Abstract

A new reconstruction algorithm for phase-object imaging is proposed that is based on the principle of diffraction tomography and utilizes the Fourier transformation property of a finite-size phase object. From the measured scattered intensity, the imaginary part of the Fourier transform of the object can be extracted, and the three-dimensional structure of the object can be reconstructed. Numerical simulations show that the algorithm also can be used for a weak absorption object if the phase shift is much larger than the absorption.

© 2001 Optical Society of America

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  1. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
    [CrossRef]
  2. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
    [CrossRef]
  3. A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus X-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
    [CrossRef]
  4. C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase contrast microimaging with coherent high-energy synchrotron x rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
    [CrossRef]
  5. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
    [CrossRef] [PubMed]
  6. T. E. Gureyev, S. W. Wilkins, “On X-ray phase retrieval from phychromatic images,” Opt. Commun. 147, 229–232 (1998).
    [CrossRef]
  7. J. Cheng, S. S. Han, “Phase imaging with partially coherent x rays,” Opt. Lett. 24, 175–177 (1999).
    [CrossRef]
  8. A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
    [CrossRef]
  9. N. Jayshree, G. Keshaa Datta, R. M. Vasu, “Optical tomographic microscope for quantitative imaging of phase objects,” Appl. Opt. 39, 277–283 (2000).
    [CrossRef]
  10. A. Barty, K. A. Nugent, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
    [CrossRef]
  11. E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. C. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.
  12. A. Devaney, “A filtered backpropogation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
    [CrossRef] [PubMed]
  13. M. Slaney, A. Kak, L. Larsen, “Limitation of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
    [CrossRef]
  14. A. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
    [CrossRef] [PubMed]
  15. X. Pan, “Unified reconstruction theory for diffraction tomography, with consideration of noise control,” J. Opt. Soc. Am. A 15, 2312–2326 (1998).
    [CrossRef]
  16. B. Chen, J. Stamnes, “Validity of diffraction tomography based on the first Born and the first Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
    [CrossRef]
  17. M. 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]
  18. K. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).
  19. M. Kang, A. Katsaggelos, “General choice of the regularization functional in regularized image restoration,” IEEE Trans. Image Process. 4, 594–602 (1995).
    [CrossRef] [PubMed]

2000 (3)

1999 (1)

1998 (4)

1997 (1)

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus X-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

1996 (3)

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase contrast microimaging with coherent high-energy synchrotron x rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

1995 (2)

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

M. Kang, A. Katsaggelos, “General choice of the regularization functional in regularized image restoration,” IEEE Trans. Image Process. 4, 594–602 (1995).
[CrossRef] [PubMed]

1989 (1)

A. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[CrossRef] [PubMed]

1984 (1)

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

1982 (1)

A. Devaney, “A filtered backpropogation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

Anastasio, M.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Barty, A.

A. Barty, K. A. Nugent, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

Castleman, K.

K. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).

Chen, B.

Cheng, J.

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Davis, T. J.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Devaney, A.

A. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[CrossRef] [PubMed]

A. Devaney, “A filtered backpropogation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

Gao, D.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus X-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, S. W. Wilkins, “On X-ray phase retrieval from phychromatic images,” Opt. Commun. 147, 229–232 (1998).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Han, S. S.

Jayshree, N.

Kak, A.

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

Kang, M.

M. Kang, A. Katsaggelos, “General choice of the regularization functional in regularized image restoration,” IEEE Trans. Image Process. 4, 594–602 (1995).
[CrossRef] [PubMed]

Katsaggelos, A.

M. Kang, A. Katsaggelos, “General choice of the regularization functional in regularized image restoration,” IEEE Trans. Image Process. 4, 594–602 (1995).
[CrossRef] [PubMed]

Keshaa Datta, G.

Kohn, V.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase contrast microimaging with coherent high-energy synchrotron x rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

Larsen, L.

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

Nugent, K. A.

A. Barty, K. A. Nugent, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Paganin, D.

A. Barty, K. A. Nugent, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Pan, X.

Pogany, A.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus X-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

Raven, C.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase contrast microimaging with coherent high-energy synchrotron x rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

Roberts, A.

Slaney, M.

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

Snigirev, A.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase contrast microimaging with coherent high-energy synchrotron x rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

Snigireva, I.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase contrast microimaging with coherent high-energy synchrotron x rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

Souvorov, A.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase contrast microimaging with coherent high-energy synchrotron x rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

Spanne, P.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase contrast microimaging with coherent high-energy synchrotron x rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

Stamnes, J.

Stevenson, A. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Vasu, R. M.

Wilkins, S. W.

T. E. Gureyev, S. W. Wilkins, “On X-ray phase retrieval from phychromatic images,” Opt. Commun. 147, 229–232 (1998).
[CrossRef]

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus X-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Wolf, E.

