Abstract

The surface integral formalism is used to derive the integral equations for the scattering of diffusive waves, which account for the contribution of object boundaries and interfaces between media and which are numerically solved without approximations. The extinction theorem and other surface integral theorems for diffusive waves are introduced to obtain the boundary values of both the photon density wave and the photon density current. We present this theory and apply it to the simulation of diffusive objects buried in diffusive media in the presence of rough interfaces.

© 1999 Optical Society of America

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  1. E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
    [CrossRef] [PubMed]
  2. A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48 (March), 38–40 (1995).
    [CrossRef]
  3. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1960).
  4. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. I.
  5. K. Michelsen, H. de Raedt, N. Garcı́a, “Time gated transillumination and reflection by biological tissues and tissuelike phantoms: simulations versus experiments,” J. Opt. Soc. Am. A 14, 1867–1871 (1997).
    [CrossRef]
  6. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  7. S. Fantini, M. A. Franceschini, E. Gratton, “Semi-infinite-geometry boundary problem for light migration in highly scattering media: a frequency-domain study in the diffusion approximation,” J. Opt. Soc. Am. B 11, 2128–2138 (1994).
    [CrossRef]
  8. J. B. Frishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” Appl. Opt. 36, 10–20 (1997).
    [CrossRef]
  9. S. Fantini, S. A. Walker, M. A. Franceschini, M. Kashke, P. M. Schlag, K. T. Moesta, “Assessment of the size, position, and optical properties of breast tumors in vivo by noninvasive optical methods,” Appl. Opt. 37, 1982–1989 (1998).
    [CrossRef]
  10. P. N. den Outer, T. M. Niewenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
    [CrossRef]
  11. X. D. Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–576 (1997).
    [CrossRef] [PubMed]
  12. Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by a Born iterative method,” J. Opt. Soc. Am. A 14, 325–342 (1997).
    [CrossRef]
  13. S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 180–213 (1997).
    [CrossRef] [PubMed]
  14. C. L. Matson, N. Clark, L. McMackin, J. S. Fender, “Three-dimensional tumor localization in thick tissue with the use of diffuse photon-density waves,” Appl. Opt. 36, 214–219 (1997).
    [CrossRef] [PubMed]
  15. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffusive photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
    [CrossRef]
  16. S. C. Feng, F. Zeng, B. Chance, “Photon migrations in the presence of a single defect: a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
    [CrossRef] [PubMed]
  17. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
    [CrossRef] [PubMed]
  18. S. A. Walker, S. Fantini, E. Gratton, “Image reconstruction by backprojection from frequency-domain optical measurements in highly scattering media,” Appl. Opt. 36, 170–179 (1997).
    [CrossRef] [PubMed]
  19. S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34, 7395–7409 (1995).
    [CrossRef] [PubMed]
  20. H. Jiang, K. D. Paulsen, U. L. Ostenberg, 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]
  21. M. R. Ostermeyer, S. L. Jacques, “Perturbation theory for diffuse light transport in complex biological tissues,” J. Opt. Soc. Am. A 14, 255–261 (1997).
    [CrossRef]
  22. J. Wu, “Convolution picture of the boundary conditions in photon migration and its implications in time-resolved optical imaging in biological tissues,” J. Opt. Soc. Am. A 14, 280–287 (1997).
    [CrossRef]
  23. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley-Interscience, New York, 1991).
  24. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  25. R. C. Haskell, L. O. Svaasand, T. Tsay, T. 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]
  26. T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–65.
  27. J. Ripoll, M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse–diffuse interfaces,” submitted to J. Opt. Soc. Am. A.
  28. Y. N. Barabanenkov, A. Y. Kargashin, “Diffusion calculation of change of backscattered light beam intensity from turbid medium owing to the existence of an inhomogeneity,” J. Mod. Phys. 40, 2243–2255 (1993).
  29. H. Stark, T. C. Lubensky, “Multiple light scattering in anisotropic random media,” Phys. Rev. Lett. 77, 2229–2232 (1996).
    [CrossRef] [PubMed]
  30. G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, New York, 1995).
  31. J. A. Sánchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
    [CrossRef]
  32. A. Madrazo, M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1859–1866 (1997).
    [CrossRef]
  33. J. Ripoll, A. Madrazo, M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a body over a random rough surface,” Opt. Commun. 142, 173–178 (1997).
    [CrossRef]
  34. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffusive photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
    [CrossRef]
  35. F. F. Jöbsis, “Noninvasive, infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198, 1264–1267 (1977).
    [CrossRef] [PubMed]

