Abstract

A condition on nonuniqueness in optical tomography is stated. The main result applies to steady-state (dc) diffusion-based optical tomography, wherein we demonstrate that simultaneous unique recovery of diffusion and absorption coefficients cannot be achieved. A specific example of two images that give identical dc data is presented. If the refractive index is considered an unknown, then nonuniqueness also occurs in frequency-domain and time-domain optical tomography, if the underlying model of the diffusion approximation is employed.

© 1998 Optical Society of America

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  1. J. C. Hebden, S. R. Arridge, and D. T. Delpy, Phys. Med. Biol. 42, 825 (1997).
    [CrossRef] [PubMed]
  2. S. R. Arridge and J. C. Hebden, Phys. Med. Biol. 42, 841 (1997).
    [CrossRef] [PubMed]
  3. H. Jiang, K. D. Paulsen, and U. L. Osterberg, Phys. Med. Biol. 41, 1483 (1996).
    [CrossRef] [PubMed]
  4. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, J. Opt. Soc. Am. A 13, 253 (1995).
    [CrossRef]
  5. S. R. Arridge and M. Schweiger, Proc. SPIE 2389, 378 (1995).
    [CrossRef]
  6. J. Sylvester and G. Uhlmann, Ann. Math. 125, 153 (1987).
    [CrossRef]
  7. V. Isakov, Inverse Problems 9, 579 (1993).
    [CrossRef]
  8. B. J. Hoenders, J. Opt. Soc. Am. A 14, 262 (1997).
    [CrossRef]
  9. M. R. Ostermeyer and S. L. Jacques, J. Opt. Soc. Am. A 14, 255 (1997).
    [CrossRef]
  10. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, Med. Phys. 22, 1779 (1995).
    [CrossRef] [PubMed]
  11. V. Isakov, Inverse Problems in Partial Differential Equations (Springer, New York, 1998).
    [CrossRef]
  12. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Deply, Med. Phys. 20, 299 (1993).
    [CrossRef] [PubMed]

1997 (4)

J. C. Hebden, S. R. Arridge, and D. T. Delpy, Phys. Med. Biol. 42, 825 (1997).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, Phys. Med. Biol. 42, 841 (1997).
[CrossRef] [PubMed]

B. J. Hoenders, J. Opt. Soc. Am. A 14, 262 (1997).
[CrossRef]

M. R. Ostermeyer and S. L. Jacques, J. Opt. Soc. Am. A 14, 255 (1997).
[CrossRef]

1996 (1)

H. Jiang, K. D. Paulsen, and U. L. Osterberg, Phys. Med. Biol. 41, 1483 (1996).
[CrossRef] [PubMed]

1995 (3)

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, J. Opt. Soc. Am. A 13, 253 (1995).
[CrossRef]

S. R. Arridge and M. Schweiger, Proc. SPIE 2389, 378 (1995).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, Med. Phys. 22, 1779 (1995).
[CrossRef] [PubMed]

1993 (2)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Deply, Med. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

V. Isakov, Inverse Problems 9, 579 (1993).
[CrossRef]

1987 (1)

J. Sylvester and G. Uhlmann, Ann. Math. 125, 153 (1987).
[CrossRef]

Arridge, S. R.

S. R. Arridge and J. C. Hebden, Phys. Med. Biol. 42, 841 (1997).
[CrossRef] [PubMed]

J. C. Hebden, S. R. Arridge, and D. T. Delpy, Phys. Med. Biol. 42, 825 (1997).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, Med. Phys. 22, 1779 (1995).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, Proc. SPIE 2389, 378 (1995).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Deply, Med. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

Delpy, D. T.

J. C. Hebden, S. R. Arridge, and D. T. Delpy, Phys. Med. Biol. 42, 825 (1997).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, Med. Phys. 22, 1779 (1995).
[CrossRef] [PubMed]

Deply, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Deply, Med. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

Hebden, J. C.

