Abstract

We extend our previously developed diffraction tomography model of diffuse photon density wave propagation in turbid media to analyze the forward problem associated with detecting and resolving both absorptive and scattering inhomogeneities. Our results assume that detection occurs in a plane but no restrictions are placed on the illumination source geometry. We then specialize these results to plane-wave illumination and derive the turbid media version of the Fourier diffraction theorem. We also develop a shot-noise-limited Fourier-domain signal-to-noise-ratio (SNR) expression to determine how background, system, and inhomogeneity parameters affect one’s ability to detect and resolve inhomogeneities. We show that, in general, scattering inhomogeneities are more easily resolved than absorbing inhomogeneities. We also show that lower temporal modulation frequencies enhance one’s ability to detect and resolve inhomogeneities. These theoretical results are compared with previously published image-domain SNR results, and qualitative agreement is demonstrated.

© 1999 Optical Society of America

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    [CrossRef]
  30. C. L. Matson, E. P. Magee, D. E. Holland, “Reflective tomography using a short-pulselength laser: system analysis for artificial satellite imaging,” Opt. Eng. 34, 2811–2820 (1995).
    [CrossRef]
  31. C. L. Matson, “Resolution, linear filtering, and positivity,” J. Opt. Soc. Am. A 15, 33–41 (1998).
    [CrossRef]
  32. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  33. A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. I. Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
    [CrossRef]
  34. C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
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1998 (1)

1997 (7)

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]

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, 214–220 (1997).
[CrossRef] [PubMed]

H. B. Jiang, K. D. Paulsen, U. L. Osterberg, M. S. Patterson, “Frequency-domain optical-image reconstruction in turbid media—an experimental study of single-target detectability,” Appl. Opt. 36, 52–63 (1997).
[CrossRef] [PubMed]

H. B. Jiang, K. D. Paulsen, U. L. Osterberg, M. S. Patterson, “Frequency-domain optical-image reconstruction in turbid media—an experimental study of single-target detectability: erratum,” Appl. Opt. 36, 2995–2996 (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]

C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express 1, 6–11 (1997).
[CrossRef] [PubMed]

1995 (3)

1994 (4)

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 91, 4887–4891 (1994).
[CrossRef] [PubMed]

J. A. Izatt, M. R. Hee, G. M. Owen, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, A. Gandjbakche, M. A. Eckhaus, “Optical coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264, 1913–1915 (1994).
[CrossRef] [PubMed]

1993 (2)

1992 (2)

A. Schatzberg, A. J. Devaney, “Super-resolution in diffraction tomography,” Inverse Probl. 8, 149–164 (1992).
[CrossRef]

C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
[CrossRef]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (2)

1987 (2)

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. I. Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

A. J. Devaney, “Linearised inverse scattering in attenuating media,” Inverse Probl. 3, 389–397 (1987).
[CrossRef]

1986 (1)

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
[CrossRef]

1983 (1)

H. Stark, M. Wengrovitz, “Comments and corrections on the use of polar sampling theorems in CT,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1329–1331 (1983).
[CrossRef]

1980 (1)

B. Ohlsen, J. Gunderson, D. M. Nilson, “Diaphanography: a method for evaluation of the female breast,” World J. Surg. 4, 701–706 (1980).
[CrossRef]

1979 (1)

1929 (1)

M. Catler, “Transillumination as an aid in the diagnosis of breast lesions. With special reference to its value in cases of bleeding nipple,” Surg. Gynecol. Obstet. 48, 721–729 (1929).

’t Hooft, G. W.

Alfano, R. R.

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264, 1913–1915 (1994).
[CrossRef] [PubMed]

B. B. Das, K. M. Yoo, R. R. Alfano, “Ultrafast time-gated imaging in thick tissues—a step towards optical mammography,” Opt. Lett. 18, 1092–1094 (1993).
[CrossRef]

Baños, A.

A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, UK, 1966).

Barry, N. P.

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]

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 91, 4887–4891 (1994).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffuse waves within the Rytov approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 320–327 (1995).
[CrossRef]

Casanove, M. J.

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. I. Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

Catler, M.

