Abstract

Inverse dynamical photon scattering (IDPS), an artificial neural network based algorithm for three-dimensional quantitative imaging in optical microscopy, is introduced. Because the inverse problem entails numerical minimization of an explicit error metric, it becomes possible to freely choose a more robust metric, to introduce regularization of the solution, and to retrieve unknown experimental settings or microscope values, while the starting guess is simply set to zero. The regularization is accomplished through an alternate directions augmented Lagrangian approach, implemented on a graphics processing unit. These improvements are demonstrated on open source experimental data, retrieving three-dimensional amplitude and phase for a thick specimen.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Comparative study of fully three-dimensional reconstruction algorithms for lens-free microscopy

Anthony Berdeu, Fabien Momey, Bastien Laperrousaz, Thomas Bordy, Xavier Gidrol, Jean-Marc Dinten, Nathalie Picollet-D’hahan, and Cédric Allier
Appl. Opt. 56(13) 3939-3951 (2017)

High-speed all-optical DNA local sequence alignment based on a three-dimensional artificial neural network

Ehsan Maleki, Hossein Babashah, Somayyeh Koohi, and Zahra Kavehvash
J. Opt. Soc. Am. A 34(7) 1173-1186 (2017)

Fast maximum-likelihood image-restoration algorithms for three- dimensional fluorescence microscopy

Joanne Markham and José-Angel Conchello
J. Opt. Soc. Am. A 18(5) 1062-1071 (2001)

References

  • View by:
  • |
  • |
  • |

  1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  2. R. W. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237 (1972).
  3. J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
    [Crossref]
  4. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
    [Crossref] [PubMed]
  5. M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
    [Crossref] [PubMed]
  6. A. M. Maiden, M. J. Humphry, and J. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. 29, 1606–1614 (2012).
    [Crossref]
  7. L. Tian and L. Waller, “3d intensity and phase imaging from light field measurements in an led array microscope,” Optica 2, 104–111 (2015).
    [Crossref]
  8. J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new Theoretical approach,” Acta Cryst. 10, 609–619 (1957).
    [Crossref]
  9. J. G. Allpress, E. A. Hewat, A. F. Moodie, and J. V. Sanders, “n-Beam lattice images. I. experimental and computed images from W4Nb26O77,” Acta Cryst. A28, 528–536 (1972).
    [Crossref]
  10. D. F. Lynch and M. A. O’Keefe, “n-Beam lattice images. II. methods of calculation,” Acta Cryst. A28, 536–548 (1972).
    [Crossref]
  11. P. Goodman and A. F. Moodie, “Numerical evaluation of N-beam wave functions in electron scattering by the multislice method,” Acta Cryst. A30, 280–290 (1974).
    [Crossref]
  12. E. J. Kirkland, Advanced Computing in Electron Microscopy (Springer, 2010).
    [Crossref]
  13. J. Van Roey, J. van der Donk, and P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
    [Crossref]
  14. A. Migukin, V. Katkovnik, and J. Astola, “Advanced multi-plane phase retrieval using graphic processing unit: augmented lagrangian technique with sparse regularization,” Proc. SPIE 8429, 84291N (2012).
    [Crossref]
  15. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [Crossref] [PubMed]
  16. A. J. D’Alfonso, A. J. Morgan, A. W. C. Yan, P. Wang, H. Sawada, A. I. Kirkland, and L. J. Allen, “Deterministic electron ptychography at atomic resolution,” Phys. Rev. B 89, 064101 (2014).
    [Crossref]
  17. W. Van den Broek and C. T. Koch, “Method for retrieval of the three-dimensional object potential by inversion of dynamical electron scattering,” Phys. Rev. Lett. 109, 245502 (2012).
    [Crossref]
  18. W. Van den Broek and C. T. Koch, “General framework for quantitative three-dimensional reconstruction from arbitrary detection geometries in tem,” Phys. Rev. B 87, 184108 (2013).
    [Crossref]
  19. C. T. Koch and W. Van den Broek, “Measuring three-dimensional positions of atoms to the highest accuracy with electrons,” C. R. Phys. 15, 119–125 (2014).
    [Crossref]
  20. C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56, 507–530 (2013).
    [Crossref]
  21. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends. Mach. Learn. 3, 1–122 (2011).
    [Crossref]
  22. C. M. Bishop, Neural Networks for Pattern Recognition (Oxford University, 1995).
  23. J. Nocedal and S. Wright, Numerical Optimization (Springer, 2010).
  24. C. T. Koch, “A flux-preserving non-linear inline holography reconstruction algorithm for partially coherent electrons,” Ultramicroscopy 108, 141–150 (2008).
    [Crossref]
  25. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Springer, 1996).
    [Crossref]
  26. A. Neumaier, “Solving ill-conditioned and singular linear systems: A tutorial on regularization,” SIAM Rev. 40, 636–666 (1998).
    [Crossref]
  27. Q. Xu, A. Sawatzky, E. Roessl, M. A. Anastasio, and C. O. Schirra, “Sparsity-regularized image reconstruction of decomposed K-edge data in spectral CT,” Phys. Med. Biol. 59, N65–N79 (2014).
    [Crossref] [PubMed]
  28. E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008).
    [Crossref] [PubMed]
  29. B. Goris, W. Van den Broek, K. Batenburg, H. H. Mezerji, and S. Bals, “Electron tomography based on a total variation minimization reconstruction technique,” Ultramicroscopy 113, 120–130 (2012).
    [Crossref]
  30. Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, and M. A. Anastasio, “Investigation of discrete imaging models and iterative image reconstruction in differential x-ray phase-contrast tomography,” Opt. Express 20, 10724–10749 (2012).
    [Crossref] [PubMed]
  31. NVIDIA, CUDA Toolkit Documentation, v6.5 ed. (CUDA2014). http://docs.nvidia.com/cuda/#axzz3EKwkzsvA .
  32. L. Tian, “3D FPM on LED array microscope,” (2015). https://sites.google.com/site/leitianoptics/open-source .
  33. X. Ou, R. Horstmeyer, G. Zheng, and C. Yang, “High numerical aperture fourier ptychography: principle, implementation and characterization,” Opt. Express 23, 3472–3491 (2015).
    [Crossref] [PubMed]
  34. L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
    [Crossref]
  35. A. Migukin, M. Agour, and V. Katkovnik, “Phase retrieval in 4f optical system: background compensation and sparse regularization of object with binary amplitude,” Appl. Opt. 52, A269–A280 (2013).
    [Crossref] [PubMed]
  36. W. Van den Broek, X. Jiang, and C. T. Koch, in Proceedings of the 2015 Microscopy Conference, (DGE – German Society for Electron Microscopy e. V., 2015), pp. 467–468.
  37. M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation,” IEEE Trans. Signal Process. 54, 4311–4322 (2006).
    [Crossref]

