Abstract

Linear reconstruction methods in diffuse optical tomography have been found to produce reasonable good images in cases in which the variation in optical properties within the medium is relatively small and a reference measurement with known background optical properties is available. In this paper we examine the correction of errors when using a first order Born approximation with an infinite space Green’s function model as the basis for linear reconstruction in diffuse optical tomography, when real data is generated on a finite domain with possibly unknown background optical properties. We consider the relationship between conventional reference measurement correction and approximation error modelling in reconstruction. It is shown that, using the approximation error modelling, linear reconstruction method can be used to produce good quality images also in situations in which the background optical properties are not known and a reference is not available.

© 2010 Optical Society of America

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  1. S. R. Arridge, "Optical tomography in medical imaging," Inv. Probl. 15, R41-R93 (1999).
  2. S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).
  3. S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, "Approximation errors and model reduction with an application in optical diffusion tomography," Inv. Probl. 22, 175-195 (2006).
  4. S. R. Arridge and J. C. Schotland, "Optical tomography: forward and inverse problems," Inv. Probl. 25, 123010 (2009).
  5. G. Bal, "Inverse transport theory and applications," Inv. Probl. 25, 053001 (2009).
  6. D. A. Boas, "A fundamental limitation of linearized algorithms for diffuse optical tomography," Opt. Express 1, 404-413 (1997).
  7. A. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical tomography," Phys. Med. Biol. 50, R1-R43 (2005).
  8. J. Heino, E. Somersalo, and J. P. Kaipio, "Compensation for geometric mismodelling by anisotropies in optical tomography," Opt. Express 13, 296-308 (2005).
  9. J. M. J. Huttunen and J. P. Kaipio, "Approximation errors in nonstationary inverse problems," Inv. Probl. Imaging 1, 77-93 (2007).
  10. J. M. J. Huttunen and J. P. Kaipio, "Model reduction in state identification problems with an application to determination of thermal parameters," Appl. Numer. Math. 59, 877-890 (2009).
  11. J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, (Springer, New York, 2005).
  12. J. Kaipio and E. Somersalo, "Statistical inverse problems: Discretization, model reduction and inverse crimes," J. Comput. Appl. Math. 198, 493-504 (2007).
  13. A. D. Klose and A. H. Hielscher, "Optical tomography with the equation of radiative transfer," Int. J. Numer. Meth. Heat Fluid Flow 18, 443-464 (2008).
  14. V. Kolehmainen, M. Schweiger, I. Nissilä, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, "Approximation errors and model reduction in three-dimensional diffuse optical tomography," J. Opt. Soc. Am. A 26, 2257-2268 (2009).
  15. S. D. Konecky, G. Y. Panasyuk, K. Lee, V. Markel, A. G. Yodh, and J. C. Schotland, "Imaging complex structures with diffuse light," Opt. Express 16, 5048-5060 (2008).
  16. A. Lehikoinen, S. Finsterle, A. Voutilainen, L. M. Heikkinen, M. Vauhkonen, and J. P. Kaipio, "Approximation errors and truncation of computational domains with application to geophysical tomography," Inv. Probl. Imaging 1, 371-389 (2007).
  17. A. Nissinen, L. M. Heikkinen, and J. P Kaipio, "The Bayesian approximation error approach for electrical impedance tomography - experimental results," Meas. Sci. Technol. 19, 015501 (2008).
  18. A. Nissinen, L. M. Heikkinen, V. Kolehmainen, and J. P Kaipio, "Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography," Meas. Sci. Technol. 20, 015504 (2009).
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  21. J. C. Schotland and V. Markel, "Inverse scattering with diffusing waves," J. Opt. Soc. Am. A 18, 2767-2777 (2001).
  22. J. Ripoll, V. Ntziachristos, and M. Nieto-Vesperinas, "The Kirchhoff approximation for diffusive waves," Phys. Rev. E 64, 1-8 (2001).
  23. N. Polydorides, "Linearization error in electical impedance tomography," Prog. Electromagn. Res., PIER 93, 323-337 (2009).
  24. T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography," Inv. Probl. 26, 015005 (2010).

