Abstract

A complex-valued wave field is reconstructed from intensity-only measurements given at multiple observation planes parallel to the object plane. The phase-retrieval algorithm is obtained from the constrained maximum likelihood approach provided that the additive noise is Gaussian. The forward propagation from the object plane to the measurement plane is treated as a constraint in the proposed variational setting of reconstruction. The developed iterative algorithm is based on an augmented Lagrangian technique. An advanced performance of the algorithm is demonstrated by numerical simulations.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  2. B. Gu and G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).
  3. G. Yang, B. Dong, B. Gu, J. Zhuang, and O. K. Ersoy, “Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
    [CrossRef] [PubMed]
  4. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  5. D. R. Luke, J. V. Burke, and R. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
    [CrossRef]
  6. J. Burke and D. R. Luke, “Variational analysis applied to the problem of optical phase retrieval,” SIAM J. Control Optim. 42, 576–595 (2003).
    [CrossRef]
  7. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  9. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
    [CrossRef] [PubMed]
  10. V. Katkovnik, A. Migukin, and J. Astola, “Backward discrete wave field propagation modeling as an inverse problem: toward perfect reconstruction of wave field distributions,” Appl. Opt. 48, 3407–3423 (2009).
    [CrossRef] [PubMed]
  11. V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wavefield distributions,” Appl. Opt. 47, 3481–3493 (2008).
    [CrossRef] [PubMed]
  12. D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
    [CrossRef]
  13. T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, “The role of support information and zero locations in phase retrieval by a quadratic approach,” J. Opt. Soc. Am. A 16, 1845–1856(1999).
    [CrossRef]
  14. R. W. Deming, “Phase retrieval from intensity-only data by relative entropy minimization,” J. Opt. Soc. Am. A 24, 3666–3679(2007).
    [CrossRef]
  15. F. Soldovieri, R. Deming, and R. Pierri, “An improved version of the relative entropy minimization approach for the phase retrieval problem,” AEU Int. J. Electron. Commun. 64, 56–65(2010).
    [CrossRef]
  16. A. Migukin, V. Katkovnik, and J. Astola, “Multiple plane phase retrieval based on inverse regularized imaging and discrete diffraction transform,” AIP Conf. Proc. 1236, 81–86 (2010).
    [CrossRef]
  17. G. Pedrini, W. Osten, and Y. Zhang, “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. 30, 833–835 (2005).
    [CrossRef] [PubMed]
  18. P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605 (2006).
    [CrossRef] [PubMed]
  19. P. Almoro, A. M. Maallo, and S. Hanson, “Fast-convergent algorithm for speckle-based phase retrieval and a design for dynamic wavefront sensing,” Appl. Opt. 48, 1485–1493 (2009).
    [CrossRef] [PubMed]
  20. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
    [CrossRef] [PubMed]
  21. X. Hu, S. Li, and Y. Wu, “Resolution-enhanced subpixel phase retrieval method,” Appl. Opt. 47, 6079–6087 (2008).
    [CrossRef]
  22. G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express 17, 624–639 (2009).
    [CrossRef] [PubMed]
  23. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259–268 (1992).
    [CrossRef]
  24. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).
    [CrossRef]
  25. A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, 1977).
  26. M. R. Hestenes, “Multiplier and gradient methods,” J. Optim. Theory Appl. 4, 303–320 (1969).
    [CrossRef]
  27. M. J. D. Powell, “A method for nonlinear constraints in minimization problems,” in Optimization, R.Fletcher, ed. (Academic, 1969), pp. 283–298.
  28. D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods. (Prentice-Hall, 1989).
  29. J. F. Cai, S. Osher, and Z. Shen, “Split Bregman methods and frame based image restoration,” Multiscale Model. Simul. 8, 337–369 (2009).
    [CrossRef]
  30. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
    [CrossRef]
  31. M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
    [CrossRef] [PubMed]
  32. M. Figueiredo, J. Bioucas-Dias, and R. Nowak, “Majorization-minimization algorithms for wavelet based image restoration,” IEEE Trans. Image Process. 16, 2980–2991 (2007).
    [CrossRef] [PubMed]
  33. Th. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).
  34. “Multi-plane phase retrieval,” http://www.cs.tut.fi/~lasip/DDT/.

