Abstract

A general theoretical formulism for in-line phase x-ray imaging was presented with a corresponding linear formula in previous works. In this report, an iterative approach is introduced for phase retrieval with a nonlinear formula. The results of simulation showed that the iterative approach can retrieve the phase map more effectively with high efficiency and flexibility.

© 2007 Optical Society of America

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References

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  1. A. Snigirev and I. Snigireva, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
    [CrossRef]
  2. S.W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A.W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335-338 (1996).
    [CrossRef]
  3. A. Pogany, D. Gao, and S.W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
    [CrossRef]
  4. F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Mammography with synchrotron radiation: Phasedetection techniques," Radiology 215, 286-293 (2000).
  5. X. Wu and H. Liu, "A general theoretical formalism for X-ray phase contrast imaging," J. X-ray Sci.Tech. 11, 33-42 (2003).
  6. X. Wu and H. Liu, "Clinical implementation of x-ray phase-contrast imaging: Theoretical foundations and design considerations," Med. Phys. 30, 2169-2179 (2003).
    [CrossRef] [PubMed]
  7. X. Wu and H. Liu, "A dual detector approach for X-ray attenuation and phase imaging," J. X-ray Sci.Tech. 12, 35-42 (2004).
  8. X. Wu and H. Liu, "Phase-space formulation for phase-contrast x-ray imaging," Appl. Opt. 44, 5847-5854 (2005).
    [CrossRef] [PubMed]
  9. X. Wu, H. Liu, and A. M. Yan, "X-ray phase-attenuation duality and phase retrieval," Opt. Lett. 30, 379-381 (2005).
    [CrossRef] [PubMed]
  10. Y. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093,706 (2005).
    [CrossRef]
  11. B. D. Arhatari, K. A. Nugent, A. G. Peele, and J. Thornton, "Phase contrast radiography. II. Imaging of complex objects," Rev. Sci. Instrum. 76, 113,704 (2005).
    [CrossRef]
  12. T. E. Gureyev, Y. L. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
    [CrossRef]
  13. T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, "Quantitative in-line phase-contrast imaging with multienergy X rays." Phys. Rev. Lett. 86, 5827-5830 (2001).
    [CrossRef] [PubMed]
  14. X. Wu and H. Liu, "A new theory of phase-contrast x-ray imaging based on Wigner distributions." Med. Phys. 31, 2378-2384 (2004).
    [CrossRef] [PubMed]
  15. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik(Stuttgart) 35, 237-246 (1972).

2006 (1)

T. E. Gureyev, Y. L. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
[CrossRef]

2005 (4)

X. Wu, H. Liu, and A. M. Yan, "X-ray phase-attenuation duality and phase retrieval," Opt. Lett. 30, 379-381 (2005).
[CrossRef] [PubMed]

X. Wu and H. Liu, "Phase-space formulation for phase-contrast x-ray imaging," Appl. Opt. 44, 5847-5854 (2005).
[CrossRef] [PubMed]

Y. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093,706 (2005).
[CrossRef]

B. D. Arhatari, K. A. Nugent, A. G. Peele, and J. Thornton, "Phase contrast radiography. II. Imaging of complex objects," Rev. Sci. Instrum. 76, 113,704 (2005).
[CrossRef]

2004 (2)

X. Wu and H. Liu, "A dual detector approach for X-ray attenuation and phase imaging," J. X-ray Sci.Tech. 12, 35-42 (2004).

X. Wu and H. Liu, "A new theory of phase-contrast x-ray imaging based on Wigner distributions." Med. Phys. 31, 2378-2384 (2004).
[CrossRef] [PubMed]

2003 (2)

X. Wu and H. Liu, "A general theoretical formalism for X-ray phase contrast imaging," J. X-ray Sci.Tech. 11, 33-42 (2003).

X. Wu and H. Liu, "Clinical implementation of x-ray phase-contrast imaging: Theoretical foundations and design considerations," Med. Phys. 30, 2169-2179 (2003).
[CrossRef] [PubMed]

2001 (1)

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, "Quantitative in-line phase-contrast imaging with multienergy X rays." Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

2000 (1)

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Mammography with synchrotron radiation: Phasedetection techniques," Radiology 215, 286-293 (2000).

1997 (1)

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

1996 (1)

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

1995 (1)

A. Snigirev and I. Snigireva, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik(Stuttgart) 35, 237-246 (1972).

Appl. Opt. (1)

Med. Phys. (2)

X. Wu and H. Liu, "A new theory of phase-contrast x-ray imaging based on Wigner distributions." Med. Phys. 31, 2378-2384 (2004).
[CrossRef] [PubMed]

X. Wu and H. Liu, "Clinical implementation of x-ray phase-contrast imaging: Theoretical foundations and design considerations," Med. Phys. 30, 2169-2179 (2003).
[CrossRef] [PubMed]

Nature (1)

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

Opt. Commun. (1)

T. E. Gureyev, Y. L. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006).
[CrossRef]

Opt. Lett. (1)

Optik(Stuttgart) (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik(Stuttgart) 35, 237-246 (1972).

