Abstract

A method is presented for fluorescence optical diffusion tomography in turbid media using multiple-frequency data. The method uses a frequency-domain diffusion equation model to reconstruct the fluorescent yield and lifetime by means of a Bayesian framework and an efficient, nonlinear optimizer. The method is demonstrated by using simulations and laboratory experiments to show that reconstruction quality can be improved in certain problems through the use of more than one frequency. A broadly applicable mutual information performance metric is also presented and used to investigate the advantages of using multiple modulation frequencies compared with using only one.

© 2004 Optical Society of America

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  1. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  2. D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
    [CrossRef]
  3. A. Pèlegrin, S. Folli, F. Buchegger, J. Mach, G. Wagnières, H. van den Bergh, “Antibody-fluorescein conjugates for photoimmunodiagnosis of human colon carcinoma in nude mice,” Cancer 67, 2529–2537 (1991).
    [CrossRef] [PubMed]
  4. B. Ballou, G. W. Fisher, T. R. Hakala, D. L. Farkas, “Tumor detection and visualization using cyanine fluorochrome-labeled antibodies,” Biotechnol. Prog. 13, 649–658 (1997).
    [CrossRef] [PubMed]
  5. R. Cubeddu, G. Canti, A. Pifferi, P. Taroni, G. Valentini, “Fluorescence lifetime imaging of experimental tumors in hematoporphyrin derivative-sensitized mice,” Photochem. Photobiol. 66, 229–236 (1997).
    [CrossRef] [PubMed]
  6. J. A. Reddy, P. S. Low, “Folate-mediated targeting of therapeutic and imaging agents to cancers,” Crit. Rev. Ther. Drug Carrier Syst. 15, 587–627 (1998).
    [CrossRef]
  7. J. S. Reynolds, T. L. Troy, R. H. Mayer, A. B. Thompson, D. J. Waters, K. K. Cornell, P. W. Snyder, E. M. Sevick-Muraca, “Imaging of spontaneous canine mammary tumors using fluorescent contrast agents,” Photochem. Photobiol. 70, 87–94 (1999).
    [CrossRef] [PubMed]
  8. U. Mahmood, C. Tung, J. A. Bogdanov, R. Weissleder, “Near-infrared optical imaging of protease activity for tumor detection,” Radiology 213, 866–870 (1999).
    [CrossRef] [PubMed]
  9. K. Licha, B. Riefke, V. Ntziachristos, A. Becker, B. Chance, W. Semmler, “Hydrophilic cyanine dyes as contrast agents for near-infrared tumor imaging: synthesis, photophysical properties and spectroscopic in vivo characterization,” Photochem. Photobiol. 72, 392–398 (2000).
    [CrossRef] [PubMed]
  10. V. Ntziachristos, A. G. Yodh, M. Schnall, B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. USA 97, 2767–2772 (2000).
    [CrossRef] [PubMed]
  11. A. Becker, C. Hessenius, K. Licha, B. Ebert, U. Sukowski, W. Semmler, B. Wiedenmann, C. Grotzinger, “Receptor-targeted optical imaging of tumors with near-infrared fluorescent ligands,” Nat. Biotechnol. 19, 327–331 (2001).
    [CrossRef] [PubMed]
  12. J. E. Bugaj, S. Achilefu, R. B. Dorshow, R. Rajagopalan, “Novel fluorescent contrast agents for optical imaging of in vivo tumors based on a receptor-targeted dye-peptide conjugate platform,” J. Biomed. Opt. 6, 122–133 (2001).
    [CrossRef] [PubMed]
  13. M. S. Patterson, B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
    [CrossRef] [PubMed]
  14. E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
    [CrossRef] [PubMed]
  15. R. Roy, E. M. Sevick-Muraca, “Three-dimensional unconstrained and constrained image-reconstruction techniques applied to fluorescence, frequency-domain photon migration,” Appl. Opt. 40, 2206–2215 (2001).
    [CrossRef]
  16. M. A. O’Leary, D. A. Boas, X. D. Li, B. Chance, A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21, 158–160 (1996).
    [CrossRef] [PubMed]
  17. D. Paithankar, A. Chen, B. Pogue, M. Patterson, E. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. 36, 2260–2272 (1997).
    [CrossRef] [PubMed]
  18. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37, 5337–5343 (1998).
    [CrossRef]
  19. J. Chang, H. L. Graber, R. L. Barbour, “Luminescence optical tomography of dense scattering media,” J. Opt. Soc. Am. A 14, 288–299 (1997).
    [CrossRef]
  20. V. Ntziachristos, R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. 26, 893–895 (2001).
    [CrossRef]
  21. A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42, 3081–3094 (2003).
    [CrossRef] [PubMed]
  22. E. Shives, Y. Xu, H. Jiang, “Fluorescence lifetime tomography of turbid media based on an oxygen-sensitive dye,” Opt. Express 10, 1557–1562 (2002).
    [CrossRef] [PubMed]
  23. J. C. Ye, K. J. Webb, C. A. Bouman, R. P. Millane, “Optical diffusion tomography using iterative coordinate descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A 16, 2400–2412 (1999).
    [CrossRef]
  24. J. C. Ye, C. A. Bouman, K. J. Webb, R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909–922 (2001).
    [CrossRef]
  25. A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography with experimental data,” Opt. Lett. 27, 95–97 (2002).
    [CrossRef]
  26. S. Oh, A. B. Milstein, R. P. Millane, C. A. Bouman, K. J. Webb, “Source–detector calibration in three-dimensional Bayesian optical diffusion tomography,” J. Opt. Soc. Am. A 19, 1983–1993 (2002).
    [CrossRef]
  27. F. Gao, H. Zhao, Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 778–791 (2002).
    [CrossRef] [PubMed]
  28. V. Ntziachristos, J. Culver, M. Holboke, A. G. Yodh, B. Chance, “Optimal selection of frequencies for diffuse optical tomography,” in Biomedical Topical Meetings, Vol. 38 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 475–477.
  29. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
    [CrossRef]
  30. T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).
  31. R. F. Wagner, D. G. Brown, M. S. Pastel, “Application of information theory to the assessment of computed tomography,” Med. Phys. 6, 83–94 (1979).
    [CrossRef] [PubMed]
  32. M. Fuderer, “The information content of MR images,” IEEE Trans. Med. Imaging 7, 368–380 (1988).
    [CrossRef] [PubMed]
  33. L. Shao, A. O. Hero, W. L. Rogers, N. H. Clinthorne, “The mutual information criterion for SPECT aperture evaluation and design,” IEEE Trans. Med. Imaging 8, 322–336 (1989).
    [CrossRef] [PubMed]
  34. A. O. Hero, L. Shao, “Information analysis of single photon emission computed tomography with count losses,” IEEE Trans. Med. Imaging 9, 117–127 (1990).
    [CrossRef] [PubMed]
  35. J. P. Culver, V. Ntziachristos, M. J. Holbrooke, A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett. 26, 701–703 (2001).
    [CrossRef]
  36. H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, J. F. Dunn, “Near infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt. 8, 102–110 (2003).
    [CrossRef] [PubMed]
  37. J. Stott, D. A. Boas, “A practical comparison between time-domain and frequency-domain diffusive optical imaging systems,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 626–628.
  38. T. Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression (Prentice Hall, Englewood Cliffs, N.J., 1971).
  39. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  40. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).
  41. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.
  42. J. S. Reynolds, C. A. Thompson, K. J. Webb, F. P. LaPlant, D. Ben-Amotz, “Frequency domain modeling of reradiation in highly scattering media,” Appl. Opt. 36, 2252–2259 (1997).
    [CrossRef] [PubMed]
  43. C. A. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
    [CrossRef] [PubMed]
  44. K. Sauer, C. A. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process. 41, 534–548 (1993).
    [CrossRef]
  45. E. K. P. Chong, S. H. Zak, An Introduction to Optimization (Wiley, New York, 1996).
  46. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  47. S. S. Saquib, C. A. Bouman, K. Sauer, “ML parameter estimation for Markov random fields with applications to Bayesian tomography,” IEEE Trans. Image Process. 7, 1029–1044 (1998).
    [CrossRef]
  48. M. L. J. Landsman, G. Kwant, G. A. Mook, W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
    [PubMed]
  49. R. C. Benson, H. A. Kues, “Fluorescence properties of indocyanine green as related to angiography,” Phys. Med. Biol. 23, 159–163 (1978).
    [CrossRef] [PubMed]
  50. M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Philadelphia, Pa., 1998).
  51. A. D. Klose, A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer—Part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transf. 72, 715–732 (2002).
    [CrossRef]
  52. D. Boas, T. Gaudette, S. Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Express 8, 263–270 (2001).
    [CrossRef] [PubMed]
  53. J. J. Stott, J. P. Culver, S. R. Arridge, D. A. Boas, “Optode positional calibration in diffuse optical tomography,” Appl. Opt. 42, 3154–3162 (2003).
    [CrossRef] [PubMed]
  54. N. Iftimia, H. Jiang, “Quantitative optical image reconstructions of turbid media by use of direct-current measurements,” Appl. Opt. 39, 5256–5261 (2000).
    [CrossRef]

