Abstract

We formulate a solution to the diffuse optical tomography (DOT) inverse problem as the minimization of an energy functional of the solution and the data. For the solution prior we introduce a local diffusion regularization potential with a threshold based on robust statistics (the Hubert function). We compare results on simulated data for the Hubert function and two other standard regularization functionals, Tikhonov and total variation.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. F. Jöbsis, Science 198, 1264 (1977).
    [CrossRef]
  2. S. Fantini, S. A. Walker, M. A. Franceschini, M. Kaschke, P. M. Schlag, and K. T. Moesta, Appl. Opt. 37, 1982 (1998).
    [CrossRef]
  3. B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, J. Biomed. Opt. 5, 185 (2000).
    [CrossRef] [PubMed]
  4. M. Cope and D. T. Delpy, Med. Biol. Eng. Comput. 26, 289 (1988).
    [CrossRef] [PubMed]
  5. S. R. Arridge, Inverse Probl. 15, R41 (1999).
    [CrossRef]
  6. S. Geman and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721 (1984).
    [CrossRef]
  7. J. Weickert, in Scale-Space Theory in Computer Vision, Vol. 1252 of Lecture Notes in Comp. Science (Springer-Verlag, 1997), p. 3.
  8. C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
    [CrossRef]
  9. M. Schweiger, S. R. Arridge, and I. Nissila, Phys. Med. Biol. (to be published).
  10. A. Tikhonov, Sov. Math. Dokl. 4, 1624 (1963).
  11. S. R. Arridge and M. Schweiger, in Information Processing in Medical Imaging (IPMI’93 Proceedings), Vol. 687 of Lecture Notes in Computer Science (Springer-Verlag, 1993), p. 259.
    [CrossRef]
  12. B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, Appl. Opt. 38, 2950 (1999).
    [CrossRef]
  13. L. I. Rudin, S. Osher, and E. Fatemi, Physica D 60, 259 (1992).
    [CrossRef]
  14. K. D. Paulsen and H. Jiang, Appl. Opt. 35, 3447 (1996).
    [CrossRef] [PubMed]
  15. T. Chan, A. Marquina, and P. Mulet, SIAM J. Sci. Comput. (USA) 22, 503 (2000).
    [CrossRef]
  16. C. A. Bouman and K. Sauer, IEEE Trans. Image Process. 2, 296 (1993).
    [CrossRef]
  17. A. H. Hielscher and S. Bartel, J. Biomed. Opt. 6, 183 (2001).
    [CrossRef] [PubMed]
  18. P. J. Hubert, Robust Statistics (Wiley, 1981).
    [CrossRef]
  19. P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intell. 12, 629 (1990).
    [CrossRef]
  20. D. S. Luo, M. A. King, and S. Glick, IEEE Trans. Nucl. Sci. 41, 2800 (1994).
    [CrossRef]
  21. F. R. Hampel, E. M. Ronchetti, and P. J. Rousseeuw, Robust Statistics: the Approach Based on Influence Functions (Wiley, 1986).
  22. M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, IEEE Trans. Image Process. 7, 421 (1998).
    [CrossRef]
  23. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, Med. Phys. 20, 299 (1993).
    [CrossRef] [PubMed]
  24. P. C. Hansen and D. P. O’Leary, SIAM J. Sci. Comput. (USA) 14, 1487 (1993).
    [CrossRef]

2001 (1)

A. H. Hielscher and S. Bartel, J. Biomed. Opt. 6, 183 (2001).
[CrossRef] [PubMed]

2000 (2)

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, J. Biomed. Opt. 5, 185 (2000).
[CrossRef] [PubMed]

T. Chan, A. Marquina, and P. Mulet, SIAM J. Sci. Comput. (USA) 22, 503 (2000).
[CrossRef]

1999 (2)

1998 (2)

S. Fantini, S. A. Walker, M. A. Franceschini, M. Kaschke, P. M. Schlag, and K. T. Moesta, Appl. Opt. 37, 1982 (1998).
[CrossRef]

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, IEEE Trans. Image Process. 7, 421 (1998).
[CrossRef]

1996 (1)

1994 (1)

D. S. Luo, M. A. King, and S. Glick, IEEE Trans. Nucl. Sci. 41, 2800 (1994).
[CrossRef]

1993 (3)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, Med. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

P. C. Hansen and D. P. O’Leary, SIAM J. Sci. Comput. (USA) 14, 1487 (1993).
[CrossRef]

C. A. Bouman and K. Sauer, IEEE Trans. Image Process. 2, 296 (1993).
[CrossRef]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, Physica D 60, 259 (1992).
[CrossRef]

1990 (1)

P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intell. 12, 629 (1990).
[CrossRef]

1988 (1)

M. Cope and D. T. Delpy, Med. Biol. Eng. Comput. 26, 289 (1988).
[CrossRef] [PubMed]

1984 (1)

S. Geman and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721 (1984).
[CrossRef]

1977 (1)

F. F. Jöbsis, Science 198, 1264 (1977).
[CrossRef]

1963 (1)

A. Tikhonov, Sov. Math. Dokl. 4, 1624 (1963).

Abele, C.

