Abstract

In diffuse optical tomography (DOT), the object with unknown optical properties is illuminated with near infrared light and the absorption and diffusion coefficient distributions of a body are estimated from the scattering and transmission data. The problem is notoriously ill-posed and complementary information concerning the optical properties needs to be used to counter-effect the ill-posedness. In this article, we propose an adaptive inhomogenous anisotropic smoothness regularization scheme that corresponds to the prior information that the unknown object has a blocky structure. The algorithm updates alternatingly the current estimate and the smoothness penalty functional, and it is demonstrated with simulated data that the algorithm is capable of locating well blocky inclusions. The dynamical range of the reconstruction is improved, compared to traditional smoothness regularization schemes, and the crosstalk between the diffusion and absorption images is clearly less. The algorithm is tested also with a three-dimensional phantom data.

© 2008 OSA

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References

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  1. S. R. Arridge, “Optical Tomography in Medical Imaging,” Inverse Probl. 15, R41–R93 (1999).
    [Crossref]
  2. J. Hebden, A. Gibson, R. Yusof, N. Everdell, E. Hillman, D. Delpy, S. Arridge, T. Austin, J. Meek, and J. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155–4166 (2002).
    [Crossref] [PubMed]
  3. J. Hebden and T. Austin, “Optical tomography of the neonatal brain,” European Radiology 17, 2926–2933 (2007).
    [Crossref] [PubMed]
  4. J. Culver, T. Durduran, D. Furya, C. Cheung, J. Greenberg, and A. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cerebral Blood Flow & Metabolism 23, 911–924 (2003).
    [Crossref]
  5. J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer Verlag, 2004).
  6. J. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inverse Probl. 15, 713–729 (1999).
    [Crossref]
  7. B. Pogue, T. McBride, J. Prewitt, U. sterberg, and K. Paulsen, “Spatially Variant Regularization Improves Diffuse Optical Tomography,” Appl. Opt. 38, 2950–2961 (1999).
    [Crossref]
  8. A. Douiri, M. Schweiger, J. Riley, and S. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18, 87–95 (2007).
    [Crossref]
  9. B. Brooksby, S. Jiang, C. Kogel, M. Duyley, H. Dehgani, J. Weaver, S. Poplack, B. Pogue, and K. Paulsen, “Magnetic resonance guided near infrared tomography of the breast,” Rev. Sci. Instrum. 75, 5262–5270 (2004).
    [Crossref]
  10. P. Yalavarthy, B. Pogue, H. Dehgani, C. Carpenter, S. Jiang, and K. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express 15, 8043–8058 (2007).
    [Crossref] [PubMed]
  11. D. Calvetti, F. Sgallari, and E. Somersalo, “Image inpainting with structural bootstrap priors,” Image and Vision Comput. 24, 782–793 (2006).
    [Crossref]
  12. D. Calvetti and E. Somersalo, “Microlocal sequential regularization in imaging,” Inverse Problems and Imaging 1, 1–11 (2007).
    [Crossref]
  13. D. Calvetti and E. Somersalo, “Gaussian hypermodels and recovery of blocky objects,” Inverse Probl. 23, 733–754 (2007).
    [Crossref]
  14. D. Calvetti and E. Somersalo, “Hypermodels in the Bayesian imaging framework,” Inverse Probl. 24, 034013 (20pp) (2008).
    [Crossref]
  15. D. Calvetti, J. P. Kaipio, and E. Somersalo, “Aristotelian prior boundary conditions,” Int. J. Math. Comp. Sci. 1, 63–81 (2006).
  16. M. Schweiger, S. Arridge, M. Hiraoka, and D. 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]
  17. D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing – Ten Lectures on Subjective Computing (Springer Verlag, 2007).
    [PubMed]
  18. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
    [Crossref]
  19. C. R. Vogel and M. E. Oman, “Fast, robust total variation-based reconstruction of noisy, blurred images,” IEEE Trans. Image Process.7, 813–824 (1998).
    [Crossref]
  20. P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell.12, 629–639 (1990).
    [Crossref]
  21. D. Calvetti and E. Somersalo, “Recovery of shapes: hypermodels and Bayesian learning,” Proc. of the Applied Inverse Problems 2007: Theoretical and Computational Aspects. J. of Physics Conference Series (to appear).
  22. “Automatically Tuned Linear Algebra Software (ATLAS),” http://math-atlas.sourceforge.net/ (19th June 2008).
  23. Y. Saad, Iterative Methods for Sparse Linear Systems (Society for Industrial Mathematics, 2003).
    [Crossref]
  24. I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
    [Crossref]
  25. T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 63, 383–405 (2006).
    [Crossref]
  26. J. Heino, E. Somersalo, and J. Kaipio, “Statistical compensation of geometric mismodeling in optical tomography,” Opt. Express 13, 296–308 (2005).
    [Crossref] [PubMed]
  27. 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]
  28. J. P. Kaipio and E. Somersalo, “Statistical inverse problems: discretization, model reduction and inverse crimes,” J. Comp. Appl. Math. 22, 493–504 (2006).
  29. C. Paige and M. Saunders, “LSQR: An Algorithm for Sparse Linear Equations And Sparse Least Squares,” ACM Trans. Math. Software 8, 43–71 (1982).
    [Crossref]

