Abstract

Numerical performance of two gradient-based methods, a truncated-Newton method with trust region (TN) and a nonlinear conjugate gradient (NCG), is studied and compared for a given data set and conditions specific for the contrast enhanced optical tomography problem. Our results suggest that the relative performance of the two methods depends upon the error functions, specific to the problem to be solved. The TN outperforms the NCG when maps of fluorescence lifetime are reconstructed while both methods performed well when the absorption coefficient constitutes the parameter set that is to be recovered.

© 2001 Optical Society of America

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  1. Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, and R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,” J. Opt. Soc. Am A,  14, .325–342, (1997).
    [Crossref]
  2. D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt,  36, 2260–2272, (1997).
    [Crossref] [PubMed]
  3. K. D. Paulsen and H. Jiang, “Enhanced frequency domain optical image reconstruction in tissues through total variation minimization,” Appl. Opt. 35, 3447–3458, (1996).
    [Crossref] [PubMed]
  4. H. B. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266, (1996).
    [Crossref]
  5. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett. 20, 426–428, (1995).
    [Crossref]
  6. S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstruction of near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1867, 372–383, (1992).
  7. R. Roy, “Image reconstruction from light measurement on biological tissue,’ Ph. D thesis, Hatfield, England, (1996).
  8. S. R. Arridge and M. Schweiger, “A gradient-based optimization scheme for optical tomography”, Opt. Express;  2, 213–226, (1998) http://www.opticsexpress.org/oearchive/source/4014.htm
    [Crossref] [PubMed]
  9. A. H. Hielscher, A. D Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans Med. Imag,  18, 262–271, (1999).
    [Crossref]
  10. A. D. Klose and A.H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phy. 26, 1698–1707, (1999).
    [Crossref]
  11. M. V. Klibanov, T. R. Lucas, and R. M. Frank, “A fast and accurate imaging algorithm in optical diffusion tomography,” Inverse Problems,  13, 1341–1361, (1997).
    [Crossref]
  12. S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, Proc. of the SPIE-The International Society for Optical Engineering,  3034, 369–380, (1997).
  13. R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express,  4, 353–371, (1999). http://www.opticsexpress.org/oearchive/source/9268.htm
    [Crossref] [PubMed]
  14. R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part II Reconstruction from synthetic measurements,” Opt. Express,  4, 372–382, (1999). http://www.opticsexpress.org/oearchive/source/10447.htm
    [Crossref] [PubMed]
  15. R. Roy and E. M. Sevick-Muraca, “Active constrained truncated Newton method for simple-bound optical tomography,” J. Opt. Soc. Am. A 17, 1627–1641, (2000).
    [Crossref]
  16. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems,  15, R41–R93, (1999).
    [Crossref]
  17. R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimisation,” Math. Programming,  26, 190–212, (1983).
    [Crossref]
  18. L. C. W. Dixon and R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” J. optimization theory and applications. 56, 245–255, (1988).
    [Crossref]
  19. T. Steihaug, “The conjugate gradient method and trust region in large scale optimisation,” SIAM J. Numerical analysis,  20, 626–637, (1983).
    [Crossref]
  20. M. R. Hestenes and E. Stiefel, ‘Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
    [Crossref]
  21. R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Computer J.,  7, 149–154 (1964).
    [Crossref]
  22. S. Bazaraa, H. D. Sherali, and C. M. Shetty, “Nonlinear programming theory and algorithms,” John Wiley & Sons Inc, New York, (1993).
  23. R. Fletcher, “Practical methods of optimisation,” Second edition, John Wiley & Sons, Chichester, (1996).
  24. M. J. D. Powell, “Restart procedures for the conjugate gradient method,” Math. Programming,  12, 241–254, (1977).
    [Crossref]
  25. A. Griewank, “On automatic differentiation,” edited M. Iri and K. Tanaka, Mathematical programming: Recent developments and application, Kluwer Academic Publishers, 83–108, (1989).
  26. B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software,  8, 53–67, (1997).
    [Crossref]
  27. A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software,  28, 217–221, (1997).
    [Crossref]
  28. A. Ishimaru, Wave propagation and scattering in random media, Academic Press, New York, (1978).
  29. O. C. Zienkiewcz and R. L. Taylor, The finite element methods in engineering science, McGraw-Hill, New York, (1989).
  30. L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Mathematics,  16, 1–3, (1966).
  31. J. C. Gilbert and J Nocedal, “Global convergence properties of conjugate gradient methods for optimization,” Report no. 1268, INRIA, (1990)
  32. P. Wolfe, “Convergence condition for ascent methods,” SIAM Rev,  11, 226–253, (1969).
    [Crossref]
  33. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37, 5337–5343, (1998).
    [Crossref]
  34. E. Polak, Optimization, algorithms and consistent approximation, Springer-Verlag, New York, (1997)
  35. M. J. D. Powell, “Nonconvex minimization calculations and the conjugate gradient methods,” Lecture Notes in Math. 1066,Spriger-Verlag, New York, 122–141, (1984).
  36. S. G. Nash, “A survey of truncated-Newton methods,” J. of Computational and Applied Math. 124, 45–59, (2000).
    [Crossref]

