Abstract

We present a model of the forward problem for diffuse photon density waves in turbid medium using a diffraction tomographic problem formulation. We consider a spatially-varying inhomogeneous structure whose absorption properties satisfy the Born approximation and whose scattering properties are identical to the homogeneous turbid media in which it is imbedded. The two-dimensional Fourier transform of the scattered field, measured in a plane, is shown to be related to the three-dimensional Fourier transform of the object evaluated on a surface which in many cases is approximately a plane.

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References

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  1. M.A. OLeary, D.A. Boas, B. Chance, and A.G. Yodh, "Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography," Opt. Lett. 20, 426-428 (1995)
    [CrossRef] [PubMed]
  2. H.B. Jiang, K.D. Paulsen, U.L. Osterberg, M.S. Patterson, "Frequency-domain optical-image reconstruction in turbid media - an experimental study of single-target detectability," Appl. Opt. 36, 52-63 (1997)
    [CrossRef] [PubMed]
  3. S.A. Walker, S. Fantini, and E. Gratton, " Image reconstruction using back-projection from frequency-domain optical measurements in highly scattering media," Appl. Opt. 36, 170-179 (1997)
    [CrossRef] [PubMed]
  4. S.B. Colak, D.G. Papioannou, G.W. Hooft, and M.B. van der Mark, "Optical image reconstruction with deconvolution in light diffusing media," in Photon Propagation in Tissues, B. Chance, D.T. Delpy, and G.J. Mueller, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2626, 306-315 (1995)
  5. X.D. Li, T. Durduran, A.G. Yodh, B. Chance, and D.N. Pattanayak, "Diffraction tomography for biochemical imaging with diffuse-photon density waves," Opt. Lett. 22, 573-575 (1997)
    [CrossRef] [PubMed]
  6. C.L. Matson, N. Clark, L. McMackin, and J.S. Fender, "Three-dimensional tumor localization in thick tissue with the use of diffuse photon-density waves," Appl. Opt. 36, 214-220 (1997)
    [CrossRef] [PubMed]
  7. A.J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Problems 2, 161-183 (1986)
    [CrossRef]
  8. A. Schatzberg and A.J. Devaney, "Super-resolution in diffraction tomography," Inverse Problems 8, 149-164 (1992)
    [CrossRef]
  9. A.J. Devaney, "Linearised inverse scattering in attenuating media," Inverse Problems 3, 389-397 (1987)
    [CrossRef]
  10. A.J. Devaney, "The limited-view problem in diffraction tomography," Inverse Problems 5, 501-521 (1989)
    [CrossRef]
  11. A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988)
  12. A. Baños, Jr., Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon Press, Oxford, 1966)
  13. K.B. Howell, "Fourier transforms," in The Transforms and Applications Handbook, A.D. Poularikas, ed. (CRC Press, Boca Raton, 1996)
  14. D.A. Boas, M.A. OLeary, B. Chance, and A.G. Yodh, " Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytic solution and applications," Proc. Natl. Acad. Sci. USA 91, 4887-4891 (1994)
    [CrossRef] [PubMed]
  15. W.F. Cheong, S.A. Prahl, and A.J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electronics 26, 2166-2185 (1990)
    [CrossRef]
  16. H. Heusmann, J. Koelzer, and G. Mitic, "Characterization of female breasts in vivo by time-resolved and spectroscopic measurements in near infrared spectroscopy," J.Biomed.Opt. 1, 425-434 (1996)
    [CrossRef]
  17. D.C. Munson, Jr. and J.L.C. Sanz, "Image reconstruction from frequency-offset Fourier data," Proc. IEEE 72, 661-669 (1984)
    [CrossRef]
  18. C.L. Matson, I.A. Delarue, T.M. Gray, and I.E. Drunzer, "Optimal Fourier spectrum estimation from the bispectrum," Computers Elect. Engng 18, 485-497 (1992)
    [CrossRef]

Other

M.A. OLeary, D.A. Boas, B. Chance, and A.G. Yodh, "Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography," Opt. Lett. 20, 426-428 (1995)
[CrossRef] [PubMed]

H.B. Jiang, K.D. Paulsen, U.L. Osterberg, M.S. Patterson, "Frequency-domain optical-image reconstruction in turbid media - an experimental study of single-target detectability," Appl. Opt. 36, 52-63 (1997)
[CrossRef] [PubMed]

S.A. Walker, S. Fantini, and E. Gratton, " Image reconstruction using back-projection from frequency-domain optical measurements in highly scattering media," Appl. Opt. 36, 170-179 (1997)
[CrossRef] [PubMed]

S.B. Colak, D.G. Papioannou, G.W. Hooft, and M.B. van der Mark, "Optical image reconstruction with deconvolution in light diffusing media," in Photon Propagation in Tissues, B. Chance, D.T. Delpy, and G.J. Mueller, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2626, 306-315 (1995)

