series B. 39, 1–38 (1977).

Devaney, A. J.

E. A. Marengo, A. J. Devaney, and R. W. Ziolkowski, “Inverse source problem and mimnimum-energy sources,” J. Opt. Soc. Am., A 17, 34–45 (2000).
[Crossref]

Durairaj, K.

M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).

Eicke, B.

B. Eicke, “Iteration methods for convexly constrained ill-posed problems in Hilbert space,” Numerical Functional Analysis and Optimization 13, 413–429 (1992).
[Crossref]

B. Eicke, “Konvex-resringierte schlechtgestellte Problems und ihr Regularisierung durch Iterationsverfahren,” Thesis, Technischen Universität Berlin (1991).

Fishbein, M. C.

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003).
[Crossref] [PubMed]

Gambhir, S. S.

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003).
[Crossref] [PubMed]

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[Crossref] [PubMed]

Gilbarg, D.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin-Heideberg-New York, 1983).

Gu, X. J.

Gupta, S.

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia 2, 491–495 (2002).
[Crossref]

Hall, D. E.

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia 2, 491–495 (2002).
[Crossref]

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[Crossref] [PubMed]

Hielscher, A. H.

A. D. Klose and A. H. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Problems 19, 387–409 (2003).
[Crossref]

Hoffman, E. A.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005).
[Crossref] [PubMed]

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Academic Radiology 11, 1029–1038 (2004).
[Crossref] [PubMed]

G. Wang, E. A. Hoffman, and G. McLennan, “Bioluminescent CT method and apparatus,” (2003). US provisional patent application.

Isakov, V.

V. Isakov, Inverse Problems for Partial Differential Equations, vol. 127 of Applied Mathematical Series (Springer, New York-Berlin-Heideberg, 1998).

Ishimaru, A.

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

Jiang, H. B.

Jiang, M.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005).
[Crossref] [PubMed]

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004).
[Crossref] [PubMed]

M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Transactions on Medical Imaging 22, 569–579 (2003).
[Crossref] [PubMed]

M. Jiang and G. Wang, “Development of iterative algorithms for image reconstruction,” J. X-Ray Sci. Technol. 10, 77–86 (2002). Invited Review.

M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of SPIE: Developments in X-Ray Tomography IV,”, vol. 5535 (2004), vol. 5535, pp. 335–351. Invited talk.

M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2004).

M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).

Jiang, S. D.

H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters 31, 365–367 (2005).
[Crossref]

Kak, A.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1987).

Kay, M. A.

A. McCaffrey, M. A. Kay, and C. H. Contag, “Advancing molecular therapies through in vivo bioluminescent imaging,” Molecuar Imaging 2, 75–86 (2003).
[Crossref]

Klose, A. D.

A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescent sources,” J. Opt. Soc. Am., A 24, 1601–1608 (2007).
[Crossref]

A. D. Klose and A. H. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Problems 19, 387–409 (2003).
[Crossref]

Kovtanyuk, A. E.

D. S. Anikonov, A. E. Kovtanyuk, and I. V. Prokhorov, Transport equation and tomography, Inverse and Ill-posed Problems Series (VSP, Utrecht, 2002).

Kress, R.

D. L. Colton and R. Kress, Inverse acoustic and elctromagnetic scattering theory (Springer, Berlin; New York, 1998), 2nd ed.

Kumar, D.

Kuo, C.

C. Kuo, O. Coquoz, T. Troy, N. Zhang, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” in “SMI Fourth Conference,” (Cologne, Germany, 2005).

Laird, N. M.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximal likelihood form incomplete data via the EM algorithm,” Journal of the Royal Statistical Society. Series B. 39, 1–38 (1977).

Larcom, L.

Leahy, R. M.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005).
[Crossref] [PubMed]

Lewis, M. A.

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33, 61–68 (2006).
[Crossref] [PubMed]

Li, C. Q.

Li, H.

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Academic Radiology 11, 1029–1038 (2004).
[Crossref] [PubMed]

Li, Y.

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004).
[Crossref] [PubMed]

Lions, J. L.

R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. I (Springer-Verlag, Berlin, 1990).

Liu, Y.

Marengo, E. A.

