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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 18, Iss. 6 — Jun. 1, 2001
  • pp: 1336–1347

Inverse problem in optical diffusion tomography. I. Fourier–Laplace inversion formulas

Vadim A. Markel and John C. Schotland  »View Author Affiliations

JOSA A, Vol. 18, Issue 6, pp. 1336-1347 (2001)

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We consider the inverse problem of reconstructing the absorption and diffusion coefficients of an inhomogeneous highly scattering medium probed by diffuse light. Inversion formulas based on the Fourier–Laplace transform are used to establish the existence and uniqueness of solutions to this problem in planar, cylindrical, and spherical geometries.

© 2001 Optical Society of America

OCIS Codes
(170.0170) Medical optics and biotechnology : Medical optics and biotechnology
(170.3010) Medical optics and biotechnology : Image reconstruction techniques
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.6960) Medical optics and biotechnology : Tomography

Original Manuscript: April 21, 2000
Revised Manuscript: September 1, 2000
Manuscript Accepted: December 13, 2000
Published: June 1, 2001

Vadim A. Markel and John C. Schotland, "Inverse problem in optical diffusion tomography. I. Fourier–Laplace inversion formulas," J. Opt. Soc. Am. A 18, 1336-1347 (2001)

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  1. G. Mueller, ed., Medical Optical Tomography: Functional Imaging and Monitoring (SPIE Press, Bellingham, Wash., 1993).
  2. R. Alfano, ed., Advances in Optical Imaging and Photon Migration (Optical Society of America, Washington, D.C., 1994).
  3. B. Chance, R. Alfano, eds., Proceedings of Optical Tomography and Spectroscopy of Tissue (SPIE Press, Bellingham, Wash., 1995).
  4. B. Chance, R. Alfano, eds., Proceedings of Optical Tomography and Spectroscopy of Tissue II (SPIE Press, Bellingham, Wash., 1997).
  5. B. Chance, R. Alfano, B. Tromberg, eds., Proceedings of Optical Tomography and Spectroscopy of Tissue III (SPIE Press, Bellingham, Wash., 1999).
  6. M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999). [CrossRef]
  7. A. B. Davis, R. F. Cahalan, D. Spinehirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B 24, 177–185 (1999). [CrossRef]
  8. L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991). [CrossRef] [PubMed]
  9. D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993). [CrossRef] [PubMed]
  10. A. Rebane, J. Feinberg, “Time-resolved holography,” Nature 351, 378–380 (1991). [CrossRef]
  11. K. M. Yoo, F. Liu, R. R. Alfano, “Imaging objects hidden in scattering media using an absorption technique,” Opt. Lett. 16, 1068–1070 (1991). [CrossRef] [PubMed]
  12. E. Leith, H. Chen, Y. Chen, D. Dilworth, J. Lopez, R. Masri, J. Rudd, J. Valdmanis, “Electronic holography and speckle methods for imaging through tissue using femtosecond gated pulses,” Appl. Opt. 30, 4204–4210 (1991). [CrossRef] [PubMed]
  13. M. R. Hee, J. A. Izatt, J. M. Jackobson, J. G. Fujimoto, E. A. Swanson, “Femtosecond transillumination optical coherence tomography,” Opt. Lett. 18, 950–952 (1993). [CrossRef] [PubMed]
  14. V. V. Lyubimov, “Image transfer in a plane layer of a scattering medium and estimation of the resolving power of optical tomography using 1st transmitted photons of ultrashort pulses,” Opt. Spectrosc. 76, 725–726 (1994).
  15. V. V. Lyubimov, “Spatial resolution in probing a strongly scattering medium with a short optical pulse,” Opt. Spectrosc. 78, 259–260 (1995).
  16. E. N. Leith, B. G. Hoover, D. S. Dilworth, P. P. Naulleau, “Ensemble-averaged Shack-Hartmann wavefront sensing for imaging through turbid media,” Appl. Opt. 37, 3643–3650 (1998). [CrossRef]
  17. V. V. Lyubimov, “On the spatial resolution of optical tomography of strongly scattering media with the use of the directly passing photons,” Opt. Spectrosc. 86, 251–252 (1999).
  18. J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990). [CrossRef] [PubMed]