2010 (3)

F. Soldovieri, R. Deming, and R. Pierri, “An improved version of the relative entropy minimization approach for the phase retrieval problem,” AEU Int. J. Electron. Commun. 64, 56–65(2010).
[CrossRef]

A. Migukin, V. Katkovnik, and J. Astola, “Multiple plane phase retrieval based on inverse regularized imaging and discrete diffraction transform,” AIP Conf. Proc. 1236, 81–86 (2010).
[CrossRef]

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef] [PubMed]

2009 (5)

2008 (3)

2007 (2)

R. W. Deming, “Phase retrieval from intensity-only data by relative entropy minimization,” J. Opt. Soc. Am. A 24, 3666–3679(2007).
[CrossRef]

M. Figueiredo, J. Bioucas-Dias, and R. Nowak, “Majorization-minimization algorithms for wavelet based image restoration,” IEEE Trans. Image Process. 16, 2980–2991 (2007).
[CrossRef] [PubMed]

2006 (2)

2005 (1)

2003 (1)

J. Burke and D. R. Luke, “Variational analysis applied to the problem of optical phase retrieval,” SIAM J. Control Optim. 42, 576–595 (2003).
[CrossRef]

2002 (2)

1999 (1)

1994 (1)

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259–268 (1992).
[CrossRef]

1982 (1)

1981 (1)

B. Gu and G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

1973 (1)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1969 (1)

M. R. Hestenes, “Multiplier and gradient methods,” J. Optim. Theory Appl. 4, 303–320 (1969).
[CrossRef]

Afonso, M. V.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef] [PubMed]

Almoro, P.

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, 1977).

Astola, J.

Bauschke, H. H.

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
[CrossRef]

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).
[CrossRef]

Bertsekas, D. P.

D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods. (Prentice-Hall, 1989).

Bioucas-Dias, J.

M. Figueiredo, J. Bioucas-Dias, and R. Nowak, “Majorization-minimization algorithms for wavelet based image restoration,” IEEE Trans. Image Process. 16, 2980–2991 (2007).
[CrossRef] [PubMed]

Bioucas-Dias, J. M.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef] [PubMed]

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).
[CrossRef]

Brady, G. R.

Burke, J.

J. Burke and D. R. Luke, “Variational analysis applied to the problem of optical phase retrieval,” SIAM J. Control Optim. 42, 576–595 (2003).
[CrossRef]

Burke, J. V.

D. R. Luke, J. V. Burke, and R. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[CrossRef]

Cai, J. F.

J. F. Cai, S. Osher, and Z. Shen, “Split Bregman methods and frame based image restoration,” Multiscale Model. Simul. 8, 337–369 (2009).
[CrossRef]

Combettes, P. L.

Deming, R.

F. Soldovieri, R. Deming, and R. Pierri, “An improved version of the relative entropy minimization approach for the phase retrieval problem,” AEU Int. J. Electron. Commun. 64, 56–65(2010).
[CrossRef]

Deming, R. W.

Dong, B.

Egiazarian, K.

Ersoy, O. K.

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259–268 (1992).
[CrossRef]

Fienup, J. R.

Figueiredo, M.

M. Figueiredo, J. Bioucas-Dias, and R. Nowak, “Majorization-minimization algorithms for wavelet based image restoration,” IEEE Trans. Image Process. 16, 2980–2991 (2007).
[CrossRef] [PubMed]

Figueiredo, M. A. T.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gu, B.

Guizar-Sicairos, M.

Hanson, S.

Hestenes, M. R.

