Abstract

An extension of the recently developed method of intensity diffraction tomography is derived that assumes that the probing field is a spherical wave produced by a point source sufficiently far from the scatterer. A discussion of the method and numerical reconstructions of a simulated three-dimensional scattering object are presented.

© 2005 Optical Society of America

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References

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  1. E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.
  2. G. Gbur, E. Wolf, “Hybrid diffraction tomography without phase information,” J. Opt. Soc. Am. A 19, 2194–2202 (2002).
    [CrossRef]
  3. G. Gbur, E. Wolf, “Diffraction tomography without phase information,” Opt. Lett. 27, 1890–1892 (2002).
    [CrossRef]
  4. A. J. Devaney, “Generalized projection-slice theorem for fan beam diffraction tomography,” Ultrason. Imaging 7, 264–275 (1985).
    [PubMed]
  5. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  6. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  7. A. J. Devaney, “Diffraction tomography,” in Inverse Methods in Electromagnetic Imaging, Part 2, W. M. Boerner, H. Brand, L. A. Cram, D. T. Gjessing, A. K. Jordan, W. Keydel, G. Schwierz, M. Vogel, eds. (Reidel, Boston, Mass., 1985), pp. 1107–1135.
  8. M. A. Anastasio, X. Pan, “Computationally efficient and statistically robust image reconstruction in three-dimensional diffraction tomography,” J. Opt. Soc. Am. A 17, 391–400 (2000).
    [CrossRef]
  9. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  10. G. Gbur, E. Wolf, “The information content of the scattered intensity in diffraction tomography,” Info. Sci. 162, 3–20 (2004).
    [CrossRef]
  11. M. Slaney, A. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984).
    [CrossRef]
  12. B. Chen, J. J. Stamnes, “Validity of diffraction tomography based on the first-Born and first-Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
    [CrossRef]
  13. M. A. Anastasio, X. Pan, “An improved reconstruction algorithm for 3D diffraction tomography using spherical-wave sources,” IEEE Trans. Biomed. Eng. 50, 517–521 (2003).
    [CrossRef] [PubMed]

2004

G. Gbur, E. Wolf, “The information content of the scattered intensity in diffraction tomography,” Info. Sci. 162, 3–20 (2004).
[CrossRef]

2003

M. A. Anastasio, X. Pan, “An improved reconstruction algorithm for 3D diffraction tomography using spherical-wave sources,” IEEE Trans. Biomed. Eng. 50, 517–521 (2003).
[CrossRef] [PubMed]

2002

2000

1998

1985

A. J. Devaney, “Generalized projection-slice theorem for fan beam diffraction tomography,” Ultrason. Imaging 7, 264–275 (1985).
[PubMed]

1984

M. Slaney, A. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984).
[CrossRef]

1983

Anastasio, M. A.

M. A. Anastasio, X. Pan, “An improved reconstruction algorithm for 3D diffraction tomography using spherical-wave sources,” IEEE Trans. Biomed. Eng. 50, 517–521 (2003).
[CrossRef] [PubMed]

M. A. Anastasio, X. Pan, “Computationally efficient and statistically robust image reconstruction in three-dimensional diffraction tomography,” J. Opt. Soc. Am. A 17, 391–400 (2000).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Chen, B.

Devaney, A. J.

A. J. Devaney, “Generalized projection-slice theorem for fan beam diffraction tomography,” Ultrason. Imaging 7, 264–275 (1985).
[PubMed]

A. J. Devaney, “Diffraction tomography,” in Inverse Methods in Electromagnetic Imaging, Part 2, W. M. Boerner, H. Brand, L. A. Cram, D. T. Gjessing, A. K. Jordan, W. Keydel, G. Schwierz, M. Vogel, eds. (Reidel, Boston, Mass., 1985), pp. 1107–1135.

Gbur, G.

Kak, A.

M. Slaney, A. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984).
[CrossRef]

Larsen, L.

