Abstract

Diffuse optical tomography is highly sensitive to measurement and modeling errors. Errors in the source and detector coupling and positions can cause significant artifacts in the reconstructed images. Recently the approximation error theory has been proposed to handle modeling errors. In this article, we investigate the feasibility of the approximation error approach to compensate for modeling errors due to inaccurately known optode locations and coupling coefficients. The approach is evaluated with simulations. The results show that the approximation error method can be used to recover from artifacts in reconstructed images due to optode coupling and position errors.

© 2013 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  28. 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]
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    [CrossRef]

2012 (1)

J. Heiskala, V. Kolehmainen, T. Tarvainen, J. P. Kaipio, and S. R. Arridge, “Approximation error method can reduce artifacts due to scalp blood flow in optical brain activation imaging,” J. Biomed. Opt.17, 096012 (2012).
[CrossRef]

2011 (2)

R. Fukuzawa, S. Okawa, S. Matsuhashi, T. Kusaka, Y. Tanikawa, Y. Hoshi, F. Gao, and Y. Yamada, “Reduction of image artifacts induced by change in the optode coupling in time-resolved diffuse optical tomography.” J. Biomed. Opt.16, 116022 (2011).
[CrossRef] [PubMed]

V. Kolehmainen, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, “Marginalization of uninteresting distributed parameters in inverse problems - application to diffuse optical tomography,” Int. J. Uncertainty Quantification1, 1–17 (2011).
[CrossRef]

2010 (3)

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “An 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).
[CrossRef]

C. Lieberman, K. Willcox, and O. Ghattas, “Parameter and state model reduction for large-scale statistical inverse problems,” SIAM J. Sci. Comput.32, 2523–2542 (2010).
[CrossRef]

T. Tarvainen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Corrections to linear methods for diffuse optical tomography using approximation error modelling.” Biomed. Opt. Express1, 209–222 (2010).
[CrossRef]

2009 (2)

2008 (1)

D. Leff, O. Warren, L. Enfield, A. Gibson, T. Athanasiou, D. Patten, J. Hebden, G. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: A systematic review,” Breast Cancer Res. Tr.108, 9–22 (2008).
[CrossRef]

2007 (2)

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).
[CrossRef]

2005 (6)

A. Gibson, J. Hebden, and S. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005). Topical Review.
[CrossRef] [PubMed]

I. Nissila, T. Noponen, K. Kotilahti, T. Katila, L. Lipiainen, T. Tarvainen, M. Schweiger, and S. Arridge, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum.76, 044302 (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]

J. Heino, E. Somersalo, and J. Kaipio, “Compensation for geometric mismodelling by anisotropies in optical tomography,” Opt. Express13, 296–308 (2005).
[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, 1879–88 (2005).
[CrossRef] [PubMed]

H. Xu, B. W. Pogue, R. Springett, and H. Dehghani, “Spectral derivative based image reconstruction provides inherent insensitivity to coupling and geometric errors,” Opt. Lett.30, 2912–2914 (2005).
[CrossRef] [PubMed]

2003 (2)

2002 (1)

2001 (1)

2000 (2)

C. H. Schmitz, H. L. Graber, H. Luo, I. Arif, J. Hira, Y. Pei, A. Bluestone, S. Zhong, R. Andronica, I. Soller, N. Ramirez, S. L. Barbour, and R. L. Barbour, “Instrumentation and calibration protocol for imaging dynamic features in dense-scattering media by optical tomography.” Appl. Opt.39, 6466–86 (2000).
[CrossRef]

E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum.71, 3415 (2000).
[CrossRef]

1999 (2)

1993 (1)

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

Andronica, R.

Arif, I.

Arridge, S.

S. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inv. Probl.25, 123010 (2009). Topical Review.
[CrossRef]

A. Gibson, J. Hebden, and S. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005). Topical Review.
[CrossRef] [PubMed]

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

D. Boas, T. Gaudette, and S. Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography.” Opt. Express8, 263–70 (2001).
[CrossRef] [PubMed]

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

Arridge, S. R.

