Abstract

Ptychographic imaging has gained popularity for its high resolving power and sensitivity as well as for its ability to map simultaneously the sample’s complex-valued refractive index and the illumination. Yet, despite significant progress that allows for reliable practical implementation, some of the technique’s fundamentals remain poorly understood, and oftentimes successful data acquisition is either overly conservative or relies more on experimenters experience than on rational data acquisition strategies. Here, we propose a theoretical framework of ptychography, which is based on Gabor’s notion of decomposition into elementary signals and the concept of frames. We demonstrate how this framework can straightforwardly be used to derive sampling requirements or to provide arguments on how to optimize the ptychographic scan. More generally, our theoretical framework can serve as a bridge between the experimental technique and the rich and well established mathematical disciplines of wavelets decomposition and spectrogram analysis.

© 2015 Optical Society of America

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    [Crossref]
  37. M. J. Bastiaans, “On optimum oversampling in the gabor scheme,” IEEE International Conference on Acoustics, Speech, and Signal Processing, 1997. ICASSP-97, (IEEE 1997), 3, pp 2101–2104 (1997).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]

2015 (1)

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] [PubMed]

2014 (4)

P. Li, T. B. Edo, and J. M. Rodenburg, “Ptychographic inversion via Wigner distribution deconvolution noise suppression and probe design,” Ultramicroscopy 147, 106–113 (2014).
[Crossref] [PubMed]

D. J. Batey, T. B. Edo, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Reciprocal-space up-sampling from real-space oversampling in x-ray ptychography,” Phys. Rev. A 89, 043812 (2014).
[Crossref]

R. Horstmeyer and C. Yang, “A phase space model of Fourier ptychographic microscopy,” Opt. Express 22, 338–358 (2014).
[Crossref] [PubMed]

X. Huang, H. Yan, R. Harder, Y. Hwu, I. K. Robinson, and Y. S. Chu, “Optimization of overlap uniformness for ptychography,” Opt. Express 22, 12634–12644 (2014).
[Crossref] [PubMed]

2013 (5)

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

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

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[Crossref]

S. Marchesini, A. Schirotzed, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

E. Şahin and L. Onural, “Calculation of the scalar diffraction field from curved surfaces by decomposing the three-dimensional field into a sum of gaussian beams,” J. Opt. Soc. Am. A 30(3), 527–536 (2013).
[Crossref]

2012 (4)

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

E. Şahin and L. Onural, “Scalar diffraction field calculation from curved surfaces via gaussian beam decomposition,” J. Opt. Soc. Am. A 29(7), 1459–1469 (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(8), 1606–1614 (2012).
[Crossref]

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, and A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86(10),100103 (2012).
[Crossref]

2009 (2)

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

2008 (3)

M. Guizar-Sicairos and J.R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16(10), 7264–7278 (2008).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

2007 (2)

J. M. Rodenburg, A. C. Hurst, 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]

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

2004 (2)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

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

1997 (1)

H. N. Chapman, “Phase-retrieval x-ray microscopy by Wigner-distribution deconvolution: signal processing,” Scan. Microsc. 11, 67–80 (1997).

1996 (1)

H. N. Chapman, “Phase-retrieval x-ray microscopy by Wigner-distribution deconvolution,” Utramicroscopy 66, 153–172 (1996).
[Crossref]

1995 (1)

P. D. Nellist, B. C. McCallum, and J. M. Rodenburg, “Resolution beyond the ’Information limit’ in transmission electron microscopy,” Nature (London) 374, 630–632 (1995).
[Crossref]

1992 (1)

J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992).
[Crossref]

1990 (1)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inform. Theory 36(5), 961–1005 (1990).
[Crossref]

1986 (1)

I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271–1283 (1986).
[Crossref]

1981 (2)