2008 (1)

D. Calvetti and E. Somersalo, “Hypermodels in the Bayesian imaging framework,” Inverse Probl. 24, 034013 (20pp) (2008).
[Crossref]

2007 (5)

P. Yalavarthy, B. Pogue, H. Dehgani, C. Carpenter, S. Jiang, and K. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express 15, 8043–8058 (2007).
[Crossref] [PubMed]

D. Calvetti and E. Somersalo, “Microlocal sequential regularization in imaging,” Inverse Problems and Imaging 1, 1–11 (2007).
[Crossref]

D. Calvetti and E. Somersalo, “Gaussian hypermodels and recovery of blocky objects,” Inverse Probl. 23, 733–754 (2007).
[Crossref]

J. Hebden and T. Austin, “Optical tomography of the neonatal brain,” European Radiology 17, 2926–2933 (2007).
[Crossref] [PubMed]

A. Douiri, M. Schweiger, J. Riley, and S. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18, 87–95 (2007).
[Crossref]

2006 (5)

D. Calvetti, F. Sgallari, and E. Somersalo, “Image inpainting with structural bootstrap priors,” Image and Vision Comput. 24, 782–793 (2006).
[Crossref]

D. Calvetti, J. P. Kaipio, and E. Somersalo, “Aristotelian prior boundary conditions,” Int. J. Math. Comp. Sci. 1, 63–81 (2006).

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 63, 383–405 (2006).
[Crossref]

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175–195 (2006).
[Crossref]

J. P. Kaipio and E. Somersalo, “Statistical inverse problems: discretization, model reduction and inverse crimes,” J. Comp. Appl. Math. 22, 493–504 (2006).

2005 (1)

2004 (1)

B. Brooksby, S. Jiang, C. Kogel, M. Duyley, H. Dehgani, J. Weaver, S. Poplack, B. Pogue, and K. Paulsen, “Magnetic resonance guided near infrared tomography of the breast,” Rev. Sci. Instrum. 75, 5262–5270 (2004).
[Crossref]

2003 (2)

J. Culver, T. Durduran, D. Furya, C. Cheung, J. Greenberg, and A. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cerebral Blood Flow & Metabolism 23, 911–924 (2003).
[Crossref]

Y. Saad, Iterative Methods for Sparse Linear Systems (Society for Industrial Mathematics, 2003).
[Crossref]

2002 (1)

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

1999 (3)

S. R. Arridge, “Optical Tomography in Medical Imaging,” Inverse Probl. 15, R41–R93 (1999).
[Crossref]

J. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inverse Probl. 15, 713–729 (1999).
[Crossref]

B. Pogue, T. McBride, J. Prewitt, U. sterberg, and K. Paulsen, “Spatially Variant Regularization Improves Diffuse Optical Tomography,” Appl. Opt. 38, 2950–2961 (1999).
[Crossref]

1995 (1)

M. Schweiger, S. Arridge, M. Hiraoka, and D. 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]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

1982 (1)

C. Paige and M. Saunders, “LSQR: An Algorithm for Sparse Linear Equations And Sparse Least Squares,” ACM Trans. Math. Software 8, 43–71 (1982).
[Crossref]

Arridge, S.

