Abstract

We introduce a hybrid tomographic method, based on recent investigations concerning the connection between computed tomography and diffraction tomography, that allows direct reconstruction of scattering objects from intensity measurements. This technique is noniterative and is intuitively easier to understand and easier to implement than some other methods described in the literature. The manner in which the new method reduces to computed tomography at short wavelengths is discussed. Numerical examples of reconstructions are presented.

© 2002 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  2. G. T. Herman, Image Reconstruction from Projections (Academic, Orlando, Fla., 1980).
  3. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
  4. E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.
  5. Some of the earliest work is described in G. N. Hounsfield, “Computerized transverse axial scanning (tomography): part I. Description of system,” Br. J. Radiol. 46, 1016–1022 (1973).
    [CrossRef] [PubMed]
  6. A historical account of the medical development of computerized tomography is given in S. Webb, From the Watching of Shadows (Hilger, Bristol, UK, 1990).
  7. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  8. C. Iaconis, I. A. Walmsley, “Direct measurement of the two-point correlation function,” Opt. Lett. 21, 1783–1785 (1996).
    [CrossRef] [PubMed]
  9. M. H. Maleki, A. J. Devaney, “Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
    [CrossRef]
  10. J. Cheng, S. Han, “Diffraction tomography reconstruction algorithms for quantitative imaging of phase objects,” J. Opt. Soc. Am. A 18, 1460–1464 (2001).
    [CrossRef]
  11. P. S. Carney, E. Wolf, G. S. Agarwal, “Diffraction tomography using power extinction measurements,” J. Opt. Soc. Am. A 16, 2643–2648 (1999).
    [CrossRef]
  12. A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Image Process. 1, 221–228 (1992).
    [CrossRef] [PubMed]
  13. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  14. G. Vdovin, “Reconstruction of an object shape from the near-field intensity of a reflected paraxial beam,” Appl. Opt. 36, 5508–5513 (1997).
    [CrossRef] [PubMed]
  15. N. Jayshree, G. K. Datta, R. M. Vasu, “Optical tomographic microscope for quantitative imaging of phase objects,” Appl. Opt. 39, 277–283 (2000).
    [CrossRef]
  16. F. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
    [CrossRef]
  17. T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  18. G. Gbur, E. Wolf, “Relation between computed tomography and diffraction tomography,” J. Opt. Soc. Am. A 18, 2132–2137 (2001).
    [CrossRef]
  19. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  20. A. J. Devaney, “Diffraction tomography,” in Inverse Methods in Electromagnetic Imaging, Part 2, W. M. Boerner, ed. (Reidel, Boston, Mass., 1985), pp. 1107–1135.
  21. 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]
  22. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985).
  23. B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables (American Mathematical Society, Providence, R.I., 1963).
  24. A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1964), Vol. 1.
  25. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

2001 (2)

2000 (2)

1999 (1)

1997 (1)

1996 (1)

1995 (1)

1994 (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

1993 (2)

1992 (1)

A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Image Process. 1, 221–228 (1992).
[CrossRef] [PubMed]

1983 (1)

1973 (1)

Some of the earliest work is described in G. N. Hounsfield, “Computerized transverse axial scanning (tomography): part I. Description of system,” Br. J. Radiol. 46, 1016–1022 (1973).
[CrossRef] [PubMed]

Agarwal, G. S.

Anastasio, M. A.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985).

Beck, M.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Born, M.

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

Carney, P. S.

Cheng, J.

Datta, G. K.

Devaney, A. J.

M. H. Maleki, A. J. Devaney, “Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
[CrossRef]

A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Image Process. 1, 221–228 (1992).
[CrossRef] [PubMed]

A. J. Devaney, “Diffraction tomography,” in Inverse Methods in Electromagnetic Imaging, Part 2, W. M. Boerner, ed. (Reidel, Boston, Mass., 1985), pp. 1107–1135.

Fuks, B. A.

B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables (American Mathematical Society, Providence, R.I., 1963).

Gbur, G.

Gori, F.

Guattari, G.

Gureyev, T. E.

Han, S.

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections (Academic, Orlando, Fla., 1980).

Hounsfield, G. N.

Some of the earliest work is described in G. N. Hounsfield, “Computerized transverse axial scanning (tomography): part I. Description of system,” Br. J. Radiol. 46, 1016–1022 (1973).
[CrossRef] [PubMed]

Iaconis, C.