M. R. Hestenes, “Multiplier and gradient methods,” J. Optim. Theory Appl. 4, 303–320 (1969).
[CrossRef]

Hu, X.

Isernia, T.

Katkovnik, V.

Kreis, Th.

Th. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

Leone, G.

Li, S.

Luke, D. R.

J. Burke and D. R. Luke, “Variational analysis applied to the problem of optical phase retrieval,” SIAM J. Control Optim. 42, 576–595 (2003).
[CrossRef]

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
[CrossRef]

D. R. Luke, J. V. Burke, and R. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[CrossRef]

Lyon, R.

D. R. Luke, J. V. Burke, and R. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[CrossRef]

Maallo, A. M.

Migukin, A.

A. Migukin, V. Katkovnik, and J. Astola, “Multiple plane phase retrieval based on inverse regularized imaging and discrete diffraction transform,” AIP Conf. Proc. 1236, 81–86 (2010).
[CrossRef]

V. Katkovnik, A. Migukin, and J. Astola, “Backward discrete wave field propagation modeling as an inverse problem: toward perfect reconstruction of wave field distributions,” Appl. Opt. 48, 3407–3423 (2009).
[CrossRef] [PubMed]

Misell, D. L.

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Nowak, R.

M. Figueiredo, J. Bioucas-Dias, and R. Nowak, “Majorization-minimization algorithms for wavelet based image restoration,” IEEE Trans. Image Process. 16, 2980–2991 (2007).
[CrossRef] [PubMed]

Osher, S.

J. F. Cai, S. Osher, and Z. Shen, “Split Bregman methods and frame based image restoration,” Multiscale Model. Simul. 8, 337–369 (2009).
[CrossRef]

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259–268 (1992).
[CrossRef]

Osten, W.

Pedrini, G.

Pierri, R.

F. Soldovieri, R. Deming, and R. Pierri, “An improved version of the relative entropy minimization approach for the phase retrieval problem,” AEU Int. J. Electron. Commun. 64, 56–65(2010).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, “The role of support information and zero locations in phase retrieval by a quadratic approach,” J. Opt. Soc. Am. A 16, 1845–1856(1999).
[CrossRef]

Powell, M. J. D.

M. J. D. Powell, “A method for nonlinear constraints in minimization problems,” in Optimization, R.Fletcher, ed. (Academic, 1969), pp. 283–298.

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259–268 (1992).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Shen, F.

Shen, Z.

J. F. Cai, S. Osher, and Z. Shen, “Split Bregman methods and frame based image restoration,” Multiscale Model. Simul. 8, 337–369 (2009).
[CrossRef]

Soldovieri, F.

F. Soldovieri, R. Deming, and R. Pierri, “An improved version of the relative entropy minimization approach for the phase retrieval problem,” AEU Int. J. Electron. Commun. 64, 56–65(2010).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, “The role of support information and zero locations in phase retrieval by a quadratic approach,” J. Opt. Soc. Am. A 16, 1845–1856(1999).
[CrossRef]

Teboulle, M.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, 1977).

Tsitsiklis, J. N.

D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods. (Prentice-Hall, 1989).

Wang, A.

Wu, Y.

Yang, G.

Zhang, Y.

Zhuang, J.

Acta Opt. Sin. (1)

B. Gu and G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

AEU Int. J. Electron. Commun. (1)

F. Soldovieri, R. Deming, and R. Pierri, “An improved version of the relative entropy minimization approach for the phase retrieval problem,” AEU Int. J. Electron. Commun. 64, 56–65(2010).
[CrossRef]

AIP Conf. Proc. (1)

A. Migukin, V. Katkovnik, and J. Astola, “Multiple plane phase retrieval based on inverse regularized imaging and discrete diffraction transform,” AIP Conf. Proc. 1236, 81–86 (2010).
[CrossRef]

Appl. Opt. (8)

