Abstract

Optical tomography is modelled as an inverse problem for the time-dependent linear transport equation. We decompose the linearized residual operator of the problem into absorption and scattering transport sensitivity functions. We show that the adjoint linearized residual operator has a similar physical meaning in optical tomography as the ‘backprojection’ operator in x-ray tomography. In this interpretation, the geometric patterns onto which the residuals are backprojected are given by the same absorption and scattering transport sensitivity functions which decompose the forward residual operator. Moreover, the ‘backtransport’ procedure, which has been introduced in an earlier paper by the author, can then be interpreted as an efficient scheme for ‘backprojecting’ all (filtered) residuals corresponding to one source position simultaneously into the parameter space by just solving one adjoint transport problem. Numerical examples of absorption and scattering transport sensitivity functions for various situations (including applications with voids) are presented.

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References

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  1. S. R. Arridge, "Photon-measurement density functions. Part I: Analytical forms," Appl. Opt. 34 (34), 7395-7409 (1995)
    [CrossRef] [PubMed]
  2. S. R. Arridge and M. Schweiger, "Photon-measurement density functions. Part II: Finite-element-method calculations," Appl. Opt. 34 (31), 8026-8037 (1995)
    [CrossRef] [PubMed]
  3. S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15 (2), R41-R93 (1999)
    [CrossRef]
  4. S. R. Arridge, H. Dehgani, M. Schweiger and E. Okada, "The Finite Element Model for the Propagation of Light in Scattering Media: A Direct Method for Domains with Non-Scattering Regions," Medical Physics, 27 (1), 252-264 (2000)
    [CrossRef] [PubMed]
  5. K. M. Case and P. F. Zweifel, "Linear Transport Theory" (Plenum Press, New York, 1967)
  6. S. Chandrasekhar, "Radiative Transfer" (Dover, New York, 1960)
  7. S. B. Colak, D. G. Papanioannou, G. W. 't. Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen and N. A. A. J. van Asten, "Tomographic image reconstruction from optical projections in light-diffusing media," Appl. Opt. 36, 180-213 (1977)
    [CrossRef]
  8. T. Dierkes, "Rekonstruktionsverfahren zur optischen Tomographie," Thesis, Preprints "Angewandte Mathematik und Informatik" 16/00-N, Münster (2000)
  9. O. Dorn, "Das inverse Transportproblem in der Lasertomographie," Thesis, Preprints "Angewandte Mathematik und Informatik" 7/97-N, Münster (1997)
  10. O. Dorn, "A transport-backtransport method for optical tomography," Inverse Problems 14, 1107-1130 (1998)
    [CrossRef]
  11. S. Feng, Z. A. Zeng and B. Chance, "Photon migration in the presence of a single defect: a perturbation analysis," Appl. Opt. 34, 3826-3837 (1995)
    [CrossRef] [PubMed]
  12. R. J. Gaudette, C. A. Brooks, M. E. Kilmer, E. L. Miller, T. Gaudette, D. Boas, "A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient," Phys. Med. Biol. 45 (4) 1051-1070 (2000)
    [CrossRef] [PubMed]
  13. J. P. Kaltenbach and M. Kaschke, "Frequency and Time-Domain Modelling of Light Transport in Random Media," in: Müller, G. (ed.): Medical Optical Tomography: Functional Imaging and Monitoring (1993)
  14. 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]
  15. V. Kolehmainen, S. R. Arridge, W. R. B. Lionheart, M. Vauhkonen and J. P. Kaipio, "Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data" Inverse Problems 15, 1375-1391 (1999)
    [CrossRef]
  16. M. Moscoso, J. B. Keller and G. Papanicolaou, "Depolarization and blurring of optical images by biological tissues," to appear in J. Opt. Soc. Am. (2000)
  17. F. Natterer, "The Mathematics of Computerized Tomography" (Stuttgart, Teubner, 1986)
  18. F. Natterer and F. Wübbeling, "A propagation-backpropagation method for ultrasound tomography," Inverse Problems 11, 1225-1232 (1995)
    [CrossRef]
  19. F. Natterer, "Numerical Solution of Bilinear Inverse Problems," Preprints "Angewandte Mathematik und Informatik" 19/96-N, Münster (1996)
  20. E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope and D. T. Delpy, "Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head," Appl. Opt. 36 (1), 21-31 (1997)
    [CrossRef] [PubMed]
  21. J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll and M. Nieto-Vesperinas, "3D Optical Tomography in the Presence of Void Regions," Opt. Express 7, 462-467 (2000), http://www.opticsexpress.org/oearchive/source/26894.htm
  22. L. Ryzhik, G. Papanicolaou and J. B. Keller, "Transport equations for elastic and other waves in random media," Wave Motion 24, 327-370 (1996)
    [CrossRef]
  23. J. M. Schmitt, A. H. Gandjbakhche and R. F. Bonner, "Use of polarized light to discriminate short-path photons in a multiply scattering medium," Appl. Opt. 31, 6535-6546 (1992)
    [CrossRef] [PubMed]
  24. J. C. Schotland, J. C. Haselgrove and J. S. Leigh, "Photon hitting density," Appl. Opt. 32, 448-453 (1993)
    [CrossRef] [PubMed]