J. Heiskala, V. Kolehmainen, T. Tarvainen, J. P. Kaipio, and S. R. Arridge, “Approximation error method can reduce artifacts due to scalp blood flow in optical brain activation imaging,” J. Biomed. Opt.17, 096012 (2012).
[CrossRef]

V. Kolehmainen, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, “Marginalization of uninteresting distributed parameters in inverse problems - application to diffuse optical tomography,” Int. J. Uncertainty Quantification1, 1–17 (2011).
[CrossRef]

T. Tarvainen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Corrections to linear methods for diffuse optical tomography using approximation error modelling.” Biomed. Opt. Express1, 209–222 (2010).
[CrossRef]

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “An 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).
[CrossRef]

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. A26, 2257–2268 (2009).
[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]

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).
[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, 1879–88 (2005).
[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]

J. J. Stott, J. P. Culver, S. R. Arridge, and D. A. Boas, “Optode positional calibration in diffuse optical tomography.” Appl. Opt.42, 3154–62 (2003).
[CrossRef] [PubMed]

E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum.71, 3415 (2000).
[CrossRef]

Athanasiou, T.

D. Leff, O. Warren, L. Enfield, A. Gibson, T. Athanasiou, D. Patten, J. Hebden, G. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: A systematic review,” Breast Cancer Res. Tr.108, 9–22 (2008).
[CrossRef]

Barbour, R. L.

Barbour, S. L.

Bluestone, A.

Boas, D.

Boas, D. A.

Bouman, C. A.

Chance, B.

V. Ntziachristos, B. Chance, and A. G. Yodh, “Differential diffuse optical tomography,” Opt. Express5, 565–570 (1999).
[CrossRef]

Culver, J. P.

Darzi, A.

D. Leff, O. Warren, L. Enfield, A. Gibson, T. Athanasiou, D. Patten, J. Hebden, G. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: A systematic review,” Breast Cancer Res. Tr.108, 9–22 (2008).
[CrossRef]

Dehghani, H.

H. Xu, B. W. Pogue, R. Springett, and H. Dehghani, “Spectral derivative based image reconstruction provides inherent insensitivity to coupling and geometric errors,” Opt. Lett.30, 2912–2914 (2005).
[CrossRef] [PubMed]

E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum.71, 3415 (2000).
[CrossRef]

Delpy, D.

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

Delpy, D. T.

E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum.71, 3415 (2000).
[CrossRef]

Enfield, L.

D. Leff, O. Warren, L. Enfield, A. Gibson, T. Athanasiou, D. Patten, J. Hebden, G. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: A systematic review,” Breast Cancer Res. Tr.108, 9–22 (2008).
[CrossRef]

Fukuzawa, R.

R. Fukuzawa, S. Okawa, S. Matsuhashi, T. Kusaka, Y. Tanikawa, Y. Hoshi, F. Gao, and Y. Yamada, “Reduction of image artifacts induced by change in the optode coupling in time-resolved diffuse optical tomography.” J. Biomed. Opt.16, 116022 (2011).
[CrossRef] [PubMed]

Gao, F.

R. Fukuzawa, S. Okawa, S. Matsuhashi, T. Kusaka, Y. Tanikawa, Y. Hoshi, F. Gao, and Y. Yamada, “Reduction of image artifacts induced by change in the optode coupling in time-resolved diffuse optical tomography.” J. Biomed. Opt.16, 116022 (2011).
[CrossRef] [PubMed]

Gaudette, T.

Ghattas, O.

C. Lieberman, K. Willcox, and O. Ghattas, “Parameter and state model reduction for large-scale statistical inverse problems,” SIAM J. Sci. Comput.32, 2523–2542 (2010).
[CrossRef]

Gibson, A.

D. Leff, O. Warren, L. Enfield, A. Gibson, T. Athanasiou, D. Patten, J. Hebden, G. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: A systematic review,” Breast Cancer Res. Tr.108, 9–22 (2008).
[CrossRef]

A. Gibson, J. Hebden, and S. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005). Topical Review.
[CrossRef] [PubMed]

Gibson, A. P.

Graber, H. L.

Hebden, J.

D. Leff, O. Warren, L. Enfield, A. Gibson, T. Athanasiou, D. Patten, J. Hebden, G. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: A systematic review,” Breast Cancer Res. Tr.108, 9–22 (2008).
[CrossRef]

A. Gibson, J. Hebden, and S. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005). Topical Review.
[CrossRef] [PubMed]

Hebden, J. C.