M. Catler, “Transillumination as an aid in the diagnosis of breast lesions. With special reference to its value in cases of bleeding nipple,” Surg. Gynecol. Obstet. 48, 721–729 (1929).

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]

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 91, 4887–4891 (1994).
[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]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffuse waves within the Rytov approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 320–327 (1995).
[CrossRef]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Clark, N.

Colak, S. B.

Dainty, J. C.

Das, B. B.

DeLarue, I. A.

C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
[CrossRef]

Devaney, A. J.

A. Schatzberg, A. J. Devaney, “Super-resolution in diffraction tomography,” Inverse Probl. 8, 149–164 (1992).
[CrossRef]

A. J. Devaney, “The limited-view problem in diffraction tomography,” Inverse Probl. 5, 501–521 (1989).
[CrossRef]

A. J. Devaney, “Linearised inverse scattering in attenuating media,” Inverse Probl. 3, 389–397 (1987).
[CrossRef]

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
[CrossRef]

Dilworth, D. S.

Drunzer, I. E.

C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
[CrossRef]

Durduran, T.

Eckhaus, M. A.

J. M. Schmitt, A. Knüttel, A. Gandjbakche, M. A. Eckhaus, “Optical coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

Fender, J. S.

Fishkin, J. B.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

French, P. M. W.

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gandjbakche, A.

J. M. Schmitt, A. Knüttel, A. Gandjbakche, M. A. Eckhaus, “Optical coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Gratton, E.

Gray, T. M.

C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gunderson, J.

B. Ohlsen, J. Gunderson, D. M. Nilson, “Diaphanography: a method for evaluation of the female breast,” World J. Surg. 4, 701–706 (1980).
[CrossRef]

Hee, M. R.

J. A. Izatt, M. R. Hee, G. M. Owen, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Ho, P. P.

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264, 1913–1915 (1994).
[CrossRef] [PubMed]

Holland, D. E.

C. L. Matson, E. P. Magee, D. E. Holland, “Reflective tomography using a short-pulselength laser: system analysis for artificial satellite imaging,” Opt. Eng. 34, 2811–2820 (1995).
[CrossRef]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hyde, S. C. W.

Izatt, J. A.

Jiang, H. B.

Jones, R.

Kak, A.

A. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

Klein, M. B.

Knüttel, A.

J. M. Schmitt, A. Knüttel, A. Gandjbakche, M. A. Eckhaus, “Optical coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

Lannes, A.

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. I. Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

Leith, E. N.

Li, X. D.

Liang, X.

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264, 1913–1915 (1994).
[CrossRef] [PubMed]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Lopez, J. L.

Magee, E. P.

C. L. Matson, E. P. Magee, D. E. Holland, “Reflective tomography using a short-pulselength laser: system analysis for artificial satellite imaging,” Opt. Eng. 34, 2811–2820 (1995).
[CrossRef]

Matson, C. L.

C. L. Matson, “Resolution, linear filtering, and positivity,” J. Opt. Soc. Am. A 15, 33–41 (1998).
[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]

C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express 1, 6–11 (1997).
[CrossRef] [PubMed]

C. L. Matson, E. P. Magee, D. E. Holland, “Reflective tomography using a short-pulselength laser: system analysis for artificial satellite imaging,” Opt. Eng. 34, 2811–2820 (1995).
[CrossRef]

C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
[CrossRef]

McMackin, L.

Melissen, J. B. M.

Nilson, D. M.

B. Ohlsen, J. Gunderson, D. M. Nilson, “Diaphanography: a method for evaluation of the female breast,” World J. Surg. 4, 701–706 (1980).
[CrossRef]

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 91, 4887–4891 (1994).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffuse waves within the Rytov approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 320–327 (1995).
[CrossRef]

Ohlsen, B.

B. Ohlsen, J. Gunderson, D. M. Nilson, “Diaphanography: a method for evaluation of the female breast,” World J. Surg. 4, 701–706 (1980).
[CrossRef]

Osterberg, U. L.