M. J. Bastiaans, “Signal description by means of a local frequency spectrum,” Proc. SPIE 373, 49–62 (1981).
[Crossref]

M. J. Bastiaans, “A sampling theorem for the complex spectrogram, and Gabor’s expansion of a signal in gaussian elementary signal,” Opt. Eng. 20(4), 594–598 (1981).
[Crossref]

1970 (1)

R. Hegerl and W. Hoppe, “Dynamische theorie der kristallstrukturanalyse durch elektronenbeugung im inhomogenen primrstrahlwellenfeld,” Ber. Bunsenges. Phys. Chem. 74, 1148–1154 (1970).
[Crossref]

1952 (1)

R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic fourier series,” Trans. Am. Math. Soc. 72, 341–366 (1952).
[Crossref]

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Elec. Eng. 93, 429–457 (1946).

1922 (1)

J. R. Carson, “Notes on the theory of modulation,” Proc. IRE 10, 57–64 (1922).
[Crossref]

Abbey, B.

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[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] [PubMed]

Bastiaans, M. J.

M. J. Bastiaans, “Signal description by means of a local frequency spectrum,” Proc. SPIE 373, 49–62 (1981).
[Crossref]

M. J. Bastiaans, “A sampling theorem for the complex spectrogram, and Gabor’s expansion of a signal in gaussian elementary signal,” Opt. Eng. 20(4), 594–598 (1981).
[Crossref]

M. J. Bastiaans, “On optimum oversampling in the gabor scheme,” IEEE International Conference on Acoustics, Speech, and Signal Processing, 1997. ICASSP-97, (IEEE 1997), 3, pp 2101–2104 (1997).
[Crossref]

M. J. Bastiaans, “Gabor’s signal expansion in optics,” Gabor Analysis and Algorithms, Applied and Numerical Harmonic Analysis, H. G. Feichtinger and T. Strohmer, eds. (BirkhüserBoston, 1998), pp. 427–451.
[Crossref]

Bates, R. H. T.

J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992).
[Crossref]

Batey, D. J.

D. J. Batey, T. B. Edo, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Reciprocal-space up-sampling from real-space oversampling in x-ray ptychography,” Phys. Rev. A 89, 043812 (2014).
[Crossref]

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[Crossref]

Bölcskei, H.

H. Bölcskei, H. G. Feichtinger, and F. Hlawatsch, “Diagonalizing the Gabor frame operator,” in Proceedings of the IEEE UK Symposium on Applications of Time-Frequency and Time-Scale Methods, August (1995), pp. 249–255.

Bunk, O.

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, and A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86(10),100103 (2012).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

J. M. Rodenburg, A. C. Hurst, 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]

Carson, J. R.

J. R. Carson, “Notes on the theory of modulation,” Proc. IRE 10, 57–64 (1922).
[Crossref]

Chapman, H. N.

H. N. Chapman, “Phase-retrieval x-ray microscopy by Wigner-distribution deconvolution: signal processing,” Scan. Microsc. 11, 67–80 (1997).

H. N. Chapman, “Phase-retrieval x-ray microscopy by Wigner-distribution deconvolution,” Utramicroscopy 66, 153–172 (1996).
[Crossref]

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] [PubMed]

Chu, Y. S.

Clark, J. N.

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[Crossref]

Cohen, L.

L. Cohen, “Time-frequency distribution - a review,” in Proceedings of the IEEE (invited paper), (IEEE, 1989) 77(7), pp. 941–981.
[Crossref]

Cullis, A. G.

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

Daubechies, I.

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inform. Theory 36(5), 961–1005 (1990).
[Crossref]

I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271–1283 (1986).
[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(5887), 379–382 (2008).
[Crossref] [PubMed]

J. M. Rodenburg, A. C. Hurst, 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]

de Jonge, M. D.

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[Crossref]

Diaz, A.

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, and A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86(10),100103 (2012).
[Crossref]

Dierolf, M.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

Dobson, B. R.

J. M. Rodenburg, A. C. Hurst, 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]

Duffin, R. J.

