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A General Framework for Iterative Reconstruction Algorithms in Optical Tomography, Using a Finite Element Method

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Computational Radiology and Imaging

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 110))

Abstract

In this paper we present several schemes for solving the inverse problem in Optical Tomography. We first set the context of Optical Tomography and discuss alternative photon transport models and measurement schemes. We develop the inverse problem as the optimisation of an objective functions and develop three classes of algorithms fors its solution: Newton methods, linearised methods, and gradient methods. We concentrate on the use numerical methods based on Finite Elements, and discuss how efficient methods may be developed using adjoint solutions. A taxonomy of algorithms is given, with an analysis of their spatial and temporal complexity.

The work of the second author was supported by Action Research Grant A/P/0503.

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References

  1. R.R. Alfano and J.G. Fujimoto, eds., OSA TOPS on Advances in Optical Imaging and Photon Migration, vol. 2, OSA, 1996.

    Google Scholar 

  2. S.R. Arridge, Photon measurement density functions. Part 1: Analytical forms, Appl. Opt., 34 (1995), pp. 7395–7409.

    Article  Google Scholar 

  3. S.R. Arridge and J.C. Hebden, Optical imaging in medicine: II. Modelling and reconstruction, Phys. Med. Biol., 42 (1997), pp. 841–853.

    Article  Google Scholar 

  4. S.R. Arridge and M. Schweiger, Direct calculation of the moments of the distribution of photon time of flight in tissue with a finite-element method, Appl. Opt., 34 (1995), pp. 2683–2687.

    Article  Google Scholar 

  5. S.R. Arridge and M. Schweiger, Photon measurement density functions. Part 2: Finite element calculations,Appl. Opt., 34 (1995), pp. 8026–8037.

    Article  Google Scholar 

  6. S.R. Arridge, M. Schweiger, and D.T. Delpy, Iterative reconstruction of near-infrared absorption images, in Inverse Problems in Scattering and Imaging, M.A. Fiddy, ed., vol. 1767, Proc. SPIE, 1992, pp. 372–383.

    Google Scholar 

  7. S.R. Arridge, M. Schweiger, M. Hiraoka, and D.T. Delpy, A finite element approach for modeling photon transport in tissue, Med. Phys., 20 (1993), pp. 299–309.

    Article  Google Scholar 

  8. S.R. Arridge, P. Van Der Zee, D.T. Delpy, and M. Cope, Reconstruction methods for infra-red absorption imaging,in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., vol. 1431, Proc. SPIE, 1991, pp. 204–215.

    Google Scholar 

  9. M.S. Bazaraa, H. D. Sherali, and C.M. Shetty, Nonlinear Programming: Theory and Algorithms, Wiley, second ed., 1993.

    Google Scholar 

  10. 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., vol. 1204, Proc. SPIE, 1990, pp. 481–491.

    Google Scholar 

  11. S.B. Colak, G.W. Hooft, D.G. Papaioannou, and M.B. Van Der Mark, 3D backprojection tomography for medical optical imaging, in Alfano and Fujimoto [1], pp. 294–298.

    Google Scholar 

  12. 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 (1988), pp. 1433–1442.

    Article  Google Scholar 

  13. 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 spectroscopy,Lancet, ii (1988), pp. 770–771.

    Article  Google Scholar 

  14. J.C. Hebden, S.R. Arridge, and D.T. Delpy, Optical imaging in medicine: I. Experimental techniques, Phys. Med. Biol., 42 (1997), pp. 825–840.

    Article  Google Scholar 

  15. J.C. Hebden, R.A. Kruger, and K.S. Wong, Time resolved imaging through a highly scattering medium, Appl. Opt., 30 (1991), pp. 788–794.

    Article  Google Scholar 

  16. D. Isaacson, Distinguishability of conductivities by electric current computed tomography,IEEE Med. Im., 5 (1986), pp. 91–95.

    Article  Google Scholar 

  17. V. Isakov, Inverse Problems in Partial Differential Equations,Springer, New York, 1998.

    Google Scholar 

  18. 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 (1995), pp. 253–266.