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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffuse waves within the Rytov approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 320–327 (1995).
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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).
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[CrossRef]

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A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
[CrossRef]

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[CrossRef]

A. J. Devaney, “Linearised inverse scattering in attenuating media,” Inverse Probl. 3, 389–397 (1987).
[CrossRef]

A. J. Devaney, “The limited-view problem in diffraction tomography,” Inverse Probl. 5, 501–521 (1989).
[CrossRef]

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A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. I. Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

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

Opt. Eng. (1)

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[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Phys. Med. Biol. (1)

J. M. Schmitt, A. Knüttel, A. Gandjbakche, M. A. Eckhaus, “Optical coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

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 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Science (2)

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264, 1913–1915 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

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[CrossRef]

Other (8)

A. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffuse waves within the Rytov approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 320–327 (1995).
[CrossRef]

A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, UK, 1966).

G. H. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Medical Optical Tomography: Functional Imaging and Monitoring, Vol. IS11 of Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991), p. 418.

M. A. O’Leary, “Imaging with diffuse photon density waves,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1996).

The PMI software was developed by D. Boas, M. A. O’Leary, X. Li, B. Chance, A. G. Yodh, M. A. Ostermeyer, S. L. Jacques. It is available on the Web or from D. Boas (email, dboas@emerald.tufts.edu).

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Figures (7)

Fig. 1
Fig. 1

Conceptual diagram showing the geometry for the diffraction tomography development. The inhomogeneity is represented by the cube, which is assumed to be embedded in an infinite homogeneous turbid medium. The center of the inhomogeneity is located at x=0, y=0, z=z2, and the detection plane is located at z0. The illumination and detection apparatuses are not shown.

Fig. 2
Fig. 2

Plots of the two-dimensional projection of the surface on which the Fourier transform of the object function is obtained with diffraction tomography. The solid line is for low-frequency DPDW’s, and the dashed curve is for high-frequency DPDW’s.

Fig. 3
Fig. 3

Plots of the Fourier-domain SNR’s using the baseline parameter values. The curves correspond to analytical predictions, and the symbols correspond to PMI predictions. The solid curve and the squares are for nonzero δμa and δμs, the dashed curve and the triangles are for δμs=0, and the dotted–dashed curve and the pluses are for δμa=0.

Fig. 4
Fig. 4

Plots of the Fourier-domain SNR’s for variations in the modulation frequency of the DPDW. The solid curve is for an absorbing inhomogeneity and ft=10 MHz, the dashed curve is for an absorbing inhomogeneity and ft=1000 MHz, the dotted–dashed curve is for a scattering inhomogeneity and ft=10 MHz, and the dotted curve is for a scattering inhomogeneity and ft=1000 MHz.

Fig. 5
Fig. 5

Plots of the Fourier-domain SNR’s for variations in the background μa. The solid curve is for an absorbing inhomogeneity and μa=0.01 cm-1, the dashed curve is for an absorbing inhomogeneity and μa=0.06 cm-1, the dotted–dashed curve is for a scattering inhomogeneity and μa=0.01 cm-1, and the dotted curve is for a scattering inhomogeneity and μa=0.06 cm-1.

Fig. 6
Fig. 6

Plots of the Fourier-domain SNR’s for variations in the background μs. The solid curve is for an absorbing inhomogeneity and μs=8 cm-1, the dashed curve is for an absorbing inhomogeneity and μs=16 cm-1, the dotted–dashed curve is for a scattering inhomogeneity and μs=8 cm-1, and the dotted curve is for a scattering inhomogeneity and μs=16 cm-1.

Fig. 7
Fig. 7

Plots of the Fourier-domain SNR’s for variations in the location of the inhomogeneity. The solid curve is for an absorbing inhomogeneity and z2=2 cm, the dashed curve is for an absorbing inhomogeneity and z2=5 cm, the dotted–dashed curve is for a scattering inhomogeneity and z2=2 cm, and the dotted curve is for a scattering inhomogeneity and z2=5 cm.

