Abstract

Optical tomography schemes using non-linear optimisation are usually based on a Newton-like method involving the construction and inversion of a large Jacobian matrix. Although such matrices can be efficiently constructed using a reciprocity principle, their inversion is still computationally difficult. In this paper we demonstrate a simple means to obtain the gradient of the objective function directly, leading to straightforward application of gradient-based optimisation methods.

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References

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  1. A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope and E. O. R. Reynolds, \Cotside measurement of cerebral blood ow in ill newborn infants by near infrared spec- troscopy," Lancet 2, 770-771 (1988).
    [CrossRef] [PubMed]
  2. S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray and E. O. R. Reynolds, \Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy," J. Appl. Physiol. 68, 1086-1091 (1990).
    [PubMed]
  3. M. Tamura, \Multichannel near-infrared optical imaging of human brain activity," in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC 1996) Vol. 2, pp. 8-10.
  4. J. C. Hebden, R. A. Kruger and K. S. Wong, \Time resolved imaging through a highly scattering medium," Appl. Opt. 30, 788-794 (1991).
    [CrossRef] [PubMed]
  5. Near-infrared spectroscopy and imaging of living systems, special issue of Philos. Trans. R. Soc. London Ser. B, Vol. 352 (1997).
  6. J. C. Hebden, S. R. Arridge and D. T. Delpy, \Optical imaging in medicine: I. Experimental techniques," Phys. Med. Biol. 42, 825-840 (1997).
    [CrossRef] [PubMed]
  7. S. R. Arridge and J. C. Hebden, \Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
    [CrossRef] [PubMed]
  8. S. B. Colak, G. W. Hooft, D. G. Papaioannou and M. B. van der Mark, \3D backprojection tomography for medical optical imaging," in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC 1996) Vol. 2, pp. 294-298.
  9. S. A. Walker, S. Fantini and E. Gratton, \Back-projection reconstructions of cylindrical inho- mogeneities from frequency domain optical measurements in turbid media," in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC 1996) Vol. 2, pp. 137-141.
  10. M. A. O Leary, D. A. Boas, B. Chance and A. G. Yodh, \Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography," Opt. Lett. 20, 426-428 (1995).
    [CrossRef]
  11. 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 1767, 372-383 (1992).
    [CrossRef]
  12. H. 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 (1995).
    [CrossRef]
  13. B. W. Pogue, M. S. Patterson, H. Jiang and K. D. Paulsen, \Initial assessment of a simple system for frequency domain diffuse optical tomography," Phys. Med. Biol. 40, 1709-1729 (1995).
    [CrossRef] [PubMed]
  14. D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray and J. Wyatt, \Estimation of optical pathlength through tissue from direct time of ight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
    [CrossRef] [PubMed]
  15. B. Chance, M. Maris, J. Sorge and M. Z. Zhang, \A phase modulation system for dual wave- length difference spectroscopy of haemoglobin deoxygenation in tissue," in Time-resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE 1204, 481-491 (1990).
    [CrossRef]
  16. J. D. Moulton, Diffusion modelling of picosecond laser pulse propagation in turbid media, M. Eng. thesis, McMaster University, Hamilton, Ontario (1990).
  17. S. R. Arridge, M. Schweiger, M. Hiraoka and D. T. Delpy, \A finite element approach for modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
    [CrossRef] [PubMed]
  18. M. Schweiger, S. R. Arridge, M. Hiraoka and D. T. Delpy, \The finite element model for the propagation of light in scattering media: Boundary and source conditions," Med. Phys. 22, 1779-1792 (1995).
    [CrossRef] [PubMed]
  19. S. R. Arridge, \Photon measurement density functions. Part 1: Analytic forms," Appl. Opt. 34, 7395-7409 (1995).
    [CrossRef] [PubMed]
  20. S. R. Arridge and M. Schweiger, \Photon measurement density functions. Part 2: Finite element calculations," Appl. Opt. 34, 8026-8037 (1995).
    [CrossRef] [PubMed]
  21. M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd edition (Wiley, New York, 1993).
  22. S. R. Arridge and M. Schweiger, \Direct calculation of the moments of the distribution of photon time of ight in tissues with a finite-element method," Appl. Opt. 34, 2683-2687 (1995).
    [CrossRef] [PubMed]
  23. M. Schweiger and S. R. Arridge, \Direct calculation of the Laplace transform of the distribution of photon time of ight in tissue with a finite-element method," Appl. Opt. 36, 9042-9049 (1997).
    [CrossRef]
  24. M. Schweiger, S. R. Arridge and D. T. Delpy, \Application of the finite-element method for the forward and inverse models in optical tomography," J. Math Imag. Vision 3, 263-283 (1993).
    [CrossRef]
  25. K. D. Paulsen and H. Jiang, \Spatially-varying optical property reconstruction using a finite element diffusion equation approximation," Med. Phys. 22, 691-701 (1995).
    [CrossRef] [PubMed]
  26. S. S. Saquib, K. M. Hanson and G. S. Cunningham, \Model-based image reconstruction from time-resolved diffusion data," in Medical Imaging: Image Processing, K. M. Hanson, ed., Proc SPIE 3034, 369-380 (1997).
    [CrossRef]
  27. O. Dorn, Das inverse Transportproblem in der Lasertomographie, Ph. D. thesis, University of M"unster, 1997.
  28. R. Roy, Image reconstruction from light measurements on biological tissue, Ph. D. thesis, Uni- versity of Hertfordshire, 1997.
  29. S. R. Arridge and M. Schweiger, \A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method," in Computational Radiology and Imaging: Therapy and Diagnosis C. Borgers and F. Natterer, eds., IMA Volumes in Mathematics and its Applications (Springer 1998, in press).
  30. S. R. Arridge, M. Hiraoka and M. Schweiger, \Statistical basis for the determination of optical pathlength in tissue," Phys. Med. Biol. 40, 1539-1558 (1995).
    [CrossRef] [PubMed]
  31. M. Schweiger and S. R. Arridge, \Optimal data types in optical tomography," in Information Processing in Medical Imaging, Lect. Notes Comput. Sci., 1230, 71-84 (1997).
    [CrossRef]

