Abstract

Ptychography is a form of phase imaging that uses iterative algorithms to reconstruct an image of a specimen from a series of diffraction patterns. It is swiftly developing into a mainstream technique, with a growing list of applications across a range of imaging modalities. As the field has advanced, numerous reconstruction algorithms have been proposed, yet the early approaches have not seen major improvement and remain popular. In this paper, we revisit the first such algorithm, the ptychographical iterative engine (PIE), and show how a simple revision and powerful extension can deliver an order of magnitude speed increase and handle difficult data sets where the original version fails completely.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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2017 (3)

M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref]

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

F. Seiboth, A. Schropp, M. Scholz, F. Wittwer, C. Rödel, M. Wünsche, T. Ullsperger, S. Nolte, J. Rahomäki, K. Parfeniukas, S. Giakoumidis, U. Vogt, U. Wagner, C. Rau, U. Boesenberg, J. Garrevoet, G. Falkenberg, E. C. Galtier, H. Ja Lee, B. Nagler, and C. G. Schroer, “Perfect X-ray focusing via fitting corrective glasses to aberrated optics,” Nat. Commun. 8, 14623 (2017).
[Crossref]

2016 (5)

2015 (5)

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457 (2015).
[Crossref]

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref]

A. M. Maiden, M. C. Sarahan, M. D. Stagg, S. M. Schramm, and M. J. Humphry, “Quantitative electron phase imaging with high sensitivity and an unlimited field of view,” Sci. Rep. 5, 14690 (2015).
[Crossref]

L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Wirtinger flow optimization,” Opt. Express 23, 4856–4866 (2015).
[Crossref]

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]

2014 (4)

2013 (7)

S. Wang, D. Shapiro, and K. Kaznatcheev, “X-ray ptychography with highly-curved wavefront,” J. Phys. 463, 012040 (2013).
[Crossref]

A. Maiden, G. Morrison, B. Kaulich, A. Gianoncelli, and J. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
[Crossref]

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

J. Marrison, L. Raty, P. Marriott, and P. O’Toole, “Ptychography—a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013).
[Crossref]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref]

F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21, 13592–13606 (2013).
[Crossref]

2012 (4)

A. Maiden, M. Humphry, M. Sarahan, B. Kraus, and J. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[Crossref]

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[Crossref]

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

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
[Crossref]

2011 (1)

2009 (1)

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

2008 (2)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
[Crossref]

2007 (2)

J. Rodenburg, A. Hurst, and A. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy 107, 227–231 (2007).
[Crossref]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

2004 (1)

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

Adams, D. E.

Aeppli, G.

M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref]

Allain, M.

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
[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]

Ames, B.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref]

Avnat, Z.

Batey, D.

Bean, R.

Berenguer, F.

Bertsekas, D. P.

D. P. Bertsekas, “Incremental gradient, subgradient, and proximal methods for convex optimization: a survey,” in Optimization for Machine Learning (2011), Vol. 3.

Bian, L.

Boesenberg, U.

F. Seiboth, A. Schropp, M. Scholz, F. Wittwer, C. Rödel, M. Wünsche, T. Ullsperger, S. Nolte, J. Rahomäki, K. Parfeniukas, S. Giakoumidis, U. Vogt, U. Wagner, C. Rau, U. Boesenberg, J. Garrevoet, G. Falkenberg, E. C. Galtier, H. Ja Lee, B. Nagler, and C. G. Schroer, “Perfect X-ray focusing via fitting corrective glasses to aberrated optics,” Nat. Commun. 8, 14623 (2017).
[Crossref]

Boyd, S.

N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim. 1, 127–239 (2014).

Bunk, O.

M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

Chamard, V.

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
[Crossref]

Chen, B.

Chen, F.

Chen, M.

Chen, Q.

Chen, R. Y.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref]

Cloetens, P.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref]

Cohen, O.

Combettes, P. L.

P. L. Combettes and J.-C. Pesquet, “Proximal splitting methods in signal processing,” in Fixed-Point Algorithms for Inverse Problems in Science and Engineering (Springer, 2011), pp. 185–212.

Cullis, A.

J. Rodenburg, A. Hurst, and A. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy 107, 227–231 (2007).
[Crossref]

Cullis, A. G.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[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]

Dahl, G. E.