Equations (26)

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(2+k2)uBa(x, y, z)=-oa(x, y, z)u0(x, y, z),
uBa(x, y, z)=oa(x, y, z)u0(x, y, z)×g(x-x, y-y, z-z)dxdydz,
uBa(x, y, z)=oa(x, y, z)u0(x, y, z)×18π2 1γαexp[i(x-x)αx+i(y-y)αy-|z-z|γα]dαxdαy×dxdydz,
γαγαr+iγαi=Re[(αx2+αy2-k2)1/2]+i Im[(αx2+αy2-k2)1/2]
UBa(ωx, ωy; z0)=18π2 1γαexp(ixαx+iyαy)exp(-ixωx-iyωy)×oa(x, y, z)u0(x, y, z)×exp(-|z0-z|γα)×exp(-ixαx-iyαy)dxdydzdαxdαydxdy.
UBa(ωx, ωy; z0)=exp(-iz0γωi)2γωoa(x, y, z)u0(x, y, z)×exp[-(z0-z)γωr]×exp[-i(xωx+yωy-zγωi)]dxdydz,
uBs(x, y, z)=os(x, y, z)u0(x, y, z)·g(x-x, y-y, z-z)dxdydz,
g(x, y, z)=xˆ x+yˆ x+zˆ x×18π21γαexp(ixαx+iyαy-|z|γα)dαxdαy=18π2 1γα(ixˆαx+iyˆαy-zˆγα)×exp(ixαx+iyαy-zγα)dαxdαy,
uBs(x, y, z)
=18π2os(x, y, z)×iαx x+iαy y-γα zu0(x, y, z)×1γαexp[i(x-x)αx+i(y-y)αy-(z-z)γα]dαxdαydxdydz.
UBs(ωx, ωy; z0)=exp(-iz0γωi)2γωos(x, y, z)×iωx x+iωy y-γω zu0(x, y, z)×exp[-(z0-z)γωr]exp[-i(xωx+yωy-zγωi)]dxdydz.
UB(ωx, ωy; z0)=exp(-iz0γωi)2γωoa(x, y, z)+os(x, y, z)iωx x+iωy y-γω z×u0(x, y, z)exp[-(z0-z)γωr]×exp[-i(xωx+yωy-zγωi)]dxdydz.
u0(x, y, z)=exp[i(z-z1)k]
UB(ωx, ωy; z0)=exp(-iz0γωi)2γωexp[-(z0-z1)ki] exp(-iz1kr)×[oa(x, y, z)-ikγωos(x, y, z)]×exp[-(z0-z)(γωr-ki)]×exp{-i[xωx+yωy+z(-γωi-kr)]}dxdydz,
UB(ωx, ωy; z0)=exp(-iz0γωi)2γωO(ωx, ωy,-γωi-kr).
SNR(ωx, ωy)=|UB(ωx, ωy)|[VarUB(ωx, ωy)]1/2,
SNR(ωx, ωy)
12γωoa(x, y, z)+os(x, y, z)iωx x+iωy y-γω z×u0(x, y, z)exp[-(z0-z)γωr]×exp[-i(xωx+yωy-zγωi)]dxdydz|u0(x, y, z0)|dxdy1/2,
u0(x, y, z)=3μsA4π|r-r1|exp(ik|r-r1|),
SNR(ωx, ωy)
3μsA8πγωoa(x, y, z)+os(x, y, z)[iωx(x-x1)+iωy(y-y1)-γω(z-z1)]ik|r-r1|-1|r-r1|2×exp(ik|r-r1|)|r-r1|exp[-(z0-z)γωr]×exp[-i(xωx+yωy-zγωi)]dxdydz|u0(x, y, z0)|dxdy1/2,
oa(x, y, z)=-3μsδμa(x, y, z),
os(x, y, z)=-δμs(x, y, z)μs+δμs(x, y, z),
SNR(ωx, ωy)3μsA8πγω-3μsδμa(x, y, z)-δμs(x, y, z)μs+δμs(x, y, z)[iωx(x-x1)+iωy(y-y1)-γω(z-z1)]×ik|r-r1|-1|r-r1|2exp(ik|r-r1|)|r-r1|×exp[-(z0-z)γωr]exp[-i(xωx+yωy-zγωi)]dxdydz|u0(x, y, z0)|dxdy1/2.
SNR(ωx, ωy)3μsAKa38πexp[-(z2-z1)ki]exp[-(z0-z2)γωr]z2-z1δμsμs+δμsik-1z2-z1-3μsδμaγω|u0(x, y, z0)|dxdy1/2,
k2=(-vμa+i2πft) 3μsv,

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