A. Douiri, M. Schweiger, J. Riley, and S. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18, 87–95 (2007).
[Crossref]

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

M. Schweiger, S. Arridge, M. Hiraoka, and D. 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]

Arridge, S. R.

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]

S. R. Arridge, “Optical Tomography in Medical Imaging,” Inverse Probl. 15, R41–R93 (1999).
[Crossref]

Austin, T.

J. Hebden and T. Austin, “Optical tomography of the neonatal brain,” European Radiology 17, 2926–2933 (2007).
[Crossref] [PubMed]

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

Brooksby, B.

B. Brooksby, S. Jiang, C. Kogel, M. Duyley, H. Dehgani, J. Weaver, S. Poplack, B. Pogue, and K. Paulsen, “Magnetic resonance guided near infrared tomography of the breast,” Rev. Sci. Instrum. 75, 5262–5270 (2004).
[Crossref]

Calvetti, D.

D. Calvetti and E. Somersalo, “Hypermodels in the Bayesian imaging framework,” Inverse Probl. 24, 034013 (20pp) (2008).
[Crossref]

D. Calvetti and E. Somersalo, “Gaussian hypermodels and recovery of blocky objects,” Inverse Probl. 23, 733–754 (2007).
[Crossref]

D. Calvetti and E. Somersalo, “Microlocal sequential regularization in imaging,” Inverse Problems and Imaging 1, 1–11 (2007).
[Crossref]

D. Calvetti, F. Sgallari, and E. Somersalo, “Image inpainting with structural bootstrap priors,” Image and Vision Comput. 24, 782–793 (2006).
[Crossref]

D. Calvetti, J. P. Kaipio, and E. Somersalo, “Aristotelian prior boundary conditions,” Int. J. Math. Comp. Sci. 1, 63–81 (2006).

D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing – Ten Lectures on Subjective Computing (Springer Verlag, 2007).
[PubMed]

D. Calvetti and E. Somersalo, “Recovery of shapes: hypermodels and Bayesian learning,” Proc. of the Applied Inverse Problems 2007: Theoretical and Computational Aspects. J. of Physics Conference Series (to appear).

Carpenter, C.

Cheung, C.

J. Culver, T. Durduran, D. Furya, C. Cheung, J. Greenberg, and A. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cerebral Blood Flow & Metabolism 23, 911–924 (2003).
[Crossref]

Culver, J.

J. Culver, T. Durduran, D. Furya, C. Cheung, J. Greenberg, and A. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cerebral Blood Flow & Metabolism 23, 911–924 (2003).
[Crossref]

Dehgani, H.

P. Yalavarthy, B. Pogue, H. Dehgani, C. Carpenter, S. Jiang, and K. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express 15, 8043–8058 (2007).
[Crossref] [PubMed]

B. Brooksby, S. Jiang, C. Kogel, M. Duyley, H. Dehgani, J. Weaver, S. Poplack, B. Pogue, and K. Paulsen, “Magnetic resonance guided near infrared tomography of the breast,” Rev. Sci. Instrum. 75, 5262–5270 (2004).
[Crossref]

Delpy, D.

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

M. Schweiger, S. Arridge, M. Hiraoka, and D. 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]

Douiri, A.

A. Douiri, M. Schweiger, J. Riley, and S. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18, 87–95 (2007).
[Crossref]

Durduran, T.

J. Culver, T. Durduran, D. Furya, C. Cheung, J. Greenberg, and A. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cerebral Blood Flow & Metabolism 23, 911–924 (2003).
[Crossref]

Duyley, M.

B. Brooksby, S. Jiang, C. Kogel, M. Duyley, H. Dehgani, J. Weaver, S. Poplack, B. Pogue, and K. Paulsen, “Magnetic resonance guided near infrared tomography of the breast,” Rev. Sci. Instrum. 75, 5262–5270 (2004).
[Crossref]

Everdell, N.

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

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

Furya, D.

J. Culver, T. Durduran, D. Furya, C. Cheung, J. Greenberg, and A. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cerebral Blood Flow & Metabolism 23, 911–924 (2003).
[Crossref]

Gibson, A.