R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic fourier series,” Trans. Am. Math. Soc. 72, 341–366 (1952).
[Crossref]

Edo, T. B.

D. J. Batey, T. B. Edo, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Reciprocal-space up-sampling from real-space oversampling in x-ray ptychography,” Phys. Rev. A 89, 043812 (2014).
[Crossref]

P. Li, T. B. Edo, and J. M. Rodenburg, “Ptychographic inversion via Wigner distribution deconvolution noise suppression and probe design,” Ultramicroscopy 147, 106–113 (2014).
[Crossref] [PubMed]

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[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]

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

Feichtinger, H. G.

H. Bölcskei, H. G. Feichtinger, and F. Hlawatsch, “Diagonalizing the Gabor frame operator,” in Proceedings of the IEEE UK Symposium on Applications of Time-Frequency and Time-Scale Methods, August (1995), pp. 249–255.

Fienup, J.R.

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Elec. Eng. 93, 429–457 (1946).

Grossmann, A.

I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271–1283 (1986).
[Crossref]

Guizar-Sicairos, M.

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, and A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86(10),100103 (2012).
[Crossref]

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

M. Guizar-Sicairos and J.R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16(10), 7264–7278 (2008).
[Crossref] [PubMed]

Harder, R.

Hegerl, R.

R. Hegerl and W. Hoppe, “Dynamische theorie der kristallstrukturanalyse durch elektronenbeugung im inhomogenen primrstrahlwellenfeld,” Ber. Bunsenges. Phys. Chem. 74, 1148–1154 (1970).
[Crossref]

Heil, C. E.

C. E. Heil, “Wavelets and frames,” Signal Processing Theory, L. Auslander, T. Kailath, and S. Mitter, eds., IMA Vol. Math. Appl. 22, (Springer-Verlag, 1990), pp 147–160.
[Crossref]

Hlawatsch, F.

H. Bölcskei, H. G. Feichtinger, and F. Hlawatsch, “Diagonalizing the Gabor frame operator,” in Proceedings of the IEEE UK Symposium on Applications of Time-Frequency and Time-Scale Methods, August (1995), pp. 249–255.

Holler, M.

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, and A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86(10),100103 (2012).
[Crossref]

Hoppe, W.

R. Hegerl and W. Hoppe, “Dynamische theorie der kristallstrukturanalyse durch elektronenbeugung im inhomogenen primrstrahlwellenfeld,” Ber. Bunsenges. Phys. Chem. 74, 1148–1154 (1970).
[Crossref]

Horstmeyer, R.

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] [PubMed]

R. Horstmeyer and C. Yang, “A phase space model of Fourier ptychographic microscopy,” Opt. Express 22, 338–358 (2014).
[Crossref] [PubMed]

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

Huang, X.

Humphry, M. J.

Hurst, A. C.

J. M. Rodenburg, A. C. Hurst, 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]

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

Hwu, Y.

Jefimovs, K.

J. M. Rodenburg, A. C. Hurst, 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]

Johnson, I.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

J. M. Rodenburg, A. C. Hurst, 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]

Kynde, S.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

Li, P.

P. Li, T. B. Edo, and J. M. Rodenburg, “Ptychographic inversion via Wigner distribution deconvolution noise suppression and probe design,” Ultramicroscopy 147, 106–113 (2014).
[Crossref] [PubMed]

Maia, F.

S. Marchesini, A. Schirotzed, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

Maiden, A. M.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[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(8), 1606–1614 (2012).
[Crossref]

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing, 3rd ed. (Academic, 2009).

Marchesini, S.

S. Marchesini, A. Schirotzed, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

Marti, O.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

McCallum, B. C.

P. D. Nellist, B. C. McCallum, and J. M. Rodenburg, “Resolution beyond the ’Information limit’ in transmission electron microscopy,” Nature (London) 374, 630–632 (1995).
[Crossref]

McNulty, I.

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[Crossref]

Menzel, A.

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

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, and A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86(10),100103 (2012).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Meyer, Y.