2000 (2)

1999 (5)

R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express,  4, 353–371, (1999). http://www.opticsexpress.org/oearchive/source/9268.htm
[Crossref] [PubMed]

R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part II Reconstruction from synthetic measurements,” Opt. Express,  4, 372–382, (1999). http://www.opticsexpress.org/oearchive/source/10447.htm
[Crossref] [PubMed]

A. H. Hielscher, A. D Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans Med. Imag,  18, 262–271, (1999).
[Crossref]

A. D. Klose and A.H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phy. 26, 1698–1707, (1999).
[Crossref]

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

1998 (2)

1997 (6)

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, and R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,” J. Opt. Soc. Am A,  14, .325–342, (1997).
[Crossref]

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt,  36, 2260–2272, (1997).
[Crossref] [PubMed]

M. V. Klibanov, T. R. Lucas, and R. M. Frank, “A fast and accurate imaging algorithm in optical diffusion tomography,” Inverse Problems,  13, 1341–1361, (1997).
[Crossref]

S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, Proc. of the SPIE-The International Society for Optical Engineering,  3034, 369–380, (1997).

B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software,  8, 53–67, (1997).
[Crossref]

A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software,  28, 217–221, (1997).
[Crossref]

1996 (2)

1995 (1)

1988 (1)

L. C. W. Dixon and R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” J. optimization theory and applications. 56, 245–255, (1988).
[Crossref]

1983 (2)

T. Steihaug, “The conjugate gradient method and trust region in large scale optimisation,” SIAM J. Numerical analysis,  20, 626–637, (1983).
[Crossref]

R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimisation,” Math. Programming,  26, 190–212, (1983).
[Crossref]

1977 (1)

M. J. D. Powell, “Restart procedures for the conjugate gradient method,” Math. Programming,  12, 241–254, (1977).
[Crossref]

1969 (1)

P. Wolfe, “Convergence condition for ascent methods,” SIAM Rev,  11, 226–253, (1969).
[Crossref]

1966 (1)

L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Mathematics,  16, 1–3, (1966).

1964 (1)

R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Computer J.,  7, 149–154 (1964).
[Crossref]

1952 (1)

M. R. Hestenes and E. Stiefel, ‘Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[Crossref]

Armijo, L.

L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Mathematics,  16, 1–3, (1966).

Arridge, S. R.

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

S. R. Arridge and M. Schweiger, “A gradient-based optimization scheme for optical tomography”, Opt. Express;  2, 213–226, (1998) http://www.opticsexpress.org/oearchive/source/4014.htm
[Crossref] [PubMed]

S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstruction of near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1867, 372–383, (1992).

Barbour, R. L.

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, and R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,” J. Opt. Soc. Am A,  14, .325–342, (1997).
[Crossref]

Bazaraa, S.

S. Bazaraa, H. D. Sherali, and C. M. Shetty, “Nonlinear programming theory and algorithms,” John Wiley & Sons Inc, New York, (1993).

Boas, D. A.

Chance, B.

Chen, A. U.

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt,  36, 2260–2272, (1997).
[Crossref] [PubMed]

Christianson, B.

B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software,  8, 53–67, (1997).
[Crossref]

A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software,  28, 217–221, (1997).
[Crossref]

Cunningham, G. S.

S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, Proc. of the SPIE-The International Society for Optical Engineering,  3034, 369–380, (1997).

Davies, A. J.

A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software,  28, 217–221, (1997).
[Crossref]

B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software,  8, 53–67, (1997).
[Crossref]

Delpy, D. T.

S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstruction of near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1867, 372–383, (1992).

Dembo, R. S.

R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimisation,” Math. Programming,  26, 190–212, (1983).
[Crossref]

Dixon, L. C. W.

B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software,  8, 53–67, (1997).
[Crossref]

A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software,  28, 217–221, (1997).
[Crossref]

L. C. W. Dixon and R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” J. optimization theory and applications. 56, 245–255, (1988).
[Crossref]

Fletcher, R.