2003 (3)

2002 (5)

2001 (8)

V. Ntziachristos, 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. C. Ye, C. A. Bouman, K. J. Webb, R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909–922 (2001).
[CrossRef]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

A. Becker, C. Hessenius, K. Licha, B. Ebert, U. Sukowski, W. Semmler, B. Wiedenmann, C. Grotzinger, “Receptor-targeted optical imaging of tumors with near-infrared fluorescent ligands,” Nat. Biotechnol. 19, 327–331 (2001).
[CrossRef] [PubMed]

J. E. Bugaj, S. Achilefu, R. B. Dorshow, R. Rajagopalan, “Novel fluorescent contrast agents for optical imaging of in vivo tumors based on a receptor-targeted dye-peptide conjugate platform,” J. Biomed. Opt. 6, 122–133 (2001).
[CrossRef] [PubMed]

R. Roy, E. M. Sevick-Muraca, “Three-dimensional unconstrained and constrained image-reconstruction techniques applied to fluorescence, frequency-domain photon migration,” Appl. Opt. 40, 2206–2215 (2001).
[CrossRef]

D. Boas, T. Gaudette, S. Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Express 8, 263–270 (2001).
[CrossRef] [PubMed]

J. P. Culver, V. Ntziachristos, M. J. Holbrooke, A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett. 26, 701–703 (2001).
[CrossRef]

2000 (3)

N. Iftimia, H. Jiang, “Quantitative optical image reconstructions of turbid media by use of direct-current measurements,” Appl. Opt. 39, 5256–5261 (2000).
[CrossRef]

K. Licha, B. Riefke, V. Ntziachristos, A. Becker, B. Chance, W. Semmler, “Hydrophilic cyanine dyes as contrast agents for near-infrared tumor imaging: synthesis, photophysical properties and spectroscopic in vivo characterization,” Photochem. Photobiol. 72, 392–398 (2000).
[CrossRef] [PubMed]

V. Ntziachristos, A. G. Yodh, M. Schnall, B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. USA 97, 2767–2772 (2000).
[CrossRef] [PubMed]

1999 (4)

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

J. S. Reynolds, T. L. Troy, R. H. Mayer, A. B. Thompson, D. J. Waters, K. K. Cornell, P. W. Snyder, E. M. Sevick-Muraca, “Imaging of spontaneous canine mammary tumors using fluorescent contrast agents,” Photochem. Photobiol. 70, 87–94 (1999).
[CrossRef] [PubMed]

U. Mahmood, C. Tung, J. A. Bogdanov, R. Weissleder, “Near-infrared optical imaging of protease activity for tumor detection,” Radiology 213, 866–870 (1999).
[CrossRef] [PubMed]

J. C. Ye, K. J. Webb, C. A. Bouman, R. P. Millane, “Optical diffusion tomography using iterative coordinate descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A 16, 2400–2412 (1999).
[CrossRef]

1998 (3)

H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37, 5337–5343 (1998).
[CrossRef]

J. A. Reddy, P. S. Low, “Folate-mediated targeting of therapeutic and imaging agents to cancers,” Crit. Rev. Ther. Drug Carrier Syst. 15, 587–627 (1998).
[CrossRef]

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

1997 (6)

B. Ballou, G. W. Fisher, T. R. Hakala, D. L. Farkas, “Tumor detection and visualization using cyanine fluorochrome-labeled antibodies,” Biotechnol. Prog. 13, 649–658 (1997).
[CrossRef] [PubMed]

R. Cubeddu, G. Canti, A. Pifferi, P. Taroni, G. Valentini, “Fluorescence lifetime imaging of experimental tumors in hematoporphyrin derivative-sensitized mice,” Photochem. Photobiol. 66, 229–236 (1997).
[CrossRef] [PubMed]

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

D. Paithankar, A. Chen, B. Pogue, M. Patterson, E. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. 36, 2260–2272 (1997).
[CrossRef] [PubMed]

J. Chang, H. L. Graber, R. L. Barbour, “Luminescence optical tomography of dense scattering media,” J. Opt. Soc. Am. A 14, 288–299 (1997).
[CrossRef]

J. S. Reynolds, C. A. Thompson, K. J. Webb, F. P. LaPlant, D. Ben-Amotz, “Frequency domain modeling of reradiation in highly scattering media,” Appl. Opt. 36, 2252–2259 (1997).
[CrossRef] [PubMed]

1996 (1)

1994 (2)

1993 (2)

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

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

1991 (1)

A. Pèlegrin, S. Folli, F. Buchegger, J. Mach, G. Wagnières, H. van den Bergh, “Antibody-fluorescein conjugates for photoimmunodiagnosis of human colon carcinoma in nude mice,” Cancer 67, 2529–2537 (1991).
[CrossRef] [PubMed]

1990 (1)

A. O. Hero, L. Shao, “Information analysis of single photon emission computed tomography with count losses,” IEEE Trans. Med. Imaging 9, 117–127 (1990).
[CrossRef] [PubMed]

1989 (1)

L. Shao, A. O. Hero, W. L. Rogers, N. H. Clinthorne, “The mutual information criterion for SPECT aperture evaluation and design,” IEEE Trans. Med. Imaging 8, 322–336 (1989).
[CrossRef] [PubMed]

1988 (1)

M. Fuderer, “The information content of MR images,” IEEE Trans. Med. Imaging 7, 368–380 (1988).
[CrossRef] [PubMed]

1979 (1)

R. F. Wagner, D. G. Brown, M. S. Pastel, “Application of information theory to the assessment of computed tomography,” Med. Phys. 6, 83–94 (1979).
[CrossRef] [PubMed]

1978 (1)

R. C. Benson, H. A. Kues, “Fluorescence properties of indocyanine green as related to angiography,” Phys. Med. Biol. 23, 159–163 (1978).
[CrossRef] [PubMed]

1976 (1)

M. L. J. Landsman, G. Kwant, G. A. Mook, W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
[PubMed]

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

Achilefu, S.