J. C. Hebden, S. R. Arridge, and D. T. Delpy, Phys. Med. Biol. 42, 825 (1997).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, Phys. Med. Biol. 42, 841 (1997).
[CrossRef] [PubMed]

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, Med. Phys. 22, 1779 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Deply, Med. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

Hoenders, B. J.

Isakov, V.

V. Isakov, Inverse Problems 9, 579 (1993).
[CrossRef]

V. Isakov, Inverse Problems in Partial Differential Equations (Springer, New York, 1998).
[CrossRef]

Jacques, S. L.

Jiang, H.

Osterberg, U. L.

Ostermeyer, M. R.

Patterson, M. S.

Paulsen, K. D.

Pogue, B. W.

Schweiger, M.

S. R. Arridge and M. Schweiger, Proc. SPIE 2389, 378 (1995).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, Med. Phys. 22, 1779 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Deply, Med. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

Sylvester, J.

J. Sylvester and G. Uhlmann, Ann. Math. 125, 153 (1987).
[CrossRef]

Uhlmann, G.

J. Sylvester and G. Uhlmann, Ann. Math. 125, 153 (1987).
[CrossRef]

Ann. Math. (1)

J. Sylvester and G. Uhlmann, Ann. Math. 125, 153 (1987).
[CrossRef]

Inverse Problems (1)

V. Isakov, Inverse Problems 9, 579 (1993).
[CrossRef]

J. Opt. Soc. Am. A (3)

Med. Phys. (2)

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, Med. Phys. 22, 1779 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Deply, Med. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

Phys. Med. Biol. (3)

J. C. Hebden, S. R. Arridge, and D. T. Delpy, Phys. Med. Biol. 42, 825 (1997).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, Phys. Med. Biol. 42, 841 (1997).
[CrossRef] [PubMed]

H. Jiang, K. D. Paulsen, and U. L. Osterberg, Phys. Med. Biol. 41, 1483 (1996).
[CrossRef] [PubMed]

Proc. SPIE (1)

S. R. Arridge and M. Schweiger, Proc. SPIE 2389, 378 (1995).
[CrossRef]

Other (1)

V. Isakov, Inverse Problems in Partial Differential Equations (Springer, New York, 1998).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Definition of Ω0 and Ω1. Ω0 is a region with every point at least a distance z0 from the domain boundary Ω. All isotropic sources are wholly contained within Ω1.

Fig. 2
Fig. 2

Mesh used for the example data, together with cross sections showing the functions in κ, μa, and μs that give rise to the same boundary data as constant values.

Fig. 3
Fig. 3

Relative difference of the intensity data Γμ˜a, κ-Γμa, κ/Γμa, κ (μa only), Γμa, κ˜-Γμa, κ/Γμa, κ (κ only), and Γμ˜a, κ˜-Γμa, κ/Γμa, κ (null space), for a source at θ=0.

Equations (15)

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

-·κΦˆω+μa+iωncΦˆω=qˆ0ω,
Γb=-cκbb·Φb,
-γ22Φˆ-2γ·γΦˆ+μaΦˆ+iωncΦˆ=qˆ0
-2Ψˆω+ηˆωΨˆω=qˆ0ωγ,
η0=2γγ+μaγ2,  ξ=ncγ2.
κ˜=κ+α,  μ˜a=μa+β.
2κ+α1/2κ+α1/2+μa+βκ+α=2κ1/2κ1/2+μaκ.
β=κ+α2κ1/2κ1/2+μaκ-2κ+α1/2κ+α1/2-μa.
n˜=n+ν.
ν=α/κn.
κ=ωncIηˆ  μa=κRηˆ-2κ1/2κ1/2.
β=14a4A+expr22a24a2Aκ+a2μ1+ακ-A+2expr22a2αr2.
-·κΦt+μaΦt+ncΦtt=q0t,
-2Ψt+η0Ψt+ξΨtt=q0tγ.
-2Ψ˜t-Ψt+η0Ψ˜t-Ψt+ξΨ˜t-Ψtt=0.

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