Abstract

Inverse problem approaches for image reconstruction can improve resolution recovery over spatial filtering methods while reducing interference artifacts in digital off-axis holography. Prior works implemented explicit regularization operators in the image space and were only able to match intensity measurements approximatively. As a consequence, convergence to a strictly compatible solution was not possible. In this paper, we replace the non-convex image reconstruction problem for a sequence of surrogate convex problems. An iterative numerical solver is designed using a simple projection operator in the data domain and a Nesterov acceleration of the simultaneous Kaczmarz method. For regularization, the complex-valued object wavefield image is represented in the multiresolution CDF 9/7 wavelet domain and an energy-weighted preconditioning promotes minimum-norm solutions. Experiments demonstrate improved resolution recovery and reduced spurious artifacts in reconstructed images. Furthermore, the method is resilient to additive Gaussian noise and subsampling of intensity measurements.

© 2017 Optical Society of America

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References

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  1. M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 8005 (2010).
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    [Crossref]
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    [Crossref] [PubMed]
  4. C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Exact complex-wave reconstruction in digital holography,” J. Opt. Soc. Am. A 28, 983–992 (2011).
    [Crossref] [PubMed]
  5. S. Sotthivirat and J. A. Fessler, “Reconstruction from digital holograms by statistical methods,” Conference Record of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers2, 1928–1932 (2003).
  6. S. Sotthivirat and J. A. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. A 21, 737–750 (2004).
    [Crossref]
  7. J. A. Fessler and S. Sotthivirat, “Simplified digital holographic reconstruction using statistical methods,” 2004 International Conference on Image Processing (ICIP’04)4, 2435–2438 (2004).
  8. A. Bourquard, N. Pavillon, E. Bostan, C. Depeursinge, and M. Unser, “A practical inverse-problem approach to digital holographic reconstruction,” Opt. Express 21, 3417–3433 (2013).
    [Crossref] [PubMed]
  9. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D: Nonlin. Phenom. 60, 259–268 (1992).
    [Crossref]
  10. Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Displ. Technol. 6, 506–509 (2010).
    [Crossref]
  11. D. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory 52, 1289–1306 (2006).
    [Crossref]
  12. E. J. Candès, M. B. Wakin, and S. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
    [Crossref]
  13. M. M. Marim, M. Atlan, E. Angelini, and J.-C. Olivo-Marin, “Compressed sensing with off-axis frequency-shifting holography,” Opt. Lett. 35, 871–873 (2010).
    [Crossref] [PubMed]
  14. A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
    [Crossref]
  15. S. Bettens, H. Yan, D. Blinder, H. Ottevaere, C. Schretter, and P. Schelkens, “Studies on the sparsifying operator in compressive digital holography,” Opt. Express (2017, submitted).
  16. M. Liebling, T. Blu, and M. Unser, “Fresnelets: New multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
    [Crossref]
  17. T.-C. Poon, Digital holography and three-dimensional display: Principles and Applications (Springer Science & Business Media, 2006).
    [Crossref]
  18. J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2004), 3rd ed.
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    [Crossref]
  20. A. A. Petrosian and F. G. Meyer, Wavelets in signal and image analysis: from theory to practice, vol. 19 (Springer Science & Business Media, 2013).
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  23. A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imag. 6, 81–94 (1984).
    [Crossref] [PubMed]
  24. S. Kaczmarz, “Angenäherte Auflösung von Systemen linearer Gleichungen,” Bulletin International de l’Académie Polonaise des Sciences et des Lettres 35, 355–357 (1937).
  25. B. T. Polyak and A. B. Juditsky, “Acceleration of stochastic approximation by averaging,” SIAM J. Contr. Optim. 30, 838–855 (1992).
    [Crossref]
  26. D. Needell, N. Srebro, and R. Ward, “Stochastic gradient descent and the randomized Kaczmarz algorithm,” arXiv preprint arXiv:1310.5715 (2013).
  27. K. Wei, “Phase retrieval via Kaczmarz methods,” arXiv preprint arXiv:1502.01822 (2015).
  28. Y. Nesterov, “A method for unconstrained convex minimization problem with the rate of convergence O(1/k2),” Doklady an SSSR 269, 543–547 (1983).
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    [Crossref]
  30. D. Kim and J. A. Fessler, “Optimized first-order methods for smooth convex minimization,” Math. Program. 159, 81–107 (2016).
    [Crossref] [PubMed]
  31. C. Schretter and H. Niederreiter, “A direct inversion method for non-uniform quasi-random point sequences,” Monte Carlo Meth. Appl. 19, 1–9 (2013).
    [Crossref]
  32. W. Sweldens, “The lifting scheme: A custom-design construction of biorthogonal wavelets,” Appl. Comput. Harmon. Anal. 3, 186–200 (1996).
    [Crossref]
  33. I. Daubechies and W. Sweldens, “Factoring wavelet transforms into lifting steps,” J. Fourier Anal. Appl. 4, 247–269 (1998).
    [Crossref]

