Abstract

Coherent diffraction imaging (CDI) is a lens-less microscopy method that extracts the complex-valued exit field from intensity measurements alone. It is of particular importance for microscopy imaging with diffraction set-ups where high quality lenses are not available. The inversion scheme allowing the phase retrieval is based on the use of an iterative algorithm. In this work, we address the question of the choice of the iterative process in the case of data corrupted by photon or electron shot noise. Several noise models are presented and further used within two inversion strategies, the ordered subset and the scaled gradient. Based on analytical and numerical analysis together with Monte-Carlo studies, we show that any physical interpretations drawn from a CDI iterative technique require a detailed understanding of the relationship between the noise model and the used inversion method. We observe that iterative algorithms often assume implicitly a noise model. For low counting rates, each noise model behaves differently. Moreover, the used optimization strategy introduces its own artefacts. Based on this analysis, we develop a hybrid strategy which works efficiently in the absence of an informed initial guess. Our work emphasises issues which should be considered carefully when inverting experimental data.

© 2012 OSA

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    [CrossRef]
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  4. D. Claus, A. M. Maiden, F. Zhang, F. G. R. Sweeney, M. Humphry, H. Schluesener, and J. M. Rodenburg, “Quantitative phase contrast optimised cancerous cell differentiation via ptychography,” Opt. Express20, 9911– 9918 (2012).
    [CrossRef] [PubMed]
  5. M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical constrasts in soft X-ray ptychography,” Phys. Rev. Lett.107, 208101 (2011).
    [CrossRef] [PubMed]
  6. 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, 34801 (2007).
    [CrossRef]
  7. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science321, 379–382 (2008).
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  8. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010).
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  9. P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 1569 (2011).
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  10. F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B82, 121415 (2010).
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  11. M. J. Humphry, B. Kraus, A. C. Hurst, A. M. Maiden, and J. M. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun.3, 730 (2012).
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    [CrossRef]
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  22. C. Ponchut, J. Clément, J.-M. Rigal, E. Papillon, J. Vallerga, D. LaMarra, and B. Mikulec, “Photon-counting X-ray imaging at kilohertz frame rates,” Nucl. Instrum. Meth. A576, 109–112 (2007).
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  25. L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag.1, 113–122 (1982).
    [CrossRef]
  26. L. B. Lucy, “An iterative technique for the rectification of observed distribution,” New Astron. Rev.79, 745–754 (1974).
  27. G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Cryst.A63, 36–42 (2007).
  28. P. Godard, M. Allain, and V. Chamard, “Imaging of highly inhomogeneous strain field in nanocrystals using x-ray Bragg ptychography: A numerical study,” Phys. Rev. B84, 144109 (2011).
    [CrossRef]
  29. J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive X-ray optics by ptychographic coherent diffractive imaging,” Opt. Express19, 21333–21344 (2011).
    [CrossRef] [PubMed]
  30. Provided that the fluctuations in one measurement are accurately described by a Poisson PDF, then this PDF is defined by a single (positive) parameter that is the mean and the variance.
  31. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett.85, 4795–4797 (2004).
    [CrossRef]
  32. M. F. Freeman and J. W. Tuckey, “Transformations related to the angular and the square root,” Ann. Math. Statist.21, 607–611 (1950).
    [CrossRef]
  33. C. A. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process.5, 480–492 (1996).
    [CrossRef] [PubMed]
  34. L. Bouchet, “A comparative-study of deconvolution methods for gamma-ray spectra,” Astron. Astrophys.113, 167–183 (1995).
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    [CrossRef]
  36. By definition, the likelihood is the PDF of the noise model seen as a function of the unknown parameters ρ. In practice, the opposite of the logarithm of the likelihood is rather considered. However, the logarithm function being a monotonic increasing function, the minimiser of the neg-loglikelihood is also the maximiser of the likelihood, i.e., the ML estimator.
  37. Following [33], it is shown that a second order Taylor expansion around hm,j = ym,j of the Poissonian fitting function ℒ𝒫 leads to (10e).
  38. M. Bertero and P. Boccacci, Introduction to inverse problems in imaging (Institute of Physics Publishing, 1998).
    [CrossRef]
  39. Since the condition hm,j > 0 is enforced if bm,j > 0 [cf. Eq. (1)], an arbitrary small background component can be introduced, hence allowing all the fitting functions and gradients to be well defined.
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    [CrossRef]
  41. In the optimization literature, OS algorithms are also known as incremental gradient methods or block iterative methods, see for instance [42, Sec. 1.5.2 ] or [43] for details.
  42. D. P. Bertsekas, Nonlinear programming, 2nd ed. (Athena Scientific, 1999).
  43. Y. Censor, D. Gordon, and R. Gordon, “BICAV: a block-iterative parallel algorithm for sparse systems with pixel-related weighting,” IEEE Trans. Med. Imag.20, 1050–1060 (2001).
    [CrossRef]
  44. H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered-subset of projection data,” IEEE Trans. Med. Imag.13, 601–609 (1994).
    [CrossRef]
  45. S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subset algorithms,” IEEE Trans. Med. Imag.22, 613–626 (2003).
    [CrossRef]
  46. The original version of the PIE introduced by Rodenburg and Faulkner in [31] considers another definition for Dj.
  47. A. M. Maiden, Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am.29, 1606–1614 (2012).
    [CrossRef]
  48. C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv:optics (2011).
  49. Since the three fitting functions ℒ𝒫, ℒ𝒢 and ℒℛ are equivalent w.r.t. a nil data, only the data such that ym,j ≠ 0 should be considered in order to discriminate the noise-models.
  50. This non-monotonic behaviour of the relative error is standard when inverse problems (e.g., image restoration or tomographic reconstruction) are solved with gradient optimization technics, see for instance [38, Chap. 6].
  51. Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuation to Information (Springer, 2004).
  52. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A20, 40–55 (2003).
    [CrossRef]
  53. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009).
    [CrossRef] [PubMed]
  54. From [52, p. 339], one notes that the constraint defined by the data set in the DM strategy takes the form of Eq. (12b), suggesting that the data fluctuations are described by the Gaussian model defined in Sec. 2.2.
  55. J. F. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika35, 246–254 (1948).

