Abstract

The transport of intensity equation (TIE) is widely applied for recovering wave fronts from an intensity measurement and a measurement of its variation along the direction of propagation. In order to get around the problem of non-uniqueness and ill-conditionedness of the solution of the TIE in the very common case of unspecified boundary conditions or noisy data, additional constraints to the solution are necessary. Although from a numerical optimization point of view, convex constraint as imposed to by total variation minimization is preferable, we will show that in many cases non-convex constraints are necessary to overcome the low-frequency artifacts so typical for convex constraints. We will provide simulated and experimental examples that demonstrate the superiority of solutions to the TIE obtained by our recently introduced gradient flipping algorithm over a total variation constrained solution.

© 2016 Optical Society of America

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References

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  1. M. R. Teague, “Deterministic phase retrieval: a green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [Crossref]
  2. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1941 (1995).
    [Crossref]
  3. V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
    [Crossref] [PubMed]
  4. M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
    [Crossref] [PubMed]
  5. A. Parvizi, W. V. den Broek, and C.T. Koch, “Recovering low spatial frequencies in wavefront sensing based on intensity measurements,” Adv. Struct. Chem. Imag. (in press).
  6. T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
    [Crossref]
  7. A. Parvizi, J. Müller, S. Funken, and C. Koch, “A practical way to resolve ambiguities in wavefront reconstructions by the transport of intensity equation,” Ultramicroscopy 154, 1–6 (2015).
    [Crossref] [PubMed]
  8. L. Tian, J. C. Petruccelli, and G. Barbastathis, “Nonlinear diffusion regularization for transport of intensity phase imaging,” Opt. Lett. 37, 4131–4133 (2012).
    [Crossref] [PubMed]
  9. A. Kostenko, K. J. Batenburg, H. Suhonen, S. E. Offerman, and L. J. van Vliet, “Phase retrieval in in-line x-ray phase contrast imaging based on total variation minimization,” Opt. Express 21, 710–723 (2013).
    [Crossref] [PubMed]
  10. G. Oszlányi and A. Süto, “The charge flipping algorithm,” Acta Crystallogr. Sect. A 64, 23–134 (2007).
  11. L. Palatinus, “The charge-flipping algorithm in crystallography,” Acta Crystallogr. Sect. B-Struct. Sci. 69, 1–16 (2013).
    [Crossref]
  12. C. Li, An Efficient Algorithm for Total Variation Regularization with Applications to the Single Pixel Camera and Compressive Sensing (Rice University, 2009).
  13. C. Li, CCompressive Sensing for 3D Data Processing Tasks: Applications, Models and Algorithms (Rice University, 2011).
  14. J. Nocedal and S. Wright, Numerical Optimization (Springer Science and Business Media, 2006).
  15. D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Comput. Math. Appl. 2, 17–40 (1976).
    [Crossref]
  16. E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
    [Crossref]
  17. E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
    [Crossref]
  18. J. M. Bioucas-Dias and M. A. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
    [Crossref] [PubMed]
  19. S. Becker, J. Bobin, and E. J. Candès, “Nesta: a fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4, 1–39 (2011).
    [Crossref]
  20. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc. 214, 51–61 (2004).
    [Crossref] [PubMed]

2015 (1)

A. Parvizi, J. Müller, S. Funken, and C. Koch, “A practical way to resolve ambiguities in wavefront reconstructions by the transport of intensity equation,” Ultramicroscopy 154, 1–6 (2015).
[Crossref] [PubMed]

2013 (2)

2012 (1)

2011 (1)

S. Becker, J. Bobin, and E. J. Candès, “Nesta: a fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4, 1–39 (2011).
[Crossref]

2007 (2)

G. Oszlányi and A. Süto, “The charge flipping algorithm,” Acta Crystallogr. Sect. A 64, 23–134 (2007).

J. M. Bioucas-Dias and M. A. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[Crossref] [PubMed]

2006 (1)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

2005 (1)

E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[Crossref]

2004 (2)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

2002 (1)

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref] [PubMed]

1997 (1)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

1995 (1)

1983 (1)

1976 (1)

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Comput. Math. Appl. 2, 17–40 (1976).
[Crossref]

Barbastathis, G.

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

Batenburg, K. J.

Becker, S.

S. Becker, J. Bobin, and E. J. Candès, “Nesta: a fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4, 1–39 (2011).
[Crossref]

Beleggia, M.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[Crossref] [PubMed]

Bobin, J.

S. Becker, J. Bobin, and E. J. Candès, “Nesta: a fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4, 1–39 (2011).
[Crossref]

Candès, E. J.