2016 (1)

D. Kim and J. A. Fessler, “Optimized first-order methods for smooth convex minimization,” Math. Program. 159, 81–107 (2016).
[Crossref] [PubMed]

2013 (2)

C. Schretter and H. Niederreiter, “A direct inversion method for non-uniform quasi-random point sequences,” Monte Carlo Meth. Appl. 19, 1–9 (2013).
[Crossref]

A. Bourquard, N. Pavillon, E. Bostan, C. Depeursinge, and M. Unser, “A practical inverse-problem approach to digital holographic reconstruction,” Opt. Express 21, 3417–3433 (2013).
[Crossref] [PubMed]

2011 (1)

2010 (3)

M. M. Marim, M. Atlan, E. Angelini, and J.-C. Olivo-Marin, “Compressed sensing with off-axis frequency-shifting holography,” Opt. Lett. 35, 871–873 (2010).
[Crossref] [PubMed]

M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 8005 (2010).

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Displ. Technol. 6, 506–509 (2010).
[Crossref]

2009 (1)

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt 48, H186–H195 (2009).
[Crossref] [PubMed]

2008 (1)

E. J. Candès, M. B. Wakin, and S. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[Crossref]

2006 (1)

D. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory 52, 1289–1306 (2006).
[Crossref]

2005 (1)

Y. Nesterov, “Smooth minimization of non-smooth functions,” Math. Program. 103, 127–152 (2005).
[Crossref]

2004 (1)

2003 (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: New multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[Crossref]

1999 (1)

1998 (1)

I. Daubechies and W. Sweldens, “Factoring wavelet transforms into lifting steps,” J. Fourier Anal. Appl. 4, 247–269 (1998).
[Crossref]

1996 (1)

W. Sweldens, “The lifting scheme: A custom-design construction of biorthogonal wavelets,” Appl. Comput. Harmon. Anal. 3, 186–200 (1996).
[Crossref]

1992 (3)

A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[Crossref]

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D: Nonlin. Phenom. 60, 259–268 (1992).
[Crossref]

B. T. Polyak and A. B. Juditsky, “Acceleration of stochastic approximation by averaging,” SIAM J. Contr. Optim. 30, 838–855 (1992).
[Crossref]

1989 (1)

S. G. Mallat, “A Theory for Multiresolution Signal Decomposition: The Wavelet Representation,” IEEE Trans. Pattern Anal. Mach. Intel. 11, 674–693 (1989).
[Crossref]

1984 (1)

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imag. 6, 81–94 (1984).
[Crossref] [PubMed]

1983 (1)

Y. Nesterov, “A method for unconstrained convex minimization problem with the rate of convergence O(1/k2),” Doklady an SSSR 269, 543–547 (1983).

1982 (1)

1937 (1)

S. Kaczmarz, “Angenäherte Auflösung von Systemen linearer Gleichungen,” Bulletin International de l’Académie Polonaise des Sciences et des Lettres 35, 355–357 (1937).

Andersen, A. H.

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imag. 6, 81–94 (1984).
[Crossref] [PubMed]

Angelini, E.

Atlan, M.

Bettens, S.

S. Bettens, H. Yan, D. Blinder, H. Ottevaere, C. Schretter, and P. Schelkens, “Studies on the sparsifying operator in compressive digital holography,” Opt. Express (2017, submitted).

Blinder, D.

S. Bettens, H. Yan, D. Blinder, H. Ottevaere, C. Schretter, and P. Schelkens, “Studies on the sparsifying operator in compressive digital holography,” Opt. Express (2017, submitted).

Blu, T.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: New multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[Crossref]

Bostan, E.

Bourquard, A.

Boyd, S.

E. J. Candès, M. B. Wakin, and S. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[Crossref]

Candès, E. J.

E. J. Candès, M. B. Wakin, and S. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[Crossref]

Cohen, A.