J. E. Bugaj, S. Achilefu, R. B. Dorshow, R. Rajagopalan, “Novel fluorescent contrast agents for optical imaging of in vivo tumors based on a receptor-targeted dye-peptide conjugate platform,” J. Biomed. Opt. 6, 122–133 (2001).
[CrossRef] [PubMed]

Arridge, S.

Arridge, S. R.

Ballou, B.

B. Ballou, G. W. Fisher, T. R. Hakala, D. L. Farkas, “Tumor detection and visualization using cyanine fluorochrome-labeled antibodies,” Biotechnol. Prog. 13, 649–658 (1997).
[CrossRef] [PubMed]

Barbour, R. L.

Becker, A.

A. Becker, C. Hessenius, K. Licha, B. Ebert, U. Sukowski, W. Semmler, B. Wiedenmann, C. Grotzinger, “Receptor-targeted optical imaging of tumors with near-infrared fluorescent ligands,” Nat. Biotechnol. 19, 327–331 (2001).
[CrossRef] [PubMed]

K. Licha, B. Riefke, V. Ntziachristos, A. Becker, B. Chance, W. Semmler, “Hydrophilic cyanine dyes as contrast agents for near-infrared tumor imaging: synthesis, photophysical properties and spectroscopic in vivo characterization,” Photochem. Photobiol. 72, 392–398 (2000).
[CrossRef] [PubMed]

Ben-Amotz, D.

Benson, R. C.

R. C. Benson, H. A. Kues, “Fluorescence properties of indocyanine green as related to angiography,” Phys. Med. Biol. 23, 159–163 (1978).
[CrossRef] [PubMed]

Berger, T.

T. Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression (Prentice Hall, Englewood Cliffs, N.J., 1971).

Bertero, M.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Philadelphia, Pa., 1998).

Boas, D.

Boas, D. A.

J. J. Stott, J. P. Culver, S. R. Arridge, D. A. Boas, “Optode positional calibration in diffuse optical tomography,” Appl. Opt. 42, 3154–3162 (2003).
[CrossRef] [PubMed]

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42, 3081–3094 (2003).
[CrossRef] [PubMed]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

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

J. Stott, D. A. Boas, “A practical comparison between time-domain and frequency-domain diffusive optical imaging systems,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 626–628.

Boccacci, P.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Philadelphia, Pa., 1998).

Bogdanov, J. A.

U. Mahmood, C. Tung, J. A. Bogdanov, R. Weissleder, “Near-infrared optical imaging of protease activity for tumor detection,” Radiology 213, 866–870 (1999).
[CrossRef] [PubMed]

Bouman, C. A.

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42, 3081–3094 (2003).
[CrossRef] [PubMed]

A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography with experimental data,” Opt. Lett. 27, 95–97 (2002).
[CrossRef]

S. Oh, A. B. Milstein, R. P. Millane, C. A. Bouman, K. J. Webb, “Source–detector calibration in three-dimensional Bayesian optical diffusion tomography,” J. Opt. Soc. Am. A 19, 1983–1993 (2002).
[CrossRef]

J. C. Ye, C. A. Bouman, K. J. Webb, R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909–922 (2001).
[CrossRef]

J. C. Ye, K. J. Webb, C. A. Bouman, R. P. Millane, “Optical diffusion tomography using iterative coordinate descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A 16, 2400–2412 (1999).
[CrossRef]

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

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

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

Brooks, D. H.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Brown, D. G.

R. F. Wagner, D. G. Brown, M. S. Pastel, “Application of information theory to the assessment of computed tomography,” Med. Phys. 6, 83–94 (1979).
[CrossRef] [PubMed]

Buchegger, F.

A. Pèlegrin, S. Folli, F. Buchegger, J. Mach, G. Wagnières, H. van den Bergh, “Antibody-fluorescein conjugates for photoimmunodiagnosis of human colon carcinoma in nude mice,” Cancer 67, 2529–2537 (1991).
[CrossRef] [PubMed]

Bugaj, J. E.

J. E. Bugaj, S. Achilefu, R. B. Dorshow, R. Rajagopalan, “Novel fluorescent contrast agents for optical imaging of in vivo tumors based on a receptor-targeted dye-peptide conjugate platform,” J. Biomed. Opt. 6, 122–133 (2001).
[CrossRef] [PubMed]

Canti, G.

R. Cubeddu, G. Canti, A. Pifferi, P. Taroni, G. Valentini, “Fluorescence lifetime imaging of experimental tumors in hematoporphyrin derivative-sensitized mice,” Photochem. Photobiol. 66, 229–236 (1997).
[CrossRef] [PubMed]

Chance, B.

K. Licha, B. Riefke, V. Ntziachristos, A. Becker, B. Chance, W. Semmler, “Hydrophilic cyanine dyes as contrast agents for near-infrared tumor imaging: synthesis, photophysical properties and spectroscopic in vivo characterization,” Photochem. Photobiol. 72, 392–398 (2000).
[CrossRef] [PubMed]

V. Ntziachristos, A. G. Yodh, M. Schnall, B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. USA 97, 2767–2772 (2000).
[CrossRef] [PubMed]

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

V. Ntziachristos, J. Culver, M. Holboke, A. G. Yodh, B. Chance, “Optimal selection of frequencies for diffuse optical tomography,” in Biomedical Topical Meetings, Vol. 38 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 475–477.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Chang, J.

Chen, A.

Chong, E. K. P.

E. K. P. Chong, S. H. Zak, An Introduction to Optimization (Wiley, New York, 1996).

Clinthorne, N. H.

L. Shao, A. O. Hero, W. L. Rogers, N. H. Clinthorne, “The mutual information criterion for SPECT aperture evaluation and design,” IEEE Trans. Med. Imaging 8, 322–336 (1989).
[CrossRef] [PubMed]

Cornell, K. K.

J. S. Reynolds, T. L. Troy, R. H. Mayer, A. B. Thompson, D. J. Waters, K. K. Cornell, P. W. Snyder, E. M. Sevick-Muraca, “Imaging of spontaneous canine mammary tumors using fluorescent contrast agents,” Photochem. Photobiol. 70, 87–94 (1999).
[CrossRef] [PubMed]

Cover, T. M.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).

Cubeddu, R.

R. Cubeddu, G. Canti, A. Pifferi, P. Taroni, G. Valentini, “Fluorescence lifetime imaging of experimental tumors in hematoporphyrin derivative-sensitized mice,” Photochem. Photobiol. 66, 229–236 (1997).
[CrossRef] [PubMed]

Culver, J.

V. Ntziachristos, J. Culver, M. Holboke, A. G. Yodh, B. Chance, “Optimal selection of frequencies for diffuse optical tomography,” in Biomedical Topical Meetings, Vol. 38 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 475–477.

