Abstract

We propose simultaneous measurement and reconstruction tailoring (SMaRT) for quantitative phase imaging; it is a joint optimization approach to inverse problems wherein minimizing the expected end-to-end error yields optimal design parameters for both the measurement and reconstruction processes. Using simulated and experimentally-collected data for a specific scenario, we demonstrate that optimizing the design of the two processes together reduces phase reconstruction error over past techniques that consider these two design problems separately. Our results suggest at times surprising design principles, and our approach can potentially inspire improved solution methods for other inverse problems in optics as well as the natural sciences.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  2. J. R. Fienup, “Phase retrieval algorithms: A comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  3. E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
    [Crossref]
  4. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [Crossref]
  5. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [Crossref]
  6. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
    [Crossref]
  7. L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18, 12552–12561 (2010).
    [Crossref] [PubMed]
  8. J. Zhong, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22, 10661–10674 (2014).
    [Crossref]
  9. J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).
  10. Y. I. Nesterets and T. E. Gureyev, “Partially coherent contrast-transfer-function approximation,” J. Opt. Soc. Am. A 33, 464–474 (2016).
    [Crossref]
  11. J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21, 14430–14441 (2013).
    [Crossref] [PubMed]
  12. T. Chakraborty and J. C. Petruccelli, “Source diversity for transport of intensity phase imaging,” Opt. Express 25, 9122–9137 (2017).
    [Crossref] [PubMed]
  13. B. Xue, S. Zheng, L. Cui, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes,” Opt. Express 19, 20244–20250 (2011).
    [Crossref] [PubMed]
  14. S. Zheng, B. Xue, W. Xue, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express 20, 972–985 (2012).
    [Crossref] [PubMed]
  15. J. Sun, C. Zuo, and Q. Chen, “Iterative optimum frequency combination method for high efficiency phase imaging of absorptive objects based on phase transfer function,” Opt. Express 23, 28031–28049 (2015).
    [Crossref] [PubMed]
  16. 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]
  17. L. Tian, J. C. Petruccelli, Q. Miao, H. Kudrolli, V. Nagarkar, and G. Barbastathis, “Compressive x-ray phase tomography based on the transport of intensity equation,” Opt. Lett. 38, 3418–3421 (2013).
    [Crossref] [PubMed]
  18. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [Crossref] [PubMed]
  19. W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” Appl. Opt. 41, 6080–6092 (2002).
    [Crossref] [PubMed]
  20. M. D. Stenner, A. Ashok, and M. A. Neifeld, “Multi-domain optimization for ultra-thin cameras,” in Frontiers in Optics, (Optical Society of America, 2006), p. FWH5.
    [Crossref]
  21. E. J. Tremblay, J. Rutkowski, I. Tamayo, P. E. X. Silveira, R. A. Stack, R. L. Morrison, M. A. Neifeld, Y. Fainman, and J. E. Ford, “Relaxing the alignment and fabrication tolerances of thin annular folded imaging systems using wavefront coding,” Appl. Opt. 46, 6751–6758 (2007).
    [Crossref] [PubMed]
  22. J. Frank, Three-Dimensional Electron Microscopy of Macromolecular Assemblies: Visualization of Biological Molecules in Their Native State (Oxford University, 2006).
    [Crossref]
  23. M. Vulović, L. M. Voortman, L. J. van Vliet, and B. Rieger, “When to use the projection assumption and the weak-phase object approximation in phase contrast cryo-EM,” Ultramicroscopy 136, 61–66 (2014).
    [Crossref]
  24. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [Crossref]
  25. L. Mandel and E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [Crossref]
  26. A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” arXiv preprint arXiv:1806.10029 (2018).
  27. Y. Nesterov, “Gradient methods for minimizing composite objective function,” CORE Discussion Papers 2007076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) (2007).
  28. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
    [Crossref]
  29. B. O’Donoghue and E. Candès, “Adaptive restart for accelerated gradient schemes,” Found. Comp. Math. 15, 715–732 (2015).
    [Crossref]
  30. C. Bao, G. Barbastathis, H. Ji, Z. Shen, and Z. Zhang, “Coherence retrieval using trace regularization,” SIAM J. Imaging Sci. 11, 679–706 (2018).
    [Crossref]
  31. G. J. Burton and I. R. Moorhead, “Color and spatial structure in natural scenes,” Appl. Opt. 26, 157–170 (1987).
    [Crossref] [PubMed]
  32. D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
    [Crossref] [PubMed]
  33. D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
    [Crossref] [PubMed]
  34. A. B. Lee, D. Mumford, and J. Huang, “Occlusion models for natural images: A statistical study of a scale-invariant dead leaves model,” Int. J. Comput. Vis. 41, 35–59 (2001).
    [Crossref]
  35. Y. LeCun, J. S. Denker, and S. A. Solla, “Optimal brain damage,” in Advances in Neural Information Processing Systems 2, D. S. Touretzky, ed. (Morgan-Kaufmann, 1990), pp. 598–605.
  36. P. Molchanov, S. Tyree, T. Karras, T. Aila, and J. Kautz, “Pruning convolutional neural networks for resource efficient inference,” in Proceedings of the International Conference on Learning Representations (ICLR), (2015).
  37. Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, (1993), pp. 40–44 vol.1.
    [Crossref]
  38. W. Karush, “Minima of functions of several variables with inequalities as side constraints,” Ph.D. thesis, University of Chicago (1939).
  39. H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, (University of California, 1951), pp. 481–492.