X.D. Li, T. Durduran, A.G. Yodh, B. Chance, and D.N. Pattanayak, "Diffraction tomography for biochemical imaging with diffuse-photon density waves," Opt. Lett. 22, 573-575 (1997)
[CrossRef] [PubMed]

C.L. Matson, N. Clark, L. McMackin, and J.S. Fender, "Three-dimensional tumor localization in thick tissue with the use of diffuse photon-density waves," Appl. Opt. 36, 214-220 (1997)
[CrossRef] [PubMed]

A.J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Problems 2, 161-183 (1986)
[CrossRef]

A. Schatzberg and A.J. Devaney, "Super-resolution in diffraction tomography," Inverse Problems 8, 149-164 (1992)
[CrossRef]

A.J. Devaney, "Linearised inverse scattering in attenuating media," Inverse Problems 3, 389-397 (1987)
[CrossRef]

A.J. Devaney, "The limited-view problem in diffraction tomography," Inverse Problems 5, 501-521 (1989)
[CrossRef]

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988)

A. Baños, Jr., Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon Press, Oxford, 1966)

K.B. Howell, "Fourier transforms," in The Transforms and Applications Handbook, A.D. Poularikas, ed. (CRC Press, Boca Raton, 1996)

D.A. Boas, M.A. OLeary, B. Chance, and A.G. Yodh, " Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytic solution and applications," Proc. Natl. Acad. Sci. USA 91, 4887-4891 (1994)
[CrossRef] [PubMed]

W.F. Cheong, S.A. Prahl, and A.J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electronics 26, 2166-2185 (1990)
[CrossRef]

H. Heusmann, J. Koelzer, and G. Mitic, "Characterization of female breasts in vivo by time-resolved and spectroscopic measurements in near infrared spectroscopy," J.Biomed.Opt. 1, 425-434 (1996)
[CrossRef]

D.C. Munson, Jr. and J.L.C. Sanz, "Image reconstruction from frequency-offset Fourier data," Proc. IEEE 72, 661-669 (1984)
[CrossRef]

C.L. Matson, I.A. Delarue, T.M. Gray, and I.E. Drunzer, "Optimal Fourier spectrum estimation from the bispectrum," Computers Elect. Engng 18, 485-497 (1992)
[CrossRef]

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Figures (1)

Figure 1
Figure 1

Plots of the two-dimensional projection of the surface on which the Fourier transform of the convolved object function is obtained with diffraction tomography. The dotted line is for real values of k, the dashed line is for Re{k2}<<Im{k2}, and the solid line is for Re{k2}>>Im{k2}. The horizontal axis is ωx and the vertical axis is ωy.

Equations (22)

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

( 2 + k 2 ) u B ( r ) = o ( r ) u o ( r )
u B ( r ) = o ( r ) u o ( r ) g ( r r ) d r′
g ( r ) = exp [ ik r ] 4 π r
= 1 8 π 2 1 α x 2 + α y 2 k 2 exp { z α x 2 + α y 2 k 2 + ix α x + iy α y } x y
u B ( r ) = 1 8 π 2 o ( r ) u o ( r ) 1 α x 2 + α y 2 k 2 exp { z z′
× α x 2 + α y 2 k 2 + i ( x x′ ) α x + i ( y y′ ) α y } x y d r′
= 1 8 π 2 o ( r ) u o ( r ) 1 γ exp { z z′ ( γ r + i γ i )
+ i ( x x′ ) α x + i ( y y′ ) α y } x y d r′
γ γ r + i γ i
= Re { α x 2 + α y 2 k 2 } + i Im { α x 2 + α y 2 k 2 }
u B ( x , y , z o ) = 1 8 π 2 exp { ix α x + iy α y iz o γ i } γ
× o ( x′ , y′ , z′ ) u o ( x′ , y′ , z′ ) exp { z o z′ γ r }
× exp { ix′ α x iy α y iz′ ( γ i ) } dx dy dz′ x y
u B ( x , y , z o ) = 1 8 π 2 exp { ix α x + iy α y i z o γ i } γ O u γ ( α x , α y , γ i ) x y
u B ( ω x , ω y , z o ) = 1 8 π 2 exp { iz o γ i } γ O u γ ( α x , α y , γ i )
× exp { i ( x α x + y α y ) } exp { i ( x + y ) } dxdyd α x d α y
U B ( ω x , ω y , z o ) = 1 2 exp { iz o γ i } γ O u γ ( ω x , ω y , γ i )
γ i = Im { ω x 2 + ω y 2 k 2 }
k 2 = ( v μ a + i 2 π f t ) 3 μ′ s v
k 2 = 15 + i 9 f t
G ( ω x , ω y , ω z ) = 4 π 2 δ ( ω x ) δ ( ω y ) 2 γ r γ r 2 + ω z 2
O u γ ( ω x , ω y , γ i ) O u γ ( ω x , ω y , 0 )

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