E. A. Marengo, A. J. Devaney, and R. W. Ziolkowski, “Inverse source problem and mimnimum-energy sources,” J. Opt. Soc. Am., A 17, 34–45 (2000).
[Crossref]

Mason, R. P.

Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, “Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden,” Molecular Imaging 3, 117–124 (2004).
[Crossref] [PubMed]

McCaffrey, A.

A. McCaffrey, M. A. Kay, and C. H. Contag, “Advancing molecular therapies through in vivo bioluminescent imaging,” Molecuar Imaging 2, 75–86 (2003).
[Crossref]

McCray, P. B.

McLennan, G.

Min, J. J.

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003).
[Crossref] [PubMed]

Moats, R. A.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005).
[Crossref] [PubMed]

Natterer, F.

F. Natterer and F. Wuübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001).
[Crossref]

F. Natterer, The Mathematics of Computerized Tomography (SIAM, Philadelphia, PA, 2001).
[Crossref]

Nelson, M. B.

B. W. Rice, M. D. Cable, and M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6, 432–440 (2001).
[Crossref] [PubMed]

Ntziachristos, V.

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

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

O’Sullivan, J. A.

D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Transactions on Signal Processing 40, 1143–1150 (1992).
[Crossref]

Paroo, Z.

Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, “Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden,” Molecular Imaging 3, 117–124 (2004).
[Crossref] [PubMed]

Patterson, M.

H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters 31, 365–367 (2005).
[Crossref]

Paulsen, K.

H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters 31, 365–367 (2005).
[Crossref]

Piana, M.

M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Problems 13, 441–463 (1997).
[Crossref]

Pogue, B.

H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters 31, 365–367 (2005).
[Crossref]

Potter, L. C.

A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative leat-squares and regularization,” IEEE Transactions on Signal Processing 46, 2345–2352 (1998).
[Crossref]

Prokhorov, I. V.

D. S. Anikonov, A. E. Kovtanyuk, and I. V. Prokhorov, Transport equation and tomography, Inverse and Ill-posed Problems Series (VSP, Utrecht, 2002).

Protter, M. H.

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Prentice-Hall, Englewood Cliffs, N. J., 1967).

Qiao, J. H.

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003).
[Crossref] [PubMed]

Rainov, N. G.

A. Soling and N. G. Rainov, “Bioluminescence imaging in vivo-application to cancer research,” Expert Opinion on Biological Therapy 3, 1163–1172 (2003).
[PubMed]

Rannou, F. R.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
[Crossref] [PubMed]

Rehemtulla, A.

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia 2, 491–495 (2002).
[Crossref]

Rice, B.

C. Kuo, O. Coquoz, T. Troy, N. Zhang, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” in “SMI Fourth Conference,” (Cologne, Germany, 2005).

Rice, B. W.

B. W. Rice, M. D. Cable, and M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6, 432–440 (2001).
[Crossref] [PubMed]

Richer, E.

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33, 61–68 (2006).
[Crossref] [PubMed]

Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, “Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden,” Molecular Imaging 3, 117–124 (2004).
[Crossref] [PubMed]

Ripoll, J.

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

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

Ross, B. D.

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia 2, 491–495 (2002).
[Crossref]

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. 16, 378–387 (2002).
[Crossref]

Rubin, D. B.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximal likelihood form incomplete data via the EM algorithm,” Journal of the Royal Statistical Society. Series B. 39, 1–38 (1977).

Rudin, W.

W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1991), 2nd ed.

Sabharwal, A.

A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative leat-squares and regularization,” IEEE Transactions on Signal Processing 46, 2345–2352 (1998).
[Crossref]

Santos, R. J.

R. J. Santos, “Equivalence of regularization and truncated iteration for general ill-posed problems,” Linear Algebra and Its applications 236, 25–33 (1996).
[Crossref]

Schulz, R. B.

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

Schulz, T. J.

D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Transactions on Signal Processing 40, 1143–1150 (1992).
[Crossref]

Scott, L. R.

S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Texts in applied mathematics; 15 (Springer-Verlag, New York, NY, 2002), 2nd ed.

Shepp, L. A.

