Abstract

We present a method for phase retrieval in propagation-based x-ray imaging, based on the contrast transfer and transport of intensity equation approaches. We show that the contrast transfer model does not coincide with the transport of intensity in the limit of small propagation distances, and we derive a new model that alleviates this problem. Using this model, we devise an algorithm to retrieve the phase from slowly varying samples that is valid beyond the limit of small distances. We show its utility by imaging in three dimensions a biological sample that causes both strong absorption and phase shift.

© 2007 Optical Society of America

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  1. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
    [CrossRef]
  2. A. Momose and J. Fukuda, Med. Phys. 22, 375 (1995).
    [CrossRef] [PubMed]
  3. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
    [CrossRef]
  4. M. R. Teague, J. Opt. Soc. Am. 72, 1199 (1982).
    [CrossRef]
  5. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
    [CrossRef] [PubMed]
  6. J.-P. Guigay, Optik (Stuttgart) 49, 121 (1977).
  7. 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]
  8. S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, Rev. Sci. Instrum. 76, 1 (2005).
    [CrossRef]
  9. X. Wu and H. Liu, J. X-Ray Sci. Technol. 11, 33 (2003).
  10. L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, Opt. Express 12, 2690 (2004).
    [CrossRef]
  11. D. M. Paganin, Coherent X-Ray Optics (Oxford U. Press, 2006).
    [CrossRef]

2006

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

2005

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, Rev. Sci. Instrum. 76, 1 (2005).
[CrossRef]

2004

L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, Opt. Express 12, 2690 (2004).
[CrossRef]

2003

X. Wu and H. Liu, J. X-Ray Sci. Technol. 11, 33 (2003).

1999

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

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

1995

A. Momose and J. Fukuda, Med. Phys. 22, 375 (1995).
[CrossRef] [PubMed]

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

1982

1977

J.-P. Guigay, Optik (Stuttgart) 49, 121 (1977).

Barnea, Z.

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

Baruchel, J.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, Rev. Sci. Instrum. 76, 1 (2005).
[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]

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Cloetens, P.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, Rev. Sci. Instrum. 76, 1 (2005).
[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]

Cookson, D. F.

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

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Dhal, B. B.

L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, Opt. Express 12, 2690 (2004).
[CrossRef]

Fukuda, J.

A. Momose and J. Fukuda, Med. Phys. 22, 375 (1995).
[CrossRef] [PubMed]

Guigay, J.-P.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, Rev. Sci. Instrum. 76, 1 (2005).
[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]

J.-P. Guigay, Optik (Stuttgart) 49, 121 (1977).

Gureyev, T. E.

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

Hayes, J. P.

L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, Opt. Express 12, 2690 (2004).
[CrossRef]

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]

Liu, H.

X. Wu and H. Liu, J. X-Ray Sci. Technol. 11, 33 (2003).

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]

Mancuso, A. P.

L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, Opt. Express 12, 2690 (2004).
[CrossRef]

Momose, A.

A. Momose and J. Fukuda, Med. Phys. 22, 375 (1995).
[CrossRef] [PubMed]

Nugent, K. A.

L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, Opt. Express 12, 2690 (2004).
[CrossRef]

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

Paganin, D.

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

Paganin, D. M.

D. M. Paganin, Coherent X-Ray Optics (Oxford U. Press, 2006).
[CrossRef]

Paterson, D.

L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, Opt. Express 12, 2690 (2004).
[CrossRef]

Peele, A. G.

L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, Opt. Express 12, 2690 (2004).
[CrossRef]

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Schelokov, I.

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

Schlenker, M.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, Rev. Sci. Instrum. 76, 1 (2005).
[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]

Scholten, R. E.

L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, Opt. Express 12, 2690 (2004).
[CrossRef]

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]

Teague, M. R.

Tran, C. Q.

L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, Opt. Express 12, 2690 (2004).
[CrossRef]

Turner, L. D.

L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, Opt. Express 12, 2690 (2004).
[CrossRef]

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]

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Wu, X.

X. Wu and H. Liu, J. X-Ray Sci. Technol. 11, 33 (2003).

Zabler, S.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, Rev. Sci. Instrum. 76, 1 (2005).
[CrossRef]

Appl. Phys. Lett.

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]

J. Opt. Soc. Am.

J. X-Ray Sci. Technol.

X. Wu and H. Liu, J. X-Ray Sci. Technol. 11, 33 (2003).

Med. Phys.