2010 (1)

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography," Inv. Probl. 26, 015005 (2010).

2009 (6)

N. Polydorides, "Linearization error in electical impedance tomography," Prog. Electromagn. Res., PIER 93, 323-337 (2009).

A. Nissinen, L. M. Heikkinen, V. Kolehmainen, and J. P Kaipio, "Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography," Meas. Sci. Technol. 20, 015504 (2009).

S. R. Arridge and J. C. Schotland, "Optical tomography: forward and inverse problems," Inv. Probl. 25, 123010 (2009).

G. Bal, "Inverse transport theory and applications," Inv. Probl. 25, 053001 (2009).

J. M. J. Huttunen and J. P. Kaipio, "Model reduction in state identification problems with an application to determination of thermal parameters," Appl. Numer. Math. 59, 877-890 (2009).

V. Kolehmainen, M. Schweiger, I. Nissilä, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, "Approximation errors and model reduction in three-dimensional diffuse optical tomography," J. Opt. Soc. Am. A 26, 2257-2268 (2009).

2008 (3)

S. D. Konecky, G. Y. Panasyuk, K. Lee, V. Markel, A. G. Yodh, and J. C. Schotland, "Imaging complex structures with diffuse light," Opt. Express 16, 5048-5060 (2008).

A. D. Klose and A. H. Hielscher, "Optical tomography with the equation of radiative transfer," Int. J. Numer. Meth. Heat Fluid Flow 18, 443-464 (2008).

A. Nissinen, L. M. Heikkinen, and J. P Kaipio, "The Bayesian approximation error approach for electrical impedance tomography - experimental results," Meas. Sci. Technol. 19, 015501 (2008).

2007 (3)

A. Lehikoinen, S. Finsterle, A. Voutilainen, L. M. Heikkinen, M. Vauhkonen, and J. P. Kaipio, "Approximation errors and truncation of computational domains with application to geophysical tomography," Inv. Probl. Imaging 1, 371-389 (2007).

J. Kaipio and E. Somersalo, "Statistical inverse problems: Discretization, model reduction and inverse crimes," J. Comput. Appl. Math. 198, 493-504 (2007).

J. M. J. Huttunen and J. P. Kaipio, "Approximation errors in nonstationary inverse problems," Inv. Probl. Imaging 1, 77-93 (2007).

2006 (1)

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, "Approximation errors and model reduction with an application in optical diffusion tomography," Inv. Probl. 22, 175-195 (2006).

2005 (2)

A. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical tomography," Phys. Med. Biol. 50, R1-R43 (2005).

J. Heino, E. Somersalo, and J. P. Kaipio, "Compensation for geometric mismodelling by anisotropies in optical tomography," Opt. Express 13, 296-308 (2005).

2001 (2)

J. C. Schotland and V. Markel, "Inverse scattering with diffusing waves," J. Opt. Soc. Am. A 18, 2767-2777 (2001).

J. Ripoll, V. Ntziachristos, and M. Nieto-Vesperinas, "The Kirchhoff approximation for diffusive waves," Phys. Rev. E 64, 1-8 (2001).

1999 (1)

S. R. Arridge, "Optical tomography in medical imaging," Inv. Probl. 15, R41-R93 (1999).

1997 (1)

1995 (1)

1992 (1)

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).

Arridge, S. R.

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography," Inv. Probl. 26, 015005 (2010).

S. R. Arridge and J. C. Schotland, "Optical tomography: forward and inverse problems," Inv. Probl. 25, 123010 (2009).

V. Kolehmainen, M. Schweiger, I. Nissilä, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, "Approximation errors and model reduction in three-dimensional diffuse optical tomography," J. Opt. Soc. Am. A 26, 2257-2268 (2009).

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, "Approximation errors and model reduction with an application in optical diffusion tomography," Inv. Probl. 22, 175-195 (2006).

A. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical tomography," Phys. Med. Biol. 50, R1-R43 (2005).