2015 (3)

2014 (4)

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

C. T. Koch and W. Van den Broek, “Measuring three-dimensional positions of atoms to the highest accuracy with electrons,” C. R. Phys. 15, 119–125 (2014).
[Crossref]

A. J. D’Alfonso, A. J. Morgan, A. W. C. Yan, P. Wang, H. Sawada, A. I. Kirkland, and L. J. Allen, “Deterministic electron ptychography at atomic resolution,” Phys. Rev. B 89, 064101 (2014).
[Crossref]

Q. Xu, A. Sawatzky, E. Roessl, M. A. Anastasio, and C. O. Schirra, “Sparsity-regularized image reconstruction of decomposed K-edge data in spectral CT,” Phys. Med. Biol. 59, N65–N79 (2014).
[Crossref] [PubMed]

2013 (3)

C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56, 507–530 (2013).
[Crossref]

A. Migukin, M. Agour, and V. Katkovnik, “Phase retrieval in 4f optical system: background compensation and sparse regularization of object with binary amplitude,” Appl. Opt. 52, A269–A280 (2013).
[Crossref] [PubMed]

W. Van den Broek and C. T. Koch, “General framework for quantitative three-dimensional reconstruction from arbitrary detection geometries in tem,” Phys. Rev. B 87, 184108 (2013).
[Crossref]

2012 (5)

Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, and M. A. Anastasio, “Investigation of discrete imaging models and iterative image reconstruction in differential x-ray phase-contrast tomography,” Opt. Express 20, 10724–10749 (2012).
[Crossref] [PubMed]

