Abstract

We give an approach for directly localizing and characterizing the properties of a compactly supported absorption coefficient perturbation as well as coarse scale structure of the background medium from a sparsely sampled, diffuse photon density wavefield. Our technique handles the problems of localization and characterization simultaneously by working directly with the data, unlike traditional techniques that require two stages. We model the unknowns as a superposition of a slowly varying perturbation on a background of unknown structure. Our model assumes that the anomaly is delineated from the background by a smooth perimeter which is modeled as a spline curve comprised of unknown control points. The algorithm proceeds by making small perturbations to the curve which are locally optimal. The result is a global, greedy-type optimization approach designed to enforce consistency with the data while requiring the solution to adhere to prior information we have concerning the likely structure of the anomaly. At each step, the algorithm adaptively determines the optimal weighting coefficients describing the characteristics of both the anomaly and the background. The success of our approach is illustrated in two simulation examples provided for a diffuse photon density wave problem arising in a bio-imaging application.

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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. 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]
  3. S. J. Norton and T. Vo Dinh, "Diffraction tomographic imaging with photon density waves: an explicit solution," J. Opt. Soc. Am. A 15, 2670 2677 (1998).
    [CrossRef]
  4. M. O'Lear , D. Boas, B. Chance and A. Yodh, "Experimental images of heterogeneous turbid media by frequency domain diffusion photon tomography," Opt. Lett. 20, 425 428, (1995).
    [CrossRef]
  5. T. J. Fareel, M. S. Patterson, B. Wilson, "A diffusion theory model of spatially resolved, stead state diffuse reflectance for the non invasive determination of tissue optical properties in vivo," Med. Phys 19, 879 888, (1992).
    [CrossRef]
  6. R. C. Haskell, L. O. Svaasand, Tsong Tseh Tsa , Ti Chen Feng, Matthew S. McAdmans and Bruce J. Tromberg, "Boundary conditions for the diffusion equation in radiative transfer," J. Opt. Soc. Am. A 11, 2722 2741, (1994).
    [CrossRef]
  7. Per Christian Hansen, Rank-Deficient and Discrete Ill-Posed Problems, (SIAM Press, Philadelphia, 1998).
    [CrossRef]
  8. R. F. Harrington, Field Computations by Moment Methods, (Macmillan, New York, 1968).
  9. 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 wave data," Proceedings of the SPIE Photonics West Meeting, Jan. 1999.
  10. G. Golub and C. Van Loan, Matrix Computations, second ed., (Johns Hopkins Press, Baltimore, 1991).

Other

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

S. J. Norton and T. Vo Dinh, "Diffraction tomographic imaging with photon density waves: an explicit solution," J. Opt. Soc. Am. A 15, 2670 2677 (1998).
[CrossRef]

M. O'Lear , D. Boas, B. Chance and A. Yodh, "Experimental images of heterogeneous turbid media by frequency domain diffusion photon tomography," Opt. Lett. 20, 425 428, (1995).
[CrossRef]

T. J. Fareel, M. S. Patterson, B. Wilson, "A diffusion theory model of spatially resolved, stead state diffuse reflectance for the non invasive determination of tissue optical properties in vivo," Med. Phys 19, 879 888, (1992).
[CrossRef]

R. C. Haskell, L. O. Svaasand, Tsong Tseh Tsa , Ti Chen Feng, Matthew S. McAdmans and Bruce J. Tromberg, "Boundary conditions for the diffusion equation in radiative transfer," J. Opt. Soc. Am. A 11, 2722 2741, (1994).
[CrossRef]

Per Christian Hansen, Rank-Deficient and Discrete Ill-Posed Problems, (SIAM Press, Philadelphia, 1998).
[CrossRef]

R. F. Harrington, Field Computations by Moment Methods, (Macmillan, New York, 1968).

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 wave data," Proceedings of the SPIE Photonics West Meeting, Jan. 1999.

G. Golub and C. Van Loan, Matrix Computations, second ed., (Johns Hopkins Press, Baltimore, 1991).

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Figures (4)

Fig. 1.
Fig. 1.

Source/receiver configuration in the transmission geometry.

Fig. 2.
Fig. 2.

Anomaly Recovery Algorithm

Fig. 3.
Fig. 3.

From left to right, top to bottom: a) Contour curves for b*(r), c 0(r), bk *(r) b) True image of the perturbation in the absorption coefficient c) TSVD reconstruction of the absorption perturbation d) our reconstruction of the absorption using h=1 and λ=.005.

Fig. 4.
Fig. 4.

From left to right, top to bottom: a) Contour curves for b*(r), c 0(r), bk *(r) b) True image of the perturbation in the absorption coefficient c) TSVD reconstruction of the absorption perturbation d) our reconstruction of the absorption using h=1 and λ=.05.

Equations (11)

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y ( r k ) = υ D v G ( r k , r ) G ( r , r s ) g ( r ) d r + n ( r k )
y i = G i g + n i , i = 1 , 2 N r or N r ,
y = Gg + n
g ( r ) S ( r ) B 1 ( r ) a 1 + ( 1 S ( r ) ) B 2 ( r ) a 2 , a 1 , a 2 R p × 1 ,
c ( s ) = [ x ( s ) , y ( s ) ] = i = 0 K 1 β k i ( s ) [ x ̂ i , y ̂ i ] , s [ 0 , L ]
g = [ SB 1 ( I S ) B 2 ] [ a 1 a 2 ] Qa
J ( a 1 , a 2 ) G [ SB 1 , ( I S ) B 2 ] [ a 1 a 2 ] y 2 .
J ( a 1 , a 2 ) + λ Ω ( c ) ,
Ω ( c ) = i = 1 K 2 ( x ̂ i x ̂ i + 1 ) 2 + ( y ̂ i y ̂ i + 1 ) 2
n sr ( ω ) = σ sr ( 0 ) υ 1
σ sr ( 0 ) = γ y ˜ s ( r ) + y ˜ s inc ( r )

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