Abstract

Stochastic reconstruction techniques are developed for mapping the interior optical properties of tissues from exterior frequency-domain photon migration measurements at the air–tissue interface. Parameter fields of absorption cross section, fluorescence lifetime, and quantum efficiency are accurately reconstructed from simulated noisy measurements of phase shift and amplitude modulation by use of a recursive, Bayesian, minimum-variance estimator known as the approximate extended Kalman filter. Parameter field updates are followed by data-driven zonation to improve the accuracy, stability, and computational efficiency of the method by moving the system from an underdetermined toward an overdetermined set of equations. These methods were originally developed by Eppstein and Dougherty [Water Resources Res. 32, 3321 (1996)] for applications in geohydrology. Estimates are constrained to within feasible ranges by modeling of parameters as β-distributed random variables. No arbitrary smoothing, regularization, or interpolation is required. Results are compared with those determined by use of Newton–Raphson-based inversions. The speed and accuracy of these preliminary Bayesian reconstructions suggest the near-future application of this inversion technology to three-dimensional biomedical imaging with frequency-domain photon migration.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
    [CrossRef] [PubMed]
  2. R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.
  3. S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
    [CrossRef]
  4. S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near-infrared absorption and scatter imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
    [CrossRef]
  5. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion photon tomography,” Opt. Lett. 20, 426–428 (1995).
    [CrossRef]
  6. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency domain data simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
    [CrossRef]
  7. M. A. O’Leary, D. A. Boas, D. X. Li, B. Chance, A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21, 158–160 (1996).
    [CrossRef] [PubMed]
  8. D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, E. M. Sevick-Muraca, “Imaging of fluorescent lifetime and yield from multiple scattered light reemitted from tissues and other random media,” Appl. Opt. 36, 2260–2272 (1997).
    [CrossRef] [PubMed]
  9. 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]
  10. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical heterogeneities with turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
    [CrossRef]
  11. Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering by a Born iterative method,” J. Opt. Soc. Am. A 14, 325–342 (1997).
    [CrossRef]
  12. W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imag. 9, 218–225 (1990).
    [CrossRef]
  13. K. D. Paulsen, H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
    [CrossRef] [PubMed]
  14. S. R. Arridge, M. Schweiger, “Image reconstruction in optical tomography,” Phil. Trans. R. Soc. London Series B 352, 717–726 (1997).
    [CrossRef]
  15. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37, 5337–5343 (1998).
    [CrossRef]
  16. N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, 2nd ed. (Wiley, New York, 1995), Vol. 1.
  17. V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
    [CrossRef]
  18. M. J. Eppstein, D. E. Dougherty, “Simultaneous estimation of transmissivity values and zonation,” Water Resources Res. 32, 3321–3336 (1996).
    [CrossRef]
  19. T. L. Troy, D. L. Page, E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–355 (1996).
    [CrossRef] [PubMed]
  20. E. M. Sevick-Muraca, C. L. Hutchinson, D. Y. Paithankar, “Optical tissue biodiagnostics using fluorescence lifetime,” Opt. Photon. News 7(1), 25–28 (1996).
  21. E. M. Sevick-Muraca, G. Lopez, T. L. Troy, J. S. Reynolds, C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
    [CrossRef] [PubMed]
  22. E. M. Sevick, C. L. Burch, “Origin of phosphorescence signals re-emitted from tissues,” Opt. Lett. 19, 1928–1930 (1994).
    [CrossRef]
  23. 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]
  24. C. L. Hutchinson, T. L. Troy, E. M. Sevick-Muraca, “Fluorescence-lifetime determination in tissues or other scattering media from measurement of excitation and emission kinetics,” Appl. Opt. 35, 2325–2332 (1996).
    [CrossRef] [PubMed]
  25. J. C. Adams, “mudpack: multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comp. 34, 133–146 (1989).
    [CrossRef]
  26. M. S. Patterson, B. Chance, B. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  27. M. J. Eppstein, D. E. Dougherty, “Optimal 3-D traveltime tomography,” Geophysics 63, 1053–1061 (1998).
    [CrossRef]
  28. M. J. Eppstein, D. E. Dougherty, “Efficient three-dimensional data inversion: soil characterization and moisture monitoring from cross-well ground-penetrating radar at a Vermont test site,” Water Resources Res. 34, 1889–1900 (1998).
    [CrossRef]
  29. M. J. Eppstein, D. E. Dougherty, “Three-dimensional stochastic tomography with upscaling,” U.S. patent application 09/110,506 (9July1998).
  30. R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. D 82, 35–45 (1960).
    [CrossRef]
  31. G. L. Smith, S. F. Schmidt, L. A. McGee, “Application of statistical filter theory to the optimal estimation of position and velocity on board a circumlunar vehicle,” (U.S. Government Printing Office, Washington, D.C., 1962).
  32. A. Gelb, ed., Applied Optimal Estimation (MIT Press, Cambridge, Mass.1974).
  33. J. S. Reynolds, T. L. Troy, E. M. Sevick-Muraca, “Multi-pixel techniques for frequency-domain photon migration imaging,” Biotech. Prog. 13, 669–680 (1997).
    [CrossRef]
  34. N. Sun, W. W.-G. Yeh, “Identification of parameter structure in groundwater inverse problem,” Water Resources Res. 21, 869–883 (1985).
    [CrossRef]
  35. F. Aschenbrenner, A. Ostin, “Automatic parameter estimation applied on a groundwater model: the problem of structure identification,” Environ. Geol. 25, 205–210 (1995).
    [CrossRef]
  36. M. J. Eppstein, D. E. Dougherty, “Optimal 3-D geophysical tomography,” in 1998 Proceedings of the Symposium on the Application of Geophysics to Environmental and Engineering Problems (SAGEEP), R. S. Bell, M. H. Powers, T. Larson, eds. (Environmental and Engineering Geophysical Society, Wheat Ridge, Colo.1998), pp. 249–256.
  37. T. L. Troy, “Biomedical optical imaging using frequency domain photon migration measurements: experiments and numerical image reconstructions,” Ph.D. dissertation (Purdue University, Lafayette, Ind., 1997).
  38. 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 mammary tumors using fluorescent contrast agents,” (submitted to Photochem. Photobiol).
  39. M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, E. M. Sevick-Muraca, “Three-dimensional optical tomography,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597 (to be published).

