Abstract

The reconstruction of the location and optical properties of objects in turbid media requires the solution of the inverse problem. Iterative solutions to this problem can require large amounts of computing time and may not converge to a unique solution. Instead, we propose a fast, simple method for approximately solving this problem in which calculated effective absorption and reduced scattering coefficients are backprojected to create an image of the objects. We reconstructed images of objects with centimeter dimensions embedded in a diffusive medium with optical characteristics similar to those of human tissue. Data were collected by a frequency-domain spectrometer operating at 120 MHz with a laser diode light source emitting at 793 nm. Intensity and phase of the incident photon density wave were collected from linear scans at different projection angles. Although the positions of the objects are correctly identified by the reconstructed images, the optical parameters of the objects are recovered only qualitatively.

© 1997 Optical Society of America

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  1. 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]
  2. S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieri, E. Gratton, “Quantitative determination of the absorption spectra of chromophores in strongly scattering media: a light-emitting-diode based technique,” Appl. Opt. 33, 5204–5213 (1994).
    [Crossref] [PubMed]
  3. B. C. Wilson, M. S. Patterson, B. W. Poque, “Instrumentation for in vivo tissue spectroscopy imaging,” in Medical Lasers and Systems II, D. M. Harris, C. M. Penney, A. Katzir, eds., Proc. SPIE1892, 132–147 (1991).
    [Crossref]
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  5. S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
    [Crossref]
  6. R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, ed., Proc. SPIE1431, 192–203 (1991).
    [Crossref]
  7. J. R. Singer, F. A. Grünbaum, P. Kohn, J. Passamani Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
    [Crossref] [PubMed]
  8. K. D. Paulsen, H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
    [Crossref] [PubMed]
  9. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson “Optical image reconstruction using frequency-domain data: simulations and experiments,” Opt. Image Sci. 13, 253–266 (1996).
    [Crossref]
  10. M. A. Oleary, 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]
  11. B. W. Pogue, M. S. Patterson, H. Jiang, K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. 40, 1709–1729 (1995).
    [Crossref] [PubMed]
  12. M. A. Franceschini, S. Fantini, S. A. Walker, J. S. Maier, E. Gratton, “Multi-channel optical instrument for near-infrared imaging of tissues,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 264–273 (1995).
    [Crossref]
  13. J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
    [Crossref] [PubMed]
  14. K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
    [Crossref]
  15. J.-M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.
  16. S. F. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
    [Crossref]
  17. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988), pp. 49–74.
  18. R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.
  19. B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorimetry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
    [Crossref]
  20. 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]
  21. D. A. Boas, M. A. Oleary, 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]
  22. P. Krämmer, H. Bartelt, H. Fischer, B. Schmauss, “Imaging in scattering media using the phase of modulated light sources,” in Photon Propagation in Tissues, B. Chance, D. Delpy, G. J. Mueller, eds., Proc. SPIE2626, 65–74 (1995).
  23. J. S. Maier, E. Gratton, “Frequency-domain methods in optical tomography: detection of localized absorbers and a backscattering reconstruction scheme,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 440–451 (1993).
    [Crossref]
  24. S. Fantini, M. A. Franceschini, S. A. Walker, J. S. Maier, E. Gratton, “Photon path distributions in turbid media: applications for imaging,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 340–349 (1995).
  25. S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, “Optical image reconstruction with deconvolution in light diffusing media,” in Photon Propagation in Tissues, B. Chance, D. Delpy, G. J. Mueller, eds., Proc. SPIE2626, 306–315 (1995).
    [Crossref]

1996 (1)

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson “Optical image reconstruction using frequency-domain data: simulations and experiments,” Opt. Image Sci. 13, 253–266 (1996).
[Crossref]

1995 (3)

M. A. Oleary, 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]

B. W. Pogue, M. S. Patterson, H. Jiang, K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. 40, 1709–1729 (1995).
[Crossref] [PubMed]

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

1994 (4)

S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieri, E. Gratton, “Quantitative determination of the absorption spectra of chromophores in strongly scattering media: a light-emitting-diode based technique,” Appl. Opt. 33, 5204–5213 (1994).
[Crossref] [PubMed]

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[Crossref]

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]

D. A. Boas, M. A. Oleary, 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]

1993 (1)

1990 (1)

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

1989 (2)

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]

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorimetry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[Crossref]

’t Hooft, G. W.

S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, “Optical image reconstruction with deconvolution in light diffusing media,” in Photon Propagation in Tissues, B. Chance, D. Delpy, G. J. Mueller, eds., Proc. SPIE2626, 306–315 (1995).
[Crossref]

Aronson, R.