E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum.71, 3415 (2000).
[CrossRef]

Heino, J.

Heiskala, J.

J. Heiskala, V. Kolehmainen, T. Tarvainen, J. P. Kaipio, and S. R. Arridge, “Approximation error method can reduce artifacts due to scalp blood flow in optical brain activation imaging,” J. Biomed. Opt.17, 096012 (2012).
[CrossRef]

Hillman, E. M. C.

E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum.71, 3415 (2000).
[CrossRef]

Hira, J.

Hiraoka, M.

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

Hoshi, Y.

R. Fukuzawa, S. Okawa, S. Matsuhashi, T. Kusaka, Y. Tanikawa, Y. Hoshi, F. Gao, and Y. Yamada, “Reduction of image artifacts induced by change in the optode coupling in time-resolved diffuse optical tomography.” J. Biomed. Opt.16, 116022 (2011).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media(Academic, New York, 1978).

Kaipio,

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

Kaipio, J.

Kaipio, J. P.

J. Heiskala, V. Kolehmainen, T. Tarvainen, J. P. Kaipio, and S. R. Arridge, “Approximation error method can reduce artifacts due to scalp blood flow in optical brain activation imaging,” J. Biomed. Opt.17, 096012 (2012).
[CrossRef]

V. Kolehmainen, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, “Marginalization of uninteresting distributed parameters in inverse problems - application to diffuse optical tomography,” Int. J. Uncertainty Quantification1, 1–17 (2011).
[CrossRef]

T. Tarvainen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Corrections to linear methods for diffuse optical tomography using approximation error modelling.” Biomed. Opt. Express1, 209–222 (2010).
[CrossRef]

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “An 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).
[CrossRef]

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. A26, 2257–2268 (2009).
[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,” Inv. 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, 1879–88 (2005).
[CrossRef] [PubMed]

Katila, T.

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

Kolehmainen, V.

J. Heiskala, V. Kolehmainen, T. Tarvainen, J. P. Kaipio, and S. R. Arridge, “Approximation error method can reduce artifacts due to scalp blood flow in optical brain activation imaging,” J. Biomed. Opt.17, 096012 (2012).
[CrossRef]

V. Kolehmainen, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, “Marginalization of uninteresting distributed parameters in inverse problems - application to diffuse optical tomography,” Int. J. Uncertainty Quantification1, 1–17 (2011).
[CrossRef]

T. Tarvainen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Corrections to linear methods for diffuse optical tomography using approximation error modelling.” Biomed. Opt. Express1, 209–222 (2010).
[CrossRef]

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “An 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).
[CrossRef]

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. A26, 2257–2268 (2009).
[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,” Inv. 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, 1879–88 (2005).
[CrossRef] [PubMed]

Kotilahti, K.

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

Kusaka, T.

R. Fukuzawa, S. Okawa, S. Matsuhashi, T. Kusaka, Y. Tanikawa, Y. Hoshi, F. Gao, and Y. Yamada, “Reduction of image artifacts induced by change in the optode coupling in time-resolved diffuse optical tomography.” J. Biomed. Opt.16, 116022 (2011).
[CrossRef] [PubMed]

Leff, D.

D. Leff, O. Warren, L. Enfield, A. Gibson, T. Athanasiou, D. Patten, J. Hebden, G. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: A systematic review,” Breast Cancer Res. Tr.108, 9–22 (2008).
[CrossRef]

Lieberman, C.

C. Lieberman, K. Willcox, and O. Ghattas, “Parameter and state model reduction for large-scale statistical inverse problems,” SIAM J. Sci. Comput.32, 2523–2542 (2010).
[CrossRef]

Lipiainen, L.

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

Luo, H.

Matsuhashi, S.

R. Fukuzawa, S. Okawa, S. Matsuhashi, T. Kusaka, Y. Tanikawa, Y. Hoshi, F. Gao, and Y. Yamada, “Reduction of image artifacts induced by change in the optode coupling in time-resolved diffuse optical tomography.” J. Biomed. Opt.16, 116022 (2011).
[CrossRef] [PubMed]

McBride, T. O.

Millane, R. P.

Milstein, A. B.

Nissila, I.