S. Becker, J. Bobin, and E. J. Candès, “Nesta: a fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4, 1–39 (2011).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[Crossref]

De Graef, M.

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref] [PubMed]

den Broek, W. V.

A. Parvizi, W. V. den Broek, and C.T. Koch, “Recovering low spatial frequencies in wavefront sensing based on intensity measurements,” Adv. Struct. Chem. Imag. (in press).

Figueiredo, M. A.

J. M. Bioucas-Dias and M. A. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[Crossref] [PubMed]

Funken, S.

A. Parvizi, J. Müller, S. Funken, and C. Koch, “A practical way to resolve ambiguities in wavefront reconstructions by the transport of intensity equation,” Ultramicroscopy 154, 1–6 (2015).
[Crossref] [PubMed]

Gabay, D.

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Comput. Math. Appl. 2, 17–40 (1976).
[Crossref]

Gureyev, T. E.

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1941 (1995).
[Crossref]

Koch, C.

A. Parvizi, J. Müller, S. Funken, and C. Koch, “A practical way to resolve ambiguities in wavefront reconstructions by the transport of intensity equation,” Ultramicroscopy 154, 1–6 (2015).
[Crossref] [PubMed]

Koch, C.T.

A. Parvizi, W. V. den Broek, and C.T. Koch, “Recovering low spatial frequencies in wavefront sensing based on intensity measurements,” Adv. Struct. Chem. Imag. (in press).

Kostenko, A.

Li, C.

C. Li, CCompressive Sensing for 3D Data Processing Tasks: Applications, Models and Algorithms (Rice University, 2011).

C. Li, An Efficient Algorithm for Total Variation Regularization with Applications to the Single Pixel Camera and Compressive Sensing (Rice University, 2009).

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

Mercier, B.

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Comput. Math. Appl. 2, 17–40 (1976).
[Crossref]

Müller, J.

A. Parvizi, J. Müller, S. Funken, and C. Koch, “A practical way to resolve ambiguities in wavefront reconstructions by the transport of intensity equation,” Ultramicroscopy 154, 1–6 (2015).
[Crossref] [PubMed]

Nocedal, J.

J. Nocedal and S. Wright, Numerical Optimization (Springer Science and Business Media, 2006).

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1941 (1995).
[Crossref]

Offerman, S. E.

Oszlányi, G.

G. Oszlányi and A. Süto, “The charge flipping algorithm,” Acta Crystallogr. Sect. A 64, 23–134 (2007).

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

Palatinus, L.

L. Palatinus, “The charge-flipping algorithm in crystallography,” Acta Crystallogr. Sect. B-Struct. Sci. 69, 1–16 (2013).
[Crossref]

Parvizi, A.

A. Parvizi, J. Müller, S. Funken, and C. Koch, “A practical way to resolve ambiguities in wavefront reconstructions by the transport of intensity equation,” Ultramicroscopy 154, 1–6 (2015).
[Crossref] [PubMed]

A. Parvizi, W. V. den Broek, and C.T. Koch, “Recovering low spatial frequencies in wavefront sensing based on intensity measurements,” Adv. Struct. Chem. Imag. (in press).

Petruccelli, J. C.

Roberts, A.

Romberg, J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Schofield, M. A.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

Suhonen, H.

Süto, A.

G. Oszlányi and A. Süto, “The charge flipping algorithm,” Acta Crystallogr. Sect. A 64, 23–134 (2007).

Tao, T.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[Crossref]

Teague, M. R.

Tian, L.

van Vliet, L. J.

Volkov, V. V.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref] [PubMed]

Wright, S.

J. Nocedal and S. Wright, Numerical Optimization (Springer Science and Business Media, 2006).

Zhu, Y.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref] [PubMed]

Acta Crystallogr. Sect. A (1)

G. Oszlányi and A. Süto, “The charge flipping algorithm,” Acta Crystallogr. Sect. A 64, 23–134 (2007).

Acta Crystallogr. Sect. B-Struct. Sci. (1)

L. Palatinus, “The charge-flipping algorithm in crystallography,” Acta Crystallogr. Sect. B-Struct. Sci. 69, 1–16 (2013).
[Crossref]

Comput. Math. Appl. (1)

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Comput. Math. Appl. 2, 17–40 (1976).
[Crossref]

IEEE Trans. Image Process. (1)

J. M. Bioucas-Dias and M. A. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[Crossref] [PubMed]

IEEE Trans. Inf. Theory (2)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[Crossref]

J. Microsc. (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. the effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

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

Micron (1)

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref] [PubMed]

Opt. Commun. (1)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

SIAM J. Imaging Sci. (1)

S. Becker, J. Bobin, and E. J. Candès, “Nesta: a fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4, 1–39 (2011).
[Crossref]

Ultramicroscopy (2)

A. Parvizi, J. Müller, S. Funken, and C. Koch, “A practical way to resolve ambiguities in wavefront reconstructions by the transport of intensity equation,” Ultramicroscopy 154, 1–6 (2015).
[Crossref] [PubMed]

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

Other (4)

A. Parvizi, W. V. den Broek, and C.T. Koch, “Recovering low spatial frequencies in wavefront sensing based on intensity measurements,” Adv. Struct. Chem. Imag. (in press).