Jayshree, N.

Kak, A. C.

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

Maleki, M. H.

Mandel, L.

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

McAlister, D. F.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Messiah, A.

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1964), Vol. 1.

Nugent, K. A.

Pan, X.

Raymer, M. G.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Roberts, A.

Santarsiero, M.

Slaney, M.

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

Teague, M. R.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Vasu, R. M.

Vdovin, G.

Walmsley, I. A.

Webb, S.

A historical account of the medical development of computerized tomography is given in S. Webb, From the Watching of Shadows (Hilger, Bristol, UK, 1990).

Wolf, E.

G. Gbur, E. Wolf, “Relation between computed tomography and diffraction tomography,” J. Opt. Soc. Am. A 18, 2132–2137 (2001).
[CrossRef]

P. S. Carney, E. Wolf, G. S. Agarwal, “Diffraction tomography using power extinction measurements,” J. Opt. Soc. Am. A 16, 2643–2648 (1999).
[CrossRef]

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.

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

Appl. Opt. (2)

Br. J. Radiol. (1)

Some of the earliest work is described in G. N. Hounsfield, “Computerized transverse axial scanning (tomography): part I. Description of system,” Br. J. Radiol. 46, 1016–1022 (1973).
[CrossRef] [PubMed]

IEEE Trans. Image Process. (1)

A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Image Process. 1, 221–228 (1992).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Other (11)

A historical account of the medical development of computerized tomography is given in S. Webb, From the Watching of Shadows (Hilger, Bristol, UK, 1990).

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

G. T. Herman, Image Reconstruction from Projections (Academic, Orlando, Fla., 1980).

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

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

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, ed. (Reidel, Boston, Mass., 1985), pp. 1107–1135.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985).

B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables (American Mathematical Society, Providence, R.I., 1963).

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1964), Vol. 1.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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Figures (7)

Fig. 1
Fig. 1

Depiction of the arrangement and notation.

Fig. 2
Fig. 2

Accessible Fourier components in the classical measurement configuration of diffraction tomography. (a) Components of F(r) accessible from measurements for one direction s0 of incidence, (b) components accessible with multiple measurements. The components for directions -s0(1) and -s0(2) are shown for comparison.

Fig. 3
Fig. 3

Vectors (u, v, w-k) and (-u, -v, w-k).

Fig. 4
Fig. 4

Two-plane measurement scheme for performing diffraction tomography with only intensity measurements.

Fig. 5
Fig. 5

(a) Real and (b) imaginary parts of the reconstructed and of the true scattering potential, F(r). The true potential is indicated by the dashed lines. The hybrid tomography method is seen to produce a good reconstruction of the scatterer.

Fig. 6
Fig. 6

Reconstruction of the imaginary part of the scattering potential by computed tomography with data from the measurement plane kd=60. The dashed line represents the true scattering potential. Computed tomography evidently is not a good method to use for this model scatterer.

Fig. 7
Fig. 7

Reconstruction of the (a) real and (b) imaginary parts of the scattering potential by using the intensity data in the traditional diffraction tomographic reconstruction scheme, Eq. (13). The dashed lines represent the actual scattering potential. It can be seen that this method does not accurately reproduce the scattering object.

Equations (52)