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, ed., Proc. SPIE1431, 192–203 (1991).
[Crossref]

Arridge, S. R.

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

Barbieri, B.

Barbour, R. L.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, ed., Proc. SPIE1431, 192–203 (1991).
[Crossref]

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.

Bartelt, H.

P. Krämmer, H. Bartelt, H. Fischer, B. Schmauss, “Imaging in scattering media using the phase of modulated light sources,” in Photon Propagation in Tissues, B. Chance, D. Delpy, G. J. Mueller, eds., Proc. SPIE2626, 65–74 (1995).

Boas, D. A.

M. A. Oleary, 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]

D. A. Boas, M. A. Oleary, 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]

Chance, B.

M. A. Oleary, 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]

D. A. Boas, M. A. Oleary, 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]

S. F. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
[Crossref]

Chang, J.

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.

Colak, S. B.

S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, “Optical image reconstruction with deconvolution in light diffusing media,” in Photon Propagation in Tissues, B. Chance, D. Delpy, G. J. Mueller, eds., Proc. SPIE2626, 306–315 (1995).
[Crossref]

Cope, M.

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

Delpy, D. T.

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

Fantini, S.

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]

S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieri, E. Gratton, “Quantitative determination of the absorption spectra of chromophores in strongly scattering media: a light-emitting-diode based technique,” Appl. Opt. 33, 5204–5213 (1994).
[Crossref] [PubMed]

M. A. Franceschini, S. Fantini, S. A. Walker, J. S. Maier, E. Gratton, “Multi-channel optical instrument for near-infrared imaging of tissues,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 264–273 (1995).
[Crossref]

S. Fantini, M. A. Franceschini, S. A. Walker, J. S. Maier, E. Gratton, “Photon path distributions in turbid media: applications for imaging,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 340–349 (1995).

Feddersen, B. A.

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorimetry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[Crossref]

Feng, S. F.

S. F. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
[Crossref]

Fischer, H.

P. Krämmer, H. Bartelt, H. Fischer, B. Schmauss, “Imaging in scattering media using the phase of modulated light sources,” in Photon Propagation in Tissues, B. Chance, D. Delpy, G. J. Mueller, eds., Proc. SPIE2626, 65–74 (1995).

Fishkin, J. B.

Franceschini, M. A.

S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieri, E. Gratton, “Quantitative determination of the absorption spectra of chromophores in strongly scattering media: a light-emitting-diode based technique,” Appl. Opt. 33, 5204–5213 (1994).
[Crossref] [PubMed]

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]

S. Fantini, M. A. Franceschini, S. A. Walker, J. S. Maier, E. Gratton, “Photon path distributions in turbid media: applications for imaging,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 340–349 (1995).

M. A. Franceschini, S. Fantini, S. A. Walker, J. S. Maier, E. Gratton, “Multi-channel optical instrument for near-infrared imaging of tissues,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 264–273 (1995).
[Crossref]

Furutsu, K.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[Crossref]

Graber, H. L.

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, ed., Proc. SPIE1431, 192–203 (1991).
[Crossref]

Gratton, E.

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]

S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieri, E. Gratton, “Quantitative determination of the absorption spectra of chromophores in strongly scattering media: a light-emitting-diode based technique,” Appl. Opt. 33, 5204–5213 (1994).
[Crossref] [PubMed]

J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[Crossref] [PubMed]

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorimetry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[Crossref]

J. S. Maier, E. Gratton, “Frequency-domain methods in optical tomography: detection of localized absorbers and a backscattering reconstruction scheme,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 440–451 (1993).
[Crossref]

M. A. Franceschini, S. Fantini, S. A. Walker, J. S. Maier, E. Gratton, “Multi-channel optical instrument for near-infrared imaging of tissues,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 264–273 (1995).
[Crossref]

S. Fantini, M. A. Franceschini, S. A. Walker, J. S. Maier, E. Gratton, “Photon path distributions in turbid media: applications for imaging,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 340–349 (1995).

Grünbaum, F. A.

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

Jiang, H.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson “Optical image reconstruction using frequency-domain data: simulations and experiments,” Opt. Image Sci. 13, 253–266 (1996).
[Crossref]

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

B. W. Pogue, M. S. Patterson, H. Jiang, K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. 40, 1709–1729 (1995).
[Crossref] [PubMed]

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988), pp. 49–74.

Kaltenbach, J.-M.

J.-M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

Kaschke, M.

J.-M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

Kohn, P.

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

Krämmer, P.

