Abstract

Diffuse optical tomography is a highly unstable problem with respect to modeling and measurement errors. During clinical measurements, the body shape is not always known, and an approximate model domain has to be employed. The use of an incorrect model domain can, however, lead to significant artifacts in the reconstructed images. Recently, the Bayesian approximation error theory has been proposed to handle model-based errors. In this work, the feasibility of the Bayesian approximation error approach to compensate for modeling errors due to unknown body shape is investigated. The approach is tested with simulations. The results show that the Bayesian approximation error method can be used to reduce artifacts in reconstructed images due to unknown domain shape.

© 2014 Optical Society of America

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2013 (2)

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, 0960121 (2012).
[CrossRef]

2011 (2)

A. Nissinen, V. P. Kolehmainen, and J. P. Kaipio, “Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography,” IEEE Trans. Med. Imaging 30, 231–242 (2011).
[CrossRef]

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

2010 (3)

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, 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,” Inverse Probl. 26, 015005 (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. Express 1, 209–222 (2010).
[CrossRef]

2009 (2)

2008 (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 optical imaging technologies for breast cancer management,” Med. Phys. 35, 2443–2451 (2008).
[CrossRef]

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).

2007 (1)

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

2006 (2)

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]

G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Technol. Cancer Res. T. 5, 351–363 (2006).

2005 (4)

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

J. Heino, E. Somersalo, and J. Kaipio, “Compensation for geometric mismodelling by anisotropies in optical tomography,” Opt. Express 13, 296–308 (2005).
[CrossRef]

V. Kolehmainen, M. Lassas, and P. Ola, “The inverse conductivity problem with an imperfectly known boundary,” SIAM J. Appl. Math. 66, 365–383 (2005).
[CrossRef]

2003 (5)

A. P. Gibson, J. Riley, M. Schweiger, J. C. Hebden, S. R. Arridge, and D. T. Delpy, “A method for generating patient-specific finite element meshes for head modelling,” Phys. Med. Biol. 48, 481–495 (2003).
[CrossRef]

J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, and A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation and metabolism in rat during focal ischemia,” J. Cereb. Blood Flow Metab. 23, 911–924 (2003).
[CrossRef]

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

A. H. Barnett, J. P. Culver, A. G. Sorensen, A. Dale, and D. A. Boas, “Robust inference of baseline optical properties of the human head with three-dimensional segmentation from magnetic resonance imaging,” Appl. Opt. 42, 3095–3108 (2003).
[CrossRef]

J. P. Culver, A. M. Siegel, J. J. Stott, and D. A. Boas, “Volumetric diffuse optical tomography of brain activity,” Opt. Lett. 28, 2061–2063 (2003).
[CrossRef]

2002 (1)

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef]

2001 (1)

2000 (1)

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, 3415–3427 (2000).
[CrossRef]

1998 (1)

1990 (1)

J. Sylvester, “An anisotropic inverse boundary value problem,” Comm. Pure Appl. Math. 43, 201–232 (1990).
[CrossRef]

Abdoulaev, G.

Arridge, S.

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

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

Arridge, S. R.

M. Mozumder, T. Tarvainen, S. R. Arridge, J. Kaipio, and V. Kolehmainen, “Compensation of optode sensitivity and position errors in diffuse optical tomography using the approximation error approach,” Biomed. Opt. Express 4, 2015–2031 (2013).
[CrossRef]

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, 0960121 (2012).
[CrossRef]

V. Kolehmainen, T. Tarvainen, and S. R. Arridge, “Marginalization of uninteresting distributed parameters in inverse problems–application to diffuse optical tomography,” Int. J. Uncertainty Quantif. 1, 1–17 (2011).
[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,” Inverse Probl. 26, 015005 (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. Express 1, 209–222 (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. A 26, 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,” 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]

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

A. P. Gibson, J. Riley, M. Schweiger, J. C. Hebden, S. R. Arridge, and D. T. Delpy, “A method for generating patient-specific finite element meshes for head modelling,” Phys. Med. Biol. 48, 481–495 (2003).
[CrossRef]

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[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, 3415–3427 (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).

Austin, T.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef]

Barbour, R.

