Abstract

We present an investigation of the effect of a 3D non-scattering gap region on image reconstruction in diffuse optical tomography. The void gap is modelled by the Radiosity-Diffusion method and the inverse problem is solved using the adjoint field method. The case of a sphere with concentric spherical gap is used as an example.

© 2000 Optical Society of America

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References

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  1. S. R. Arridge, “Optical Tomography in Medical Imaging,” Inverse Problems 15, R41–R93 (1999).
    [CrossRef]
  2. S. R. Arridge and J. C. Hebden, “Optical Imaging in Medicine: II. Modelling and Reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
    [CrossRef] [PubMed]
  3. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of Finite-Difference Transport and Diffusion Calculations for Photon Migration in Homogeneous and Hetergeneous Tissue,” Phys. Med. Biol. 43, 1285–1302 (1998).
    [CrossRef] [PubMed]
  4. O. Dorn, “A Transport-BackTransport Method for Optical Tomography,” Inverse Problems 14, 1107–1130 (1998).
    [CrossRef]
  5. A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 261698–1707 (1999).
    [CrossRef] [PubMed]
  6. O. Dorn, “Scattering and absorption transport sensitivity functions for optical tomography,” Opt. Express This issue (2000).
  7. M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
    [CrossRef] [PubMed]
  8. S. R. Arridge, H. Dehghani, 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,” Med. Phys. 27, 252–264 (2000).
    [CrossRef] [PubMed]
  9. J. Ripoll, S. R. Arridge, H. Dehghani, and M. Nieto-Vesperinas, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
    [CrossRef]
  10. H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical Tomography in the Presence of Void Regions,” J. Opt. Soc. Am. A 17, 1659–1670 (2000).
    [CrossRef]
  11. M. Schweiger and S. R. Arridge, “Comparison of 2D and 3D reconstruction algorithms in Optical Tomography,” Appl. Opt. 37, 7419–7428 (1998).
    [CrossRef]
  12. J. Ripoll, Ph.D. thesis, University Autónoma of Madrid, 2000.
  13. S. R. Arridge and M. Schweiger, “The Finite Element Model for the Propagation of Light in Scattering Media: Boundary and Source Conditions,” Med. Phys. 22, 1779–1792, (1995).
    [CrossRef] [PubMed]
  14. F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001).
    [CrossRef]
  15. J. Schoberl, “NetGen”, http://www.sfb013.uni-linz.ac.at/joachim/netgen/
  16. M. F. Cohen and J. R. Wallace, Radiosity and Realistic Image Synthesis (Academic, London, 1993).
  17. H. R. Zatz, Master’s thesis, Cornell University, 1993.
  18. S. R. Arridge and M. Schweiger, “Photon Measurement Density Functions. Part 2: Finite Element Calculations,” Appl. Opt. 34, 8026–8037 (1995).
    [CrossRef] [PubMed]
  19. V. Kolehmainen, M. Vaukhonen, J. P. Kaipio, and S. R. Arridge, “Recovery of piecewise constant coefficients in optical diffusion tomography,” Opt. Express This issue (2000).

2000 (3)

1999 (2)

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 261698–1707 (1999).
[CrossRef] [PubMed]

S. R. Arridge, “Optical Tomography in Medical Imaging,” Inverse Problems 15, R41–R93 (1999).
[CrossRef]

1998 (3)

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of Finite-Difference Transport and Diffusion Calculations for Photon Migration in Homogeneous and Hetergeneous Tissue,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

O. Dorn, “A Transport-BackTransport Method for Optical Tomography,” Inverse Problems 14, 1107–1130 (1998).
[CrossRef]

M. Schweiger and S. R. Arridge, “Comparison of 2D and 3D reconstruction algorithms in Optical Tomography,” Appl. Opt. 37, 7419–7428 (1998).
[CrossRef]

1997 (1)

S. R. Arridge and J. C. Hebden, “Optical Imaging in Medicine: II. Modelling and Reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

1996 (1)

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

1995 (2)

S. R. Arridge and M. Schweiger, “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 and M. Schweiger, “Photon Measurement Density Functions. Part 2: Finite Element Calculations,” Appl. Opt. 34, 8026–8037 (1995).
[CrossRef] [PubMed]

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of Finite-Difference Transport and Diffusion Calculations for Photon Migration in Homogeneous and Hetergeneous Tissue,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

Arridge, S. R.