1998 (4)

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, “Optimal 3-D traveltime tomography,” Geophysics 63, 1053–1061 (1998).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, “Efficient three-dimensional data inversion: soil characterization and moisture monitoring from cross-well ground-penetrating radar at a Vermont test site,” Water Resources Res. 34, 1889–1900 (1998).
[CrossRef]

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

1997 (6)

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

E. M. Sevick-Muraca, G. Lopez, T. L. Troy, J. S. Reynolds, 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, T. L. Troy, E. M. Sevick-Muraca, “Multi-pixel techniques for frequency-domain photon migration imaging,” Biotech. Prog. 13, 669–680 (1997).
[CrossRef]

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]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering by a Born iterative method,” J. Opt. Soc. Am. A 14, 325–342 (1997).
[CrossRef]

S. R. Arridge, M. Schweiger, “Image reconstruction in optical tomography,” Phil. Trans. R. Soc. London Series B 352, 717–726 (1997).
[CrossRef]

1996 (6)

M. J. Eppstein, D. E. Dougherty, “Simultaneous estimation of transmissivity values and zonation,” Water Resources Res. 32, 3321–3336 (1996).
[CrossRef]

T. L. Troy, D. L. Page, E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–355 (1996).
[CrossRef] [PubMed]

E. M. Sevick-Muraca, C. L. Hutchinson, D. Y. Paithankar, “Optical tissue biodiagnostics using fluorescence lifetime,” Opt. Photon. News 7(1), 25–28 (1996).