Barnett, A. H.

Birgul, O.

G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Technol. Cancer Res. T. 5, 351–363 (2006).

Bluestone, A.

Boas, D. A.

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 optical imaging technologies for breast cancer management,” Med. Phys. 35, 2443–2451 (2008).
[CrossRef]

Cheung, C.

J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, and A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation and metabolism in rat during focal ischemia,” J. Cereb. Blood Flow Metab. 23, 911–924 (2003).
[CrossRef]

Culver, J. P.

Dale, A.

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).

Dehghani, H.

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[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, 3415–3427 (2000).
[CrossRef]

Delpy, D. T.

A. P. Gibson, J. Riley, M. Schweiger, J. C. Hebden, S. R. Arridge, and D. T. Delpy, “A method for generating patient-specific finite element meshes for head modelling,” Phys. Med. Biol. 48, 481–495 (2003).
[CrossRef]

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[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, 3415–3427 (2000).
[CrossRef]

Durduran, T.

J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, and A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation and metabolism in rat during focal ischemia,” J. Cereb. Blood Flow Metab. 23, 911–924 (2003).
[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).

Everdell, N.

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef]

Furuya, D.

J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, and A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation and metabolism in rat during focal ischemia,” J. Cereb. Blood Flow Metab. 23, 911–924 (2003).
[CrossRef]

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).

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

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef]

Gibson, A. P.

A. P. Gibson, J. Riley, M. Schweiger, J. C. Hebden, S. R. Arridge, and D. T. Delpy, “A method for generating patient-specific finite element meshes for head modelling,” Phys. Med. Biol. 48, 481–495 (2003).
[CrossRef]

Greenberg, J. H.

J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, and A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation and metabolism in rat during focal ischemia,” J. Cereb. Blood Flow Metab. 23, 911–924 (2003).
[CrossRef]

Gulsen, G.

G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Technol. Cancer Res. T. 5, 351–363 (2006).

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).

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

Hebden, J. C.

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

A. P. Gibson, J. Riley, M. Schweiger, J. C. Hebden, S. R. Arridge, and D. T. Delpy, “A method for generating patient-specific finite element meshes for head modelling,” Phys. Med. Biol. 48, 481–495 (2003).
[CrossRef]

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[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, 3415–3427 (2000).
[CrossRef]

Heino, J.

Heiskala, J.

T. Näsi, H. Mäki, P. Hiltunen, J. Heiskala, I. Nissilä, K. Kotilahti, and R. J. Ilmoniemi, “Effect of task-related extracerebral circulation on diffuse optical tomography: experimental data and simulations on the forehead,” Biomed. Opt. Express 4, 412–426 (2013).
[CrossRef]

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, 0960121 (2012).
[CrossRef]

Held, L.

H. Rue and L. Held, Gaussian Markov Random Fields: Theory and Applications, Vol. 104 of Monographs on Statistics and Applied Probability (Chapman & Hall, 2005).

Hielscher, A.

Hillman, E. M. C.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[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, 3415–3427 (2000).
[CrossRef]

Hiltunen, P.

Ilmoniemi, R. J.

Ishimaru, A.

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

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, 0960121 (2012).
[CrossRef]

A. Nissinen, V. P. Kolehmainen, and J. P. Kaipio, “Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography,” IEEE Trans. Med. Imaging 30, 231–242 (2011).
[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,” Inverse Probl. 26, 015005 (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. Express 1, 209–222 (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. A 26, 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,” Inverse Probl. 22, 175–195 (2006).
[CrossRef]

Kolehmainen, V.

M. Mozumder, T. Tarvainen, S. R. Arridge, J. Kaipio, and V. Kolehmainen, “Compensation of optode sensitivity and position errors in diffuse optical tomography using the approximation error approach,” Biomed. Opt. Express 4, 2015–2031 (2013).
[CrossRef]

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, 0960121 (2012).
[CrossRef]

V. Kolehmainen, T. Tarvainen, and S. R. Arridge, “Marginalization of uninteresting distributed parameters in inverse problems–application to diffuse optical tomography,” Int. J. Uncertainty Quantif. 1, 1–17 (2011).
[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,” Inverse Probl. 26, 015005 (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. Express 1, 209–222 (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. A 26, 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,” Inverse Probl. 22, 175–195 (2006).
[CrossRef]

V. Kolehmainen, M. Lassas, and P. Ola, “The inverse conductivity problem with an imperfectly known boundary,” SIAM J. Appl. Math. 66, 365–383 (2005).
[CrossRef]

Kolehmainen, V. P.