Culver, J. P.

Dehghani, H.

H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, J. F. Dunn, “Near infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt. 8, 102–110 (2003).
[CrossRef] [PubMed]

DiMarzio, C. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Dorshow, R. B.

J. E. Bugaj, S. Achilefu, R. B. Dorshow, R. Rajagopalan, “Novel fluorescent contrast agents for optical imaging of in vivo tumors based on a receptor-targeted dye-peptide conjugate platform,” J. Biomed. Opt. 6, 122–133 (2001).
[CrossRef] [PubMed]

Duderstadt, J. J.

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

Dunn, J. F.

H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, J. F. Dunn, “Near infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt. 8, 102–110 (2003).
[CrossRef] [PubMed]

Ebert, B.

A. Becker, C. Hessenius, K. Licha, B. Ebert, U. Sukowski, W. Semmler, B. Wiedenmann, C. Grotzinger, “Receptor-targeted optical imaging of tumors with near-infrared fluorescent ligands,” Nat. Biotechnol. 19, 327–331 (2001).
[CrossRef] [PubMed]

Farkas, D. L.

B. Ballou, G. W. Fisher, T. R. Hakala, D. L. Farkas, “Tumor detection and visualization using cyanine fluorochrome-labeled antibodies,” Biotechnol. Prog. 13, 649–658 (1997).
[CrossRef] [PubMed]

Feng, T.-C.

Fisher, G. W.

B. Ballou, G. W. Fisher, T. R. Hakala, D. L. Farkas, “Tumor detection and visualization using cyanine fluorochrome-labeled antibodies,” Biotechnol. Prog. 13, 649–658 (1997).
[CrossRef] [PubMed]

Folli, S.

A. Pèlegrin, S. Folli, F. Buchegger, J. Mach, G. Wagnières, H. van den Bergh, “Antibody-fluorescein conjugates for photoimmunodiagnosis of human colon carcinoma in nude mice,” Cancer 67, 2529–2537 (1991).
[CrossRef] [PubMed]

Fuderer, M.

M. Fuderer, “The information content of MR images,” IEEE Trans. Med. Imaging 7, 368–380 (1988).
[CrossRef] [PubMed]

Gao, F.

Gaudette, R. J.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Gaudette, T.

Graber, H. L.

Grotzinger, C.

A. Becker, C. Hessenius, K. Licha, B. Ebert, U. Sukowski, W. Semmler, B. Wiedenmann, C. Grotzinger, “Receptor-targeted optical imaging of tumors with near-infrared fluorescent ligands,” Nat. Biotechnol. 19, 327–331 (2001).
[CrossRef] [PubMed]

Hakala, T. R.

B. Ballou, G. W. Fisher, T. R. Hakala, D. L. Farkas, “Tumor detection and visualization using cyanine fluorochrome-labeled antibodies,” Biotechnol. Prog. 13, 649–658 (1997).
[CrossRef] [PubMed]

Hamilton, L. J.

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

Haskell, R. C.

Hero, A. O.

A. O. Hero, L. Shao, “Information analysis of single photon emission computed tomography with count losses,” IEEE Trans. Med. Imaging 9, 117–127 (1990).
[CrossRef] [PubMed]

L. Shao, A. O. Hero, W. L. Rogers, N. H. Clinthorne, “The mutual information criterion for SPECT aperture evaluation and design,” IEEE Trans. Med. Imaging 8, 322–336 (1989).
[CrossRef] [PubMed]

Hessenius, C.

A. Becker, C. Hessenius, K. Licha, B. Ebert, U. Sukowski, W. Semmler, B. Wiedenmann, C. Grotzinger, “Receptor-targeted optical imaging of tumors with near-infrared fluorescent ligands,” Nat. Biotechnol. 19, 327–331 (2001).
[CrossRef] [PubMed]

Hielscher, A. H.

A. D. Klose, A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer—Part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transf. 72, 715–732 (2002).
[CrossRef]

Holboke, M.

V. Ntziachristos, J. Culver, M. Holboke, A. G. Yodh, B. Chance, “Optimal selection of frequencies for diffuse optical tomography,” in Biomedical Topical Meetings, Vol. 38 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 475–477.

Holbrooke, M. J.

Hutchinson, C. L.

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

Iftimia, N.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Jiang, H.

Kilmer, M.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Klose, A. D.

A. D. Klose, A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer—Part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transf. 72, 715–732 (2002).
[CrossRef]

Kues, H. A.

R. C. Benson, H. A. Kues, “Fluorescence properties of indocyanine green as related to angiography,” Phys. Med. Biol. 23, 159–163 (1978).
[CrossRef] [PubMed]

Kwant, G.

M. L. J. Landsman, G. Kwant, G. A. Mook, W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
[PubMed]

Landsman, M. L. J.

M. L. J. Landsman, G. Kwant, G. A. Mook, W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
[PubMed]

LaPlant, F. P.

Li, X. D.

Licha, K.

A. Becker, C. Hessenius, K. Licha, B. Ebert, U. Sukowski, W. Semmler, B. Wiedenmann, C. Grotzinger, “Receptor-targeted optical imaging of tumors with near-infrared fluorescent ligands,” Nat. Biotechnol. 19, 327–331 (2001).
[CrossRef] [PubMed]

K. Licha, B. Riefke, V. Ntziachristos, A. Becker, B. Chance, W. Semmler, “Hydrophilic cyanine dyes as contrast agents for near-infrared tumor imaging: synthesis, photophysical properties and spectroscopic in vivo characterization,” Photochem. Photobiol. 72, 392–398 (2000).
[CrossRef] [PubMed]

Lopez, G.

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

Low, P. S.

J. A. Reddy, P. S. Low, “Folate-mediated targeting of therapeutic and imaging agents to cancers,” Crit. Rev. Ther. Drug Carrier Syst. 15, 587–627 (1998).
[CrossRef]

Mach, J.

A. Pèlegrin, S. Folli, F. Buchegger, J. Mach, G. Wagnières, H. van den Bergh, “Antibody-fluorescein conjugates for photoimmunodiagnosis of human colon carcinoma in nude mice,” Cancer 67, 2529–2537 (1991).
[CrossRef] [PubMed]

Mahmood, U.

U. Mahmood, C. Tung, J. A. Bogdanov, R. Weissleder, “Near-infrared optical imaging of protease activity for tumor detection,” Radiology 213, 866–870 (1999).
[CrossRef] [PubMed]

Mayer, R. H.

J. S. Reynolds, T. L. Troy, R. H. Mayer, A. B. Thompson, D. J. Waters, K. K. Cornell, P. W. Snyder, E. M. Sevick-Muraca, “Imaging of spontaneous canine mammary tumors using fluorescent contrast agents,” Photochem. Photobiol. 70, 87–94 (1999).
[CrossRef] [PubMed]

McAdams, M. S.

Millane, R. P.

Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Milstein, A. B.

Mook, G. A.

M. L. J. Landsman, G. Kwant, G. A. Mook, W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
[PubMed]

Ntziachristos, V.