I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271–1283 (1986).
[Crossref]

Nellist, P. D.

P. D. Nellist, B. C. McCallum, and J. M. Rodenburg, “Resolution beyond the ’Information limit’ in transmission electron microscopy,” Nature (London) 374, 630–632 (1995).
[Crossref]

Nugent, K. A.

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[Crossref]

Onural, L.

Ou, X.

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] [PubMed]

Peele, A. G.

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[Crossref]

Pešic, Z. D.

D. J. Batey, T. B. Edo, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Reciprocal-space up-sampling from real-space oversampling in x-ray ptychography,” Phys. Rev. A 89, 043812 (2014).
[Crossref]

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[Crossref]

Pfeifer, M. A.

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[Crossref]

Pfeiffer, F.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

J. M. Rodenburg, A. C. Hurst, 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]

Rau, C.

D. J. Batey, T. B. Edo, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Reciprocal-space up-sampling from real-space oversampling in x-ray ptychography,” Phys. Rev. A 89, 043812 (2014).
[Crossref]

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[Crossref]

Robinson, I. K.

Rodenburg, J. M.

P. Li, T. B. Edo, and J. M. Rodenburg, “Ptychographic inversion via Wigner distribution deconvolution noise suppression and probe design,” Ultramicroscopy 147, 106–113 (2014).
[Crossref] [PubMed]

D. J. Batey, T. B. Edo, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Reciprocal-space up-sampling from real-space oversampling in x-ray ptychography,” Phys. Rev. A 89, 043812 (2014).
[Crossref]

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[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(8), 1606–1614 (2012).
[Crossref]

J. M. Rodenburg, A. C. Hurst, 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]

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

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

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

P. D. Nellist, B. C. McCallum, and J. M. Rodenburg, “Resolution beyond the ’Information limit’ in transmission electron microscopy,” Nature (London) 374, 630–632 (1995).
[Crossref]

J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992).
[Crossref]

Sahin, E.

Schaeffer, A. C.

R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic fourier series,” Trans. Am. Math. Soc. 72, 341–366 (1952).
[Crossref]

Schirotzed, A.

S. Marchesini, A. Schirotzed, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

Thibault, P.

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

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

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Tropp, J. A.

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] [PubMed]

Vila-Comamala, J.

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, and A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86(10),100103 (2012).
[Crossref]

Vine, D. J.

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[Crossref]

Wagner, U.

D. J. Batey, T. B. Edo, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Reciprocal-space up-sampling from real-space oversampling in x-ray ptychography,” Phys. Rev. A 89, 043812 (2014).
[Crossref]

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[Crossref]

Waigh, T. A.

D. J. Batey, T. B. Edo, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Reciprocal-space up-sampling from real-space oversampling in x-ray ptychography,” Phys. Rev. A 89, 043812 (2014).
[Crossref]

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[Crossref]

Williams, G. J.

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[Crossref]

Wu, H.-T.

S. Marchesini, A. Schirotzed, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

Yan, H.

Yang, C.

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] [PubMed]

R. Horstmeyer and C. Yang, “A phase space model of Fourier ptychographic microscopy,” Opt. Express 22, 338–358 (2014).
[Crossref] [PubMed]

S. Marchesini, A. Schirotzed, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

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

Zheng, G.

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

Appl. Phys. Lett. (1)

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

Ber. Bunsenges. Phys. Chem. (1)

R. Hegerl and W. Hoppe, “Dynamische theorie der kristallstrukturanalyse durch elektronenbeugung im inhomogenen primrstrahlwellenfeld,” Ber. Bunsenges. Phys. Chem. 74, 1148–1154 (1970).
[Crossref]

IEEE Trans. Inform. Theory (1)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inform. Theory 36(5), 961–1005 (1990).
[Crossref]

Inverse Probl. (1)

