Abstract

We present a method for phase retrieval from x-ray Fresnel diffraction patterns for multimaterial objects. Previously, homogeneous object assumptions have been used and have been introduced in the Radon domain. Here, we apply prior knowledge in the object domain, which permits the introduction of multiple materials. This is achieved first by a tomographic reconstruction of an attenuation scan and then introduction of the prior followed by a forward projection step to yield the a priori phase maps. The method is applied to the reconstruction of an object of known composition consisting of both soft and hard materials and is shown to perform better than previously proposed algorithms.

© 2012 Optical Society of America

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  1. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, Appl. Phys. Lett. 75, 2912 (1999).
    [CrossRef]
  2. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
    [CrossRef]
  3. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
    [CrossRef]
  4. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, J. Microsc. 206, 33 (2002).
    [CrossRef]
  5. J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, Opt. Lett. 32, 1617 (2007).
    [CrossRef]
  6. M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, Med. Phys. 35, 4556 (2008).
    [CrossRef]
  7. X. Wu, H. Liu, and A. Yan, Opt. Lett. 30, 379 (2005).
    [CrossRef]
  8. L. J. Allen and M. P. Oxley, Opt. Commun. 199, 65 (2001).
    [CrossRef]
  9. V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, Opt. Express 19, 22809 (2011).
    [CrossRef]
  10. J. Moosmann, R. Hofmann, A. Bronnikov, and T. Baumbach, Opt. Express 18, 25771 (2010).
    [CrossRef]
  11. M. Langer, A. Pacureanu, P. Cloetens, Q. Grimal, and F. Peyrin, “X-ray phase nanotomography resolves the 3D human bone ultrastructure,” PLoS One (to be published).
  12. M. Langer, P. Cloetens, and F. Peyrin, IEEE Trans. Image Process. 19, 2428 (2010).
    [CrossRef]
  13. M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, Opt. Express 18, 6423 (2010).
    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).
  15. J. Banhart, Advanced Tomographic Methods in Materials Research and Engineering (Oxford University, 2008).

2011 (1)

2010 (3)

2008 (1)

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, Med. Phys. 35, 4556 (2008).
[CrossRef]

2007 (1)

2005 (1)

2002 (1)

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, J. Microsc. 206, 33 (2002).
[CrossRef]

2001 (1)

L. J. Allen and M. P. Oxley, Opt. Commun. 199, 65 (2001).
[CrossRef]

1999 (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, Appl. Phys. Lett. 75, 2912 (1999).
[CrossRef]

1996 (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

1995 (1)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Allen, L. J.

L. J. Allen and M. P. Oxley, Opt. Commun. 199, 65 (2001).
[CrossRef]

Banhart, J.

J. Banhart, Advanced Tomographic Methods in Materials Research and Engineering (Oxford University, 2008).

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, Appl. Phys. Lett. 75, 2912 (1999).
[CrossRef]

Baumbach, T.

Beltran, M. A.

Boistel, R.

Bronnikov, A.

Cloetens, P.

M. Langer, P. Cloetens, and F. Peyrin, IEEE Trans. Image Process. 19, 2428 (2010).
[CrossRef]

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, Med. Phys. 35, 4556 (2008).
[CrossRef]

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, Opt. Lett. 32, 1617 (2007).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, Appl. Phys. Lett. 75, 2912 (1999).
[CrossRef]

M. Langer, A. Pacureanu, P. Cloetens, Q. Grimal, and F. Peyrin, “X-ray phase nanotomography resolves the 3D human bone ultrastructure,” PLoS One (to be published).

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

Davidoiu, V.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Grimal, Q.

M. Langer, A. Pacureanu, P. Cloetens, Q. Grimal, and F. Peyrin, “X-ray phase nanotomography resolves the 3D human bone ultrastructure,” PLoS One (to be published).

Guigay, J. P.

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, Med. Phys. 35, 4556 (2008).
[CrossRef]

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, Opt. Lett. 32, 1617 (2007).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, Appl. Phys. Lett. 75, 2912 (1999).
[CrossRef]

Gureyev, T. E.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, J. Microsc. 206, 33 (2002).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

Hofmann, R.

Kitchen, M. J.

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Langer, M.

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, Opt. Express 19, 22809 (2011).
[CrossRef]

M. Langer, P. Cloetens, and F. Peyrin, IEEE Trans. Image Process. 19, 2428 (2010).
[CrossRef]

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, Med. Phys. 35, 4556 (2008).
[CrossRef]

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, Opt. Lett. 32, 1617 (2007).
[CrossRef]

M. Langer, A. Pacureanu, P. Cloetens, Q. Grimal, and F. Peyrin, “X-ray phase nanotomography resolves the 3D human bone ultrastructure,” PLoS One (to be published).

Liu, H.

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, Appl. Phys. Lett. 75, 2912 (1999).
[CrossRef]

Mayo, S. C.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, J. Microsc. 206, 33 (2002).
[CrossRef]

Miller, P. R.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, J. Microsc. 206, 33 (2002).
[CrossRef]

Moosmann, J.

Nugent, K. A.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

Oxley, M. P.

L. J. Allen and M. P. Oxley, Opt. Commun. 199, 65 (2001).
[CrossRef]

Pacureanu, A.

