Abstract

We propose an approach for the estimation of the optical absorption coefficient in medical optical tomography in the presence of geometric mismodelling. We focus on cases in which the boundaries of the measurement domain or the optode positions are not accurately known. In general, geometric distortion of the domain produces anisotropic changes for the material parameters in the model. Hence, geometric mismodelling in an isotropic case may correspond to an anisotropic model. We seek to approximate the errors due to geometric mismodelling as extraneous additive noise and to pose a simple anisotropic model for the diffusion coefficient. We show that while geometric mismodelling may deteriorate the estimates of the absorption coefficient significantly, using the proposed model enables the recovery of the main features.

© 2005 Optical Society of America

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  1. S. R.  Arridge, “Optical tomography in medical imaging,” Inv. Probl. 15, R41–R93 (1999).
    [CrossRef]
  2. I.  Nissilä, T.  Noponen, J.  Heino, T.  Kajava, T.  Katila “Diffuse optical imaging,” in Advances in Electromagnetic Fields in Living Systems 4, J.  Lin, ed. (Kluwer, New York, to be published)
  3. H.  Dehghani, B. W.  Pogue, S. P.  Poplack, K. D.  Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42, 135–145 (2003).
    [CrossRef] [PubMed]
  4. A.  Li, E. L.  Miller, M. E.  Kilmer, T. J.  Brukilacchio, T.  Chaves, J.  Stott, Q.  Zhang, T.  Wu, M.  Chorlton, R. H.  Moore, D. B.  Kopans, D. A.  Boas, “Tomographic optical breast imaging guided by three-dimensional mammography,” Appl. Opt. 42, 5181–5190 (2003).
    [CrossRef] [PubMed]
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    [PubMed]
  6. M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
    [CrossRef] [PubMed]
  7. B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
    [CrossRef]
  8. M. A.  Franceschini, V.  Toronov, M. E.  Filiaci, E.  Gratton, S.  Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Exp. 6, 49–57 (2000).
    [CrossRef]
  9. A. Y.  Bluestone, G.  Abdoulaev, C. H.  Schmitz, R.  Barbour, A. H.  Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Exp. 9, 272–286 (2001).
    [CrossRef]
  10. 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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
    [CrossRef] [PubMed]
  11. J. C.  Hebden, A.  Gibson, T.  Austin, R.  Yusof, N.  Everdell, D. T.  Delpy, S. R.  Arridge, J. H.  Meek, J. S.  Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49, 1117–1130 (2004).
    [CrossRef] [PubMed]
  12. J. J.  Stott, J. P.  Culver, S. R.  Arridge, D. A.  Boas, “Optode positional calibration in diffuse optical tomography,” Appl. Opt. 42, 3154–3162 (2003).
    [CrossRef] [PubMed]
  13. D. A.  Boas, T.  Gaudette, S. R.  Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Exp. 8, 263–270 (2001).
    [CrossRef]
  14. T.  Tarvainen, V.  Kolehmainen, M.  Vauhkonen, A.  Vanne, J. P.  Kaipio, A. P.  Gibson, S. R.  Arridge, M.  Schweiger, “Computational calibration method for optical tomography,” Applied Optics, 2005, in press.
    [CrossRef] [PubMed]
  15. J. P.  Kaipio, E.  Somersalo, Computational and Statistical Methods for Inverse Problems (Applied Mathematical Sciences 160, Springer-Verlag, New York, ISBN 0-387-22073-9, 2004).
  16. S. R.  Arridge, M.  Schweiger, M.  Hiraoka, D. T.  Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
    [CrossRef] [PubMed]
  17. M.  Schweiger, S. R.  Arridge, M.  Hiraoka, D. T.  Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
    [CrossRef] [PubMed]
  18. J.  Heino, E.  Somersalo, “Estimation of optical absorption in anisotropic background,” Inv. Probl. 18, 559–73 (2002).
    [CrossRef]
  19. J.  Sylvester, “An anisotropic inverse boundary value problem,” Comm. Pure Appl. Math. 43, 201–232 (1990).
    [CrossRef]
  20. V.  Kolehmainen, Department of Applied Physics, University of Kuopio, P.O.B 1627, FIN-70211 Kuopio, Finland, M. Lassas, and P. Ola are preparing a manuscript to be called “Inverse conductivity problem with an imperfectly known boundary.”
  21. S.  Järvenpää, Implementation of PML Absorbing Boundary Condition for Solving Maxwell’s Equations with Whitney Elements (PhD Thesis, University of Helsinki, 2001).
    [PubMed]
  22. J.  Heino, E.  Somersalo, “A modelling error approach for the estimation of optical absorption in the presence of anisotropies,” Phys. Med. Biol. 49, 4785–4798 (2004) .
    [CrossRef] [PubMed]
  23. Åke  Björck, Numerical Methods for Least Squares Problems (SIAM, Philadelphia, 1996).
    [CrossRef]
  24. J. E.  Dennis, R. B.  Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM, Philadelphia, 1996).
    [CrossRef]