V. Ntziachristos, 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. P. Culver, V. Ntziachristos, M. J. Holbrooke, A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett. 26, 701–703 (2001).
[CrossRef]

K. Licha, B. Riefke, V. Ntziachristos, A. Becker, B. Chance, W. Semmler, “Hydrophilic cyanine dyes as contrast agents for near-infrared tumor imaging: synthesis, photophysical properties and spectroscopic in vivo characterization,” Photochem. Photobiol. 72, 392–398 (2000).
[CrossRef] [PubMed]

V. Ntziachristos, A. G. Yodh, M. Schnall, B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. USA 97, 2767–2772 (2000).
[CrossRef] [PubMed]

V. Ntziachristos, J. Culver, M. Holboke, A. G. Yodh, B. Chance, “Optimal selection of frequencies for diffuse optical tomography,” in Biomedical Topical Meetings, Vol. 38 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 475–477.

O’Leary, M. A.

Oh, S.

Paithankar, D.

Pastel, M. S.

R. F. Wagner, D. G. Brown, M. S. Pastel, “Application of information theory to the assessment of computed tomography,” Med. Phys. 6, 83–94 (1979).
[CrossRef] [PubMed]

Patterson, M.

Patterson, M. S.

Paulsen, K. D.

H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, J. F. Dunn, “Near infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt. 8, 102–110 (2003).
[CrossRef] [PubMed]

Pèlegrin, A.

A. Pèlegrin, S. Folli, F. Buchegger, J. Mach, G. Wagnières, H. van den Bergh, “Antibody-fluorescein conjugates for photoimmunodiagnosis of human colon carcinoma in nude mice,” Cancer 67, 2529–2537 (1991).
[CrossRef] [PubMed]

Pifferi, A.

R. Cubeddu, G. Canti, A. Pifferi, P. Taroni, G. Valentini, “Fluorescence lifetime imaging of experimental tumors in hematoporphyrin derivative-sensitized mice,” Photochem. Photobiol. 66, 229–236 (1997).
[CrossRef] [PubMed]

Pogue, B.

Pogue, B. W.

H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, J. F. Dunn, “Near infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt. 8, 102–110 (2003).
[CrossRef] [PubMed]

M. S. Patterson, B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
[CrossRef] [PubMed]

Rajagopalan, R.

J. E. Bugaj, S. Achilefu, R. B. Dorshow, R. Rajagopalan, “Novel fluorescent contrast agents for optical imaging of in vivo tumors based on a receptor-targeted dye-peptide conjugate platform,” J. Biomed. Opt. 6, 122–133 (2001).
[CrossRef] [PubMed]

Reddy, J. A.

J. A. Reddy, P. S. Low, “Folate-mediated targeting of therapeutic and imaging agents to cancers,” Crit. Rev. Ther. Drug Carrier Syst. 15, 587–627 (1998).
[CrossRef]

Reynolds, J. S.

A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography with experimental data,” Opt. Lett. 27, 95–97 (2002).
[CrossRef]

J. S. Reynolds, T. L. Troy, R. H. Mayer, A. B. Thompson, D. J. Waters, K. K. Cornell, P. W. Snyder, E. M. Sevick-Muraca, “Imaging of spontaneous canine mammary tumors using fluorescent contrast agents,” Photochem. Photobiol. 70, 87–94 (1999).
[CrossRef] [PubMed]

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

J. S. Reynolds, C. A. Thompson, K. J. Webb, F. P. LaPlant, D. Ben-Amotz, “Frequency domain modeling of reradiation in highly scattering media,” Appl. Opt. 36, 2252–2259 (1997).
[CrossRef] [PubMed]

Riefke, B.

K. Licha, B. Riefke, V. Ntziachristos, A. Becker, B. Chance, W. Semmler, “Hydrophilic cyanine dyes as contrast agents for near-infrared tumor imaging: synthesis, photophysical properties and spectroscopic in vivo characterization,” Photochem. Photobiol. 72, 392–398 (2000).
[CrossRef] [PubMed]

Rogers, W. L.

L. Shao, A. O. Hero, W. L. Rogers, N. H. Clinthorne, “The mutual information criterion for SPECT aperture evaluation and design,” IEEE Trans. Med. Imaging 8, 322–336 (1989).
[CrossRef] [PubMed]

Roy, R.

Saquib, S. S.

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

Sauer, K.

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

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

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

Schnall, M.

V. Ntziachristos, A. G. Yodh, M. Schnall, B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. USA 97, 2767–2772 (2000).
[CrossRef] [PubMed]

Semmler, W.

A. Becker, C. Hessenius, K. Licha, B. Ebert, U. Sukowski, W. Semmler, B. Wiedenmann, C. Grotzinger, “Receptor-targeted optical imaging of tumors with near-infrared fluorescent ligands,” Nat. Biotechnol. 19, 327–331 (2001).
[CrossRef] [PubMed]

K. Licha, B. Riefke, V. Ntziachristos, A. Becker, B. Chance, W. Semmler, “Hydrophilic cyanine dyes as contrast agents for near-infrared tumor imaging: synthesis, photophysical properties and spectroscopic in vivo characterization,” Photochem. Photobiol. 72, 392–398 (2000).
[CrossRef] [PubMed]

Sevick-Muraca, E.

Sevick-Muraca, E. M.

R. Roy, E. M. Sevick-Muraca, “Three-dimensional unconstrained and constrained image-reconstruction techniques applied to fluorescence, frequency-domain photon migration,” Appl. Opt. 40, 2206–2215 (2001).
[CrossRef]

J. S. Reynolds, T. L. Troy, R. H. Mayer, A. B. Thompson, D. J. Waters, K. K. Cornell, P. W. Snyder, E. M. Sevick-Muraca, “Imaging of spontaneous canine mammary tumors using fluorescent contrast agents,” Photochem. Photobiol. 70, 87–94 (1999).
[CrossRef] [PubMed]

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

Shao, L.

A. O. Hero, L. Shao, “Information analysis of single photon emission computed tomography with count losses,” IEEE Trans. Med. Imaging 9, 117–127 (1990).
[CrossRef] [PubMed]

L. Shao, A. O. Hero, W. L. Rogers, N. H. Clinthorne, “The mutual information criterion for SPECT aperture evaluation and design,” IEEE Trans. Med. Imaging 8, 322–336 (1989).
[CrossRef] [PubMed]

Shives, E.

Snyder, P. W.

J. S. Reynolds, T. L. Troy, R. H. Mayer, A. B. Thompson, D. J. Waters, K. K. Cornell, P. W. Snyder, E. M. Sevick-Muraca, “Imaging of spontaneous canine mammary tumors using fluorescent contrast agents,” Photochem. Photobiol. 70, 87–94 (1999).
[CrossRef] [PubMed]

Springett, R.

H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, J. F. Dunn, “Near infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt. 8, 102–110 (2003).
[CrossRef] [PubMed]

Stott, J.

J. Stott, D. A. Boas, “A practical comparison between time-domain and frequency-domain diffusive optical imaging systems,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 626–628.

Stott, J. J.

Sukowski, U.

A. Becker, C. Hessenius, K. Licha, B. Ebert, U. Sukowski, W. Semmler, B. Wiedenmann, C. Grotzinger, “Receptor-targeted optical imaging of tumors with near-infrared fluorescent ligands,” Nat. Biotechnol. 19, 327–331 (2001).
[CrossRef] [PubMed]

Svaasand, L. O.

Taroni, P.