R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Computer J.,  7, 149–154 (1964).
[Crossref]

R. Fletcher, “Practical methods of optimisation,” Second edition, John Wiley & Sons, Chichester, (1996).

Frank, R. M.

M. V. Klibanov, T. R. Lucas, and R. M. Frank, “A fast and accurate imaging algorithm in optical diffusion tomography,” Inverse Problems,  13, 1341–1361, (1997).
[Crossref]

Gilbert, J. C.

J. C. Gilbert and J Nocedal, “Global convergence properties of conjugate gradient methods for optimization,” Report no. 1268, INRIA, (1990)

Griewank, A.

A. Griewank, “On automatic differentiation,” edited M. Iri and K. Tanaka, Mathematical programming: Recent developments and application, Kluwer Academic Publishers, 83–108, (1989).

Hanson, K. M.

A. H. Hielscher, A. D Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans Med. Imag,  18, 262–271, (1999).
[Crossref]

S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, Proc. of the SPIE-The International Society for Optical Engineering,  3034, 369–380, (1997).

Hestenes, M. R.

M. R. Hestenes and E. Stiefel, ‘Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[Crossref]

Hielscher, A. H.

A. H. Hielscher, A. D Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans Med. Imag,  18, 262–271, (1999).
[Crossref]

Hielscher, A.H.

A. D. Klose and A.H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phy. 26, 1698–1707, (1999).
[Crossref]

Ishimaru, A.

A. Ishimaru, Wave propagation and scattering in random media, Academic Press, New York, (1978).

Jiang, H.

Jiang, H. B.

Klibanov, M. V.

M. V. Klibanov, T. R. Lucas, and R. M. Frank, “A fast and accurate imaging algorithm in optical diffusion tomography,” Inverse Problems,  13, 1341–1361, (1997).
[Crossref]

Klose, A. D

A. H. Hielscher, A. D Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans Med. Imag,  18, 262–271, (1999).
[Crossref]

Klose, A. D.

A. D. Klose and A.H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phy. 26, 1698–1707, (1999).
[Crossref]

Lucas, T. R.

M. V. Klibanov, T. R. Lucas, and R. M. Frank, “A fast and accurate imaging algorithm in optical diffusion tomography,” Inverse Problems,  13, 1341–1361, (1997).
[Crossref]

Nash, S. G.

S. G. Nash, “A survey of truncated-Newton methods,” J. of Computational and Applied Math. 124, 45–59, (2000).
[Crossref]

Nocedal, J

J. C. Gilbert and J Nocedal, “Global convergence properties of conjugate gradient methods for optimization,” Report no. 1268, INRIA, (1990)

O’Leary, M. A.

Osterberg, U. L.

Paithankar, D. Y.

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt,  36, 2260–2272, (1997).
[Crossref] [PubMed]

Patterson, M. S.

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt,  36, 2260–2272, (1997).
[Crossref] [PubMed]

H. B. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266, (1996).
[Crossref]

Paulsen, K. D.

Pei, Y. L.

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, and R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,” J. Opt. Soc. Am A,  14, .325–342, (1997).
[Crossref]

Pogue, B. W.

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt,  36, 2260–2272, (1997).
[Crossref] [PubMed]

H. B. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency domain data: Simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266, (1996).
[Crossref]

Polak, E.

E. Polak, Optimization, algorithms and consistent approximation, Springer-Verlag, New York, (1997)

Powell, M. J. D.

M. J. D. Powell, “Restart procedures for the conjugate gradient method,” Math. Programming,  12, 241–254, (1977).
[Crossref]

M. J. D. Powell, “Nonconvex minimization calculations and the conjugate gradient methods,” Lecture Notes in Math. 1066,Spriger-Verlag, New York, 122–141, (1984).

Price, R. C.

L. C. W. Dixon and R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” J. optimization theory and applications. 56, 245–255, (1988).
[Crossref]

Reeves, C. M.

R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Computer J.,  7, 149–154 (1964).
[Crossref]

Roy, R.

Saquib, S. S.

S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, Proc. of the SPIE-The International Society for Optical Engineering,  3034, 369–380, (1997).

Schweiger, M.

S. R. Arridge and M. Schweiger, “A gradient-based optimization scheme for optical tomography”, Opt. Express;  2, 213–226, (1998) http://www.opticsexpress.org/oearchive/source/4014.htm
[Crossref] [PubMed]

S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstruction of near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1867, 372–383, (1992).

Sevick-Muraca, E. M.

Sherali, H. D.