2004 (2)

J. C.  Hebden, A.  Gibson, T.  Austin, R.  Yusof, N.  Everdell, D. T.  Delpy, S. R.  Arridge, J. H.  Meek, J. S.  Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49, 1117–1130 (2004).
[CrossRef] [PubMed]

J.  Heino, E.  Somersalo, “A modelling error approach for the estimation of optical absorption in the presence of anisotropies,” Phys. Med. Biol. 49, 4785–4798 (2004) .
[CrossRef] [PubMed]

2003 (3)

2002 (2)

J.  Heino, E.  Somersalo, “Estimation of optical absorption in anisotropic background,” Inv. Probl. 18, 559–73 (2002).
[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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

2001 (3)

A. Y.  Bluestone, G.  Abdoulaev, C. H.  Schmitz, R.  Barbour, A. H.  Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Exp. 9, 272–286 (2001).
[CrossRef]

D. A.  Boas, T.  Gaudette, S. R.  Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Exp. 8, 263–270 (2001).
[CrossRef]

B. W.  Pogue, S. P.  Poplack, T. O.  McBride, W. A.  Welss, K. S.  Osterman, U. L.  Ostenberg, K. D.  Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: Pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

2000 (1)

M. A.  Franceschini, V.  Toronov, M. E.  Filiaci, E.  Gratton, S.  Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Exp. 6, 49–57 (2000).
[CrossRef]

1999 (1)

S. R.  Arridge, “Optical tomography in medical imaging,” Inv. Probl. 15, R41–R93 (1999).
[CrossRef]

1998 (1)

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

1997 (1)

M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef] [PubMed]

1995 (1)

M.  Schweiger, S. R.  Arridge, M.  Hiraoka, D. T.  Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

1993 (1)

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

1990 (1)

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

Abdoulaev, G.

A. Y.  Bluestone, G.  Abdoulaev, C. H.  Schmitz, R.  Barbour, A. H.  Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Exp. 9, 272–286 (2001).
[CrossRef]

Anday, E.

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

Arridge, S. R.

J. C.  Hebden, A.  Gibson, T.  Austin, R.  Yusof, N.  Everdell, D. T.  Delpy, S. R.  Arridge, J. H.  Meek, J. S.  Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49, 1117–1130 (2004).
[CrossRef] [PubMed]

J. J.  Stott, J. P.  Culver, S. R.  Arridge, D. A.  Boas, “Optode positional calibration in diffuse optical tomography,” Appl. Opt. 42, 3154–3162 (2003).
[CrossRef] [PubMed]

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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

D. A.  Boas, T.  Gaudette, S. R.  Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Exp. 8, 263–270 (2001).
[CrossRef]

S. R.  Arridge, “Optical tomography in medical imaging,” Inv. Probl. 15, R41–R93 (1999).
[CrossRef]

M.  Schweiger, S. R.  Arridge, M.  Hiraoka, D. T.  Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

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

T.  Tarvainen, V.  Kolehmainen, M.  Vauhkonen, A.  Vanne, J. P.  Kaipio, A. P.  Gibson, S. R.  Arridge, M.  Schweiger, “Computational calibration method for optical tomography,” Applied Optics, 2005, in press.
[CrossRef] [PubMed]

Austin, T.

J. C.  Hebden, A.  Gibson, T.  Austin, R.  Yusof, N.  Everdell, D. T.  Delpy, S. R.  Arridge, J. H.  Meek, J. S.  Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49, 1117–1130 (2004).
[CrossRef] [PubMed]

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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Barbour, R.

A. Y.  Bluestone, G.  Abdoulaev, C. H.  Schmitz, R.  Barbour, A. H.  Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Exp. 9, 272–286 (2001).
[CrossRef]

Björck, Åke

Åke  Björck, Numerical Methods for Least Squares Problems (SIAM, Philadelphia, 1996).
[CrossRef]

Bluestone, A. Y.