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

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
[Crossref]

Greenberg, J.

J. Culver, T. Durduran, D. Furya, C. Cheung, J. Greenberg, and A. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cerebral Blood Flow & Metabolism 23, 911–924 (2003).
[Crossref]

Hebden, J.

J. Hebden and T. Austin, “Optical tomography of the neonatal brain,” European Radiology 17, 2926–2933 (2007).
[Crossref] [PubMed]

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

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
[Crossref]

Heino, J.

J. Heino, E. Somersalo, and J. Kaipio, “Statistical compensation of geometric mismodeling in optical tomography,” Opt. Express 13, 296–308 (2005).
[Crossref] [PubMed]

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
[Crossref]

Hillman, E.

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

Hiraoka, M.

M. Schweiger, S. Arridge, M. Hiraoka, and D. 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]

Järvenpää, S.

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
[Crossref]

Jennions, D.

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
[Crossref]

Jiang, S.

P. Yalavarthy, B. Pogue, H. Dehgani, C. Carpenter, S. Jiang, and K. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express 15, 8043–8058 (2007).
[Crossref] [PubMed]

B. Brooksby, S. Jiang, C. Kogel, M. Duyley, H. Dehgani, J. Weaver, S. Poplack, B. Pogue, and K. Paulsen, “Magnetic resonance guided near infrared tomography of the breast,” Rev. Sci. Instrum. 75, 5262–5270 (2004).
[Crossref]

Kaipio, J.

J. Heino, E. Somersalo, and J. Kaipio, “Statistical compensation of geometric mismodeling in optical tomography,” Opt. Express 13, 296–308 (2005).
[Crossref] [PubMed]

J. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inverse Probl. 15, 713–729 (1999).
[Crossref]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer Verlag, 2004).

Kaipio, J. P.

D. Calvetti, J. P. Kaipio, and E. Somersalo, “Aristotelian prior boundary conditions,” Int. J. Math. Comp. Sci. 1, 63–81 (2006).

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 63, 383–405 (2006).
[Crossref]

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175–195 (2006).
[Crossref]

J. P. Kaipio and E. Somersalo, “Statistical inverse problems: discretization, model reduction and inverse crimes,” J. Comp. Appl. Math. 22, 493–504 (2006).

Katila, T.

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
[Crossref]

Kogel, C.

B. Brooksby, S. Jiang, C. Kogel, M. Duyley, H. Dehgani, J. Weaver, S. Poplack, B. Pogue, and K. Paulsen, “Magnetic resonance guided near infrared tomography of the breast,” Rev. Sci. Instrum. 75, 5262–5270 (2004).
[Crossref]

Kolehmainen, V.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 63, 383–405 (2006).
[Crossref]

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175–195 (2006).
[Crossref]

J. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inverse Probl. 15, 713–729 (1999).
[Crossref]

Kotilahti, K.

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
[Crossref]

Lipiäinen, L.

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
[Crossref]

Malik, J.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell.12, 629–639 (1990).
[Crossref]

McBride, T.

Meek, J.

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

Nissilä, I.

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
[Crossref]

Noponen, T.

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
[Crossref]

Oman, M. E.

C. R. Vogel and M. E. Oman, “Fast, robust total variation-based reconstruction of noisy, blurred images,” IEEE Trans. Image Process.7, 813–824 (1998).
[Crossref]

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

Paige, C.

C. Paige and M. Saunders, “LSQR: An Algorithm for Sparse Linear Equations And Sparse Least Squares,” ACM Trans. Math. Software 8, 43–71 (1982).
[Crossref]

Paulsen, K.

Perona, P.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell.12, 629–639 (1990).
[Crossref]

Pogue, B.

Poplack, S.

B. Brooksby, S. Jiang, C. Kogel, M. Duyley, H. Dehgani, J. Weaver, S. Poplack, B. Pogue, and K. Paulsen, “Magnetic resonance guided near infrared tomography of the breast,” Rev. Sci. Instrum. 75, 5262–5270 (2004).
[Crossref]

Prewitt, J.

Riley, J.

A. Douiri, M. Schweiger, J. Riley, and S. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18, 87–95 (2007).
[Crossref]

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

Saad, Y.