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

Nissilä, I.

Noponen, T.

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

Ntziachristos, V.

V. Ntziachristos, B. Chance, and A. G. Yodh, “Differential diffuse optical tomography,” Opt. Express5, 565–570 (1999).
[CrossRef]

Oh, S.

Okawa, S.

R. Fukuzawa, S. Okawa, S. Matsuhashi, T. Kusaka, Y. Tanikawa, Y. Hoshi, F. Gao, and Y. Yamada, “Reduction of image artifacts induced by change in the optode coupling in time-resolved diffuse optical tomography.” J. Biomed. Opt.16, 116022 (2011).
[CrossRef] [PubMed]

Osterberg, U. L.

Patten, D.

D. Leff, O. Warren, L. Enfield, A. Gibson, T. Athanasiou, D. Patten, J. Hebden, G. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: A systematic review,” Breast Cancer Res. Tr.108, 9–22 (2008).
[CrossRef]

Paulsen, K. D.

Pei, Y.

Pogue, B. W.

Prewitt, J.

Pulkkinen, A.

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “An 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).
[CrossRef]

Ramirez, N.

Schmidt, F. E. W.

E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum.71, 3415 (2000).
[CrossRef]

Schmitz, C. H.

Schotland, J.

S. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inv. Probl.25, 123010 (2009). Topical Review.
[CrossRef]

Schweiger, M.

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “An 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).
[CrossRef]

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. A26, 2257–2268 (2009).
[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]

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).
[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, 1879–88 (2005).
[CrossRef] [PubMed]

I. Nissila, T. Noponen, K. Kotilahti, T. Katila, L. Lipiainen, T. Tarvainen, M. Schweiger, and S. Arridge, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum.76, 044302 (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]

E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum.71, 3415 (2000).
[CrossRef]

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

Siegel, A. M.

Soller, I.

Somersalo, E.

Kaipio and E. Somersalo, “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,” Inv. Probl.22, 175–195 (2006).
[CrossRef]

J. Heino, E. Somersalo, and J. Kaipio, “Compensation for geometric mismodelling by anisotropies in optical tomography,” Opt. Express13, 296–308 (2005).
[CrossRef] [PubMed]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer, New York, 2005).

Springett, R.

Stott, J. J.

Tanikawa, Y.

R. Fukuzawa, S. Okawa, S. Matsuhashi, T. Kusaka, Y. Tanikawa, Y. Hoshi, F. Gao, and Y. Yamada, “Reduction of image artifacts induced by change in the optode coupling in time-resolved diffuse optical tomography.” J. Biomed. Opt.16, 116022 (2011).
[CrossRef] [PubMed]

Tarvainen, T.

J. Heiskala, V. Kolehmainen, T. Tarvainen, J. P. Kaipio, and S. R. Arridge, “Approximation error method can reduce artifacts due to scalp blood flow in optical brain activation imaging,” J. Biomed. Opt.17, 096012 (2012).
[CrossRef]

V. Kolehmainen, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, “Marginalization of uninteresting distributed parameters in inverse problems - application to diffuse optical tomography,” Int. J. Uncertainty Quantification1, 1–17 (2011).
[CrossRef]

T. Tarvainen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Corrections to linear methods for diffuse optical tomography using approximation error modelling.” Biomed. Opt. Express1, 209–222 (2010).
[CrossRef]

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “An 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).
[CrossRef]

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. A26, 2257–2268 (2009).
[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,” Inv. 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, 1879–88 (2005).
[CrossRef] [PubMed]

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

Vanne, A.

Vauhkonen, M.

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “An 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).
[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,” Inv. 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, 1879–88 (2005).
[CrossRef] [PubMed]

Warren, O.

D. Leff, O. Warren, L. Enfield, A. Gibson, T. Athanasiou, D. Patten, J. Hebden, G. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: A systematic review,” Breast Cancer Res. Tr.108, 9–22 (2008).
[CrossRef]

Webb, K. J.

Willcox, K.

C. Lieberman, K. Willcox, and O. Ghattas, “Parameter and state model reduction for large-scale statistical inverse problems,” SIAM J. Sci. Comput.32, 2523–2542 (2010).
[CrossRef]

Xu, H.

Yamada, Y.