A. Nissinen, V. P. Kolehmainen, and J. P. Kaipio, “Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography,” IEEE Trans. Med. Imaging 30, 231–242 (2011).
[CrossRef]

Kotilahti, K.

Lassas, M.

V. Kolehmainen, M. Lassas, and P. Ola, “The inverse conductivity problem with an imperfectly known boundary,” SIAM J. Appl. Math. 66, 365–383 (2005).
[CrossRef]

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).

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]

Mäki, H.

Meek, J. H.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef]

Mozumder, M.

Nalcioglu, O.

G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Technol. Cancer Res. T. 5, 351–363 (2006).

Näsi, T.

Nissilä, I.

Nissinen, A.

A. Nissinen, V. P. Kolehmainen, and J. P. Kaipio, “Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography,” IEEE Trans. Med. Imaging 30, 231–242 (2011).
[CrossRef]

Ola, P.

V. Kolehmainen, M. Lassas, and P. Ola, “The inverse conductivity problem with an imperfectly known boundary,” SIAM J. Appl. Math. 66, 365–383 (2005).
[CrossRef]

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).

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 optical imaging technologies for breast cancer management,” Med. Phys. 35, 2443–2451 (2008).
[CrossRef]

B. W. Pogue and K. D. Paulsen, “High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of a priori magnetic resonance imaging structural information,” Opt. Lett. 23, 1716–1718 (1998).
[CrossRef]

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 optical imaging technologies for breast cancer management,” Med. Phys. 35, 2443–2451 (2008).
[CrossRef]

B. W. Pogue and K. D. Paulsen, “High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of a priori magnetic resonance imaging structural information,” Opt. Lett. 23, 1716–1718 (1998).
[CrossRef]

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,” Inverse Probl. 26, 015005 (2010).
[CrossRef]

Richards, R.

Riley, J.

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

A. P. Gibson, J. Riley, M. Schweiger, J. C. Hebden, S. R. Arridge, and D. T. Delpy, “A method for generating patient-specific finite element meshes for head modelling,” Phys. Med. Biol. 48, 481–495 (2003).
[CrossRef]

Rue, H.

H. Rue and L. Held, Gaussian Markov Random Fields: Theory and Applications, Vol. 104 of Monographs on Statistics and Applied Probability (Chapman & Hall, 2005).

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, 3415–3427 (2000).
[CrossRef]

Schmitz, C.

Schotland, J.

S. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[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,” Inverse 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. A 26, 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,” 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]

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

A. P. Gibson, J. Riley, M. Schweiger, J. C. Hebden, S. R. Arridge, and D. T. Delpy, “A method for generating patient-specific finite element meshes for head modelling,” Phys. Med. Biol. 48, 481–495 (2003).
[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, 3415–3427 (2000).
[CrossRef]

Shafiiha, R.

G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Technol. Cancer Res. T. 5, 351–363 (2006).

Siegel, A. M.

Somersalo, E.

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

J. Heino, E. Somersalo, and J. Kaipio, “Compensation for geometric mismodelling by anisotropies in optical tomography,” Opt. Express 13, 296–308 (2005).
[CrossRef]

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

Sorensen, A. G.

Stott, J. J.

Sylvester, J.

J. Sylvester, “An anisotropic inverse boundary value problem,” Comm. Pure Appl. Math. 43, 201–232 (1990).
[CrossRef]

Tarvainen, T.