B. Goris, W. Van den Broek, K. Batenburg, H. H. Mezerji, and S. Bals, “Electron tomography based on a total variation minimization reconstruction technique,” Ultramicroscopy 113, 120–130 (2012).
[Crossref]

A. Migukin, V. Katkovnik, and J. Astola, “Advanced multi-plane phase retrieval using graphic processing unit: augmented lagrangian technique with sparse regularization,” Proc. SPIE 8429, 84291N (2012).
[Crossref]

W. Van den Broek and C. T. Koch, “Method for retrieval of the three-dimensional object potential by inversion of dynamical electron scattering,” Phys. Rev. Lett. 109, 245502 (2012).
[Crossref]

A. M. Maiden, M. J. Humphry, and J. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. 29, 1606–1614 (2012).
[Crossref]

2011 (1)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends. Mach. Learn. 3, 1–122 (2011).
[Crossref]

2009 (1)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

2008 (2)

C. T. Koch, “A flux-preserving non-linear inline holography reconstruction algorithm for partially coherent electrons,” Ultramicroscopy 108, 141–150 (2008).
[Crossref]

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008).
[Crossref] [PubMed]

2006 (1)

M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation,” IEEE Trans. Signal Process. 54, 4311–4322 (2006).
[Crossref]

2004 (1)

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

1998 (1)

A. Neumaier, “Solving ill-conditioned and singular linear systems: A tutorial on regularization,” SIAM Rev. 40, 636–666 (1998).
[Crossref]

1993 (1)

1982 (1)

1981 (1)

1974 (1)

P. Goodman and A. F. Moodie, “Numerical evaluation of N-beam wave functions in electron scattering by the multislice method,” Acta Cryst. A30, 280–290 (1974).
[Crossref]

1972 (3)

J. G. Allpress, E. A. Hewat, A. F. Moodie, and J. V. Sanders, “n-Beam lattice images. I. experimental and computed images from W4Nb26O77,” Acta Cryst. A28, 528–536 (1972).
[Crossref]

D. F. Lynch and M. A. O’Keefe, “n-Beam lattice images. II. methods of calculation,” Acta Cryst. A28, 536–548 (1972).
[Crossref]

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

1957 (1)

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new Theoretical approach,” Acta Cryst. 10, 609–619 (1957).
[Crossref]

Agour, M.

Aharon, M.

M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation,” IEEE Trans. Signal Process. 54, 4311–4322 (2006).
[Crossref]

Allen, L. J.

A. J. D’Alfonso, A. J. Morgan, A. W. C. Yan, P. Wang, H. Sawada, A. I. Kirkland, and L. J. Allen, “Deterministic electron ptychography at atomic resolution,” Phys. Rev. B 89, 064101 (2014).
[Crossref]

Allpress, J. G.

J. G. Allpress, E. A. Hewat, A. F. Moodie, and J. V. Sanders, “n-Beam lattice images. I. experimental and computed images from W4Nb26O77,” Acta Cryst. A28, 528–536 (1972).
[Crossref]

Anastasio, M. A.

Q. Xu, A. Sawatzky, E. Roessl, M. A. Anastasio, and C. O. Schirra, “Sparsity-regularized image reconstruction of decomposed K-edge data in spectral CT,” Phys. Med. Biol. 59, N65–N79 (2014).
[Crossref] [PubMed]

Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, and M. A. Anastasio, “Investigation of discrete imaging models and iterative image reconstruction in differential x-ray phase-contrast tomography,” Opt. Express 20, 10724–10749 (2012).
[Crossref] [PubMed]

Astola, J.

A. Migukin, V. Katkovnik, and J. Astola, “Advanced multi-plane phase retrieval using graphic processing unit: augmented lagrangian technique with sparse regularization,” Proc. SPIE 8429, 84291N (2012).
[Crossref]

Bals, S.

B. Goris, W. Van den Broek, K. Batenburg, H. H. Mezerji, and S. Bals, “Electron tomography based on a total variation minimization reconstruction technique,” Ultramicroscopy 113, 120–130 (2012).
[Crossref]

Batenburg, K.

B. Goris, W. Van den Broek, K. Batenburg, H. H. Mezerji, and S. Bals, “Electron tomography based on a total variation minimization reconstruction technique,” Ultramicroscopy 113, 120–130 (2012).
[Crossref]

Bishop, C. M.

