Abstract

Phase objects can be characterized using well-known methods such as shear interferometry and deflectometry, which provide information on the partial derivatives of the phase. It is often believed that for phase retrieval it is strictly necessary to have knowledge of two partial derivatives in orthogonal directions. In the praxis, this implies that the measurements have to be performed along two dimensions, which often requires a rotation of the object or rotation of the shear direction. This is time consuming and errors can be easily generated from the process of rotation, especially for image registration in the axial direction. In the present Letter, we will demonstrate that only one partial derivative often suffices to recover the phase, and we will discuss under which conditions that is possible. Simulations and validation experiments are presented.

© 2013 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing (Wiley, 2007).
  2. G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, Opt. Commun. 257, 16 (2006).
    [CrossRef]
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    [CrossRef]
  4. M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, J. Microsc. 199, 79 (2000).
    [CrossRef]
  5. M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
    [CrossRef]
  6. M. Shribak and S. Inoué, Appl. Opt. 45, 460 (2006).
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  7. L. Miccio, A. Finizio, R. Puglisi, D. Balduzzi, A. Galli, and P. Ferraro, Biomed. Opt. Express 2, 331 (2011).
    [CrossRef]
  8. C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
    [CrossRef]
  9. F. Pfeiffer, J. Herzen, M. Willner, M. Chabior, S. Auweter, M. Reiser, and F. Bamberg, Z. Med. Phys. 23, 176 (2013).
    [CrossRef]
  10. T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, Opt. Commun. 285, 4763 (2012).
    [CrossRef]
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    [CrossRef]
  12. S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, J. Biomed. Opt. 13, 024020 (2008).
    [CrossRef]
  13. B. Bhaduri and G. Popescu, Opt. Lett. 37, 1868 (2012).
    [CrossRef]
  14. http://micro.magnet.fsu.edu/primer/techniques/dic/dicgallery/chineseliverflukesmall.html .

2013 (1)

F. Pfeiffer, J. Herzen, M. Willner, M. Chabior, S. Auweter, M. Reiser, and F. Bamberg, Z. Med. Phys. 23, 176 (2013).
[CrossRef]

2012 (2)

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, Opt. Commun. 285, 4763 (2012).
[CrossRef]

B. Bhaduri and G. Popescu, Opt. Lett. 37, 1868 (2012).
[CrossRef]

2011 (1)

2008 (1)

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, J. Biomed. Opt. 13, 024020 (2008).
[CrossRef]

2007 (3)

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
[CrossRef]

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, Nucl. Instrum. Methods Phys. Res. A 580, 925 (2007).
[CrossRef]

R. Legarda-Saenz, Opt. Lasers Eng. 45, 915 (2007).
[CrossRef]

2006 (2)

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, Opt. Commun. 257, 16 (2006).
[CrossRef]

M. Shribak and S. Inoué, Appl. Opt. 45, 460 (2006).
[CrossRef]

2004 (1)

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
[CrossRef]

2000 (1)

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, J. Microsc. 199, 79 (2000).
[CrossRef]

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
[CrossRef]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, J. Microsc. 199, 79 (2000).
[CrossRef]

Auweter, S.

F. Pfeiffer, J. Herzen, M. Willner, M. Chabior, S. Auweter, M. Reiser, and F. Bamberg, Z. Med. Phys. 23, 176 (2013).
[CrossRef]

Balduzzi, D.

Bamberg, F.

F. Pfeiffer, J. Herzen, M. Willner, M. Chabior, S. Auweter, M. Reiser, and F. Bamberg, Z. Med. Phys. 23, 176 (2013).
[CrossRef]

Bhaduri, B.

Bruder, J.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, Opt. Commun. 285, 4763 (2012).
[CrossRef]

Bunk, O.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, Opt. Commun. 285, 4763 (2012).
[CrossRef]

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, Nucl. Instrum. Methods Phys. Res. A 580, 925 (2007).
[CrossRef]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
[CrossRef]

Chabior, M.

F. Pfeiffer, J. Herzen, M. Willner, M. Chabior, S. Auweter, M. Reiser, and F. Bamberg, Z. Med. Phys. 23, 176 (2013).
[CrossRef]

Cloetens, P.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, Opt. Commun. 285, 4763 (2012).
[CrossRef]

Cogswell, C. J.

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, J. Biomed. Opt. 13, 024020 (2008).
[CrossRef]

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
[CrossRef]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, J. Microsc. 199, 79 (2000).
[CrossRef]

David, C.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, Opt. Commun. 285, 4763 (2012).
[CrossRef]

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, Nucl. Instrum. Methods Phys. Res. A 580, 925 (2007).
[CrossRef]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
[CrossRef]

Fekete, P. W.

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, J. Microsc. 199, 79 (2000).
[CrossRef]

Ferraro, P.

Finizio, A.

Flores, J. L.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, Opt. Commun. 257, 16 (2006).
[CrossRef]

Galli, A.

García-Torales, G.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, Opt. Commun. 257, 16 (2006).
[CrossRef]

González-Álvarez, A.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, Opt. Commun. 257, 16 (2006).
[CrossRef]

Herzen, J.

F. Pfeiffer, J. Herzen, M. Willner, M. Chabior, S. Auweter, M. Reiser, and F. Bamberg, Z. Med. Phys. 23, 176 (2013).
[CrossRef]

Inoué, S.

King, S. V.

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, J. Biomed. Opt. 13, 024020 (2008).
[CrossRef]

Kottler, C.