Other

A. D. Edwards, J. S. Wyatt, C. E. Richardson, D. T. Delpy, M. Cope and E. O. R. Reynolds, \Cotside measurement of cerebral blood ow in ill newborn infants by near infrared spec- troscopy," Lancet 2, 770-771 (1988).
[CrossRef] [PubMed]

S. Wyatt, M. Cope, D. T. Delpy, C. E. Richardson, A. D. Edwards, S. C. Wray and E. O. R. Reynolds, \Quantitation of cerebral blood volume in newborn infants by near infrared spectroscopy," J. Appl. Physiol. 68, 1086-1091 (1990).
[PubMed]

M. Tamura, \Multichannel near-infrared optical imaging of human brain activity," in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC 1996) Vol. 2, pp. 8-10.

J. C. Hebden, R. A. Kruger and K. S. Wong, \Time resolved imaging through a highly scattering medium," Appl. Opt. 30, 788-794 (1991).
[CrossRef] [PubMed]

Near-infrared spectroscopy and imaging of living systems, special issue of Philos. Trans. R. Soc. London Ser. B, Vol. 352 (1997).

J. C. Hebden, S. R. Arridge and D. T. Delpy, \Optical imaging in medicine: I. Experimental techniques," Phys. Med. Biol. 42, 825-840 (1997).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, \Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

S. B. Colak, G. W. Hooft, D. G. Papaioannou and M. B. van der Mark, \3D backprojection tomography for medical optical imaging," in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC 1996) Vol. 2, pp. 294-298.

S. A. Walker, S. Fantini and E. Gratton, \Back-projection reconstructions of cylindrical inho- mogeneities from frequency domain optical measurements in turbid media," in OSA Trends in Optics and Photonics on Advances in Optical Imaging and Photon Migration, R. R. Alfano and J. G. Fujimoto, eds. (Optical Society of America, Washington, DC 1996) Vol. 2, pp. 137-141.

M. A. O Leary, D. A. Boas, B. Chance and A. G. Yodh, \Experimental images of heterogeneous turbid media by frequency-domain diffusing-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 1767, 372-383 (1992).
[CrossRef]

H. 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 (1995).
[CrossRef]

B. W. Pogue, M. S. Patterson, H. Jiang and K. D. Paulsen, \Initial assessment of a simple system for frequency domain diffuse optical tomography," Phys. Med. Biol. 40, 1709-1729 (1995).
[CrossRef] [PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray and J. Wyatt, \Estimation of optical pathlength through tissue from direct time of ight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

B. Chance, M. Maris, J. Sorge and M. Z. Zhang, \A phase modulation system for dual wave- length difference spectroscopy of haemoglobin deoxygenation in tissue," in Time-resolved Laser Spectroscopy in Biochemistry II, J. R. Lakowicz, ed., Proc. SPIE 1204, 481-491 (1990).
[CrossRef]

J. D. Moulton, Diffusion modelling of picosecond laser pulse propagation in turbid media, M. Eng. thesis, McMaster University, Hamilton, Ontario (1990).