A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[Crossref]

Cuche, E.

Daubechies, I.

I. Daubechies and W. Sweldens, “Factoring wavelet transforms into lifting steps,” J. Fourier Anal. Appl. 4, 247–269 (1998).
[Crossref]

A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[Crossref]

Depeursinge, C.

Donoho, D.

D. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory 52, 1289–1306 (2006).
[Crossref]

Ebrahimi, T.

P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG 2000 Suite (Wiley Publishing, 2009).
[Crossref]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D: Nonlin. Phenom. 60, 259–268 (1992).
[Crossref]

Feauveau, J.-C.

A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[Crossref]

Fessler, J. A.

D. Kim and J. A. Fessler, “Optimized first-order methods for smooth convex minimization,” Math. Program. 159, 81–107 (2016).
[Crossref] [PubMed]

S. Sotthivirat and J. A. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. A 21, 737–750 (2004).
[Crossref]

J. A. Fessler and S. Sotthivirat, “Simplified digital holographic reconstruction using statistical methods,” 2004 International Conference on Image Processing (ICIP’04)4, 2435–2438 (2004).

S. Sotthivirat and J. A. Fessler, “Reconstruction from digital holograms by statistical methods,” Conference Record of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers2, 1928–1932 (2003).

Fienup, J. R.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2004), 3rd ed.

Javidi, B.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Displ. Technol. 6, 506–509 (2010).
[Crossref]

Juditsky, A. B.

B. T. Polyak and A. B. Juditsky, “Acceleration of stochastic approximation by averaging,” SIAM J. Contr. Optim. 30, 838–855 (1992).
[Crossref]

Kaczmarz, S.

S. Kaczmarz, “Angenäherte Auflösung von Systemen linearer Gleichungen,” Bulletin International de l’Académie Polonaise des Sciences et des Lettres 35, 355–357 (1937).

Kak, A. C.

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imag. 6, 81–94 (1984).
[Crossref] [PubMed]

Kim, D.

D. Kim and J. A. Fessler, “Optimized first-order methods for smooth convex minimization,” Math. Program. 159, 81–107 (2016).
[Crossref] [PubMed]

Kim, M. K.

M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 8005 (2010).

Kühn, J.

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt 48, H186–H195 (2009).
[Crossref] [PubMed]

Liebling, M.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: New multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[Crossref]

Mallat, S. G.

S. G. Mallat, “A Theory for Multiresolution Signal Decomposition: The Wavelet Representation,” IEEE Trans. Pattern Anal. Mach. Intel. 11, 674–693 (1989).
[Crossref]

Marim, M. M.

Marquet, P.

Meyer, F. G.

A. A. Petrosian and F. G. Meyer, Wavelets in signal and image analysis: from theory to practice, vol. 19 (Springer Science & Business Media, 2013).

Needell, D.

D. Needell, N. Srebro, and R. Ward, “Stochastic gradient descent and the randomized Kaczmarz algorithm,” arXiv preprint arXiv:1310.5715 (2013).

Nesterov, Y.

Y. Nesterov, “Smooth minimization of non-smooth functions,” Math. Program. 103, 127–152 (2005).
[Crossref]

Y. Nesterov, “A method for unconstrained convex minimization problem with the rate of convergence O(1/k2),” Doklady an SSSR 269, 543–547 (1983).

Niederreiter, H.

C. Schretter and H. Niederreiter, “A direct inversion method for non-uniform quasi-random point sequences,” Monte Carlo Meth. Appl. 19, 1–9 (2013).
[Crossref]

Olivo-Marin, J.-C.

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D: Nonlin. Phenom. 60, 259–268 (1992).
[Crossref]

Ottevaere, H.

S. Bettens, H. Yan, D. Blinder, H. Ottevaere, C. Schretter, and P. Schelkens, “Studies on the sparsifying operator in compressive digital holography,” Opt. Express (2017, submitted).

Pavillon, N.

Petrosian, A. A.

A. A. Petrosian and F. G. Meyer, Wavelets in signal and image analysis: from theory to practice, vol. 19 (Springer Science & Business Media, 2013).

Polyak, B. T.

B. T. Polyak and A. B. Juditsky, “Acceleration of stochastic approximation by averaging,” SIAM J. Contr. Optim. 30, 838–855 (1992).
[Crossref]

Poon, T.-C.