2018 (1)

C. Bao, G. Barbastathis, H. Ji, Z. Shen, and Z. Zhang, “Coherence retrieval using trace regularization,” SIAM J. Imaging Sci. 11, 679–706 (2018).
[Crossref]

2017 (1)

2016 (1)

2015 (2)

2014 (2)

M. Vulović, L. M. Voortman, L. J. van Vliet, and B. Rieger, “When to use the projection assumption and the weak-phase object approximation in phase contrast cryo-EM,” Ultramicroscopy 136, 61–66 (2014).
[Crossref]

J. Zhong, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22, 10661–10674 (2014).
[Crossref]

2013 (3)

2012 (2)

2011 (1)

2010 (1)

2009 (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]

2007 (1)

2002 (1)

2001 (1)

A. B. Lee, D. Mumford, and J. Huang, “Occlusion models for natural images: A statistical study of a scale-invariant dead leaves model,” Int. J. Comput. Vis. 41, 35–59 (2001).
[Crossref]

1998 (1)

1995 (1)

1992 (1)

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[Crossref] [PubMed]

1987 (2)

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[Crossref]

1983 (1)

1982 (2)

1977 (1)

J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

1976 (1)

1972 (1)

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

Aila, T.

P. Molchanov, S. Tyree, T. Karras, T. Aila, and J. Kautz, “Pruning convolutional neural networks for resource efficient inference,” in Proceedings of the International Conference on Learning Representations (ICLR), (2015).

Arthur, K.

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” arXiv preprint arXiv:1806.10029 (2018).

Ashok, A.

M. D. Stenner, A. Ashok, and M. A. Neifeld, “Multi-domain optimization for ultra-thin cameras,” in Frontiers in Optics, (Optical Society of America, 2006), p. FWH5.
[Crossref]

Bai, X.

Bao, C.

C. Bao, G. Barbastathis, H. Ji, Z. Shen, and Z. Zhang, “Coherence retrieval using trace regularization,” SIAM J. Imaging Sci. 11, 679–706 (2018).
[Crossref]

Barbastathis, G.

Barty, A.

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]

Burton, G. J.

Candès, E.

B. O’Donoghue and E. Candès, “Adaptive restart for accelerated gradient schemes,” Found. Comp. Math. 15, 715–732 (2015).
[Crossref]

Candès, E. J.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
[Crossref]

Cathey, W. T.

Chakraborty, T.

Chao, T.

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[Crossref] [PubMed]

Chen, Q.

Claus, R. A.

Cui, L.

Dauwels, J.

Denker, J. S.

Y. LeCun, J. S. Denker, and S. A. Solla, “Optimal brain damage,” in Advances in Neural Information Processing Systems 2, D. S. Touretzky, ed. (Morgan-Kaufmann, 1990), pp. 598–605.

Dowski, E. R.

Eldar, Y. C.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
[Crossref]

Fainman, Y.

Field, D. J.

Fienup, J. R.

Ford, J. E.

Frank, J.

J. Frank, Three-Dimensional Electron Microscopy of Macromolecular Assemblies: Visualization of Biological Molecules in Their Native State (Oxford University, 2006).
[Crossref]

Gerchberg, R. W.

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

Goy, A.

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” arXiv preprint arXiv:1806.10029 (2018).

Guigay, J.-P.

J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Gureyev, T. E.