S. R. Arridge, M. Schweiger, M. Hiraoka and D. T. Delpy, \A finite element approach for modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka and D. T. Delpy, \The finite element model for the propagation of light in scattering media: Boundary and source conditions," Med. Phys. 22, 1779-1792 (1995).
[CrossRef] [PubMed]

S. R. Arridge, \Photon measurement density functions. Part 1: Analytic forms," Appl. Opt. 34, 7395-7409 (1995).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, \Photon measurement density functions. Part 2: Finite element calculations," Appl. Opt. 34, 8026-8037 (1995).
[CrossRef] [PubMed]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd edition (Wiley, New York, 1993).

S. R. Arridge and M. Schweiger, \Direct calculation of the moments of the distribution of photon time of ight in tissues with a finite-element method," Appl. Opt. 34, 2683-2687 (1995).
[CrossRef] [PubMed]

M. Schweiger and S. R. Arridge, \Direct calculation of the Laplace transform of the distribution of photon time of ight in tissue with a finite-element method," Appl. Opt. 36, 9042-9049 (1997).
[CrossRef]

M. Schweiger, S. R. Arridge and D. T. Delpy, \Application of the finite-element method for the forward and inverse models in optical tomography," J. Math Imag. Vision 3, 263-283 (1993).
[CrossRef]

K. D. Paulsen and H. Jiang, \Spatially-varying optical property reconstruction using a finite element diffusion equation approximation," Med. Phys. 22, 691-701 (1995).
[CrossRef] [PubMed]

S. S. Saquib, K. M. Hanson and G. S. Cunningham, \Model-based image reconstruction from time-resolved diffusion data," in Medical Imaging: Image Processing, K. M. Hanson, ed., Proc SPIE 3034, 369-380 (1997).
[CrossRef]

O. Dorn, Das inverse Transportproblem in der Lasertomographie, Ph. D. thesis, University of M"unster, 1997.

R. Roy, Image reconstruction from light measurements on biological tissue, Ph. D. thesis, Uni- versity of Hertfordshire, 1997.

S. R. Arridge and M. Schweiger, \A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method," in Computational Radiology and Imaging: Therapy and Diagnosis C. Borgers and F. Natterer, eds., IMA Volumes in Mathematics and its Applications (Springer 1998, in press).

S. R. Arridge, M. Hiraoka and M. Schweiger, \Statistical basis for the determination of optical pathlength in tissue," Phys. Med. Biol. 40, 1539-1558 (1995).
[CrossRef] [PubMed]

M. Schweiger and S. R. Arridge, \Optimal data types in optical tomography," in Information Processing in Medical Imaging, Lect. Notes Comput. Sci., 1230, 71-84 (1997).
[CrossRef]

Supplementary Material (8)

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Figures (4)

Figure 1.
Figure 1.

Target image (col. 1), reconstruction after 100 iterations with conjugate gradient method (col. 2), with steepest descent method (col. 3) and with ART method (col. 4). In all cases, the top image is μa , and the bottom image is μs . The animations linked to the figure show the first 50 reconstruction iterations, and the final 50 in steps of 10. [Media 1] [Media 2] [Media 3] [Media 4] [Media 5] [Media 6]

Figure 2.
Figure 2.

L2 data norms as a function of iteration number, for different algorithms, as a function of iteration number (left) and of runtime (right).

Figure 3.
Figure 3.

L2 solution norms as a function of iteration number, for different algorithms. Left: absorption, right: scatter.

Figure 4.
Figure 4.

Target image (left col.), reconstruction after 100 iterations with conjugate gradient method (right col.). Top row is μa , bottom row is μs . The animations linked to the figure show the first 50 reconstruction iterations, and the final 50 in steps of 10. [Media 7] [Media 8]

Tables (2)

Tables Icon

Table 1. Optical parameters of circular test object.

Tables Icon

Table 2. Optical parameters of neonatal head model.

Equations (44)