    Article  Google Scholar 

  19. M.V. Klibanov, T.R. Lucas, and R.M. Frank, A fast and accurate imaging algorithm in optical diffusion tomography, Inverse Problems, 13 (1997), pp. 1341–1361.

    Article  MathSciNet  MATH  Google Scholar 

  20. D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters,J. SIAM, 11 (1963), pp. 431–441.

    MathSciNet  MATH  Google Scholar 

  21. R. Model, M. Orlt, M. Walzel, and R. Hunlich, Reconstruction algorithm for near-infrared imaging in turbid media by means of time-domain data, Appl. Opt., 14 (1997), pp. 313–324.

    Google Scholar 

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

    Google Scholar 

  23. K. Mueller, R. Yagel, and F. Cornhill, The weighted-distance scheme: A globally optimizing projection ordering method for ART, IEEE Med. Im., 16 (1997), pp. 223–230.

    Article  Google Scholar 

  24. F. Natterer, The Mathematics of Computerised Tomography, Wiley, New York, 1986.

    Google Scholar 

  25. F. Natterer and F. Wubbeling, A propagation-backpropagation method for ultrasound tomography, Inverse Problems, 11 (1995), pp. 1225–1232.

    Article  MathSciNet  MATH  Google Scholar 

  26. 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 (1995), pp. 426–428.

    Article  Google Scholar 

  27. K.D. Paulsen and H. Jiang, Spatially-varying optical property reconstruction using a finite element diffusion equation approximation, Med. Phys., 22 (1995), pp. 691–701.

    Article  Google Scholar 

  28. K.D. Paulsen, P.M. Meaney, M.J. Moskowitz, and J.J.M. Sullivan, A dual mesh scheme for finite element based reconstruction algorithms, IEEE Med. Im., 14 (1995), pp. 504–514.

    Article  Google Scholar 

  29. PH. I. Trans, Royal Soc. B, Near-infrared spectroscopy and imaging of living systems,vol. 352, 1997.

    Google Scholar 

  30. 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 (1995), pp. 1709–1729.

    Article  Google Scholar 

  31. 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 (1997), pp. 369–380.

    Google Scholar 

  32. 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 (1997), pp. 9042–9049.

    Article  Google Scholar 

  33. M. Schweiger and S.R. Arridge, The finite element model for the propagation of light in scattering media: Frequency domain case, Med. Phys., 24 (1997), pp. 895–902.

    Article  Google Scholar 

  34. 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 (1993), pp. 263–283.

    Article  MATH  Google Scholar 

  35. 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 (1995), pp. 1779–1792.

    Article  Google Scholar 

  36. M. Tamura, Multichannel near-infrared optical imaging of human brain activity,in Advances in Optical Imaging and Photon Migration, vol. 2, Proc. OSA, Proc. OSA, 1996, pp. 8–10.

    Google Scholar 

  37. S.A. Walker, S. Fantini, and E. Gratton, Back-projection reconstructions of cylindrical inhomogeneities from frequency domain optical measurements in turbid media, in Alfano and Fujimoto [1], pp. 137–141.

    Google Scholar 

  38. 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 (1990), pp. 1086–1091.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University College London, WC1E 6BT, UK

    Simon R. Arridge

  2. Department of Medical Physics, University College London, WC1E 6JA, UK

    Martin Schweiger

Authors
  1. Simon R. Arridge
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  2. Martin Schweiger
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Tufts University, Medford, MA, 02155, USA

    Christoph Börgers

  2. Institut für Numerische und instrumentelle Mathematik, Universität Münster, Einsteinstrasse 62, D-48149, Münster, Germany

    Frank Natterer

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© 1999 Springer Science+Business Media New York

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Arridge, S.R., Schweiger, M. (1999). A General Framework for Iterative Reconstruction Algorithms in Optical Tomography, Using a Finite Element Method. In: Börgers, C., Natterer, F. (eds) Computational Radiology and Imaging. The IMA Volumes in Mathematics and its Applications, vol 110. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1550-9_4

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  • DOI: https://doi.org/10.1007/978-1-4612-1550-9_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7189-5

  • Online ISBN: 978-1-4612-1550-9

  • eBook Packages: Springer Book Archive

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