S. R. Arridge, "Optical tomography in medical imaging," Inv. Probl. 15, R41-R93 (1999).

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).

Bal, G.

G. Bal, "Inverse transport theory and applications," Inv. Probl. 25, 053001 (2009).

Boas, D. A.

Chance, B.

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).

Delpy, D. T.

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).

Finsterle, S.

A. Lehikoinen, S. Finsterle, A. Voutilainen, L. M. Heikkinen, M. Vauhkonen, and J. P. Kaipio, "Approximation errors and truncation of computational domains with application to geophysical tomography," Inv. Probl. Imaging 1, 371-389 (2007).

Gibson, A.

A. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical tomography," Phys. Med. Biol. 50, R1-R43 (2005).

Hebden, J. C.

A. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical tomography," Phys. Med. Biol. 50, R1-R43 (2005).

Heikkinen, L. M.

A. Nissinen, L. M. Heikkinen, V. Kolehmainen, and J. P Kaipio, "Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography," Meas. Sci. Technol. 20, 015504 (2009).

A. Nissinen, L. M. Heikkinen, and J. P Kaipio, "The Bayesian approximation error approach for electrical impedance tomography - experimental results," Meas. Sci. Technol. 19, 015501 (2008).

A. Lehikoinen, S. Finsterle, A. Voutilainen, L. M. Heikkinen, M. Vauhkonen, and J. P. Kaipio, "Approximation errors and truncation of computational domains with application to geophysical tomography," Inv. Probl. Imaging 1, 371-389 (2007).

Heino, J.

Hielscher, A. H.

A. D. Klose and A. H. Hielscher, "Optical tomography with the equation of radiative transfer," Int. J. Numer. Meth. Heat Fluid Flow 18, 443-464 (2008).

Huttunen, J. M. J.

J. M. J. Huttunen and J. P. Kaipio, "Model reduction in state identification problems with an application to determination of thermal parameters," Appl. Numer. Math. 59, 877-890 (2009).

J. M. J. Huttunen and J. P. Kaipio, "Approximation errors in nonstationary inverse problems," Inv. Probl. Imaging 1, 77-93 (2007).

Kaipio, J.

J. Kaipio and E. Somersalo, "Statistical inverse problems: Discretization, model reduction and inverse crimes," J. Comput. Appl. Math. 198, 493-504 (2007).

Kaipio, J. P

A. Nissinen, L. M. Heikkinen, V. Kolehmainen, and J. P Kaipio, "Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography," Meas. Sci. Technol. 20, 015504 (2009).

A. Nissinen, L. M. Heikkinen, and J. P Kaipio, "The Bayesian approximation error approach for electrical impedance tomography - experimental results," Meas. Sci. Technol. 19, 015501 (2008).

Kaipio, J. P.

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography," Inv. Probl. 26, 015005 (2010).

J. M. J. Huttunen and J. P. Kaipio, "Model reduction in state identification problems with an application to determination of thermal parameters," Appl. Numer. Math. 59, 877-890 (2009).

V. Kolehmainen, M. Schweiger, I. Nissilä, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, "Approximation errors and model reduction in three-dimensional diffuse optical tomography," J. Opt. Soc. Am. A 26, 2257-2268 (2009).

A. Lehikoinen, S. Finsterle, A. Voutilainen, L. M. Heikkinen, M. Vauhkonen, and J. P. Kaipio, "Approximation errors and truncation of computational domains with application to geophysical tomography," Inv. Probl. Imaging 1, 371-389 (2007).

J. M. J. Huttunen and J. P. Kaipio, "Approximation errors in nonstationary inverse problems," Inv. Probl. Imaging 1, 77-93 (2007).

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, "Approximation errors and model reduction with an application in optical diffusion tomography," Inv. Probl. 22, 175-195 (2006).

J. Heino, E. Somersalo, and J. P. Kaipio, "Compensation for geometric mismodelling by anisotropies in optical tomography," Opt. Express 13, 296-308 (2005).

Klose, A. D.

