Abstract

We describe a direct quantitative phase reconstruction approach using an X-ray laboratory-based source. Using a single phase-contrast image from each tomographic projection we show that it is possible to modify the filter term in a filtered back projection reconstruction to take account of the broad spectrum from a laboratory source. The accessibility of conventional X-ray laboratory sources makes this method very useful for quantitative phase imaging of homogeneous and weakly absorbing objects.

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References

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  1. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9(7), 686–698 (1942).
    [Crossref]
  2. E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980).
    [Crossref]
  3. D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
  4. S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11(19), 2289–2302 (2003).
    [Crossref] [PubMed]
  5. A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19(3), 472–480 (2002).
    [Crossref] [PubMed]
  6. A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography,” Opt. Express 14(18), 8103–8110 (2006).
    [Crossref] [PubMed]
  7. B. D. Arhatari, F. De Carlo, and A. G. Peele, “Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects,” Rev. Sci. Instrum. 78(5), 053701 (2007).
    [Crossref] [PubMed]
  8. T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase and amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006).
    [Crossref]
  9. R. C. Chen, H. L. Xie, L. Rigon, R. Longo, E. Castelli, and T. Q. Xiao, “Phase retrieval in quantitative x-ray microtomography with a single sample-to-detector distance,” Opt. Lett. 36(9), 1719–1721 (2011).
    [Crossref] [PubMed]
  10. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
    [Crossref]
  11. B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75(12), 5271–5276 (2004).
    [Crossref]
  12. B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic x-ray laboratory source,” Opt. Express 16(24), 19950–19956 (2008).
    [Crossref] [PubMed]
  13. K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
    [Crossref] [PubMed]
  14. A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun. 171(4-6), 239–244 (1999).
    [Crossref]
  15. R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000).
    [Crossref]
  16. G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76(4), 045804 (2007).
    [Crossref]

2011 (1)

2008 (1)

2007 (2)

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76(4), 045804 (2007).
[Crossref]

B. D. Arhatari, F. De Carlo, and A. G. Peele, “Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects,” Rev. Sci. Instrum. 78(5), 053701 (2007).
[Crossref] [PubMed]

2006 (2)

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase and amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006).
[Crossref]

A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography,” Opt. Express 14(18), 8103–8110 (2006).
[Crossref] [PubMed]

2004 (1)

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75(12), 5271–5276 (2004).
[Crossref]

2003 (1)

2002 (1)

2000 (1)

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000).
[Crossref]

1999 (1)

A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun. 171(4-6), 239–244 (1999).
[Crossref]

1996 (2)

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

1980 (1)

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980).
[Crossref]

1942 (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9(7), 686–698 (1942).
[Crossref]

Abela, R.

Arhatari, B. D.

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic x-ray laboratory source,” Opt. Express 16(24), 19950–19956 (2008).
[Crossref] [PubMed]

B. D. Arhatari, F. De Carlo, and A. G. Peele, “Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects,” Rev. Sci. Instrum. 78(5), 053701 (2007).
[Crossref] [PubMed]

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75(12), 5271–5276 (2004).
[Crossref]

Balaur, E.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Bronnikov, A. V.

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19(3), 472–480 (2002).
[Crossref] [PubMed]

A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun. 171(4-6), 239–244 (1999).
[Crossref]

Castelli, E.

Chen, R. C.

Cookson, D. J.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Davis, T. J.

De Carlo, F.

B. D. Arhatari, F. De Carlo, and A. G. Peele, “Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects,” Rev. Sci. Instrum. 78(5), 053701 (2007).
[Crossref] [PubMed]

Fitzgerald, R.

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000).
[Crossref]

Förster, E.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980).
[Crossref]

Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

Goetz, K.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980).
[Crossref]

Groso, A.

Gureyev, T. E.

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76(4), 045804 (2007).
[Crossref]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase and amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006).
[Crossref]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11(19), 2289–2302 (2003).
[Crossref] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Hannah, K.

Longo, R.

Mancuso, A. P.

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75(12), 5271–5276 (2004).
[Crossref]

Mayo, S.

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76(4), 045804 (2007).
[Crossref]

Mayo, S. C.

Miller, P. R.

Myers, G. R.

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76(4), 045804 (2007).
[Crossref]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase and amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006).
[Crossref]

Nesterets, Y. I.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase and amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006).
[Crossref]

Nugent, K. A.

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75(12), 5271–5276 (2004).
[Crossref]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Paganin, D.