Y. Saad, Iterative Methods for Sparse Linear Systems (Society for Industrial Mathematics, 2003).
[Crossref]

Saunders, M.

C. Paige and M. Saunders, “LSQR: An Algorithm for Sparse Linear Equations And Sparse Least Squares,” ACM Trans. Math. Software 8, 43–71 (1982).
[Crossref]

Schweiger, M.

A. Douiri, M. Schweiger, J. Riley, and S. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18, 87–95 (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]

M. Schweiger, S. Arridge, M. Hiraoka, and D. 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]

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
[Crossref]

Sgallari, F.

D. Calvetti, F. Sgallari, and E. Somersalo, “Image inpainting with structural bootstrap priors,” Image and Vision Comput. 24, 782–793 (2006).
[Crossref]

Somersalo, E.

D. Calvetti and E. Somersalo, “Hypermodels in the Bayesian imaging framework,” Inverse Probl. 24, 034013 (20pp) (2008).
[Crossref]

D. Calvetti and E. Somersalo, “Gaussian hypermodels and recovery of blocky objects,” Inverse Probl. 23, 733–754 (2007).
[Crossref]

D. Calvetti and E. Somersalo, “Microlocal sequential regularization in imaging,” Inverse Problems and Imaging 1, 1–11 (2007).
[Crossref]

D. Calvetti, F. Sgallari, and E. Somersalo, “Image inpainting with structural bootstrap priors,” Image and Vision Comput. 24, 782–793 (2006).
[Crossref]

D. Calvetti, J. P. Kaipio, and E. Somersalo, “Aristotelian prior boundary conditions,” Int. J. Math. Comp. Sci. 1, 63–81 (2006).

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. P. Kaipio and E. Somersalo, “Statistical inverse problems: discretization, model reduction and inverse crimes,” J. Comp. Appl. Math. 22, 493–504 (2006).

J. Heino, E. Somersalo, and J. Kaipio, “Statistical compensation of geometric mismodeling in optical tomography,” Opt. Express 13, 296–308 (2005).
[Crossref] [PubMed]

J. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inverse Probl. 15, 713–729 (1999).
[Crossref]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer Verlag, 2004).

D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing – Ten Lectures on Subjective Computing (Springer Verlag, 2007).
[PubMed]

D. Calvetti and E. Somersalo, “Recovery of shapes: hypermodels and Bayesian learning,” Proc. of the Applied Inverse Problems 2007: Theoretical and Computational Aspects. J. of Physics Conference Series (to appear).

sterberg, U.

Tarvainen, T.

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, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 63, 383–405 (2006).
[Crossref]

Vauhkonen, M.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 63, 383–405 (2006).
[Crossref]

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175–195 (2006).
[Crossref]

J. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inverse Probl. 15, 713–729 (1999).
[Crossref]

Vogel, C. R.

C. R. Vogel and M. E. Oman, “Fast, robust total variation-based reconstruction of noisy, blurred images,” IEEE Trans. Image Process.7, 813–824 (1998).
[Crossref]

Weaver, J.

B. Brooksby, S. Jiang, C. Kogel, M. Duyley, H. Dehgani, J. Weaver, S. Poplack, B. Pogue, and K. Paulsen, “Magnetic resonance guided near infrared tomography of the breast,” Rev. Sci. Instrum. 75, 5262–5270 (2004).
[Crossref]

Wyatt, J.

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

Yalavarthy, P.

Yodh, A.

J. Culver, T. Durduran, D. Furya, C. Cheung, J. Greenberg, and A. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cerebral Blood Flow & Metabolism 23, 911–924 (2003).
[Crossref]

Yusof, R.

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

ACM Trans. Math. Software (1)

C. Paige and M. Saunders, “LSQR: An Algorithm for Sparse Linear Equations And Sparse Least Squares,” ACM Trans. Math. Software 8, 43–71 (1982).
[Crossref]

Appl. Opt. (1)

European Radiology (1)

J. Hebden and T. Austin, “Optical tomography of the neonatal brain,” European Radiology 17, 2926–2933 (2007).
[Crossref] [PubMed]

Image and Vision Comput. (1)

D. Calvetti, F. Sgallari, and E. Somersalo, “Image inpainting with structural bootstrap priors,” Image and Vision Comput. 24, 782–793 (2006).
[Crossref]

Int. J. Math. Comp. Sci. (1)

D. Calvetti, J. P. Kaipio, and E. Somersalo, “Aristotelian prior boundary conditions,” Int. J. Math. Comp. Sci. 1, 63–81 (2006).

