Abstract

A method for solving the two-dimensional inverse problems of optical diffusion tomography is proposed. The method is especially designed for the imaging of small inclusions embedded in the backgrounds of strongly scattering media. Numerical simulations show that the results are stable with respect to external noise at the boundary of the sample. The location of an inclusion is obtained with an accuracy of the order of several photon transport mean-free paths in the medium in cases both with and without noise in the scattering data used for the solution of the inverse problem.

© 1998 Optical Society of America

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References

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  1. D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
    [CrossRef] [PubMed]
  2. A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, (3) 34–40 (1995).
    [CrossRef]
  3. See B. Chance, R. Alfano, eds., Optical Tomography: Photon Migration and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE 2389 (1995), and references therein.
  4. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1990).
  5. R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1989).
  6. J. Chang, H. L. Graber, R. Barbour, “Luminescence optical tomography of dense scattering media,” J. Opt. Soc. Am. A 14, 288–299 (1997), and references therein.
  7. G. R. Carrier, M. Crook, C. E. Pearson, Functions of a Complex Variable: Theory and Techniques (McGraw-Hill, New York, 1966).
  8. F. D. Gakhov, Boundary Value Problems (Dover, New York, 1990).
  9. C. A. Brebbia, J. C. F. Telles, L. C. Wrobel, Boundary Element Techniques (Springer-Verlag, New York, 1984).
    [CrossRef]

1997 (1)

1995 (2)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, (3) 34–40 (1995).
[CrossRef]

See B. Chance, R. Alfano, eds., Optical Tomography: Photon Migration and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE 2389 (1995), and references therein.

1993 (1)

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Barbour, R.

Benaron, D. A.

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Brebbia, C. A.

C. A. Brebbia, J. C. F. Telles, L. C. Wrobel, Boundary Element Techniques (Springer-Verlag, New York, 1984).
[CrossRef]

Carrier, G. R.

G. R. Carrier, M. Crook, C. E. Pearson, Functions of a Complex Variable: Theory and Techniques (McGraw-Hill, New York, 1966).

Chance, B.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, (3) 34–40 (1995).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1990).

Chang, J.

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1989).

Crook, M.

G. R. Carrier, M. Crook, C. E. Pearson, Functions of a Complex Variable: Theory and Techniques (McGraw-Hill, New York, 1966).

Gakhov, F. D.

F. D. Gakhov, Boundary Value Problems (Dover, New York, 1990).

Graber, H. L.

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1989).

Pearson, C. E.

G. R. Carrier, M. Crook, C. E. Pearson, Functions of a Complex Variable: Theory and Techniques (McGraw-Hill, New York, 1966).

Stevenson, D. K.

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Telles, J. C. F.

C. A. Brebbia, J. C. F. Telles, L. C. Wrobel, Boundary Element Techniques (Springer-Verlag, New York, 1984).
[CrossRef]

Wrobel, L. C.

C. A. Brebbia, J. C. F. Telles, L. C. Wrobel, Boundary Element Techniques (Springer-Verlag, New York, 1984).
[CrossRef]

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, (3) 34–40 (1995).
[CrossRef]

J. Opt. Soc. Am. A (1)

Optical Tomography: Photon Migration and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation (1)

See B. Chance, R. Alfano, eds., Optical Tomography: Photon Migration and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE 2389 (1995), and references therein.

Phys. Today (1)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, (3) 34–40 (1995).
[CrossRef]

Science (1)

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Other (5)

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1990).

R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1989).

G. R. Carrier, M. Crook, C. E. Pearson, Functions of a Complex Variable: Theory and Techniques (McGraw-Hill, New York, 1966).

F. D. Gakhov, Boundary Value Problems (Dover, New York, 1990).

C. A. Brebbia, J. C. F. Telles, L. C. Wrobel, Boundary Element Techniques (Springer-Verlag, New York, 1984).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Geometry of the problem with one inclusion (not drawn to scale). The actual diameter of the inclusion is 2% of the emitter radius.

Fig. 2
Fig. 2

Geometry of the problem with two inclusions (not drawn to scale).

Fig. 3
Fig. 3

Angular flux distribution with one inclusion. The inclusion is located at α = π/2 (solid line), α = π (dotted line), or α = 3π/2 (dashed line).

Fig. 4
Fig. 4

Angular flux distribution with two inclusions. The curves correspond to various angles between inclusions. Thick solid curve, one inclusion; long-dashed curve, Δϕ = π/6; dotted curve, Δϕ = π/4; dashed–dotted curve, Δϕ = π/3; dashed–double-dotted curve; Δϕ = π/2; thin solid curve, Δϕ = π.

Fig. 5
Fig. 5

Angular flux distribution with noisy data. Solid curve, the background solution with one inclusion and no noise. Dotted curve, the distribution with white noise at the emitter. The magnitude of the random component was taken as ∼10% of the total signal.

Tables (3)

Tables Icon

Table 1 Results of Solving the Inverse Problem with One Inclusion

Tables Icon

Table 2 Results of Solving the Inverse Problem with Two Inclusions Located a Distance a from the Center

Tables Icon

Table 3 Results of Solving the Inverse Problem with One Inclusion and Noisy Data a

Equations (17)

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- i ω u ω = D Δ u ω - a r u ω ,
u ω r =   d r   a r D   G ω r ,   r u ω r + Γ d S G ω r ,   r n   u ω r - u ω r n   G ω r ,   r ,
L D = 2 A 0 D / ω ,
a r = k   a k δ r - r k ,
u r = k a k D   G r ,   r k u r k + Γ d S G r ,   r n   u r - u r n   G r ,   r ,
G r ,   r = - 1 2 π ln | r - r | .
w z = - k   α k ln z - z k w z k - 1 2 π i Γ d ξ   w ξ ξ - z ,
w z = J cot γ / 2 1 z - 1 + 1 2 ,
J φ = J 2   cot γ / 2 1 sin 2 ϕ / 2 .
δ j ϕ = 2 a 2 π 1 - r 2 1 + r 2 - 2 r   cos ϕ - α ,
a J   tan γ / 2 2   ln   d 1 - r 2 1 + r 2 - 2 r   cos α ,
J inc ϕ = J ϕ + J   tan γ / 2 2 π   ln d 1 - r 2 1 + r 2 - 2 r   cos ϕ - α × 1 - r 2 1 + r 2 - 2 r   cos α .
δ u r = Γ d S G r ,   r n   δ u r - δ u r n   G r ,   r ,
δ u r 2 δ u r 2 Γ Γ d S n   G r ,   r 2 .
δ j s   v s = - 1 4 π 0 2 π d σ u σ 1 sin 2 σ - s 2 ,
δ j 2 u 2   1 8 π 2 Δ / 2 π / 2 d ξ sin 4 ξ .
l t / l a 1 ,

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