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.

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  1. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, 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, "Imagingof 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. SPIE 1867, 372-383, (1992).
  7. R. Roy, "Image reconstruction from light measurement on biological tissue,' Ph. D thesis, Hatfield, England, (1996).
  8. S. R. Arridge, 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, 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 Iri, M. and Tanaka, K., 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.vanderZee, "Givingreverse 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, 1984) pp. 122-141.
  36. S. G. Nash, " A survey of truncated-Newton methods," J. of Computational and Applied Math. 124, 45-59 (2000).
    [CrossRef]

Other (36)

Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, 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, "Imagingof fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media," Appl. Opt, 36, 2260-2272, (1997).
[CrossRef] [PubMed]

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]

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]

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]

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. SPIE 1867, 372-383, (1992).

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

S. R. Arridge, 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]

A. H. Hielscher, A. D Klose, 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]

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).

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]

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]

S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems, 15, R41-R93, (1999).
[CrossRef]

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

T. Steihaug, "The conjugate gradient method and trust region in large scale optimisation," SIAM J. Numerical analysis 20, 626-637, (1983).
[CrossRef]

M. R. Hestenes and E. Stiefel, 'Methods of conjugate gradients for solving linear systems," J. Res. Nat. Bur. Stand. 49, 409-436 (1952).
[CrossRef]

R. Fletcher and C. M. Reeves, "Function minimization by conjugate gradients," Computer J. 7, 149-154 (1964).
[CrossRef]

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).

M. J. D. Powell, "Restart procedures for the conjugate gradient method," Math. Programming 12, 241-254, (1977).
[CrossRef]

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

B. Christianson, A.J., Davies, L.C.W. Dixon, R. Roy, and P.vanderZee, "Givingreverse 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]

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).

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

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

P. Wolfe, "Convergence condition for ascent methods," SIAM Rev. 11, 226-253, (1969).
[CrossRef]

H. Jiang, "Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations," Appl. Opt. 37, 5337-5343, (1998).
[CrossRef]

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, 1984) pp. 122-141.

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

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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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