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76(4), 045804 (2007).
[Crossref]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11(19), 2289–2302 (2003).
[Crossref] [PubMed]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Paganin, D. M.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase and amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006).
[Crossref]

Peele, A. G.

B. D. Arhatari, K. Hannah, E. Balaur, and A. G. Peele, “Phase imaging using a polychromatic x-ray laboratory source,” Opt. Express 16(24), 19950–19956 (2008).
[Crossref] [PubMed]

B. D. Arhatari, F. De Carlo, and A. G. Peele, “Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects,” Rev. Sci. Instrum. 78(5), 053701 (2007).
[Crossref] [PubMed]

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75(12), 5271–5276 (2004).
[Crossref]

Pogany, A.

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11(19), 2289–2302 (2003).
[Crossref] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

Rigon, L.

Stampanoni, M.

Stevenson, A. W.

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11(19), 2289–2302 (2003).
[Crossref] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

Wilkins, S. W.

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76(4), 045804 (2007).
[Crossref]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase and amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006).
[Crossref]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11(19), 2289–2302 (2003).
[Crossref] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

Xiao, T. Q.

Xie, H. L.

Zaumseil, P.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980).
[Crossref]

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9(7), 686–698 (1942).
[Crossref]

Appl. Phys. Lett. (1)

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase and amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006).
[Crossref]

J. Opt. Soc. Am. A (1)

Krist. Tech. (1)

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980).
[Crossref]

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(6607), 335–338 (1996).
[Crossref]

Opt. Commun. (1)

A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun. 171(4-6), 239–244 (1999).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (1)

G. R. Myers, S. Mayo, T. E. Gureyev, D. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76(4), 045804 (2007).
[Crossref]

Phys. Rev. Lett. (1)

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Phys. Today (1)

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000).
[Crossref]

Physica (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9(7), 686–698 (1942).
[Crossref]

Rev. Sci. Instrum. (2)

B. D. Arhatari, F. De Carlo, and A. G. Peele, “Direct quantitative tomographic reconstruction for weakly absorbing homogeneous phase objects,” Rev. Sci. Instrum. 78(5), 053701 (2007).
[Crossref] [PubMed]

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75(12), 5271–5276 (2004).
[Crossref]

Other (1)

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).

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Figures (4)

Fig. 1
Fig. 1

The linear attenuation of polystyrene as a function of energy and the calculated effective value µpoly for the measured source spectrum is shown in (a). Similarly δ and δpoly are shown on (b). The combined detector response function and X-ray source spectrum for a Tungsten target at 40kV are shown in both Fig (a) and (b) (dotted lines).

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Fig. 2
Fig. 2

(a). The recorded intensity image of polystyrene micro-spheres using a polychromatic X-ray source at 40kV. (b). The distribution of Eq. (8) for polystyrene using the set-up described in the text.

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Fig. 3
Fig. 3

(a) The rendering based on the reconstruction result of polystyrene micro-spheres using Eq. (7). (b). One of the reconstructed slices. (c). The corresponding plot along the dash line shown in (b).

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Fig. 4
Fig. 4

(a) Comparison of the nominal bulk density distribution of polystyrene spheres determined by different approaches. The solid line is calculated using Eq. (7) and 8), the dotted line is obtained by using an effective energy, and the dashed line is obtained by using an energy corresponding to maximum intensity in the source spectrum. (b). The reconstruction slice of a combination of polyimide film and polystyrene spheres. (c). The plot along the dashed horizontal line in Fig. 4(b).

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Equations (8)

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

I θ,λ z ( x,y )= I 0 [ 1+ g θ,λ ( x,y ) ].
g θ,λ ( x,y )=− μ λ t ^ θ ( x,y )+z δ λ ∇ 2 t ^ θ ( x,y ),
t ^ θ (x,y)= ∫ sampl e θ ρ f (x,y,z)dz
∫ I θ,λ z ( x,y ) D λ dλ ∫ D λ dλ = ∫ I 0 [ 1+ g θ,λ ( x,y ) ] D λ dλ ∫ D λ dλ
I θ,poly z ( x,y )= I 0 [ 1+ g θ,poly ( x,y ) ].
g θ,poly ( x,y )=− μ poly t ^ θ ( x,y )+z δ poly ∇ 2 t ^ θ ( x,y )
ρ f ( x,y,z )=− 1 δ poly ∫ 0 π q θ ( xcosθ+zsinθ,y ) dθ,
S( ξ,η )= |ξ| μ poly δ poly +4 π 2 z( ξ 2 + η 2 ) ,

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