Abstract

Image reconstruction in diffuse optical tomography (DOT) is, in general, posed as a model-based, nonlinear optimization problem, which requires repeated use of the three-dimensional (3D) forward and inverse solvers. To cope with the computation and storage problem for some applications, such as breast tumor diagnosis, it is preferable to develop a subdomain-based parallel computation scheme. In this study, we propose a two-level image reconstruction scheme for 3D DOT, which combines the Schwarz-type domain-decomposition (DD)-based forward calculation and the matrix-decomposition (MD)-based inversion. In the forward calculation, the solution to the diffusion equation is initially obtained using a whole-domain finite difference method at a coarse grid, and then updated with a parallel DD scheme at a fine grid. The inversion procedure starts with the wavelet-decomposition-based reconstruction at a coarse grid, and then follows with a Levenberg–Marquardt least-squares solution at a fine grid, where an MD strategy is adopted for the relevant linear inversion. It is demonstrated that the combination of the DD-based forward solver and MD-based inversion allows for coarse-grain parallel implementation of both the forward and inverse issues and effectively reduces computation and storage loads for the large-scale problem. Also, both numerical simulations and phantom experiments show that MD-based linear inversion is superior to the row-fashioned algebraic reconstruction technique.

© 2010 Optical Society of America

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  1. F. Gao, Y. Xue, and H. Zhao, “Two-dimensional optical tomography of hemodynamic changes in a preterm infant brain,” Chin. Opt. Lett. 5, 472–474 (2007).
  2. G. Strangman, D. Boas, and J. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry 52, 679–693 (2002).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. W. Mo, T. Chan, L. Chen, and N. Chen, “Quantitative characterization of optical and physiological parameters in normal breasts using time-resolved spectroscopy: in vivo results of 19 Singapore women,” J. Biomed. Opt. 14, 064004 (2009).
    [CrossRef]
  7. X. Intes, J. Ripoll, and Y. Chen, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30, 1039–1047 (2003).
    [CrossRef] [PubMed]
  8. B. Kanmani, “Diffuse optical tomography through solving a system of quadratic equations: theory and simulations,” Phys. Med. Biol. 51, 981–998 (2006).
    [CrossRef] [PubMed]
  9. S. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  10. H. A. Schwarz, “über einen Grenz bergang durch alternirendes Verfahren,” Ges. Math. Abhandlungen 1, 133–143(1870).
  11. M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Servick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imaging 20, 147–163 (2001).
    [CrossRef] [PubMed]
  12. I.-Y. Son and X. Intes, “A 2-level domain decomposition algorithm for inverse diffuse optical tomography,” in Proceedings of the International Conference on Image Processing (IEEE, 2004), pp. 3315–3318.
  13. K. Kwon, I. Son, and B. Yazici, “Two-level domain decomposition algorithm for a nonlinear inverse DOT problem,” Proc. SPIE 5693, 459–468 (2005).
    [CrossRef]
  14. K. Kwon, B. Yazici, and M. Guven, “Two-level domain decomposition methods for diffuse tomography,” Inverse Probl. 22, 1533–1559 (2006).
    [CrossRef]
  15. X. Gu and A. Hielscher, “Parallelization of transport-theory based optical tomography algorithms by domain decomposition,” in Biomedical Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), p. BSuE40.
  16. U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid(Academic, 2001).
  17. W. L. Briggs, A Multigrid Tutorial (SIAM, 1987).
  18. W. Hackbusch, Multi-Grid Methods and Applications(Springer-Verlag, 1985).
  19. F. Gao, H. Zhao, Y. Yanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49, 1055–1078 (2004).
    [CrossRef] [PubMed]
  20. F. Gao, H. Zhao, and Y. Tanikawa, “Time-resolved diffuse optical tomography using a modified generalized spectrum technique,” IEICE Trans. Inf. & Syst. E85-D, 133–142 (2002).
  21. J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 1999).
    [CrossRef]
  22. S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674–693 (1989).
    [CrossRef]
  23. W. Zhu and Y. Wang, “Total least-squares reconstruction with wavelets for optical tomography,” J. Opt. Soc. Am. A 15, 2639–2650 (1998).
    [CrossRef]
  24. W. Zhu and Y. Wang, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217(1997).
    [CrossRef] [PubMed]
  25. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, 1998).
    [CrossRef]
  26. A. H. Hielscher and S. Bartel, “Parallel programming of gradient-based iterative image reconstruction schemes for optical tomography,” Comput. Methods Programs Biomed. 73, 101–113 (2004).
    [CrossRef] [PubMed]
  27. F. Gao, H. Zhao, Y. Tanikawa, K. Homma, and Y. Yamada, “Influences of target size and contrast on near infrared diffuse optical tomography-a comparison between featured-data and full time-resolved schemes,” Opt. Quantum Electron. 37, 1287–1304 (2005).
    [CrossRef]