R. Cubeddu, G. Canti, A. Pifferi, P. Taroni, G. Valentini, “Fluorescence lifetime imaging of experimental tumors in hematoporphyrin derivative-sensitized mice,” Photochem. Photobiol. 66, 229–236 (1997).
[CrossRef] [PubMed]

Thomas, J. A.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).

Thompson, A. B.

J. S. Reynolds, T. L. Troy, R. H. Mayer, A. B. Thompson, D. J. Waters, K. K. Cornell, P. W. Snyder, E. M. Sevick-Muraca, “Imaging of spontaneous canine mammary tumors using fluorescent contrast agents,” Photochem. Photobiol. 70, 87–94 (1999).
[CrossRef] [PubMed]

Thompson, C. A.

Tromberg, B. J.

Troy, T. L.

J. S. Reynolds, T. L. Troy, R. H. Mayer, A. B. Thompson, D. J. Waters, K. K. Cornell, P. W. Snyder, E. M. Sevick-Muraca, “Imaging of spontaneous canine mammary tumors using fluorescent contrast agents,” Photochem. Photobiol. 70, 87–94 (1999).
[CrossRef] [PubMed]

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

Tsay, T.-T.

Tung, C.

U. Mahmood, C. Tung, J. A. Bogdanov, R. Weissleder, “Near-infrared optical imaging of protease activity for tumor detection,” Radiology 213, 866–870 (1999).
[CrossRef] [PubMed]

Valentini, G.

R. Cubeddu, G. Canti, A. Pifferi, P. Taroni, G. Valentini, “Fluorescence lifetime imaging of experimental tumors in hematoporphyrin derivative-sensitized mice,” Photochem. Photobiol. 66, 229–236 (1997).
[CrossRef] [PubMed]

van den Bergh, H.

A. Pèlegrin, S. Folli, F. Buchegger, J. Mach, G. Wagnières, H. van den Bergh, “Antibody-fluorescein conjugates for photoimmunodiagnosis of human colon carcinoma in nude mice,” Cancer 67, 2529–2537 (1991).
[CrossRef] [PubMed]

Wagner, R. F.

R. F. Wagner, D. G. Brown, M. S. Pastel, “Application of information theory to the assessment of computed tomography,” Med. Phys. 6, 83–94 (1979).
[CrossRef] [PubMed]

Wagnières, G.

A. Pèlegrin, S. Folli, F. Buchegger, J. Mach, G. Wagnières, H. van den Bergh, “Antibody-fluorescein conjugates for photoimmunodiagnosis of human colon carcinoma in nude mice,” Cancer 67, 2529–2537 (1991).
[CrossRef] [PubMed]

Waters, D. J.

J. S. Reynolds, T. L. Troy, R. H. Mayer, A. B. Thompson, D. J. Waters, K. K. Cornell, P. W. Snyder, E. M. Sevick-Muraca, “Imaging of spontaneous canine mammary tumors using fluorescent contrast agents,” Photochem. Photobiol. 70, 87–94 (1999).
[CrossRef] [PubMed]

Webb, K. J.

Weissleder, R.

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

U. Mahmood, C. Tung, J. A. Bogdanov, R. Weissleder, “Near-infrared optical imaging of protease activity for tumor detection,” Radiology 213, 866–870 (1999).
[CrossRef] [PubMed]

Wiedenmann, B.

A. Becker, C. Hessenius, K. Licha, B. Ebert, U. Sukowski, W. Semmler, B. Wiedenmann, C. Grotzinger, “Receptor-targeted optical imaging of tumors with near-infrared fluorescent ligands,” Nat. Biotechnol. 19, 327–331 (2001).
[CrossRef] [PubMed]

Xu, H.

H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, J. F. Dunn, “Near infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt. 8, 102–110 (2003).
[CrossRef] [PubMed]

Xu, Y.

Yamada, Y.

Ye, J. C.

J. C. Ye, C. A. Bouman, K. J. Webb, R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909–922 (2001).
[CrossRef]

J. C. Ye, K. J. Webb, C. A. Bouman, R. P. Millane, “Optical diffusion tomography using iterative coordinate descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A 16, 2400–2412 (1999).
[CrossRef]

Yodh, A. G.

J. P. Culver, V. Ntziachristos, M. J. Holbrooke, A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett. 26, 701–703 (2001).
[CrossRef]

V. Ntziachristos, A. G. Yodh, M. Schnall, B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. USA 97, 2767–2772 (2000).
[CrossRef] [PubMed]

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

V. Ntziachristos, J. Culver, M. Holboke, A. G. Yodh, B. Chance, “Optimal selection of frequencies for diffuse optical tomography,” in Biomedical Topical Meetings, Vol. 38 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 475–477.

Zak, S. H.

E. K. P. Chong, S. H. Zak, An Introduction to Optimization (Wiley, New York, 1996).

Zhang, Q.

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42, 3081–3094 (2003).
[CrossRef] [PubMed]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Zhao, H.

Zijlstra, W. G.

M. L. J. Landsman, G. Kwant, G. A. Mook, W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
[PubMed]

Appl. Opt. (9)

M. S. Patterson, B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
[CrossRef] [PubMed]

D. Paithankar, A. Chen, B. Pogue, M. Patterson, E. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. 36, 2260–2272 (1997).
[CrossRef] [PubMed]

H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37, 5337–5343 (1998).
[CrossRef]

R. Roy, E. M. Sevick-Muraca, “Three-dimensional unconstrained and constrained image-reconstruction techniques applied to fluorescence, frequency-domain photon migration,” Appl. Opt. 40, 2206–2215 (2001).
[CrossRef]

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42, 3081–3094 (2003).
[CrossRef] [PubMed]

F. Gao, H. Zhao, Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 778–791 (2002).
[CrossRef] [PubMed]

J. S. Reynolds, C. A. Thompson, K. J. Webb, F. P. LaPlant, D. Ben-Amotz, “Frequency domain modeling of reradiation in highly scattering media,” Appl. Opt. 36, 2252–2259 (1997).
[CrossRef] [PubMed]

J. J. Stott, J. P. Culver, S. R. Arridge, D. A. Boas, “Optode positional calibration in diffuse optical tomography,” Appl. Opt. 42, 3154–3162 (2003).
[CrossRef] [PubMed]

N. Iftimia, H. Jiang, “Quantitative optical image reconstructions of turbid media by use of direct-current measurements,” Appl. Opt. 39, 5256–5261 (2000).
[CrossRef]

Bell Syst. Tech. J. (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

Biotechnol. Prog. (1)

B. Ballou, G. W. Fisher, T. R. Hakala, D. L. Farkas, “Tumor detection and visualization using cyanine fluorochrome-labeled antibodies,” Biotechnol. Prog. 13, 649–658 (1997).
[CrossRef] [PubMed]

Cancer (1)

A. Pèlegrin, S. Folli, F. Buchegger, J. Mach, G. Wagnières, H. van den Bergh, “Antibody-fluorescein conjugates for photoimmunodiagnosis of human colon carcinoma in nude mice,” Cancer 67, 2529–2537 (1991).
[CrossRef] [PubMed]

Crit. Rev. Ther. Drug Carrier Syst. (1)