L. A. Shepp and Y. Vardi, “Maximum likelihood restoration for emission tomography,” IEEE Transactions on Medical Imaging 1, 113–122 (1982).
[Crossref] [PubMed]

Slaney, M.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1987).

Slavine, N. V.

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33, 61–68 (2006).
[Crossref] [PubMed]

Smith, D. J.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005).
[Crossref] [PubMed]

Snyder, D. L.

D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Transactions on Signal Processing 40, 1143–1150 (1992).
[Crossref]

Soling, A.

A. Soling and N. G. Rainov, “Bioluminescence imaging in vivo-application to cancer research,” Expert Opinion on Biological Therapy 3, 1163–1172 (2003).
[PubMed]

Stegman, L. D.

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia 2, 491–495 (2002).
[Crossref]

Sundaresan, G.

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003).
[Crossref] [PubMed]

Tian, J.

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Academic Radiology 11, 1029–1038 (2004).
[Crossref] [PubMed]

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems (W. H. Winston, Washington, D. C., 1977).

Troy, T.

C. Kuo, O. Coquoz, T. Troy, N. Zhang, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” in “SMI Fourth Conference,” (Cologne, Germany, 2005).

Trudinger, N. S.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin-Heideberg-New York, 1983).

Vardi, Y.

L. A. Shepp and Y. Vardi, “Maximum likelihood restoration for emission tomography,” IEEE Transactions on Medical Imaging 1, 113–122 (1982).
[Crossref] [PubMed]

Wang, G.

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, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005).
[Crossref] [PubMed]

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Academic Radiology 11, 1029–1038 (2004).
[Crossref] [PubMed]

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004).
[Crossref] [PubMed]

M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Transactions on Medical Imaging 22, 569–579 (2003).
[Crossref] [PubMed]

M. Jiang and G. Wang, “Development of iterative algorithms for image reconstruction,” J. X-Ray Sci. Technol. 10, 77–86 (2002). Invited Review.

A. Cong and G. Wang, “Multispectral bioluminescence tomography: Methodology and simulation,” International Journal of Biomedical Imaging 2006 (2006). Article ID 57614. doi:10.1155/IJBI/2006/57614.

M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of SPIE: Developments in X-Ray Tomography IV,”, vol. 5535 (2004), vol. 5535, pp. 335–351. Invited talk.

M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2004).

G. Wang, E. A. Hoffman, and G. McLennan, “Bioluminescent CT method and apparatus,” (2003). US provisional patent application.

G. Wang et al, “Development of the first bioluminescent tomography system,” Radiology Suppl. (Proceedings of the RSNA) 229(P) (2003).

M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).

Wang, L. H. V.

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

Wang, L. V.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13, 6756–6771 (2005).
[Crossref] [PubMed]

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Academic Radiology 11, 1029–1038 (2004).
[Crossref] [PubMed]

Weinberger, H. F.

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Prentice-Hall, Englewood Cliffs, N. J., 1967).

Weissleder, R.

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

Wu, J. C.

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003).
[Crossref] [PubMed]

Wuübbeling, F.

F. Natterer and F. Wuübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001).
[Crossref]

Zabner, J.

Zhang, N.

C. Kuo, O. Coquoz, T. Troy, N. Zhang, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” in “SMI Fourth Conference,” (Cologne, Germany, 2005).

Zhang, Q. H.

Zhou, T.

M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).

Zhu, F.

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Academic Radiology 11, 1029–1038 (2004).
[Crossref] [PubMed]

Ziolkowski, R. W.

E. A. Marengo, A. J. Devaney, and R. W. Ziolkowski, “Inverse source problem and mimnimum-energy sources,” J. Opt. Soc. Am., A 17, 34–45 (2000).
[Crossref]

Zwarg, D.

C. Kuo, O. Coquoz, T. Troy, N. Zhang, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” in “SMI Fourth Conference,” (Cologne, Germany, 2005).