T.-C. Poon, Digital holography and three-dimensional display: Principles and Applications (Springer Science & Business Media, 2006).
[Crossref]

Rivenson, Y.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Displ. Technol. 6, 506–509 (2010).
[Crossref]

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D: Nonlin. Phenom. 60, 259–268 (1992).
[Crossref]

Schelkens, P.

P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG 2000 Suite (Wiley Publishing, 2009).
[Crossref]

S. Bettens, H. Yan, D. Blinder, H. Ottevaere, C. Schretter, and P. Schelkens, “Studies on the sparsifying operator in compressive digital holography,” Opt. Express (2017, submitted).

Schretter, C.

C. Schretter and H. Niederreiter, “A direct inversion method for non-uniform quasi-random point sequences,” Monte Carlo Meth. Appl. 19, 1–9 (2013).
[Crossref]

S. Bettens, H. Yan, D. Blinder, H. Ottevaere, C. Schretter, and P. Schelkens, “Studies on the sparsifying operator in compressive digital holography,” Opt. Express (2017, submitted).

Seelamantula, C. S.

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Exact complex-wave reconstruction in digital holography,” J. Opt. Soc. Am. A 28, 983–992 (2011).
[Crossref] [PubMed]

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt 48, H186–H195 (2009).
[Crossref] [PubMed]

Skodras, A.

P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG 2000 Suite (Wiley Publishing, 2009).
[Crossref]

Sotthivirat, S.

S. Sotthivirat and J. A. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. A 21, 737–750 (2004).
[Crossref]

J. A. Fessler and S. Sotthivirat, “Simplified digital holographic reconstruction using statistical methods,” 2004 International Conference on Image Processing (ICIP’04)4, 2435–2438 (2004).

S. Sotthivirat and J. A. Fessler, “Reconstruction from digital holograms by statistical methods,” Conference Record of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers2, 1928–1932 (2003).

Srebro, N.

D. Needell, N. Srebro, and R. Ward, “Stochastic gradient descent and the randomized Kaczmarz algorithm,” arXiv preprint arXiv:1310.5715 (2013).

Stern, A.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Displ. Technol. 6, 506–509 (2010).
[Crossref]

Sweldens, W.

I. Daubechies and W. Sweldens, “Factoring wavelet transforms into lifting steps,” J. Fourier Anal. Appl. 4, 247–269 (1998).
[Crossref]

W. Sweldens, “The lifting scheme: A custom-design construction of biorthogonal wavelets,” Appl. Comput. Harmon. Anal. 3, 186–200 (1996).
[Crossref]

Unser, M.

A. Bourquard, N. Pavillon, E. Bostan, C. Depeursinge, and M. Unser, “A practical inverse-problem approach to digital holographic reconstruction,” Opt. Express 21, 3417–3433 (2013).
[Crossref] [PubMed]

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Exact complex-wave reconstruction in digital holography,” J. Opt. Soc. Am. A 28, 983–992 (2011).
[Crossref] [PubMed]

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt 48, H186–H195 (2009).
[Crossref] [PubMed]

M. Liebling, T. Blu, and M. Unser, “Fresnelets: New multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[Crossref]

Wakin, M. B.

E. J. Candès, M. B. Wakin, and S. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[Crossref]

Ward, R.

D. Needell, N. Srebro, and R. Ward, “Stochastic gradient descent and the randomized Kaczmarz algorithm,” arXiv preprint arXiv:1310.5715 (2013).

Wei, K.

K. Wei, “Phase retrieval via Kaczmarz methods,” arXiv preprint arXiv:1502.01822 (2015).

Yan, H.

S. Bettens, H. Yan, D. Blinder, H. Ottevaere, C. Schretter, and P. Schelkens, “Studies on the sparsifying operator in compressive digital holography,” Opt. Express (2017, submitted).

Appl. Comput. Harmon. Anal. (1)

W. Sweldens, “The lifting scheme: A custom-design construction of biorthogonal wavelets,” Appl. Comput. Harmon. Anal. 3, 186–200 (1996).
[Crossref]

Appl. Opt (1)

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt 48, H186–H195 (2009).
[Crossref] [PubMed]

Appl. Opt. (2)

Bulletin International de l’Académie Polonaise des Sciences et des Lettres (1)

S. Kaczmarz, “Angenäherte Auflösung von Systemen linearer Gleichungen,” Bulletin International de l’Académie Polonaise des Sciences et des Lettres 35, 355–357 (1937).