A. Y.  Bluestone, G.  Abdoulaev, C. H.  Schmitz, R.  Barbour, A. H.  Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Exp. 9, 272–286 (2001).
[CrossRef]

Boas, D. A.

Brukilacchio, T. J.

Chance, B.

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

Chaves, T.

Chorlton, M.

Culver, J. P.

Dehghani, H.

Delpy, D. T.

J. C.  Hebden, A.  Gibson, T.  Austin, R.  Yusof, N.  Everdell, D. T.  Delpy, S. R.  Arridge, J. H.  Meek, J. S.  Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49, 1117–1130 (2004).
[CrossRef] [PubMed]

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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

M.  Schweiger, S. R.  Arridge, M.  Hiraoka, D. T.  Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

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

Dennis, J. E.

J. E.  Dennis, R. B.  Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM, Philadelphia, 1996).
[CrossRef]

Everdell, N.

J. C.  Hebden, A.  Gibson, T.  Austin, R.  Yusof, N.  Everdell, D. T.  Delpy, S. R.  Arridge, J. H.  Meek, J. S.  Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49, 1117–1130 (2004).
[CrossRef] [PubMed]

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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Fantini, S.

M. A.  Franceschini, V.  Toronov, M. E.  Filiaci, E.  Gratton, S.  Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Exp. 6, 49–57 (2000).
[CrossRef]

M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef] [PubMed]

Filiaci, M. E.

M. A.  Franceschini, V.  Toronov, M. E.  Filiaci, E.  Gratton, S.  Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Exp. 6, 49–57 (2000).
[CrossRef]

Franceschini, M. A.

M. A.  Franceschini, V.  Toronov, M. E.  Filiaci, E.  Gratton, S.  Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Exp. 6, 49–57 (2000).
[CrossRef]

M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef] [PubMed]

Gaida, G.

M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef] [PubMed]

Gaudette, T.

D. A.  Boas, T.  Gaudette, S. R.  Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Exp. 8, 263–270 (2001).
[CrossRef]

Gibson, A.

J. C.  Hebden, A.  Gibson, T.  Austin, R.  Yusof, N.  Everdell, D. T.  Delpy, S. R.  Arridge, J. H.  Meek, J. S.  Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49, 1117–1130 (2004).
[CrossRef] [PubMed]

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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Gibson, A. P.

T.  Tarvainen, V.  Kolehmainen, M.  Vauhkonen, A.  Vanne, J. P.  Kaipio, A. P.  Gibson, S. R.  Arridge, M.  Schweiger, “Computational calibration method for optical tomography,” Applied Optics, 2005, in press.
[CrossRef] [PubMed]

Gratton, E.

M. A.  Franceschini, V.  Toronov, M. E.  Filiaci, E.  Gratton, S.  Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Exp. 6, 49–57 (2000).
[CrossRef]

M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef] [PubMed]

Hebden, J. C.

J. C.  Hebden, A.  Gibson, T.  Austin, R.  Yusof, N.  Everdell, D. T.  Delpy, S. R.  Arridge, J. H.  Meek, J. S.  Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49, 1117–1130 (2004).
[CrossRef] [PubMed]

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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Heino, J.

J.  Heino, E.  Somersalo, “A modelling error approach for the estimation of optical absorption in the presence of anisotropies,” Phys. Med. Biol. 49, 4785–4798 (2004) .
[CrossRef] [PubMed]

J.  Heino, E.  Somersalo, “Estimation of optical absorption in anisotropic background,” Inv. Probl. 18, 559–73 (2002).
[CrossRef]

I.  Nissilä, T.  Noponen, J.  Heino, T.  Kajava, T.  Katila “Diffuse optical imaging,” in Advances in Electromagnetic Fields in Living Systems 4, J.  Lin, ed. (Kluwer, New York, to be published)

Hielscher, A. H.

A. Y.  Bluestone, G.  Abdoulaev, C. H.  Schmitz, R.  Barbour, A. H.  Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Exp. 9, 272–286 (2001).
[CrossRef]

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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Hiraoka, M.

M.  Schweiger, S. R.  Arridge, M.  Hiraoka, D. T.  Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

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

Hong, L.

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

Järvenpää, S.

S.  Järvenpää, Implementation of PML Absorbing Boundary Condition for Solving Maxwell’s Equations with Whitney Elements (PhD Thesis, University of Helsinki, 2001).
[PubMed]

Jess, H.

