Abstract

Model reduction is often required in diffuse optical tomography (DOT), typically because of limited available computation time or computer memory. In practice, this means that one is bound to use coarse mesh and truncated computation domain in the model for the forward problem. We apply the (Bayesian) approximation error model for the compensation of modeling errors caused by domain truncation and a coarse computation mesh in DOT. The approach is tested with a three-dimensional example using experimental data. The results show that when the approximation error model is employed, it is possible to use mesh densities and computation domains that would be unacceptable with a conventional measurement error model.

© 2009 Optical Society of America

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References

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  1. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, 41-93 (1999).
    [CrossRef]
  2. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1-43 (2005).
    [CrossRef]
  3. B. J. Tromberg, B. W. Pogue, K. D. Paulsen, A. G. Yodh, D. A. Boas, and A. E. Cerussi, “Assessing the future of diffuse imaging technologies for breast cancer management,” Med. Phys. 35, 2443-2451 (2008).
    [CrossRef] [PubMed]
  4. 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,” Inverse Probl. 22, 175-195 (2006).
    [CrossRef]
  5. J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Applied Mathematical Sciences, Springer, 2005).
  6. J. Kaipio and E. Somersalo, “Statistical inverse problems: Discretization, model reduction and inverse crimes,” J. Comput. Appl. Math. 198, 493-504 (2007).
    [CrossRef]
  7. 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]
  8. J. Heino and E. Somersalo, “Estimation of optical absorption in anisotropic background,” Inverse Probl. 18, 559-573 (2002).
    [CrossRef]
  9. I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,”Rev. Sci. Instrum. 76, 1-10 (2005) article number 044302.
    [CrossRef]
  10. M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365-2386 (2005).
    [CrossRef] [PubMed]
  11. M. Schweiger and S. R. Arridge, “Application of temporal filters to time-resolved data in optical tomography,” Phys. Med. Biol. 44, 1699-1717 (1999).
    [CrossRef] [PubMed]
  12. D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing (Springer, 2007).
  13. M. Schweiger and S. R. Arridge, “Optical tomography with local basis functions,” J. Electron. Imaging 12, 583-593 (2003).
    [CrossRef]
  14. 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).
    [CrossRef]
  15. M. Schweiger, I. Nissilä, D. A. Boas, and S. R. Arridge, “Image reconstruction in optical tomography in the presence of coupling errors,” Appl. Opt. 46, 2743-2756 (2007).
    [CrossRef] [PubMed]
  16. T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “Computational calibration method for optical tomography,” Appl. Opt. 44, 1879-1888 (2006).
    [CrossRef]
  17. R. P. Brent, Algorithms for Minimization without Derivatives (Dover, 2002).
  18. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

2008 (1)

B. J. Tromberg, B. W. Pogue, K. D. Paulsen, A. G. Yodh, D. A. Boas, and A. E. Cerussi, “Assessing the future of diffuse imaging technologies for breast cancer management,” Med. Phys. 35, 2443-2451 (2008).
[CrossRef] [PubMed]

2007 (3)

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

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).
[CrossRef]

M. Schweiger, I. Nissilä, D. A. Boas, and S. R. Arridge, “Image reconstruction in optical tomography in the presence of coupling errors,” Appl. Opt. 46, 2743-2756 (2007).
[CrossRef] [PubMed]

2006 (2)

T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “Computational calibration method for optical tomography,” Appl. Opt. 44, 1879-1888 (2006).
[CrossRef]

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,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

2005 (3)

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1-43 (2005).
[CrossRef]

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,”Rev. Sci. Instrum. 76, 1-10 (2005) article number 044302.
[CrossRef]

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365-2386 (2005).
[CrossRef] [PubMed]

2003 (1)

M. Schweiger and S. R. Arridge, “Optical tomography with local basis functions,” J. Electron. Imaging 12, 583-593 (2003).
[CrossRef]

2002 (1)

J. Heino and E. Somersalo, “Estimation of optical absorption in anisotropic background,” Inverse Probl. 18, 559-573 (2002).
[CrossRef]

1999 (2)

M. Schweiger and S. R. Arridge, “Application of temporal filters to time-resolved data in optical tomography,” Phys. Med. Biol. 44, 1699-1717 (1999).
[CrossRef] [PubMed]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, 41-93 (1999).
[CrossRef]

1995 (1)

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]

Arridge, S. R.