J. A. Reddy, P. S. Low, “Folate-mediated targeting of therapeutic and imaging agents to cancers,” Crit. Rev. Ther. Drug Carrier Syst. 15, 587–627 (1998).
[CrossRef]

IEEE Signal Process. Mag. (1)

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

IEEE Trans. Image Process. (3)

J. C. Ye, C. A. Bouman, K. J. Webb, R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909–922 (2001).
[CrossRef]

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

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

IEEE Trans. Med. Imaging (3)

M. Fuderer, “The information content of MR images,” IEEE Trans. Med. Imaging 7, 368–380 (1988).
[CrossRef] [PubMed]

L. Shao, A. O. Hero, W. L. Rogers, N. H. Clinthorne, “The mutual information criterion for SPECT aperture evaluation and design,” IEEE Trans. Med. Imaging 8, 322–336 (1989).
[CrossRef] [PubMed]

A. O. Hero, L. Shao, “Information analysis of single photon emission computed tomography with count losses,” IEEE Trans. Med. Imaging 9, 117–127 (1990).
[CrossRef] [PubMed]

IEEE Trans. Signal Process. (1)

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

Inverse Probl. (1)

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

J. Appl. Physiol. (1)

M. L. J. Landsman, G. Kwant, G. A. Mook, W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
[PubMed]

J. Biomed. Opt. (2)

H. Xu, H. Dehghani, B. W. Pogue, R. Springett, K. D. Paulsen, J. F. Dunn, “Near infrared imaging in the small animal brain: optimization of fiber positions,” J. Biomed. Opt. 8, 102–110 (2003).
[CrossRef] [PubMed]

J. E. Bugaj, S. Achilefu, R. B. Dorshow, R. Rajagopalan, “Novel fluorescent contrast agents for optical imaging of in vivo tumors based on a receptor-targeted dye-peptide conjugate platform,” J. Biomed. Opt. 6, 122–133 (2001).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (4)

J. Quant. Spectrosc. Radiat. Transf. (1)

A. D. Klose, A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer—Part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transf. 72, 715–732 (2002).
[CrossRef]

Med. Phys. (1)

R. F. Wagner, D. G. Brown, M. S. Pastel, “Application of information theory to the assessment of computed tomography,” Med. Phys. 6, 83–94 (1979).
[CrossRef] [PubMed]

Nat. Biotechnol. (1)

A. Becker, C. Hessenius, K. Licha, B. Ebert, U. Sukowski, W. Semmler, B. Wiedenmann, C. Grotzinger, “Receptor-targeted optical imaging of tumors with near-infrared fluorescent ligands,” Nat. Biotechnol. 19, 327–331 (2001).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (4)

Photochem. Photobiol. (4)

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

R. Cubeddu, G. Canti, A. Pifferi, P. Taroni, G. Valentini, “Fluorescence lifetime imaging of experimental tumors in hematoporphyrin derivative-sensitized mice,” Photochem. Photobiol. 66, 229–236 (1997).
[CrossRef] [PubMed]

J. S. Reynolds, T. L. Troy, R. H. Mayer, A. B. Thompson, D. J. Waters, K. K. Cornell, P. W. Snyder, E. M. Sevick-Muraca, “Imaging of spontaneous canine mammary tumors using fluorescent contrast agents,” Photochem. Photobiol. 70, 87–94 (1999).
[CrossRef] [PubMed]

K. Licha, B. Riefke, V. Ntziachristos, A. Becker, B. Chance, W. Semmler, “Hydrophilic cyanine dyes as contrast agents for near-infrared tumor imaging: synthesis, photophysical properties and spectroscopic in vivo characterization,” Photochem. Photobiol. 72, 392–398 (2000).
[CrossRef] [PubMed]

Phys. Med. Biol. (1)

R. C. Benson, H. A. Kues, “Fluorescence properties of indocyanine green as related to angiography,” Phys. Med. Biol. 23, 159–163 (1978).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

V. Ntziachristos, A. G. Yodh, M. Schnall, B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. USA 97, 2767–2772 (2000).
[CrossRef] [PubMed]

Radiology (1)

U. Mahmood, C. Tung, J. A. Bogdanov, R. Weissleder, “Near-infrared optical imaging of protease activity for tumor detection,” Radiology 213, 866–870 (1999).
[CrossRef] [PubMed]

Other (9)

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).

V. Ntziachristos, J. Culver, M. Holboke, A. G. Yodh, B. Chance, “Optimal selection of frequencies for diffuse optical tomography,” in Biomedical Topical Meetings, Vol. 38 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 475–477.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Philadelphia, Pa., 1998).

J. Stott, D. A. Boas, “A practical comparison between time-domain and frequency-domain diffusive optical imaging systems,” in Biomedical Topical Meetings, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 626–628.

T. Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression (Prentice Hall, Englewood Cliffs, N.J., 1971).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

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

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

E. K. P. Chong, S. H. Zak, An Introduction to Optimization (Wiley, New York, 1996).

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Figures (12)

Fig. 1
Fig. 1

Measurement approach for reconstructing all unknowns, showing appropriate source and detector wavelengths for reconstructing [μax, Dx], [μam, Dm], and [η, τ].

Fig. 2
Fig. 2

Reconstruction of fluorescent yield using simulated data, showing the improvement due to use of multiple modulation frequencies: (a) source/detector geometry, (b) true image cross section, (c) reconstruction using 78.4-MHz data, (d) reconstruction using 314-MHz data, (e) reconstruction using 78.4- and 314-MHz data.

Fig. 3
Fig. 3

(a) Schematic of the experimental setup, showing the box and the tissue phantom and a glass sphere filled with ICG/Intralipid, rubber tubes, and Intralipid suspension. (b) Source fiber positions. The same positions were selected as detection regions from the camera images.

Fig. 4
Fig. 4

Reconstruction of μax (cm-1), obtained by using 78-, 314-, and 627-MHz data.

Fig. 5
Fig. 5

Reconstruction of μam (cm-1), obtained by using 78-, 314-, and 627-MHz data.

Fig. 6
Fig. 6

Reconstruction of η (in 10-4 cm-1), obtained by using 78-, 314-, and 627-MHz data.  

Fig. 7
Fig. 7

Reconstruction of τ (in 10-10 s), obtained by using 78-, 314-, and 627-MHz data.

Fig. 8
Fig. 8

Reconstruction of η (in 10-4 cm-1), obtained by using only 78-MHz data.

Fig. 9
Fig. 9

Reconstruction of τ (in 10-10 s), obtained by using only 78-MHz data.

Fig. 10
Fig. 10

Reconstructions of η (in 10-4 cm-1) using various values of σ, showing a progression from overregularization to underregularization. The z=2.85 cm cross sections are shown. The τ model used σ=1×10-10 s in all cases.

Fig. 11
Fig. 11

(a)–(e) z=2.85 cm cross sections of reconstructed τ (in 10-10 s) for various σ, showing a progression from overregularization to underregularization. (f) τ^avg as a function of σ for the τ reconstruction. The × symbol represents the value of σ that was used in generating the data of Fig. 7. The η model used σ=2.5×10-5 cm-1 in all cases.

Fig. 12
Fig. 12

Mutual information versus α for (a) the simulation model and (b) the experiment model. In (a), the + symbols mark the results for the true value of α used in the simulation. In (b), the + symbol marks the result for the estimated value of α in the experiment. The units of information are nats rather than bits, as the base e logarithm was used.