F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
[CrossRef] [PubMed]

V. Katkovnik, A. Migukin, and J. Astola, “Backward discrete wave field propagation modeling as an inverse problem: toward perfect reconstruction of wave field distributions,” Appl. Opt. 48, 3407–3423 (2009).
[CrossRef] [PubMed]

V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wavefield distributions,” Appl. Opt. 47, 3481–3493 (2008).
[CrossRef] [PubMed]

G. Yang, B. Dong, B. Gu, J. Zhuang, and O. K. Ersoy, “Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
[CrossRef] [PubMed]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

X. Hu, S. Li, and Y. Wu, “Resolution-enhanced subpixel phase retrieval method,” Appl. Opt. 47, 6079–6087 (2008).
[CrossRef]

P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605 (2006).
[CrossRef] [PubMed]

P. Almoro, A. M. Maallo, and S. Hanson, “Fast-convergent algorithm for speckle-based phase retrieval and a design for dynamic wavefront sensing,” Appl. Opt. 48, 1485–1493 (2009).
[CrossRef] [PubMed]

IEEE Trans. Image Process. (2)

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef] [PubMed]

M. Figueiredo, J. Bioucas-Dias, and R. Nowak, “Majorization-minimization algorithms for wavelet based image restoration,” IEEE Trans. Image Process. 16, 2980–2991 (2007).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

J. Optim. Theory Appl. (1)

M. R. Hestenes, “Multiplier and gradient methods,” J. Optim. Theory Appl. 4, 303–320 (1969).
[CrossRef]

J. Phys. D (1)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Multiscale Model. Simul. (1)

J. F. Cai, S. Osher, and Z. Shen, “Split Bregman methods and frame based image restoration,” Multiscale Model. Simul. 8, 337–369 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Physica D (Amsterdam) (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259–268 (1992).
[CrossRef]

SIAM J. Control Optim. (1)

J. Burke and D. R. Luke, “Variational analysis applied to the problem of optical phase retrieval,” SIAM J. Control Optim. 42, 576–595 (2003).
[CrossRef]

SIAM J. Imaging Sci. (1)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
[CrossRef]

SIAM Rev. (1)

D. R. Luke, J. V. Burke, and R. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[CrossRef]

Other (7)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Th. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

“Multi-plane phase retrieval,” http://www.cs.tut.fi/~lasip/DDT/.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).
[CrossRef]

A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, 1977).

M. J. D. Powell, “A method for nonlinear constraints in minimization problems,” in Optimization, R.Fletcher, ed. (Academic, 1969), pp. 283–298.

D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods. (Prentice-Hall, 1989).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Multiple plane wave field reconstruction scenario: u 0 [ k ] and u r [ k ] are wave field distributions at the object and the rth parallel measurement (sensor) plane, respectively, r = 1 , ... K .

Fig. 2
Fig. 2

Reconstructions of the amplitude for the AM object distribution with different numbers of sensor planes K and distances z 1 between the object and the first sensor plane. Logo test-image, noiseless data σ = 0 , μ ˜ = 5 · 10 3 .

Fig. 3
Fig. 3

Accuracy of the phase reconstruction with respect to the distance between measurement planes Δ z . PM object, chessboard test image, noisy data σ = 0.05 .

Fig. 4
Fig. 4

Comparison of the amplitude and phase reconstructions obtained by AL (left column) and SBMIR (right column) algorithms. The top row demonstrates the amplitude reconstructions (a) by AL, RMSE ( | u 0 | ) = 0.041 , and (b) by SBMIR, RMSE ( | u 0 | ) = 0.08 . The bottom row shows the phase reconstructions, obtained (c) by AL, RMSE ( ϕ 0 ) = 0.091 , and (d) by SBMIR, RMSE ( ϕ 0 ) = 0.28 . AM object, lena test image, K = 5 , noiseless data σ = 0 .