Huang, J.

A. B. Lee, D. Mumford, and J. Huang, “Occlusion models for natural images: A statistical study of a scale-invariant dead leaves model,” Int. J. Comput. Vis. 41, 35–59 (2001).
[Crossref]

Ji, H.

C. Bao, G. Barbastathis, H. Ji, Z. Shen, and Z. Zhang, “Coherence retrieval using trace regularization,” SIAM J. Imaging Sci. 11, 679–706 (2018).
[Crossref]

Karras, T.

P. Molchanov, S. Tyree, T. Karras, T. Aila, and J. Kautz, “Pruning convolutional neural networks for resource efficient inference,” in Proceedings of the International Conference on Learning Representations (ICLR), (2015).

Karush, W.

W. Karush, “Minima of functions of several variables with inequalities as side constraints,” Ph.D. thesis, University of Chicago (1939).

Kautz, J.

P. Molchanov, S. Tyree, T. Karras, T. Aila, and J. Kautz, “Pruning convolutional neural networks for resource efficient inference,” in Proceedings of the International Conference on Learning Representations (ICLR), (2015).

Krishnaprasad, P. S.

Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, (1993), pp. 40–44 vol.1.
[Crossref]

Kudrolli, H.

Kuhn, H. W.

H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, (University of California, 1951), pp. 481–492.

LeCun, Y.

Y. LeCun, J. S. Denker, and S. A. Solla, “Optimal brain damage,” in Advances in Neural Information Processing Systems 2, D. S. Touretzky, ed. (Morgan-Kaufmann, 1990), pp. 598–605.

Lee, A. B.

A. B. Lee, D. Mumford, and J. Huang, “Occlusion models for natural images: A statistical study of a scale-invariant dead leaves model,” Int. J. Comput. Vis. 41, 35–59 (2001).
[Crossref]

Li, S.

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” arXiv preprint arXiv:1806.10029 (2018).

Mandel, L.

Miao, Q.

Molchanov, P.

P. Molchanov, S. Tyree, T. Karras, T. Aila, and J. Kautz, “Pruning convolutional neural networks for resource efficient inference,” in Proceedings of the International Conference on Learning Representations (ICLR), (2015).

Moorhead, I. R.

Morrison, R. L.

Mumford, D.

A. B. Lee, D. Mumford, and J. Huang, “Occlusion models for natural images: A statistical study of a scale-invariant dead leaves model,” Int. J. Comput. Vis. 41, 35–59 (2001).
[Crossref]

Nagarkar, V.

Neifeld, M. A.

Nesterets, Y. I.

Nesterov, Y.

Y. Nesterov, “Gradient methods for minimizing composite objective function,” CORE Discussion Papers 2007076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) (2007).

Nugent, K. A.

O’Donoghue, B.

B. O’Donoghue and E. Candès, “Adaptive restart for accelerated gradient schemes,” Found. Comp. Math. 15, 715–732 (2015).
[Crossref]

Paganin, D.

Pati, Y. C.

Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, (1993), pp. 40–44 vol.1.
[Crossref]

Petruccelli, J. C.

Rezaiifar, R.

Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, (1993), pp. 40–44 vol.1.
[Crossref]

Rieger, B.

M. Vulović, L. M. Voortman, L. J. van Vliet, and B. Rieger, “When to use the projection assumption and the weak-phase object approximation in phase contrast cryo-EM,” Ultramicroscopy 136, 61–66 (2014).
[Crossref]

Roberts, A.

Rutkowski, J.

Saxton, W. O.

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

Shen, Z.

C. Bao, G. Barbastathis, H. Ji, Z. Shen, and Z. Zhang, “Coherence retrieval using trace regularization,” SIAM J. Imaging Sci. 11, 679–706 (2018).
[Crossref]

Silveira, P. E. X.

Solla, S. A.

Y. LeCun, J. S. Denker, and S. A. Solla, “Optimal brain damage,” in Advances in Neural Information Processing Systems 2, D. S. Touretzky, ed. (Morgan-Kaufmann, 1990), pp. 598–605.

Stack, R. A.

Stenner, M. D.

M. D. Stenner, A. Ashok, and M. A. Neifeld, “Multi-domain optimization for ultra-thin cameras,” in Frontiers in Optics, (Optical Society of America, 2006), p. FWH5.
[Crossref]

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[Crossref]

Strohmer, T.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
[Crossref]

Sun, J.

Tadmor, Y.