1998 (1)

1997 (12)

J. B. Frishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” Appl. Opt. 36, 10–20 (1997).
[CrossRef]

K. Michelsen, H. de Raedt, N. Garcı́a, “Time gated transillumination and reflection by biological tissues and tissuelike phantoms: simulations versus experiments,” J. Opt. Soc. Am. A 14, 1867–1871 (1997).
[CrossRef]

X. D. Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–576 (1997).
[CrossRef] [PubMed]

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

S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 180–213 (1997).
[CrossRef] [PubMed]

C. L. Matson, N. Clark, L. McMackin, J. S. Fender, “Three-dimensional tumor localization in thick tissue with the use of diffuse photon-density waves,” Appl. Opt. 36, 214–219 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

S. A. Walker, S. Fantini, E. Gratton, “Image reconstruction by backprojection from frequency-domain optical measurements in highly scattering media,” Appl. Opt. 36, 170–179 (1997).
[CrossRef] [PubMed]

M. R. Ostermeyer, S. L. Jacques, “Perturbation theory for diffuse light transport in complex biological tissues,” J. Opt. Soc. Am. A 14, 255–261 (1997).
[CrossRef]

J. Wu, “Convolution picture of the boundary conditions in photon migration and its implications in time-resolved optical imaging in biological tissues,” J. Opt. Soc. Am. A 14, 280–287 (1997).
[CrossRef]

A. Madrazo, M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1859–1866 (1997).
[CrossRef]

J. Ripoll, A. Madrazo, M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a body over a random rough surface,” Opt. Commun. 142, 173–178 (1997).
[CrossRef]

1996 (3)

H. Stark, T. C. Lubensky, “Multiple light scattering in anisotropic random media,” Phys. Rev. Lett. 77, 2229–2232 (1996).
[CrossRef] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Ostenberg, 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]

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

1995 (4)

1994 (3)

1993 (2)

P. N. den Outer, T. M. Niewenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
[CrossRef]

Y. N. Barabanenkov, A. Y. Kargashin, “Diffusion calculation of change of backscattered light beam intensity from turbid medium owing to the existence of an inhomogeneity,” J. Mod. Phys. 40, 2243–2255 (1993).

1992 (1)

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffusive photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

1991 (1)

1989 (1)

1977 (1)

F. F. Jöbsis, “Noninvasive, infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198, 1264–1267 (1977).
[CrossRef] [PubMed]

’t Hooft, G. W.

Anderson, E. R.

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, New York, 1995).

Aronson, R.

Arridge, S. R.

Barabanenkov, Y. N.

Y. N. Barabanenkov, A. Y. Kargashin, “Diffusion calculation of change of backscattered light beam intensity from turbid medium owing to the existence of an inhomogeneity,” J. Mod. Phys. 40, 2243–2255 (1993).

Barbour, R. L.

Boas, D. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffusive photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffusive photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–65.

Brenner, M.

Chance, B.

X. D. Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–576 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

S. C. Feng, F. Zeng, B. Chance, “Photon migrations in the presence of a single defect: a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
[CrossRef] [PubMed]

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48 (March), 38–40 (1995).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffusive photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffusive photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

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

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–65.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1960).

Clark, N.

Colak, S. B.

Coquoz, O.

de Haller, E. B.

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

de Raedt, H.

den Outer, P. N.

Durduran, T.

X. D. Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–576 (1997).
[CrossRef] [PubMed]

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–65.

Fantini, S.

Fender, J. S.

Feng, S. C.

Feng, T.

Franceschini, M. A.

Frishkin, J. B.

Garci´a, N.