Phys. Rev. Lett. (1)

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, "Quantitative in-line phase-contrast imaging with multienergy X rays." Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

Radiology (1)

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Mammography with synchrotron radiation: Phasedetection techniques," Radiology 215, 286-293 (2000).

Rev. Sci. Instrum. (4)

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

Y. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093,706 (2005).
[CrossRef]

B. D. Arhatari, K. A. Nugent, A. G. Peele, and J. Thornton, "Phase contrast radiography. II. Imaging of complex objects," Rev. Sci. Instrum. 76, 113,704 (2005).
[CrossRef]

A. Snigirev and I. Snigireva, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Tech. (2)

X. Wu and H. Liu, "A dual detector approach for X-ray attenuation and phase imaging," J. X-ray Sci.Tech. 12, 35-42 (2004).

X. Wu and H. Liu, "A general theoretical formalism for X-ray phase contrast imaging," J. X-ray Sci.Tech. 11, 33-42 (2003).

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Figures (6)

Fig. 1.
Fig. 1.

The attenuation and phase maps of the virtual sample for the simulation. The size of both maps is 1024 by 1024. The attenuation map is of average value 0.6455 and standard deviation 0.2257. The phase map is of average value 9.3203 and standard deviation 4.8829.The RZP is chosen as the 100 by 100 square on the upper-left corner of the sample. In the RZP, ϕ = 5.

Fig. 2.
Fig. 2.

The simulated attenuation-based and phase-contrast images of the virtual sample. Due to the short wavelength of the x-ray and hence weak diffraction, the phase contrast effects are almost unnoticeable in I 2, but can be seen in the difference image M 2 I 2 -I 1.

Fig. 3.
Fig. 3.

The retrieved phase map using the linearized formula. The retrieved map of A 2 ϕ is clearly not the product of the two. And due to the ambiguity in the retrieval, the phase map cannot be uniquely determined. Direct division of (a) gives a phase map (b).

Fig. 4.
Fig. 4.

The retrieved phase map using the iterative algorithm. The small remnants of attenuation on ϕ is due to the deficiency of the algorithm for calculating the gradient of the attenuation map and the violation of the assumption Eq. (3).

Fig. 5.
Fig. 5.

The standard deviation of the difference map versus the iteration count. The STD increases at first because of the improperly chosen initial distribution A2f = 0, but converges quickly and levels off after iteration 13.

Fig. 6.
Fig. 6.

One dimensional tests of the iterative retrieval algorithm. The attenuation and phase maps of the “virtual object” are taken from the 500th row of the images shown in Fig. 1(a) and (b), respectively.

Equations (8)

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

̂ [ I bp ( x ) ] = I 0 { cos ( πλ R 2 u 2 M ) ̂ [ A 2 ( η ) ] + 2 sin ( πλ R 2 u 2 M ) ̂ [ A 2 ( η ) ϕ ( η ) ] + i λ R 2 u M sin ( πλ R 2 u 2 M ) ̂ [ A ( η ) d A ( η ) ] i 2 λ R 2 u M cos ( πλ R 2 u 2 M ) ̂ [ A ( η ) d A ( η ) ϕ ( η ) ] } ,
exp ( ϕ ( η ) ϕ ( η λ R 2 u M ) ) 1 + ϕ ( η ) ϕ ( η λ R 2 u M )
A ( η ± λ R 2 u M ) A ( η ) ± λ R 2 u M d A ( η ) ,
̂ [ I bp ( x ) ] = I 0 { cos ( πλ R 2 u 2 M ) ̂ [ A 2 ( η ) ] + 2 sin ( πλ R 2 u 2 M ) ̂ [ A 2 ( η ) ϕ ( η ) ] }
T 1 T 2 T 4 T 3 .
0 2 sin ( πλ R 2 u 2 M ) ̂ [ A 2 ( η ) ] i 2 λ R 2 u M cos ( πλ R 2 u 2 M ) ̂ [ A ( η ) d A ( η ) d η ]
̂ [ I bp ( x ) ] = I 0 OTF G . U . ( u M ) OTF det ( u M )
{ cos ( πλ R 2 u 2 M ) ̂ [ A 2 ( η ) ] + 2 sin ( πλ R 2 u 2 M ) ̂ [ A 2 ( η ) ϕ ( η ) ] i λ R 2 u M cos ( πλ R 2 u 2 M ) ̂ [ d ln A 2 ( η ) A 2 ( η ) ϕ ( η ) ] }

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