Abstract

We extend the backpropagation algorithm of standard diffraction tomography to backpropagation in turbid media. We analyze the behavior of the backpropagation algorithm both for a single-view geometry, as is common in mammography, and for multiple views. The most general form of the algorithm permits arbitrary placement of sources and detectors in the background medium. In addition, we specialize the algorithm for the case of a planar array of detectors, which permits the backpropagation algorithm to be implemented with fast-Fourier-domain noniterative algebraic methods. In this case the algorithm can be used to reconstruct three-dimensional images in a minute or less, depending on the number of views. We demonstrate the theoretical results with computer simulations.

© 1999 Optical Society of America

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References

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  1. 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]
  2. 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]
  3. 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]
  4. 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 tumors in vivo by noninvasive optical methods,” Appl. Opt. 37, 1982–1989 (1998).
    [CrossRef]
  5. 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]
  6. W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. L. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (1997).
    [CrossRef]
  7. W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217 (1997).
    [CrossRef] [PubMed]
  8. Advances in Optical Imaging and Photon Migration, J. G. Fujimoto, M. S. Patterson, eds., Vol. 21 of 1998 OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998).
  9. A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
    [CrossRef]
  10. C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express 1, 6–11 (1997).
    [CrossRef] [PubMed]
  11. C. L. Matson, H. Liu, “Analysis of the forward problem with diffuse photon density waves in turbid media by use of a diffraction tomography model,” J. Opt. Soc. Am. A 16, 455–466 (1999).
    [CrossRef]
  12. 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]
  13. 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]
  14. X. D. Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biomedical imaging with diffuse photon density waves: errata,” Opt. Lett. 22, 1198 (1997).
    [CrossRef]
  15. X. Cheng, D. A. Boas, “Diffuse optical reflection tomography with continuous-wave illumination,” Opt. Express 3, 118–123 (1998).
    [CrossRef] [PubMed]
  16. A. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998).
  17. S. J. Norton, Tuan Vo-Dinh, “Diffraction tomographic imaging with photon density waves: an explicit solution,” J. Opt. Soc. Am. A 15, 2670–2677 (1998).
    [CrossRef]
  18. 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]
  19. C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Signal Process. 42, 156–163 (1994).
    [CrossRef]
  20. E. L. Kosarev, “On the limit of superresolution while restoring signals,” Radiotekh. Elektron. 35, 68–87 (1990).
  21. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953).
  22. G. Barton, Elements of Green’s Functions and Propagation (Oxford U. Press, Oxford, UK, 1989).
  23. A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, UK, 1966).
  24. A. Schatzberg, A. J. Devaney, A. J. Witten, “Estimating target location from scattered field data,” Signal Process. 40, 227–237 (1994).
    [CrossRef]
  25. A. J. Devaney, G. A. Tsihrintzis, “Maximum likelihood estimation of object location in diffraction tomography, part II: strongly scattering objects,” IEEE Trans. Signal Process. 39, 1466–1470 (1991).
    [CrossRef]
  26. G. A. Tsihrintzis, A. J. Devaney, “Maximum likelihood estimation of object location in diffraction tomography,” IEEE Trans. Signal Process. 39, 672–682 (1991).
    [CrossRef]
  27. The Photon Migration Imaging (PMI) software was developed by D. A. 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 David Boas, whose e-mail address is dboas@nmr.mgh.harvard.edu.
  28. V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
    [CrossRef] [PubMed]
  29. H. Liu, C. L. Matson, K. Lau, R. R. Mapakshi, “Experimental validation of a backpropagation algorithm for three-dimensional breast tumor localization,” submitted to IEEE Sel. Top. Quantum Electron.
  30. H. Stark, “Sampling theorems in polar coordinates,” J. Opt. Soc. Am. 69, 1519–1525 (1979).
    [CrossRef]
  31. 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]
  32. 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]
  33. A. J. Devaney, “The limited-view problem in diffraction tomography,” Inverse Probl. 5, 501–521 (1989).
    [CrossRef]
  34. The Interactive Data Language software package is available from Research Systems in Boulder, Colo.

1999

1998

1997

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

W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. L. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (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, 573–575 (1997).
[CrossRef] [PubMed]

X. D. Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biomedical imaging with diffuse photon density waves: errata,” Opt. Lett. 22, 1198 (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]

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]

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217 (1997).
[CrossRef] [PubMed]

1995

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. 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]

1994

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Signal Process. 42, 156–163 (1994).
[CrossRef]

A. Schatzberg, A. J. Devaney, A. J. Witten, “Estimating target location from scattered field data,” Signal Process. 40, 227–237 (1994).
[CrossRef]

1991

A. J. Devaney, G. A. Tsihrintzis, “Maximum likelihood estimation of object location in diffraction tomography, part II: strongly scattering objects,” IEEE Trans. Signal Process. 39, 1466–1470 (1991).
[CrossRef]

G. A. Tsihrintzis, A. J. Devaney, “Maximum likelihood estimation of object location in diffraction tomography,” IEEE Trans. Signal Process. 39, 672–682 (1991).
[CrossRef]

1990

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

E. L. Kosarev, “On the limit of superresolution while restoring signals,” Radiotekh. Elektron. 35, 68–87 (1990).

1989

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

1986

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

1983

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]

1979

Baños, A.