M. Langer, A. Pacureanu, P. Cloetens, Q. Grimal, and F. Peyrin, “X-ray phase nanotomography resolves the 3D human bone ultrastructure,” PLoS One (to be published).

Paganin, D.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, J. Microsc. 206, 33 (2002).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

Paganin, D. M.

Peyrin, F.

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, Opt. Express 19, 22809 (2011).
[CrossRef]

M. Langer, P. Cloetens, and F. Peyrin, IEEE Trans. Image Process. 19, 2428 (2010).
[CrossRef]

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, Med. Phys. 35, 4556 (2008).
[CrossRef]

M. Langer, A. Pacureanu, P. Cloetens, Q. Grimal, and F. Peyrin, “X-ray phase nanotomography resolves the 3D human bone ultrastructure,” PLoS One (to be published).

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Schlenker, M.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, Appl. Phys. Lett. 75, 2912 (1999).
[CrossRef]

Sixou, B.

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Uesugi, K.

Van Dyck, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, Appl. Phys. Lett. 75, 2912 (1999).
[CrossRef]

Van Landuyt, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, Appl. Phys. Lett. 75, 2912 (1999).
[CrossRef]

Wilkins, S. W.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, J. Microsc. 206, 33 (2002).
[CrossRef]

Wu, X.

Yan, A.

Appl. Phys. Lett. (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, Appl. Phys. Lett. 75, 2912 (1999).
[CrossRef]

IEEE Trans. Image Process. (1)

M. Langer, P. Cloetens, and F. Peyrin, IEEE Trans. Image Process. 19, 2428 (2010).
[CrossRef]

J. Microsc. (1)

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, J. Microsc. 206, 33 (2002).
[CrossRef]

Med. Phys. (1)

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, Med. Phys. 35, 4556 (2008).
[CrossRef]

Opt. Commun. (1)

L. J. Allen and M. P. Oxley, Opt. Commun. 199, 65 (2001).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

Rev. Sci. Instrum. (1)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

J. Banhart, Advanced Tomographic Methods in Materials Research and Engineering (Oxford University, 2008).

M. Langer, A. Pacureanu, P. Cloetens, Q. Grimal, and F. Peyrin, “X-ray phase nanotomography resolves the 3D human bone ultrastructure,” PLoS One (to be published).

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Figures (3)

Fig. 1.
Fig. 1.

Experimental setup. Monochromatic x rays are selected from undulator radiation with a multilayer monochromator. The sample is mounted on a translation–rotation stage for tomographic imaging. A fluorescent screen CCD-based detector is mounted on a translation stage in the beam direction to allow for variation of the free space propagation distance. Tomographic scans are then recorded at several sample-to-detector distances.

Fig. 2.
Fig. 2.

Results. (a) Tomographic slice reconstructed with standard Tikhonov regularization [5,6]. Gray-scale window [10002200]. Note the low frequency artifacts present in the image. (b) Tomographic slice reconstructed with a prior homogeneous object [11], with δ/β=310 (corresponding to aluminum); the low frequency artifact is alleviated and values are well reconstructed in the Al but not in the other materials. (c) Reconstruction with δ/β=1938 (corresponding to PET); here the low-frequency artifact is also removed and values are correctly reconstructed in the PET but not otherwise. (d) Tomographic slice reconstructed with the proposed method, two δ/β ratios are used [δ/β=310 (Al) and δ/β=1938 (PET)] the low-frequency artifact is alleviated, and values are well reconstructed in the two materials chosen. Gray-scale window in (b)–(d) [01700].

Fig. 3.
Fig. 3.

Creation of the a priori phase maps. (a) Flat and dark field corrected attenuation projection image (D0). Gray-scale window [0.81.0]. (b) Transverse slice of the standard (attenuation) tomogram. Gray-scale window [410]. (c) Slice through the a priori volume. This is generated by applying a δ/β map to the attenuation tomogram in (b), calculated by thresholding the attenuation tomogram. Gray-scale window [10002000]. (c) A priori phase map calculated from the a priori volume in (d) by projection. Gray-scale window [700].

Tables (1)

Tables Icon

Table 1. Reconstructed μ and 2πδn/λ

Equations (13)

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n(r)=1δn(r)+iβ(r),
Tθ(x)=aθ(x)exp[iφθ(x)]=exp[Bθ(x)+iφθ(x)],
Bθ(x)=(2πλ)(θ,x)lineβn(r)dz,
φθ(x)=(2πλ)(θ,x)lineδn(r)dz,
Tθ,D(x)=FrD[T(x)].
ID(x)=|Tθ,D(x)|2+γ(x),
I0(x)=aθ2(x)+γ(x),
φ^θ(x)=argminD|I˜θ,D,φ(x)Iθ,D(x)|2+α|φθ(x)φθ,0(x)|2,
φθ,0(x)=f(x)*δn2βln[Iθ,0(x)],
β^(r)=β(r)+Γ(r),
m(r)={δaβag[β^(r)]>tδbβbotherwise,
δ0(r)=m(r)β^(r).
φθ,0(x)=(2πλ)(θ,x)lineδ0(r)dz.

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