C. M. Bishop, Neural Networks for Pattern Recognition (Oxford University, 1995).

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends. Mach. Learn. 3, 1–122 (2011).
[Crossref]

Bruckstein, A.

M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation,” IEEE Trans. Signal Process. 54, 4311–4322 (2006).
[Crossref]

Bunk, O.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

Chen, M.

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends. Mach. Learn. 3, 1–122 (2011).
[Crossref]

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new Theoretical approach,” Acta Cryst. 10, 609–619 (1957).
[Crossref]

D’Alfonso, A. J.

A. J. D’Alfonso, A. J. Morgan, A. W. C. Yan, P. Wang, H. Sawada, A. I. Kirkland, and L. J. Allen, “Deterministic electron ptychography at atomic resolution,” Phys. Rev. B 89, 064101 (2014).
[Crossref]

Diaz, A.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

Dong, J.

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends. Mach. Learn. 3, 1–122 (2011).
[Crossref]

Elad, M.

M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation,” IEEE Trans. Signal Process. 54, 4311–4322 (2006).
[Crossref]

Engl, H. W.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Springer, 1996).
[Crossref]

Färm, E.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

Faulkner, H. M.

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

Fienup, J. R.

Gerchberg, R. W.

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

Goodman, P.

P. Goodman and A. F. Moodie, “Numerical evaluation of N-beam wave functions in electron scattering by the multislice method,” Acta Cryst. A30, 280–290 (1974).
[Crossref]

Goris, B.

B. Goris, W. Van den Broek, K. Batenburg, H. H. Mezerji, and S. Bals, “Electron tomography based on a total variation minimization reconstruction technique,” Ultramicroscopy 113, 120–130 (2012).
[Crossref]

Guizar-Sicairos, M.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

Hanke, M.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Springer, 1996).
[Crossref]

Härkönen, E.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

Hewat, E. A.

J. G. Allpress, E. A. Hewat, A. F. Moodie, and J. V. Sanders, “n-Beam lattice images. I. experimental and computed images from W4Nb26O77,” Acta Cryst. A28, 528–536 (1972).
[Crossref]

Holler, M.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

Horstmeyer, R.

Humphry, M. J.

A. M. Maiden, M. J. Humphry, and J. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. 29, 1606–1614 (2012).
[Crossref]

Jiang, H.

C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56, 507–530 (2013).
[Crossref]

Jiang, X.

W. Van den Broek, X. Jiang, and C. T. Koch, in Proceedings of the 2015 Microscopy Conference, (DGE – German Society for Electron Microscopy e. V., 2015), pp. 467–468.

Karvinen, P.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

Katkovnik, V.

A. Migukin, M. Agour, and V. Katkovnik, “Phase retrieval in 4f optical system: background compensation and sparse regularization of object with binary amplitude,” Appl. Opt. 52, A269–A280 (2013).
[Crossref] [PubMed]

A. Migukin, V. Katkovnik, and J. Astola, “Advanced multi-plane phase retrieval using graphic processing unit: augmented lagrangian technique with sparse regularization,” Proc. SPIE 8429, 84291N (2012).
[Crossref]

Kirkland, A. I.

A. J. D’Alfonso, A. J. Morgan, A. W. C. Yan, P. Wang, H. Sawada, A. I. Kirkland, and L. J. Allen, “Deterministic electron ptychography at atomic resolution,” Phys. Rev. B 89, 064101 (2014).
[Crossref]

Kirkland, E. J.

E. J. Kirkland, Advanced Computing in Electron Microscopy (Springer, 2010).
[Crossref]

Koch, C. T.

C. T. Koch and W. Van den Broek, “Measuring three-dimensional positions of atoms to the highest accuracy with electrons,” C. R. Phys. 15, 119–125 (2014).
[Crossref]

W. Van den Broek and C. T. Koch, “General framework for quantitative three-dimensional reconstruction from arbitrary detection geometries in tem,” Phys. Rev. B 87, 184108 (2013).
[Crossref]

W. Van den Broek and C. T. Koch, “Method for retrieval of the three-dimensional object potential by inversion of dynamical electron scattering,” Phys. Rev. Lett. 109, 245502 (2012).
[Crossref]

C. T. Koch, “A flux-preserving non-linear inline holography reconstruction algorithm for partially coherent electrons,” Ultramicroscopy 108, 141–150 (2008).
[Crossref]

W. Van den Broek, X. Jiang, and C. T. Koch, in Proceedings of the 2015 Microscopy Conference, (DGE – German Society for Electron Microscopy e. V., 2015), pp. 467–468.