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, Nucl. Instrum. Methods Phys. Res. A 580, 925 (2007).
[CrossRef]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
[CrossRef]

Larkin, K. G.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
[CrossRef]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, J. Microsc. 199, 79 (2000).
[CrossRef]

Legarda-Saenz, R.

R. Legarda-Saenz, Opt. Lasers Eng. 45, 915 (2007).
[CrossRef]

Libertun, A.

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, J. Biomed. Opt. 13, 024020 (2008).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 2007).

Miccio, L.

Paez, G.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, Opt. Commun. 257, 16 (2006).
[CrossRef]

Pfeiffer, F.

F. Pfeiffer, J. Herzen, M. Willner, M. Chabior, S. Auweter, M. Reiser, and F. Bamberg, Z. Med. Phys. 23, 176 (2013).
[CrossRef]

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, Opt. Commun. 285, 4763 (2012).
[CrossRef]

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, Nucl. Instrum. Methods Phys. Res. A 580, 925 (2007).
[CrossRef]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, Opt. Express 15, 1175 (2007).
[CrossRef]

Piestun, R.

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, J. Biomed. Opt. 13, 024020 (2008).
[CrossRef]

Popescu, G.

Preza, C.

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, J. Biomed. Opt. 13, 024020 (2008).
[CrossRef]

Puglisi, R.

Reiser, M.

F. Pfeiffer, J. Herzen, M. Willner, M. Chabior, S. Auweter, M. Reiser, and F. Bamberg, Z. Med. Phys. 23, 176 (2013).
[CrossRef]

Sheppard, C. J. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
[CrossRef]

Shribak, M.

Smith, N. I.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
[CrossRef]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, J. Microsc. 199, 79 (2000).
[CrossRef]

Strojnik, M.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, Opt. Commun. 257, 16 (2006).
[CrossRef]

Villa, J.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, Opt. Commun. 257, 16 (2006).
[CrossRef]

Weitkamp, T.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, Opt. Commun. 285, 4763 (2012).
[CrossRef]

Willner, M.

F. Pfeiffer, J. Herzen, M. Willner, M. Chabior, S. Auweter, M. Reiser, and F. Bamberg, Z. Med. Phys. 23, 176 (2013).
[CrossRef]

Appl. Opt. (1)

Biomed. Opt. Express (1)

J. Biomed. Opt. (1)

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, J. Biomed. Opt. 13, 024020 (2008).
[CrossRef]

J. Microsc. (2)

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, J. Microsc. 199, 79 (2000).
[CrossRef]

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, J. Microsc. 214, 7 (2004).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (1)

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, Nucl. Instrum. Methods Phys. Res. A 580, 925 (2007).
[CrossRef]

Opt. Commun. (2)

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, Opt. Commun. 285, 4763 (2012).
[CrossRef]

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, Opt. Commun. 257, 16 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

R. Legarda-Saenz, Opt. Lasers Eng. 45, 915 (2007).
[CrossRef]

Opt. Lett. (1)

Z. Med. Phys. (1)

F. Pfeiffer, J. Herzen, M. Willner, M. Chabior, S. Auweter, M. Reiser, and F. Bamberg, Z. Med. Phys. 23, 176 (2013).
[CrossRef]

Other (2)

http://micro.magnet.fsu.edu/primer/techniques/dic/dicgallery/chineseliverflukesmall.html .

D. Malacara, Optical Shop Testing (Wiley, 2007).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Lena’s picture, (b) partial derivative in the horizontal direction, and (c) reconstructed image obtained by integrating the partial derivative.

Fig. 2.
Fig. 2.

(a) Partial derivative (of Lena’s image) in the horizontal direction with noise, (b) reconstructed image obtained by integrating the partial derivative, and (c) image obtained after filtering using the algorithm [see Eq. (12)] discussed in the present Letter.

Fig. 3.
Fig. 3.

(a) Partial derivative in the horizontal direction of an Ambystoma mexicanum obtained by x-ray differential phase contrast and (b) reconstructed image obtained by integrating the partial derivative with λ=0.07.

Fig. 4.
Fig. 4.

(a) Partial derivative in the horizontal direction of Chinese liver fluke (Clonorchis sinensis) obtained by DIC and (b) reconstructed image obtained by integrating the partial derivative with λ=0.1.

Equations (14)

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

g(x,y)=φ(x,y)x.
φ(x,y)=g(x,y)dx+h(y),
φrec(x,y)=g(x,y)dx.
Φ(ωx,ωy)Φrec(ωx,ωy)=δ(ωx)H(ωy),
MSE=|φ(x,y)φrec(x,y)|2dxdy|φ(x,y)|2dxdy.
MSE=|Φ(ωx,ωy)Φrec(ωx,ωy)|2dωxdωy|Φ(ωx,ωy)|2dωxdωy.
2φrec(x,y)x2=g(x,y)x,
φrec(x,y)=k=1Ln=1Lcknsin(πkxL)sin(πnyL)
g(x,y)x=k=1Ln=1Ldknsin(πkxL)sin(πnyL),
dkn=1L202L02Lg(x,y)xsin(πkxL)sin(πnyL)dxdy.
φrec(x,y)=(Lπ)2k=1Ln=1Ldknk2sin(πkxL)sin(πnyL).
φrec(x,y)=(Lπ)2k=1Ln=1Ldkn(k2+λn2)×sin(πkxL)sin(πnyL),
2φrecx2+λ2φrecy2=gx.
[|φrecxg|2+λ|φrecy|2]dxdy.

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