Gratton, E.

Haskell, R. C.

Ishimaru, A.

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

Jacques, S. L.

Jiang, H.

Jöbsis, F. F.

F. F. Jöbsis, “Noninvasive, infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198, 1264–1267 (1977).
[CrossRef] [PubMed]

Kargashin, A. Y.

Y. N. Barabanenkov, A. Y. Kargashin, “Diffusion calculation of change of backscattered light beam intensity from turbid medium owing to the existence of an inhomogeneity,” J. Mod. Phys. 40, 2243–2255 (1993).

Kashke, M.

Lagendijk, A.

Li, X. D.

Lubensky, T. C.

H. Stark, T. C. Lubensky, “Multiple light scattering in anisotropic random media,” Phys. Rev. Lett. 77, 2229–2232 (1996).
[CrossRef] [PubMed]

Madrazo, A.

A. Madrazo, M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1859–1866 (1997).
[CrossRef]

J. Ripoll, A. Madrazo, M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a body over a random rough surface,” Opt. Commun. 142, 173–178 (1997).
[CrossRef]

Matson, C. L.

McAdams, M. S.

McMackin, L.

Melissen, J. B. M.

Michelsen, K.

Moesta, K. T.

Nieto-Vesperinas, M.

J. Ripoll, A. Madrazo, M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a body over a random rough surface,” Opt. Commun. 142, 173–178 (1997).
[CrossRef]

A. Madrazo, M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1859–1866 (1997).
[CrossRef]

J. A. Sánchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
[CrossRef]

J. Ripoll, M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse–diffuse interfaces,” submitted to J. Opt. Soc. Am. A.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley-Interscience, New York, 1991).

Niewenhuizen, T. M.

O’Leary, M. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffusive photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffusive photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

Ostenberg, U. L.

Ostermeyer, M. R.

Paasschens, J. C. J.

Papaioannou, D. G.

Pattanayak, D. N.

Patterson, M. S.

Paulsen, K. D.

Pei, Y.

Pogue, B. W.

Ripoll, J.

J. Ripoll, A. Madrazo, M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a body over a random rough surface,” Opt. Commun. 142, 173–178 (1997).
[CrossRef]

J. Ripoll, M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse–diffuse interfaces,” submitted to J. Opt. Soc. Am. A.

Sánchez-Gil, J. A.

Schlag, P. M.

Schomberg, H.

Stark, H.

H. Stark, T. C. Lubensky, “Multiple light scattering in anisotropic random media,” Phys. Rev. Lett. 77, 2229–2232 (1996).
[CrossRef] [PubMed]

Svaasand, L. O.

Tromberg, B. J.

Tsay, T.

van Asten, N. A. A. J.

van der Mark, M. B.

Walker, S. A.

Wang, Y.

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, New York, 1995).

Wilson, B. C.

Wu, J.

Yao, Y.

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48 (March), 38–40 (1995).
[CrossRef]

Yodh, A. G.

X. D. Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–576 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffusive photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffusive photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef]

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–65.

Zeng, F.

Zhu, W.

Appl. Opt. (9)

J. B. Frishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” Appl. Opt. 36, 10–20 (1997).
[CrossRef]

S. Fantini, S. A. Walker, M. A. Franceschini, M. Kashke, P. M. Schlag, K. T. Moesta, “Assessment of the size, position, and optical properties of breast tumors in vivo by noninvasive optical methods,” Appl. Opt. 37, 1982–1989 (1998).
[CrossRef]

S. C. Feng, F. Zeng, B. Chance, “Photon migrations in the presence of a single defect: a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

S. A. Walker, S. Fantini, E. Gratton, “Image reconstruction by backprojection from frequency-domain optical measurements in highly scattering media,” Appl. Opt. 36, 170–179 (1997).
[CrossRef] [PubMed]

S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34, 7395–7409 (1995).
[CrossRef] [PubMed]

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

S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, N. A. A. J. van Asten, “Tomographic image reconstruction from optical projections in light-diffusing media,” Appl. Opt. 36, 180–213 (1997).
[CrossRef] [PubMed]

C. L. Matson, N. Clark, L. McMackin, J. S. Fender, “Three-dimensional tumor localization in thick tissue with the use of diffuse photon-density waves,” Appl. Opt. 36, 214–219 (1997).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

J. Mod. Phys. (1)

Y. N. Barabanenkov, A. Y. Kargashin, “Diffusion calculation of change of backscattered light beam intensity from turbid medium owing to the existence of an inhomogeneity,” J. Mod. Phys. 40, 2243–2255 (1993).