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical Tomography in the Presence of Void Regions,” J. Opt. Soc. Am. A 17, 1659–1670 (2000).
[CrossRef]

S. R. Arridge, H. Dehghani, 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,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

J. Ripoll, S. R. Arridge, H. Dehghani, and M. Nieto-Vesperinas, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
[CrossRef]

S. R. Arridge, “Optical Tomography in Medical Imaging,” Inverse Problems 15, R41–R93 (1999).
[CrossRef]

M. Schweiger and S. R. Arridge, “Comparison of 2D and 3D reconstruction algorithms in Optical Tomography,” Appl. Opt. 37, 7419–7428 (1998).
[CrossRef]

S. R. Arridge and J. C. Hebden, “Optical Imaging in Medicine: II. Modelling and Reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[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]

S. R. Arridge and M. Schweiger, “The Finite Element Model for the Propagation of Light in Scattering Media: Boundary and Source Conditions,” Med. Phys. 22, 1779–1792, (1995).
[CrossRef] [PubMed]

V. Kolehmainen, M. Vaukhonen, J. P. Kaipio, and S. R. Arridge, “Recovery of piecewise constant coefficients in optical diffusion tomography,” Opt. Express This issue (2000).

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of Finite-Difference Transport and Diffusion Calculations for Photon Migration in Homogeneous and Hetergeneous Tissue,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

Cohen, M. F.

M. F. Cohen and J. R. Wallace, Radiosity and Realistic Image Synthesis (Academic, London, 1993).

Dehghani, H.

Delpy, D. T.

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical Tomography in the Presence of Void Regions,” J. Opt. Soc. Am. A 17, 1659–1670 (2000).
[CrossRef]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Dorn, O.

O. Dorn, “A Transport-BackTransport Method for Optical Tomography,” Inverse Problems 14, 1107–1130 (1998).
[CrossRef]

O. Dorn, “Scattering and absorption transport sensitivity functions for optical tomography,” Opt. Express This issue (2000).

Firbank, M.

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Hebden, J. C.

S. R. Arridge and J. C. Hebden, “Optical Imaging in Medicine: II. Modelling and Reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Hielscher, A. H.

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 261698–1707 (1999).
[CrossRef] [PubMed]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of Finite-Difference Transport and Diffusion Calculations for Photon Migration in Homogeneous and Hetergeneous Tissue,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

Kaipio, J. P.

V. Kolehmainen, M. Vaukhonen, J. P. Kaipio, and S. R. Arridge, “Recovery of piecewise constant coefficients in optical diffusion tomography,” Opt. Express This issue (2000).

Klose, A. D.

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 261698–1707 (1999).
[CrossRef] [PubMed]

Kolehmainen, V.

V. Kolehmainen, M. Vaukhonen, J. P. Kaipio, and S. R. Arridge, “Recovery of piecewise constant coefficients in optical diffusion tomography,” Opt. Express This issue (2000).

Natterer, F.

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001).
[CrossRef]

Nieto-Vesperinas, M.

Okada, E.

S. R. Arridge, H. Dehghani, 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,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

Ripoll, J.

Schoberl, J.

J. Schoberl, “NetGen”, http://www.sfb013.uni-linz.ac.at/joachim/netgen/

Schweiger, M.

S. R. Arridge, H. Dehghani, 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,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, “Optical Tomography in the Presence of Void Regions,” J. Opt. Soc. Am. A 17, 1659–1670 (2000).
[CrossRef]

M. Schweiger and S. R. Arridge, “Comparison of 2D and 3D reconstruction algorithms in Optical Tomography,” Appl. Opt. 37, 7419–7428 (1998).
[CrossRef]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[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]

S. R. Arridge and M. Schweiger, “The Finite Element Model for the Propagation of Light in Scattering Media: Boundary and Source Conditions,” Med. Phys. 22, 1779–1792, (1995).
[CrossRef] [PubMed]

Vaukhonen, M.

V. Kolehmainen, M. Vaukhonen, J. P. Kaipio, and S. R. Arridge, “Recovery of piecewise constant coefficients in optical diffusion tomography,” Opt. Express This issue (2000).

Wallace, J. R.

M. F. Cohen and J. R. Wallace, Radiosity and Realistic Image Synthesis (Academic, London, 1993).

Wübbeling, F.

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001).
[CrossRef]

Zatz, H. R.