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[Crossref] [PubMed]

Tamayo, I.

Teague, M. R.

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]

Tian, L.

Tolhurst, D. J.

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[Crossref] [PubMed]

Tremblay, E. J.

Tucker, A. W.

H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, (University of California, 1951), pp. 481–492.

Tyree, S.

P. Molchanov, S. Tyree, T. Karras, T. Aila, and J. Kautz, “Pruning convolutional neural networks for resource efficient inference,” in Proceedings of the International Conference on Learning Representations (ICLR), (2015).

van Vliet, L. J.

M. Vulović, L. M. Voortman, L. J. van Vliet, and B. Rieger, “When to use the projection assumption and the weak-phase object approximation in phase contrast cryo-EM,” Ultramicroscopy 136, 61–66 (2014).
[Crossref]

Voortman, L. M.

M. Vulović, L. M. Voortman, L. J. van Vliet, and B. Rieger, “When to use the projection assumption and the weak-phase object approximation in phase contrast cryo-EM,” Ultramicroscopy 136, 61–66 (2014).
[Crossref]

Voroninski, V.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
[Crossref]

Vulovic, M.

M. Vulović, L. M. Voortman, L. J. van Vliet, and B. Rieger, “When to use the projection assumption and the weak-phase object approximation in phase contrast cryo-EM,” Ultramicroscopy 136, 61–66 (2014).
[Crossref]

Waller, L.

Wolf, E.

Xue, B.

Xue, W.

Zhang, Z.

C. Bao, G. Barbastathis, H. Ji, Z. Shen, and Z. Zhang, “Coherence retrieval using trace regularization,” SIAM J. Imaging Sci. 11, 679–706 (2018).
[Crossref]

Zheng, S.

Zhong, J.

Zhou, F.

Zuo, C.

Appl. Opt. (5)

Found. Comp. Math. (1)

B. O’Donoghue and E. Candès, “Adaptive restart for accelerated gradient schemes,” Found. Comp. Math. 15, 715–732 (2015).
[Crossref]

Int. J. Comput. Vis. (1)

A. B. Lee, D. Mumford, and J. Huang, “Occlusion models for natural images: A statistical study of a scale-invariant dead leaves model,” Int. J. Comput. Vis. 41, 35–59 (2001).
[Crossref]

J. Opt. Soc. Am. (3)

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

Ophthalmic Physiol. Opt. (1)

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[Crossref] [PubMed]

Opt. Commun. (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[Crossref]

Opt. Express (7)

Opt. Lett. (3)

Optik (2)

J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

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

SIAM J. Imaging Sci. (3)

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
[Crossref]

C. Bao, G. Barbastathis, H. Ji, Z. Shen, and Z. Zhang, “Coherence retrieval using trace regularization,” SIAM J. Imaging Sci. 11, 679–706 (2018).
[Crossref]

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

Ultramicroscopy (1)

M. Vulović, L. M. Voortman, L. J. van Vliet, and B. Rieger, “When to use the projection assumption and the weak-phase object approximation in phase contrast cryo-EM,” Ultramicroscopy 136, 61–66 (2014).
[Crossref]

Other (9)

Y. LeCun, J. S. Denker, and S. A. Solla, “Optimal brain damage,” in Advances in Neural Information Processing Systems 2, D. S. Touretzky, ed. (Morgan-Kaufmann, 1990), pp. 598–605.

P. Molchanov, S. Tyree, T. Karras, T. Aila, and J. Kautz, “Pruning convolutional neural networks for resource efficient inference,” in Proceedings of the International Conference on Learning Representations (ICLR), (2015).

Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, (1993), pp. 40–44 vol.1.
[Crossref]

W. Karush, “Minima of functions of several variables with inequalities as side constraints,” Ph.D. thesis, University of Chicago (1939).

H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, (University of California, 1951), pp. 481–492.

J. Frank, Three-Dimensional Electron Microscopy of Macromolecular Assemblies: Visualization of Biological Molecules in Their Native State (Oxford University, 2006).
[Crossref]

M. D. Stenner, A. Ashok, and M. A. Neifeld, “Multi-domain optimization for ultra-thin cameras,” in Frontiers in Optics, (Optical Society of America, 2006), p. FWH5.
[Crossref]

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” arXiv preprint arXiv:1806.10029 (2018).

Y. Nesterov, “Gradient methods for minimizing composite objective function,” CORE Discussion Papers 2007076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) (2007).