J. Opt. Soc. Am. A (10)

J. A. Sánchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
[CrossRef]

A. Madrazo, M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1859–1866 (1997).
[CrossRef]

K. Michelsen, H. de Raedt, N. Garcı́a, “Time gated transillumination and reflection by biological tissues and tissuelike phantoms: simulations versus experiments,” J. Opt. Soc. Am. A 14, 1867–1871 (1997).
[CrossRef]

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

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[CrossRef]

R. C. Haskell, L. O. Svaasand, T. Tsay, T. 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]

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Figures (10)

Fig. 1
Fig. 1

Scattering geometry.

Fig. 2
Fig. 2

Different geometries corresponding to two values of the correlation length, T=0.5 cm (left column) and T=1.0 cm (right column), for three rms heights, σ=0.05 cm (top row), σ=0.1 cm (middle row), and σ=0.2 cm (bottom row). In all cases the mean radius ρ was 5 cm.

Fig. 3
Fig. 3

(a) Relative amplitude |ϕ|/|ϕ(inc)| and (b) total phase Δ for a smooth cylinder of radius 5 cm. The source position was r0=(0, 6 cm). Other parameters were μaout=0.02 cm-1, Dout=0.0333 cm, μain=0.035 cm-1, and Din=0.019004 cm.

Fig. 4
Fig. 4

Relative amplitude |ϕ|/|ϕ(inc)| for the following rough-cylinder parameters: (a) T=0.5 cm and σ=0.1 cm, (b) T=0.5 cm and σ=0.2 cm, (c) T=1.0 cm and σ=0.1 cm, (d) T=1.0 cm and σ=0.2 cm. In all cases the mean radius ρ was 5 cm, and r0=(0, 6 cm). Other parameters were μaout=0.02 cm-1, Dout=0.0333 cm, μain=0.035 cm-1, and Din=0.019004 cm.

Fig. 5
Fig. 5

Total phase Δ for the following object sizes and positions: radius R=1 cm: (a) (x, y)=(0, 0), (c) (x, y)=(0, 3 cm), (e) (x, y)=(0,-3 cm), (g) (x, y)=(3 cm, 0); radius R=0.2 cm: (b) (x, y)=(0, 0), (d) (x, y)=(0, 3 cm), (f) (x, y)=(0,-3 cm), (h) (x, y)=(3 cm, 0). In all cases a smooth cylinder of radius 5 cm was used, and r0=(0, 6 cm). Other parameters were μaout=0.02 cm-1, Dout=0.0333 cm, μain=0.035 cm-1, Din=0.019004 cm, μaobj=1.0 cm-1, and Dobj=0.019608 cm.

Fig. 6
Fig. 6

Relative contribution of the object, i.e., |ϕ(obj)|/|ϕ(no obj)| for the following object sizes and positions: radius R=1 cm: (a) (x, y)=(0, 0), (c) (x, y)=(0, 3 cm), (e) (x, y)=(0,-3 cm), (g) (x, y)=(3 cm, 0); radius R=0.2 cm: (b) (x, y)=(0, 0), (d) (x, y)=(0, 3 cm), (f) (x, y)=(0,-3 cm), (h) (x, y)=(3 cm, 0). In all cases a smooth cylinder of radius 5 cm was used, and r0=(0, 6 cm). Other parameters were as in Fig. 5.