Fig. 5
Fig. 5

Convergence rates of the AL and SBMIR algorithms for the test presented in Fig. 4.

Fig. 6
Fig. 6

Comparison of the amplitude and phase reconstructions obtained by AL (left column) and SBMIR (right column) algorithms. The top row demonstrates the amplitude reconstructions (a) by AL, RMSE ( | u 0 | ) = 0.23 , and (b) by SBMIR, RMSE ( | u 0 | ) = 0.35 . The bottom row illustrates the phase reconstructions, obtained (c) by AL, RMSE ( ϕ 0 ) = 0.26 , and (d) by SBMIR, RMSE ( ϕ 0 ) = 0.58 . PM object, chessboard test image, K = 5 , noisy data σ = 0.05 .

Fig. 7
Fig. 7

Convergence rates of the AL and SBMIR algorithms for the test presented in Fig. 6.

Fig. 8
Fig. 8

Cross sections of the true phase (dotted curve) and phase reconstructions obtained for the Mexican Hat test image. The solid curve corresponds to AL, RMSE ( ϕ 0 ) = 0.187 , and the dashed curve corresponds to SBMIR, RMSE ( ϕ 0 ) = 0.511 . PM object, 1000 iterations, K = 5 , noisy data σ = 0.05 .

Fig. 9
Fig. 9

Convergence rates of the AL and SBMIR algorithms for the phase reconstruction (Mexican Hat) of the test presented in Fig. 8.

Tables (2)

Tables Icon

Table 1 Quantitative Comparison of the Amplitude Reconstruction, RMSE ( | u 0 | ) for the Test, Presented in Fig. 2

Tables Icon

Table 2 Computational Time (in Seconds) for 100 Iterations of the AL Algorithm

Equations (30)