P. Krämmer, H. Bartelt, H. Fischer, B. Schmauss, “Imaging in scattering media using the phase of modulated light sources,” in Photon Propagation in Tissues, B. Chance, D. Delpy, G. J. Mueller, eds., Proc. SPIE2626, 65–74 (1995).

Lubowsky, J.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, ed., Proc. SPIE1431, 192–203 (1991).
[Crossref]

Maier, J. S.

M. A. Franceschini, S. Fantini, S. A. Walker, J. S. Maier, E. Gratton, “Multi-channel optical instrument for near-infrared imaging of tissues,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 264–273 (1995).
[Crossref]

J. S. Maier, E. Gratton, “Frequency-domain methods in optical tomography: detection of localized absorbers and a backscattering reconstruction scheme,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 440–451 (1993).
[Crossref]

S. Fantini, M. A. Franceschini, S. A. Walker, J. S. Maier, E. Gratton, “Photon path distributions in turbid media: applications for imaging,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 340–349 (1995).

Oleary, M. A.

M. A. Oleary, 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]

D. A. Boas, M. A. Oleary, 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]

Osterberg, U. L.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson “Optical image reconstruction using frequency-domain data: simulations and experiments,” Opt. Image Sci. 13, 253–266 (1996).
[Crossref]

Papaioannou, D. G.

S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, “Optical image reconstruction with deconvolution in light diffusing media,” in Photon Propagation in Tissues, B. Chance, D. Delpy, G. J. Mueller, eds., Proc. SPIE2626, 306–315 (1995).
[Crossref]

Passamani Zubelli, J.

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

Patterson, M. S.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson “Optical image reconstruction using frequency-domain data: simulations and experiments,” Opt. Image Sci. 13, 253–266 (1996).
[Crossref]

B. W. Pogue, M. S. Patterson, H. Jiang, K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. 40, 1709–1729 (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]

B. C. Wilson, M. S. Patterson, B. W. Poque, “Instrumentation for in vivo tissue spectroscopy imaging,” in Medical Lasers and Systems II, D. M. Harris, C. M. Penney, A. Katzir, eds., Proc. SPIE1892, 132–147 (1991).
[Crossref]

M. S. Patterson, B. W. Pogue, B. C. Wilson, “Computer simulation and experimental studies of optical imaging with photon density waves,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

Paulsen, K. D.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson “Optical image reconstruction using frequency-domain data: simulations and experiments,” Opt. Image Sci. 13, 253–266 (1996).
[Crossref]

B. W. Pogue, M. S. Patterson, H. Jiang, K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. 40, 1709–1729 (1995).
[Crossref] [PubMed]

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

Piston, D. W.

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorimetry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[Crossref]

Pogue, B. W.

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson “Optical image reconstruction using frequency-domain data: simulations and experiments,” Opt. Image Sci. 13, 253–266 (1996).
[Crossref]

B. W. Pogue, M. S. Patterson, H. Jiang, K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. 40, 1709–1729 (1995).
[Crossref] [PubMed]

M. S. Patterson, B. W. Pogue, B. C. Wilson, “Computer simulation and experimental studies of optical imaging with photon density waves,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

Poque, B. W.

B. C. Wilson, M. S. Patterson, B. W. Poque, “Instrumentation for in vivo tissue spectroscopy imaging,” in Medical Lasers and Systems II, D. M. Harris, C. M. Penney, A. Katzir, eds., Proc. SPIE1892, 132–147 (1991).
[Crossref]

Schmauss, B.

P. Krämmer, H. Bartelt, H. Fischer, B. Schmauss, “Imaging in scattering media using the phase of modulated light sources,” in Photon Propagation in Tissues, B. Chance, D. Delpy, G. J. Mueller, eds., Proc. SPIE2626, 65–74 (1995).

Singer, J. R.

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

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988), pp. 49–74.

van der Mark, M. B.

S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, “Optical image reconstruction with deconvolution in light diffusing media,” in Photon Propagation in Tissues, B. Chance, D. Delpy, G. J. Mueller, eds., Proc. SPIE2626, 306–315 (1995).
[Crossref]

van der Zee, P.

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

Walker, S. A.

M. A. Franceschini, S. Fantini, S. A. Walker, J. S. Maier, E. Gratton, “Multi-channel optical instrument for near-infrared imaging of tissues,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 264–273 (1995).
[Crossref]

S. Fantini, M. A. Franceschini, S. A. Walker, J. S. Maier, E. Gratton, “Photon path distributions in turbid media: applications for imaging,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 340–349 (1995).

Wang, Y.

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.

Wilson, B. C.