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

Barbour, R. L.

W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. L. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (1997).
[CrossRef]

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217 (1997).
[CrossRef] [PubMed]

Barton, G.

G. Barton, Elements of Green’s Functions and Propagation (Oxford U. Press, Oxford, UK, 1989).

Boas, D. A.

X. Cheng, D. A. Boas, “Diffuse optical reflection tomography with continuous-wave illumination,” Opt. Express 3, 118–123 (1998).
[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]

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]

Chance, B.

Chang, J.

Cheng, X.

Clark, N.

Deng, Y.

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217 (1997).
[CrossRef] [PubMed]

Devaney, A. J.

A. Schatzberg, A. J. Devaney, A. J. Witten, “Estimating target location from scattered field data,” Signal Process. 40, 227–237 (1994).
[CrossRef]

A. J. Devaney, G. A. Tsihrintzis, “Maximum likelihood estimation of object location in diffraction tomography, part II: strongly scattering objects,” IEEE Trans. Signal Process. 39, 1466–1470 (1991).
[CrossRef]

G. A. Tsihrintzis, A. J. Devaney, “Maximum likelihood estimation of object location in diffraction tomography,” IEEE Trans. Signal Process. 39, 672–682 (1991).
[CrossRef]

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

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

Durduran, T.

Fantini, S.

Fender, J. S.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953).

Franceschini, M. A.

Frank, G. L.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Graber, H. L.

Gratton, E.

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]

Jiang, H. B.

Kak, A.

A. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998).

Kaschke, M.

Kosarev, E. L.

E. L. Kosarev, “On the limit of superresolution while restoring signals,” Radiotekh. Elektron. 35, 68–87 (1990).

Lau, K.

H. Liu, C. L. Matson, K. Lau, R. R. Mapakshi, “Experimental validation of a backpropagation algorithm for three-dimensional breast tumor localization,” submitted to IEEE Sel. Top. Quantum Electron.

Li, X. D.

Liu, H.

C. L. Matson, H. Liu, “Analysis of the forward problem with diffuse photon density waves in turbid media by use of a diffraction tomography model,” J. Opt. Soc. Am. A 16, 455–466 (1999).
[CrossRef]

H. Liu, C. L. Matson, K. Lau, R. R. Mapakshi, “Experimental validation of a backpropagation algorithm for three-dimensional breast tumor localization,” submitted to IEEE Sel. Top. Quantum Electron.

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]

Mapakshi, R. R.

H. Liu, C. L. Matson, K. Lau, R. R. Mapakshi, “Experimental validation of a backpropagation algorithm for three-dimensional breast tumor localization,” submitted to IEEE Sel. Top. Quantum Electron.

Matson, C. L.

C. L. Matson, H. Liu, “Analysis of the forward problem with diffuse photon density waves in turbid media by use of a diffraction tomography model,” J. Opt. Soc. Am. A 16, 455–466 (1999).
[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, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Signal Process. 42, 156–163 (1994).
[CrossRef]

H. Liu, C. L. Matson, K. Lau, R. R. Mapakshi, “Experimental validation of a backpropagation algorithm for three-dimensional breast tumor localization,” submitted to IEEE Sel. Top. Quantum Electron.

McMackin, L.

Moesta, K. T.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953).

Norton, S. J.

O’Leary, M. A.

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]

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]

Osterberg, U. L.

Pattanayak, D. N.

Patterson, M. S.

Paulsen, K. D.

Peters, V. G.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Schatzberg, A.

A. Schatzberg, A. J. Devaney, A. J. Witten, “Estimating target location from scattered field data,” Signal Process. 40, 227–237 (1994).
[CrossRef]

Schlag, P. M.

Slaney, M.

A. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998).

Stark, H.

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]

H. Stark, “Sampling theorems in polar coordinates,” J. Opt. Soc. Am. 69, 1519–1525 (1979).
[CrossRef]

Tsihrintzis, G. A.

A. J. Devaney, G. A. Tsihrintzis, “Maximum likelihood estimation of object location in diffraction tomography, part II: strongly scattering objects,” IEEE Trans. Signal Process. 39, 1466–1470 (1991).
[CrossRef]

G. A. Tsihrintzis, A. J. Devaney, “Maximum likelihood estimation of object location in diffraction tomography,” IEEE Trans. Signal Process. 39, 672–682 (1991).
[CrossRef]

Vo-Dinh, Tuan

Walker, S. A.