Other

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

S. R. Arridge and M. Schweiger, "Photon-measurement density functions. Part II: Finite-element-method calculations," Appl. Opt. 34 (31), 8026-8037 (1995)
[CrossRef] [PubMed]

S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15 (2), R41-R93 (1999)
[CrossRef]

S. R. Arridge, H. Dehgani, M. Schweiger and E. Okada, "The Finite Element Model for the Propagation of Light in Scattering Media: A Direct Method for Domains with Non-Scattering Regions," Medical Physics, 27 (1), 252-264 (2000)
[CrossRef] [PubMed]

K. M. Case and P. F. Zweifel, "Linear Transport Theory" (Plenum Press, New York, 1967)

S. Chandrasekhar, "Radiative Transfer" (Dover, New York, 1960)

S. B. Colak, D. G. Papanioannou, G. W. 't. Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen and N. A. A. J. van Asten, "Tomographic image reconstruction from optical projections in light-diffusing media," Appl. Opt. 36, 180-213 (1977)
[CrossRef]

T. Dierkes, "Rekonstruktionsverfahren zur optischen Tomographie," Thesis, Preprints "Angewandte Mathematik und Informatik" 16/00-N, Münster (2000)

O. Dorn, "Das inverse Transportproblem in der Lasertomographie," Thesis, Preprints "Angewandte Mathematik und Informatik" 7/97-N, Münster (1997)

O. Dorn, "A transport-backtransport method for optical tomography," Inverse Problems 14, 1107-1130 (1998)
[CrossRef]

S. Feng, Z. A. Zeng and B. Chance, "Photon migration in the presence of a single defect: a perturbation analysis," Appl. Opt. 34, 3826-3837 (1995)
[CrossRef] [PubMed]

R. J. Gaudette, C. A. Brooks, M. E. Kilmer, E. L. Miller, T. Gaudette, D. Boas, "A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient," Phys. Med. Biol. 45 (4) 1051-1070 (2000)
[CrossRef] [PubMed]

J. P. Kaltenbach and M. Kaschke, "Frequency and Time-Domain Modelling of Light Transport in Random Media," in: Müller, G. (ed.): Medical Optical Tomography: Functional Imaging and Monitoring (1993)

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]

V. Kolehmainen, S. R. Arridge, W. R. B. Lionheart, M. Vauhkonen and J. P. Kaipio, "Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data" Inverse Problems 15, 1375-1391 (1999)
[CrossRef]

M. Moscoso, J. B. Keller and G. Papanicolaou, "Depolarization and blurring of optical images by biological tissues," to appear in J. Opt. Soc. Am. (2000)