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, J. Biomed. Opt. 5, 185 (2000).
[CrossRef] [PubMed]

Arridge, S. R.

S. R. Arridge, Inverse Probl. 15, R41 (1999).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, Med. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, in Information Processing in Medical Imaging (IPMI’93 Proceedings), Vol. 687 of Lecture Notes in Computer Science (Springer-Verlag, 1993), p. 259.
[CrossRef]

M. Schweiger, S. R. Arridge, and I. Nissila, Phys. Med. Biol. (to be published).

Bartel, S.

A. H. Hielscher and S. Bartel, J. Biomed. Opt. 6, 183 (2001).
[CrossRef] [PubMed]

Black, M. J.

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, IEEE Trans. Image Process. 7, 421 (1998).
[CrossRef]

Bouman, C. A.

C. A. Bouman and K. Sauer, IEEE Trans. Image Process. 2, 296 (1993).
[CrossRef]

Chan, T.

T. Chan, A. Marquina, and P. Mulet, SIAM J. Sci. Comput. (USA) 22, 503 (2000).
[CrossRef]

Cope, M.

M. Cope and D. T. Delpy, Med. Biol. Eng. Comput. 26, 289 (1988).
[CrossRef] [PubMed]

Delpy, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, Med. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

M. Cope and D. T. Delpy, Med. Biol. Eng. Comput. 26, 289 (1988).
[CrossRef] [PubMed]

Fantini, S.

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, Physica D 60, 259 (1992).
[CrossRef]

Franceschini, M. A.

Geman, D.

S. Geman and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721 (1984).
[CrossRef]

Geman, S.

S. Geman and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721 (1984).
[CrossRef]

Glick, S.

D. S. Luo, M. A. King, and S. Glick, IEEE Trans. Nucl. Sci. 41, 2800 (1994).
[CrossRef]

Hampel, F. R.

F. R. Hampel, E. M. Ronchetti, and P. J. Rousseeuw, Robust Statistics: the Approach Based on Influence Functions (Wiley, 1986).

Hansen, P. C.

P. C. Hansen and D. P. O’Leary, SIAM J. Sci. Comput. (USA) 14, 1487 (1993).
[CrossRef]

Heeger, D.

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, IEEE Trans. Image Process. 7, 421 (1998).
[CrossRef]

Hielscher, A. H.

A. H. Hielscher and S. Bartel, J. Biomed. Opt. 6, 183 (2001).
[CrossRef] [PubMed]

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, Med. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

Hubert, P. J.

P. J. Hubert, Robust Statistics (Wiley, 1981).
[CrossRef]

Jiang, H.

Jöbsis, F. F.

F. F. Jöbsis, Science 198, 1264 (1977).
[CrossRef]

Kaschke, M.

Kaufman, H.

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, J. Biomed. Opt. 5, 185 (2000).
[CrossRef] [PubMed]

King, M. A.

D. S. Luo, M. A. King, and S. Glick, IEEE Trans. Nucl. Sci. 41, 2800 (1994).
[CrossRef]

Luo, D. S.

D. S. Luo, M. A. King, and S. Glick, IEEE Trans. Nucl. Sci. 41, 2800 (1994).
[CrossRef]

Malik, J.

P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intell. 12, 629 (1990).
[CrossRef]

Marimont, D. H.

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, IEEE Trans. Image Process. 7, 421 (1998).
[CrossRef]

Marquina, A.

T. Chan, A. Marquina, and P. Mulet, SIAM J. Sci. Comput. (USA) 22, 503 (2000).
[CrossRef]

McBride, T. O.

Moesta, K. T.

Mulet, P.

T. Chan, A. Marquina, and P. Mulet, SIAM J. Sci. Comput. (USA) 22, 503 (2000).
[CrossRef]

Nissila, I.

M. Schweiger, S. R. Arridge, and I. Nissila, Phys. Med. Biol. (to be published).

O’Leary, D. P.

P. C. Hansen and D. P. O’Leary, SIAM J. Sci. Comput. (USA) 14, 1487 (1993).
[CrossRef]

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, Physica D 60, 259 (1992).
[CrossRef]

Osterberg, U. L.

Paulsen, K. D.

Perona, P.

P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intell. 12, 629 (1990).
[CrossRef]

Pogue, B. W.

Prewitt, J.

Ronchetti, E. M.

F. R. Hampel, E. M. Ronchetti, and P. J. Rousseeuw, Robust Statistics: the Approach Based on Influence Functions (Wiley, 1986).

Rousseeuw, P. J.

F. R. Hampel, E. M. Ronchetti, and P. J. Rousseeuw, Robust Statistics: the Approach Based on Influence Functions (Wiley, 1986).

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, Physica D 60, 259 (1992).
[CrossRef]

Sapiro, G.

