Abstract

On comparison with the usual propagating scalar waves, the attenuation of diffuse photon density waves gives rise to important differences in structural information, such as higher spatial resolution in detection at short distances from objects and deviation from the Rayleigh limit at larger distances. This damping also establishes a minimum spatial resolution threshold for diffusive waves, which occurs by illumination in continuous mode, and demonstrates that in most cases spatial resolution is not improved by increasing the modulation frequency. Assessments of this formulation with numerical simulations of scattering and wave-front reconstruction in the presence of noise are given.

© 1999 Optical Society of America

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  1. A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34 (1995), and references therein.
    [CrossRef]
  2. E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
    [CrossRef] [PubMed]
  3. See related studies 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).
  4. S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for near infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
    [CrossRef]
  5. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
    [CrossRef] [PubMed]
  6. C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
    [CrossRef]
  7. 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, 214–220 (1997).
    [CrossRef] [PubMed]
  8. 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, 573–575 (1997).
    [CrossRef] [PubMed]
  9. H. Wabnitz, H. Rinneberg, “Imaging in turbid media by photon density waves: spatial resolution and scaling relations,” Appl. Opt. 36, 64–74 (1997).
    [CrossRef] [PubMed]
  10. 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]
  11. 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]
  12. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Simultaneous reconstruction of optical absorption and scattering maps in turbid media from near-infrared frequency-domain data,” Opt. Lett. 20, 2128–2130 (1995).
    [CrossRef] [PubMed]
  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. P. N. den Outer, T. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
    [CrossRef]
  15. S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 35, 3826–3837 (1995).
    [CrossRef]
  16. S. R. Arridge, J. C. Hebden, “Optical imaging in medicine. II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
    [CrossRef] [PubMed]
  17. J. C. Schotland, “Continuous-wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997).
    [CrossRef]
  18. S. R. Arridge, M. Schweiger, “A gradient-based optimization scheme for optical tomography,” Opt. Express 2, 213–226 (1998).
    [CrossRef] [PubMed]
  19. S. J. Norton, T. Vo-Dinh, “Diffraction tomographic imaging with photon density waves: an explicit solution,” J. Opt. Soc. Am. A 15, 2670–2677 (1998).
    [CrossRef]
  20. S. A. Walker, D. A. Boas, E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37, 1935–1944 (1998).
    [CrossRef]
  21. S. Fantini, S. A. Walker, M. A. Franceschini, M. Kaschke, P. M. Schlag, K. T. Moesta, “Assessment of the size, position, and optical properties of breast tumor in vivo by noninvasive optical methods,” Appl. Opt. 37, 1982–1989 (1998).
    [CrossRef]
  22. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Imaging with diffusing light: an experimental study of the effect of background optical properties,” Appl. Opt. 37, 3564–3573 (1998).
    [CrossRef]
  23. B. DeBecker, A. Bulatov, J. L. Birman, “Two-dimensional inverse problem of diffusion tomography: the approach applicable for small inclusions,” Appl. Opt. 37, 4294–4299 (1998).
    [CrossRef]
  24. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).
  25. J. A. Moon, R. Mahon, M. D. Duncan, J. Reintjes, “Resolution limits for imaging through turbid media with diffuse light,” Opt. Lett. 18, 1591–1593 (1993).
    [CrossRef] [PubMed]
  26. S. R. Arridge, W. R. B. Lionheart, “Nonuniqueness in diffusion-bases optical tomography,” Opt. Lett. 23, 882–884 (1998).
    [CrossRef]
  27. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.
  28. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995), Chap. 3.
  29. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley—Interscience, New York, 1991), Chap. 2.
  30. For near-field optical methods see, e.g., D. W. Pohl, D. Courjon, eds., Near Field Optics (Kluwer Academic, Dordrecht, The Netherlands, 1993) or M. Nieto-Vesperinas, N. Garcı́a, eds., Optics at the Nanometer Scale (Kluwer Academic, Dordrecht, The Netherlands, 1996). For a recent review on the subject see J. J. Greffet, R. Carminati, Image Formation in Near Field Optics, Prog. Surf. Sci. 56, 133 (1997).
    [CrossRef]
  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. J. Ripoll, M. Nieto-Vesperinas, “Scattering integral equations for diffusive waves. Detection of objects buried in diffusive media in the presence of rough interfaces,” J. Opt. Soc. Am. A (to be published).
  35. J. Ripoll, M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse–diffuse interfaces,” J. Opt. Soc. Am. A (to be published).
  36. See, for example, R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  37. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
    [CrossRef]
  38. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
    [CrossRef] [PubMed]
  39. 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]

1998 (7)

1997 (11)

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]

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]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine. II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

J. C. Schotland, “Continuous-wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997).
[CrossRef]

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, 214–220 (1997).
[CrossRef] [PubMed]