Wang, Y.

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217 (1997).
[CrossRef] [PubMed]

W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. L. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (1997).
[CrossRef]

Wengrovitz, M.

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]

Witten, A. J.

A. Schatzberg, A. J. Devaney, A. J. Witten, “Estimating target location from scattered field data,” Signal Process. 40, 227–237 (1994).
[CrossRef]

Wyman, D. R.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Yao, Y.

W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. L. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (1997).
[CrossRef]

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217 (1997).
[CrossRef] [PubMed]

Yodh, A. G.

Zhu, W.

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217 (1997).
[CrossRef] [PubMed]

W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. L. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (1997).
[CrossRef]

Appl. Opt.

IEEE Trans. Acoust., Speech, Signal Process.

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]

IEEE Trans. Med. Imaging

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217 (1997).
[CrossRef] [PubMed]

IEEE Trans. Signal Process.

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Signal Process. 42, 156–163 (1994).
[CrossRef]

A. J. Devaney, G. A. Tsihrintzis, “Maximum likelihood estimation of object location in diffraction tomography, part II: strongly scattering objects,” IEEE Trans. Signal Process. 39, 1466–1470 (1991).
[CrossRef]

G. A. Tsihrintzis, A. J. Devaney, “Maximum likelihood estimation of object location in diffraction tomography,” IEEE Trans. Signal Process. 39, 672–682 (1991).
[CrossRef]

Inverse Probl.

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

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

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]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Radiotekh. Elektron.

E. L. Kosarev, “On the limit of superresolution while restoring signals,” Radiotekh. Elektron. 35, 68–87 (1990).

Signal Process.

A. Schatzberg, A. J. Devaney, A. J. Witten, “Estimating target location from scattered field data,” Signal Process. 40, 227–237 (1994).
[CrossRef]

Other

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953).

G. Barton, Elements of Green’s Functions and Propagation (Oxford U. Press, Oxford, UK, 1989).

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

A. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998).

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]

Advances in Optical Imaging and Photon Migration, J. G. Fujimoto, M. S. Patterson, eds., Vol. 21 of 1998 OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998).

H. Liu, C. L. Matson, K. Lau, R. R. Mapakshi, “Experimental validation of a backpropagation algorithm for three-dimensional breast tumor localization,” submitted to IEEE Sel. Top. Quantum Electron.

The Photon Migration Imaging (PMI) software was developed by D. A. 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 David Boas, whose e-mail address is dboas@nmr.mgh.harvard.edu.

The Interactive Data Language software package is available from Research Systems in Boulder, Colo.

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Figures (12)

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 apparatus are not shown.

Fig. 2
Fig. 2

Slices of the frequency-domain responses of low-pass filters used to regularize the reconstruction algorithm. Solid line, pillbox filter; dashed curve, modified Hamming filter; dotted curve, Hanning filter.

Fig. 3
Fig. 3

Slices of the PSF’s corresponding to the low-pass filters shown in Fig. 2. Solid curve, PSF corresponding to the pillbox filter; dashed curve, PSF corresponding to the modified Hamming filter; dotted curve, PSF corresponding to the Hanning filter.

Fig. 4
Fig. 4

Planar slices of the reconstructed DPDW in a turbid-medium volume with a 1-cm-diameter inhomogeneity located at z=2 cm. The background material properties are μa=0.015 cm-1 and μs=14 cm-1, and the inhomogeneity material properties are μa=0.5 cm-1 and μs=12 cm-1. The illumination source is at the bottom of each slice, the detection plane is at the top of each slice perpendicular to each slice, and the slice shown contains the inhomogeneity center. The image size is 8 cm by 8 cm. Clockwise from the upper left: true inhomogeneity location, reconstructed DPDW made with a pillbox filter, reconstructed DPDW made with a Hanning filter, and reconstructed DPDW made with a modified Hamming filter. All three filters have a cutoff frequency of 15 pixels.

Fig. 5
Fig. 5

Planar slices of the reconstructed DPDW in a turbid-medium volume with a 1-cm-diameter inhomogeneity located at z=2 cm. The material and system properties are as described in Fig. 4. Clockwise from the upper left: true inhomogeneity location, reconstructed DPDW made with a pillbox filter, reconstructed DPDW made with a Hanning filter, and reconstructed DPDW made with a modified Hamming filter. All three filters have a cutoff frequency of 40 pixels.

Fig. 6
Fig. 6

Estimated inhomogeneity locations in the z direction for a 1-cm-diameter inhomogeneity located at z=2 cm. The material and system properties are as described in Fig. 4. Solid curve, pillbox filter; dotted curve, Hanning filter; dashed curve, modified Hamming filter.