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

C. L. Hutchinson, T. L. Troy, E. M. Sevick-Muraca, “Fluorescence-lifetime determination in tissues or other scattering media from measurement of excitation and emission kinetics,” Appl. Opt. 35, 2325–2332 (1996).
[CrossRef] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency domain data simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[CrossRef]

1995 (3)

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef]

F. Aschenbrenner, A. Ostin, “Automatic parameter estimation applied on a groundwater model: the problem of structure identification,” Environ. Geol. 25, 205–210 (1995).
[CrossRef]

K. D. Paulsen, H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
[CrossRef] [PubMed]

1994 (3)

E. M. Sevick, C. L. Burch, “Origin of phosphorescence signals re-emitted from tissues,” Opt. Lett. 19, 1928–1930 (1994).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical heterogeneities with turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

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]

1990 (2)

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imag. 9, 218–225 (1990).
[CrossRef]

1989 (2)

J. C. Adams, “mudpack: multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comp. 34, 133–146 (1989).
[CrossRef]

M. S. Patterson, B. Chance, B. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

1985 (1)

N. Sun, W. W.-G. Yeh, “Identification of parameter structure in groundwater inverse problem,” Water Resources Res. 21, 869–883 (1985).
[CrossRef]

1960 (1)

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. D 82, 35–45 (1960).
[CrossRef]

Adams, J. C.

J. C. Adams, “mudpack: multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comp. 34, 133–146 (1989).
[CrossRef]

Aronson, R.

R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

Arridge, S. R.

S. R. Arridge, M. Schweiger, “Image reconstruction in optical tomography,” Phil. Trans. R. Soc. London Series B 352, 717–726 (1997).
[CrossRef]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near-infrared absorption and scatter imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Aschenbrenner, F.

F. Aschenbrenner, A. Ostin, “Automatic parameter estimation applied on a groundwater model: the problem of structure identification,” Environ. Geol. 25, 205–210 (1995).
[CrossRef]

Balakrishnan, N.

N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, 2nd ed. (Wiley, New York, 1995), Vol. 1.

Barbour, R. L.

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]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering by a Born iterative method,” J. Opt. Soc. Am. A 14, 325–342 (1997).
[CrossRef]

R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

Boas, D. A.

Burch, C. L.

Chance, B.

Chang, J.

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]

R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

Chen, A. U.

Chew, W. C.

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imag. 9, 218–225 (1990).
[CrossRef]

Cope, M.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

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 mammary tumors using fluorescent contrast agents,” (submitted to Photochem. Photobiol).

Delpy, D. T.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near-infrared absorption and scatter imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Dougherty, D. E.

M. J. Eppstein, D. E. Dougherty, “Optimal 3-D traveltime tomography,” Geophysics 63, 1053–1061 (1998).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, “Efficient three-dimensional data inversion: soil characterization and moisture monitoring from cross-well ground-penetrating radar at a Vermont test site,” Water Resources Res. 34, 1889–1900 (1998).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, “Simultaneous estimation of transmissivity values and zonation,” Water Resources Res. 32, 3321–3336 (1996).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, “Optimal 3-D geophysical tomography,” in 1998 Proceedings of the Symposium on the Application of Geophysics to Environmental and Engineering Problems (SAGEEP), R. S. Bell, M. H. Powers, T. Larson, eds. (Environmental and Engineering Geophysical Society, Wheat Ridge, Colo.1998), pp. 249–256.

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, E. M. Sevick-Muraca, “Three-dimensional optical tomography,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597 (to be published).

M. J. Eppstein, D. E. Dougherty, “Three-dimensional stochastic tomography with upscaling,” U.S. patent application 09/110,506 (9July1998).

Eppstein, M. J.

M. J. Eppstein, D. E. Dougherty, “Optimal 3-D traveltime tomography,” Geophysics 63, 1053–1061 (1998).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, “Efficient three-dimensional data inversion: soil characterization and moisture monitoring from cross-well ground-penetrating radar at a Vermont test site,” Water Resources Res. 34, 1889–1900 (1998).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, “Simultaneous estimation of transmissivity values and zonation,” Water Resources Res. 32, 3321–3336 (1996).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, “Optimal 3-D geophysical tomography,” in 1998 Proceedings of the Symposium on the Application of Geophysics to Environmental and Engineering Problems (SAGEEP), R. S. Bell, M. H. Powers, T. Larson, eds. (Environmental and Engineering Geophysical Society, Wheat Ridge, Colo.1998), pp. 249–256.

