Abstract

Optical tomography schemes using non-linear optimisation are usually based on a Newton-like method involving the construction and inversion of a large Jacobian matrix. Although such matrices can be efficiently constructed using a reciprocity principle, their inversion is still computationally difficult. In this paper we demonstrate a simple means to obtain the gradient of the objective function directly, leading to straightforward application of gradient-based optimisation methods.

© 1998 Optical Society of America

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  1. A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, and E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spec-troscopy,” Lancet 2, 770–771 (1988).
    [CrossRef] [PubMed]
  2. J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, and E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
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  4. J. C. Hebden, R. A. Kruger, and K. S. Wong, “Time resolved imaging through a highly scattering medium,” Appl. Opt. 30, 788–794 (1991).
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    [CrossRef] [PubMed]
  7. S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
    [CrossRef] [PubMed]
  8. S. B. Colak, G. W. Hooft, D. G. Papaioannou, and M. B. van der Mark, “3D backprojection tomography for medical optical imaging,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 294–298.
  9. S. A. Walker, S. Fantini, and E. Gratton, “Back-projection reconstructions of cylindrical inho-mogeneities from frequency domain optical measurements in turbid media,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 137–141.
  10. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
    [CrossRef]
  11. S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstructionof near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE 1767, 372–383 (1992).
    [CrossRef]
  12. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1995).
    [CrossRef]
  13. B. W. Pogue, M. S. Patterson, H. Jiang, and K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. 40, 1709–1729 (1995).
    [CrossRef] [PubMed]
  14. D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
    [CrossRef] [PubMed]
  15. B. Chance, M. Maris, J. Sorge, and M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE 1204, 481–491 (1990).
    [CrossRef]
  16. J. D. Moulton, Diffusion modelling of picosecond laser pulse propagation in turbid media, M. Eng. thesis, McMaster University, Hamilton, Ontario (1990).
  17. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
    [CrossRef] [PubMed]
  18. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
    [CrossRef] [PubMed]
  19. S. R. Arridge, “Photon measurement density functions. Part 1: Analytic forms,” Appl. Opt. 34, 7395–7409 (1995).
    [CrossRef] [PubMed]
  20. S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite element calculations,” Appl. Opt. 34, 8026–8037 (1995).
    [CrossRef] [PubMed]
  21. M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd edition (Wiley, New York, 1993).
  22. S. R. Arridge and M. Schweiger, “Direct calculation of the moments of the distribution of photon time of flight in tissues with a finite-element method,” Appl. Opt. 34, 2683–2687 (1995).
    [CrossRef] [PubMed]
  23. M. Schweiger and S. R. Arridge, “Direct calculation of the Laplace transform of the distribution of photon time of flight in tissue with a finite-element method,” Appl. Opt. 36, 9042–9049 (1997).
    [CrossRef]
  24. M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math Imag. Vision 3, 263–283 (1993).
    [CrossRef]
  25. K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
    [CrossRef] [PubMed]
  26. S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, K. M. Hanson, ed., Proc SPIE 3034, 369–380 (1997).
    [CrossRef]
  27. O. Dorn, Das inverse Transportproblem in der Lasertomographie, Ph. D. thesis, University of Münster, 1997.
  28. R. Roy, Image reconstruction from light measurements on biological tissue, Ph. D. thesis, University of Hertfordshire, 1997.
  29. S. R. Arridge and M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method,” in Computational Radiology and Imaging: Therapy and DiagnosisC. Borgers and F. Natterer, eds., IMA Volumes in Mathematics and its Applications (Springer1998, in press).
  30. S. R. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539–1558 (1995).
    [CrossRef] [PubMed]
  31. M. Schweiger and S. R. Arridge, “Optimal data types in optical tomography,” in Information Processing in Medical Imaging, Lect. Notes Comput. Sci.,  1230, 71–84 (1997).
    [CrossRef]

1997 (5)

J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

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

M. Schweiger and S. R. Arridge, “Optimal data types in optical tomography,” in Information Processing in Medical Imaging, Lect. Notes Comput. Sci.,  1230, 71–84 (1997).
[CrossRef]

M. Schweiger and S. R. Arridge, “Direct calculation of the Laplace transform of the distribution of photon time of flight in tissue with a finite-element method,” Appl. Opt. 36, 9042–9049 (1997).
[CrossRef]

S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, K. M. Hanson, ed., Proc SPIE 3034, 369–380 (1997).
[CrossRef]

1995 (9)