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, 573–575 (1997).
[CrossRef] [PubMed]

H. Wabnitz, H. Rinneberg, “Imaging in turbid media by photon density waves: spatial resolution and scaling relations,” Appl. Opt. 36, 64–74 (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. 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. 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]

1996 (1)

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

1995 (6)

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

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Simultaneous reconstruction of optical absorption and scattering maps in turbid media from near-infrared frequency-domain data,” Opt. Lett. 20, 2128–2130 (1995).
[CrossRef] [PubMed]

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34 (1995), and references therein.
[CrossRef]

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 35, 3826–3837 (1995).
[CrossRef]

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

1994 (1)

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

1993 (2)

1992 (1)

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

1991 (1)

’t Hooft, G. W.

Aronson, R.

Arridge, S. R.

S. R. Arridge, W. R. B. Lionheart, “Nonuniqueness in diffusion-bases optical tomography,” Opt. Lett. 23, 882–884 (1998).
[CrossRef]

S. R. Arridge, M. Schweiger, “A gradient-based optimization scheme for optical tomography,” Opt. Express 2, 213–226 (1998).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine. II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

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

Barbour, R. L.

Birman, J. L.

Boas, D. A.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).

Bulatov, A.

Chance, B.

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]

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, 573–575 (1997).
[CrossRef] [PubMed]

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

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34 (1995), and references therein.
[CrossRef]

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 35, 3826–3837 (1995).
[CrossRef]

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

Clark, N.

Colak, S. B.

Cope, M.

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

Cubeddu, R.

de Haller, E. B.

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

DeBecker, B.

Delpy, D. T.

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

den Outer, P. N.

Duncan, M. D.

Durduran, T.

Fantini, S.

Fender, J. S.

Feng, S.

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 35, 3826–3837 (1995).
[CrossRef]

Franceschini, M. A.

Gonatas, C. P.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

Gratton, E.

Hebden, J. C.

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine. II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Ishii, M.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

Jiang, H.

Kaschke, M.

Lagendijk, A.

Leigh, J. S.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

Li, X. D.

Lionheart, W. R. B.

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]

Mahon, R.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995), Chap. 3.

Matson, C. L.

McMackin, L.

Melissen, J. B. M.

Moesta, K. T.

Moon, J. A.

Nieto-Vesperinas, M.

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]

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, “Scattering integral equations for diffusive waves. Detection of objects buried in diffusive media in the presence of rough interfaces,” J. Opt. Soc. Am. A (to be published).

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

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley—Interscience, New York, 1991), Chap. 2.

Nieuwenhuizen, T. M.

Norton, S. J.

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]

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

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

Osterberg, U. L.

Paasschens, J. C. J.

Papaioannou, D. G.

Pattanayak, D. N.

Patterson, M. S.

Paulsen, K. D.

Pei, Y.

Pifferi, A.

Pogue, B. W.

Reintjes, J.

Rinneberg, H.

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,” J. Opt. Soc. Am. A (to be published).

J. Ripoll, M. Nieto-Vesperinas, “Scattering integral equations for diffusive waves. Detection of objects buried in diffusive media in the presence of rough interfaces,” J. Opt. Soc. Am. A (to be published).

Sánchez-Gil, J. A.

Schlag, P. M.

Schomberg, H.

Schotland, J. C.

J. C. Schotland, “Continuous-wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997).
[CrossRef]

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

Schweiger, M.

Taroni, P.

Torricelli, A.

Valentini, G.

van Asten, N. A. A. J.

van der Mark, M. B.

van der Zee, P.

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

Vo-Dinh, T.

Wabnitz, H.

Walker, S. A.

Wang, Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995), Chap. 3.

Yao, Y.

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34 (1995), and references therein.
[CrossRef]

Yodh, A. G.

Zeng, F.

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 35, 3826–3837 (1995).
[CrossRef]

Zhu, W.

Appl. Opt. (10)

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, 214–220 (1997).
[CrossRef] [PubMed]

H. Wabnitz, H. Rinneberg, “Imaging in turbid media by photon density waves: spatial resolution and scaling relations,” Appl. Opt. 36, 64–74 (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. 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]

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 35, 3826–3837 (1995).
[CrossRef]

S. A. Walker, D. A. Boas, E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37, 1935–1944 (1998).
[CrossRef]

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

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Imaging with diffusing light: an experimental study of the effect of background optical properties,” Appl. Opt. 37, 3564–3573 (1998).
[CrossRef]

B. DeBecker, A. Bulatov, J. L. Birman, “Two-dimensional inverse problem of diffusion tomography: the approach applicable for small inclusions,” Appl. Opt. 37, 4294–4299 (1998).
[CrossRef]

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]

J. Biomed. Opt. (1)

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

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

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]

Opt. Express (1)

Opt. Lett. (5)

Phys. Med. Biol. (1)

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine. II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Phys. Rev. E (1)

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

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

Phys. Today (1)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34 (1995), and references therein.
[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 inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
[CrossRef]

Other (9)

J. Ripoll, M. Nieto-Vesperinas, “Scattering integral equations for diffusive waves. Detection of objects buried in diffusive media in the presence of rough interfaces,” J. Opt. Soc. Am. A (to be published).