H. R. Zatz, Master’s thesis, Cornell University, 1993.

Appl. Opt. (2)

Inverse Problems (2)

S. R. Arridge, “Optical Tomography in Medical Imaging,” Inverse Problems 15, R41–R93 (1999).
[CrossRef]

O. Dorn, “A Transport-BackTransport Method for Optical Tomography,” Inverse Problems 14, 1107–1130 (1998).
[CrossRef]

J. Opt. Soc. Am. A (2)

Med. Phys. (3)

S. R. Arridge, H. Dehghani, 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,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 261698–1707 (1999).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, “The Finite Element Model for the Propagation of Light in Scattering Media: Boundary and Source Conditions,” Med. Phys. 22, 1779–1792, (1995).
[CrossRef] [PubMed]

Phys. Med. Biol. (3)

S. R. Arridge and J. C. Hebden, “Optical Imaging in Medicine: II. Modelling and Reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of Finite-Difference Transport and Diffusion Calculations for Photon Migration in Homogeneous and Hetergeneous Tissue,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Other (7)

J. Ripoll, Ph.D. thesis, University Autónoma of Madrid, 2000.

O. Dorn, “Scattering and absorption transport sensitivity functions for optical tomography,” Opt. Express This issue (2000).

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001).
[CrossRef]

J. Schoberl, “NetGen”, http://www.sfb013.uni-linz.ac.at/joachim/netgen/

M. F. Cohen and J. R. Wallace, Radiosity and Realistic Image Synthesis (Academic, London, 1993).

H. R. Zatz, Master’s thesis, Cornell University, 1993.

V. Kolehmainen, M. Vaukhonen, J. P. Kaipio, and S. R. Arridge, “Recovery of piecewise constant coefficients in optical diffusion tomography,” Opt. Express This issue (2000).

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Figures (4)

Fig. 1.
Fig. 1.

Left : cutaway of spherical mesh. Right : location of sources and detectors on the sphere surface

Fig. 2.
Fig. 2.

distribution of photon density, photons/mm3 (top row) and mean time of photon flight, picoseconds (bottom row) over sphere surface. Left to right, solid sphere (no gap), gap widths 3mm, 4mm, 5mm.

Fig. 3.
Fig. 3.

Sensitivity functions for the 3mm gap case. Left intensity (photons/mm2), right mean time(picoseconds mm). The functions plotted are cross-sections through the equatorial plane of the sphere. Also available as a QuickTime movie, pmdf.mov. (3.8MB)

Fig. 4.
Fig. 4.

Target images (top row) and reconstructions (bottom row) for the 3mm gap case. The images are transverse, sagittal and coronal slices through the true centre of the blob, orientated according to the diagram in the top right panel. Bottom right shows a profile along the equatorial diameter through the blob centre. A movie showing a rotating orthographic view is attached rotating3d.mov (3.8MB).

Equations (14)

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

· κ ( r ) Φ ( r ; ω ) + ( µ a ( r ) + i ω c ) Φ ( r ; ω ) = 0 r Ω d ( Ξ d Ω d )
Φ ( m ; ω ) + 2 A κ ( m ) Φ ( m ; ω ) ν = η ( m ; ω ) m Ω 1 +
Φ ( m ; ω ) + 2 A κ Φ ( m ; ω ) ν = 1 π Ξ cos θ cos θ Φ ( m ; ω ) 2 A h ( m , m ) ×
exp [ ( μ a + i ω c ) m m ] m m 2 d m m , m Ξ
cos θ = ν ̂ ( m ) · m m m m , cos θ = ν ̂ ( m ) · m m m m
y η ( m ; ω ) = 𝓟 η ( μ a κ ) κ ( m ) Φ η ( m ; ω ) ν , m Ω 1 +
C = 1 2 j = 1 s g j 𝓟 j μ a κ , g j 𝓟 j ( μ a κ ) L 2 ( Ω )
( α β ) = ( Re ( Φ j - Ψ j ) Re ( Φ j - · Ψ j ) )
· κ ( r ) Ψ j ( r ; ω ) + ( μ a ( r ) i ω c ) Ψ j ( r ; ω ) = 0 r Ω d \ ( Ξ d Ω d )
Ψ j ( m ; ω ) + 2 A κ ( m ) Ψ j ( m ; ω ) ν = g j ( m ; ω ) 𝓟 j ( μ a κ ) m Ω 1 +
Ψ j ( m ; ω ) + 2 A κ Ψ j ( m ; ω ) ν = 1 π Ξ cos θ cos θ ' Ψ j ( m ; ω ) 2 A h ( m , m ) x
exp [ ( μ a i ω c ) m m ] m m 2 d m m , m Ξ
f ( m , m ) = cos θ ( m ) cos θ ( m ) π m m 2
f τ α , τ α ( m , m ) = k = 1 N k = 1 N u n ( k ) ( m ) u n ( k ) ( m ) cos θ n ( k ) cos θ n ( k ) π N n ( k ) N n ( k ) 2

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