S. R. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539–1558 (1995).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, “Direct calculation of the moments of the distribution of photon time of flight in tissues with a finite-element method,” Appl. Opt. 34, 2683–2687 (1995).
[CrossRef] [PubMed]

S. R. Arridge, “Photon measurement density functions. Part 1: Analytic forms,” Appl. Opt. 34, 7395–7409 (1995).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite element calculations,” Appl. Opt. 34, 8026–8037 (1995).
[CrossRef] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1995).
[CrossRef]

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

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

K. D. Paulsen and 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, and K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. 40, 1709–1729 (1995).
[CrossRef] [PubMed]

1993 (2)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math Imag. Vision 3, 263–283 (1993).
[CrossRef]

1992 (1)

S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstructionof near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE 1767, 372–383 (1992).
[CrossRef]

1991 (1)

1990 (2)

B. Chance, M. Maris, J. Sorge, and M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE 1204, 481–491 (1990).
[CrossRef]

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, and E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

1988 (2)

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, and E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spec-troscopy,” Lancet 2, 770–771 (1988).
[CrossRef] [PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Arridge, S. R.

J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

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

M. Schweiger and S. R. Arridge, “Optimal data types in optical tomography,” in Information Processing in Medical Imaging, Lect. Notes Comput. Sci.,  1230, 71–84 (1997).
[CrossRef]

M. Schweiger and S. R. Arridge, “Direct calculation of the Laplace transform of the distribution of photon time of flight in tissue with a finite-element method,” Appl. Opt. 36, 9042–9049 (1997).
[CrossRef]

S. R. Arridge and M. Schweiger, “Direct calculation of the moments of the distribution of photon time of flight in tissues with a finite-element method,” Appl. Opt. 34, 2683–2687 (1995).
[CrossRef] [PubMed]

S. R. Arridge, “Photon measurement density functions. Part 1: Analytic forms,” Appl. Opt. 34, 7395–7409 (1995).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite element calculations,” Appl. Opt. 34, 8026–8037 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539–1558 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math Imag. Vision 3, 263–283 (1993).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstructionof near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE 1767, 372–383 (1992).
[CrossRef]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method,” in Computational Radiology and Imaging: Therapy and DiagnosisC. Borgers and F. Natterer, eds., IMA Volumes in Mathematics and its Applications (Springer1998, in press).

Bazaraa, M. S.

M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd edition (Wiley, New York, 1993).

Boas, D. A.

Chance, B.

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

B. Chance, M. Maris, J. Sorge, and M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE 1204, 481–491 (1990).
[CrossRef]

Colak, S. B.

S. B. Colak, G. W. Hooft, D. G. Papaioannou, and M. B. van der Mark, “3D backprojection tomography for medical optical imaging,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 294–298.

Cope, M.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, and E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, and E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spec-troscopy,” Lancet 2, 770–771 (1988).
[CrossRef] [PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Cunningham, G. S.

S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, K. M. Hanson, ed., Proc SPIE 3034, 369–380 (1997).
[CrossRef]

Delpy, D. T.

J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math Imag. Vision 3, 263–283 (1993).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstructionof near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE 1767, 372–383 (1992).
[CrossRef]

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, and E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, and E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spec-troscopy,” Lancet 2, 770–771 (1988).
[CrossRef] [PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Dorn, O.

O. Dorn, Das inverse Transportproblem in der Lasertomographie, Ph. D. thesis, University of Münster, 1997.

Edwards, A. D.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, and E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, and E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spec-troscopy,” Lancet 2, 770–771 (1988).
[CrossRef] [PubMed]

Fantini, S.

S. A. Walker, S. Fantini, and E. Gratton, “Back-projection reconstructions of cylindrical inho-mogeneities from frequency domain optical measurements in turbid media,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 137–141.

Gratton, E.

S. A. Walker, S. Fantini, and E. Gratton, “Back-projection reconstructions of cylindrical inho-mogeneities from frequency domain optical measurements in turbid media,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 137–141.

Hanson, K. M.

S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, K. M. Hanson, ed., Proc SPIE 3034, 369–380 (1997).
[CrossRef]

Hebden, J. C.

J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

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

J. C. Hebden, R. A. Kruger, and K. S. Wong, “Time resolved imaging through a highly scattering medium,” Appl. Opt. 30, 788–794 (1991).
[CrossRef] [PubMed]

Hiraoka, M.

S. R. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539–1558 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Hooft, G. W.

S. B. Colak, G. W. Hooft, D. G. Papaioannou, and M. B. van der Mark, “3D backprojection tomography for medical optical imaging,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 294–298.

Jiang, H.

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

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1995).
[CrossRef]

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

Kruger, R. A.