I. Sutskever, J. Martens, G. E. Dahl, and G. E. Hinton, “On the importance of initialization and momentum in deep learning,” in Proceedings of the 30th International Conference on International Conference on Machine Learning (2013), Vol. 28, pp. 1139–1147.

Dai, Q.

Daurer, B. J.

S. Marchesini, H. Krishnan, B. J. Daurer, D. A. Shapiro, T. Perciano, J. A. Sethian, and F. R. Maia, “SHARP: a distributed GPU-based ptychographic solver,” J. Appl. Crystallogr. 49, 1245–1252 (2016).
[Crossref]

David, C.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

Diaz, A.

Dierolf, M.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

Dinapoli, R.

M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref]

Dobson, B. R.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

Dong, J.

Edo, T.

Enders, B.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref]

Falkenberg, G.

F. Seiboth, A. Schropp, M. Scholz, F. Wittwer, C. Rödel, M. Wünsche, T. Ullsperger, S. Nolte, J. Rahomäki, K. Parfeniukas, S. Giakoumidis, U. Vogt, U. Wagner, C. Rau, U. Boesenberg, J. Garrevoet, G. Falkenberg, E. C. Galtier, H. Ja Lee, B. Nagler, and C. G. Schroer, “Perfect X-ray focusing via fitting corrective glasses to aberrated optics,” Nat. Commun. 8, 14623 (2017).
[Crossref]

Faulkner, H. M. L.

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

Fienup, J. R.

Fuoss, P. H.

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

Galtier, E. C.

F. Seiboth, A. Schropp, M. Scholz, F. Wittwer, C. Rödel, M. Wünsche, T. Ullsperger, S. Nolte, J. Rahomäki, K. Parfeniukas, S. Giakoumidis, U. Vogt, U. Wagner, C. Rau, U. Boesenberg, J. Garrevoet, G. Falkenberg, E. C. Galtier, H. Ja Lee, B. Nagler, and C. G. Schroer, “Perfect X-ray focusing via fitting corrective glasses to aberrated optics,” Nat. Commun. 8, 14623 (2017).
[Crossref]

Gardner, D. F.

Garrevoet, J.

F. Seiboth, A. Schropp, M. Scholz, F. Wittwer, C. Rödel, M. Wünsche, T. Ullsperger, S. Nolte, J. Rahomäki, K. Parfeniukas, S. Giakoumidis, U. Vogt, U. Wagner, C. Rau, U. Boesenberg, J. Garrevoet, G. Falkenberg, E. C. Galtier, H. Ja Lee, B. Nagler, and C. G. Schroer, “Perfect X-ray focusing via fitting corrective glasses to aberrated optics,” Nat. Commun. 8, 14623 (2017).
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Supplementary Material (2)

NameDescription
» Supplement 1: PDF (1870 KB)      A description of the error metric used in the primary manuscript, and details of two minor improvements to the algorithm.
» Visualization 1: AVI (46713 KB)      A visualisation showing convergence of a single pixel from a ptychographic reconstruction using different algorithms.

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Figures (8)