Commun. Pure Appl. Math. (1)

A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[Crossref]

Doklady an SSSR (1)

Y. Nesterov, “A method for unconstrained convex minimization problem with the rate of convergence O(1/k2),” Doklady an SSSR 269, 543–547 (1983).

IEEE Trans. Image Process. (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: New multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[Crossref]

IEEE Trans. Inform. Theory (1)

D. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory 52, 1289–1306 (2006).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intel. (1)

S. G. Mallat, “A Theory for Multiresolution Signal Decomposition: The Wavelet Representation,” IEEE Trans. Pattern Anal. Mach. Intel. 11, 674–693 (1989).
[Crossref]

J. Displ. Technol. (1)

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Displ. Technol. 6, 506–509 (2010).
[Crossref]

J. Fourier Anal. Appl. (2)

I. Daubechies and W. Sweldens, “Factoring wavelet transforms into lifting steps,” J. Fourier Anal. Appl. 4, 247–269 (1998).
[Crossref]

E. J. Candès, M. B. Wakin, and S. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[Crossref]

J. Opt. Soc. Am. A (2)

Math. Program. (2)

Y. Nesterov, “Smooth minimization of non-smooth functions,” Math. Program. 103, 127–152 (2005).
[Crossref]

D. Kim and J. A. Fessler, “Optimized first-order methods for smooth convex minimization,” Math. Program. 159, 81–107 (2016).
[Crossref] [PubMed]

Monte Carlo Meth. Appl. (1)

C. Schretter and H. Niederreiter, “A direct inversion method for non-uniform quasi-random point sequences,” Monte Carlo Meth. Appl. 19, 1–9 (2013).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. D: Nonlin. Phenom. (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D: Nonlin. Phenom. 60, 259–268 (1992).
[Crossref]

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Figures (12)

Fig. 1
Fig. 1 Geometry of an off-axis holographic acquisition setup. The intensity of the interference between a slightly tilted reference wavefield and the object wavefield is recorded by the digital CCD sensor. The phase information of the observed sample is then encoded in slight local displacements of fringe patterns appearing on the detector plane.
Fig. 2
Fig. 2 Direct image reconstruction in the Fourier domain. The complex-valued object wavefield image O is recovered from the intensity hologram I = |O + R|2 by demodulation and band-pass filtering of higher frequencies for removing the fringe modulation pattern of the reference beam R. Finally, the wavefield is refocused on the sample plane.
Fig. 3
Fig. 3 Forward image formation model used in inverse image reconstruction. CDF 9/7 wavelet coefficients represent the amplitude and phase components of the sample wavefield Od. Angular spectrum light transport and the interference between the propagated wavefield O and the near-uniform off-axis reference beam R generates a fringe pattern in I.
Fig. 4
Fig. 4 Exact (left) and relaxed (right) projection operators infering complex-valued measurements from the current solution O. The closest value from O is selected such that addition to the reference beam R intersects the circular region of all compatible intensity values.
Fig. 5
Fig. 5 Comparison of the overall increasing norm as a function of the iteration number for the reference Kaczmarz method and the two Nesterov accelerated methods discussed in this work. Nesterov methods reach quadratic convergence speed instead of the much slower linear convergence of the Kaczmarz method plotted in light gray.
Fig. 6
Fig. 6 Progressive sharpening of the solution until convergence. The regularization prior in the inverse reconstruction method selects the minimum-norm solution that is compatible with the intensity measurements. Stopping early the iterative process yields a blurry intermediate solution that may already be of sufficient quality for the operator.
Fig. 7
Fig. 7 The four simulated and optically acquired intensity hologram data used in experiments (top row) and their frequency power spectrum (second row). The blank scan for recording the reference beam amplitude are shown as well (third row) along with the amplitude (fourth row) and phase shift (bottom row) reference results with direct Fourier reconstruction.
Fig. 8
Fig. 8 Impact of the relaxed data projection operator for avoiding the introduction of noise artifacts into the solution. A bias-variance trade-off is driven by the radius of the intensity tolerance region, expressed as a fraction of the standard deviation of the noise realization.
Fig. 9
Fig. 9 Progressive quality improvement with increasing amount of intensity measurements used for constraining the solution. In comparison to a full-data reference reconstruction, a high quality phase image is recovered from one quarter of all detector pixels only.
Fig. 10
Fig. 10 Side-by-side comparison of resolution recovery in a similar region of interest in both the simulated and acquired USAF-1951 test targets. Parallel bars are crisper with the inverse reconstruction technique, while spurious low-frequency background noise is reduced.
Fig. 11
Fig. 11 Amplitude and phase shift images recovered by iterative reconstruction (top row). A side-by-side comparison in a region of interest (bottom row) shows a slightly crisper recovery in comparison to a Fourier reconstruction. Note that in this case, a bandlimiting filter was used in the optical line for optimizing quality of the direct reconstruction solution.
Fig. 12
Fig. 12 Lifting scheme block diagram where an input signal x is split into even and odd samples (xe and xo). Then a series of convolution-accumulate operations is applied alternately on the two divided signals, using prediction (si) and update (ti) filters. Finally, the channels are scaled with constants Ki, resulting in the final approximation signal λ and detail signal γ.