Int. J. Numer. Meth. Eng. (1)

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Meth. Eng. 63, 383–405 (2006).
[Crossref]

Inverse Probl. (5)

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]

D. Calvetti and E. Somersalo, “Gaussian hypermodels and recovery of blocky objects,” Inverse Probl. 23, 733–754 (2007).
[Crossref]

D. Calvetti and E. Somersalo, “Hypermodels in the Bayesian imaging framework,” Inverse Probl. 24, 034013 (20pp) (2008).
[Crossref]

S. R. Arridge, “Optical Tomography in Medical Imaging,” Inverse Probl. 15, R41–R93 (1999).
[Crossref]

J. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inverse Probl. 15, 713–729 (1999).
[Crossref]

Inverse Problems and Imaging (1)

D. Calvetti and E. Somersalo, “Microlocal sequential regularization in imaging,” Inverse Problems and Imaging 1, 1–11 (2007).
[Crossref]

Iterative Methods for Sparse Linear Systems (1)

Y. Saad, Iterative Methods for Sparse Linear Systems (Society for Industrial Mathematics, 2003).
[Crossref]

J. Cerebral Blood Flow & Metabolism (1)

J. Culver, T. Durduran, D. Furya, C. Cheung, J. Greenberg, and A. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cerebral Blood Flow & Metabolism 23, 911–924 (2003).
[Crossref]

J. Comp. Appl. Math. (1)

J. P. Kaipio and E. Somersalo, “Statistical inverse problems: discretization, model reduction and inverse crimes,” J. Comp. Appl. Math. 22, 493–504 (2006).

Meas. Sci. Technol. (1)

A. Douiri, M. Schweiger, J. Riley, and S. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18, 87–95 (2007).
[Crossref]

Med. Phys. (1)

M. Schweiger, S. Arridge, M. Hiraoka, and D. 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. Express (2)

Phys. Med. Biol. (1)

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

Physica D (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

Rev. Sci. Instrum. (1)

B. Brooksby, S. Jiang, C. Kogel, M. Duyley, H. Dehgani, J. Weaver, S. Poplack, B. Pogue, and K. Paulsen, “Magnetic resonance guided near infrared tomography of the breast,” Rev. Sci. Instrum. 75, 5262–5270 (2004).
[Crossref]

Other (7)

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer Verlag, 2004).

C. R. Vogel and M. E. Oman, “Fast, robust total variation-based reconstruction of noisy, blurred images,” IEEE Trans. Image Process.7, 813–824 (1998).
[Crossref]

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell.12, 629–639 (1990).
[Crossref]

D. Calvetti and E. Somersalo, “Recovery of shapes: hypermodels and Bayesian learning,” Proc. of the Applied Inverse Problems 2007: Theoretical and Computational Aspects. J. of Physics Conference Series (to appear).

“Automatically Tuned Linear Algebra Software (ATLAS),” http://math-atlas.sourceforge.net/ (19th June 2008).

D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing – Ten Lectures on Subjective Computing (Springer Verlag, 2007).
[PubMed]

I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography.” J. Biomed Opt.11, 064015 (2006).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Solid lines presents the triangular mesh. Dashed line is a voronoi cell Ωi connected to node xi.

Fig. 2
Fig. 2

Images of the absorption coefficient μa (top) and diffusion coefficient κ (bottom) of the two dimensional simulated targets. The left hand side images correspond to the blocky inclusion model and the right hand side images to the smooth inclusion model. The parameter values are as in Table 1.

Fig. 3
Fig. 3

Reconstructions of optical parameters μa (a) and κ (b) from simulated data from the blocky inclusions model. The reconstructions on the left of each panel are those obtained after the first Gauss-Newton iteration, thus corresponding to a homogenous smoothness penalty without adaptation, while on the right are those obtained after 20 iteration sweeps. Panels (c) and (d) show the respective profiles along the circular contour indicated in Fig. 2, traversed counterclockwise. The solid line is the true target, the dotted line the solution after the first iteration and the dashed line the final reconstruction.