Lagasse, P. E.

Li, C.

C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56, 507–530 (2013).
[Crossref]

Lynch, D. F.

D. F. Lynch and M. A. O’Keefe, “n-Beam lattice images. II. methods of calculation,” Acta Cryst. A28, 536–548 (1972).
[Crossref]

Maiden, A. M.

A. M. Maiden, M. J. Humphry, and J. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. 29, 1606–1614 (2012).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

Menzel, A.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

Mezerji, H. H.

B. Goris, W. Van den Broek, K. Batenburg, H. H. Mezerji, and S. Bals, “Electron tomography based on a total variation minimization reconstruction technique,” Ultramicroscopy 113, 120–130 (2012).
[Crossref]

Migukin, A.

A. Migukin, M. Agour, and V. Katkovnik, “Phase retrieval in 4f optical system: background compensation and sparse regularization of object with binary amplitude,” Appl. Opt. 52, A269–A280 (2013).
[Crossref] [PubMed]

A. Migukin, V. Katkovnik, and J. Astola, “Advanced multi-plane phase retrieval using graphic processing unit: augmented lagrangian technique with sparse regularization,” Proc. SPIE 8429, 84291N (2012).
[Crossref]

Modregger, P.

Moodie, A. F.

P. Goodman and A. F. Moodie, “Numerical evaluation of N-beam wave functions in electron scattering by the multislice method,” Acta Cryst. A30, 280–290 (1974).
[Crossref]

J. G. Allpress, E. A. Hewat, A. F. Moodie, and J. V. Sanders, “n-Beam lattice images. I. experimental and computed images from W4Nb26O77,” Acta Cryst. A28, 528–536 (1972).
[Crossref]

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new Theoretical approach,” Acta Cryst. 10, 609–619 (1957).
[Crossref]

Morgan, A. J.

A. J. D’Alfonso, A. J. Morgan, A. W. C. Yan, P. Wang, H. Sawada, A. I. Kirkland, and L. J. Allen, “Deterministic electron ptychography at atomic resolution,” Phys. Rev. B 89, 064101 (2014).
[Crossref]

Neubauer, A.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Springer, 1996).
[Crossref]

Neumaier, A.

A. Neumaier, “Solving ill-conditioned and singular linear systems: A tutorial on regularization,” SIAM Rev. 40, 636–666 (1998).
[Crossref]

Nocedal, J.

J. Nocedal and S. Wright, Numerical Optimization (Springer, 2010).

O’Keefe, M. A.

D. F. Lynch and M. A. O’Keefe, “n-Beam lattice images. II. methods of calculation,” Acta Cryst. A28, 536–548 (1972).
[Crossref]

Ou, X.

Pan, X.

Parikh, N.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends. Mach. Learn. 3, 1–122 (2011).
[Crossref]

Peleato, B.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends. Mach. Learn. 3, 1–122 (2011).
[Crossref]

Raabe, J.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

Ritala, M.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

Rodenburg, J.

A. M. Maiden, M. J. Humphry, and J. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. 29, 1606–1614 (2012).
[Crossref]

Rodenburg, J. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

Roessl, E.

Q. Xu, A. Sawatzky, E. Roessl, M. A. Anastasio, and C. O. Schirra, “Sparsity-regularized image reconstruction of decomposed K-edge data in spectral CT,” Phys. Med. Biol. 59, N65–N79 (2014).
[Crossref] [PubMed]

Sanders, J. V.

J. G. Allpress, E. A. Hewat, A. F. Moodie, and J. V. Sanders, “n-Beam lattice images. I. experimental and computed images from W4Nb26O77,” Acta Cryst. A28, 528–536 (1972).
[Crossref]

Sawada, H.