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]

B. C. Wilson, M. S. Patterson, B. W. Poque, “Instrumentation for in vivo tissue spectroscopy imaging,” in Medical Lasers and Systems II, D. M. Harris, C. M. Penney, A. Katzir, eds., Proc. SPIE1892, 132–147 (1991).
[Crossref]

M. S. Patterson, B. W. Pogue, B. C. Wilson, “Computer simulation and experimental studies of optical imaging with photon density waves,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

Yamada, Y.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[Crossref]

Yodh, A. G.

M. A. Oleary, 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]

D. A. Boas, M. A. Oleary, 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]

Zeng, F.

S. F. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
[Crossref]

Appl. Opt. (2)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Med. Phys. (1)

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

Opt. Image Sci. (1)

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson “Optical image reconstruction using frequency-domain data: simulations and experiments,” Opt. Image Sci. 13, 253–266 (1996).
[Crossref]

Opt. Lett. (1)

Phys. Med. Biol. (1)

B. W. Pogue, M. S. Patterson, H. Jiang, K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. 40, 1709–1729 (1995).
[Crossref] [PubMed]

Phys. Rev. E (1)

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[Crossref]

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

D. A. Boas, M. A. Oleary, 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]

Rev. Sci. Instrum. (1)

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorimetry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[Crossref]

Science (1)

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

Other (13)

P. Krämmer, H. Bartelt, H. Fischer, B. Schmauss, “Imaging in scattering media using the phase of modulated light sources,” in Photon Propagation in Tissues, B. Chance, D. Delpy, G. J. Mueller, eds., Proc. SPIE2626, 65–74 (1995).

J. S. Maier, E. Gratton, “Frequency-domain methods in optical tomography: detection of localized absorbers and a backscattering reconstruction scheme,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 440–451 (1993).
[Crossref]

S. Fantini, M. A. Franceschini, S. A. Walker, J. S. Maier, E. Gratton, “Photon path distributions in turbid media: applications for imaging,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 340–349 (1995).

S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, “Optical image reconstruction with deconvolution in light diffusing media,” in Photon Propagation in Tissues, B. Chance, D. Delpy, G. J. Mueller, eds., Proc. SPIE2626, 306–315 (1995).
[Crossref]

J.-M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

S. F. Feng, F. Zeng, B. Chance, “Monte Carlo simulations of photon migration path distributions in multiple scattering media,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. Alfano, eds., Proc. SPIE1888, 78–89 (1993).
[Crossref]

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988), pp. 49–74.

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.

M. A. Franceschini, S. Fantini, S. A. Walker, J. S. Maier, E. Gratton, “Multi-channel optical instrument for near-infrared imaging of tissues,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. Alfano, eds., Proc. SPIE2389, 264–273 (1995).
[Crossref]

B. C. Wilson, M. S. Patterson, B. W. Poque, “Instrumentation for in vivo tissue spectroscopy imaging,” in Medical Lasers and Systems II, D. M. Harris, C. M. Penney, A. Katzir, eds., Proc. SPIE1892, 132–147 (1991).
[Crossref]

M. S. Patterson, B. W. Pogue, B. C. Wilson, “Computer simulation and experimental studies of optical imaging with photon density waves,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. J. Mueller, B. Chance, R. Alfano, S. Arridge, J. Bleuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

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

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, ed., Proc. SPIE1431, 192–203 (1991).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1

X-ray weight function is filtered to account for the size of the sampling period d. Filtered values are then projected back along the source–detector ray for a projection at angle θ. After this step the projections at all angles θ are summed.

Fig. 2
Fig. 2

Block diagram of the frequency-domain instrumentation. A 120-MHz radio frequency signal generated by synthesizer S2 is amplified by amplifier A2 and sent to the 793-nm laser diode, producing 50 mW of output light that is then coupled with a 20% efficiency into a fiber-optic conduit, giving an output power in the medium of 10 mW. A small fraction of the light is sent to a reference photomultiplier tube (PMT) to correct for source intensity fluctuations. The output light produces a photon density wave that is perturbed by the objects. The multiply scattered light is collected by a fiber-optic bundle that is connected to a sample PMT. A cross-correlation electronics system processes the PMT signal by using a 1-KHz signal generated by synthesizer S1 and amplified by A1. The measurement is repeated 41 times as the source and detector are scanned across the area of interest in the direction of the arrows. After scan #1 the scanning direction is rotated 30° relative to its original position and the scan is repeated. The data for each slice consist of seven projections at 30° increments from 0° to 180°.