Academic Radiology (1)

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method,” Academic Radiology 11, 1029–1038 (2004).
[Crossref] [PubMed]

Annu. Rev. Biomed. Eng. (1)

C. Contag and M. H. Bachmann, “Advances in bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. 4, 235–260 (2002).
[Crossref] [PubMed]

Circulation (1)

J. C. Wu, I. Y. Chen, G. Sundaresan, J. J. Min, A. De, J. H. Qiao, M. C. Fishbein, and S. S. Gambhir, “Molecular imaging of cardiac cell transplantation in living animals using optical bioluminescence and positron emission tomography,” Circulation 108, 1302–1305 (2003).
[Crossref] [PubMed]

Expert Opinion on Biological Therapy (1)

A. Soling and N. G. Rainov, “Bioluminescence imaging in vivo-application to cancer research,” Expert Opinion on Biological Therapy 3, 1163–1172 (2003).
[PubMed]

IEEE Transactions on Medical Imaging (3)

L. A. Shepp and Y. Vardi, “Maximum likelihood restoration for emission tomography,” IEEE Transactions on Medical Imaging 1, 113–122 (1982).
[Crossref] [PubMed]

M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Transactions on Medical Imaging 22, 569–579 (2003).
[Crossref] [PubMed]

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

IEEE Transactions on Signal Processing (2)

D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Transactions on Signal Processing 40, 1143–1150 (1992).
[Crossref]

A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative leat-squares and regularization,” IEEE Transactions on Signal Processing 46, 2345–2352 (1998).
[Crossref]

Inverse Problems (3)

M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Problems 13, 441–463 (1997).
[Crossref]

A. D. Klose and A. H. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Problems 19, 387–409 (2003).
[Crossref]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93 (1999).
[Crossref]

J. Biomed. Opt. (1)

B. W. Rice, M. D. Cable, and M. B. Nelson, “In vivo imaging of light-emitting probes,” J. Biomed. Opt. 6, 432–440 (2001).
[Crossref] [PubMed]

J. Magn. Reson. (1)

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. 16, 378–387 (2002).
[Crossref]

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

A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescent sources,” J. Opt. Soc. Am., A 24, 1601–1608 (2007).
[Crossref]

E. A. Marengo, A. J. Devaney, and R. W. Ziolkowski, “Inverse source problem and mimnimum-energy sources,” J. Opt. Soc. Am., A 17, 34–45 (2000).
[Crossref]

J. X-Ray Sci. Technol. (1)

M. Jiang and G. Wang, “Development of iterative algorithms for image reconstruction,” J. X-Ray Sci. Technol. 10, 77–86 (2002). Invited Review.

Journal of the Royal Statistical Society. Series B. (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximal likelihood form incomplete data via the EM algorithm,” Journal of the Royal Statistical Society. Series B. 39, 1–38 (1977).

Linear Algebra and Its applications (1)

R. J. Santos, “Equivalence of regularization and truncated iteration for general ill-posed problems,” Linear Algebra and Its applications 236, 25–33 (1996).
[Crossref]

Med. Phys. (2)

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33, 61–68 (2006).
[Crossref] [PubMed]

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems for bioluminescent tomography,” Med. Phys. 31, 2289–2299 (2004).
[Crossref] [PubMed]

Molecuar Imaging (1)

A. McCaffrey, M. A. Kay, and C. H. Contag, “Advancing molecular therapies through in vivo bioluminescent imaging,” Molecuar Imaging 2, 75–86 (2003).
[Crossref]

Molecular Imaging (1)

Z. Paroo, R. A. Bollinger, D. A. Braasch, E. Richer, D. R. Corey, P. P. Antich, and R. P. Mason, “Validating bioluminescence imaging as a high-throughput, quantitative modality for assessing tumor burden,” Molecular Imaging 3, 117–124 (2004).
[Crossref] [PubMed]

Nat. Biotech. (1)

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

Neoplasia (1)

A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia 2, 491–495 (2002).
[Crossref]

Numerical Functional Analysis and Optimization (1)

B. Eicke, “Iteration methods for convexly constrained ill-posed problems in Hilbert space,” Numerical Functional Analysis and Optimization 13, 413–429 (1992).
[Crossref]

Opt. Express (4)

Optics Letters (1)

H. Dehghani, S. Davis, S. D. Jiang, B. Pogue, K. Paulsen, and M. Patterson, “Spectrally resolved bioluminescence optical tomography,” Optics Letters 31, 365–367 (2005).
[Crossref]

Phys. Med. Biol. (3)

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50, 5421–5441 (2005).
[Crossref] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[Crossref] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
[Crossref] [PubMed]

Other (23)

M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of SPIE: Developments in X-Ray Tomography IV,”, vol. 5535 (2004), vol. 5535, pp. 335–351. Invited talk.