Maris, M.

B. Chance, M. Maris, J. Sorge, and M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE 1204, 481–491 (1990).
[CrossRef]

Moulton, J. D.

J. D. Moulton, Diffusion modelling of picosecond laser pulse propagation in turbid media, M. Eng. thesis, McMaster University, Hamilton, Ontario (1990).

O’Leary, M. A.

Osterberg, U. L.

Papaioannou, D. G.

S. B. Colak, G. W. Hooft, D. G. Papaioannou, and M. B. van der Mark, “3D backprojection tomography for medical optical imaging,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 294–298.

Patterson, M. S.

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

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1995).
[CrossRef]

Paulsen, K. D.

B. W. Pogue, M. S. Patterson, H. Jiang, and 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 and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
[CrossRef] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1995).
[CrossRef]

Pogue, B. W.

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

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1995).
[CrossRef]

Reynolds, E. O. R.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, and E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, and E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spec-troscopy,” Lancet 2, 770–771 (1988).
[CrossRef] [PubMed]

Richardson, C. E.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, and E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, and E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spec-troscopy,” Lancet 2, 770–771 (1988).
[CrossRef] [PubMed]

Roy, R.

R. Roy, Image reconstruction from light measurements on biological tissue, Ph. D. thesis, University of Hertfordshire, 1997.

Saquib, S. S.

S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, K. M. Hanson, ed., Proc SPIE 3034, 369–380 (1997).
[CrossRef]

Schweiger, M.

M. Schweiger and S. R. Arridge, “Optimal data types in optical tomography,” in Information Processing in Medical Imaging, Lect. Notes Comput. Sci.,  1230, 71–84 (1997).
[CrossRef]

M. Schweiger and S. R. Arridge, “Direct calculation of the Laplace transform of the distribution of photon time of flight in tissue with a finite-element method,” Appl. Opt. 36, 9042–9049 (1997).
[CrossRef]

S. R. Arridge and M. Schweiger, “Photon measurement density functions. Part 2: Finite element calculations,” Appl. Opt. 34, 8026–8037 (1995).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, “Direct calculation of the moments of the distribution of photon time of flight in tissues with a finite-element method,” Appl. Opt. 34, 2683–2687 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539–1558 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math Imag. Vision 3, 263–283 (1993).
[CrossRef]

S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstructionof near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE 1767, 372–383 (1992).
[CrossRef]

S. R. Arridge and M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method,” in Computational Radiology and Imaging: Therapy and DiagnosisC. Borgers and F. Natterer, eds., IMA Volumes in Mathematics and its Applications (Springer1998, in press).

Sherali, H. D.

M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd edition (Wiley, New York, 1993).

Shetty, C. M.

M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd edition (Wiley, New York, 1993).

Sorge, J.

B. Chance, M. Maris, J. Sorge, and M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE 1204, 481–491 (1990).
[CrossRef]

Tamura, M.

M. Tamura, “Multichannel near-infrared optical imaging of human brain activity,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 8–10.

van der Mark, M. B.

S. B. Colak, G. W. Hooft, D. G. Papaioannou, and M. B. van der Mark, “3D backprojection tomography for medical optical imaging,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 294–298.

van der Zee, P.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Walker, S. A.

S. A. Walker, S. Fantini, and E. Gratton, “Back-projection reconstructions of cylindrical inho-mogeneities from frequency domain optical measurements in turbid media,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 137–141.

Wong, K. S.

Wray, S.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Wray, S. C.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, and E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

Wyatt, J.

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Wyatt, J. S.

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, and E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, and E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spec-troscopy,” Lancet 2, 770–771 (1988).
[CrossRef] [PubMed]

Yodh, A. G.

Zhang, M. Z.

B. Chance, M. Maris, J. Sorge, and M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE 1204, 481–491 (1990).
[CrossRef]

Appl. Opt. (5)

J. Appl. Physiol. (1)

J. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray, and E. O. R. Reynolds, “Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy,” J. Appl. Physiol. 68, 1086–1091 (1990).
[PubMed]

J. Math Imag. Vision (1)

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math Imag. Vision 3, 263–283 (1993).
[CrossRef]

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

Lancet (1)

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope, and E. O. R. Reynolds, “Cotside measurement of cerebral blood flow in ill newborn infants by near infrared spec-troscopy,” Lancet 2, 770–771 (1988).
[CrossRef] [PubMed]

Lect. Notes Comput. Sci. (1)

M. Schweiger and S. R. Arridge, “Optimal data types in optical tomography,” in Information Processing in Medical Imaging, Lect. Notes Comput. Sci.,  1230, 71–84 (1997).
[CrossRef]