A. D. Klose and A. H. Hielscher, "Optical tomography with the equation of radiative transfer," Int. J. Numer. Meth. Heat Fluid Flow 18, 443-464 (2008).

Kolehmainen, V.

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography," Inv. Probl. 26, 015005 (2010).

V. Kolehmainen, M. Schweiger, I. Nissilä, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, "Approximation errors and model reduction in three-dimensional diffuse optical tomography," J. Opt. Soc. Am. A 26, 2257-2268 (2009).

A. Nissinen, L. M. Heikkinen, V. Kolehmainen, and J. P Kaipio, "Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography," Meas. Sci. Technol. 20, 015504 (2009).

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, "Approximation errors and model reduction with an application in optical diffusion tomography," Inv. Probl. 22, 175-195 (2006).

Konecky, S. D.

Lee, K.

Lehikoinen, A.

A. Lehikoinen, S. Finsterle, A. Voutilainen, L. M. Heikkinen, M. Vauhkonen, and J. P. Kaipio, "Approximation errors and truncation of computational domains with application to geophysical tomography," Inv. Probl. Imaging 1, 371-389 (2007).

Markel, V.

Nieto-Vesperinas, M.

J. Ripoll, V. Ntziachristos, and M. Nieto-Vesperinas, "The Kirchhoff approximation for diffusive waves," Phys. Rev. E 64, 1-8 (2001).

Nissilä, I.

Nissinen, A.

A. Nissinen, L. M. Heikkinen, V. Kolehmainen, and J. P Kaipio, "Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography," Meas. Sci. Technol. 20, 015504 (2009).

A. Nissinen, L. M. Heikkinen, and J. P Kaipio, "The Bayesian approximation error approach for electrical impedance tomography - experimental results," Meas. Sci. Technol. 19, 015501 (2008).

Ntziachristos, V.

J. Ripoll, V. Ntziachristos, and M. Nieto-Vesperinas, "The Kirchhoff approximation for diffusive waves," Phys. Rev. E 64, 1-8 (2001).

O’Leary, M. A.

Panasyuk, G. Y.

Polydorides, N.

N. Polydorides, "Linearization error in electical impedance tomography," Prog. Electromagn. Res., PIER 93, 323-337 (2009).

Pulkkinen, A.

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography," Inv. Probl. 26, 015005 (2010).

Ripoll, J.

J. Ripoll, V. Ntziachristos, and M. Nieto-Vesperinas, "The Kirchhoff approximation for diffusive waves," Phys. Rev. E 64, 1-8 (2001).

Schotland, J. C.

Schweiger, M.

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography," Inv. Probl. 26, 015005 (2010).

V. Kolehmainen, M. Schweiger, I. Nissilä, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, "Approximation errors and model reduction in three-dimensional diffuse optical tomography," J. Opt. Soc. Am. A 26, 2257-2268 (2009).

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, "Approximation errors and model reduction with an application in optical diffusion tomography," Inv. Probl. 22, 175-195 (2006).

Somersalo, E.

J. Kaipio and E. Somersalo, "Statistical inverse problems: Discretization, model reduction and inverse crimes," J. Comput. Appl. Math. 198, 493-504 (2007).

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, "Approximation errors and model reduction with an application in optical diffusion tomography," Inv. Probl. 22, 175-195 (2006).

J. Heino, E. Somersalo, and J. P. Kaipio, "Compensation for geometric mismodelling by anisotropies in optical tomography," Opt. Express 13, 296-308 (2005).

Tarvainen, T.

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography," Inv. Probl. 26, 015005 (2010).

V. Kolehmainen, M. Schweiger, I. Nissilä, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, "Approximation errors and model reduction in three-dimensional diffuse optical tomography," J. Opt. Soc. Am. A 26, 2257-2268 (2009).

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, "Approximation errors and model reduction with an application in optical diffusion tomography," Inv. Probl. 22, 175-195 (2006).

Vauhkonen, M.

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography," Inv. Probl. 26, 015005 (2010).