F. Natterer, "The Mathematics of Computerized Tomography" (Stuttgart, Teubner, 1986)

F. Natterer and F. Wübbeling, "A propagation-backpropagation method for ultrasound tomography," Inverse Problems 11, 1225-1232 (1995)
[CrossRef]

F. Natterer, "Numerical Solution of Bilinear Inverse Problems," Preprints "Angewandte Mathematik und Informatik" 19/96-N, Münster (1996)

E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope and D. T. Delpy, "Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head," Appl. Opt. 36 (1), 21-31 (1997)
[CrossRef] [PubMed]

J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll and M. Nieto-Vesperinas, "3D Optical Tomography in the Presence of Void Regions," Opt. Express 7, 462-467 (2000), http://www.opticsexpress.org/oearchive/source/26894.htm

L. Ryzhik, G. Papanicolaou and J. B. Keller, "Transport equations for elastic and other waves in random media," Wave Motion 24, 327-370 (1996)
[CrossRef]

J. M. Schmitt, A. H. Gandjbakhche and R. F. Bonner, "Use of polarized light to discriminate short-path photons in a multiply scattering medium," Appl. Opt. 31, 6535-6546 (1992)
[CrossRef] [PubMed]

J. C. Schotland, J. C. Haselgrove and J. S. Leigh, "Photon hitting density," Appl. Opt. 32, 448-453 (1993)
[CrossRef] [PubMed]

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Figures (9)

Fig. 1.
Fig. 1.

The geometries of the four experiments. Displayed is the distribution of the scattering cross section b. The values are b=1.0 cm-1 inside the voids, and b=100.0 cm-1 in the background. Top left: experiment 1; Top right: experiment 2; Bottom left: experiment 3; Bottom right: experiment 4. Indicated in the top left image are also the boundary points P1, P2 and P3, where the sources and receivers are located.

Fig. 2.
Fig. 2.

Left: Data corresponding to a source at boundary point P1=(40, 0) and a receiver at P3=(60, 120). Right: Data corresponding to a source at boundary point P2=(120, 40) and a receiver at P3=(60, 120). In both images the curves correspond to (from bottom to top) experiment 1 (blue, dash-dotted), 4 (red, solid), 2 (green, stars), and 3 (magenta, dashed). The vertical lines in the figures indicate at which time steps (T1=50 and T2=120) the sensitivity functions displayed in this paper have been calculated.

Fig. 3.
Fig. 3.

Transport absorption sensitivity functions -ψ a (xr , tr ; xsc ) for a source located at P1 and a receiver at P3. The receiving time is tr =10 s, which corresponds to the time step T1=50. Top left: experiment 1 (N=99.9); Top right: experiment 2 (N=664.1); Bottom left: experiment 3 (N=1000.0); Bottom right: experiment 4 (N=221.8).

Fig. 4.
Fig. 4.

Transport absorption sensitivity functions -ψ a (xr , tr ; xsc ) for a source located at P1 and a receiver at P3. The receiving time is tr =24 s, which corresponds to the time step T2=120. Top left: experiment 1 (N=3.0×104); Top right: experiment 2 (N=5.4×104); Bottom left: experiment 3 (N=7.3×104); Bottom right: experiment 4 (N=4.1×104).

Fig. 5.
Fig. 5.

Transport absorption sensitivity functions -ψ a (xr , tr ; xsc ) for a source located at P2 and a receiver at P3. The receiving time is tr =10 s, which corresponds to the time step T1=50. Top left: experiment 1 (N=5.6×103); Top right: experiment 2 (N=8.5×104); Bottom left: experiment 3 (N=1.0×105); Bottom right: experiment 4 (N=1.3×104).

Fig. 6.
Fig. 6.