R. Fukuzawa, S. Okawa, S. Matsuhashi, T. Kusaka, Y. Tanikawa, Y. Hoshi, F. Gao, and Y. Yamada, “Reduction of image artifacts induced by change in the optode coupling in time-resolved diffuse optical tomography.” J. Biomed. Opt.16, 116022 (2011).
[CrossRef] [PubMed]

Yang, G.

D. Leff, O. Warren, L. Enfield, A. Gibson, T. Athanasiou, D. Patten, J. Hebden, G. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: A systematic review,” Breast Cancer Res. Tr.108, 9–22 (2008).
[CrossRef]

Yodh, A. G.

V. Ntziachristos, B. Chance, and A. G. Yodh, “Differential diffuse optical tomography,” Opt. Express5, 565–570 (1999).
[CrossRef]

Zhong, S.

Appl. Opt. (5)

Biomed. Opt. Express (1)

Breast Cancer Res. Tr. (1)

D. Leff, O. Warren, L. Enfield, A. Gibson, T. Athanasiou, D. Patten, J. Hebden, G. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: A systematic review,” Breast Cancer Res. Tr.108, 9–22 (2008).
[CrossRef]

Int. J. Uncertainty Quantification (1)

V. Kolehmainen, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, “Marginalization of uninteresting distributed parameters in inverse problems - application to diffuse optical tomography,” Int. J. Uncertainty Quantification1, 1–17 (2011).
[CrossRef]

Inv. Probl. (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).
[CrossRef]

T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “An 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).
[CrossRef]

S. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inv. Probl.25, 123010 (2009). Topical Review.
[CrossRef]

J. Biomed. Opt. (2)

R. Fukuzawa, S. Okawa, S. Matsuhashi, T. Kusaka, Y. Tanikawa, Y. Hoshi, F. Gao, and Y. Yamada, “Reduction of image artifacts induced by change in the optode coupling in time-resolved diffuse optical tomography.” J. Biomed. Opt.16, 116022 (2011).
[CrossRef] [PubMed]

J. Heiskala, V. Kolehmainen, T. Tarvainen, J. P. Kaipio, and S. R. Arridge, “Approximation error method can reduce artifacts due to scalp blood flow in optical brain activation imaging,” J. Biomed. Opt.17, 096012 (2012).
[CrossRef]

J. Comput. Appl. Math. (1)

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

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

Med. Phys. (1)

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

Opt. Express (3)

Opt. Lett. (2)

Phys. Med. Biol. (2)

A. Gibson, J. Hebden, and S. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005). Topical Review.
[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]

Rev. Sci. Instrum. (2)

E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum.71, 3415 (2000).
[CrossRef]

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

SIAM J. Sci. Comput. (1)

C. Lieberman, K. Willcox, and O. Ghattas, “Parameter and state model reduction for large-scale statistical inverse problems,” SIAM J. Sci. Comput.32, 2523–2542 (2010).
[CrossRef]

Other (2)

A. Ishimaru, Wave Propagation and Scattering in Random Media(Academic, New York, 1978).

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer, New York, 2005).

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Figures (5)

Fig. 1
Fig. 1

Covariance matrices for optode coupling approximation error ε1 (top row) and optode position approximation error ε2 (bottom row). (a) Covariance matrix of coupling coefficient errors, Γε1. (b) and (c) are Log Amplitude and Phase blocks of Γε1. (d) Covariance matrix of optode position errors, Γε2. (e) and (f) are Log Amplitude and Phase blocks of Γε2. The colors in a) and d) were scaled to highlight the different correlation structures of the approximation errors ε1 and ε2.

Fig. 2
Fig. 2

From left: (a) First column: Target optical properties (top: scattering, bottom: absorption coefficients). (b) Second column: Reconstructions using CEM with no modeling errors (tCPU = 431s). (c) Third column: Reconstructions using CEM with incorrect optode coupling coefficients (rows 1 and 2, tCPU = 514s), incorrect optode locations (rows 3 and 4, tCPU = 677s) and a combination of these both (rows 5 and 6, tCPU = 2321s). (d) Fourth column: Reconstructions using AEM with incorrect optode coupling coefficient (tCPU = 490s), optode locations (tCPU = 519s) and a combination of these both (tCPU = 500s).