Fig. 7
Fig. 7

Total phase Δ for a rough cylinder with parameters T=0.5 cm, σ=0.2 cm, and ρ=5 cm for the following object sizes and positions: radius R=1 cm: (a) (x, y)=(0, 0), (c) (x, y)=(0, 3 cm), (e) (x, y)=(0,-3 cm), (g) (x, y)=(3 cm, 0); radius R=0.2 cm: (b) (x, y)=(0, 0), (d) (x, y)=(0, 3 cm), (f) (x, y)=(0,-3 cm), (h) (x, y)=(3 cm, 0). Other parameters were as in Fig. 5.

Fig. 8
Fig. 8

Relative contribution of the object to the scattered wave in percent, {[|ϕ(sc obj)|-|ϕ(sc no obj)|]/|ϕ(sc obj)|}×100, for an object of radius R=1 cm placed at positions (a) (x, y)=(0, 0) and (b) (x, y)=(0, 3.5 cm) and for an object of radius R=0.2 cm placed at positions (c) (x, y)=(0, 0) and (d) (x, y)=(0, 3.5 cm). In all cases the scan was performed at a constant radial distance of 6 cm, equal to the source position distance r0=(0, 6 cm). The scan was performed in the case of a rough cylinder of statistics T=0.5 cm, σ=0.2 cm, and ρ=5 cm (solid curves) and in the case of a smooth cylinder (dotted curves). Other parameters were as in Fig. 5.

Fig. 9
Fig. 9

Relative contribution of the object to the scattered wave in percent, {[|ϕ(sc obj)|-|ϕ(sc no obj)|]/|ϕ(sc obj)|}×100, for an object of radius R=1 cm placed at positions (a) (x, y)=(3 cm, 0) and (b) (x, y)=(0,-3 cm) and for an object of radius R=0.2 cm placed at positions (c) (x, y)=(3 cm, 0) and (d) (x, y)=(0,-3 cm). In all cases the scan was performed at a constant radial distance of 6 cm, equal to the source position distance r0=(0, 6 cm). The scan was performed in the case of a rough cylinder of statistics T=0.5 cm, σ=0.2 cm, and ρ=5 cm (solid curves) and in the case of a smooth cylinder (dotted curves). Other parameters were as in Fig. 5.

Fig. 10
Fig. 10

(a) Scattered amplitude for a rough cylinder with statistics T=0.5 cm, σ=0.2 cm, and ρ=5 cm, (b) scattered amplitude for a smooth cylinder of radius 5 cm, (c) relative phase [Δ-Δ(no obj)] for a rough cylinder with statistics T=0.5 cm, σ=0.2 cm, and ρ=5 cm, (d) relative phase [Δ-Δ(no obj)] for a smooth cylinder of radius 5 cm. All cases were performed for the following positions of an R=1 cm object: no object (dotted–dashed curves) (x, y)=(0, 2.5 cm) [dotted curves and (1)], and (x, y)=(0, 3.5 cm) [solid curves and (2)]. The scan was performed at a constant radial distance of 6 cm, equal to the source position distance r0=(0, 6 cm). Other parameters were as in Fig. 5.

Equations (39)