  19. S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “New results for the development of infrared absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1991). [CrossRef]
  20. 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, ed., Proc. SPIE1431, 192–203 (1991). [CrossRef]
  21. J. Schotland, J. S. Leigh, “Photon diffusion imaging,” Biophys. J. 61, 446 (1992).
  22. C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995). [CrossRef]
  23. M. Ishii, J. S. Leigh, J. C. Schotland, “Photon diffusion imaging of model and biological systems,” in Optical Tomography, Photon Migration and Spectroscopy of Tissue and Model Media: Theory, Human Studies, B. Chance, ed., Proc. SPIE2389, 312–317 (1995).
  24. M. O’Leary, D. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing photon tomography,” Opt. Lett. 20, 426–429 (1995). [CrossRef]
  25. V. V. Lyubimov, “Optics of photon density waves in strongly scattering media and spatial resolution in tomography,” Opt. Spectrosc. 81, 299–301 (1996).
  26. J. C. Schotland, “Continuous wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997). [CrossRef]
  27. V. B. Volkonskii, O. V. Kravtsenyuk, V. V. Lyubimov, E. P. Mironov, A. G. Murzin, “The use of statistical characteristics of photon trajectories for the tomographic studies of optical macroheterogeneities in strongly scattering objects,” Opt. Spectrosc. 86, 253–260 (1999).
  28. O. V. Kravtsenyuk, V. V. Lyubimov, “Specific features of statistical characteristics of photon trajectories in a strongly scattering medium near an object surface,” Opt. Spectrosc. 88, 608–614 (2000). [CrossRef]
  29. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999). [CrossRef]
  30. S. R. Arridge, W. R. B. Lionhart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23, 882–884 (1998). [CrossRef]
  31. C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon density waves,” Opt. Express 1, 6–11 (1997), http://www.opticsexpress.org . [CrossRef] [PubMed]
  32. 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]
  33. D. N. Pattanayak, A. G. Yodh, “Diffuse optical 3D-slice imaging of bounded turbid media using a new integro-differential equation,” Opt. Express 4, 231–240 (1999) http://www.opticsexpress.org . [CrossRef] [PubMed]
  34. X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, A. G. Yodh, “Near-field diffraction tomography with diffuse photon-density waves,” Phys. Rev. E 61, 4295–4309 (2000). [CrossRef]
  35. The right-hand side of this equation is the same in different versions of perturbation theory, such as the first Born approximation or the first Rytov approximation, but the definition of the data function ϕ is different. Definition (10) is obtained in the first Born approximation; reexponentiation according to the Rytov expansion leads to ϕ(r1, r2)=-G0(r1, r2)ln[G(r1, r2)/G0(r1, r2)].
  36. B. Chu, E. Gulari, E. Gulari, “Photon correlation measurements of colloidal size distributions. II. Details of histogram approach and comparison of methods of data analysis,” Phys. Scr. 19, 476–485 (1979). [CrossRef]
  37. I. Ersh, L. S. Muratov, S. Y. Novozhilov, B. M. Stockman, M. I. Stockman, “Kinetics of the immunological reaction of agglutination and rapid determination of bacteria using an automated laser photon-correlation spectrometer,” Dok. Biochem. 287, 125–129 (1986).
  38. R. Bauer, M. Hansen, S. Hansen, L. Ogendal, S. Lomholt, K. Qvist, “The structure of casein aggregates during renneting studied by indirect Fourier transformation and inverse Laplace transformation of static and dynamic light scattering data, respectively,” J. Chem. Phys. 103, 2725–2737 (1995). [CrossRef]
  39. P. K. Venkatesh, R. W. Carr, M. H. Cohen, A. M. Dean, “Microcanonical transition state theory rate coefficients from thermal rate constants via inverse Laplace transformation,” J. Phys. Chem. A 102, 8104–8115 (1998). [CrossRef]
  40. E. Schnedermann, “The inverse Laplace transform as the ultimate tool for transverse mass spectroscopy,” Z. Phys. C 64, 85–90 (1994). [CrossRef]

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