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

o r [ k ] = | u r [ k ] | 2 + ε r [ k ] , r = 1 , ... K ,
u r = A r · u 0 ,
o r = | u r | 2 + ε r , r = 1 , ... K ,
u r = A r , s · u s ,
u ^ r ( p ) = A r · u ^ 0 ( p ) , p = 0 , 1 , ... , u ^ 0 ( p + 1 ) = 1 K r = 1 K A r H · [ o r | u ^ r ( p ) | u ^ r ( p ) ] ,
KL = r = 1 K R 2 o r ( x ) lg o r ( x ) | u r ( x ) | 2 d x .
u ^ 0 ( x ) = min u 0     KL provided     R 2 | o r ( x ) | d x = R 2 | u r ( x ) | 2 d x .
u ^ r ( p ) = A r · u ^ 0 ( p ) , p = 0 , 1 , ... , u ^ 0 ( p + 1 ) = 1 K K r = 1 A r H · [ o r | u ^ r ( p ) | 2 u ^ r ( p ) ] ,
u ^ r ( p ) = A r · u ^ 0 ( p ) , p = 0 , 1 , ... , u ^ 0 ( p + 1 ) = ( r = 1 K A r H A r + μ · I n × n ) 1 × r = 1 K A r H · [ o r | u ^ r ( p ) | u ^ r ( p ) ] ,
1.     Repeat for     p = 0 , 1 , ... , 2.       For     r = 1 , ... K 1 , u ^ r + 1 ( p ) = A r + 1 , r · [ o r | u ^ r ( p ) | u ^ r ( p ) ] , 3.       For     r = K , u ^ 1 ( p + 1 ) = A 1 , K · [ o K | u ^ K ( p ) | u ^ K ( p ) ] , 4.       End on     r , 5.       End for     p .
J = r = 1 K 1 2 σ r 2 | | o r | u r | 2 | | 2 2 + μ · pen ( u 0 ) ,
u ^ 0 = arg min u 0 r = 1 K 1 2 σ r 2 | | o r | u r | 2 | | 2 2 + μ · pen ( u 0 ) subject to     u r = A r u 0 , r = 1 , ... , K .
pen ( u 0 ) = | | u 0 | | 2 2 .
u ^ 0 = arg min u 0     J , J = r = 1 K 1 2 σ r 2 | | o r | A r · u 0 | 2 | | 2 2 + μ · pen ( u 0 ) .
L ( u 0 , { u r } , { Λ r } ) = r = 1 K 1 σ r 2 [ 1 2 | | o r | u r | 2 | | 2 + 1 γ r | | u r A r · u 0 | | 2 + 2 γ r Re { Λ r H ( u r A r · u 0 ) } ] + μ | | u 0 | | 2 2 ,
( u 0 t + 1 , { u r t + 1 } ) arg min u 0 , { u r } L ( u 0 , { u r } , { Λ r t } ) ,
Λ r t + 1 = Λ r t + α r · ( u r t + 1 A r · u 0 t ) , r = 1 , ... , K .
For    t = 0 , 1 , ... , For    r = 1 , ... K , u r t + 1 arg min { u r }     L ( u 0 t , { u r } , { Λ r t } ) ,
Λ r t + 1 = Λ r t + α r · ( u r t + 1 A r · u 0 t ) , End on   r ,
u 0 t + 1 arg min u 0       L ( u 0 , { u r t + 1 } , { Λ r t } ) , End on   t .
AL Algorithm 1.     Set t = 0 ( initialization ) , u 0 0 , Λ r 0 , 2.     Repeat for     t = 0 , 1 , ... , 3.     Repeat for   r = 1 , ... K , 4.     u r t + 1 / 2 = A r · u 0 t , 5.     u r t + 1 [ k ] = G ( o r [ k ] , u r t + 1 / 2 [ k ] , Λ r t [ k ] ) , 6.     Λ r t + 1 = Λ r t + α r · ( u r t + 1 u r t + 1 / 2 ) , 7.     End on   r , 8.     u 0 t + 1 = ( r = 1 K 1 γ r σ r 2 A r H A r + μ · I n × n ) 1 × r = 1 K 1 γ r σ r 2 A r H ( u r t + 1 + Λ r t ) , 9.     End on   t .
u 0 t + 1 = [ r = 1 K A r H A r + μ ˜ γ · I n × n ] 1 r = 1 K A r H ( u r t + 1 + Λ r t ) ,
L ( u 0 , { u r } , { Λ r } ) = r = 1 K 1 σ r 2 { 1 2 | | o r | u r | 2 | | 2 + 1 γ r | | u r A r · u 0 | | 2 + 1 γ r [ Λ r H ( u r A r · u 0 ) + ( u r A r · u 0 ) H Λ r ] } + μ | | u 0 | | 2 2 .
L u r * [ k ] = 1 σ r 2 ( | u r [ k ] | 2 o r [ k ] ) · u r [ k ] + 1 γ r σ r 2 ( u r [ k ] ( A r · u 0 ) [ k ] + Λ r [ k ] ) = 0 .
u r [ k ] = ( A r · u 0 ) [ k ] Λ r [ k ] γ r ( | u r [ k ] | 2 o r [ k ] ) + 1 = η r [ k ] κ r [ k ] .
| u r [ k ] | 3 + | u r [ k ] | · ( 1 γ r o r [ k ] ) sgn ( κ r [ k ] ) · | η r [ k ] | γ r = 0 .
u ^ r [ k ] = ( A r · u 0 ) [ k ] Λ r [ k ] γ k ( | u ˜ r [ k ] | 2 o r [ k ] ) + 1 .
u ^ r [ k ] = G ( o r [ k ] , u r [ k ] , Λ r [ k ] ) ,
u ^ 0 = ( r = 1 K 1 γ r σ r 2 A r H A r + μ · I n × n ) 1 r = 1 K 1 γ r σ r 2 A r H ( u r + Λ r ) .
Λ r t + 1 = Λ r t + α r · ( u r t + 1 A r · u 0 t ) .

Metrics