C. Li, An Efficient Algorithm for Total Variation Regularization with Applications to the Single Pixel Camera and Compressive Sensing (Rice University, 2009).

C. Li, CCompressive Sensing for 3D Data Processing Tasks: Applications, Models and Algorithms (Rice University, 2011).

J. Nocedal and S. Wright, Numerical Optimization (Springer Science and Business Media, 2006).

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Figures (10)

Fig. 1
Fig. 1 Periodic piece-wise constant phase object: a) Head-phantom original phase. b) Reconstructed phase by GFA. c) Reconstructed phase by TVAL3 d) Intensity variation along the optical axis. e) Phase cross sections taken along the red-line in (a).
Fig. 2
Fig. 2 Non-periodic piece-wise constant phase object: a) Head-phantom original phase. b) Reconstructed phase by GFA. c) Reconstructed phase by TVAL3. d) Laplacian of the phase. e) Phase cross sections taken along the red-line in (a).
Fig. 3
Fig. 3 a) Reconstruction in the presence of 10dB noise by GFA. b) Reconstruction in the presence of 30dB noise by GFA. c) Reconstruction in the presence of 10dB noise by TVAL3. d) Reconstruction in the presence of 30dB noise by TVAL3. e) Original phase.
Fig. 4
Fig. 4 Results of the Tikhonov-regularized FFT-solver. a) Reconstructed phase for 10 dB noise, RMSE = 10.6. d) Reconstruction for 30 dB noise, RMSE = 11.4.
Fig. 5
Fig. 5 Partially piece-wise constant phase object: a) Original phase. b) Reconstructed phase by GFA (RMSE = 0.22). c) Reconstructed phase by TVAL3 (RMSE = 5.4) d) Intensity variation in transverse plane.
Fig. 6
Fig. 6 Piece-wise linear phase object: a) Original phase. b) Reconstructed phase by TVAL3 (RMSE = 1.7). c) Reconstructed phase by GFA (RMSE = 0.95) d) Graphical representation of the measurement.
Fig. 7
Fig. 7 Fresnel-propagation based measurement. a) Finite difference of two images at 0.1 μm over- and under-focus. b) Phase reconstructed by GFA (RMSE = 1.50). c) Phase Reconstructed by TVAL3 (RMSE = 1.95). d) Plot of the radially averaged power spectrum. e) Line profile of the data extracted along the red line.
Fig. 8
Fig. 8 Schematic of optical setup.
Fig. 9
Fig. 9 Experimental images: a) Under-focused, b) Over-focused, c) Intensity variation along the optical axis.
Fig. 10
Fig. 10 Reconstructed phase by a) Tikhonov regularization FFT-based, b) GFA method, c) TV minimization approach. (d),(e) and (f) Line profiles extracted along the red lines in each of the phase maps graphically depicted above.

Tables (1)

Tables Icon

Table 1 RMSE of the TV-regularization, ∥∇∥0-regularization and Tikhonov regularization as applied to the various test-cases in this paper. *Periodic boundary conditions. **Non-periodic boundary conditions.

Equations (14)

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

( I φ ) = k I z .
2 φ = k d I z .
min TV ( φ ) i D i φ 1 , s.t. 2 φ = k d I z exp ,
1 4 ( 1 2 1 2 12 2 1 2 1 )
φ 0 = ( 1 ε ) N 2 and ( I φ ) = k I z ,
G = φ = k I 2 I z and
D = I z = 1 k ( I φ ) ,
G ( ) = k I 2 D ( 1 ) ;
G ( ) = { G ( ) if G ( ) 1 > δ and β G ( ) if G ( ) 1 > δ ,
D ( ) = 1 k ( I G ( ) ) ;
D ( ) = { D ( ) within the padding , 1 [ h ( D ( ) ) + ( 1 h ) ( d I z exp ) ] within the measured area ,
h ( q ) = exp ( R L P 2 q 2 ) .
χ 2 = Σ [ I sim ( R L P ) I exp ] 2 Σ I exp ,
φ = k 2 1 I 2 D ( ) .

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