Fig. 3
Fig. 3

Reconstructions using approximation error model when no modelling errors are present. Top: absorption, Bottom: scattering. First column: reconstructions using pure optode coupling approximation error model (y = A(x, ξ) + ε1 + e). Second column: reconstructions using optode location approximation error model (y = A(x, ξ0) + ε2(ξ) + e). Third column: reconstructions using combined optode coupling and location approximation error model (y = A(x, ξ0) + ε1 + ε2 + e).

Fig. 4
Fig. 4

AEM reconstructions using different optode coupling coefficient errors. In each of the image pairs, μa is on the left and μs on the right. The data at each of the four rows was generated using the different prior distributions given in Table 1. The AE statistics at each of the four columns was trained using the prior distributions in Table 1. The arrows denote pairs (μa, μs) where the approximation error statistics was trained using the same prior that was used in simulation of the data with optode coupling errors.

Fig. 5
Fig. 5

AEM reconstructions using different optode position errors. In each of the image pairs, μa is on the left and μs on the right. The data at each of the four rows was generated using the prior distributions given in Table 1. The AE statistics in each of the four columns was trained using the prior distributions given in Table 1. The arrows denote pairs (μa, μs) where the approximation error statistics was trained using the same prior distibution that was used in simulation of the data with misplaced optodes.

Tables (1)

Tables Icon

Table 1 Prior models for coupling coefficients and optode positions. U(a, b) denotes the uniform density between [a, b]. The perturbation δθ in the location is given in degrees. The width of one source or detector fiber corresponds to 2.3° and the separation of adjacent optodes in the equiangular case ζ0 is 11.25°. (1° is equivalent to 0.44mm along the domain boundary Ω)

Equations (51)