2009 (1)

W. Mo, T. Chan, L. Chen, and N. Chen, “Quantitative characterization of optical and physiological parameters in normal breasts using time-resolved spectroscopy: in vivo results of 19 Singapore women,” J. Biomed. Opt. 14, 064004 (2009).
[CrossRef]

2007 (2)

2006 (2)

B. Kanmani, “Diffuse optical tomography through solving a system of quadratic equations: theory and simulations,” Phys. Med. Biol. 51, 981–998 (2006).
[CrossRef] [PubMed]

K. Kwon, B. Yazici, and M. Guven, “Two-level domain decomposition methods for diffuse tomography,” Inverse Probl. 22, 1533–1559 (2006).
[CrossRef]

2005 (2)

K. Kwon, I. Son, and B. Yazici, “Two-level domain decomposition algorithm for a nonlinear inverse DOT problem,” Proc. SPIE 5693, 459–468 (2005).
[CrossRef]

F. Gao, H. Zhao, Y. Tanikawa, K. Homma, and Y. Yamada, “Influences of target size and contrast on near infrared diffuse optical tomography-a comparison between featured-data and full time-resolved schemes,” Opt. Quantum Electron. 37, 1287–1304 (2005).
[CrossRef]

2004 (2)

A. H. Hielscher and S. Bartel, “Parallel programming of gradient-based iterative image reconstruction schemes for optical tomography,” Comput. Methods Programs Biomed. 73, 101–113 (2004).
[CrossRef] [PubMed]

F. Gao, H. Zhao, Y. Yanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49, 1055–1078 (2004).
[CrossRef] [PubMed]

2003 (2)

X. Intes, J. Ripoll, and Y. Chen, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30, 1039–1047 (2003).
[CrossRef] [PubMed]

D. Grosenick, T. Moesta, and H. Wabnitz, “Time-domain optical mammography: initial clinical results on detection and characterization of breast tumor,” Appl. Opt. 42, 3170–3186(2003).
[CrossRef] [PubMed]

2002 (2)

G. Strangman, D. Boas, and J. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry 52, 679–693 (2002).
[CrossRef] [PubMed]

F. Gao, H. Zhao, and Y. Tanikawa, “Time-resolved diffuse optical tomography using a modified generalized spectrum technique,” IEICE Trans. Inf. & Syst. E85-D, 133–142 (2002).

2001 (1)

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Servick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imaging 20, 147–163 (2001).
[CrossRef] [PubMed]

1999 (1)

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

1998 (1)

1997 (2)

A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. 20, 435–442 (1997).
[CrossRef] [PubMed]

W. Zhu and Y. Wang, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217(1997).
[CrossRef] [PubMed]

1989 (1)

S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674–693 (1989).
[CrossRef]

1870 (1)

H. A. Schwarz, “über einen Grenz bergang durch alternirendes Verfahren,” Ges. Math. Abhandlungen 1, 133–143(1870).

Arridge, S.

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

Bartel, S.

A. H. Hielscher and S. Bartel, “Parallel programming of gradient-based iterative image reconstruction schemes for optical tomography,” Comput. Methods Programs Biomed. 73, 101–113 (2004).
[CrossRef] [PubMed]

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, 1998).
[CrossRef]

Boas, D.

G. Strangman, D. Boas, and J. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry 52, 679–693 (2002).
[CrossRef] [PubMed]

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, 1998).
[CrossRef]

Briggs, W. L.

W. L. Briggs, A Multigrid Tutorial (SIAM, 1987).

Chan, T.

W. Mo, T. Chan, L. Chen, and N. Chen, “Quantitative characterization of optical and physiological parameters in normal breasts using time-resolved spectroscopy: in vivo results of 19 Singapore women,” J. Biomed. Opt. 14, 064004 (2009).
[CrossRef]

Chance, B.