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, IEEE Trans. Image Process. 7, 421 (1998).
[CrossRef]

Sauer, K.

C. A. Bouman and K. Sauer, IEEE Trans. Image Process. 2, 296 (1993).
[CrossRef]

Schlag, P. M.

Schweiger, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, Med. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, in Information Processing in Medical Imaging (IPMI’93 Proceedings), Vol. 687 of Lecture Notes in Computer Science (Springer-Verlag, 1993), p. 259.
[CrossRef]

M. Schweiger, S. R. Arridge, and I. Nissila, Phys. Med. Biol. (to be published).

Tikhonov, A.

A. Tikhonov, Sov. Math. Dokl. 4, 1624 (1963).

Vogel, C. R.

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
[CrossRef]

Walker, S. A.

Weickert, J.

J. Weickert, in Scale-Space Theory in Computer Vision, Vol. 1252 of Lecture Notes in Comp. Science (Springer-Verlag, 1997), p. 3.

Appl. Opt. (3)

IEEE Trans. Image Process. (2)

C. A. Bouman and K. Sauer, IEEE Trans. Image Process. 2, 296 (1993).
[CrossRef]

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, IEEE Trans. Image Process. 7, 421 (1998).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

D. S. Luo, M. A. King, and S. Glick, IEEE Trans. Nucl. Sci. 41, 2800 (1994).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

S. Geman and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721 (1984).
[CrossRef]

P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intell. 12, 629 (1990).
[CrossRef]

Inverse Probl. (1)

S. R. Arridge, Inverse Probl. 15, R41 (1999).
[CrossRef]

J. Biomed. Opt. (2)

A. H. Hielscher and S. Bartel, J. Biomed. Opt. 6, 183 (2001).
[CrossRef] [PubMed]

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, J. Biomed. Opt. 5, 185 (2000).
[CrossRef] [PubMed]

Med. Biol. Eng. Comput. (1)

M. Cope and D. T. Delpy, Med. Biol. Eng. Comput. 26, 289 (1988).
[CrossRef] [PubMed]

Med. Phys. (1)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, Med. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

Physica D (1)

L. I. Rudin, S. Osher, and E. Fatemi, Physica D 60, 259 (1992).
[CrossRef]

Science (1)

F. F. Jöbsis, Science 198, 1264 (1977).
[CrossRef]

SIAM J. Sci. Comput. (USA) (2)

P. C. Hansen and D. P. O’Leary, SIAM J. Sci. Comput. (USA) 14, 1487 (1993).
[CrossRef]

T. Chan, A. Marquina, and P. Mulet, SIAM J. Sci. Comput. (USA) 22, 503 (2000).
[CrossRef]

Sov. Math. Dokl. (1)

A. Tikhonov, Sov. Math. Dokl. 4, 1624 (1963).

Other (6)

S. R. Arridge and M. Schweiger, in Information Processing in Medical Imaging (IPMI’93 Proceedings), Vol. 687 of Lecture Notes in Computer Science (Springer-Verlag, 1993), p. 259.
[CrossRef]

J. Weickert, in Scale-Space Theory in Computer Vision, Vol. 1252 of Lecture Notes in Comp. Science (Springer-Verlag, 1997), p. 3.

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
[CrossRef]

M. Schweiger, S. R. Arridge, and I. Nissila, Phys. Med. Biol. (to be published).

P. J. Hubert, Robust Statistics (Wiley, 1981).
[CrossRef]

F. R. Hampel, E. M. Ronchetti, and P. J. Rousseeuw, Robust Statistics: the Approach Based on Influence Functions (Wiley, 1986).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Illustration of the choice of σ for the Hubert function with respect to the diffusion function.

Fig. 2
Fig. 2

2D phantom: (a) Adapted forward mesh, 2572 nodes; (b) regular backward mesh, 3062 nodes; (c) schematic diagram; (d) target μ a ; (e) target μ s ; (f) optical properties.

Fig. 3
Fig. 3

Comparison of the reconstructions of μ a (top) and μ s (bottom): Tikhonov, NHE = 0.012 (left), TV, NHE = 0.011 (middle), and Hubert, NHE = 0.008 (right).

Equations (10)

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

( κ ( r ) + μ a ( r ) + i ω c ) Φ ( r , ω ) = q ( r , ω ) ,
Φ ( m , ω ) + 2 κ ( m ) A Φ ( m , ω ) n = 0 .
F ( x ) = y ,
g y δ , where δ > 0 .
E ( x ) 1 2 g F ( x ) R 2 + τ Ω ψ ( x ) min ,
grad E ( x ) = F T ( x ) R ( F ( x ) g ) + τ L ( x ) x ,
L ( x ) = ( ψ ( x ) x ) .
Hess ˜ E ( x ) = F T ( x ) R F ( x ) + τ L ( x ) .
ψ ( x ) { x 2 2 if x σ σ x σ 2 2 otherwise , }
σ = 1.4826 MAD ( x )

Metrics