S. Bazaraa, H. D. Sherali, and C. M. Shetty, “Nonlinear programming theory and algorithms,” John Wiley & Sons Inc, New York, (1993).

Shetty, C. M.

S. Bazaraa, H. D. Sherali, and C. M. Shetty, “Nonlinear programming theory and algorithms,” John Wiley & Sons Inc, New York, (1993).

Steihaug, T.

T. Steihaug, “The conjugate gradient method and trust region in large scale optimisation,” SIAM J. Numerical analysis,  20, 626–637, (1983).
[Crossref]

R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimisation,” Math. Programming,  26, 190–212, (1983).
[Crossref]

Stiefel, E.

M. R. Hestenes and E. Stiefel, ‘Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[Crossref]

Taylor, R. L.

O. C. Zienkiewcz and R. L. Taylor, The finite element methods in engineering science, McGraw-Hill, New York, (1989).

van der Zee, P.

A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software,  28, 217–221, (1997).
[Crossref]

B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software,  8, 53–67, (1997).
[Crossref]

Wang, Y.

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, and R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,” J. Opt. Soc. Am A,  14, .325–342, (1997).
[Crossref]

Wolfe, P.

P. Wolfe, “Convergence condition for ascent methods,” SIAM Rev,  11, 226–253, (1969).
[Crossref]

Yao, Y. Q.

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, and R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,” J. Opt. Soc. Am A,  14, .325–342, (1997).
[Crossref]

Yodh, A. G.

Zhu, W. W.

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, and R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,” J. Opt. Soc. Am A,  14, .325–342, (1997).
[Crossref]

Zienkiewcz, O. C.

O. C. Zienkiewcz and R. L. Taylor, The finite element methods in engineering science, McGraw-Hill, New York, (1989).

Adv. In Eng. Software (1)

A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software,  28, 217–221, (1997).
[Crossref]

Appl. Opt (1)

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt,  36, 2260–2272, (1997).
[Crossref] [PubMed]

Appl. Opt. (2)

Computer J. (1)

R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Computer J.,  7, 149–154 (1964).
[Crossref]

IEEE Trans Med. Imag (1)

A. H. Hielscher, A. D Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans Med. Imag,  18, 262–271, (1999).
[Crossref]

in Medical Imaging: Image Processing, Proc. of the SPIE-The International Society for Optical Engineering (1)

S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, Proc. of the SPIE-The International Society for Optical Engineering,  3034, 369–380, (1997).

Inverse Problems (2)

M. V. Klibanov, T. R. Lucas, and R. M. Frank, “A fast and accurate imaging algorithm in optical diffusion tomography,” Inverse Problems,  13, 1341–1361, (1997).
[Crossref]

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

J. of Computational and Applied Math. (1)

S. G. Nash, “A survey of truncated-Newton methods,” J. of Computational and Applied Math. 124, 45–59, (2000).
[Crossref]

J. Opt. Soc. Am A (1)

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, and R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,” J. Opt. Soc. Am A,  14, .325–342, (1997).
[Crossref]

J. Opt. Soc. Am. A (2)

J. optimization theory and applications. (1)

L. C. W. Dixon and R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” J. optimization theory and applications. 56, 245–255, (1988).
[Crossref]

J. Res. Nat. Bur. Stand. (1)

M. R. Hestenes and E. Stiefel, ‘Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[Crossref]

Math. Programming (2)

M. J. D. Powell, “Restart procedures for the conjugate gradient method,” Math. Programming,  12, 241–254, (1977).
[Crossref]

R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimisation,” Math. Programming,  26, 190–212, (1983).
[Crossref]

Med. Phy. (1)

A. D. Klose and A.H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phy. 26, 1698–1707, (1999).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Opt. Meth. And Software (1)

B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software,  8, 53–67, (1997).
[Crossref]

Pacific J. Mathematics (1)

L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Mathematics,  16, 1–3, (1966).

SIAM J. Numerical analysis (1)

T. Steihaug, “The conjugate gradient method and trust region in large scale optimisation,” SIAM J. Numerical analysis,  20, 626–637, (1983).
[Crossref]

SIAM Rev (1)

P. Wolfe, “Convergence condition for ascent methods,” SIAM Rev,  11, 226–253, (1969).
[Crossref]

Other (10)

J. C. Gilbert and J Nocedal, “Global convergence properties of conjugate gradient methods for optimization,” Report no. 1268, INRIA, (1990)

E. Polak, Optimization, algorithms and consistent approximation, Springer-Verlag, New York, (1997)

M. J. D. Powell, “Nonconvex minimization calculations and the conjugate gradient methods,” Lecture Notes in Math. 1066,Spriger-Verlag, New York, 122–141, (1984).