A. J. D’Alfonso, A. J. Morgan, A. W. C. Yan, P. Wang, H. Sawada, A. I. Kirkland, and L. J. Allen, “Deterministic electron ptychography at atomic resolution,” Phys. Rev. B 89, 064101 (2014).
[Crossref]

Sawatzky, A.

Q. Xu, A. Sawatzky, E. Roessl, M. A. Anastasio, and C. O. Schirra, “Sparsity-regularized image reconstruction of decomposed K-edge data in spectral CT,” Phys. Med. Biol. 59, N65–N79 (2014).
[Crossref] [PubMed]

Schirra, C. O.

Q. Xu, A. Sawatzky, E. Roessl, M. A. Anastasio, and C. O. Schirra, “Sparsity-regularized image reconstruction of decomposed K-edge data in spectral CT,” Phys. Med. Biol. 59, N65–N79 (2014).
[Crossref] [PubMed]

Sidky, E. Y.

Soltanolkotabi, M.

Stampanoni, M.

Tang, G.

Tian, L.

Van den Broek, W.

C. T. Koch and W. Van den Broek, “Measuring three-dimensional positions of atoms to the highest accuracy with electrons,” C. R. Phys. 15, 119–125 (2014).
[Crossref]

W. Van den Broek and C. T. Koch, “General framework for quantitative three-dimensional reconstruction from arbitrary detection geometries in tem,” Phys. Rev. B 87, 184108 (2013).
[Crossref]

B. Goris, W. Van den Broek, K. Batenburg, H. H. Mezerji, and S. Bals, “Electron tomography based on a total variation minimization reconstruction technique,” Ultramicroscopy 113, 120–130 (2012).
[Crossref]

W. Van den Broek and C. T. Koch, “Method for retrieval of the three-dimensional object potential by inversion of dynamical electron scattering,” Phys. Rev. Lett. 109, 245502 (2012).
[Crossref]

W. Van den Broek, X. Jiang, and C. T. Koch, in Proceedings of the 2015 Microscopy Conference, (DGE – German Society for Electron Microscopy e. V., 2015), pp. 467–468.

van der Donk, J.

Van Roey, J.

Waller, L.

Wang, P.

A. J. D’Alfonso, A. J. Morgan, A. W. C. Yan, P. Wang, H. Sawada, A. I. Kirkland, and L. J. Allen, “Deterministic electron ptychography at atomic resolution,” Phys. Rev. B 89, 064101 (2014).
[Crossref]

Wright, S.

J. Nocedal and S. Wright, Numerical Optimization (Springer, 2010).

Xu, Q.

Q. Xu, A. Sawatzky, E. Roessl, M. A. Anastasio, and C. O. Schirra, “Sparsity-regularized image reconstruction of decomposed K-edge data in spectral CT,” Phys. Med. Biol. 59, N65–N79 (2014).
[Crossref] [PubMed]

Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, and M. A. Anastasio, “Investigation of discrete imaging models and iterative image reconstruction in differential x-ray phase-contrast tomography,” Opt. Express 20, 10724–10749 (2012).
[Crossref] [PubMed]

Yan, A. W. C.

A. J. D’Alfonso, A. J. Morgan, A. W. C. Yan, P. Wang, H. Sawada, A. I. Kirkland, and L. J. Allen, “Deterministic electron ptychography at atomic resolution,” Phys. Rev. B 89, 064101 (2014).
[Crossref]

Yang, C.

Yeh, L.-H.

Yin, W.

C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56, 507–530 (2013).
[Crossref]

Zhang, Y.

C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56, 507–530 (2013).
[Crossref]

Zheng, G.

Zhong, J.

Acta Cryst. (4)

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new Theoretical approach,” Acta Cryst. 10, 609–619 (1957).
[Crossref]

J. G. Allpress, E. A. Hewat, A. F. Moodie, and J. V. Sanders, “n-Beam lattice images. I. experimental and computed images from W4Nb26O77,” Acta Cryst. A28, 528–536 (1972).
[Crossref]

D. F. Lynch and M. A. O’Keefe, “n-Beam lattice images. II. methods of calculation,” Acta Cryst. A28, 536–548 (1972).
[Crossref]

P. Goodman and A. F. Moodie, “Numerical evaluation of N-beam wave functions in electron scattering by the multislice method,” Acta Cryst. A30, 280–290 (1974).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