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1vϕ(r, t)t+·J(r, t)+μa(r)ϕ(r, t)=q0(r, t).
J(r, t)=-D(r)ϕ(r, t).
1vϕ(r, t)t-·[D(r)ϕ(r, t)]
+μa(r)ϕ(r, t)=q0(r, t).
2ϕ(r)+D(r)D(r)·ϕ(r)+κ2(r)ϕ(r)=-S0(r)D(r),
κ2(r)=-μa(r)D(r)+iωvD(r).
D(r)=D0,rV˜D1(r),rV,
Δκ2(r)=0,rV˜κ02-κ12(r),rV.
2ϕ(r)+κ02ϕ(r)=-S0(r)D(r)+Δκ2(r)ϕ(r)-D(r)D(r)·ϕ(r),
2G(κ0|r-r|)+κ02G(κ0|r-r|)=-4πδ(r-r).
V˜δ(r-r)ϕ(r)d3r
=-14πV˜-S0(r)D0G(κ0|r-r|)d3r-14πS˜[ϕ(r)rG(κ0|r-r|)-G(κ0|r-r|)rϕ(r)]·dS,
Vδ(r-r)ϕ(r)d3r
=-14πV-S0(r)D1(r)G(κ0|r-r|)d3r-14πVG(κ0|r-r|)Δκ2(r)ϕ(r)d3r-14πVG(κ0|r-r|)-rD1(r)D1(r)·rϕ(r)d3r-14πS[ϕ(r)rG(κ0|r-r|)-G(κ0|r-r|)r|ϕ(r)]·dS.
ϕ+(r)|S=ϕ-(r)|S,
D0·nˆ(r)·ϕ+(r)|S=D1(r)·nˆ(r)·ϕ-(r)|S,
S˜dS=S()dS+S[-nˆ(r)]dS,
ϕ+(r>)=ϕ(inc)(r>)+14πΣS(+)(r>)+14πΣ()(r>),
0=ϕ(inc)(r<)+14πΣS(+)(r<)+14πΣ()(r<),
ϕ-(r<)=14πVG(κ0|r<-r|)D1(r)D1(r)·ϕ-(r)d3r-14πVG(κ0|r<-r|)Δκ2(r)ϕ-(r)d3r-14πΣS(-)(r<),
0=14πVG(κ0|r>-r|)D1(r)D1(r)·ϕ-(r)d3r-14πVG(κ0|r>-r|)Δκ2(r)ϕ-(r)d3r-14πΣS(-)(r>),
ΣS(+,-)(r)=Sϕ(+,-)(r)G(κ0|r-r|)n-G(κ0|r-r|)ϕ(+,-)(r)ndS.
ΣS(-)(r)=ΣS(+)(r)+S1-D0D(r)|S×G(κ0|r-r|)ϕ+(r)ndS;
Σ()(r)=S()ϕ+(r)G(κ0|r-r|)n-G(κ0|r-r|)ϕ+(r)ndS=0.
14πV˜-S0(r)D0G(κ0|r-r)d3r,
ϕ+(r>)
=ϕ(inc)(r>)-14πVG(κ0|r>-r|)Δκ2(r)ϕ-(r)d3r+14πVG(κ0|r>-r|)D1(r)D1(r)·ϕ-(r)d3r-14πS1-D0D(r)|SG(κ0|r>-r|)ϕ+(r)ndS,
ϕ-(r<)
=ϕ(inc)(r<)-14πVG(κ0|r<-r|)Δκ2(r)ϕ-(r)d3r+14πVG(κ0|r<-r|)D1(r)D1(r)·ϕ-(r)d3r+14πS1-D(r)|SD0G(κ0|r<-r|)ϕ-(r)ndS.
[ϕ(out)(r)]R(θ)=[ϕ(in)(r)]R(θ),
[ϕ(in)(r)]ξ=[ϕ(obj)(r)]ξ,
Doutϕ(out)(r)mR(θ)=Dinϕ(in)(r)mR(θ),
Dinϕ(in)(r)nξ=Dobjϕ(obj)(r)nξ,
ϕ(out)(r)=ϕ(inc)(r-r0)+14π×R(θ)ϕ(out)(r)G(κout|r-r|)m-G(κout|r-r|)ϕ(out)(r)mdS,
0=ϕ(inc)(r-r0)+14πR(θ)ϕ(out)(r)G(κout|r-r|)m-G(κout|r-r|)ϕ(out)(r)mdS,
ϕ(in)(r)=-14πR(θ)ϕ(in)(r)G(κin|r-r|)m-G(κin|r-r|)ϕ(in)(r)mdS+14πξϕ(in)(r)G(κin|r-r|)n-G(κin|r-r|)ϕ(in)(r)ndS,
0=-14πR(θ)ϕ(in)(r)G(κin|r-r|)m-G(κin|r-r|)ϕ(in)(r)mdS+14πξϕ(r)G(κin|r-r|)n-G(κin|r-r|)ϕ(in)(r)ndS,
ϕ(obj)(r)=-14πξϕ(obj)(r)G(κobj|r-r|)n-G(κobj|r-r|)ϕ(obj)(r)ndS,
0=-14πξϕ(obj)(r)G(κobj|r-r|)n-G(κobj|r-r|)ϕ(obj)(r)ndS.

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