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

F = 𝛲 [ μ , κ ]
Ψ = 1 2 j = 1 S i = 1 M j ( y j , i 𝛲 j , i [ μ , κ ] σ j , i ) 2
Ψ = 1 2 ( y F ) T R 2 ( y F ) = 1 2 b T b
R = diag R 1 R 2 R S = diag ( σ 1,1 , σ 1,2 , σ j , i , σ S , M s )
κ ( r ) Φ ̂ r ω + μ ( r ) Φ ̂ ( r , ω ) + ιω c Φ ̂ r ω = q ̂ 0 r t ,
κ ( r ) Φ r t + μ ( r ) Φ ( r , t ) + 1 c Φ ( r , t ) t = q 0 r t ,
Γ ( ξ ) = ( ξ ) n ̂ Φ ( ξ ) ,
Φ ( ξ ) + κ α n ̂ Φ ( ξ ) = 0 ,
( K ( κ ) + C ( μ ) + αA + ιω B ) Φ ( ω ) = Q ( ω )
( K ( κ ) + C ( μ ) + αA ) Φ ( t ) + B Φ ( t ) t = Q ( t )
K ij = Ω κ ( r ) u i ( r ) u j ( r ) d n r
C ij = Ω μ ( r ) u i ( r ) u j ( r ) d n r
B ij = 1 c Ω u i ( r ) u j ( r ) d n r
A ij = Ω u i ( r ) u j ( r ) d ( Ω )
F j = 𝚳 [ Φ j ]
𝚳 ̅ = τ ε
time - gated intensity : 𝚳 E ¯ ( T ) [ Φ ( t ) ] = 1 ε [ Φ ( t ) ] 0 T B [ Φ ( t ) ] dt ,
n - th temporal moment : 𝚳 t n [ Φ ( t ) ] = 1 ε [ Φ ( t ) ] 0 t n B [ Φ ( t ) ] dt ,
n - th central moment : 𝚳 c n [ Φ ( t ) ] = 1 ε [ Φ ( t ) ] 0 ( t t ) n B [ Φ ( t ) ] dt ,
normalised Laplace transform : 𝚳 L ̅ ( s ) [ Φ ( t ) ] = 1 ε [ Φ ( t ) ] 0 e st B [ Φ ( t ) ] dt .
Ψ x k = j = 1 S i = 1 M j ( y j , i P j , i [ μ , κ ] σ j , i 2 ) ( P j , i [ μ , κ ] x k )
z = j = 1 S P ' j T R j 1 b j = P ' T R 1 b
= j = 1 S J j T b j = J T b
b 1 b 2 . . . b S = J 1 , ( μ ) J 1 , ( κ ) J 2 , ( μ ) J 2 , ( κ ) . . . . . . J S , ( μ ) J S , ( κ ) Δ μ Δ κ
J j i = P MD F ( j , i ) T = [ P MD F ( j , i ) , ( μ ) T P MD F ( j , i ) , ( κ ) T ]
z = J T b
= j = 1 S J j T b j
= j = 1 S j = 1 M j P MD F ( j , i ) T b j , i
( K + C + αA ι ω B ) Φ m + ( ω ) = Q i + ( ω )
z ( μ ) = j = 1 S i = 1 M j b j , i σ j , i Φ i + ( ω ) × Φ j ( ω )
z ( κ ) = j = 1 S i = 1 M j b j , i σ j , i Φ i + ( ω ) × Φ j ( ω )
ν j + ( ω ) = i = 1 M j b j , i σ j , i Q i + ( ω )
( K + C + αA ι ω B ) η j + ( ω ) = ν j + ( ω )
z ( μ ) = j = 1 S η j + ( ω ) × Φ j ( ω )
z ( κ ) = j = 1 S η j + ( ω ) × Φ j ( ω )
J j i 𝚳 ̅ = 1 F j , i ε ( J j i 𝚳 F j , i 𝚳 ̅ J j i ε )
J 𝚳 ̅ T b 𝚳 ̅ = j S i M j J j i 𝚳 ̅ b j , i 𝚳 ̅
= j S i M j 1 F j , i ε ( J j i 𝚳 F j , i 𝚳 ̅ J j i ε ) b j , i 𝚳 ̅
z ( μ ) 𝚳 ̅ = j S i M j b j , i 𝚳 ̅ σ j , i 𝚳 ̅ F j , i ε ( τ [ Φ i + × Φ j ] F j , i 𝚳 ̅ Φ i + × Φ j )
z ( κ ) 𝚳 ̅ = j S i M j b j , i 𝚳 ̅ σ j , i 𝚳 ̅ F j , i ε ( τ [ Φ i + × Φ j ] F j , i 𝚳 ̅ Φ i + × Φ j )
ν j 𝚳 ̅ + ( 0 ) = i = 1 M j b j , i 𝚳 ̅ F j , i 𝚳 ̅ σ j , i 𝚳 ¯ F j , i ε Q i +
ν j 𝚳 ̅ + ( 1 ) = i = 1 M j b j , i 𝚳 ̅ σ j , i 𝚳 ¯ F j , i ε Q i +
z ( μ ) 𝚳 ̅ = j S τ [ η j 𝚳 ̅ + ( 1 ) × Φ j ] η j 𝚳 ¯ + ( 0 ) × Φ j
z ( κ ) 𝚳 ̅ = j S τ [ η j 𝚳 ̅ + ( 1 ) × Φ j ] η j 𝚳 ¯ + ( 0 ) × Φ j

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