M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef] [PubMed]

Kaipio, J. P.

T.  Tarvainen, V.  Kolehmainen, M.  Vauhkonen, A.  Vanne, J. P.  Kaipio, A. P.  Gibson, S. R.  Arridge, M.  Schweiger, “Computational calibration method for optical tomography,” Applied Optics, 2005, in press.
[CrossRef] [PubMed]

J. P.  Kaipio, E.  Somersalo, Computational and Statistical Methods for Inverse Problems (Applied Mathematical Sciences 160, Springer-Verlag, New York, ISBN 0-387-22073-9, 2004).

Kajava, T.

I.  Nissilä, T.  Noponen, J.  Heino, T.  Kajava, T.  Katila “Diffuse optical imaging,” in Advances in Electromagnetic Fields in Living Systems 4, J.  Lin, ed. (Kluwer, New York, to be published)

Kaschke, M.

M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef] [PubMed]

Katila, T.

I.  Nissilä, T.  Noponen, J.  Heino, T.  Kajava, T.  Katila “Diffuse optical imaging,” in Advances in Electromagnetic Fields in Living Systems 4, J.  Lin, ed. (Kluwer, New York, to be published)

Kilmer, M. E.

Kolehmainen, V.

V.  Kolehmainen, Department of Applied Physics, University of Kuopio, P.O.B 1627, FIN-70211 Kuopio, Finland, M. Lassas, and P. Ola are preparing a manuscript to be called “Inverse conductivity problem with an imperfectly known boundary.”

T.  Tarvainen, V.  Kolehmainen, M.  Vauhkonen, A.  Vanne, J. P.  Kaipio, A. P.  Gibson, S. R.  Arridge, M.  Schweiger, “Computational calibration method for optical tomography,” Applied Optics, 2005, in press.
[CrossRef] [PubMed]

Kopans, D. B.

Li, A.

Li, C.

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

Mantulin, W. W.

M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef] [PubMed]

McBride, T. O.

B. W.  Pogue, S. P.  Poplack, T. O.  McBride, W. A.  Welss, K. S.  Osterman, U. L.  Ostenberg, K. D.  Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: Pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Meek, J. H.

J. C.  Hebden, A.  Gibson, T.  Austin, R.  Yusof, N.  Everdell, D. T.  Delpy, S. R.  Arridge, J. H.  Meek, J. S.  Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49, 1117–1130 (2004).
[CrossRef] [PubMed]

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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Miller, E. L.

Moesta, K. T.

M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef] [PubMed]

Moore, R. H.

Murray, T.

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

Nioka, S.

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

Nissilä, I.

I.  Nissilä, T.  Noponen, J.  Heino, T.  Kajava, T.  Katila “Diffuse optical imaging,” in Advances in Electromagnetic Fields in Living Systems 4, J.  Lin, ed. (Kluwer, New York, to be published)

Noponen, T.

I.  Nissilä, T.  Noponen, J.  Heino, T.  Kajava, T.  Katila “Diffuse optical imaging,” in Advances in Electromagnetic Fields in Living Systems 4, J.  Lin, ed. (Kluwer, New York, to be published)

Ostenberg, U. L.

B. W.  Pogue, S. P.  Poplack, T. O.  McBride, W. A.  Welss, K. S.  Osterman, U. L.  Ostenberg, K. D.  Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: Pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Osterman, K. S.

B. W.  Pogue, S. P.  Poplack, T. O.  McBride, W. A.  Welss, K. S.  Osterman, U. L.  Ostenberg, K. D.  Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: Pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Ovetsky, Y.

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

Paulsen, K. D.

H.  Dehghani, B. W.  Pogue, S. P.  Poplack, K. D.  Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42, 135–145 (2003).
[CrossRef] [PubMed]

B. W.  Pogue, S. P.  Poplack, T. O.  McBride, W. A.  Welss, K. S.  Osterman, U. L.  Ostenberg, K. D.  Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: Pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Pidikiti, D.

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

Pogue, B. W.

H.  Dehghani, B. W.  Pogue, S. P.  Poplack, K. D.  Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42, 135–145 (2003).
[CrossRef] [PubMed]

B. W.  Pogue, S. P.  Poplack, T. O.  McBride, W. A.  Welss, K. S.  Osterman, U. L.  Ostenberg, K. D.  Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: Pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Poplack, S. P.