2012 (6)

M. J. Humphry, B. Kraus, A. C. Hurst, A. M. Maiden, and J. M. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun.3, 730 (2012).
[CrossRef] [PubMed]

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett.108, 73901 (2012).
[CrossRef]

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy120, 64–72 (2012).
[CrossRef] [PubMed]

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

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

D. Claus, A. M. Maiden, F. Zhang, F. G. R. Sweeney, M. Humphry, H. Schluesener, and J. M. Rodenburg, “Quantitative phase contrast optimised cancerous cell differentiation via ptychography,” Opt. Express20, 9911– 9918 (2012).
[CrossRef] [PubMed]

2011 (5)

J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive X-ray optics by ptychographic coherent diffractive imaging,” Opt. Express19, 21333–21344 (2011).
[CrossRef] [PubMed]

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv:optics (2011).

P. Godard, M. Allain, and V. Chamard, “Imaging of highly inhomogeneous strain field in nanocrystals using x-ray Bragg ptychography: A numerical study,” Phys. Rev. B84, 144109 (2011).
[CrossRef]

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 1569 (2011).
[CrossRef]

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical constrasts in soft X-ray ptychography,” Phys. Rev. Lett.107, 208101 (2011).
[CrossRef] [PubMed]

2010 (4)

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010).
[CrossRef] [PubMed]

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B82, 121415 (2010).
[CrossRef]

V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett.104, 165501 (2010).
[CrossRef] [PubMed]

K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys.59, 1–100 (2010).
[CrossRef]

2009 (2)

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009).
[CrossRef] [PubMed]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy109, 1256–1262 (2009).
[CrossRef] [PubMed]

2008 (2)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science321, 379–382 (2008).
[CrossRef] [PubMed]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express16, 7264–7276 (2008).
[CrossRef] [PubMed]

2007 (4)

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, 34801 (2007).
[CrossRef]

C. Ponchut, J. Clément, J.-M. Rigal, E. Papillon, J. Vallerga, D. LaMarra, and B. Mikulec, “Photon-counting X-ray imaging at kilohertz frame rates,” Nucl. Instrum. Meth. A576, 109–112 (2007).
[CrossRef]

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy107, 227–231 (2007).
[CrossRef]

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Cryst.A63, 36–42 (2007).

2006 (2)

M. Allain and J.-P. Roques, “High resolution techniques for gamma-ray diffuse emission: application to INTEGRAL/SPI,” Astron. Astrophys.43, 1175–1187 (2006).
[CrossRef]

C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad.13, 120–130 (2006).
[CrossRef]

2004 (2)

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectroholography,” Nature432, 885–888 (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]

2003 (2)

S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subset algorithms,” IEEE Trans. Med. Imag.22, 613–626 (2003).
[CrossRef]

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A20, 40–55 (2003).
[CrossRef]

2001 (1)

Y. Censor, D. Gordon, and R. Gordon, “BICAV: a block-iterative parallel algorithm for sparse systems with pixel-related weighting,” IEEE Trans. Med. Imag.20, 1050–1060 (2001).
[CrossRef]

2000 (1)

F. Livet, F. Bley, J. Mainville, R. Caudron, S. G. J. Mochrie, E. Geissler, G. Dolino, D. Abernathy, G. Grübel, and M. Sutton, “Using direct illumination CCDs as high resolution area detector for X-ray scattering,” Nucl. Instr. Meth. A451, 596–609 (2000).
[CrossRef]

1996 (1)

C. A. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process.5, 480–492 (1996).
[CrossRef] [PubMed]

1995 (1)

L. Bouchet, “A comparative-study of deconvolution methods for gamma-ray spectra,” Astron. Astrophys.113, 167–183 (1995).

1994 (1)

H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered-subset of projection data,” IEEE Trans. Med. Imag.13, 601–609 (1994).
[CrossRef]

1986 (1)

1984 (1)

K. Lange and R. Carson, “EM reconstruction algorithm for emission and transmission tomography,” IEEE Trans. Med. Imag.8, 306–316 (1984).