A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. 20, 435–442 (1997).
[CrossRef] [PubMed]

Chen, L.

W. Mo, T. Chan, L. Chen, and N. Chen, “Quantitative characterization of optical and physiological parameters in normal breasts using time-resolved spectroscopy: in vivo results of 19 Singapore women,” J. Biomed. Opt. 14, 064004 (2009).
[CrossRef]

Chen, N.

W. Mo, T. Chan, L. Chen, and N. Chen, “Quantitative characterization of optical and physiological parameters in normal breasts using time-resolved spectroscopy: in vivo results of 19 Singapore women,” J. Biomed. Opt. 14, 064004 (2009).
[CrossRef]

Chen, Y.

X. Intes, J. Ripoll, and Y. Chen, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30, 1039–1047 (2003).
[CrossRef] [PubMed]

Dougherty, D. E.

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Servick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imaging 20, 147–163 (2001).
[CrossRef] [PubMed]

Eppstein, M. J.

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Servick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imaging 20, 147–163 (2001).
[CrossRef] [PubMed]

Gao, F.

F. Gao, Y. Xue, and H. Zhao, “Two-dimensional optical tomography of hemodynamic changes in a preterm infant brain,” Chin. Opt. Lett. 5, 472–474 (2007).

F. Gao, H. Zhao, Y. Tanikawa, K. Homma, and Y. Yamada, “Influences of target size and contrast on near infrared diffuse optical tomography-a comparison between featured-data and full time-resolved schemes,” Opt. Quantum Electron. 37, 1287–1304 (2005).
[CrossRef]

F. Gao, H. Zhao, Y. Yanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49, 1055–1078 (2004).
[CrossRef] [PubMed]

F. Gao, H. Zhao, and Y. Tanikawa, “Time-resolved diffuse optical tomography using a modified generalized spectrum technique,” IEICE Trans. Inf. & Syst. E85-D, 133–142 (2002).

Grosenick, D.

Gu, X.

X. Gu and A. Hielscher, “Parallelization of transport-theory based optical tomography algorithms by domain decomposition,” in Biomedical Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), p. BSuE40.

Guven, M.

K. Kwon, B. Yazici, and M. Guven, “Two-level domain decomposition methods for diffuse tomography,” Inverse Probl. 22, 1533–1559 (2006).
[CrossRef]

Hackbusch, W.

W. Hackbusch, Multi-Grid Methods and Applications(Springer-Verlag, 1985).

Hawrysz, D. J.

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Servick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imaging 20, 147–163 (2001).
[CrossRef] [PubMed]

Hielscher, A.

X. Gu and A. Hielscher, “Parallelization of transport-theory based optical tomography algorithms by domain decomposition,” in Biomedical Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), p. BSuE40.

Hielscher, A. H.

A. H. Hielscher and S. Bartel, “Parallel programming of gradient-based iterative image reconstruction schemes for optical tomography,” Comput. Methods Programs Biomed. 73, 101–113 (2004).
[CrossRef] [PubMed]

Homma, K.

F. Gao, H. Zhao, Y. Tanikawa, K. Homma, and Y. Yamada, “Influences of target size and contrast on near infrared diffuse optical tomography-a comparison between featured-data and full time-resolved schemes,” Opt. Quantum Electron. 37, 1287–1304 (2005).
[CrossRef]

Intes, X.

X. Intes, J. Ripoll, and Y. Chen, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30, 1039–1047 (2003).
[CrossRef] [PubMed]

I.-Y. Son and X. Intes, “A 2-level domain decomposition algorithm for inverse diffuse optical tomography,” in Proceedings of the International Conference on Image Processing (IEEE, 2004), pp. 3315–3318.

Jiang, H.

Kanmani, B.

B. Kanmani, “Diffuse optical tomography through solving a system of quadratic equations: theory and simulations,” Phys. Med. Biol. 51, 981–998 (2006).
[CrossRef] [PubMed]

Kwon, K.

K. Kwon, B. Yazici, and M. Guven, “Two-level domain decomposition methods for diffuse tomography,” Inverse Probl. 22, 1533–1559 (2006).
[CrossRef]

K. Kwon, I. Son, and B. Yazici, “Two-level domain decomposition algorithm for a nonlinear inverse DOT problem,” Proc. SPIE 5693, 459–468 (2005).
[CrossRef]

Mallat, S. G.