S. Marchesini, A. Schirotzed, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

J. Inst. Elec. Eng. (1)

D. Gabor, “Theory of communication,” J. Inst. Elec. Eng. 93, 429–457 (1946).

J. Math. Phys. (1)

I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271–1283 (1986).
[Crossref]

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

Nat. Photonics (1)

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

Nature (1)

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

Nature (London) (1)

P. D. Nellist, B. C. McCallum, and J. M. Rodenburg, “Resolution beyond the ’Information limit’ in transmission electron microscopy,” Nature (London) 374, 630–632 (1995).
[Crossref]

New J. Phys. (2)

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

Opt. Eng. (1)

M. J. Bastiaans, “A sampling theorem for the complex spectrogram, and Gabor’s expansion of a signal in gaussian elementary signal,” Opt. Eng. 20(4), 594–598 (1981).
[Crossref]

Opt. Express (3)

Phil. Trans. R. Soc. Lond. A (1)

J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992).
[Crossref]

Phys. Rev. A (3)

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[Crossref]

D. J. Batey, T. B. Edo, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Reciprocal-space up-sampling from real-space oversampling in x-ray ptychography,” Phys. Rev. A 89, 043812 (2014).
[Crossref]

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A 80(6), 063823 (2009).
[Crossref]

Phys. Rev. B (1)

M. Guizar-Sicairos, M. Holler, A. Diaz, J. Vila-Comamala, O. Bunk, and A. Menzel, “Role of the illumination spatial-frequency spectrum for ptychography,” Phys. Rev. B 86(10),100103 (2012).
[Crossref]

Phys. Rev. Lett. (2)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

J. M. Rodenburg, A. C. Hurst, 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]

Proc. IRE (1)

J. R. Carson, “Notes on the theory of modulation,” Proc. IRE 10, 57–64 (1922).
[Crossref]

Proc. SPIE (1)

M. J. Bastiaans, “Signal description by means of a local frequency spectrum,” Proc. SPIE 373, 49–62 (1981).
[Crossref]

Scan. Microsc. (1)

H. N. Chapman, “Phase-retrieval x-ray microscopy by Wigner-distribution deconvolution: signal processing,” Scan. Microsc. 11, 67–80 (1997).

Science (1)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Trans. Am. Math. Soc. (1)

R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic fourier series,” Trans. Am. Math. Soc. 72, 341–366 (1952).
[Crossref]

Ultramicroscopy (4)

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

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

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

P. Li, T. B. Edo, and J. M. Rodenburg, “Ptychographic inversion via Wigner distribution deconvolution noise suppression and probe design,” Ultramicroscopy 147, 106–113 (2014).
[Crossref] [PubMed]

Utramicroscopy (1)

H. N. Chapman, “Phase-retrieval x-ray microscopy by Wigner-distribution deconvolution,” Utramicroscopy 66, 153–172 (1996).
[Crossref]

Other (6)

S. Mallat, A Wavelet Tour of Signal Processing, 3rd ed. (Academic, 2009).

L. Cohen, “Time-frequency distribution - a review,” in Proceedings of the IEEE (invited paper), (IEEE, 1989) 77(7), pp. 941–981.
[Crossref]

M. J. Bastiaans, “Gabor’s signal expansion in optics,” Gabor Analysis and Algorithms, Applied and Numerical Harmonic Analysis, H. G. Feichtinger and T. Strohmer, eds. (BirkhüserBoston, 1998), pp. 427–451.
[Crossref]

M. J. Bastiaans, “On optimum oversampling in the gabor scheme,” IEEE International Conference on Acoustics, Speech, and Signal Processing, 1997. ICASSP-97, (IEEE 1997), 3, pp 2101–2104 (1997).
[Crossref]

C. E. Heil, “Wavelets and frames,” Signal Processing Theory, L. Auslander, T. Kailath, and S. Mitter, eds., IMA Vol. Math. Appl. 22, (Springer-Verlag, 1990), pp 147–160.
[Crossref]

H. Bölcskei, H. G. Feichtinger, and F. Hlawatsch, “Diagonalizing the Gabor frame operator,” in Proceedings of the IEEE UK Symposium on Applications of Time-Frequency and Time-Scale Methods, August (1995), pp. 249–255.