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Figures (11)

Fig. 1
Fig. 1 A standard optical microscope, wherein an area light source at the front focal plane of a condenser lens illuminates a thin specimen (blue) coincident with the back focal plane. An objective–tube lens 4f system forms a magnified image (blue dotted) of the specimen at the intermediate image plane. Rays passing through the specimen’s center and edge are shown in gray and dark red, respectively. For clarity, relative sizes are not to scale.
Fig. 2
Fig. 2 Positions and exposure times computed using SMaRT are shown in dark green, with those for GP-TIE shown in tan. Total exposure time for the two methods are the same.
Fig. 3
Fig. 3 Estimated transfer function for the real (α, left) and imaginary (β, right) components of the specimen s(x, y) are shown in green. The dashed tan line indicates the effective transfer function for comparison images generated using the GP-TIE and TIE methods. A normalized frequency of 1.0 corresponds to the maximum frequency allowable by the numerical aperture and magnification of the system.
Fig. 4
Fig. 4 Ground truth and reconstructed phase images of a simulated Siemens star target; root-mean-square error in radians is given in brackets for each reconstruction. Crops “A” and “B” are shown magnified in Fig. 5 and Fig. 6, respectively. Cross sections of the reconstructed phase and reconstruction error along the dashed line “C” are visualized in Fig. 7 and Fig. 8, respectively. The scale bar indicates 500 μm on the intermediate image plane and 100 μm in the specimen.
Fig. 5
Fig. 5 Magnified crops of the region indicated by “A” in Fig. 4.
Fig. 6
Fig. 6 Magnified crops of the region indicated by “B” in Fig. 4. The center in the ground truth is a flat circle because of the diffraction limit.
Fig. 7
Fig. 7 Reconstructed phase of simulated Siemens star along the dashed line “C” in Fig. 4. The ground truth is shown by the gray gradient.
Fig. 8
Fig. 8 Reconstruction error for simulated Siemens star along the dashed line “C” in Fig. 4.
Fig. 9
Fig. 9 Reconstructed phase images of actual Siemens star. Crops “A” and “B” are shown magnified in Fig. 10. Cross sections of the reconstructed phase along the dashed line “C” are visualized in Fig. 11. The scale bar indicates 500 μm on the intermediate image plane and 100 μm in the specimen.
Fig. 10
Fig. 10 Crops of reconstructed Siemens star for regions indicated by “A” and “B” in Fig. 9.
Fig. 11
Fig. 11 Reconstructed phase for the actual Siemens star along the dashed line “C” in Fig. 9.

Tables (1)

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Algorithm 1 Proximal operator for squared (weighted) sum of norms.

Equations (65)