Fig. 1.
Fig. 1. Flowchart showing the operation of the PIE family of algorithms. One iteration of each algorithm consists of J loops through this process, so that each diffraction pattern is used, in the sequence sj, to update the object and probe estimates.
Fig. 2.
Fig. 2. Operation of the various object update functions. (a) Shows how the different algorithms weight the previous and new object estimates as a function of probe modulus [cf. Eq. (9)]. (b) Shows how they penalize changes to the object as a function of probe modulus [cf. Eq. (13)].
Fig. 3.
Fig. 3. Addition of momentum to the object update. (a) Shows how momentum is implemented within the flowchart of Fig. 1. (b) Visualizes two ways in which the momentum term can be added to the object estimate for the case T=5: one is the conventional momentum scheme and the second is due to Nesterov.
Fig. 4.
Fig. 4. Simulation setup. The specimen (a) modulus and (b) phase used in the two simulations; these template images were scaled as listed in Table 2. The red square indicates the field of view and the red circle indicates the size of the probes. (c) Shows the highly diffuse probe used in the optical simulation, and (d) shows the convergent beam probe used in the x-ray simulation. Both are displayed on the colorwheel scale shown in the inset to (c), which is also used in probe images of later figures. (e) and (f) give example diffraction patterns from the optical and x-ray simulations, respectively (contrast boosted).
Fig. 5.
Fig. 5. Simulation results for the case of a strongly scattering specimen and a highly structured probe. (a) Convergence over 200 iterations of the ePIE, PIE, and rPIE algorithms. The solid traces indicate the median reconstruction from ten trials of each algorithm using different diffraction pattern orders, and the shaded regions indicate the range of convergence results over these ten trials. (b)–(d) The final probe (top) and specimen modulus (middle) and phase (bottom) for ePIE, PIE, and rPIE, respectively. Probes are shown on the colorwheel scale of Fig. 4.
Fig. 6.
Fig. 6. Including momentum significantly improves performance. (a) Tuning the momentum part of the update does require some effort. However, (b) shows how effective it can be for difficult data sets, in this case, an increasingly strong phase object and a highly diffuse probe. Solid traces indicate the median final error over ten trials of each algorithm, and shading indicates the range of final errors over these trials.
Fig. 7.
Fig. 7. Performance of the different algorithms in a simulated x-ray experiment. (a) Their convergence over 300 iterations—solid traces are the median and shading is the range of results from ten trials of each algorithm; (b)–(e) final probe (top row) and specimen modulus (middle) and phase (bottom) images of ePIE, PIE, rPIE, and mPIE, respectively. Probes are shown on the colorwheel scale of Fig. 4.
Fig. 8.
Fig. 8. Diffuse probe and an intricate specimen (lily pollen in this optical experiment) make for a difficult ptychographic reconstruction. (a) Progress of the diffraction error [Eq. (23)] over 100 iterations of four ptychographic algorithms. Ten runs of each algorithm were carried out with different position orders. The median run is shown as a solid trace, while all other outcomes were within the correspondingly colored boundaries. ePIE and PIE give the phase images shown in (b) and (c), respectively. rPIE give reasonable results in most cases, and mPIE converges reliably within 25 iterations. Both give excellent reconstructions, such as that shown in (d), where the phase has been unwrapped. Insets to the images show the reconstructed probe displayed using the colorwheel of Fig. 4. Scale bar 200 μm.

Tables (3)

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Table 1. Three Different Interpretations of the Update Functions in PIE-Type Ptychographic Algorithms

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Table 2. Parameters Used in Simulations and Experiment

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Table 3. Suggested Parameter Ranges for mPIE

Equations (23)

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

ojr=Ojxj,
xj=Rs(j)Rj(min)Δr+r.
Δr=(λzMΔc,λzNΔc),
Δr=(Δc,Δc).
ψjr=Pjrojr,
Ojxj=ojr.
ojr=ojr+αPjr*|Pjr|max2(ψjrψjr),
ojr=ψjrPjr=Pjr*ψjr|Pjr|2.
ojr=(1wjr)ojr+wjrPjr*ψjr|Pjr|2=ojr+wjrPjr*(ψjrψjr)|Pjr|2,
Ej(obj)=r|Pjrojrψjr|2.
Ejr(obj)=2Pjr*(Pjrojrψjr).
ojr=ojrγ2Ejr(obj)=ojr+γPjr*(ψjrψjr).
Ej(reg)=r|Pjrorψjr|2+rur|oror|2.
Ejr(reg)=2Pjr*(Pjrojrψjr)+2ujr(ojrojr).
ojr=ojr+Pjr*(ψjrψjr)|Pjr|2+ujr.
wjr=γjr|Pjr|2=|Pjr|2|Pjr|2+ujr.
ujr=|Pjr|max2α|Pjr|2.
ojr=ojr+Pjr*(ψjrψjr)(1α)|Pjr|2+α|Pjr|max2.
vjx=ηobjv(jT)x+(OjxO(j+1T)x),
O(j+1)x=O(j+1T)x+vjx.
O(j+1)x=Ojx+ηobjvjx.
ojr=ojr+γobjPjr*(ψjrψjr)(1α)|Pjr|2+α|Pjr|max2.
Ediff=u|Isju|I[ψjr]|2|2uIsju2,

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