Fig. 7
Fig. 7

Slice of the detected scattered wave’s Fourier amplitude for a 1-cm-diameter inhomogeneity located at z=2 cm. The material and system properties are as described in Fig. 4.

Fig. 8
Fig. 8

Estimated inhomogeneity locations in the z direction for a 1-cm-diameter inhomogeneity located at z=6 cm. The material and system properties are as described in Fig. 4. Solid curve, pillbox filter; dotted curve, Hanning filter; the dashed curve, modified Hamming filter.

Fig. 9
Fig. 9

Slice of the detected scattered wave’s Fourier amplitude for an inhomogeneity located at z=6 cm. The material and system properties are as described in Fig. 4.

Fig. 10
Fig. 10

Schematic of a system and target used for the multiple-view backpropagation reconstructions. The background material properties are μa=0.015 cm-1 and μs=14 cm-1, the upper inhomogeneity material properties are μa=0.5 cm-1 and μs=12 cm-1, and the lower inhomogeneity material properties are μa=0.7 cm-1 and μs=9 cm-1. Both spheres have a diameter of 1 cm. The upper inhomogeneity is located at x=2 cm, y=0, and z=6 cm; the lower inhomogeneity is located at x=0,y=0, and z=2 cm; the illumination source is located at x=y=z=0; and the detection plane is located at z=8 cm.

Fig. 11
Fig. 11

Reconstruction of the two-sphere inhomogeneity, described in Fig. 10, made with the pillbox filter.

Fig. 12
Fig. 12

Reconstruction of the two-sphere inhomogeneity, described in Fig. 10, made with the modified Hamming filter.

Equations (21)

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u(x, y, z)=-3µsδμa(x, y, z)u0(x, y, z)×g(x-x, y-y, z-z)+δμs(x, y, z)μs+δμs(x, y, z)u0(x, y, z)·g(x-x, y-y, z-z)dxdydz,
U(ωx, ωy; z0)=-exp(-iz0γωi)2γω3µsδμa(x, y, z)+δμs(x, y, z)μs+δμs(x, y, z)×iωxx+iωyy-γωz×u0(x, y, z)exp[-(z0-z)γωr]×exp[-i(xωx+yωy-zγωi)]dxdydz,
γωγωr+iγωi=Re[(ωx2+ωy2-k2)1/2]+i Im[(ωx2+ωy2-k2)1/2].
kkr+iki=Re(-vμa+i2πft)3µsv1/2+i Im(-vμa+i2πft)3µsv1/2,
u0(x, y, z)=exp[i(z-z1)k],
Upw(ωx, ωy; z0)
=-exp(-iz0γωi)2γωexp[-(z0-z1)ki]×exp(-iz1kr)3µsδμa(x, y, z)-ikγωδμs(x, y, z)μs+δμs(x, y, z)×exp[-(z0-z)(γωr-ki)]×exp{-i[xωx+yωy+z×(-γωi-kr)]}dxdydz,
urec(x, y, z)=u(x, y, z)nˆ×gb(x-x, y-y, z-z)dxdydz,
gb(x-x, y-y, z-z)
=18π2F(ωx, ωy)γω×{exp[(z-z)γω]-exp[(2z0-z-z)γω]}×exp[(x-x)ωx+(y-y)ωy]dωxdωy,
urec(x, y, z)=u(x, y, z0)14π2F(ωx, ωy)×exp[(z0-z)γω]exp[(x-x)ωx+(y-y)ωy]dωxdωydxdy.
Urec(ωx, ωy; z)=F(ωx, ωy)exp[(z0-z)γω]×U(ωx, ωy; z0).
Hb(ωx, ωy; z0-z)=exp[(z0-z)γω].
Urec(ωx, ωy; z3)
=-F(ωx, ωy)K1(z3)exp(-iz3γωi)2γω×3μsδμa(x, y, z)+δμs(x, y, z)μs+δμs(x, y, z)×iωxx+iωyy-γωzu0(x, y, z)×exp[-(z3-z)γωr]×exp[-i(xωx+yωy-zγωi)]dxdydz,
H1(ρ)=10ρR00otherwise,
H2(ρ)=0.625+0.375 cosπρR00ρR0,0otherwise,
H3(ρ)=0.5+0.5 cosπρR00ρR00otherwise,
Urec,pw(ωx, ωy; z3)
=-F(ωx, ωy)K1(z3)exp(-iz3γωi)2γω×exp[-(z3-z1)ki]exp(-iz1kr)×3µsδμa(x, y, z)-ikγωδμs(x, y, z)μs+δμs(x, y, z)×exp[-(z3-z)(γωr-ki)]exp{-i[xωx+yωy+z(-γωi-kr)]}dxdydz.
I(x, y, z)=ϕurec(x, y, z; ϕ),

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