M. Schweiger, I. Nissilä, D. A. Boas, and S. R. Arridge, “Image reconstruction in optical tomography in the presence of coupling errors,” Appl. Opt. 46, 2743-2756 (2007).
[CrossRef] [PubMed]

T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “Computational calibration method for optical tomography,” Appl. Opt. 44, 1879-1888 (2006).
[CrossRef]

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,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1-43 (2005).
[CrossRef]

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,”Rev. Sci. Instrum. 76, 1-10 (2005) article number 044302.
[CrossRef]

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365-2386 (2005).
[CrossRef] [PubMed]

M. Schweiger and S. R. Arridge, “Optical tomography with local basis functions,” J. Electron. Imaging 12, 583-593 (2003).
[CrossRef]

M. Schweiger and S. R. Arridge, “Application of temporal filters to time-resolved data in optical tomography,” Phys. Med. Biol. 44, 1699-1717 (1999).
[CrossRef] [PubMed]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, 41-93 (1999).
[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]

Boas, D. A.

B. J. Tromberg, B. W. Pogue, K. D. Paulsen, A. G. Yodh, D. A. Boas, and A. E. Cerussi, “Assessing the future of diffuse imaging technologies for breast cancer management,” Med. Phys. 35, 2443-2451 (2008).
[CrossRef] [PubMed]

M. Schweiger, I. Nissilä, D. A. Boas, and S. R. Arridge, “Image reconstruction in optical tomography in the presence of coupling errors,” Appl. Opt. 46, 2743-2756 (2007).
[CrossRef] [PubMed]

Brent, R. P.

R. P. Brent, Algorithms for Minimization without Derivatives (Dover, 2002).

Calvetti, D.

D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing (Springer, 2007).

Cerussi, A. E.

B. J. Tromberg, B. W. Pogue, K. D. Paulsen, A. G. Yodh, D. A. Boas, and A. E. Cerussi, “Assessing the future of diffuse imaging technologies for breast cancer management,” Med. Phys. 35, 2443-2451 (2008).
[CrossRef] [PubMed]

Delpy, D. T.

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]

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).
[CrossRef]

Gibson, A. P.

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1-43 (2005).
[CrossRef]

Heikkinen, L. M.

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).
[CrossRef]

Heino, J.

J. Heino and E. Somersalo, “Estimation of optical absorption in anisotropic background,” Inverse Probl. 18, 559-573 (2002).
[CrossRef]

Hiraoka, M.

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]

Kaipio, J.

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

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Applied Mathematical Sciences, Springer, 2005).

Kaipio, J. P.

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).
[CrossRef]

T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “Computational calibration method for optical tomography,” Appl. Opt. 44, 1879-1888 (2006).
[CrossRef]

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,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

Katila, T.

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,”Rev. Sci. Instrum. 76, 1-10 (2005) article number 044302.
[CrossRef]

Kolehmainen, V.

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,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “Computational calibration method for optical tomography,” Appl. Opt. 44, 1879-1888 (2006).
[CrossRef]

Kotilahti, K.

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,”Rev. Sci. Instrum. 76, 1-10 (2005) article number 044302.
[CrossRef]

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).
[CrossRef]

Lipiäinen, L.

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,”Rev. Sci. Instrum. 76, 1-10 (2005) article number 044302.
[CrossRef]

Nissilä, I.

M. Schweiger, I. Nissilä, D. A. Boas, and S. R. Arridge, “Image reconstruction in optical tomography in the presence of coupling errors,” Appl. Opt. 46, 2743-2756 (2007).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365-2386 (2005).
[CrossRef] [PubMed]

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,”Rev. Sci. Instrum. 76, 1-10 (2005) article number 044302.
[CrossRef]

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

Noponen, T.

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,”Rev. Sci. Instrum. 76, 1-10 (2005) article number 044302.
[CrossRef]

Paulsen, K. D.

B. J. Tromberg, B. W. Pogue, K. D. Paulsen, A. G. Yodh, D. A. Boas, and A. E. Cerussi, “Assessing the future of diffuse imaging technologies for breast cancer management,” Med. Phys. 35, 2443-2451 (2008).
[CrossRef] [PubMed]

Pogue, B. W.

B. J. Tromberg, B. W. Pogue, K. D. Paulsen, A. G. Yodh, D. A. Boas, and A. E. Cerussi, “Assessing the future of diffuse imaging technologies for breast cancer management,” Med. Phys. 35, 2443-2451 (2008).
[CrossRef] [PubMed]

Schweiger, M.