Transport absorption sensitivity functions -ψ a (xr , tr ; xsc ) for a source located at P2 and a receiver at P3. The receiving time is tr =24 s, which corresponds to the time step T2=120. Top left: experiment 1(N=9.8×104); Top right: experiment 2(N=1.7×105); Bottom left: experiment 3(N=2.3×105); Bottom right: experiment 4 (N=1.4×105).

Fig. 7.
Fig. 7.

Transport scattering sensitivity functions ψ b (xr , tr ; xsc ) for a source located at P1 and a receiver at P3. The receiving time is tr =10 s, which corresponds to the time step T1=50. Top left: experiment 1 (N=1.3); Top right: experiment 2 (N=166.6); Bottom left: experiment 3 (N=164.3); Bottom right: experiment 4 (N=3.45).

Fig. 8.
Fig. 8.

Displayed is the same transport scattering sensitivity function as in example 3 of Figure 7, but calculated with different numerical discretizations. Left: 12 direction vectors are used as in Figure 7, but a finer spatial grid of 240×240 pixels instead of 120×120 pixels. Right: A 120×120 grid is used as in Figure 7, but 24 discretized directions instead of 12 directions. Shown are ψ b (xr , tr ; xsc ) for a source located at P1 and a receiver at P3. The receiving time step is T1=50.

Fig. 9.
Fig. 9.

Transport scattering sensitivity functions ψb(xr , tr ; xsc ) for a source located at P1 and a receiver at P3. The receiving time is tr =24 s, which corresponds to the time step T2=120. Top left: experiment 1 (N=144.3); Top right: experiment 2 (N=443.0); Bottom left: experiment 3 (N=619.1); Bottom right: experiment 4 (N=204.0).

Equations (46)