Med. Phys. (3)

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

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Med. Biol. (5)

S. R. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539–1558 (1995).
[CrossRef] [PubMed]

J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

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

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

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Proc SPIE (1)

S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, K. M. Hanson, ed., Proc SPIE 3034, 369–380 (1997).
[CrossRef]

Proc. SPIE (2)

B. Chance, M. Maris, J. Sorge, and M. Z. Zhang, “A phase modulation system for dual wavelength difference spectroscopy of haemoglobin deoxygenation in tissue,” in Time-resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE 1204, 481–491 (1990).
[CrossRef]

S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstructionof near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE 1767, 372–383 (1992).
[CrossRef]

Other (9)

M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd edition (Wiley, New York, 1993).

J. D. Moulton, Diffusion modelling of picosecond laser pulse propagation in turbid media, M. Eng. thesis, McMaster University, Hamilton, Ontario (1990).

S. B. Colak, G. W. Hooft, D. G. Papaioannou, and M. B. van der Mark, “3D backprojection tomography for medical optical imaging,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 294–298.

S. A. Walker, S. Fantini, and E. Gratton, “Back-projection reconstructions of cylindrical inho-mogeneities from frequency domain optical measurements in turbid media,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 137–141.

O. Dorn, Das inverse Transportproblem in der Lasertomographie, Ph. D. thesis, University of Münster, 1997.

R. Roy, Image reconstruction from light measurements on biological tissue, Ph. D. thesis, University of Hertfordshire, 1997.

S. R. Arridge and M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method,” in Computational Radiology and Imaging: Therapy and DiagnosisC. Borgers and F. Natterer, eds., IMA Volumes in Mathematics and its Applications (Springer1998, in press).

M. Tamura, “Multichannel near-infrared optical imaging of human brain activity,” in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC1996) Vol. 2, pp. 8–10.

Near-infrared spectroscopy and imaging of living systems, special issue of Philos. Trans. R. Soc. London Ser. B, Vol. 352 (1997).

Supplementary Material (8)

» Media 1: MOV (415 KB)     
» Media 2: MOV (611 KB)     
» Media 3: MOV (279 KB)     
» Media 4: MOV (602 KB)     
» Media 5: MOV (744 KB)     
» Media 6: MOV (485 KB)     
» Media 7: MOV (880 KB)     
» Media 8: MOV (436 KB)     

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Figures (4)

Figure 1.
Figure 1.

Target image (col. 1), reconstruction after 100 iterations with conjugate gradient method (col. 2), with steepest descent method (col. 3) and with ART method (col. 4). In all cases, the top image is μa , and the bottom image is μs . The animations linked to the figure show the first 50 reconstruction iterations, and the final 50 in steps of 10. [Media 1] [Media 2] [Media 3] [Media 4] [Media 5] [Media 6]

Figure 2.
Figure 2.

L2 data norms as a function of iteration number, for different algorithms, as a function of iteration number (left) and of runtime (right).

Figure 3.
Figure 3.

L2 solution norms as a function of iteration number, for different algorithms. Left: absorption, right: scatter.

Figure 4.
Figure 4.

Target image (left col.), reconstruction after 100 iterations with conjugate gradient method (right col.). Top row is μa , bottom row is μs . The animations linked to the figure show the first 50 reconstruction iterations, and the final 50 in steps of 10. [Media 7] [Media 8]

Tables (2)

Tables Icon

Table 1. Optical parameters of circular test object.

Tables Icon

Table 2. Optical parameters of neonatal head model.

Equations (44)