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, E. M. Sevick-Muraca, “Three-dimensional optical tomography,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597 (to be published).

M. J. Eppstein, D. E. Dougherty, “Three-dimensional stochastic tomography with upscaling,” U.S. patent application 09/110,506 (9July1998).

Graber, H.

R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

Graber, H. L.

Grunbaum, F. A.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Hawrysz, D. J.

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, E. M. Sevick-Muraca, “Three-dimensional optical tomography,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597 (to be published).

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near-infrared absorption and scatter imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Hutchinson, C. L.

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

E. M. Sevick-Muraca, C. L. Hutchinson, D. Y. Paithankar, “Optical tissue biodiagnostics using fluorescence lifetime,” Opt. Photon. News 7(1), 25–28 (1996).

C. L. Hutchinson, T. L. Troy, E. M. Sevick-Muraca, “Fluorescence-lifetime determination in tissues or other scattering media from measurement of excitation and emission kinetics,” Appl. Opt. 35, 2325–2332 (1996).
[CrossRef] [PubMed]

Jiang, H.

Johnson, N. L.

N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, 2nd ed. (Wiley, New York, 1995), Vol. 1.

Kalman, R. E.

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. D 82, 35–45 (1960).
[CrossRef]

Kohn, P.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Kotz, S.

N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, 2nd ed. (Wiley, New York, 1995), Vol. 1.

Li, D. X.

Lopez, G.

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

McGee, L. A.

G. L. Smith, S. F. Schmidt, L. A. McGee, “Application of statistical filter theory to the optimal estimation of position and velocity on board a circumlunar vehicle,” (U.S. Government Printing Office, Washington, D.C., 1962).

O’Leary, M. A.

Osterberg, U. L.

Ostin, A.

F. Aschenbrenner, A. Ostin, “Automatic parameter estimation applied on a groundwater model: the problem of structure identification,” Environ. Geol. 25, 205–210 (1995).
[CrossRef]

Page, D. L.

T. L. Troy, D. L. Page, E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–355 (1996).
[CrossRef] [PubMed]

Paithankar, D. Y.

Patterson, M. S.

Paulsen, K. D.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency domain data simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[CrossRef]

K. D. Paulsen, H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
[CrossRef] [PubMed]

Pei, Y.

Pogue, B. W.

Reynolds, J. S.

J. S. Reynolds, T. L. Troy, E. M. Sevick-Muraca, “Multi-pixel techniques for frequency-domain photon migration imaging,” Biotech. Prog. 13, 669–680 (1997).
[CrossRef]

E. M. Sevick-Muraca, G. Lopez, T. L. Troy, J. S. Reynolds, 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, 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 mammary tumors using fluorescent contrast agents,” (submitted to Photochem. Photobiol).

Schmidt, S. F.

G. L. Smith, S. F. Schmidt, L. A. McGee, “Application of statistical filter theory to the optimal estimation of position and velocity on board a circumlunar vehicle,” (U.S. Government Printing Office, Washington, D.C., 1962).

Schweiger, M.

S. R. Arridge, M. Schweiger, “Image reconstruction in optical tomography,” Phil. Trans. R. Soc. London Series B 352, 717–726 (1997).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near-infrared absorption and scatter imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

Sevick, E. M.

Sevick-Muraca, E. M.

E. M. Sevick-Muraca, G. Lopez, T. L. Troy, J. S. Reynolds, 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, T. L. Troy, E. M. Sevick-Muraca, “Multi-pixel techniques for frequency-domain photon migration imaging,” Biotech. Prog. 13, 669–680 (1997).
[CrossRef]

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

C. L. Hutchinson, T. L. Troy, E. M. Sevick-Muraca, “Fluorescence-lifetime determination in tissues or other scattering media from measurement of excitation and emission kinetics,” Appl. Opt. 35, 2325–2332 (1996).
[CrossRef] [PubMed]

E. M. Sevick-Muraca, C. L. Hutchinson, D. Y. Paithankar, “Optical tissue biodiagnostics using fluorescence lifetime,” Opt. Photon. News 7(1), 25–28 (1996).