M. Slaney, A. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Pan, X.

M. A. Anastasio, X. Pan, “An improved reconstruction algorithm for 3D diffraction tomography using spherical-wave sources,” IEEE Trans. Biomed. Eng. 50, 517–521 (2003).
[CrossRef] [PubMed]

M. A. Anastasio, X. Pan, “Computationally efficient and statistically robust image reconstruction in three-dimensional diffraction tomography,” J. Opt. Soc. Am. A 17, 391–400 (2000).
[CrossRef]

Slaney, M.

M. Slaney, A. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984).
[CrossRef]

Stamnes, J. J.

Teague, M. R.

Wolf, E.

G. Gbur, E. Wolf, “The information content of the scattered intensity in diffraction tomography,” Info. Sci. 162, 3–20 (2004).
[CrossRef]

G. Gbur, E. Wolf, “Diffraction tomography without phase information,” Opt. Lett. 27, 1890–1892 (2002).
[CrossRef]

G. Gbur, E. Wolf, “Hybrid diffraction tomography without phase information,” J. Opt. Soc. Am. A 19, 2194–2202 (2002).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.

Appl. Opt.

IEEE Trans. Biomed. Eng.

M. A. Anastasio, X. Pan, “An improved reconstruction algorithm for 3D diffraction tomography using spherical-wave sources,” IEEE Trans. Biomed. Eng. 50, 517–521 (2003).
[CrossRef] [PubMed]

IEEE Trans. Microwave Theory Tech.

M. Slaney, A. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984).
[CrossRef]

Info. Sci.

G. Gbur, E. Wolf, “The information content of the scattered intensity in diffraction tomography,” Info. Sci. 162, 3–20 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Ultrason. Imaging

A. J. Devaney, “Generalized projection-slice theorem for fan beam diffraction tomography,” Ultrason. Imaging 7, 264–275 (1985).
[PubMed]

Other

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

A. J. Devaney, “Diffraction tomography,” in Inverse Methods in Electromagnetic Imaging, Part 2, W. M. Boerner, H. Brand, L. A. Cram, D. T. Gjessing, A. K. Jordan, W. Keydel, G. Schwierz, M. Vogel, eds. (Reidel, Boston, Mass., 1985), pp. 1107–1135.

E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.

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Figures (11)

Fig. 1
Fig. 1

Illustration of the configuration for weak scattering and diffraction tomography.

Fig. 2
Fig. 2

Illustration of the configuration for plane-wave I-DT.

Fig. 3
Fig. 3

Illustration of the configuration for spherical-wave DT.

Fig. 4
Fig. 4

Fourier components available when spherical-wave tomography is used for different values of the parameter α. Here Kz is the z-component of the wavevector in Fourier space, and KT is the magnitude of the transverse part of the wavevector in Fourier space, i.e., KT(Kx2+Ky2)1/2.

Fig. 5
Fig. 5

Illustration of the configuration for spherical-wave I-DT.  

Fig. 6
Fig. 6

The z=1.91, 0, and -1.69 planes of the simulated 3D scattering potential.

Fig. 7
Fig. 7

Noiseless reconstructions of the z=1.91, 0, and -1.69 planes obtained by use of the plane-wave I-DT method (α=1).

Fig. 8
Fig. 8

Noiseless reconstructions of the z=1.91, 0, and -1.69 planes obtained by use of the spherical-wave I-DT method with geometry one (α=0.7, Δ=4).

Fig. 9
Fig. 9

Noisy reconstructions of the z=1.91, 0, and -1.69 planes obtained by use of the spherical-wave I-DT method with geometry one (α=0.7, Δ=4).

Fig. 10
Fig. 10

Noisy reconstructions of the z=1.91, 0, and -1.69 planes obtained by use of the spherical-wave I-DT method with geometry two (α=0.7, Δ=16).

Fig. 11
Fig. 11

Illustration of (a) the wave vectors of the scattering potential, that are mixed together in a single intensity measurement and (b) a four-measurement arrangement used to extract a component of the scattering potential.