Fig. 4
Fig. 4

Reconstructions of optical parameters μa (a) and κ (b) from simulated data from the smooth inclusions model. The reconstructions on the left of each panel are those obtained after the first Gauss-Newton iteration, thus corresponding to a homogenous smoothness penalty without adaptation, while on the right are those obtained after 20 iteration sweeps. Panels (c) and (d) show the respective profiles along the circular contour indicated in Fig. 2, traversed counterclockwise. The solid line is the true target, the dotted line the solution after the first iteration and the dashed line the final reconstruction.

Fig. 5
Fig. 5

The geometry of the phantom with the diameter of 69.25 mm and height of 110 mm. The 16 sources (marked with ×) and 16 detectors (marked with ○) are located in two rows on the cylindrical surface. A two dimensional slice of the reconstructions are shown along the plane of intersection at z = 0 (red). The locations of the inclusions are projected on the bottom.

Fig. 6
Fig. 6

Histograms of the logarithm of the amplitude (left) the phase (right) from a repeated measurement with one source/detector pair. The red line shows a Gaussian with the same mean and variance as the sample. The values of the abscissae are arbitrary, as the data are not calibrated.

Fig. 7
Fig. 7

Cross sections of the reconstructed absorption (top) and scattering (bottom) coefficient through the plane z=0 (see Fig. 5). The reconstructions on the left are before adaptation, corresponding to a homogenous smoothness penalty, while those on the right are after eight adaptation sweeps.

Tables (1)

Tables Icon

Table 1 Optical parameters of the simulated object.The numbering of the inclusion is in agreement with Fig. 2.

Equations (61)