A. Ishimaru, Wave propagation and scattering in random media, Academic Press, New York, (1978).

O. C. Zienkiewcz and R. L. Taylor, The finite element methods in engineering science, McGraw-Hill, New York, (1989).

A. Griewank, “On automatic differentiation,” edited M. Iri and K. Tanaka, Mathematical programming: Recent developments and application, Kluwer Academic Publishers, 83–108, (1989).

S. Bazaraa, H. D. Sherali, and C. M. Shetty, “Nonlinear programming theory and algorithms,” John Wiley & Sons Inc, New York, (1993).

R. Fletcher, “Practical methods of optimisation,” Second edition, John Wiley & Sons, Chichester, (1996).

S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstruction of near-infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1867, 372–383, (1992).

R. Roy, “Image reconstruction from light measurement on biological tissue,’ Ph. D thesis, Hatfield, England, (1996).

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Figures (6)

Figure 1.
Figure 1.

A two-dimensional square with three target (heterogeneities) regions with different optical properties. Size of each heterogeneity is 0.0625cm2.

Figure 2.
Figure 2.

a) Actual distribution of absorption coefficients, (b) reconstruction of absorption coefficients by the TN method (c) reconstruction of absorption coefficients by the NCG method.

Figure 3.
Figure 3.

(a) Actual spatial distribution of lifetimes (background 1 ns), (b) reconstruction of lifetimes by the TN method.

Figure 4.
Figure 4.

a) Actual spatial distribution of lifetimes (background 10 ns) (b) reconstruction of lifetimes by the TN method (c) reconstruction of lifetimes by the NCG method

Figure 5.
Figure 5.

Error function E as a function of the absorption coefficient of three target regions at 150 MHz.

Figure 6.
Figure 6.

Error equation E as a function of the lifetimes on three target regions

Tables (12)

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Table 1 Background optical parameters used for the optimization problems (Eqns 1 & 2)

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Table 2 Target optical parameters used for the optimization problems (Eqns 1 & 2)

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Table 3. Recovery of absorption coefficient (Problem 1)

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Table 4. Recovery of fluorescent lifetime (background 1 ns, Problem 2)

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Table 5 Recovery of fluorescent lifetime (background 10 ns, Problem 3)

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Table 6 Curvature directions of problems 1, 2, and 3 at modulation frequencies of 150 and 500 M

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Table 7 Gradient vectors for problems 1, 2, and 3 at modulation frequency of 150 and 500 MHz.

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Table 8 α calculated in the TN optimization method

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Table 9 α calculated in the NCG optimization method

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Table 10 Direction of the gradient vectors at initial guess

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Table 11 Values of error function E at different step lengths for Problem 2 for NCG

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Table 12 Newton directions calculated by the truncated Newton method of problems 1, 2, and 3

Equations (19)

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

· [ D x ( r ) Φ x ( r , ω ) ] + [ c + μ a xi ( r ) + μ a xf ( r ) ] Φ x ( r , ω ) = 0 on Ω
· [ D m ( r ) Φ m ( r , ω ) ] + [ c + μ a m ( r ) ] Φ m ( r , ω ) = ϕμ a xf 1 1 iωτ Φ x ( r , ω ) on Ω
2 D x Φ x n + Φ x + ( r ¯ , r ¯ s ) = 0 on d Ω
K Φ ¯ x , m = b
E ( μ ¯ a xf , τ ¯ ) = 1 2 1 = 1 N q j = 1 j 1 N B ( ( Φ m ) c ( Φ m ) me ( Φ m ) me ) i , j ( ( Φ m * ) c ( Φ m * ) me ( Φ m * ) me ) i , j
E ( ( μ a ) k + d ) = E ( ( μ a ) k ) + g k T d + 1 2 d T G k d
G k d = g k
r i g k min ( 1 k , g k )
G ( μ a ) d = 1 σ [ g ( μ a + σ d ) g ( μ a ) ]
σ = machineprecision d
( μ a ) k + 1 = ( μ a ) k + α k d k , ( μ a ) k + 1 > 0
d k = g k + β k d k 1
β k = g k T ( g k g k 1 ) g k 1 2
d k T g k ε 1 d k g k > 0
E ( μ a + α d ) E ( μ a ) ε 2 α d T g
E ( μ a + 2 α d ) E ( μ a ) > 2 ε 3 α d T g
m d 2 d , G ( x ) d M d 2
E ( x k ) , d k m M E ( x k ) d k
α k = ( τ ) k d k α k > 0

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