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Figures (4)

Fig. 1
Fig. 1 Windowed discrete Fourier transform basis with frequencies fx and fy from −2 to 2. A Gaussian window function with sigma equals to 0.25 × image size was used to “window” the Fourier basis.
Fig. 2
Fig. 2 Representation of ptychography phase space. Two Heisenberg boxes and their spread in the real and reciprocal space are located at (qn, Rn) and (qm, Rm). The projection of these boxes onto the real space represents the |PR,q(r)| (in solid and dashed blue lines), here illustrated by Gaussian functions, and onto the reciprocal space represents |{PR,q(r)}| (in solid and dashed green lines).
Fig. 3
Fig. 3 The coverage of the ptychography phase space. The boxes are illustrated by blue boxes of size Di in real space and size qi in reciprocal space. Critical sampling with αi = βi = 1 is illustrated in the top-right corner. “Oversampling,” i.e., αiβi < 1, is illustrated in the bottom-left corner where there is overlap of the windows in both real and reciprocal space.
Fig. 4
Fig. 4 Illustration of the role of the overlap for 1D signal (a) The square magnitude of the window p(x). (b) “Scanning” the window function. (c) The sum of the square magnitude contributions of the window ∑|p(xxn)|2 and the windowed real and imaginary parts of the signal.

Equations (18)

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ψ ( r ; R ) = O ( r ) P ( r R ) ,
I ( q ; R ) = | Ψ ( q ; R ) | 2 = | + O ( r ) P ( r R ) exp ( i q r ) d r | 2 .
P R , q ( r ) P * ( r R ) exp ( i q r ) ,
I ( q ; R ) = | Ψ ( q ; R ) | 2 = | + O ( r ) P R , q * ( r ) d r | 2 ,
| Ψ ( q ; R ) | 2 = + 𝒲 O ( r , q ) 𝒲 P ( r R , q q ) d r d q ,
O ( r ) = + Ψ ( q ; R ) P * ( r R ) exp ( i q r ) d q d R 2 π + | P ( r R ) | 2 d R ,
P ( r ) = + Ψ ( q ; R ) O * ( r + R ) exp [ i q ( r + R ) ] d q d R 2 π + | O ( r + R ) | 2 d R .
O ( r ) = + ψ ( r ; R ) P * ( r R ) d R + | P ( r R ) | 2 d R ,
P ( r ) = + ψ ( r + R ; R ) O * ( r + R ) d R + | O ( r + R ) | 2 d R ,
a k x , k y , n x , n y = Ψ ( k x Q x , k y Q y , n x R x , n y R y ) ,
O ( x , y ) k x k y n x n y a k x , k y , n x , n y P * ( x n x R x , y n y R y ) exp [ i ( k x Q x x N x + k y Q y y N y ) ] ,
( α i D i ) ( β i q i ) 2 π ,
n 1 n 2 | P * ( x n 1 R x , y n 2 R y ) | 2 = const .
ψ ( r + R ; R ) = O ( r + R ) P ( r ) .
O ( r + R ) P ( r ) = 1 { Ψ ( q ; R ) exp ( i q R ) } .
P ( r ) + | O ( r + R ) | 2 d R = 1 2 π + Ψ ( q ; R ) O * ( r + R ) exp [ i q ( r + R ) ] d q d R ,
O ( r ) = 1 2 π + | P ( r R ) | 2 d R ψ ( q ; R ) exp [ i q r ] d q = 2 π ψ ( r ; R ) P * ( r R ) d R ,
P ( r ) = 1 2 π + | O ( r + R ) | 2 d R Ψ ( q ; R ) exp [ i q ( r + R ) ] d q = 2 π ψ ( r + R ; R ) O * ( r + R ) d R ,

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