M. Schweiger, I. Nissilä, D. A. Boas, and S. R. Arridge, “Image reconstruction in optical tomography in the presence of coupling errors,” Appl. Opt. 46, 2743-2756 (2007).
[CrossRef] [PubMed]

T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “Computational calibration method for optical tomography,” Appl. Opt. 44, 1879-1888 (2006).
[CrossRef]

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,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365-2386 (2005).
[CrossRef] [PubMed]

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,”Rev. Sci. Instrum. 76, 1-10 (2005) article number 044302.
[CrossRef]

M. Schweiger and S. R. Arridge, “Optical tomography with local basis functions,” J. Electron. Imaging 12, 583-593 (2003).
[CrossRef]

M. Schweiger and S. R. Arridge, “Application of temporal filters to time-resolved data in optical tomography,” Phys. Med. Biol. 44, 1699-1717 (1999).
[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]

Somersalo, E.

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

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,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

J. Heino and E. Somersalo, “Estimation of optical absorption in anisotropic background,” Inverse Probl. 18, 559-573 (2002).
[CrossRef]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Applied Mathematical Sciences, Springer, 2005).

D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing (Springer, 2007).

Tarvainen, T.

T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “Computational calibration method for optical tomography,” Appl. Opt. 44, 1879-1888 (2006).
[CrossRef]

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,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,”Rev. Sci. Instrum. 76, 1-10 (2005) article number 044302.
[CrossRef]

Tromberg, B. J.

B. J. Tromberg, B. W. Pogue, K. D. Paulsen, A. G. Yodh, D. A. Boas, and A. E. Cerussi, “Assessing the future of diffuse imaging technologies for breast cancer management,” Med. Phys. 35, 2443-2451 (2008).
[CrossRef] [PubMed]

Vanne, A.

Vauhkonen, M.

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).
[CrossRef]

T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “Computational calibration method for optical tomography,” Appl. Opt. 44, 1879-1888 (2006).
[CrossRef]

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,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

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).
[CrossRef]

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

Yodh, A. G.

B. J. Tromberg, B. W. Pogue, K. D. Paulsen, A. G. Yodh, D. A. Boas, and A. E. Cerussi, “Assessing the future of diffuse imaging technologies for breast cancer management,” Med. Phys. 35, 2443-2451 (2008).
[CrossRef] [PubMed]

Appl. Opt. (2)

Inv. Probl. Imaging (1)

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).
[CrossRef]

Inverse Probl. (3)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, 41-93 (1999).
[CrossRef]

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,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

J. Heino and E. Somersalo, “Estimation of optical absorption in anisotropic background,” Inverse Probl. 18, 559-573 (2002).
[CrossRef]

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).
[CrossRef]

J. Electron. Imaging (1)

M. Schweiger and S. R. Arridge, “Optical tomography with local basis functions,” J. Electron. Imaging 12, 583-593 (2003).
[CrossRef]

Med. Phys. (2)

B. J. Tromberg, B. W. Pogue, K. D. Paulsen, A. G. Yodh, D. A. Boas, and A. E. Cerussi, “Assessing the future of diffuse imaging technologies for breast cancer management,” Med. Phys. 35, 2443-2451 (2008).
[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]

Phys. Med. Biol. (3)

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1-43 (2005).
[CrossRef]

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365-2386 (2005).
[CrossRef] [PubMed]

M. Schweiger and S. R. Arridge, “Application of temporal filters to time-resolved data in optical tomography,” Phys. Med. Biol. 44, 1699-1717 (1999).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,”Rev. Sci. Instrum. 76, 1-10 (2005) article number 044302.
[CrossRef]

Other (4)

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Applied Mathematical Sciences, Springer, 2005).

D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing (Springer, 2007).

R. P. Brent, Algorithms for Minimization without Derivatives (Dover, 2002).

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

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Figures (7)

Fig. 1
Fig. 1

Top: Measurement domain Ω. The dots denote the location of the sources and detectors. The (circles) lines above and below the sources and detectors denote the top and bottom of the truncated model domain Ω ̃ . The middle of the three circles at the center of the cylinder denotes the central slice of the domain Ω. Bottom: Central slice of the target ( μ a left, μ s right).

Fig. 2
Fig. 2

Two random samples from the prior density (28). The images display the cross section of the 3D parameters at the central slice of the cylinder domain Ω. Left: Absorption μ a . Right: Scatter μ s .

Fig. 3
Fig. 3

Modeling error between the accurate model A Ω , δ and target model A Ω , h in test case 1; see Table 1. Left: Covariance structure of the approximation error ε. The displayed quantity is the signed standard deviation sign ( Γ ε ) | Γ ε | , where the product refers to the element-by-element (array) multiplication. Right: Normalized eigenvalues λ λ max of Γ ε .