S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674–693 (1989).
[CrossRef]

Mo, W.

W. Mo, T. Chan, L. Chen, and N. Chen, “Quantitative characterization of optical and physiological parameters in normal breasts using time-resolved spectroscopy: in vivo results of 19 Singapore women,” J. Biomed. Opt. 14, 064004 (2009).
[CrossRef]

Moesta, T.

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 1999).
[CrossRef]

Oosterlee, C.

U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid(Academic, 2001).

Ripoll, J.

X. Intes, J. Ripoll, and Y. Chen, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30, 1039–1047 (2003).
[CrossRef] [PubMed]

Schüller, A.

U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid(Academic, 2001).

Schwarz, H. A.

H. A. Schwarz, “über einen Grenz bergang durch alternirendes Verfahren,” Ges. Math. Abhandlungen 1, 133–143(1870).

Servick-Muraca, E. M.

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Servick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imaging 20, 147–163 (2001).
[CrossRef] [PubMed]

Son, I.

K. Kwon, I. Son, and B. Yazici, “Two-level domain decomposition algorithm for a nonlinear inverse DOT problem,” Proc. SPIE 5693, 459–468 (2005).
[CrossRef]

Son, I.-Y.

I.-Y. Son and X. Intes, “A 2-level domain decomposition algorithm for inverse diffuse optical tomography,” in Proceedings of the International Conference on Image Processing (IEEE, 2004), pp. 3315–3318.

Strangman, G.

G. Strangman, D. Boas, and J. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry 52, 679–693 (2002).
[CrossRef] [PubMed]

Sutton, J.

G. Strangman, D. Boas, and J. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry 52, 679–693 (2002).
[CrossRef] [PubMed]

Tanikawa, Y.

F. Gao, H. Zhao, Y. Tanikawa, K. Homma, and Y. Yamada, “Influences of target size and contrast on near infrared diffuse optical tomography-a comparison between featured-data and full time-resolved schemes,” Opt. Quantum Electron. 37, 1287–1304 (2005).
[CrossRef]

F. Gao, H. Zhao, and Y. Tanikawa, “Time-resolved diffuse optical tomography using a modified generalized spectrum technique,” IEICE Trans. Inf. & Syst. E85-D, 133–142 (2002).

Trottenberg, U.

U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid(Academic, 2001).

Villringer, A.

A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. 20, 435–442 (1997).
[CrossRef] [PubMed]

Wabnitz, H.

Wang, Y.

W. Zhu and Y. Wang, “Total least-squares reconstruction with wavelets for optical tomography,” J. Opt. Soc. Am. A 15, 2639–2650 (1998).
[CrossRef]

W. Zhu and Y. Wang, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217(1997).
[CrossRef] [PubMed]

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 1999).
[CrossRef]

Xue, Y.

Yamada, Y.

F. Gao, H. Zhao, Y. Tanikawa, K. Homma, and Y. Yamada, “Influences of target size and contrast on near infrared diffuse optical tomography-a comparison between featured-data and full time-resolved schemes,” Opt. Quantum Electron. 37, 1287–1304 (2005).
[CrossRef]

F. Gao, H. Zhao, Y. Yanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49, 1055–1078 (2004).
[CrossRef] [PubMed]

Yanikawa, Y.

F. Gao, H. Zhao, Y. Yanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49, 1055–1078 (2004).
[CrossRef] [PubMed]

Yazici, B.

K. Kwon, B. Yazici, and M. Guven, “Two-level domain decomposition methods for diffuse tomography,” Inverse Probl. 22, 1533–1559 (2006).
[CrossRef]

K. Kwon, I. Son, and B. Yazici, “Two-level domain decomposition algorithm for a nonlinear inverse DOT problem,” Proc. SPIE 5693, 459–468 (2005).
[CrossRef]

Yuan, Z.

Zhao, H.

F. Gao, Y. Xue, and H. Zhao, “Two-dimensional optical tomography of hemodynamic changes in a preterm infant brain,” Chin. Opt. Lett. 5, 472–474 (2007).