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( κ ( r ) + μ a ( r ) + i ω c ) Φ i ( r ) = q 0 , i ( r ) r Ω ,
Φ i ( r ) + 1 2 γ κ α Φ i ( r ) n ^ = { q i γ r m i 0 r Ω \ m i ,
Γ i , j = n j κ Φ i ( r ) n ^ d S = n j 2 γ α Φ i ( r ) d S
Γ k = Γ ( i 1 ) N d + j : = Γ i , j .
y = ( Re log ( Γ ) Im log ( Γ ) )
ξ = ( m , n ) T N s + N d , m = ( m 1 , , m N s ) , n = ( n 1 , , n N d )
y = A ( x , ξ ) + e
Φ ˜ i ( r ) = s ^ i Φ i ( r ) ,
s ^ i = s i exp ( i δ i )
d ^ j = d j exp ( i η j ) ,
Γ ˜ i , j = s ^ i d ^ j n j 2 γ α Φ i ( r ) d S = s ^ i d ^ j Γ i , j
d = ( d 1 , , d N d ) , η = ( η 1 , , η N d )
s = ( s 1 , , s N s ) , δ = ( δ 1 , , δ N s )
ζ = ( s , δ , d , η ) T 2 ( N d + N s )
g k ( ζ ) : = s ^ i d ^ j = d i s j exp ( i ( η i + δ j ) ) , k = ( i 1 ) N d + j
y = ( Re log ( Γ ( x , ξ ) ) Im log ( Γ ( x , ξ ) ) ) A ( x , ξ ) + ( Re log ( g ( ζ ) ) Im log ( g ( ζ ) ) ) ε 1 ( ζ ) + e ,
y = A ( x , ξ ) + ε 1 ( ζ ) + e
π ( x , ζ , ξ , e | y ) = π ( y | x , e , ζ , ξ ) π ( x , ζ , ξ , e ) π ( y ) .
π ( x , ζ , ξ | y ) = π ( x , ζ , ξ , e | y ) d e .
( x , ξ , ζ ) MAP = arg max x , ξ , ζ π ( x , ξ , ζ | y ) .
π ( x | y ) = π ( x , ξ , ζ | y ) d ζ d ξ
π ( x , e | y , ζ = ζ 0 , ξ = ξ 0 ) = π ( y | x , e , ζ = ζ 0 , ξ = ξ 0 ) π ( x ) π ( e ) π ( y )
x ~ N ( x ¯ , Γ x ) e ~ N ( e ¯ , Γ e )
π ( x | y , ζ = ζ 0 , ξ = ξ 0 ) = π ( x , e | y , ζ = ζ 0 , ξ = ξ 0 ) d e
π ( x | y , ξ = ξ 0 , ζ = ζ 0 ) exp { 1 2 ( y A ( x , ξ 0 ) ε 1 ( ζ 0 ) e ¯ ) T Γ e 1 ( y A ( x , ξ 0 ) ε 1 ( ζ 0 ) e ¯ ) 1 2 ( x x ¯ ) T Γ x 1 ( x x ¯ ) }
x MAP = arg max x π ( x | y , ζ = ζ 0 , ξ = ξ 0 ) = arg min x { L e ( y A ( x , ξ 0 ) ε 1 ( ζ 0 ) e ¯ ) 2 + L x ( x x ¯ ) 2 } ,
x MAP = arg min x { L e ( y A ( x , ξ 0 ) ) 2 + L x ( x x ¯ ) 2 } ,
y = A ( x , ξ ) + ε 1 ( ζ ) + e = A ( x , ξ 0 ) + ε 1 ( ζ 0 ) + [ A ( x , ξ ) + ε 1 ( ζ ) ( A ( x , ξ 0 ) + ε 1 ( ζ 0 ) ) ] + e = A ( x , ξ 0 ) + { A ( x , ξ ) A ( x , ξ 0 ) } ε 2 ( x , ξ ) + ε 1 ( ζ ) + e = A ( x , ξ 0 ) + ε 1 ( ζ ) + ε 2 ( x , ξ ) + e
n = ε 1 ( ζ ) + ε 2 ( x , ξ ) + e .
π ˜ ( x | y ) exp { 1 2 ( y A ( x , ξ 0 ) n ¯ ) T Γ n 1 ( y A ( x , ξ 0 ) n ¯ ) 1 2 ( x x ¯ ) T Γ x 1 ( x x ¯ ) }
n ¯ = ε ¯ 1 + ε ¯ 2 + e ¯ , Γ n = Γ ε 1 + Γ ε 2 + Γ e
ε 1 ~ N ( ε ¯ 1 , Γ ε 1 ) , ε 2 ~ N ( ε ¯ 2 , Γ ε 2 ) .
x MAP = arg min x { L n ( y A ( x , ξ 0 ) n ¯ ) 2 + L x ( x x ¯ ) 2 }
ζ ( ) = ( s ( ) , δ ( ) , d ( ) , η ( ) ) T
ε ¯ 1 = 1 N = 1 N ε 1 ( )
Γ ε 1 = 1 N 1 = 1 N ε 1 ( ) ε 1 ( ) T ε ¯ 1 ε ¯ 1 T
{ x ( ) , = 1 , 2 , , M } , { ξ ( ) , = 1 , 2 , , M }
ε 2 ( ) = A ( x ( ) , ξ ( ) ) A ( x ( ) , ξ 0 )
ε ¯ 2 = 1 M = 1 M ε 2 ( )
Γ ε 2 = 1 M 1 = 1 M ε 2 ( ) ε 2 ( ) T ε ¯ 2 ε ¯ 2 T
π ( e ) = N ( 0 , Γ e ) , Γ e = diag ( σ e , 1 2 , , σ e , 2 N s N d 2 )
π ( x ) exp { 1 2 L x ( x x ¯ ) 2 } , L x T L x = Γ x 1
x ¯ = ( μ ¯ a μ ¯ s ) , Γ x = ( Γ μ a 0 0 Γ μ s ) .
f = f in + f bg
f in ~ N ( 0 , Γ in , f )
f ¯ = f * I , Γ f = Γ in , f + σ bg , f 2 I I T
π ( ζ ) = π ( s ) π ( δ ) π ( d ) π ( η ) ,
π ( s ) = i N s π ( s i ) , π ( δ ) = i N s π ( δ i ) , π ( d ) = j N d π ( d j ) , π ( η ) = i N d π ( η j ) .
π ( s i ) = U ( s min , 1 ) , π ( d j ) = U ( d min , 1 )
π ( δ i ) = U ( 0 , δ max ) , π ( η j ) = U ( 0 , η max ) .
ξ k = ξ 0 , k + δ θ k

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