Fig. 4
Fig. 4

Pure discretization errors. Central horizontal slice from the 3D reconstructions of absorption μ a and scattering μ s . Top row: MAP estimate with the conventional error model (MAP-CEM) using the accurate forward model A Ω , δ (number of nodes in the FEM mesh N n = 148,276 ). Left: μ a , CEM . Right: μ a , EEM . Bottom row: MAP estimates with the conventional (MAP-CEM) and enhanced error models (MAP-EEM) using the target model A Ω , h (the number of nodes N n = 2,413 ). Correct model domain Ω ̃ = Ω is used in the target model A Ω , h . Columns from left to right: μ a , CEM , μ a , EEM , μ s , CEM , and μ s , EEM . The number of unknowns x = ( μ a , μ s ) T in the estimation with both models A Ω , δ and A Ω , h was 15,336.

Fig. 5
Fig. 5

Pure discretization errors. Vertical slices from the 3D reconstructions of absorption μ a and scattering μ s . The slices have been chosen such that the inclusion in the parameter is visible. The arrangement of the images is equivalent to that of Fig. 4.

Fig. 6
Fig. 6

Discretization and domain truncation errors. The model domain Ω ̃ in the target models A Ω ̃ , δ and A Ω ̃ , h is a truncated cylinder with radius 35 mm and height 14.5 mm; see Fig. 1. Central horizontal slice from the 3D reconstructions of absorption μ a and scattering μ s . Top row: Pure domain truncation errors. MAP estimates with the conventional (MAP-CEM) and enhanced error models (MAP-EEM) using the target model A Ω ̃ , δ ( N n = 23,413 ) . Columns from left to right: μ a , CEM , μ a , EEM , μ s , CEM , μ s , EEM . Bottom row: Combined domain truncation and discretization errors. MAP-CEM and MAP-EEM estimates using the target model A Ω ̃ , h ( N n = 1085 ) . Columns from left to right: μ a , CEM , μ a , EEM , μ s , CEM , μ s , EEM . The number of unknowns x = ( μ a , μ s ) T in the estimation was 2816 for both models A Ω ̃ , δ and A Ω ̃ , h .

Fig. 7
Fig. 7

Discretization and domain truncation errors. Vertical slice from the 3D reconstructions of absorption μ a and scattering μ s . Arrangement of the reconstructions is the same as in Fig. 6.

Tables (7)

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Table 1 Algorithm 1: Construction of the enhanced error model

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Table 1 Mesh Details for Test Case 1 a

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Table 2 Mesh Details for Test Case 2 a

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Table 3 Results of Initial Estimation (29) for Test Case 1 (Figs. 4, 5)

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Table 4 Reconstruction Times for Test Case 1 (Figs. 4, 5) a

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Table 5 Results of Initial Estimation (29) for Test Case 2 (Figs. 6, 7)

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Table 6 Reconstruction Times for Test Case 2 (Figs. 6, 7) a

Equations (50)