F. Gao, H. Zhao, Y. Tanikawa, K. Homma, and Y. Yamada, “Influences of target size and contrast on near infrared diffuse optical tomography-a comparison between featured-data and full time-resolved schemes,” Opt. Quantum Electron. 37, 1287–1304 (2005).
[CrossRef]

F. Gao, H. Zhao, Y. Yanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49, 1055–1078 (2004).
[CrossRef] [PubMed]

F. Gao, H. Zhao, and Y. Tanikawa, “Time-resolved diffuse optical tomography using a modified generalized spectrum technique,” IEICE Trans. Inf. & Syst. E85-D, 133–142 (2002).

Zhu, W.

W. Zhu and Y. Wang, “Total least-squares reconstruction with wavelets for optical tomography,” J. Opt. Soc. Am. A 15, 2639–2650 (1998).
[CrossRef]

W. Zhu and Y. Wang, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217(1997).
[CrossRef] [PubMed]

Appl. Opt. (2)

Biol. Psychiatry (1)

G. Strangman, D. Boas, and J. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry 52, 679–693 (2002).
[CrossRef] [PubMed]

Chin. Opt. Lett. (1)

Comput. Methods Programs Biomed. (1)

A. H. Hielscher and S. Bartel, “Parallel programming of gradient-based iterative image reconstruction schemes for optical tomography,” Comput. Methods Programs Biomed. 73, 101–113 (2004).
[CrossRef] [PubMed]

Ges. Math. Abhandlungen (1)

H. A. Schwarz, “über einen Grenz bergang durch alternirendes Verfahren,” Ges. Math. Abhandlungen 1, 133–143(1870).

IEEE Trans. Med. Imaging (2)

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Servick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imaging 20, 147–163 (2001).
[CrossRef] [PubMed]

W. Zhu and Y. Wang, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16, 210–217(1997).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Machine Intell. (1)

S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674–693 (1989).
[CrossRef]

IEICE Trans. Inf. & Syst. (1)

F. Gao, H. Zhao, and Y. Tanikawa, “Time-resolved diffuse optical tomography using a modified generalized spectrum technique,” IEICE Trans. Inf. & Syst. E85-D, 133–142 (2002).

Inverse Probl. (2)

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

K. Kwon, B. Yazici, and M. Guven, “Two-level domain decomposition methods for diffuse tomography,” Inverse Probl. 22, 1533–1559 (2006).
[CrossRef]

J. Biomed. Opt. (1)

W. Mo, T. Chan, L. Chen, and N. Chen, “Quantitative characterization of optical and physiological parameters in normal breasts using time-resolved spectroscopy: in vivo results of 19 Singapore women,” J. Biomed. Opt. 14, 064004 (2009).
[CrossRef]

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

Med. Phys. (1)

X. Intes, J. Ripoll, and Y. Chen, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30, 1039–1047 (2003).
[CrossRef] [PubMed]

Opt. Quantum Electron. (1)

F. Gao, H. Zhao, Y. Tanikawa, K. Homma, and Y. Yamada, “Influences of target size and contrast on near infrared diffuse optical tomography-a comparison between featured-data and full time-resolved schemes,” Opt. Quantum Electron. 37, 1287–1304 (2005).
[CrossRef]

Phys. Med. Biol. (2)

B. Kanmani, “Diffuse optical tomography through solving a system of quadratic equations: theory and simulations,” Phys. Med. Biol. 51, 981–998 (2006).
[CrossRef] [PubMed]

F. Gao, H. Zhao, Y. Yanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49, 1055–1078 (2004).
[CrossRef] [PubMed]

Proc. SPIE (1)

K. Kwon, I. Son, and B. Yazici, “Two-level domain decomposition algorithm for a nonlinear inverse DOT problem,” Proc. SPIE 5693, 459–468 (2005).
[CrossRef]

Trends Neurosci. (1)

A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. 20, 435–442 (1997).
[CrossRef] [PubMed]

Other (7)

J. Nocedal and S. J. Wright, Numerical Optimization(Springer, 1999).
[CrossRef]

I.-Y. Son and X. Intes, “A 2-level domain decomposition algorithm for inverse diffuse optical tomography,” in Proceedings of the International Conference on Image Processing (IEEE, 2004), pp. 3315–3318.

X. Gu and A. Hielscher, “Parallelization of transport-theory based optical tomography algorithms by domain decomposition,” in Biomedical Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), p. BSuE40.