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995), Chap. 3.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley—Interscience, New York, 1991), Chap. 2.

For near-field optical methods see, e.g., D. W. Pohl, D. Courjon, eds., Near Field Optics (Kluwer Academic, Dordrecht, The Netherlands, 1993) or M. Nieto-Vesperinas, N. Garcı́a, eds., Optics at the Nanometer Scale (Kluwer Academic, Dordrecht, The Netherlands, 1996). For a recent review on the subject see J. J. Greffet, R. Carminati, Image Formation in Near Field Optics, Prog. Surf. Sci. 56, 133 (1997).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).

See related studies 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).

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

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Figures (12)

Fig. 1
Fig. 1

Geometry used for the angular-spectrum representation.

Fig. 2
Fig. 2

Values of (a) qr and (b) qi, normalized to κr: solid circles, PSW’s; open circles, DPDW’s.

Fig. 3
Fig. 3

Amplitude [(a) PSW’s and (b) DPDW’s] and phase [(c) PSW’s and (d) DPDW’s] of F(K, z) for different values of z: solid circles, z=λ; open circles, z=0.5λ; stars, z=0.1λ; squares, z=0.05λ.

Fig. 4
Fig. 4

Amplitude of H(R, z) for (a) PSW’s and (b) DPDW’s for different values of z: solid circles, z=λ; open circles, z=0.5λ; stars, z=0.1λ; squares, z=0.05λ.

Fig. 5
Fig. 5

Spatial resolution limit Δd in centimeters as z increases, for the following cases: DPDW’s in breast tissue (μa=0.035 cm-1, μs=15 cm-1): solid curve, ω=0 [dc]; dotted curve, ω=100 MHz (λ0=13.38 cm); dashed curve, ω=200 MHz (λ0=7.53 cm); dotted-dashed curve, ω=300 MHz (λ0=5.60 cm). PSW’s: squares, dc; solid circles, λ0=13.38 cm; open circles, λ0=7.53 cm. In all cases n=1.333.

Fig. 6
Fig. 6

DPDW’s spatial resolution limit Δd in centimeters as we increment z, in dc regime (ω=0), for the following cases: solid curve, breast parameters μa=0.035 cm-1, μs=15 cm-1; dotted curve, abdomen parameters μa=0.09 cm-1, μs=9.5 cm-1; short-dashed curve, back parameters μa=0.09 cm-1, μs=10.5 cm-1; long-dashed curve, white matter μa=0.22 cm-1, μs=9.1 cm-1; dotted-dashed curve, grey matter μa=0.27 cm-1, μs=20.6 cm-1. In all cases n=1.333.

Fig. 7
Fig. 7

Scattering geometry.

Fig. 8
Fig. 8

Scattered amplitude corresponding to two diffuse cylinders of R=0.1 cm with breast tumor parameters μa=0.24 cm-1, μs=10 cm-1, embedded in breast tissue (μa=0.035 cm-1, μs=15 cm-1), with the source located at rsource=(0, 2.0 cm) with modulation frequency ω=200 MHz, separated by distances (a) d=1 cm, (b) d=1.5 cm, (c) d=2.0 cm, (d) d=2.5 cm for the following Z detector distances: solid curve, zdetect=0.2 cm; dotted curve, zdetect=0.4 cm; short-dashed curve, zdetect=0.6 cm; long-dashed curve, zdetect=0.8 cm; dotted-dashed curve, zdetect=1.0 cm. In all cases n=1.333.

Fig. 9
Fig. 9

Values of the noise-free contrast Cnf(%) as we vary the detector-plane distance zdetect in the case of two cylinders of radius R=0.1 cm, with breast tumor parameters μa=0.24 cm-1, μs=10 cm-1, embedded in breast tissue (μa=0.035 cm-1, μs=15 cm-1), with the source located at rsource=(0, 2.0 cm) with modulation frequency ω=200 MHz, separated by the following distances: solid curve, d=1 cm; dotted curve, d=1.5 cm; short-dashed curve, d=2.0 cm; long-dashed curve, d=2.5 cm. In all cases n=1.333.