Tables (1)

Tables Icon

Table 1 Numerical values of the prediction and update filters for the CDF 9/7 DWT.

Equations (36)

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I = | O + R | 2
= | O | 2 + | R | 2 + R O * + O R *
O = Ψ d O d = F 1 ( K d F ( O d ) ) and O d = Ψ d O = F 1 ( K d F ( O ) ) .
K d = 1 K d = exp ( 2 i π d λ D )
D ( x , y ) = 1 ( λ p x N x / 2 1 2 N x ) 2 ( λ p y N y / 2 1 2 N y ) 2 ,
I ˜ = I | R | 2
= | O | 2 + R O * + O R * ,
O ˜ = I ˜ R *
= | O | 2 R * + R O * R * + O
C ( x , y ) = min ( 1 , max ( 0 , r ( x r 0.5 ) 2 + ( y r 0.5 ) 2 ) / w ) ,
C H ( x , y ) = C ( x , y ) 2 ( 3 2 C ( x , y ) ) .
O ˜ d = F 1 ( K d ( O ˜ C H ) ) .
argmin x x subject to I = | R + Ψ d W 1 x | 2 ,
argmin x x subject to O = Ψ d W 1 x ,
O = Ψ d W 1 x t ,
P ( O ) = I ( O + R ) | O + R | R .
I = | P ( O ) + R | 2 .
P ˜ ( O ) = I ( O + R ) | O + R | R with I = { max ( I 2 , 0 ) if | O + R | 2 < max ( I 2 , 0 ) I + 2 if | O + R | 2 > I + 2 | O + R | 2 otherwise
max ( I 2 , 0 ) | P ˜ ( O ) + R | I + 2 .
P ( O ) = R I | R | | R | .
x = ( A * A ) 1 A * y .
x t + 1 = x t + Δ t
Δ t = 2 M i = 1 M [ P ( O ) ] i [ A ] i x t [ A ] i 2 [ A * ] i
y t + 1 = x t + Δ t ,
x t + 1 = y t + 1 + λ t 1 λ t + 1 ( y t + 1 y t ) with λ t + 1 = 1 + 1 + 4 λ t 2 2 .
u t + 1 = u t + λ t Δ t
x t + 1 = ( 1 1 λ t + 1 ) y t + 1 + 1 λ t + 1 u t + 1 with λ t + 1 = 1 + 1 + 4 λ t 2 2 .
x ( z ) = k x k z k ,
h ( z ) = k = k b k e h k z k .
x e ( z 2 ) = x ( z ) + x ( z ) 2 and x o ( z 2 ) = z 2 [ x ( z ) x ( z ) ] .
( λ ( z ) γ ( z ) ) = P ( z ) ( x e ( z ) x o ( z ) ) .
P ( z ) = ( h e ( z ) h o ( z ) g e ( z ) g o ( z ) ) = ( K 1 0 0 K 2 ) i = 1 m ( 1 s i ( z ) 0 1 ) ( 1 0 t i ( z ) 1 ) .
P ( z ) 1 = ( i = m 1 ( 1 0 t i ( z ) 1 ) ( 1 s i ( z ) 0 1 ) ) ( 1 / K 1 0 0 1 / K 2 ) .
( P ( z ) 1 ) = ( 1 / K 1 0 0 1 / K 2 ) i = 1 m ( 1 0 s i ( z ) 1 ) ( 1 t i ( z ) 0 1 ) .
α = i = 1 N I i i = 1 N | R i + [ Ψ d W 1 x ] i | 2 1 ,
P ( O ) = I ( O + α R ) | O + α R | α R .

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