M. Mozumder, T. Tarvainen, S. R. Arridge, J. Kaipio, and V. Kolehmainen, “Compensation of optode sensitivity and position errors in diffuse optical tomography using the approximation error approach,” Biomed. Opt. Express 4, 2015–2031 (2013).
[CrossRef]

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, 0960121 (2012).
[CrossRef]

V. Kolehmainen, T. Tarvainen, and S. R. Arridge, “Marginalization of uninteresting distributed parameters in inverse problems–application to diffuse optical tomography,” Int. J. Uncertainty Quantif. 1, 1–17 (2011).
[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,” Inverse Probl. 26, 015005 (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. Express 1, 209–222 (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. A 26, 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,” Inverse Probl. 22, 175–195 (2006).
[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 optical imaging technologies for breast cancer management,” Med. Phys. 35, 2443–2451 (2008).
[CrossRef]

Unlu, M. B.

G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Technol. Cancer Res. T. 5, 351–363 (2006).

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

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).

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]

Wyatt, J. S.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef]

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).

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 optical imaging technologies for breast cancer management,” Med. Phys. 35, 2443–2451 (2008).
[CrossRef]

J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, and A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation and metabolism in rat during focal ischemia,” J. Cereb. Blood Flow Metab. 23, 911–924 (2003).
[CrossRef]

Yusof, R. M.

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef]

Appl. Opt. (2)

Biomed. Opt. Express (3)

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).

Comm. Pure Appl. Math. (1)

J. Sylvester, “An anisotropic inverse boundary value problem,” Comm. Pure Appl. Math. 43, 201–232 (1990).
[CrossRef]

IEEE Trans. Med. Imaging (1)

A. Nissinen, V. P. Kolehmainen, and J. P. Kaipio, “Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography,” IEEE Trans. Med. Imaging 30, 231–242 (2011).
[CrossRef]

Int. J. Uncertainty Quantif. (1)

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

Inverse 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,” Inverse 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,” Inverse Probl. 26, 015005 (2010).
[CrossRef]

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

J. Biomed. Opt. (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, 0960121 (2012).
[CrossRef]

J. Cereb. Blood Flow Metab. (1)

J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, and A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation and metabolism in rat during focal ischemia,” J. Cereb. Blood Flow Metab. 23, 911–924 (2003).
[CrossRef]

J. Comput. Appl. Math. (1)

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

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

Med. Phys. (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 optical imaging technologies for breast cancer management,” Med. Phys. 35, 2443–2451 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Med. Biol. (4)

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef]

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

A. P. Gibson, J. Riley, M. Schweiger, J. C. Hebden, S. R. Arridge, and D. T. Delpy, “A method for generating patient-specific finite element meshes for head modelling,” Phys. Med. Biol. 48, 481–495 (2003).
[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]

Rev. Sci. Instrum. (1)

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, 3415–3427 (2000).
[CrossRef]

SIAM J. Appl. Math. (1)

V. Kolehmainen, M. Lassas, and P. Ola, “The inverse conductivity problem with an imperfectly known boundary,” SIAM J. Appl. Math. 66, 365–383 (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]

Technol. Cancer Res. T. (1)

G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Technol. Cancer Res. T. 5, 351–363 (2006).

Other (3)

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

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

H. Rue and L. Held, Gaussian Markov Random Fields: Theory and Applications, Vol. 104 of Monographs on Statistics and Applied Probability (Chapman & Hall, 2005).

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Figures (3)

Fig. 1.
Fig. 1.

Mapping of optical properties according to the deformation T [see Eqs. (19), (20)]. Left, optical properties in a sample domain Ω(l). Right, mapped optical properties in domain Ω˜. Top, deformation of coordinate axes from domain Ω(l) to domain Ω˜. Bottom, a draw of μa from π(x) in domain Ω(l) mapped to domain Ω˜.

Fig. 2.
Fig. 2.

Pure domain modeling error, from left to right. First column, actual target domain is shown with black solid line. Grey patch is the mean ±s.t.d of head shapes for prior π(γ). Black dashed line is the elliptical model domain. Second column, target optical property μa (top) and μs (bottom) pairs for each head shape. Third column, reconstructions using CEM with no modeling errors [i.e., forward model used is Aδ(x¯,γ)]. Fourth column, reconstructions using CEM in the model domain [i.e., forward model used is Aδ(x,γ˜)]. Fifth column, reconstructions using AEM in the model domain [i.e., forward model used is Aδ(x,γ˜)].

Fig. 3.
Fig. 3.