A. Lehikoinen, S. Finsterle, A. Voutilainen, L. M. Heikkinen, M. Vauhkonen, and J. P. Kaipio, "Approximation errors and truncation of computational domains with application to geophysical tomography," Inv. Probl. Imaging 1, 371-389 (2007).

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, "Approximation errors and model reduction with an application in optical diffusion tomography," Inv. Probl. 22, 175-195 (2006).

Voutilainen, A.

A. Lehikoinen, S. Finsterle, A. Voutilainen, L. M. Heikkinen, M. Vauhkonen, and J. P. Kaipio, "Approximation errors and truncation of computational domains with application to geophysical tomography," Inv. Probl. Imaging 1, 371-389 (2007).

Yodh, A. G.

Appl. Numer. Math. (1)

J. M. J. Huttunen and J. P. Kaipio, "Model reduction in state identification problems with an application to determination of thermal parameters," Appl. Numer. Math. 59, 877-890 (2009).

Int. J. Numer. Meth. Heat Fluid Flow (1)

A. D. Klose and A. H. Hielscher, "Optical tomography with the equation of radiative transfer," Int. J. Numer. Meth. Heat Fluid Flow 18, 443-464 (2008).

Inv. Probl. (5)

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography," Inv. Probl. 26, 015005 (2010).

S. R. Arridge, "Optical tomography in medical imaging," Inv. Probl. 15, R41-R93 (1999).

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, "Approximation errors and model reduction with an application in optical diffusion tomography," Inv. Probl. 22, 175-195 (2006).

S. R. Arridge and J. C. Schotland, "Optical tomography: forward and inverse problems," Inv. Probl. 25, 123010 (2009).

G. Bal, "Inverse transport theory and applications," Inv. Probl. 25, 053001 (2009).

Inv. Probl. Imaging (2)

J. M. J. Huttunen and J. P. Kaipio, "Approximation errors in nonstationary inverse problems," Inv. Probl. Imaging 1, 77-93 (2007).

A. Lehikoinen, S. Finsterle, A. Voutilainen, L. M. Heikkinen, M. Vauhkonen, and J. P. Kaipio, "Approximation errors and truncation of computational domains with application to geophysical tomography," Inv. Probl. Imaging 1, 371-389 (2007).

J. Comput. Appl. Math. (1)

J. Kaipio and E. Somersalo, "Statistical inverse problems: Discretization, model reduction and inverse crimes," J. Comput. Appl. Math. 198, 493-504 (2007).

J. Opt. Soc. Am. A (2)

Meas. Sci. Technol. (2)

A. Nissinen, L. M. Heikkinen, and J. P Kaipio, "The Bayesian approximation error approach for electrical impedance tomography - experimental results," Meas. Sci. Technol. 19, 015501 (2008).

A. Nissinen, L. M. Heikkinen, V. Kolehmainen, and J. P Kaipio, "Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography," Meas. Sci. Technol. 20, 015504 (2009).

Opt. Express (3)

Opt. Lett. (1)

Phys. Med. Biol. (2)

A. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical tomography," Phys. Med. Biol. 50, R1-R43 (2005).

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).

Phys. Rev. E (1)

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Figures (6)

Fig. 1.
Fig. 1.

Simulation domain with two inclusions. Locations of sources and detectors are marked with black dots.

Fig. 2.
Fig. 2.

Horizontal slices at the heights z = 59mm (two columns from the left) and z = 50mm (two columns from the right) from 3D reconstructions of absorption (left) and scattering (right) distributions with known background. The images from top to bottom are: target distribution (top row), and reconstructions using RM (second row), EM-1 (third row) and EM-2 (fourth row).

Fig. 3.
Fig. 3.

Vertical slices at the depths x = 21mm (two columns from the left) and x = 45mm (two columns from the right) from 3D reconstructions of absorption (left) and scattering (right) distributions with known background. The images from top to bottom are: target distribution (top row), and reconstructions using RM (second row), EM-1 (third row), and EM-2 (fourth row).

Fig. 4.
Fig. 4.