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minimize M 0 , R 0 f 𝒮 0 ( s ) s r R ( p M ( s ) ) 2 2 d s ,
minimize x 1 , , x L b l = 1 L A l x l 2 2 + γ ( l = 1 L x l ) 2 .
Θ ( x , y ) = Θ ¯ + s ( x , y ) = Θ ¯ + α ( x , y ) + i β ( x , y ) ,
s ( x , y ) m = 1 M n = 1 N s ˜ m n exp ( i 2 π ξ m x ) exp ( i 2 π η n y ) = m , n s ˜ m n ψ m n ( x , y ) ,
U in ( x , y ) = exp [ i 2 π ( ξ x + η y ) ] ,
U prop ( x , y , z ) = ϕ z U in ( x , y ) [ 1 + s prop ( x λ z ξ , y λ z η , z ; λ ) ] ,
s prop ( x , y , z ; λ ) = m , n s ˜ m n ψ m n ( x , y ) exp [ i π λ z ( ξ m 2 + η n 2 ) ] .
I coherent ( x , y , z ; ξ , η , λ ) = 1 + 2 Re { s prop ( x λ z ξ , y λ z η , z ; λ ) } + | s prop ( x λ z ξ , y λ z η , z ; λ ) | 2 .
I coherent ( x , y , z ; ξ , η , λ ) 1 + 2 Re [ s prop ( x λ z ξ , y λ z η , z ; λ ) ] .
m , n { α ˜ m n cos [ π λ z ( ξ m 2 + η n 2 ) ] + β ˜ m n sin [ π λ z ( ξ m 2 + η n 2 ) ] } δ ( ξ ξ m , η η n ) .
W ( x 1 , y 1 , x 2 , y 2 , ω ) = I 0 μ ( x 1 x 2 , y 1 y 2 , ω ) ,
μ ( Δ x , Δ y , ω ) = ρ ( a , b , ω ) exp [ i k ( a Δ x + b Δ y ) ] d a d b .
I ( x , y , z ) = I 0 ρ ( a , b , ω ) I coherent ( x , y , z ; k a , k b , 2 π c / ω ) d a d b d ω .
I ( x , y , z ) I 0 + 2 I 0 Re { ρ ( a , b , ω ) s prop ( x + 2 π z a , y + 2 π z b , z ; 2 π c / ω ) d a d b d ω }
= I 0 + [ I 0 4 π 2 z 2 ρ ( x 2 π z , y 2 π z , ω ) ] 2 Re { s prop ( x , y , z ; 2 π c / ω ) } d ω ,
I ˜ ( ξ , η , z ) I 0 δ ( ξ , η ) + I ˜ real ( ξ , η , z ) + I ˜ imag ( ξ , η , z ) ,
I ˜ real ( ξ , η , z ) = I 0 m , n α ˜ m n δ ( ξ ξ m , η η m ) P ˜ real ( ξ m , η n , z ) ,
I ˜ imag ( ξ , η , z ) = I 0 m , n β ˜ m n δ ( ξ ξ m , η η m ) P ˜ imag ( ξ m , η n , z ) ,
P ˜ real ( ξ m , η n , z ) = 2 cos [ λ π z ( ξ m 2 + η n 2 ) ] μ ( λ z ξ m , λ z η n , ω ) d ω ,
P ˜ imag ( ξ m , η n , z ) = 2 sin [ λ π z ( ξ m 2 + η n 2 ) ] μ ( λ z ξ m , λ z η n , ω ) d ω .
pixel l m n ( noiseless ) = 𝒞 1 τ l χ ( x x m , y y n ) I ( x , y , z l ) d x d y ,
pixel l m n ( noisy ) ~ 𝒩 ( pixel l m n ( noiseless ) , 𝒞 2 τ l I 0 ) ,
pixel ˜ l m n ( noisy ) = 1 MN m ^ = 1 M n ^ = 1 N pixel l m ^ n ^ ( noisy ) exp { i 2 π [ ( x m ^ x m ¯ ) ξ m + ( y n ^ y n ¯ ) η n ] }
~ 𝒩 ( 𝒞 1 τ l I ^ ( ξ m , η n , z l ) , 𝒞 2 τ l I 0 ) ,
I ^ ( ξ m , η n , z l ) = I 0 χ ˜ ( ξ m , η n ) [ P ˜ real ( ξ m , η n , z l ) α ˜ m n + P ˜ imag ( ξ m , η n , z l ) β ˜ m n ] .
y = p M ( s ) = Ps + n ,
P ( l m n , p q r ) = { p l m n 1 = χ ˜ ( ξ m , η n ) P ˜ real ( ξ m , η n , z l ) if m = p , n = q , r = 1 p l m n 2 = χ ˜ ( ξ m , η n ) P ˜ imag ( ξ m , η n , z l ) if m = p , n = q , r = 2 0 otherwise .
R ( p q r , l m n ) = { x m n r l if m = p , n = q 0 otherwise .
minimize x , τ 1 , , τ L E s 𝒮 0 , n [ s R ( Ps + n ) 2 2 ] subject to τ 0 l τ l .