T. L. Troy, D. L. Page, E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–355 (1996).
[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 mammary tumors using fluorescent contrast agents,” (submitted to Photochem. Photobiol).

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, E. M. Sevick-Muraca, “Three-dimensional optical tomography,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597 (to be published).

Singer, J. R.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Smith, G. L.

G. L. Smith, S. F. Schmidt, L. A. McGee, “Application of statistical filter theory to the optimal estimation of position and velocity on board a circumlunar vehicle,” (U.S. Government Printing Office, Washington, D.C., 1962).

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 mammary tumors using fluorescent contrast agents,” (submitted to Photochem. Photobiol).

Sun, N.

N. Sun, W. W.-G. Yeh, “Identification of parameter structure in groundwater inverse problem,” Water Resources Res. 21, 869–883 (1985).
[CrossRef]

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 mammary tumors using fluorescent contrast agents,” (submitted to Photochem. Photobiol).

Tromberg, B. J.

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

Troy, T. L.

E. M. Sevick-Muraca, G. Lopez, T. L. Troy, J. S. Reynolds, 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, T. L. Troy, E. M. Sevick-Muraca, “Multi-pixel techniques for frequency-domain photon migration imaging,” Biotech. Prog. 13, 669–680 (1997).
[CrossRef]

T. L. Troy, D. L. Page, E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–355 (1996).
[CrossRef] [PubMed]

C. L. Hutchinson, T. L. Troy, E. M. Sevick-Muraca, “Fluorescence-lifetime determination in tissues or other scattering media from measurement of excitation and emission kinetics,” Appl. Opt. 35, 2325–2332 (1996).
[CrossRef] [PubMed]

T. L. Troy, “Biomedical optical imaging using frequency domain photon migration measurements: experiments and numerical image reconstructions,” Ph.D. dissertation (Purdue University, Lafayette, Ind., 1997).

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 mammary tumors using fluorescent contrast agents,” (submitted to Photochem. Photobiol).

van der Zee, P.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

Venugopalan, V.

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

Wang, Y.

Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering by a Born iterative method,” J. Opt. Soc. Am. A 14, 325–342 (1997).
[CrossRef]

R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

Wang, Y. M.

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imag. 9, 218–225 (1990).
[CrossRef]

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 mammary tumors using fluorescent contrast agents,” (submitted to Photochem. Photobiol).

Wilson, B.

Yao, Y.

Yeh, W. W.-G.

N. Sun, W. W.-G. Yeh, “Identification of parameter structure in groundwater inverse problem,” Water Resources Res. 21, 869–883 (1985).
[CrossRef]

Yodh, A. G.

You, J. S.

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

Zhu, W.

Zubelli, J. P.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Appl. Math. Comp. (1)

J. C. Adams, “mudpack: multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comp. 34, 133–146 (1989).
[CrossRef]

Appl. Opt. (5)

Biotech. Prog. (1)

J. S. Reynolds, T. L. Troy, E. M. Sevick-Muraca, “Multi-pixel techniques for frequency-domain photon migration imaging,” Biotech. Prog. 13, 669–680 (1997).
[CrossRef]

Environ. Geol. (1)

F. Aschenbrenner, A. Ostin, “Automatic parameter estimation applied on a groundwater model: the problem of structure identification,” Environ. Geol. 25, 205–210 (1995).
[CrossRef]

Geophysics (1)

M. J. Eppstein, D. E. Dougherty, “Optimal 3-D traveltime tomography,” Geophysics 63, 1053–1061 (1998).
[CrossRef]

IEEE Trans. Med. Imag. (1)

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imag. 9, 218–225 (1990).
[CrossRef]

J. Basic Eng. D (1)

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. D 82, 35–45 (1960).
[CrossRef]

J. Biomed. Opt. (1)

T. L. Troy, D. L. Page, E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–355 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

Med. Phys. (1)

K. D. Paulsen, H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
[CrossRef] [PubMed]

Opt. Lett. (3)

Opt. Photon. News (1)

E. M. Sevick-Muraca, C. L. Hutchinson, D. Y. Paithankar, “Optical tissue biodiagnostics using fluorescence lifetime,” Opt. Photon. News 7(1), 25–28 (1996).