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

( κ ( x ) ϕ ( x ) ) + ( μ a ( x ) + i ω c ) ϕ ( x ) = 0 , x Ω ,
ϕ ( x ) + 2 ζ κ ( x ) ϕ n ( x ) = q ( x ) , x Ω ,
ζ = 1 + R 1 R , R 1.4399 n 2 + 0.7099 n 1 + 0.6681 + 0.0636 n ,
Γ ( x ) = Γ q ( x ) = κ ( x ) ϕ n ( x ) x Ω .
F ( μ a , κ ; q ) = [ Relog ( Γ q ( x ) ) Imlog ( Γ q ( x ) ) ] = [ log | Γ q ( x ) | arg Γ q ( x ) ] .
D = { ( log | Γ ( r j ) | , arg ( Γ ( r j ) ) ) | 1 L , 1 j m } .
y = f ( p ) , f : 2 N 2 M , M = L m .
y = f ( p ) + e , e π noise ( e ) ,
π ( y | p ) π noise ( y f ( p ) ) ,
π ( p , y ) = π ( y | p ) π prior ( p ) π noise ( y f ( p ) ) π prior ( p ) .
π post ( p ) = π ( p | y ) = π ( p , y ) π ( y ) π noise ( y f ( p ) ) π prior ( p ) , y = y measured .
π noise ( e ) exp ( 1 2 ( e e * ) T C n 1 ( e e * ) ) , π prior ( p ) exp ( 1 2 ( p p * ) T C p 1 ( p p * ) ) ,
π post ( p ) exp ( 1 2 ( y e * f ( p ) ) T C n 1 ( y e * f ( p ) ) 1 2 ( p p * ) T C p 1 ( p p * ) ) .
C n 1 = S T S , C p 1 = L T L ,
π post ( p ) exp ( 1 2 S ( y e * f ( p ) ) 2 1 2 L ( p p * ) 2 ) ,
p MAP = argmin ( S ( y e * f ( p ) ) 2 + L ( p p * ) 2 ) ,
J ( p ) = L ( p p * ) 2
𝒥 int ( u ; λ ) = Ω ( ( λ u ) ) 2 d x .
Ω i = { x Ω | | x x i | < | x x j | , i j } ,
( λ ( x i ) u ( x i ) ) 1 | Ω i | Ω i ( λ u ) d x = 1 | Ω i | Ω i λ n u d S ,
1 | Ω i | Ω i λ n u d S j = 1 p | Ω i j | | Ω i | λ ( y i j ) ( u ( x i j ) u ( x i ) ) .
( λ ( x i ) u ( x i ) ) ( L int p ) i , L int N int × N ,
J int ( p ; λ ) = L int p 2 .
π ( p int | p bdry ) exp ( 1 2 L int p 2 ) .
𝒥 bdry ( u ) = Ω ( Δ τ u ) 2 d S ,
Δ τ u ( x i ) 1 | Ω i Ω | Ω i Ω Δ τ u d S = 1 | Ω i Ω | C i ν u d ,
J bdry ( p bdry ) = L bdry p bdry 2 , L bdry N bdry × N bdry ,
π ( p bdry ) exp ( 1 2 α L bdry p 2 ) ,
π prior ( p ) = π ( p int | p bdry ) π ( p bdry ) .
π prior ( p ) exp ( 1 2 [ L int L int ] [ p int p brdy ] 2 1 2 α L bdry p 2 ) = exp ( 1 2 L p 2 ) ,
L = [ L int L int 0 α L bdry ] .
var ( p i ) = ( L 1 L T ) i i = L T e i 2 ,
λ ( y i j ) = 1 1 + | τ D v i j u ^ ( y i j ) | k ,
v i j = 1 h i j ( x i j x i ) , h i j = | x i j x i | .
W = W κ ^ , μ ^ a = [ γ 1 L κ ^ γ 2 L μ ^ a ] 2 N × 2 N ,
p = argmin p 2 N { S ( y e * f ( p ) ) 2 + δ W ( p p * ) 2 } , δ > 0.
κ ^ = 1 max   κ ( x i ) κ , μ ^ a = 1 max μ a ( x i ) μ a .
κ ^ ( y i j ) = 1 2 ( κ ^ ( x i ) + κ ^ ( x i j ) ) , μ ^ a ( y i j ) = 1 2 ( μ ^ a ( x i ) + μ ^ a ( x i j ) ) ,
S = [ σ log | Γ | 1 I 0 0 σ φ 1 I ] .
y y nh y h + f ( p h ) ,
y = f ( p ) + e , e 𝒩 ( 0 , ( diag ( y ) ) 2 ) .
S = diag ( 1 / | y 1 | , 1 / | y 2 | , , 1 / | y 2 M | ) .
C μ a = μ a ¯ inside inclusion μ a ¯ outside inclusion ,
C = [ 1 1 1 1 1 ] N bdry × ( N bdry 1 ) ,
p bdry = C z + μ 1 , z N bdry 1 .
L p = [ L int ' L int α L bdry ] [ p int C z + μ 1 ] = [ L int ' L int C α L bdry C ] [ p int z ] + [ μ L int 1 0 ] = K α p + r ,
K α = [ L int ' L int C α L bdry C ] = [ I α I ] [ L int ' L int C α L bdry C ] = [ I α I ] K 1 .
π ( p | μ = 0 ) exp ( 1 2 L p 2 ) = exp ( 1 2 K α p 2 ) ,
p j = e j T p int = [ e j 0 ] T p ,
var ( p j ) = E { p j 2 } = [ e j 0 ] T E { p ( p ) T } [ e j 0 ] = [ e j 0 ] T ( K α T K α ) 1 [ e j 0 ] = v 2 ,
K α T v = [ e j 0 ] or v = ( K α T ) + [ e j 0 ] ,
K α K α T v = K α [ e j 0 ] ,
K 1 K 1 T [ 1 α 1 ] v = K 1 [ e j 0 ] ,
v = [ 1 α 1 1 ] ( K 1 T ) + [ e j 0 ] = [ v 1 α 1 v 2 ] ,
var ( p j ) = v 1 2 + 1 α 2 v 2 2 .
p k = n k T ( C z ) = [ 0 C T n k ] T p ,
var ( p k ) = u 2 ,
K α T u = [ 0 C T n k ] ,
u = [ 1 α 1 1 ] ( K 1 T ) + [ 0 C T n k ] = [ u 1 α 1 u 2 ] ,
var ( p k ) = u 1 2 + 1 α 2 u 2 2 .
α 2 = u 2 2 v 2 2 u 1 2 v 1 2 .

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