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

F = 𝛲 [ μ , κ ]
Ψ = 1 2 j = 1 S i = 1 M j ( y j , i 𝛲 j , i [ μ , κ ] σ j , i ) 2
Ψ = 1 2 ( y F ) T R 2 ( y F ) = 1 2 b T b
R = diag R 1 R 2 R S = diag ( σ 1,1 , σ 1,2 , σ j , i , σ S , M s )
κ ( r ) Φ ̂ r ω + μ ( r ) Φ ̂ ( r , ω ) + ιω c Φ ̂ r ω = q ̂ 0 r t ,
κ ( r ) Φ r t + μ ( r ) Φ ( r , t ) + 1 c Φ ( r , t ) t = q 0 r t ,
Γ ( ξ ) = ( ξ ) n ̂ Φ ( ξ ) ,
Φ ( ξ ) + κ α n ̂ Φ ( ξ ) = 0 ,
( K ( κ ) + C ( μ ) + αA + ιω B ) Φ ( ω ) = Q ( ω )
( K ( κ ) + C ( μ ) + αA ) Φ ( t ) + B Φ ( t ) t = Q ( t )
K ij = Ω κ ( r ) u i ( r ) u j ( r ) d n r
C ij = Ω μ ( r ) u i ( r ) u j ( r ) d n r
B ij = 1 c Ω u i ( r ) u j ( r ) d n r
A ij = Ω u i ( r ) u j ( r ) d ( Ω )
F j = 𝚳 [ Φ j ]
𝚳 ̅ = τ ε
time - gated intensity : 𝚳 E ¯ ( T ) [ Φ ( t ) ] = 1 ε [ Φ ( t ) ] 0 T B [ Φ ( t ) ] dt ,
n - th temporal moment : 𝚳 t n [ Φ ( t ) ] = 1 ε [ Φ ( t ) ] 0 t n B [ Φ ( t ) ] dt ,
n - th central moment : 𝚳 c n [ Φ ( t ) ] = 1 ε [ Φ ( t ) ] 0 ( t t ) n B [ Φ ( t ) ] dt ,
normalised Laplace transform : 𝚳 L ̅ ( s ) [ Φ ( t ) ] = 1 ε [ Φ ( t ) ] 0 e st B [ Φ ( t ) ] dt .
Ψ x k = j = 1 S i = 1 M j ( y j , i P j , i [ μ , κ ] σ j , i 2 ) ( P j , i [ μ , κ ] x k )
z = j = 1 S P ' j T R j 1 b j = P ' T R 1 b
= j = 1 S J j T b j = J T b
b 1 b 2 . . . b S = J 1 , ( μ ) J 1 , ( κ ) J 2 , ( μ ) J 2 , ( κ ) . . . . . . J S , ( μ ) J S , ( κ ) Δ μ Δ κ
J j i = P MD F ( j , i ) T = [ P MD F ( j , i ) , ( μ ) T P MD F ( j , i ) , ( κ ) T ]
z = J T b
= j = 1 S J j T b j
= j = 1 S j = 1 M j P MD F ( j , i ) T b j , i
( K + C + αA ι ω B ) Φ m + ( ω ) = Q i + ( ω )
z ( μ ) = j = 1 S i = 1 M j b j , i σ j , i Φ i + ( ω ) × Φ j ( ω )
z ( κ ) = j = 1 S i = 1 M j b j , i σ j , i Φ i + ( ω ) × Φ j ( ω )
ν j + ( ω ) = i = 1 M j b j , i σ j , i Q i + ( ω )
( K + C + αA ι ω B ) η j + ( ω ) = ν j + ( ω )
z ( μ ) = j = 1 S η j + ( ω ) × Φ j ( ω )
z ( κ ) = j = 1 S η j + ( ω ) × Φ j ( ω )
J j i 𝚳 ̅ = 1 F j , i ε ( J j i 𝚳 F j , i 𝚳 ̅ J j i ε )
J 𝚳 ̅ T b 𝚳 ̅ = j S i M j J j i 𝚳 ̅ b j , i 𝚳 ̅
= j S i M j 1 F j , i ε ( J j i 𝚳 F j , i 𝚳 ̅ J j i ε ) b j , i 𝚳 ̅
z ( μ ) 𝚳 ̅ = j S i M j b j , i 𝚳 ̅ σ j , i 𝚳 ̅ F j , i ε ( τ [ Φ i + × Φ j ] F j , i 𝚳 ̅ Φ i + × Φ j )
z ( κ ) 𝚳 ̅ = j S i M j b j , i 𝚳 ̅ σ j , i 𝚳 ̅ F j , i ε ( τ [ Φ i + × Φ j ] F j , i 𝚳 ̅ Φ i + × Φ j )
ν j 𝚳 ̅ + ( 0 ) = i = 1 M j b j , i 𝚳 ̅ F j , i 𝚳 ̅ σ j , i 𝚳 ¯ F j , i ε Q i +
ν j 𝚳 ̅ + ( 1 ) = i = 1 M j b j , i 𝚳 ̅ σ j , i 𝚳 ¯ F j , i ε Q i +
z ( μ ) 𝚳 ̅ = j S τ [ η j 𝚳 ̅ + ( 1 ) × Φ j ] η j 𝚳 ¯ + ( 0 ) × Φ j
z ( κ ) 𝚳 ̅ = j S τ [ η j 𝚳 ̅ + ( 1 ) × Φ j ] η j 𝚳 ¯ + ( 0 ) × Φ j

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