Phil. Trans. R. Soc. London Series B (1)

S. R. Arridge, M. Schweiger, “Image reconstruction in optical tomography,” Phil. Trans. R. Soc. London Series B 352, 717–726 (1997).
[CrossRef]

Photochem. Photobiol. (1)

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

Phys. Rev. E (1)

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

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

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical heterogeneities with turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Science (1)

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Water Resources Res. (3)

M. J. Eppstein, D. E. Dougherty, “Simultaneous estimation of transmissivity values and zonation,” Water Resources Res. 32, 3321–3336 (1996).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, “Efficient three-dimensional data inversion: soil characterization and moisture monitoring from cross-well ground-penetrating radar at a Vermont test site,” Water Resources Res. 34, 1889–1900 (1998).
[CrossRef]

N. Sun, W. W.-G. Yeh, “Identification of parameter structure in groundwater inverse problem,” Water Resources Res. 21, 869–883 (1985).
[CrossRef]

Other (11)

G. L. Smith, S. F. Schmidt, L. A. McGee, “Application of statistical filter theory to the optimal estimation of position and velocity on board a circumlunar vehicle,” (U.S. Government Printing Office, Washington, D.C., 1962).

A. Gelb, ed., Applied Optimal Estimation (MIT Press, Cambridge, Mass.1974).

M. J. Eppstein, D. E. Dougherty, “Three-dimensional stochastic tomography with upscaling,” U.S. patent application 09/110,506 (9July1998).

N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, 2nd ed. (Wiley, New York, 1995), Vol. 1.

R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near-infrared absorption and scatter imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, “Optimal 3-D geophysical tomography,” in 1998 Proceedings of the Symposium on the Application of Geophysics to Environmental and Engineering Problems (SAGEEP), R. S. Bell, M. H. Powers, T. Larson, eds. (Environmental and Engineering Geophysical Society, Wheat Ridge, Colo.1998), pp. 249–256.

T. L. Troy, “Biomedical optical imaging using frequency domain photon migration measurements: experiments and numerical image reconstructions,” Ph.D. dissertation (Purdue University, Lafayette, Ind., 1997).

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 mammary tumors using fluorescent contrast agents,” (submitted to Photochem. Photobiol).

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, E. M. Sevick-Muraca, “Three-dimensional optical tomography,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597 (to be published).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

(a) Actual absorption μ axf at the excitation wavelength owing to fluorophores in a 16 cm2 synthetic 2-D domain. The domain is discretized onto a 33 × 33 node grid and contains a 3 × 3 node heterogeneity with a 10:1 uptake ratio of light-absorbing dye. The locations of the four light sources (numbered circles) and 120 detectors (small dots) are indicated. The gray scale (in inverse centimeters) applies to (a)–(f) here as well as in Fig. 4 below. (b) Sample reconstruction of the absorption parameter field from modulation phase and amplitude measurements at the excitation wavelength by the Newton-Raphson method.36 (c) Sample reconstruction of absorption from modulation phase measurements at the excitation frequency by the AEKF alone (without DDZ). (d)–(f) sample reconstructions of absorption from modulation phase measurements at the excitation frequency by the APPRIZE method.

Fig. 2
Fig. 2

Evolution of residual errors and dimensionality of parameterization for the sample APPRIZE reconstruction of absorption shown in Fig. 1(d). (a) Reduction in output rms error of modulation phase at the excitation frequency for all 480 measurements. (b) Reduction in output rms error of the modulation log amplitude at the excitation frequency for all 480 measurements. (c) Reduction in rms error of the absorption μ axf parameter estimate at all 1089 nodes. (d) Reduction in the number of zones used in the parameterization. The symbols depict the steps of the process and show how the AEKF is alternated with DDZ for each of the four sources during each iteration of APPRIZE.