C. R. Phys. (1)

C. T. Koch and W. Van den Broek, “Measuring three-dimensional positions of atoms to the highest accuracy with electrons,” C. R. Phys. 15, 119–125 (2014).
[Crossref]

Comput. Optim. Appl. (1)

C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56, 507–530 (2013).
[Crossref]

Found. Trends. Mach. Learn. (1)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends. Mach. Learn. 3, 1–122 (2011).
[Crossref]

IEEE Trans. Signal Process. (1)

M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation,” IEEE Trans. Signal Process. 54, 4311–4322 (2006).
[Crossref]

J. Opt. Soc. Am. (2)

J. Van Roey, J. van der Donk, and P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
[Crossref]

A. M. Maiden, M. J. Humphry, and J. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. 29, 1606–1614 (2012).
[Crossref]

Opt. Express (3)

Optica (1)

Optik (1)

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

Phys. Med. Biol. (2)

Q. Xu, A. Sawatzky, E. Roessl, M. A. Anastasio, and C. O. Schirra, “Sparsity-regularized image reconstruction of decomposed K-edge data in spectral CT,” Phys. Med. Biol. 59, N65–N79 (2014).
[Crossref] [PubMed]

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008).
[Crossref] [PubMed]

Phys. Rev. B (2)

W. Van den Broek and C. T. Koch, “General framework for quantitative three-dimensional reconstruction from arbitrary detection geometries in tem,” Phys. Rev. B 87, 184108 (2013).
[Crossref]

A. J. D’Alfonso, A. J. Morgan, A. W. C. Yan, P. Wang, H. Sawada, A. I. Kirkland, and L. J. Allen, “Deterministic electron ptychography at atomic resolution,” Phys. Rev. B 89, 064101 (2014).
[Crossref]

Phys. Rev. Lett. (1)

W. Van den Broek and C. T. Koch, “Method for retrieval of the three-dimensional object potential by inversion of dynamical electron scattering,” Phys. Rev. Lett. 109, 245502 (2012).
[Crossref]

Proc. SPIE (1)

A. Migukin, V. Katkovnik, and J. Astola, “Advanced multi-plane phase retrieval using graphic processing unit: augmented lagrangian technique with sparse regularization,” Proc. SPIE 8429, 84291N (2012).
[Crossref]

Sci. Rep. (1)

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014).
[Crossref] [PubMed]

SIAM Rev. (1)

A. Neumaier, “Solving ill-conditioned and singular linear systems: A tutorial on regularization,” SIAM Rev. 40, 636–666 (1998).
[Crossref]

Ultramicroscopy (3)

C. T. Koch, “A flux-preserving non-linear inline holography reconstruction algorithm for partially coherent electrons,” Ultramicroscopy 108, 141–150 (2008).
[Crossref]

B. Goris, W. Van den Broek, K. Batenburg, H. H. Mezerji, and S. Bals, “Electron tomography based on a total variation minimization reconstruction technique,” Ultramicroscopy 113, 120–130 (2012).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

Other (7)

E. J. Kirkland, Advanced Computing in Electron Microscopy (Springer, 2010).
[Crossref]

C. M. Bishop, Neural Networks for Pattern Recognition (Oxford University, 1995).

J. Nocedal and S. Wright, Numerical Optimization (Springer, 2010).

NVIDIA, CUDA Toolkit Documentation, v6.5 ed. (CUDA2014). http://docs.nvidia.com/cuda/#axzz3EKwkzsvA .

L. Tian, “3D FPM on LED array microscope,” (2015). https://sites.google.com/site/leitianoptics/open-source .

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Springer, 1996).
[Crossref]

W. Van den Broek, X. Jiang, and C. T. Koch, in Proceedings of the 2015 Microscopy Conference, (DGE – German Society for Electron Microscopy e. V., 2015), pp. 467–468.

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 (10)

Fig. 1
Fig. 1

One step of the multislice algorithm. The incoming wave ψj in layer j is multiplied with the transmission function tj that is a function of Vj: either the projected potential or the difference of the refraction index with the medium. Next, Huygens-waves emanate from all points (for clarity, only a few are drawn) and propagate to the next slice where they add coherently to produce ψj+1; this is computed as a convolution of ψjtj with the Fresnel propagator p.

