Abstract

In optical diffusion tomography the reconstruction of the absorbtion and scattering coefficients is conventionally carried out in a pixel basis. The resulting number of unknowns makes the associated inverse problem severely ill-posed. We have recently proposed a new approach in which the goal is to reconstruct boundaries of piece-wise constant tissue regions as well as the diffusion and absorption coefficients within these regions. This method assumes that there is a feasible initial guess on the domain boundaries. In this paper we propose an extension to this approach in which the initial estimate for the boundary and coefficient estimation is extracted from a conventional pixel based reconstruction using standard image processing operations. In the computation of the pixel based reconstruction the output least squares problem is augmented with an approximated total variation prior. The performance of the proposed approach is evaluated using simulated frequency domain data. It is shown that since the total variation type approach favors domains with constant coefficients it is well suited for the fixing of the starting point for the actual boundary and coefficient reconstruction method.

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References

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  1. 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]
  2. V. Kolehmainen, S. R. Arridge, M. Vauhkonen, and J. P. Kaipio, "Simultaneous reconstruction of internal tissue region boundaries and coefficients in optical diffusion tomography," Phys Med Biol (2000), in Press.
  3. D. Dobson and F. Santosa, "An image enhancement technique for electrical impedance tomography," Inverse Problems 10, 317-334 (1994).
    [CrossRef]
  4. J. P. Kaipio, V. Kolehmainen, E. Somersalo, and M. Vauhkonen, "Statistical inversion methods in electrical impedance tomography," Inverse Problems (2000), in Press.
  5. K. D. Paulsen and H. Jiang, "Enhanced frequency-domain optical image reconstruction in tissues through total variation minimization," Appl. Opt. 35, 3447-3458 (1996).
    [CrossRef] [PubMed]
  6. M. C. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, New York, 1967).
  7. S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15, R41-R93 (1999).
    [CrossRef]
  8. 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]
  9. J. P. Kaltenbach and M. Kaschke, "Frequency- and Time-domain Modelling of Light Transport in Random Media," in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, and P. van der Zee, eds., (SPIE, Bellingham, WA, 1993), pp. 65-86.
  10. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. T. Bosch, "Scattering and Absorption of Turbid Materials Determined from Reflection Measurements. Part 1: Theory," Appl. Opt. 22, 2456-2462 (1983).
    [CrossRef] [PubMed]
  11. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, "Boundary conditions for the diffusion equation in radiative transfer," J. Opt. Soc. Am. A 11, 2727-2741 (1994).
    [CrossRef]
  12. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A Finite Element Approach for Modeling Photon Transport in Tissue," Med. Phys. 20, 299-309 (1993).
    [CrossRef] [PubMed]
  13. 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]
  14. 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]
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    [CrossRef] [PubMed]
  16. A. Fiacco and G. McCormick, Nonlinear Programming (SIAM, 1990).
    [CrossRef]
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  19. M. Cheney, D. Isaacson, J. Newell, S. Simske, and J. Goble, "NOSER: An algorithm for solving the inverse conductivity problem," Int J Imaging Systems and Technology 2, 66-75 (1990).
    [CrossRef]
  20. J. S. Lim, Two-dimensional signal and image processing (Prentice Hall, Englewood Cliffs, NJ, 1990).
  21. Matlab: Image Processing toolbox user's guide, 2 ed., The MathWorks Inc., 24 Prime Park Way, Natick, MA 01760-1500, USA.
  22. D. W. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," J. Soc. Indust. Appl. Math. 11, 431-441 (1963).
    [CrossRef]
  23. M. E. Kilmer, E. L. Miller, D. A. Boas, D. H. Brooks, C. A. DiMarzio and R. J. Gaudette, "Direct object localization and characterization from diffuse photon density data," Proceedings of the SPIE Photonics West Meeting, Jan. 1999.
  24. M. Schweiger and S. R. Arridge, "Comparison of 2D and 3D reconstruction algorithms in Optical Tomography," Appl. Opt. 37, 7419-7428 (1998).
    [CrossRef]
  25. S. R. Arridge, J. C. Hebden, M. Schweiger, F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, H. Dehghani, and D. T. Delby, "A method for three-dimensional time-resolved optical tomography," Int. J. Imaging Syst. Technol. 11, 2-11 (2000).
    [CrossRef]
  26. C. Brechbühler, G. Gerig, and O. Kübler, "Parametrization of closed surfaces for 3-D shape description," Computer Vision and Image Understanding 61, 154-170 (1995).
    [CrossRef]
  27. A. Kelemen, G. Szekely, and G. Gerig, "Three-dimensional model-based segmentation," IEEE Trans Med Imaging 18, 828-839 (1995).
    [CrossRef]

Other

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]

V. Kolehmainen, S. R. Arridge, M. Vauhkonen, and J. P. Kaipio, "Simultaneous reconstruction of internal tissue region boundaries and coefficients in optical diffusion tomography," Phys Med Biol (2000), in Press.