Domain modeling and discretization errors, from left. First column, the actual measurement domain is shown with black solid line. Gray patch is the mean ±s.t.d of head shapes for prior π(γ). Black dashed line is the elliptical model domain. Second column, target optical property μa (top) and μs (bottom) pairs for each head shape. Third column, reconstructions using CEM using coarse discretization [i.e., forward model used is Ah(x¯,γ)]. Fourth column, reconstructions using CEM with model domain and coarse discretization [i.e., forward model used is Ah(x,γ˜)]. Fifth column, reconstructions using AEM with model domain and coarse discretization [i.e., forward model used is Ah(x,γ˜)].

Tables (4)

Tables Icon

Table 1. 2D Finite Element Meshes for the Simulation Cases

Tables Icon

Table 2. Parameter Values Used in This Study

Tables Icon

Table 3. Offline Computation Times (tCPU) for the Estimation of the AE Statistics Using Ns=128 Samples

Tables Icon

Table 4. TCPU(s) for the MAP Estimations in Test Cases 1 and 2

Equations (42)

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

(·κ(r)+μa(r)+jωc)Φ(r)=0,rΩ,
Φ(r)+12ζκαΦ(r)n^={qiζrsi0rΩsi,
Γij(r)=κΦij(r)n^=2ζαΦij(r)rdj,
Φh=k=1Nnϕkψk(r),
μa(r)=l=1Npμa,lχl(r),
μs(r)=l=1Npμs,lχl(r),
y=(Relog(Γ)Imlog(Γ)),
y=Ah(x,γ)+e,
π(x,γ,e|y)=π(y|x,e,γ)π(x,γ,e)π(y),
π(x,γ|y)=π(x,γ,e|y)de.
(x,γ)MAP=argmaxx,γπ(x,γ|y).
π(x|y)=π(x,γ|y)dγ,
π(x,e|y,γ=γ˜)=π(y|x,e,γ=γ˜)π(x)π(e)π(y).
y=Ah(x,γ˜)+e.
xN(x*,Γx),eN(e*,Γe),
π(x|y,γ=γ˜)=π(x,e|y,γ=γ˜)de,
π(x|y,γ=γ˜)exp{12(xx*)TΓx1(xx*)12(yAh(x,γ˜)e*)TΓe1(yAh(x,γ˜)e*)}.
xcem=argmaxxπ(x|y,γ=γ˜)=argminx{Le(yAh(x,γ˜)e*)2+Lx(xx*)2},
y=Aδ(x¯,γ)+e
x¯(r)=x(T(r)),
T(Ω,Ω˜):ΩΩ˜
Px¯=x,
y=Ah(x,γ˜)+{Aδ(x¯,γ)Ah(x,γ˜)}+e=Ah(x,γ˜)+ε(x¯,γ)+e.
n(x¯,γ)=ε(x¯,γ)+e.
n*ε*+e*,ΓnΓε+Γe,
π˜(x|y)exp{12(yAh(x,γ˜)n*)TΓn1(yAh(x,γ˜)n*)12(xx*)TΓx1(xx*)}.
xaem=argminx{Ln(yAh(x,γ˜)n*)2+Lx(xx*)2},
r(θ)=k=1pγkvk(θ),θ[0,2π],
γ*=1Ns=1Nsγ(),
Γγ=1Ns1=1Ns(γ()γ*)(γ()γ*)T.
Δ()=(γ()γ*)TΓγ1(γ()γ*)=γ()γ*Γγ1.
π(x)exp{12Lx(xx*)2},LxTLx=Γx1,
x*=(μa*μs*),Γx=(Γμa00Γμs).
f=fin+fbg,
finN(0,Γin,f),
f*=cI,Γf=Γin,f+σbg,f2IIT.
S={x¯(l),l=1,2,…,Ns}
x(l)=P(l)x¯,P(l):Ω(l)Ω˜,
ε()=Aδ(x¯(),γ())Aδ(x(),γ˜),
ε()=Aδ(x¯(),γ())Ah(x(),γ˜).
ε*=1Ns=1Nsε(),
Γε=1Ns1=1Ns(ε()ε*)(ε()ε*)T.

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