M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2004).

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1987).

F. Natterer, The Mathematics of Computerized Tomography (SIAM, Philadelphia, PA, 2001).
[Crossref]

G. Wang, E. A. Hoffman, and G. McLennan, “Bioluminescent CT method and apparatus,” (2003). US provisional patent application.

G. Wang et al, “Development of the first bioluminescent tomography system,” Radiology Suppl. (Proceedings of the RSNA) 229(P) (2003).

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

D. S. Anikonov, A. E. Kovtanyuk, and I. V. Prokhorov, Transport equation and tomography, Inverse and Ill-posed Problems Series (VSP, Utrecht, 2002).

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin-Heideberg-New York, 1983).

R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. I (Springer-Verlag, Berlin, 1990).

V. Isakov, Inverse Problems for Partial Differential Equations, vol. 127 of Applied Mathematical Series (Springer, New York-Berlin-Heideberg, 1998).

W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1991), 2nd ed.

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Prentice-Hall, Englewood Cliffs, N. J., 1967).

B. Eicke, “Konvex-resringierte schlechtgestellte Problems und ihr Regularisierung durch Iterationsverfahren,” Thesis, Technischen Universität Berlin (1991).

A. Cong and G. Wang, “Multispectral bioluminescence tomography: Methodology and simulation,” International Journal of Biomedical Imaging 2006 (2006). Article ID 57614. doi:10.1155/IJBI/2006/57614.

M. Jiang, T. Zhou, J. T. Cheng, W. Cong, K. Durairaj, and G. Wang, “Image reconstruction for bioluminescence tomography,” in “Proceedings of the RSNA,” (2005).

F. Natterer and F. Wuübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, PA, 2001).
[Crossref]

S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Texts in applied mathematics; 15 (Springer-Verlag, New York, NY, 2002), 2nd ed.

D. L. Colton and R. Kress, Inverse acoustic and elctromagnetic scattering theory (Springer, Berlin; New York, 1998), 2nd ed.

M. D. Buhmann, Radial basis functions: theory and implementations, vol. 12 of Cambridge Monographs on Applied and Computational Mathematics (Cambridge University Press, Cambridge, 2003).
[Crossref]

C. Kuo, O. Coquoz, T. Troy, N. Zhang, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” in “SMI Fourth Conference,” (Cologne, Germany, 2005).

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems (W. H. Winston, Washington, D. C., 1977).

M. Bertero and P. Boccacci, Inverse Problems in Imaging (Institute of Physical Publishing, Bristol and Philadelphia, 1998).
[Crossref]

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Figures (4)

Fig. 1.
Fig. 1.

(a) A heterogeneous mouse phantom consisting of bone (B), heart (H), lungs (L), and muscle (M). (b) A cross-section through two luminescent sources in the left lung and another source in the right lung. The four arrows show the directions of the CCD camera for data measurement.

Fig. 2.
Fig. 2.

Reconstructed results by the CL method and a cross-section at z=0cm. (a) and (b) are results from data measured at the four views. (c) and (d) are from data measured at the front view only.

Fig. 3.
Fig. 3.

(a) A cross-section through two hollow cylinders for hosting luminescent sources in the left lung. The four arrows show the direction of the CCD camera during data acquisition. (b) The measurements at the four views combined along the phantom side surface with unit µW/cm2

Fig. 4.
Fig. 4.

Representative results reconstructed by the EM algorithm. (a) and (c) are the sources reconstructed by the EMalgorithm from the data measured in the four views and in the front view only, respectively. (b) and (d) are cross-sections at z=0 cm of the sources in (a) and (c), respectively.

Tables (8)

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Table 1. Optical parameters for the phantom

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Table 2. Finite element information for the simulation

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Table 3. Quantitative results for the reconstructed locations of the three sources at S1=(-0.90, 0.25, 0), S2=(-0.90, -0.25, 0) and S3=(0.90, 0.25, 0), respectively. The unit is cm.