Equations (63)

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  [D(r)ϕ(r, ω)]-[μa(r)+jω/c]ϕ(r, ω)
=-δ(r-rsk),
  [Dx(r)ϕx(r, ω)]-[μax(r)+jω/c]ϕx(r, ω)
=-δ(r-rsk),
  [Dm(r)ϕm(r, ω)]-[μam(r)+jω/c]ϕm(r, ω)
=-ϕx(r, ω)ημaf(r) 1-jωτ(r)1+[ωτ(r)]2,
xx=[xxAT, xxBT]T=[μax(r1),, μax(rN), Dx(r1),, Dx(rN)]T,
xm=[xmAT, xmBT]T=[μam(r1),, μam(rN), Dm(r1),, Dm(rN)]T,
xf=[xfAT, xfBT]T=[η(r1),, η(rN), τ(r1),, τ(rN)]T,
x^MAP=arg maxx0 [pX|Y(x|y)]
=arg maxx0 [log pY|X(y|x)+log pX(x)],
pY|X(y|x)=1(πα)P|Λ|-1exp-y-f(x)Λ2α,
α2 Λ-1=α2diag(|y1|, |y2|,, |yP|).
pX(x)=pXA(xA)pXB(xB)
=1σANζ(ρA)×exp-1ρAσAρA{i,j}NAbi-j|xi-xj|ρA×1σBNζ(ρB)×exp-1ρBσBρB{i,j}NBbi-j|xi-xj|ρB,
x^x=arg maxxx0,αx0[pXx|Yx(xx|yx, αx)],
x^m=arg maxxm0,αm0[pXm|Ym(xm|ym, αm)],
x^f=arg maxxf0,αf0[pXf|Yf(xf|yf, αf, x^x, x^m)].
l(x)=-P logy-f(x)Λ2-1ρAσAρA{i,j}NAbi-j|xi-xj|ρA-1ρBσBρB{i,j}NBbi-j|xi-xj|ρB.
αˆ1P y-f(xˆ)Λ2,
xˆarg updatex0[log pY|X(y|x, αˆ)+log pX(x|αˆ)],
c(xx, α^x)=1α^x yx-fx(xx)Λx2+1ρxAσxAρxA{i,j}NxAbi-j|xxAi-xxAj|ρxA+1ρxBσxBρxB{i,j}NxBbi-j|xxBi-xxBj|ρxB,
c(xm, α^m)=1α^m ym-fm(xm)Λm2+1ρmAσmAρmA{i,j}NmAbi-j|xmAi-xmAj|ρmA+1ρmBσmBρmB{i,j}NmBbi-j|xmBi-xmBj|ρmB,
c(xf, x^x, x^m, α^f)
=1α^f yf-ff(xf, x^x, x^m)Λf2+1ρfAσfAρfA{i,j}NfAbi-j|xfAi-xfAj|ρfA+1ρfBσfBρfB{i,j}NfBbi-j|xfBi-xfBj|ρfB,
h(xf, r, ω)=η(r) 1-jωτ(r)1+[ωτ(r)]2.
ϕf(rsk, rdm; ω, xf)=h(xf, r, ω)gx(rsk, r; ω, xx)×gm(r, rdm; ω, xm)d3r,
h(xf, r, ω)gx(rsk, r; ω, xx)=ϕx(r, ω) η(r)[1-jωτ(r)]1+[ωτ(r)]2.
fω(xf)=ϕf(rs1, rd1; ω, xf)ϕf(rs1, rd2; ω, xf)ϕf(rs1, rdM; ω, xf)ϕf(rs2, rd1; ω, xf)ϕf(rsK, rdM; ω, xf).
f(xf)=[[fω1(xf)]T, [fω2(xf)]T,, [fωQ(xf)]T]T.
y=[yω1T, yω2T,, yωQT]T,
Gx(ω)
=gx(rs1, r1; ω, xx)gx(rs1, rN; ω, xx)gx(rsK, r1; ω, xx)gx(rsK, rN; ω, xx),
Gm(ω)
=gm(rd1, r1; ω, xm)gm(rd1, rN; ω, xm)gm(rdM, r1; ω, xm)gm(rdM, rN; ω, xm),
Jω=VG1,1x(ω)G1,1m(ω)G1,Nx(ω)G1,Nm(ω)G1,1x(ω)GM,1m(ω)G1,Nx(ω)GM,Nm(ω)G2,1x(ω)G1,1m(ω)G2,Nx(ω)G1,Nm(ω)GK,1x(ω)GM,1m(ω)GK,Nx(ω)GM,Nm(ω),
hω(xf)=[h(xf, r1, ω),, h(xf, rN, ω)]T.
fω(xf)=Jωhω(xf).
c(xf, α^f)=1α^fq=1Qyfωq-Jωqhωq(xf)Λfωq2+1ρfAσfAρfA{i,j}NfAbi-j|xfAi-xfAj|ρfA+1ρfBσfBρfB{i,j}NfBbi-j|xfBi-xfBj|ρfB.
x^iarg minxi01αˆq=1Qyωq-[Jωq]*(i)h(x, ri, ωq)Λωq2+1ρσρjNibi-j|xi-x^j|ρ,
x^iarg minxi01αˆq=1Qzωq-[Jωq]*(i)[h(x, ri, ωq)-h(x˜, ri, ωq)]Λωq2+1ρσρjNibi-j|xi-x^j|ρ,
=arg minxi01αˆq=1Qθ1,ωq[h(x, ri, ωq)-h(x˜, ri, ωq)]+θ2,ωq2 [h(x, ri, ωq)-h(x˜, ri, ωq)]2+1ρσρjNibi-j|xi-x˜j|ρ,
θ1,ωq=-2 Re([Jωq]*iHΛωqzωq),
θ2,ωq=2[Jωq]*iHΛωq[Jωq]*i.
τ^avg=i=1Nηˆ(ri)τˆ(ri)i=1Nηˆ(ri).
I(X; Y)=H(Y)-H(Y|X),
H(Y)=E[-log pY(Y)],
H(Y|X)=E[-log pY|X(Y|X)],
pX(x)=1(2π)Nσ2 |C|-1/2exp-12σ2 xHCx,
Ci,j=2ifi=j2bi-jifij ,
E[Y|X]=JX,
I(X; Y)=12logI+2σ2α ΛJC-1JH,
E[X-Xˆ2]DX(I(X; Y)).
E[Y]=E[E[Y|X]]=E[JX]=JE[X]=0.
E[YYH]=E[(JX+Z)(JX+Z)H]
=E[ZZH]+E[(JX)ZH]+E[Z(JX)H]+E[JXXHJH]
=E[ZZH]+JE[XXH]JH
=α2 Λ-1+σ2JC-1JH.
pY(y)=1[(2π)P|Υ|]1/2exp-12 yΥ-12,
Υ=α2 Λ-1+σ2JC-1JH.
H(Y)=12log[(2π)P]+P2+12log|Υ|.
H(Y|X)=12log[(2π)P]+P2+12logα2 Λ-1.
I(X; Y)=H(Y)-H(Y|X)=12log|Υ||α2Λ-1|=12log|α2Λ-1+σ2JC-1JH||α2Λ-1|=12logI+2σ2α ΛJC-1JH,

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