D. Dobson and F. Santosa, "An image enhancement technique for electrical impedance tomography," Inverse Problems 10, 317-334 (1994).
[CrossRef]

J. P. Kaipio, V. Kolehmainen, E. Somersalo, and M. Vauhkonen, "Statistical inversion methods in electrical impedance tomography," Inverse Problems (2000), in Press.

K. D. Paulsen and H. Jiang, "Enhanced frequency-domain optical image reconstruction in tissues through total variation minimization," Appl. Opt. 35, 3447-3458 (1996).
[CrossRef] [PubMed]

M. C. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, New York, 1967).

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

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]

J. P. Kaltenbach and M. Kaschke, "Frequency- and Time-domain Modelling of Light Transport in Random Media," in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, and P. van der Zee, eds., (SPIE, Bellingham, WA, 1993), pp. 65-86.

R. A. J. Groenhuis, H. A. Ferwerda, and J. J. T. Bosch, "Scattering and Absorption of Turbid Materials Determined from Reflection Measurements. Part 1: Theory," Appl. Opt. 22, 2456-2462 (1983).
[CrossRef] [PubMed]

R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, "Boundary conditions for the diffusion equation in radiative transfer," J. Opt. Soc. Am. A 11, 2727-2741 (1994).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A Finite Element Approach for Modeling Photon Transport in Tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

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]

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, "Application of temporal filters to time resolved data in optical tomography," Phys. Med. Biol. 44, 1699-1717 (1999).
[CrossRef] [PubMed]

A. Fiacco and G. McCormick, Nonlinear Programming (SIAM, 1990).
[CrossRef]

S. Järvenpää, "A finite element model for the inverse conductivity problem,", Phil. Lic. thesis, University of Helsinki, Finland (1996).

V. Kolehmainen, E. Somersalo, P. J. Vauhkonen, M. Vauhkonen, and J. P. Kaipio, "A Bayesian approach and total variation priors in 3D electrical impedance tomography," In Proc 20th Ann Int Conf IEEE Eng Med Biol Soc, pp. 1028-1031 (Hong Kong, China, 1998).

M. Cheney, D. Isaacson, J. Newell, S. Simske, and J. Goble, "NOSER: An algorithm for solving the inverse conductivity problem," Int J Imaging Systems and Technology 2, 66-75 (1990).
[CrossRef]

J. S. Lim, Two-dimensional signal and image processing (Prentice Hall, Englewood Cliffs, NJ, 1990).

Matlab: Image Processing toolbox user's guide, 2 ed., The MathWorks Inc., 24 Prime Park Way, Natick, MA 01760-1500, USA.

D. W. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," J. Soc. Indust. Appl. Math. 11, 431-441 (1963).
[CrossRef]

M. E. Kilmer, E. L. Miller, D. A. Boas, D. H. Brooks, C. A. DiMarzio and R. J. Gaudette, "Direct object localization and characterization from diffuse photon density data," Proceedings of the SPIE Photonics West Meeting, Jan. 1999.

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, J. C. Hebden, M. Schweiger, F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, H. Dehghani, and D. T. Delby, "A method for three-dimensional time-resolved optical tomography," Int. J. Imaging Syst. Technol. 11, 2-11 (2000).
[CrossRef]

C. Brechbühler, G. Gerig, and O. Kübler, "Parametrization of closed surfaces for 3-D shape description," Computer Vision and Image Understanding 61, 154-170 (1995).
[CrossRef]

A. Kelemen, G. Szekely, and G. Gerig, "Three-dimensional model-based segmentation," IEEE Trans Med Imaging 18, 828-839 (1995).
[CrossRef]

Supplementary Material (1)

» Media 1: GIF (349 KB)     

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Figures (3)

Fig. 1.
Fig. 1.