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Table 4. Quantitative results for the reconstructed source integrals of the sources. The sources are listed in the order as in Table 3. Their true values are 105.1, 97.4 and 105.1, respectively. The unit is nW.

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Table 5. Quantitative results for the reconstructed source moments of the sources. The sources are listed in the order as in Table 3. Their true values are 125.5, 116.5 and 125.5, respectively. The unit is nW.

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Table 6. Quantitative results for the reconstructed locations of the two sources at S1=(-0.90,0.15,0) and S2=(-0.90,-0.15,0), respectively. The unit is cm.

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Table 7. Quantitative results for the reconstructed source integrals of the sources. The sources are listed in the order as in Table 6. Their true values are 105.1 and 97.4, respectively. The unit is nW.

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Table 8. Finite element information for the physical phantom experiment

Equations (71)

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1 c u t ( x , θ , t ) + θ · x u ( x , θ , t ) + μ ( x ) u ( x , θ , t ) = μ s ( x ) S 2 η ( θ · θ ) u ( x , θ , t ) d θ + q ( x , θ , t )
u ( x , θ , 0 ) = 0 , x Ω , θ S 2 ,
u ( x , θ , t ) = g ( x , θ , t ) , x Γ , θ S 2 , ν ( x ) · θ < 0 , t > 0 ,
g ( x , t ) = S 2 ν ( x ) · θ u ( x , θ , t ) d θ , x = Γ , t 0 .
u 0 ( x , t ) = 1 4 π S 2 u ( x , θ , t ) d θ .
μ s = ( 1 η ¯ ) μ s ,
D ( x ) = 1 3 ( μ a ( x ) + μ s ( x ) ) ,
q 0 ( x , t ) = 1 4 π S 2 q ( x , θ , t ) d θ .
1 c u 0 ( x , t ) t · ( D ( x ) u 0 ( x , t ) ) + μ a ( x ) u 0 ( x , t ) = q 0 ( x , t ) , x Ω , t > 0
u 0 ( x , 0 ) = 0 , x Ω ,
u 0 ( x , t ) + 2 D ( x ) u 0 ν ( x , t ) = g ( x , t ) , x Γ , t > 0 .
g ( x , t ) = D ( x ) u 0 ν ( x , t ) .
· ( D ( x ) u 0 ( x ) ) + μ a ( x ) u 0 ( x ) = q 0 ( x ) , x Ω ,
u 0 ( x ) + 2 D ( x ) u 0 ν ( x ) = g ( x ) , x Γ .
BLT ( P ) { · ( D u 0 ) + μ a u 0 = q 0 , in Ω , u 0 + 2 D u 0 ν = g , on Γ , D u 0 ν = g , on Γ P .
γ 0 [ u ] = u Γ , and γ 1 [ u ] = D u ν Γ
L [ u ] = · ( D u ) + μ a u .
L [ w 1 ] = 0 , in Ω ,
γ 0 [ w 1 ] = f , on Γ P
γ 0 [ w 1 ] + 2 γ 1 [ w 1 ] = g , on Γ Γ P .
N Γ P [ f ] = γ 1 [ w 1 ] Γ P .
L [ w 2 ] = q 0 , in Ω ,