Fig. 2
Fig. 2

The multislice algorithm as an ANN. The incoming wave ψ1 is multiplied with the transmission function t1. The next layer of edges and nodes encodes a real-space convolution with the Fresnel propagator p, resulting in ψ2. This repeats until the exit wave ψN+1 is reached. Then, a real space convolution with LF produces ψN+2, and the intensity I is computed. The last layers compute E from the measurements J.

Fig. 3
Fig. 3

One image selected from the open source dataset [32], with two stacked USAF resolution test charts as specimen. The magnified area in the red box shows the region of interest (ROI) that is reconstructed in Sec. 3.

Fig. 4
Fig. 4

Reconstructed amplitudes, exp(−w), of both slices with the SSD error metric and without regularization. A resolution of 2.76 μm is achieved.

Fig. 5
Fig. 5

Plot of R-factor and error metrics E, without regularization. (a) Case 1: Error metric set to SSD, no nuisance parameter optimization. (b) Error metric set to SNAD. Case 2: Solid line, no nuisance parameter optimization. Case 3: dashed line, C1 and α optimization.

Fig. 6
Fig. 6

Reconstructed amplitudes, exp(−w), of both slices with the SNAD error metric and without regularization. A resolution of 1.38 μm is achieved.

Fig. 7
Fig. 7

Reconstructed amplitudes, exp(−w), of both slices with the SNAD error metric, without regularization and with estimation of the nuisance parameters C1 and α. A resolution of 1.38 μm is achieved. Note how the artifacts in the upper left corner of the first slice are considerably reduced.

Fig. 8
Fig. 8

Reconstructed amplitudes, exp(−w), the ADAL method. A resolution of 1.38 μm is achieved.

Fig. 9
Fig. 9

Plot of the R-factor and error metric E for Cases 3 (dashed line) and 4 (solid line).

Fig. 10
Fig. 10

The reconstructed phase, v, at each slice, with the ADAL method.

Tables (1)

Tables Icon

Table 1 Summary of the results. From left to right, the columns respectively list: the optimization scenario as listed on p. 8; whether regularization was applied; the error metric E; the optimization algorithm; the simultaneous optimization of C1 and α; and finally the reconstruction’s resolution, R-factor, SSD and SNAD. *: C1 value taken from 3.

Equations (20)

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

ψ j + 1 = p ( ψ j t j ) , with t j = exp ( i V j ) ,
ψ N + 2 = LF ψ N + 1 , I = | ψ N + 2 | 2 ,
E = m k f ( I m k , J m k ) ,
f ( I m k , J m k ) = 1 2 ( I m k J m k ) 2 ,
E V j k = 2 Im ( ψ j k t j k E ψ j k t j k ) ,
h = 1 [ exp ( ( π α C 1 ν ) 2 ) ] ,
E α = m k E [ I m h ] k [ I m h ] k α ,
= m k E [ I m h ] k [ I m h α ] k , with ,
h α = 2 π 2 C 1 2 α 1 [ ν 2 exp ( ( π α C 1 ν ) 2 ) ] .
arg min V μ i D i V 1 + E ( V )
arg min U , V μ i U i 1 + E ( V ) , s . t . D i V = U i i
A ( V , U , υ ) = μ i ( U i 1 υ i T ( D i V U i ) + β 2 D i V U i 2 2 ) + E
A V = μ [ D T υ + β D T ( D V U ) ] + E V
U i ( + 1 ) = max ( | D i V ( ) υ i ( ) β ( ) | 1 β ( ) , 0 ) sgn ( D i V ( ) υ i ( ) β ( ) ) ,
V ( + 1 ) = V ( ) α ( + 1 ) A V ( V ( ) , U ( + 1 ) , υ ( ) ) ,
υ i ( + 1 ) = υ i ( ) β ( ) ( D i V ( + 1 ) U i ( + 1 ) ) .
β ( + 1 ) = 1.01 β ( ) , with β ( 0 ) 1 / 2 .
R = i | I i J i | i J i .
J J 1 2 ( J 0.25 ( J 0.25 ) 2 + 0.0375 2 + 1 ) ,
f ( I i , J i ) = ( I i J i ) 2 + ε 1 2 J i + ε 2

Metrics