Equations (34)

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U(r)Ui(r)exp[ψ(r)],
ψ(r)=1Ui(r) VF(r) exp[ik|r-r|]|r-r| Ui(r)d3r
F(r)=k24π [n2(r)-1]
F˜(K)=1(2π)3 VF(r)exp[-iKr]d3r
F˜[u, v, (w-k)]F˜(us1+vs2+(w-k)s0).
ψ(x, y, z)=i(2π)2 1wF˜[u, v, (w-k)]exp[i(w-k)z]exp[i(ux+vy)]dudv,
w=(k2-u2-v2)1/2whenu2+v2k2i(u2+v2-k2)1/2whenu2+v2>k2.
ψˆ(u, v; z)=1(2π)2  ψ(x, y, z)exp[-i(ux+vy)]dxdy.
ψˆ(u, v; z)=(2π)2iw F˜[u, v, (w-k)]exp[i(w-k)z].
DI(r)log[I(r)/Ii(r)]=ψ(r)+ψ*(r),
DˆI(u, v; z)=i(2π)2|w|2{w*F˜[u, v, (w-k)]exp[i(w-k)z]-w[F˜[-u, -v, (w-k)]]* exp[-i(w*-k)z]}.
DˆΔ(u, v; d)DˆI(u, v; d)-DˆI(u, v; d+Δ)exp[i(w-k)Δ]Δ,
limΔ0 DˆΔ(u, v; d)=-z DˆI(u, v; z)|z=d-i(w-k)DˆI(u, v; d).
DˆΔ(u, v; d)=(2π)2iwΔF˜[u, v, (w-k)]exp[i(w-k)d]×{1-exp[2i(w-k)Δ]}.
Ui(r)=exp[ik|r-r0|]|r-r0|,
ψ(r)1Ui(r)exp[ikz0]z0exp[ikρ2/2(z+z0)]×VF(r)exp[ikz]exp[ik|ra-rb|]|ra-rb|d3r,
ra=(αρ, z),
rb=(ρ/α, z),
αz0z+z01/2<1.
Ui(r)1|r-r0|exp[ik(z+z0)]exp[ikρ2/2(z+z0)],
ψ(r)=|r-r0|z0exp[-ikz]×VF(r)exp[ikz]exp[ik|ra-rb|]|ra-rb|d3r.
Q(r)=ψ(r)|r-r0|.
Qˆ[u, v; z]=1z0wα(2π)2iα2F˜[u/α2, v/α2, wα-k]×exp[i(wα-k)z],
wα[k2-(u/α)2-(v/α)2]1/2
DS(r, z0)1|r-r0|log[I(r)/Ii(r)],
DˆS(u, v; d, z0)=(2π)2iz0wαα2{F˜[u/α2, v/α2, (wα-k)]exp[i(wα-k)d]-[F˜[-u/α2, -v/α2, (wα-k)]]* exp[-i(wα-k)d]}.
z0=(z+Δ)z0z.
DˆΔS(u, v; d)=z0DˆS(u, v; d, z0)-z0DˆS(u, v; d+Δ, z0)exp[i(wα-k)Δ]Δ.
DˆΔS(u, v; d)=(2π)2iα2wαΔF˜[u/α2, v/α2, (wα-k)]{1-exp[2i(wα-k)Δ]}exp[i(wα-k)d].
2(wα-k)Δ=2nπ,
Var{F˜[u/α2, v/α2, (wα-k)]}=α4wα2Δ2(2π)4 Var{DˆΔS(u, v; d)}2[1-cos(2(wα-k)Δ)],
θi=tan-1ui2+vi2/αj2k2-(ui/αj)2-(vi/αj)2-k,
Li=ui2/αj4+vi2/αj4+(wαj-k)2.
Det=2(2π)8z04wα12wα22α14α24×sin[2(wα1-wα2)d+2(wα2-k)Δ].

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