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1 c u t + θ · u ( x , θ , t ) + ( a ( x ) + b ( x ) ) u ( x , θ , t ) b ( x ) S n 1 η ( θ · θ ) u ( x , θ , t ) d θ
= δ ( t ) δ ( x x s ) δ ( θ θ s ) in Ω × S n 1 × [ 0 , T ] .
u ( x , θ , 0 ) = 0 in Ω × S n 1 ,
u ( x , θ , t ) = 0 on Γ .
η ( θ · θ ) = 1 g 2 2 ( 1 + g 2 2 g cos ϑ ) 3 / 2
q ( x , θ , t ) = δ ( t ) δ ( x x s ) δ ( θ θ s ) .
M a , b ( x r , t r ) = S + n -1 ν ( x r ) · θ u ( x r , θ , t r ) on Ω × [ 0 , T ] ,
R : P × P D , R ( a , b ) ( x r , t r ) = M a , b ( x r , t r ) G ˜ ( x r , t r ) .
R a , b : P × P D , R a , b ( δa , δb ) ( x r , t r ) = S + n 1 ν ( x r ) · θω ( x r , θ , t r )
ω t + θ · ω ( x , θ , t ) + ( a ( x ) + b ( x ) ) ω ( x , θ , t ) b ( x ) S n 1 η ( θ · θ ) ω ( x , θ , t ) )
= Q δa ( x , θ , t ) + Q δb ( x , θ , t )
ω ( x , θ , 0 ) = 0 in Ω × S n 1 ,
ω ( x , θ , t ) = 0 on Γ .
Q δa ( x , θ , t ) = δ a ( x ) u ( x , θ , t )
Q δb ( x , θ , t ) = δb ( x ) ( u ( x , θ , t ) S n 1 η ( θ · θ ) u ( x , θ , t ) )
R a , b ( δa , δb ) , ς D = ( δa , δb ) , R a , b * ς P × P
z t θ · z ( x , θ , t ) + ( a ( x ) + b ( x ) ) z ( x , θ , t ) b ( x ) S n 1 η ( θ · θ ) z ( x , θ , t )
= 0 in Ω × S n 1 × [ 0 , T ]
z ( x , θ , T ) = 0 in Ω × S n 1 ,
z ( x , θ , t ) = ς ( x , t ) uniformly in θ on Γ + ,
( R a , b * ς ) ( x ) = ( Δ a ( x ) , Δ b ( x ) )
Δ a ( x ) = [ 0 , T ] S n 1 u ( x , θ , t ) z ( x , θ , t ) dt ,
Δ a ( x ) = [ 0 , T ] S n 1 [ S n 1 η ( θ · θ ) u ( x , θ , t ) u ( x , θ , t ) ] z ( x , θ , t ) dt .
( R a , b ( δa , δb ) ( x r , t r ) =
Ω dx sc Ψ a ( x r , t r ; x sc ) δa ( x sc ) + Ω dx sc Ψ b ( x r , t r ; x sc ) δb ( x sc )
( R a , b * ς ) ( x sc ) =
( Ω dx r [ 0 , T ] dt r Ψ a ( x r , t r ; x sc ) ς ( x r , t r ) , Ω dx r [ 0 , T ] dt r Ψ b ( x r , t r ; x sc ) ς ( x r , t r ) )
R a , b ( δa , δb ) = R ( a , b ) .
( δa , δb ) = R a , b * ( R a , b R a , b * ) 1 R ( a , b ) .
G t + θ · G ( x , θ , t ) + ( a ( x ) + b ( x ) ) G ( x , θ , t ) b ( x ) S n 1 η ( θ · θ ) G ( x , θ , t )
= δ ( x x sc ) δ ( θ θ sc ) δ ( t t sc )
G ( x , θ , 0 ) = 0 in Ω × s n 1 ,
G ( x , θ , t ) = 0 on Γ .
G ̂ t θ · G ̂ ( x , θ , t ) + ( a ( x ) + b ( x ) ) G ̂ ( x , θ , t ) b ( x ) S n 1 η ( θ · θ ) G ̂ ( x , θ , t )
= δ ( x x r ) δ ( θ θ r ) δ ( t t r )
G ̂ ( x , θ , T ) = 0 in Ω × S n 1 ,
G ̂ ( x , θ , t ) = 0 on Γ + .
Φ a [ x r , θ r , t r ] ( x sc , θ sc , t sc ) = G ̂ [ x r , θ r , t r ] ( x sc , θ sc , t sc ) G [ x s , θ s , 0 ] ( x sc , θ sc , t sc ) .
( R a , b ( δa , 0 ) ) ( x r , t r ) = - S + n 1 r Ω dx sc S n 1 sc [ 0 , T ] dt sc ν ( x r ) · θ r
× G [ x s , θ s , 0 ] ( x sc , θ sc , t sc ) G [ x sc , θ sc , t sc ] ( x r , θ r , t r ) δa ( x sc )
G [ x sc , θ sc , t sc ] ( x r , θ r , t r ) = G ̂ [ x r , θ r , t r ] ( x sc , θ sc , t sc ) ,
Ψ a ( x r , t r ; x sc ) = S + n 1 r ν ( x r ) · θ r [ 0 , T ] dt sc S n 1 sc Φ a [ x r , θ r , t r ] ( x sc , θ sc , t sc ) ,
( R a , b ( δa , 0 ) ) ( x r , t r ) = Ω dx sc Ψ a ( x r , t r ; x sc ) δa ( x sc ) ,
Φ b [ x r , θ r , t r ] ( x sc , θ sc , t sc ) = G ̂ [ x r , θ r , t r ] ( x sc , θ sc , t sc ) ×
[ G [ x s , θ s , 0 ] ( x sc , θ sc , t sc ) S n 1 η ( θ sc · θ ) G [ x s , θ s , 0 ] ( x sc , θ , t sc ) ] .
Ψ b ( x r , t r , x sc ) = S + n 1 r ν ( x r ) . θ r [ 0 , T ] dt sc S n 1 sc Φ b [ x r , θ r , t r ] ( x sc , θ sc , t sc ) ,

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