Horizontal slices at the heights z = 59mm (two columns from the left) and z = 50mm (two columns from the right) from 3D reconstructions of absorption (left) and scattering (right) distributions with mismodelled background. The images from top to bottom are: target distribution (top row), and reconstructions using RM (second row), EM-1 (third row), and EM-2 (fourth row).

Fig. 5.
Fig. 5.

Vertical slices at the depths x = 21mm (two columns from the left) and x = 45mm (two columns from the right) from 3D reconstructions of absorption (left) and scattering (right) distributions with mismodelled background. The images from top to bottom are: target distribution (top row), and reconstructions using RM (second row), EM-1 (third row), and EM-2 (fourth row).

Fig. 6.
Fig. 6.

Relative errors between the target optical properties and estimated optical properties against the difference in background value calculated with the RM (∗), EM-1 (∘) and EM-2 (+).

Tables (1)

Tables Icon

Table 1. The absorption coefficient μa(mm-1) and the reduced scattering coefficient μs(mm-1) of the background medium and the inclusions.

Equations (40)

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y=A(x)+e,A:𝕏m
xxhn,AAh
(·D+μa+iωc)Φ=0,inΩ
Φ+2ζDΦν=J,onΩ
𝓛(x)Φ=J.
A(x)=𝓜𝓖(x)J
A(x)=(𝓜1𝓖(x)J1𝓜2𝓖(x)J2𝓜S𝓖(x)JS).
A(x0+δx)=𝓜[𝓘𝓖(x0)𝓥(δx)]1𝓖(x0)
=A(x0)+𝓜[𝓖(x0)𝓥(δx)+(𝓖(x0)𝓥(δx))2+]𝓖(x0),
A(x0+δx)=A(x0)+𝖩(x0)δx,
gobsy0=𝖩(x0)δx,
y0|rm=𝓜ΩG(rm,rs;x0)J(rs)drs
(yy0)|rm=𝓜ΩG(rm,r;x0)𝓥(δx)G(r,rs;x0)J(rs)drs.
G(r1,r2)=eσr1r2r1r2
x̂=argminx{12gobsA(x)Γe2+Ψ(x)}
x̂=x0+argminδx{12gobsA(x0)𝖩(x0)δxΓe2+Ψ(δx)}.
y˜h=Ah(xh)+e+ε0.
A˜h(xh)=Ah(xh)+grefAh(xh,0).
ε0=grefAh(xh,0)gref=Ah(xh,0)+ε0.
gobsgref=𝖩(xh,0)δxh
x̂=argminxh{12gobsAh(xh)gref+Ah(xh,0)Γe2+Ψ(xh)}
x̂=xh,0+argminδxh{12gobsgref𝖩(xh,0)δxhΓe2+Ψ(δxh)}.
ε(xh)=A(x)Ah(xh)
A˜h(xh)=Ah(xh)+ε(xh).
x̂=xh,0+argminδxh{12gobsAh(xh,0)𝖩(xh,0)δxhε¯Γe2+Ψ(δxh)}
x̂=argminxh{12gobsAh(xh)ε¯Γe+ε2+Ψ(xh)}
x̂=xh,0+argminδxh{12gobsAh(xh,0)𝖩(xh,0)δxhε¯Γe+ε2+Ψ(δxh)}.
ε grefAh(xh,0),Γε𝖨.
Alog(x)=𝓜log𝒢(x)+e
Alog(x0+δx)=Alog(x0)+𝖲𝖩(x0)δx+e
logA˜h(xh)=logAh(xh,0)+logA(x)logA(x0)+e
=log[y0grefA(x)]+e
diag[1y0]𝖩(xh,0)diag[1gref]𝖩(x0).
Ah(xh)=G(xh,0)+𝖩(xh,0)δxh
𝖩(μa)=(G*(rk,rd,i)·G(rk,rs,j))
𝖩(D)=(G*(rk,rd,i)·G(rk,rs,j))
ε=A(x())Ah(xh())
ε¯=1r=1rεh()
Γε=1r1=1rε()ε()TεεT.
êx=(xx̂)2x2·100

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