E s 𝒮 0 , n [ s R ( Ps + n ) 2 2 ] = E s 𝒮 0 , n { m , n R m n [ P m n ( α ˜ m n β ˜ m n ) + n m n ] ( α ˜ m n β ˜ m n ) 2 2 } ,
R m n = ( x m n 11 x m n 12 x m n 1 L x m n 21 x m n 22 x m n 2 L ) , P m n = ( p 1 m n 1 p 1 m n 2 p L m n 1 p L m n 2 ) .
B m n B m n H = C m n = [ E s 𝒮 0 ( | α ˜ m n | 2 ) E s 𝒮 0 ( α ˜ m n β ˜ m n * ) E s 𝒮 0 ( β ˜ m n α ˜ m n * ) E s 𝒮 0 ( | β ˜ m n | 2 ) ] , B m n = ( b m n 1 b m n 3 b m n 2 b m n 4 ) ,
m , n ( R m n P m n I ) B m n 2 2 + m , n tr [ R m n H R m n E n ( n m n H n m n ) ] .
minimize x , τ 1 , , τ L b l = 1 L A l x l 2 2 + l = 1 L σ l 2 x l 2 2 subject to τ 0 l τ l ,
A l m n = [ ( p l m n 1 b m n 1 + p l m n 2 b m n 2 ) I ( p l m n 1 b m n 3 + p l m n 2 b m n 4 ) I ] .
minimize τ 1 , , τ L l = 1 L σ l 2 x l 2 2 subject to τ 0 l τ l .
minimize x 1 , , x L b l = 1 L A l x l 2 2 + γ ( l = 1 L x l 2 ) 2 ,
α ( x , y ) = ζ cos [ ϕ ( x , y ) ] 1 , and β ( x , y ) = ζ sin [ ϕ ( x , y ) ] ,
C m n = 2 π Δ 2 [ 1 + 4 π 2 Δ 2 ( ξ m 2 + η n 2 ) ] 3 / 2 ( ζ 2 / 2 + ζ / 2 1 0 0 ζ 2 / 2 ζ / 2 ) .
ρ 2 ( a , b , ω ) = ρ 1 ( a a 0 , b b 0 , ω ) .
T ( m n , k ) = { 1 + t m n t m n if k = t m n t m n t m n if k = t m n + 1 0 otherwise .
b ^ = [ I 4 ( Q T ) ] b , A ^ l = ( I 4 Q ) [ ( I 4 T ) A l ( I 2 T ) ] ( I 2 Q ) ,
minimize v 1 , , v K k = 1 K γ k / v k subject to k = 1 K v k v 0 ,
v k = v 0 γ k / γ total , where γ total = k γ k .
l ( r ; s , θ ) = p ( r cos θ s sin θ , r sin θ + s cos θ ) .
C i j ( ρ , θ ) = E x y [ p i ( x , y ) p j ( x + ρ cos θ , y + ρ sin θ ) ] = E r s [ l i ( r ; s , θ ) l j ( r + ρ ; s , θ ) ] .
E r s [ l i ( r ; s , θ ) l j ( r + ρ ; s , θ ) ] = E ( v i v j ) exp ( ρ / Δ ) + E ( v i ) E ( v j ) [ 1 exp ( ρ / Δ ) ]
= E ( v i ) E ( v j ) + [ E ( v i v j ) E ( v i ) E ( v j ) ] exp ( ρ / Δ ) .
E ( v i ) E ( v j ) δ ( f x , f y ) + [ E ( v i v j ) E ( v i ) E ( v j ) ] 2 π Δ 2 [ 1 + 4 π 2 Δ 2 ( f x 2 + f y 2 ) ] 3 / 2 .
minimize x 1 2 Ax b 2 2 + 1 2 ( i = 1 n w i x i 2 ) 2 ,
Prox g α ( x ) = arg min y { 1 2 y x 2 2 + α 2 ( i = 1 n w i y i 2 ) 2 } ,
minimize y ( i = 1 n w i y i 2 ) 2 subject to y i x i 2 2 i 2 , i { 1 , , n } .
minimize y i = 1 n w i y i 2 subject to y i x i 2 2 i 2 , i { 1 , , n } .
minimize y i y i 2 subject to y i x i 2 i ,
minimize β 1 , , β n i = 1 n 1 2 ( β i 1 ) 2 x i 2 2 + α 2 ( i = 1 n w i β i x i 2 ) 2 subject to β i 0 , i { 1 , , n } .
minimize γ γ γ 2 ξ γ + α γ w w γ subject to γ 0 .
γ ^ = ξ ^ s w ^ , where s = ( w ^ ξ ^ ) / ( α 1 + w ^ w ^ ) .
l ( γ ^ ) = γ ^ γ ^ 2 ξ ^ γ ^ + α γ ^ w ^ w ^ γ ^ = γ ^ W γ ^ 2 ξ ^ γ ^
= W 1 / 2 γ ^ W 1 / 2 ξ ^ 2 2 + 𝒪 ( 1 ) ,
γ ^ = ( I α w ^ w ^ 1 + α w ^ w ^ ) ξ ^ ,
γ i = ξ i s w i , i 𝒫 .
ξ i s w i , i 𝒫 .
γ ^ = α w ^ w γ = ξ ^ , α w ¯ w γ ξ ¯ ,
0 = ( I + α w w ) γ ξ u ,
ξ 1 / w 1 ξ 2 / w 2 ξ n / w n ,

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