H.  Dehghani, B. W.  Pogue, S. P.  Poplack, K. D.  Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42, 135–145 (2003).
[CrossRef] [PubMed]

B. W.  Pogue, S. P.  Poplack, T. O.  McBride, W. A.  Welss, K. S.  Osterman, U. L.  Ostenberg, K. D.  Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: Pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Schlag, P. M.

M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef] [PubMed]

Schmitz, C. H.

A. Y.  Bluestone, G.  Abdoulaev, C. H.  Schmitz, R.  Barbour, A. H.  Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Exp. 9, 272–286 (2001).
[CrossRef]

Schnabel, R. B.

J. E.  Dennis, R. B.  Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM, Philadelphia, 1996).
[CrossRef]

Schweiger, M.

M.  Schweiger, S. R.  Arridge, M.  Hiraoka, D. T.  Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

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

T.  Tarvainen, V.  Kolehmainen, M.  Vauhkonen, A.  Vanne, J. P.  Kaipio, A. P.  Gibson, S. R.  Arridge, M.  Schweiger, “Computational calibration method for optical tomography,” Applied Optics, 2005, in press.
[CrossRef] [PubMed]

Seeber, M.

M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef] [PubMed]

Somersalo, E.

J.  Heino, E.  Somersalo, “A modelling error approach for the estimation of optical absorption in the presence of anisotropies,” Phys. Med. Biol. 49, 4785–4798 (2004) .
[CrossRef] [PubMed]

J.  Heino, E.  Somersalo, “Estimation of optical absorption in anisotropic background,” Inv. Probl. 18, 559–73 (2002).
[CrossRef]

J. P.  Kaipio, E.  Somersalo, Computational and Statistical Methods for Inverse Problems (Applied Mathematical Sciences 160, Springer-Verlag, New York, ISBN 0-387-22073-9, 2004).

Stott, J.

Stott, J. J.

Sylvester, J.

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

Tarvainen, T.

T.  Tarvainen, V.  Kolehmainen, M.  Vauhkonen, A.  Vanne, J. P.  Kaipio, A. P.  Gibson, S. R.  Arridge, M.  Schweiger, “Computational calibration method for optical tomography,” Applied Optics, 2005, in press.
[CrossRef] [PubMed]

Thomas, R.

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

Toronov, V.

M. A.  Franceschini, V.  Toronov, M. E.  Filiaci, E.  Gratton, S.  Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Exp. 6, 49–57 (2000).
[CrossRef]

Vanne, A.

T.  Tarvainen, V.  Kolehmainen, M.  Vauhkonen, A.  Vanne, J. P.  Kaipio, A. P.  Gibson, S. R.  Arridge, M.  Schweiger, “Computational calibration method for optical tomography,” Applied Optics, 2005, in press.
[CrossRef] [PubMed]

Vauhkonen, M.

T.  Tarvainen, V.  Kolehmainen, M.  Vauhkonen, A.  Vanne, J. P.  Kaipio, A. P.  Gibson, S. R.  Arridge, M.  Schweiger, “Computational calibration method for optical tomography,” Applied Optics, 2005, in press.
[CrossRef] [PubMed]

Welss, W. A.

B. W.  Pogue, S. P.  Poplack, T. O.  McBride, W. A.  Welss, K. S.  Osterman, U. L.  Ostenberg, K. D.  Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: Pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Worden, K.

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

Wu, T.

Wyatt, J. S.

J. C.  Hebden, A.  Gibson, T.  Austin, R.  Yusof, N.  Everdell, D. T.  Delpy, S. R.  Arridge, J. H.  Meek, J. S.  Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49, 1117–1130 (2004).
[CrossRef] [PubMed]

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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Yusof, R.

J. C.  Hebden, A.  Gibson, T.  Austin, R.  Yusof, N.  Everdell, D. T.  Delpy, S. R.  Arridge, J. H.  Meek, J. S.  Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49, 1117–1130 (2004).
[CrossRef] [PubMed]

Yusof, R. M.

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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

Zhang, Q.

Zhou, S.

