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.

©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. 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’Leary, 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, and B. Wilson, “A diffusion theory model of spatially resolved, steady 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 Tsay, 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).

1999 (2)

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]

1998 (1)

1995 (1)

M. O’Leary, 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]

1994 (1)

R. C. Haskell, L. O. Svaasand, Tsong-Tseh Tsay, 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]

1992 (1)

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

Arridge, S. R.

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. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, R41–R93, (1999).
[Crossref]

Boas, D.

M. O’Leary, 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]

Boas, D. A.

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.

Brooks, D. H.

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.

Chance, B.

M. O’Leary, 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]

DiMarzio, C. A.

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.

Fareel, T. J.

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

Feng, Ti-Chen

R. C. Haskell, L. O. Svaasand, Tsong-Tseh Tsay, 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]

Gaudette, R. J.

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.

Golub, G.

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

Hansen, Per Christian

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

Harrington, R. F.

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

Haskell, R. C.

R. C. Haskell, L. O. Svaasand, Tsong-Tseh Tsay, 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]

Kaipio, J. P.

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]

Kilmer, M. E.

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.

Kolehmainen, V.

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]

Lionheart, W. R. B.

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]

McAdmans, Matthew S.

R. C. Haskell, L. O. Svaasand, Tsong-Tseh Tsay, 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]

Miller, E. L.

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.

Norton, S. J.

O’Leary, M.

M. O’Leary, 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]

Patterson, M. S.

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

Svaasand, L. O.

R. C. Haskell, L. O. Svaasand, Tsong-Tseh Tsay, 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]

Tromberg, Bruce J.

R. C. Haskell, L. O. Svaasand, Tsong-Tseh Tsay, 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]

Tsay, Tsong-Tseh

R. C. Haskell, L. O. Svaasand, Tsong-Tseh Tsay, 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]

Van Loan, C.

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

Vauhkonen, M.

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]

Vo-Dinh, T.

Wilson, B.

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

Yodh, A.

M. O’Leary, 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]

Inverse Problems (2)

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]

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

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]

R. C. Haskell, L. O. Svaasand, Tsong-Tseh Tsay, 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]

Med. Phys (1)

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

Opt. Lett. (1)

M. O’Leary, 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]

Other (4)

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.

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Fig. 2.
Fig. 2. Anomaly Recovery Algorithm

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Fig. 3.
Fig. 3. From left to right, top to bottom: a) Contour curves for b*(r), c 0(r), b k *(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.

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Fig. 4.
Fig. 4. From left to right, top to bottom: a) Contour curves for b*(r), c 0(r), b k *(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.

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Equations (11)

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

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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