Fig. 3
Fig. 3

Comparison of directly measured parameters, phase, and dc, with calculated values of effective μa and μs for 1.5-cm-diameter cylinders immersed in a highly scattering background medium (μs = 7.9 ± 0.1cm-1, μa0 = 0.079 ± 0.005cm-1). (a) Type A - cylinder is less absorbing than the background, (b) type A + cylinder is more absorbing than the background. The parameters dc, effective μa, and effective μs are shown in normalized units to illustrate the differences in contrast and sharpness of the perturbation caused by the cylinder. The two peaks in the phase plots are due to the distortion of the photon density wave front by each cylinder.

Fig. 4
Fig. 4

Reconstructed images of calculated parameters, effective μa and μs, for a 1.0-cm-diameter, type A - cylinder (μa = 0.045 ± 0.03 cm-1) immersed in a highly scattering background medium (μs = 7.9 ± 0.1cm-1, μa = 0.079 ± 0.005 cm-1). (a) Effective μa image, (b) effective μs image. Each image is reconstructed from seven projections taken at 30° intervals from 0° to 180°. The reconstruction region is shown by the thick black line. All images are scaled by calibration factor c = 25 [μimage = (μ* - μbackground) c + μbackground].

Fig. 5
Fig. 5

Reconstructed images of calculated parameters, effective μ(a and μs, for a 1.0-cm-diameter, type A + cylinder (μa = 0.13 ± 0.01 cm-1) immersed in a highly scattering background medium (μs = 7.9 ± 0.1 cm-1, μa = 0.079 ±0.005 cm-1). (a) Effective μa image, (b) effective μs image. Each image is reconstructed from seven projections taken at 30° intervals from 0° to 180°. The reconstruction region is shown by the thick black line.

Fig. 6
Fig. 6

Reconstructed images of calculated values, effective μa and μs, for two 1.0-cm-diameter, type A - cylinders (μa = 0.045 ± 0.03 cm-1), centers located at (-1 cm, 0 cm) and (1 cm, 0 cm). Both cylinders were immersed in a highly scattering background medium (μs = 7.9 ± 0.1 cm-1, μa = 0.079 ± 0.005 cm-1). (a) Effective μa image, (b) effective μs image. In the effective μs images the two cylinders are resolved whereas in the effective μa images they are not.

Fig. 7
Fig. 7

Reconstructed images of calculated values, effective μa and μs, for two 1.0-cm-diameter, type A + cylinders (μa = 0.13 ± 0.01 cm-1), centers located at (-1 cm, 0 cm) and (1 cm, 0 cm). Both cylinders were immersed in a highly scattering background medium (μs = 7.9 ± 0.1 cm-1, μa = 0.079 ± 0.005 cm-1). (a) Effective μa image, (b) effective μs image. In the effective μs images the two cylinders are resolved whereas in the effective μa images they are not.

Fig. 8
Fig. 8

Reconstructed images of calculated values, effective μa and μs, for two 1.0-cm-diameter cylinders, centers located at (-1 cm, 0 cm) and (1 cm, 0 cm). (a) The left cylinder is less absorbing than the background medium (type A -) whereas the right cylinder is more absorbing than the background medium (type A +). The values of the objects are recovered qualitatively correctly in the effective μa map. (b) The increased resolution of the effective μs images allows a more accurate determination of the cylinder positions.

Fig. 9
Fig. 9

Experimental setup. (a) Side view of the 16-L glass container filled with Intralipid/India ink mixture. Fiber optics carry light from the source and to the detector. The fiber tips are separated by 5 cm inside the tank and scanned by means of an XYZ positioning scanner. The object is suspended in the scattering medium by a 3-mm-diameter glass rod. (b) Top view of cylindrical object. The source and the detector make a one-dimensional scan perpendicular to the object across the volume of interest. (c) Top view of multiple projections measured on a single object. Each square denotes a separate projection. In our measurements, projection angles ranged from 0° to 180° in 30° steps. (d) Top view of multiple projections measured on two cylindrical objects.

Tables (3)

Tables Icon

Table 1 Size and Optical Properties of Background Medium and Objects.

Tables Icon

Table 2 Contrast and FWHM of Single-Cylinder Calibrated Images with a Diameter of 1.0cm

Tables Icon

Table 3 Contrast and Relative Resolution of Two-Cylinder Images with a Diameter of 1.0 cm.

Equations (10)

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

Udc=S4πνDexp-rμaD1/2r,
Uac=SA4πνDexprRe kr,
Φ=r Imk+Φs,
k=-μaD-iωνD1/2,
D=1/3μsRef. 14.
ht=12dsin pp-14dsinp2p22,
μθft=-μθmtht-tdt.
μ*x, y=0πμθfx cos θ+y sin θdθ.
Cx=xpeak-xbackgroundxbackground.
Rx=xpeak1+xpeak2/2-xminxmmCx,

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