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

Appl. Opt. (3)

Comm. Pure Appl. Math. (1)

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

Inv. Probl. (2)

S. R.  Arridge, “Optical tomography in medical imaging,” Inv. Probl. 15, R41–R93 (1999).
[CrossRef]

J.  Heino, E.  Somersalo, “Estimation of optical absorption in anisotropic background,” Inv. Probl. 18, 559–73 (2002).
[CrossRef]

Med. Phys. (2)

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

M.  Schweiger, S. R.  Arridge, M.  Hiraoka, D. T.  Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

Opt. Exp. (4)

B.  Chance, E.  Anday, S.  Nioka, S.  Zhou, L.  Hong, K.  Worden, C.  Li, T.  Murray, Y.  Ovetsky, D.  Pidikiti, R.  Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Exp. 2, 411–423 (1998).
[CrossRef]

M. A.  Franceschini, V.  Toronov, M. E.  Filiaci, E.  Gratton, S.  Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Exp. 6, 49–57 (2000).
[CrossRef]

A. Y.  Bluestone, G.  Abdoulaev, C. H.  Schmitz, R.  Barbour, A. H.  Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Exp. 9, 272–286 (2001).
[CrossRef]

D. A.  Boas, T.  Gaudette, S. R.  Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Exp. 8, 263–270 (2001).
[CrossRef]

Phys. Med. Biol. (3)

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, J. S.  Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
[CrossRef] [PubMed]

J. C.  Hebden, A.  Gibson, T.  Austin, R.  Yusof, N.  Everdell, D. T.  Delpy, S. R.  Arridge, J. H.  Meek, J. S.  Wyatt, “Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,” Phys. Med. Biol. 49, 1117–1130 (2004).
[CrossRef] [PubMed]

J.  Heino, E.  Somersalo, “A modelling error approach for the estimation of optical absorption in the presence of anisotropies,” Phys. Med. Biol. 49, 4785–4798 (2004) .
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

M. A.  Franceschini, K. T.  Moesta, S.  Fantini, G.  Gaida, E.  Gratton, H.  Jess, W. W.  Mantulin, M.  Seeber, P. M.  Schlag, M.  Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad. Sci. USA 94, 6468–6473 (1997).
[CrossRef] [PubMed]

Radiology (1)

B. W.  Pogue, S. P.  Poplack, T. O.  McBride, W. A.  Welss, K. S.  Osterman, U. L.  Ostenberg, K. D.  Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: Pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Other (7)

T.  Tarvainen, V.  Kolehmainen, M.  Vauhkonen, A.  Vanne, J. P.  Kaipio, A. P.  Gibson, S. R.  Arridge, M.  Schweiger, “Computational calibration method for optical tomography,” Applied Optics, 2005, in press.
[CrossRef] [PubMed]

J. P.  Kaipio, E.  Somersalo, Computational and Statistical Methods for Inverse Problems (Applied Mathematical Sciences 160, Springer-Verlag, New York, ISBN 0-387-22073-9, 2004).

Åke  Björck, Numerical Methods for Least Squares Problems (SIAM, Philadelphia, 1996).
[CrossRef]

J. E.  Dennis, R. B.  Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM, Philadelphia, 1996).
[CrossRef]

I.  Nissilä, T.  Noponen, J.  Heino, T.  Kajava, T.  Katila “Diffuse optical imaging,” in Advances in Electromagnetic Fields in Living Systems 4, J.  Lin, ed. (Kluwer, New York, to be published)

V.  Kolehmainen, Department of Applied Physics, University of Kuopio, P.O.B 1627, FIN-70211 Kuopio, Finland, M. Lassas, and P. Ola are preparing a manuscript to be called “Inverse conductivity problem with an imperfectly known boundary.”

S.  Järvenpää, Implementation of PML Absorbing Boundary Condition for Solving Maxwell’s Equations with Whitney Elements (PhD Thesis, University of Helsinki, 2001).
[PubMed]

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Figures (5)

Fig. 1.
Fig. 1.

Schematic representation of the mapping Φ : Ω'→Ω and its inverse Φ-1 : Ω→Ω′.

Fig. 2.
Fig. 2.

Transformation defined in Eq. (16). (a) The domain Ω′ used in the inversion. The 16 optodes are located at equal distances on the boundary. The figure also depicts two inclusions inside the domain. (b) The actual domain Ω. The data used in the numerical inversion example is generated using this model with constant isotropic κ=κ 0=0.05 cm-1 and background absorption µ a,bg=0.1 cm-1. In the inclusions, µ a=0.2 cm-1. The simulated data set was collected by placing the source at each optode location at turn, and using the rest of the optodes for measurement, resulting in 240 measurements of amplitude and phase each.

Fig. 3.
Fig. 3.