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[2+k2]U(r)=-4πF(r)U(r),
F(r)=k24π [n2(r)-1],
U(r)Ui(r)exp[ψ(r)],
ψ(r)=1Ui(r)VF(r) exp[ik|r-r|]|r-r| Ui(r)d3r.
exp[ik|r-r|]|r-r|=i2π1wexp{i[us1+vs2+ws0](r-r)}dudv,
w=k2-u2-v2whenu2+v2k2iu2+v2-k2whenu2+v2>k2,
ψ(x, y; d)=i2πVd3r1w F(r)×exp[i(w-k)(d-z)]×exp{i[u(x-x)+v(y-y)]}dudv.
ψ(x, y; d)=(2π)2i1w F˜[us1+vs2+(w-k)s0]×exp[i(w-k)d]exp[i(ux+vy)]dudv,
F˜(K)=1(2π)3VF(r)exp(-iKr)d3r
Dψ(x, y; d)ψ(x, y; d)=logU(x, y; d)Ui(x, y; d).
D^ψ(u, v; d)=1(2π)2Dψ(x, y; d)×exp[-i(ux+vy)]dxdy
δ(u-u)=12πexp[±i(u-u)x]dx,
D^ψ(u, v; d)=(2π)2iw F˜[us1+vs2+(w-k)s0]×exp[i(w-k)d].
I(x, y; d)|U(x, y; d)|2=exp[ψ(x, y; d)+ψ*(x, y; d)],
DI(x, y; d)=log[I(x, y; d)]=ψ(x, y; d)+ψ*(x, y; d).
D^I(u, v; d)=1(2π)2DI(x, y; d)×exp[-i(ux+vy)]dxdy.
D^I(u, v; d)=i(2π)2|w|2 {w*F˜[us1+vs2+(w-k)s0]×exp[i(w-k)d]-w[F˜[-us1-vs2+(w-k)s0]]*exp[-i(w*-k)d]}.
F˜[K]0,|K|  2π/σ.
w-k-12k (u2+v2).
Dψ(x, y; d)
=(2π)2i1w F˜us1+vs2-12k (u2+v2)s0×exp-12ki(u2+v2)d×exp[i(ux+vy)]dudv.
Dψ(x, y; d)2πikVF(r)δ(x-x)δ(y-y)d3r.
DI(x, y; d)-4πkVIm[F(r)]δ(x-x)×δ(y-y)d3r,
2ik z+T2ψ(x, y, z)=0,
-2ik z+T2ψ*(x, y, z)=0.
-2k z ψi(x, y, z)+T2ψr(x, y, z)=0,
2k z ψr(x, y, z)+T2ψi(x, y, z)=0,
ψr(x, y, z)=12log[I(x, y, z)].
T2ψi(x, y, z)=-k 1I(x, y, z)I(x, y, z)z.
ψi(r)=-2kG(r,r) z ψr(r)d2r,
G(r, r)=12πlog(|r-r|/Λ)
DH(x, y; d)=ψr(x, y; d)+iψi(x, y; d),
ψ^i(u, v; z)=k(u2+v2)z D^I(u, v; z).
D^I(u, v; z)=i(2π)2w {F˜[u, v]exp[-iz(u2+v2)/2k]-[F˜[-u, -v]]*exp[iz(u2+v2)/2k]},
F˜us1+vs2-12k (u2+v2)s0=F˜[u, v]
z D^I(u, v; d)=D^I(u, v; d+Δ)-D^I(u, v; d)Δ.
ψ^i(u, v; d)=kΔ(u2+v2) [D^I(u, v; d+Δ)-D^I(u, v; d)].
D^H(u, v; d)=i(2π)2w {F˜[u, v]exp[-id(u2+v2)/2k]×[1+exp[-iΔ(u2+v2)/4k]×j0[Δ(u2+v2)/4k]]-[F˜[-u, -v]]*exp[id(u2+v2)/2k]×[1-exp[-iΔ(u2+v2)/4k]×j0[Δ(u2+v2)/4k]]}
Δ(u2+v2)/4k<π/2
Δ<σ2k2π.
D^H(u, v; d)=i(2π)2w F˜[u, v]exp[-id(u2+v2)/2k],
D^Δ(u, v; d, Δ)D^I(u, v; d)-D^I(u, v; d+Δ)exp[-iΔ(u2+v2)/2k]Δ.
D^Δ(u, v; d, Δ)=(2π)2iwΔ F˜[u, v]exp[-id(u2+v2)/2k]×{1-exp[-iΔ(u2+v2)/k]}.
Δk (u2+v2)=2nπ(n=0, 1, 2,).
D^I(0, 0; d)=(2π)2ik {F˜[0, 0]-[F˜[0, 0]]*}=-2(2π)2k F˜i(0, 0),
Us(r, θ)=l=0αlhl(1)(kr)Yl0(θ),
αl=il4π(2l+1)×jl(ka)jl(kna)-njl(ka)jl(kna)nhl(1)(ka)jl(kna)-hl(1)(ka)jl(kna),
U(r)=Ui(r)+Us(r),
|n-1|1,
2ka|n-1|1.
F˜[K]=k2a3[n2-1](2π)3j1(Ka)Ka.
F˜[K]A+BK2,

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