γ 0 [ w 2 ] = 0 , on Γ P ,
γ 0 [ w 2 ] + 2 γ 1 [ w 2 ] = 0 , on Γ Γ P .
Λ Γ P [ q 0 ] = γ 1 [ w 2 ] Γ P .
L [ u ] = q 0 , in Ω ,
γ 0 [ u ] = g + 2 g , on Γ P ,
γ 0 [ u ] + 2 γ 1 [ u ] = g , on Γ Γ P .
L [ v ] = q 0 , in Ω .
γ 0 [ v ] = g + 2 g , on Γ P ,
γ 0 [ v ] + 2 γ 1 [ v ] = g , on Γ Γ P .
g = γ 1 [ u ] = γ 1 [ w 1 ] + γ 1 [ w 2 ] = N Γ P [ g + 2 g ] Λ Γ P [ q 0 ] , on Γ P .
Λ Γ P [ q 0 ] = N Γ P [ g + 2 g ] + g , on Γ P .
Ω [ v · L [ w ] w · L [ v ] ] dx = Γ [ v γ 1 [ w ] w γ 1 [ v ] ] d Γ .
ϕ = T Γ P [ ψ ] ,
L [ ϕ ] = 0 , in Ω ,
γ 0 [ ϕ ] = ψ , on Γ P ,
γ 0 [ ϕ ] + 2 γ 1 [ ϕ ] = 0 , on Γ Γ P .
q 0 , T Γ P [ ψ ] L 2 ( Ω ) = Λ Γ P [ q 0 ] , ψ L 2 ( Γ P ) ,
Λ Γ P * = T Γ P .
𝒩 [ Λ Γ P ] = 𝓡 [ Λ Γ P * ] = 𝓡 [ T Γ P ] .
H 0 , Γ P 2 ( Ω ) = { p H 2 ( Ω ) : γ 0 [ p ] Γ P = 0 , γ 1 [ p ] Γ P = 0 , and γ 0 [ p ] + 2 γ 1 [ p ] Γ Γ P = 0 } .
𝓡 [ T Γ P ] = L [ H 0 , Γ P 2 ( Ω ) ] .
q , v L 2 ( Ω ) = Ω v L [ p ] d x = Γ [ v γ 1 [ p ] p γ 1 [ v ] ] d Γ + Ω L [ v ] p d x = 0 ,
L [ w 2 ] = q , in Ω γ 0 [ w 2 ] = 0 , on Γ P γ 0 [ w 2 ] + 2 γ 1 [ w 2 ] = 0 , on Γ \ Γ P , γ 1 [ w 2 ] = 0 , on Γ P .
N Γ P [ g + 2 g ] + g H 1 2 ( Γ P ) ,
q 0 ( y ) = s = 1 S a s δ ( y y s ) .
q 0 ( y ) = s = 1 S g s ( y x s ) χ B r 0 s , r 1 s ( x s )
φ ( r ) = 1 ,
φ ( r ) = sinh ( μ a D r ) μ a D r .
r 0 s r 1 s r N 1 φ C ( s ) ( r ) g s ( r ) dr = R 0 τ ( s ) R 1 τ ( s ) r N 1 φ C ( s ) ( r ) G τ ( s ) ( r ) dr , for s = 1 , , S ,
Λ Γ P [ q 0 ] = b .
F [ q 0 ] = Γ P { b log Λ Γ P [ q 0 ] Λ Γ P [ q 0 ] } d Γ ,
arg max F q 0 0 [ q 0 ] .
f ( t ) = F [ q 0 + tv ] , for t around 0 ,
F [ q 0 ] = Λ Γ P * [ b Λ Γ P [ q 0 ] 1 ] L 2 ( Ω ) .
q 0 · Λ Γ P * [ b Λ Γ P [ q 0 ] 1 ] = 0 .
L [ ϕ 1 ] = 0 , in Ω ,
γ 0 [ ϕ 1 ] = 1 , on Γ P ,
γ 0 [ ϕ 1 ] + 2 γ 1 [ ϕ 1 ] = 0 , on Γ Γ P .
q 0 = 1 ϕ 1 q 0 · T Γ P [ b Λ Γ P [ q 0 ] ] .
q 0 ( n + 1 ) = 1 ϕ 1 q 0 ( n ) · T Γ P [ b Λ Γ P [ q 0 ( n ) ] ] .
𝓒 = { q 0 : q 0 satisfies some convex constraints . } ,
q 0 ( n + 1 ) = P 𝓒 { q 0 ( n ) + λ n Λ Γ P * [ b Λ Γ P [ q 0 ( n ) ] ] } ,
arg min q 0 𝓒 1 2 b Λ Γ P [ q 0 ] L 2 ( Γ ) 2 .
Ω [ μ a u q 0 ] dx = Γ g d Γ .
Ω q 0 ( 0 ) dx = Γ g d Γ + Ω μ a u dx .
Ω q 0 ( 0 ) dx Γ g d Γ + Ω μ a w 1 dx .
q 0 ( 0 ) = Q 0 χ Ω 0
Q 0 Ω 0 Γ P g d Γ + Ω μ a w 1 dx ,
q i ( x ) = A i χ Ω i ( x ) , Ω = { x x x 0 < r } ,

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