U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid(Academic, 2001).

W. L. Briggs, A Multigrid Tutorial (SIAM, 1987).

W. Hackbusch, Multi-Grid Methods and Applications(Springer-Verlag, 1985).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, 1998).
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Overlapping DD of 3D slab model.

Fig. 2
Fig. 2

Coarse-grid correction of the two-level DD for the DOT forward problem.

Fig. 3
Fig. 3

Wavelet coarse-grid correction of the two- level MD for the DOT inverse problem.

Fig. 4
Fig. 4

Fine-grid correction of the two-level MD for the DOT inverse problem.

Fig. 5
Fig. 5

(a) L 2 error versus number of iterations with the overlapping region width of ω = 0 , 1, 2, 3, respectively, for the two-level DD forward calculation in the iterative form of Gauss–Seidel. (b) Execution time of each case with 11 iterations; (c), (d) execution time of cases with ω = 1 , 2, 3, by which the L 2 ( Ω ) error of each case is down to 10 5 and 10 8 , respectively.

Fig. 6
Fig. 6

L 2 error versus number of iterations with ω = 1 for two-level DD forward calculation in the iteration formats of Jacobi and Gauss–Seidel, respectively.

Fig. 7
Fig. 7

(a) 3D representation of the modeled medium where a couple of 10 mm × 10 mm × 10 mm cubes are located within. (b) Sources and detectors are arranged in the same way on the surface plane of z = 0 mm and z = 32 mm .

Fig. 8
Fig. 8

Reconstructed (a) μ a images and (b) μ s images of the slab test object using ART algorithm at a noise level of SNR = 35 dB with 50 iterations.

Fig. 9
Fig. 9

Reconstructed (a) μ a images and (b) μ s images of the slab test object using a wavelet-based MD algorithm at a noise level of SNR = 35 dB with five iterations in the coarse grid and an additional 20 iterations in the fine grid.

Fig. 10
Fig. 10

Profiles of the reconstructed (a) μ a images and (b) μ s images along the y axis at the plane z = 16 mm .

Fig. 11
Fig. 11

(a) Solid slab phantom and (b) its geometry sketch.

Fig. 12
Fig. 12

Multichannel time-correlated single-photon counting system.

Fig. 13
Fig. 13

Normalized temporal profile spread functions (TPSFs) measured by the 16 detectors at a wavelength of 830 nm on the solid phantom with a homogeneous background (rod 2 and rod 3 are inserted, blue lines) for the seventh source and the profiles measured by same detector arrays on the solid phantom with a heterogeneous background (rods 8 and 9 are inserted, red lines).

Fig. 14
Fig. 14

Absorption (left column) and scattering (right column) images reconstructed from the measured data by ART: (a), (b) the vertical slice at z = 25 mm ; (c), (d) the horizontal slice at y = 55 mm ; and (e), (f) the horizontal slice at y = 35 mm (units: mm 1 ). The rectangular and circular lines indicate the original locations and sizes of the targets.

Fig. 15
Fig. 15

Absorption (left column) and scattering (right column) images reconstructed from the measured data by the MD method: (a), (b) the vertical slice at z = 25 mm ; (c), (d) the horizontal slice at y = 55 mm ; and (e), (f) the horizontal slice at y = 35 mm (units: mm 1 ). The rectangular and circular lines indicate the original locations and sizes of the targets.

Tables (5)

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Table 1 Notation Used in Domain-Decomposition-Based Forward Calculation

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Table 2 Execution Time of Forward Problem Using Non-Domain-Decomposition and Domain-Decomposition-Based Parallel-Computing Methods, Respectively

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Table 3 Optical Properties and Geometry Scale of Slab Test Object

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Table 4 Average Value of Recovered Absorption and Scattering Coefficients ( mm 1 ) of Targets and Target Size (mm) for Algebraic Reconstruction Technique and Matrix-Decomposition Method

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Table 5 Optical Properties of Solid Phantom Measured at 830 nm and Its Geometry Scale

Equations (30)