An example of a piecewise constant object domain Ω=∪ k , Ak with constant coefficients {κk , µ a,k , k=0, 1, 2, 3}. The outer boundaries of regions A 1, A 2 and A 3 are denoted by C 1, C 2 and C 3, respectively. The background region A 0 has Ω as the outer boundary.

Fig. 2.
Fig. 2.

A test case with three boundaries {C }. Region A 1 is located up, A 2 is located on the lower left side of Ω and A 3 on the lower right side. Left column shows images of κ([mm]) and right column shows images of µ a ([mm-1]). Rows from top to bottom: True distributions, pixelwise reconstructions with the approximated TV prior, initial estimates for the shape estimation and the bottom row shows the final estimates with the shape estimation method. The animation (349kB) shows the progress of the inversion approach step by step.

Fig. 3.
Fig. 3.

Left: The reconstructed µ a in a regular grid. Right: The edges that were detected from the µ a image.

Tables (1)

Tables Icon

Table 1. Values of {κk , µ a,k } in the initial and in the final estimates of the shape estimation in Fig. 2. The initial estimate is shown in the 3rd row in Fig. 2, and the final estimate in the bottom row, respectively. The dimension of κk is [mm] and the dimension of µ a,k is [mm-1].

Equations (26)

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Ω = k = 0 L A k ,
{ C ( S ) } = ( x ( s ) y ( s ) ) = n = 1 N θ ( γ n x θ n x ( s ) γ n y θ n y ( s ) ) , = 1 , . . . , L
θ 1 α ( s ) = 1
θ n α ( s ) = sin ( 2 π n 2 ( s + ϕ s ) ) , n = 2 , 4 , 6 ,
θ n α ( s ) = cos ( 2 π ( n 1 2 ) ( s + ϕ s ) ) , n = 3 , 5 , 7 ,
γ = ( γ 1 x 1 , , γ N θ x 1 , γ 1 y 1 , , γ N θ y 1 , , , γ 1 x L , , γ N θ x L , γ 1 y L , , γ N θ y L ) T .
· κ Φ j ( ω ) + μ a Φ j ( ω ) + i ω c Φ j ( ω ) = q 0 j , r Ω
Φ j ( ω ) + 2 κ ϑ Φ j ( ω ) ν = g s j r Ω ,
Γ i , j ( ω ) = κ ( ζ i ) ν · Φ j ( ω ) .
z = 𝓟 ( f ) .
z = ( re ( Γ 1 ( ω ) ) , , re ( Γ s ( ω ) ) , im ( Γ 1 ( ω ) ) , , im ( Γ s ( ω ) ) T ,
f ̂ = arg min f z meas 𝓟 ( f ) z meas 2 2 ,
f ( l + 1 ) = f ( l ) + ( J ˜ ( l ) T J ˜ ( l ) ) 1 J ˜ l T z ˜ ( l ) ,
κ = m = 1 N p κ m χ m , μ a = m = 1 N p μ a , m χ m ,
f = ( κ μ a ) N p × N p .
f ̂ = arg min f { z meas 𝓟 ( f ) z meas 2 2 + Ψ ( f ) } ,
Ψ ( f ) = β κ TV ( κ ) + β μ a TV ( μ a ) ,
TV ( α ) = j = 1 J d j Δ j T α ,
t h τ ( t ) = 1 τ log ( cosh ( τt ) ) ,
W ( f ( l ) ) = ς l m = 1 2 N p 1 f m ( l ) ,
f ( l + 1 ) = f ( l ) + ( J ˜ ( l ) T J ˜ ( l ) + H TV τ ( l ) + H W ( l ) ) 1 ( J ˜ ( l ) T z ˜ l g TV τ ( l ) g W ( l ) ) ,
f ̂ = arg min f z meas 𝓟 ( f ) z meas 2 2 ,
f = ( γ κ μ a ) 2 L N θ x L + 1 x L + 1 .
f ( l + 1 ) = f ( l ) + s ( l ) d ( l )
d ( l ) = ( J ˜ ( l ) T J ˜ ( l ) + λ I ) 1 J ˜ ( l ) T z ˜ ( l ) ,
Ξ ( l ) ( s ) z meas 𝓟 ( f ( l ) + s d ( l ) ) z meas 2 2

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