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y = A Ω ( x ) + e ,
y = A Ω ̃ , h ( x h ) + e ,
y = A Ω ̃ , h ( x h ) + [ A Ω ( x ) A Ω ̃ , h ( x h ) ] + e = A Ω ̃ , h ( x h ) + ε ( x ) + e ,
π ( x h | y ) = π ( x h ) π ( y | x h ) π ( y ) , y = y measured ,
π ( y | x h ) = π noise ( y A Ω ̃ , h ( x h ) ) ,
x h N ( x h * , Γ x h ) , e N ( e * , Γ e ) ,
π ( x h | y ) π + ( x h ) exp ( 1 2 ( x h x h * ) T Γ x h 1 ( x h x h * ) 1 2 ( y A Ω ̃ , h ( x h ) e * ) T Γ e 1 ( y A Ω ̃ , h ( x h ) e * ) ) ,
π + ( x h ) = k = 1 n θ ( x h , k ) , θ ( t ) = { 1 , t > 0 0 , otherwise } ,
x h , CEM = arg min x h > 0 { L e ( y A Ω ̃ , h ( x h ) e * ) 2 + L x h ( x h x h * ) 2 } .
R N R m , x δ A Ω , δ ( x δ ) , δ > 0 small .
y = A Ω , δ ( x δ ) + e , x δ R N , N = N δ .
P : R N R n , x δ x h .
A Ω ̃ , h : R n R m
y = A Ω ̃ , h ( x h ) + [ A Ω , δ ( x δ ) A Ω ̃ , h ( x h ) ] + e = A Ω ̃ , h ( x h ) + ε ( x δ ) + e .
S = { x δ ( 1 ) , x δ ( 2 ) , , x δ ( K ) } ,
n = ε ( x δ ) + e = [ A Ω , δ ( x δ ) A Ω ̃ , h ( P x δ ) ] + e ,
π ( n ) = R N π ( n | x δ ) π ( x δ ) d x δ = R N π noise ( n [ A Ω , δ ( x δ ) A Ω ̃ , h ( P x δ ) ] ) π ( x δ ) d x δ .
π ( n ) 1 K l = 1 K π noise ( n [ A Ω , δ ( x δ ( l ) ) A Ω ̃ , h ( P x δ ( l ) ) ] ) .
cov ( n ) = Γ n = Γ e + Γ ε ,
Γ ε 1 K 1 l = 1 K ξ ε ( l ) ξ ε ( l ) T , ξ ε ( l ) = [ A Ω , δ ( x δ ( l ) ) A Ω ̃ , h ( P x δ ( l ) ) ε * ] .
π ( n ) exp ( 1 2 ( n ε * e * ) T Γ n 1 ( n ε * e * ) ) .
π ( x h | y ) exp ( 1 2 ( x h x h * ) T Γ x h 1 ( x h x h * ) 1 2 ( y A Ω ̃ , h ( x h ) ε * e * ) T Γ n 1 ( y A Ω ̃ , h ( x h ) ε * e * ) ) ,
L x h T L x h = Γ x h 1 , L e + ε T L e + ε = Γ n 1 ,
x h , EEM = arg min x h > 0 { L e + ε ( y A Ω ̃ , h ( x h ) ε * e * ) 2 + L x h ( x h x h * ) 2 } .
x h 1 σ 2 y A Ω ̃ , h ( x h ) 2 + α x h 2 .
κ ( r ) Φ k ( r , ω ) + μ a ( r ) Φ k ( r , ω ) + i ω c Φ k ( r , ω ) = 0 , r Ω ̃ ,
Φ k ( r , ω ) + 2 ζ κ ( r ) Φ k ( r , ω ) ν = g k ( r , ω ) , r Ω ̃ ,
g k ( r , ω ) = { 2 f s , r s k 0 , r Ω ̃ \ s k . } .
z i , k ( r i , ω ) = ( 1 2 ζ ) Φ k ( r i , ω ) .
Φ h = i = 1 N n ϕ i φ i ( r ) ,
μ a ( r ) = j = 1 n p μ a , j χ j ( r ) ,
μ s ( r ) = j = 1 n p μ s , j χ j ( r ) ,
y = ( re ( log ( z ) ) im ( log ( z ) ) ) R m .
x h = ( μ a μ s ) R n , n = 2 n p .
y = A Ω ̃ , h ( x h ) ,
y = ( re ( log ( z ) ) im ( log ( z ) ) ) R 384
e N ( 0 , Γ e ) ,
L e ( y y 0 ) 2 = 2 c 2 , Γ e 1 = L e T L e
π ( x δ ) exp { 1 2 L x δ ( x δ x δ * ) 2 } , L x δ T L x δ = Γ x δ 1 ,
Γ x δ = [ Γ μ a 0 0 Γ μ s ] .
Γ μ a = Γ μ a + σ μ a , bg 2 1 , Γ μ s = Γ μ s + σ μ s , bg 2 1 ,
A Ω , h A Ω , h A Ω , h
SNR = 10 log 10 ( A Ω , δ 2 ε * 2 + trace ( Γ ε ) ) ,
{ a , s , η , ϕ } = arg min a # , s # , η # , ϕ # { L ̃ ( ( y meas + Δ y ) A Ω , h ( exp ( a # ) , exp ( s # ) ) e ̃ ) 2 }
a = log ( μ a , 0 ) R , s = log ( μ s , 0 ) R , Δ y = ( η # ϕ # ) R m ,
e ̃ = e * , L ̃ T L ̃ = Γ e 1 ,
e ̃ = e * + ε * , L ̃ T L ̃ = ( Γ e + Γ ε ) 1 .
log ( μ a μ a , 0 ) , log ( μ s μ s , 0 )
ε * = 1 K l = 1 K ε ( l ) ,
Γ ε = 1 K 1 l = 1 K ( ε ( l ) ε * ) ( ε ( l ) ε * ) T .

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