Fig. 10
Fig. 10

Values of (a) U˜noise(K, z=1 cm), (b) U(SC)(K, z=1 cm), (c) U˜filt(K, z=1 cm)=U(SC)(K, z=1 cm)N(K) for a detector-plane distance zdetect=1 cm in the case of two cylinders of radius R=0.1 cm, with breast tumor parameters μa=0.24 cm-1, μs=10 cm-1, embedded in breast tissue (μa=0.035 cm-1, μs=15 cm-1), with the source located at rsource=(0, 2.0 cm) with modulation frequency ω=200 MHz, separated by a distance of d=2.5 cm. N(K) is a Hanning filter with Kcut=15κr. Noise parameters: η=10% and σξ=10°. In all cases n=1.333.

Fig. 11
Fig. 11

Normalized scattered amplitude in the case of two cylinders of radius R=0.1 cm, with breast tumor parameters μa=0.24 cm-1, μs=10 cm-1, embedded in breast tissue (μa=0.035 cm-1, μs=15 cm-1), with the source located at rsource=(0, 2.0 cm) with modulation frequency ω=200 MHz, separated by a distance d=2.5 cm in the following cases. (a) Measured at a plane-detection distance of zdetect=1 cm with noise parameters η=10% and σξ=10°. (b) Measured at a plane-detection distance of zdetect=1.5 cm with noise parameters η=30% and σξ=10°. (c) Solid curve, after filtering by a Hanning filter with Kcut=15κr, the image obtained in (a); dotted curve, direct measurement without noise at zdetect=1 cm; solid circle, after filtering by a Hanning filter with Kcut=15κr, an image with noise parameters η=30% and σξ=10°. (d) Solid curve, after filtering by a Hanning filter with Kcut=15κr, an image with noise parameters η=10% and σξ=10°; dotted curve, direct measurement without noise at zdetect=1.5 cm; solid circle, after filtering by a Hanning filter with Kcut=15κr, the image obtained in (b). In all cases n=1.333.

Fig. 12
Fig. 12

Normalized scattered amplitude backpropagated onto z=0.2 cm in the case of two cylinders of radius R=0.1 cm, with breast tumor parameters (μa=0.24 cm-1, μs=10 cm-1), embedded in breast tissue (μa=0.035 cm-1, μs=15 cm-1), with the source located at rsource=(0, 2.0 cm) with modulation frequency ω=200 MHz, separated by a distance d=2.5 cm, for the following. (a) Solid line, noise-free image taken at zdetect=1.0 cm backpropagated with Kcut=10κr; dotted curve, direct measurement at zdetect=0.2 cm. (b) Solid curve, noise-free image taken at zdetect=1.5 cm backpropagated with Kcut=10κr; dotted-curve, direct measurement at zdetect=0.2 cm. (c) Solid curve, image taken at zdetect=1.0 cm with noise parameters σξ=10°, η=10% backpropagated with Kcut=10κr; dotted curve, image taken at zdetect=1.0 cm with noise parameters σξ=10°, η=30% backpropagated with Kcut=8κr. (d) Solid curve, image taken at zdetect=1.5 cm with noise parameters σξ=10°, η=10% backpropagated with Kcut=7κr; dotted curve, image taken at zdetect=1.5 cm with noise parameters σξ=10°, η=30% backpropagated with Kcut=6κr. In all cases n=1.333.

Equations (20)

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κ0=(-μa/D+iωn/cD)1/2,
U(R, z)=-+A(K)exp[iK·R+iq(K)z]dK,
A(K)exp[iq(K)z]=14π2-+U(R, z)exp(-iK·R)dR.
H(R, z)=-+F(K, z)exp(iK·R)dK.
U(R, z)=-+H(R-R, z)U(R, z=0)dR.
|F(K, z)|=exp[-(|K|2-κ02z)1/2]=1/2,
|K|2-κ02=(ln 2/z)2.
Δ|K|=2[κ02+(ln 2/z)2]1/2,
Δd/λ0=1/2{1+[(ln 2/2πz/λ0)]2}-1/2,
|F(K, z)||F(K=0, z)|=exp{-[qi(K)-κi]z}=12,
Δ|K|=2κi+ln 2z2+κr2-κi2-κi2κr2(κi+ln 2/z)21/2,
Δdλ0=12λ0la+ln 22πz/λ02-λ0la2+1-1+ln 22πz/la-2-1/2.
Δd=(π/ln 2)z.
Δd=121la+ln 22πz2-1la2-1/2.
Cnf(%)=|Umax(SC)|-|Umin(SC)||Umax(SC)|+|Umin(SC)|×100.
Unoise(R, z)=[|U(SC)(R, z)|+N(R)]×exp{i[ϕ(R, z)+ξ(R)]},
η(%)=σN/|Umax(SC)|×100,
C(%)=Cnf(%)-η(%).
Ufilt(R, z)=-Kcut+KcutU˜noise(K, z)N(K)exp(iK·R)dK,
N(K)=12+12cosKxKcutxπ·12+12cosKyKcutyπ,

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