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

Appl. Opt. (2)

Appl. Phys. Lett. (1)

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase contrast microimaging with coherent high-energy synchrotron x rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

IEEE Trans. Image Process. (1)

M. Kang, A. Katsaggelos, “General choice of the regularization functional in regularized image restoration,” IEEE Trans. Image Process. 4, 594–602 (1995).
[CrossRef] [PubMed]

IEEE Trans. Microwave Theory Tech. (1)

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

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

Nature (2)

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

Opt. Commun. (2)

A. Barty, K. A. Nugent, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

T. E. Gureyev, S. W. Wilkins, “On X-ray phase retrieval from phychromatic images,” Opt. Commun. 147, 229–232 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (2)

A. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus X-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Ultrason. Imaging (1)

A. Devaney, “A filtered backpropogation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

Other (2)

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

K. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).

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Figures (4)

Fig. 1
Fig. 1

Geometry for the diffraction tomography experiment. The incident wave is along the s0 direction, and the scattering direction is s.

Fig. 2
Fig. 2

Original and reconstructed images for a square disk phase object. (a) Original object. (b) Reconstruction from the noiseless scattered intensity by our algorithm with α=0.001. (c) Reconstruction from the noise scattered intensity by our algorithm with α=0.1; the noise-to-signal ratio is 2%. (d) Reconstruction from the noise scattered intensity by our algorithm with α=0.001; the noise-to-signal ratio is 2%. From (a) to (d), the vertical axis is the position in the x direction, and the horizontal axis is the position in the y direction. (e) Cross-sectional cuts along x=5λ for the original object as in (a) (solid curve) and the reconstructed object as in (b) (dotted-dashed curve), (c) (dashed curve), and (d) (dotted curve) from our algorithm. The reconstruction from the noiseless data is so good that the solid curve and the dotted-dashed curve are not distinguishable in the figure. The horizontal axis is the position in the y direction, and the vertical axis is the refractive-index perturbation.

Fig. 3
Fig. 3

Original and reconstructed image for a square ring low-absorption object. (a) Original object. (b) Phase reconstruction from the noiseless scattered intensity by our algorithm with α=0.01. The vertical axis is the position in the x direction, and the horizontal axis is the position in the y direction.

Fig. 4
Fig. 4

Reconstructed sections of a 3D cube pure phase object. (a) Reconstructed xy section. (b) Reconstructed yz section; α=0.001. The vertical axis is the position in the x direction, and the horizontal axis is the position in the y direction.

Equations (24)

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φ(r)exp(-jωt)=[φi(r)+φs(r, ω)]exp(-jωt).
φs(r, ω)=V(r, ω)×exp(jks0r)exp(jk|r-r|)|r-r|d3r,
V(r, ω)=k24π [n2(r)-1].
φs(rs, s0, ω)=f(s, s0, ω) exp(jkr)r,
I(rs, s0, ω)=|φ(rs, s0, ω)|2.
I(rs, s0, ω)=|φ(rs, s0, ω)|2=|exp(jkrs0s)+exp(jkr)r×V˜[k(s-s0)]|2.
I(rs, s0, ω)=exp-jkrξ22+1r V˜(kξ)2=1+1r2 [V˜r(kξ)2+V˜i(kξ)2]+2rcoskrξ22V˜r(kξ)-sinkrξ22V˜i(kξ),
V˜r(kξ)=V˜r(-kξ),
V˜i(kξ)=-V˜i(-kξ).
I(rs, s0, ω)-I(-rs, -s0, ω)
=1r2 [V˜r(kξ)2+V˜i(kξ)2]+2rcoskrξ22V˜r(kξ)-sinkrξ22V˜i(kξ)-1r2 [V˜r(-kξ)2+V˜i(-kξ)2]-2rcoskrξ22V˜r(-kξ)-sinkrξ22V˜i(-kξ)=2r-sinkrξ22V˜i(kξ)-2r-sinkrξ22V˜i(-kξ)=-4rsinkrξ22V˜i(kξ).
V˜i(kξ)=-r4 sinkrξ22 [I(rs, s0, ω)-I(-rs, -s0, ω)].
V(r)=V(x, y, z)=0
if(x<0)or(y<0)or(z<0).
f(x, y, z)=2j|η|<2kdηxdηydηzV˜i(ηx, ηy, ηz)exp[j(xηx+yηy+zηz)]=|η|<2kdηxdηydηzexp[j(xηx+yηy+zηz)]×[V˜(ηx, ηy, ηz)-V˜*(ηx, ηy, ηz)]=[Vl(x, y, z)-Vl(-x, -y, -z)],
Vl(x, y, z)=|η|<2kdηxdηydηzV˜(ηx, ηy, ηz)×exp[j(xηx+yηy+zηz)].
I(rs, s0, ω)-I(-rs, -s0, ω)
=-4rsinkrξ22V˜i(kξ)+N,
F(ξ)=I(rs, s0, ω)-I(-rs, -s0, ω),
H(ξ)=-4rsinkrξ22,
G(ξ)=V˜i(kξ),
F(ξ)=H(ξ)×G(ξ)+N.
G¯(ξ)=H*(ξ)×F(ξ)|H(ξ)|2+α,
F(ξ)-H(ξ)×G¯(ξ)=N,

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