Fig. 3
Fig. 3

Sample final estimates of the variance the transformed absorption ξ(μ axf ). (a) Estimated variance of ξ(μ axf ) for the case 1 estimate of μ axf shown in Fig. 1(c) generated by the AEKF alone. (b) Estimated variance of ξ(μ axf ) for the case 1 estimate of μ axf shown in Fig. 1(d) generated by APPRIZE. (c) Estimated variance of ξ(μ axf ) for the case 2 estimate of μ axf shown in Fig. 4(f) generated by APPRIZE.

Fig. 4
Fig. 4

Spectroscopic estimates generated by APPRIZE of homogeneous absorption (true μ axf = 0.01 cm-1) at the excitation wavelength in a synthetic 2-D domain with the same dimensions and measurement configuration as in Fig. 1(a). (a) True absorption map and a (perfect) reconstruction starting from an initial estimate of μ axf = 0.005 cm-1. (b), (c), (d), (e), (f) Sample reconstructions from initial estimates of μ axf = 0.02, 0.04, 0.06, 0.08, 0.1 cm-1, respectively. The gray scale is the same as in Fig. 1(a).

Fig. 5
Fig. 5

(a) Actual fluorescence lifetime (τ) in a synthetic 2-D domain with the same dimensions and measurement configuration as in Fig. 1(a). The domain contains a single heterogeneity with a 1:10 contrast in the upper left-hand quadrant. The gray scale (in nanoseconds) applies to (a)–(c). (b) Reconstructed τ from modulation phase and amplitude measurements at the emission frequency by the Newton–Raphson method.37 (c) (Perfect) reconstruction of τ from modulation phase measurements at the emission frequency by the APPRIZE method described in this paper. (d)–(g) Evolution of residual errors and dimensionality of parameterization for the sample APPRIZE reconstruction of fluorescence lifetime shown in (c): (d) reduction in output rms error of modulation phase at the excitation frequency for all 480 measurements, (e) reduction in output rms error of the modulation log amplitude at the excitation frequency for all 480 measurements, (f) reduction in rms error of the fluorescence lifetime τ parameter estimate at all 1089 nodes, (g) reduction in number of zones used in the parameterization.

Fig. 6
Fig. 6

(a) Actual fluorescence quantum efficiency (ϕ) in a synthetic 2-D domain with the same dimensions and measurement configuration as in Fig. 1(a). The domain contains a single heterogeneity with a 10:1 contrast in the upper left-hand quadrant. The gray scale applies to (a)–(c). (b) Reconstructed ϕ from modulation phase and amplitude measurements at the emission frequency with the Newton–Raphson method.37 (c) (Perfect) reconstruction of ϕ from modulation phase and amplitude measurements at the emission frequency by the APPRIZE method described in this paper. (d)–(g) Evolution of residual errors and dimensionality of parameterization for the sample APPRIZE reconstruction of fluorescence quantum efficiency shown in (c): (d) reduction in output rms error of modulation phase at the excitation frequency for all 480 measurements, (e) reduction in output rms error of the modulation log amplitude at the excitation frequency for all 480 measurements, (f) reduction in rms error of the fluorescence quantum efficiency ϕ parameter estimate at all 1089 nodes, (g) reduction in number of zones used in the parameterization.

Tables (1)

Tables Icon

Table 1 Optical Properties of the Four True Synthetic Domainsa

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

-·DxrΦxr, ω+iωcx+μaxir+μaxfr×Φxr, ω=Sr, ω,
-·DmrΦmr, ω+iωcm+μamir+μamfr×Φmr, ω=ϕμaxf1-iωτ1+ωτ2 Φxr, ω,
Dx,mr=13μax,mr+1-gμsx,mr.
LOOP for a subset of measurements zJ=xyform JacobianPxy=J·Pyyestimate cross covariancePxx=Pxy·JT+Qestimate state covarianceK=PxyT·R+Pxx-1compute gain matrixy=y+K·z-xupdate parameter estimatePyy=Pyy-K·Pxy(update parameter covariance estimate)END LOOP,
ξy=lnB-Apqy-A-pq,
y=1expξy+pqB-Apq+A.
max|yij-cj|maxη, ρ·maxc-minc,

Metrics