Diffusion tensor generated by the transformation defined in Eq. (16). The tensor distribution is illustrated by ellipses. The axes of an ellipse correspond to the directions of the diffusion tensor eigenvectors. The diffusion tensor is written as κ=UΛUT , where U=[1 2] contains the eigenvectors i , i=1, 2, and Λ=diag(λ 1,λ 2) the eigenvalues. The axis corresponding to 1 and the larger eigenvalue λ 1 is given a constant length, and the second axis is scaled by λ 2/λ 1. The colour reflects the value of the larger eigenvalue λ 1.

Fig. 4.
Fig. 4.

(a) Estimate of the absorption coefficient using the isotropic LM iteration in domain Ω′ used for inversion, and (b) the estimate in (a) mapped onto the actual domain Ω.

Fig. 5.
Fig. 5.

Estimates for the optical absorption using the modelling error approach combined with LM iteration. (a) Case 1 with θ∗=π/2 in the domain Ω′, and (b) the estimate in (a) mapped into the real domain Ω. (c) Case 2 with θ∗=0 in Ω′, and (d) the estimate in (c) mapped into Ω.

Tables (1)

Tables Icon

Table 1. The parameter values used in the modelling error approach.

Equations (36)

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

div ( κ u ) ( μ a ik ) u = q ,
( γ u + 1 2 n · κ u ) Ω = f .
x = Φ ( x ) Ω , x = Φ 1 ( x ) Ω .
u ( x ) = u ( Φ ( x ) ) u Φ ( x ) u ( x ) .
Ω ψ n · κ u d S Ω ( ψ · κ u + ψ μ ˜ u ) d x = Ω ψ q d x ,
2 γ Ω ψ u d S + Ω ( ψ · κ u + ψ μ ˜ u ) d x = Ω 2 ψ f d S Ω ψ q d x .
= j = 1 n x j e j .
u = [ D Φ 1 ] T u ,
d x = 1 det ( D Φ 1 ) d x .
2 Ω γ ψ u d S + Ω ' ( ψ · κ u + ψ μ ˜ u ) d x = 2 Ω ' ψ f d S Ω ' ψ q d x ,
κ = 1 det ( D Φ 1 ) [ D Φ 1 ] κ [ D Φ 1 ] T ,
μ ˜ = 1 det ( D Φ 1 ) μ ˜ ,
q = 1 det ( D Φ 1 ) q ,
f = 1 det ( D ϕ 1 ) f ,
γ = 1 det ( D ϕ 1 ) γ ,
{ r = r ( 1 + 0.15 ( cos 2 θ 0.3 sin 6 θ ) ) θ = θ + 0.04 r sin 5 θ ,
( D Φ 1 ) j , k = x j x k ,
x j x k = i , x j y y y i y i x k .
y = G ( μ a , κ ) + ν ,
y = G ( x , z ) + ν .
y = G ( x , z * ) + ( G ( x , z ) G ( x , z * ) ) + ν = G ( x , z * ) + ε ( x , z ) + ν .
ε ( x , z ) D z G ( x c , z * ) δ z , δ z = z z * ,
cov ( z ) = E { δ z δ z T } = Γ z ,
Γ ε = E { ε ( x , z ) ε ( x , z ) T } D z G ( x c , z * ) Γ z D z G ( x c , z * ) T .
y = G ( x , z * ) + e , cov ( e ) = Γ ε + Γ ν ,
y = G ( x ) + e ,
π ( y x ) exp ( 1 2 ( y G ( x ) ) T Γ 1 ( y G ( x ) ) ) .
0 ( x ) = 1 2 ( y G ( x ) ) T Γ 1 ( y G ( x ) ) = 1 2 y G ( x ) Γ 1 2 .
G ( x ) = G ( x c + δ x ) G ( x c ) + D G ( x c ) δ x .
Minimize y ( G ( x c ) + D G ( x c ) δ x ) Γ 1 2 subject to δ x r .
y ( G ( x c ) + D G ( x c ) δ x ) Γ 1 2 + λ δ x 2 .
δ x = ( D G ( x c ) T Γ 1 D G ( x c ) + λ I ) 1 D G ( x c ) T Γ 1 ( y G ( x c ) ) .
π ( x y ) π ( y x ) π pr ( x )
exp ( 1 2 y G ( x ) Γ 1 2 V ( x ) ) ,
V ( x ) = log π pr ( x ) .
( x ) = 0 ( x ) + V ( x ) .

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