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{ [ · κ ( r ) μ a ( r ) c p ] Φ ( r , r s , p ) = δ ( r r s ) c Φ ( r , r s , p ) = κ α n ^ · Φ ( r , r s , p ) r Ω ,
Ψ ( ξ d , ζ s , p ) = κ ( r ) n ^ · Φ ( r , ζ s , p ) r = ξ d .
χ d , s ( p 1 , p 2 ) = Ψ ( ξ d , ζ s , p 2 ) Ψ ( ξ d , ζ s , p 1 ) .
μ = def ( μ a μ s ) , μ 2 N ,
μ ^ = argmin μ 1 2 s = 1 s d = 1 D [ χ d , s ( p 1 , p 2 ) F d , s ( μ ; p 1 p 2 ) ] 2 ,
{ μ ( k ) = ( ( J ( k ) ) T J ( k ) + λ I ) 1 ( J ( k ) ) T b ( k ) μ ( k + 1 ) = μ ( k ) + δ μ ( k ) ,
J ( k ) = F ( μ ( k ) ; p 1 , p 2 ) μ ( k ) .
J ^ ( k ) δ μ ( k ) = B ( k ) ,
K Φ = q ,
Φ 2 h = ( K 2 h ) 1 q 2 h .
Φ ( 0 ) = P 2 h h ( Φ 2 h ) ,
r e s h = q h K h Φ ( i 1 ) .
r e s 2 h = h 2 h ( r e s h ) ,
K 2 h e 2 h = r e s 2 h .
Φ ( i , 0 ) = Φ ( i 1 ) + P 2 h h ( e 2 h ) .
{ [ · κ μ a c p ] Φ ( i , m ) = q m , in Ω m c Φ ( i , m ) + κ α · Φ ( i , m ) = 0 , on Ω m Ω Φ ( i , m ) = φ , otherwise ,
φ = { Φ ( i , 0 ) in the Jacobi format Φ ( i , m 1 ) in the Gauss–Seidel format .
A = [ 0 K 1 1 K 2 K 1 1 K M K 2 1 K 1 0 K 2 1 K M K M 1 K 1 K M 1 K 2 0 ] ,
Q = [ K 1 1 q 1 K m 1 q m K M 1 q M ] ,
{ Φ ( i , m ) = A Φ ( i , 0 ) + Q Φ ( i ) = Φ ( i , M ) .
{ Φ ( i , m ) = L Φ ( i , m 1 ) + U Φ ( i , 0 ) + Q Φ ( i ) = Φ ( i , M ) ,
μ ˜ = W 3 D ( μ ) ,
J ˜ j T = W 3 D ( J j T ) , ( j = 1 , , M TOT ) .
[ J ˜ A λ I ] δ μ ˜ A = [ b 0 ] ,
[ J ^ 1 ( k ) J ^ 2 ( k ) J ^ m ( k ) J ^ m ( k ) ] [ δ μ 1 ( k ) δ μ 2 ( k ) δ μ m ( k ) δ μ M ( k ) ] = B ( k ) ,
J ^ m ( k ) = ( J ^ 1 , N 1 + + N ( m 1 ) + 1 μ a J ^ 1 , N 1 + + N ( m 1 ) + N m μ a J ^ 1 , N 1 + + N ( m 1 ) + 1 μ s J ^ 1 , N 1 + + N ( m 1 ) + N m μ s J ^ 2 , N 1 + + N ( m 1 ) + 1 μ a J ^ 2 , N 1 + + N ( m 1 ) + N m μ a J ^ 2 , N 1 + + N ( m 1 ) + 1 μ s J ^ 2 , N 1 + + N ( m 1 ) + N m μ s J ^ M TOT + 2 N , N 1 + + N ( m 1 ) + 1 μ a J ^ M TOT + 2 N , N 1 + + N ( m 1 ) + N m μ a J ^ M TOT + 2 N , N 1 + + N ( m 1 ) + 1 μ s J ^ M TOT + 2 N , N 1 + + N ( m 1 ) + N m μ s ) ,
J ^ m ( k ) δ μ m ( k ) = B ( k ) l = 1 m 1 J ^ l ( k ) δ μ l ( k ) l = m + 1 M J ^ l ( k ) δ μ l ( k 1 ) .
ε = K Φ q 2 ,
Ψ = Ψ ¯ ( 1 + 10 SNR 20 R noise ) ,
Γ d , s ( p 1 , p 2 ) = χ d , s after ( p 1 , p 2 ) χ d , s pre ( p 1 , p 2 ) ,

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