A. Momose and J. Fukuda, Med. Phys. 22, 375 (1995).
[CrossRef] [PubMed]

Nat. Phys.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Opt. Express

L. D. Turner, B. B. Dhal, J. P. Hayes, A. P. Mancuso, K. A. Nugent, D. Paterson, R. E. Scholten, C. Q. Tran, and A. G. Peele, Opt. Express 12, 2690 (2004).
[CrossRef]

Optik (Stuttgart)

J.-P. Guigay, Optik (Stuttgart) 49, 121 (1977).

Phys. Rev. Lett.

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

Rev. Sci. Instrum.

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

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, Rev. Sci. Instrum. 76, 1 (2005).
[CrossRef]

Other

D. M. Paganin, Coherent X-Ray Optics (Oxford U. Press, 2006).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic of the experimental setup. Phase contrast is formed by moving the detector away a distance D from the sample. Tomographic scans are then recorded at several distances, typically two using the TIE and four using the CTF. Absorption contrast images are recorded by moving the detector to D = 0 .

Fig. 2
Fig. 2

Amplitude and phase contrast factors for the CTF and TIE.

Fig. 3
Fig. 3

(a) Absorption radiograph of a Crossodactylodes bokermani frog embedded in a liquid-filled cylindrical polypropylene tube sample holder. (b) Phase map retrieved from radiographs at sample-to-detector distances 18, 83, 283 and 973 mm . The bright areas are due to air bubbles in the liquid. (c) Tomographic slice through the frog head using the absorption radiographs and (d) the retrieved phase maps. Fine structures in the soft tissue are visible in (d) despite the presence of strongly absorbing structures (bone).

Equations (17)

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n ( x , y , z ) = 1 δ r ( x , y , z ) + i β ( x , y , z ) ,
T ( x ) = a ( x ) exp [ i φ ( x ) ] = exp [ B ( x ) + i φ ( x ) ] ,
I ̃ D ( f ) = T ( x λ D f 2 ) T * ( x + λ D f 2 ) exp ( i 2 π x f ) d x ,
T ( x + λ D f 2 ) = T ( x ) + 1 2 λ D f T ( x ) .
I D ( x ) = I 0 ( x ) λ D 2 π [ I 0 ( x ) φ ( x ) ] .
[ I 0 ( x ) φ ( x ) ] = 2 π λ z I 0 ( x ) .
T ( x ) 1 B ( x ) i φ ( x ) .
I ̃ D ( f ) = δ ( f ) 2 cos ( π λ D f 2 ) B ̃ ( f ) + 2 sin ( π λ D f 2 ) φ ̃ ( f ) ,
B ( x ) 1 , φ ( x ) φ ( x + λ D f ) 1 .
I ̃ D 0 ( f ) = δ ( f ) 2 B ̃ ( f ) + 2 π λ D f 2 φ ̃ ( f ) ,
I D 0 ( x ) = I 0 ( x ) λ D 2 π 2 φ ( x ) ,
I ̃ D ( f ) = exp ( i 2 π x f ) a ( x λ D f 2 ) a ( x + λ D f 2 ) × [ 1 + i φ ( x λ D f 2 ) i φ ( x + λ D f 2 ) ] d x .
I ̃ D ( f ) = I ̃ D φ = 0 + sin ( π λ D f 2 ) a ( x ) φ ( x ) exp ( i 2 π x f ) × [ a ( x + λ D f ) + a ( x λ D f ) ] d x + i cos ( π λ D f 2 ) a ( x ) φ ( x ) exp ( i 2 π x f ) × [ a ( x + λ D f ) a ( x λ D f ) ] d x .
I ̃ D ( f ) = I ̃ D φ = 0 ( f ) + 2 sin ( π λ D f 2 ) × exp ( i 2 π x f ) φ ( x ) a 2 ( x ) d x + i 2 cos ( π λ D f 2 ) λ D f × exp ( i 2 π x f ) φ ( x ) a ( x ) a ( x ) d x ,
I ̃ D ( f ) = I ̃ D φ = 0 ( f ) + 2 sin ( π λ D f 2 ) F { I 0 φ } ( f ) + cos ( π λ D f 2 ) λ D 2 π F { ( φ I 0 ) } ( f ) .
I ̃ D ( f ) = I ̃ D φ = 0 + 2 [ sin ( π λ D f 2 ) π λ D f 2 cos ( π λ D f 2 ) ] × F { I 0 φ } ( f ) cos ( π λ D f 2 ) λ D 2 π F { ( I 0 φ ) } ( f ) .
F { I 0 φ ( n + 1 ) } ( f ) = D A D ( f ) [ I ̃ D ( f ) I ̃ D φ = 0 ( f ) Δ D ( f ) ] D A D 2 ( f ) ,

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