1982 (1)

L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag.1, 113–122 (1982).
[CrossRef]

1974 (1)

L. B. Lucy, “An iterative technique for the rectification of observed distribution,” New Astron. Rev.79, 745–754 (1974).

1950 (1)

M. F. Freeman and J. W. Tuckey, “Transformations related to the angular and the square root,” Ann. Math. Statist.21, 607–611 (1950).
[CrossRef]

1948 (1)

J. F. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika35, 246–254 (1948).

Abernathy, D.

F. Livet, F. Bley, J. Mainville, R. Caudron, S. G. J. Mochrie, E. Geissler, G. Dolino, D. Abernathy, G. Grübel, and M. Sutton, “Using direct illumination CCDs as high resolution area detector for X-ray scattering,” Nucl. Instr. Meth. A451, 596–609 (2000).
[CrossRef]

Ahn, S.

S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subset algorithms,” IEEE Trans. Med. Imag.22, 613–626 (2003).
[CrossRef]

Alfonso, C.

V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett.104, 165501 (2010).
[CrossRef] [PubMed]

Allain, M.

P. Godard, M. Allain, and V. Chamard, “Imaging of highly inhomogeneous strain field in nanocrystals using x-ray Bragg ptychography: A numerical study,” Phys. Rev. B84, 144109 (2011).
[CrossRef]

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 1569 (2011).
[CrossRef]

M. Allain and J.-P. Roques, “High resolution techniques for gamma-ray diffuse emission: application to INTEGRAL/SPI,” Astron. Astrophys.43, 1175–1187 (2006).
[CrossRef]

Allen, L. J.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett.108, 73901 (2012).
[CrossRef]

Anscombe, J. F.

J. F. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika35, 246–254 (1948).

Bauer, G.

V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett.104, 165501 (2010).
[CrossRef] [PubMed]

Beckers, M.

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical constrasts in soft X-ray ptychography,” Phys. Rev. Lett.107, 208101 (2011).
[CrossRef] [PubMed]

Bertero, M.

M. Bertero and P. Boccacci, Introduction to inverse problems in imaging (Institute of Physics Publishing, 1998).
[CrossRef]

Bertsekas, D. P.

D. P. Bertsekas, Nonlinear programming, 2nd ed. (Athena Scientific, 1999).

Bley, F.

F. Livet, F. Bley, J. Mainville, R. Caudron, S. G. J. Mochrie, E. Geissler, G. Dolino, D. Abernathy, G. Grübel, and M. Sutton, “Using direct illumination CCDs as high resolution area detector for X-ray scattering,” Nucl. Instr. Meth. A451, 596–609 (2000).
[CrossRef]

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to inverse problems in imaging (Institute of Physics Publishing, 1998).
[CrossRef]

Bouchet, L.

L. Bouchet, “A comparative-study of deconvolution methods for gamma-ray spectra,” Astron. Astrophys.113, 167–183 (1995).

Bouman, C. A.

C. A. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process.5, 480–492 (1996).
[CrossRef] [PubMed]

Bourgeois, L.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett.108, 73901 (2012).
[CrossRef]

Broennimann, C.

C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad.13, 120–130 (2006).
[CrossRef]

Bunk, O.

J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive X-ray optics by ptychographic coherent diffractive imaging,” Opt. Express19, 21333–21344 (2011).
[CrossRef] [PubMed]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 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,” Science321, 379–382 (2008).
[CrossRef] [PubMed]

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V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett.104, 165501 (2010).
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P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 1569 (2011).
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K. Lange and R. Carson, “EM reconstruction algorithm for emission and transmission tomography,” IEEE Trans. Med. Imag.8, 306–316 (1984).

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Y. Censor, D. Gordon, and R. Gordon, “BICAV: a block-iterative parallel algorithm for sparse systems with pixel-related weighting,” IEEE Trans. Med. Imag.20, 1050–1060 (2001).
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P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 1569 (2011).
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V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett.104, 165501 (2010).
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P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 1569 (2011).
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V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett.104, 165501 (2010).
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J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy107, 227–231 (2007).
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J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive X-ray optics by ptychographic coherent diffractive imaging,” Opt. Express19, 21333–21344 (2011).
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P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science321, 379–382 (2008).
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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, 34801 (2007).
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J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive X-ray optics by ptychographic coherent diffractive imaging,” Opt. Express19, 21333–21344 (2011).
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V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett.104, 165501 (2010).
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M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010).
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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, 34801 (2007).
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C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett.108, 73901 (2012).
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Humphry, M. J.

M. J. Humphry, B. Kraus, A. C. Hurst, A. M. Maiden, and J. M. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun.3, 730 (2012).
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M. J. Humphry, B. Kraus, A. C. Hurst, A. M. Maiden, and J. M. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun.3, 730 (2012).
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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, 34801 (2007).
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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, 34801 (2007).
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C. Ponchut, J. Clément, J.-M. Rigal, E. Papillon, J. Vallerga, D. LaMarra, and B. Mikulec, “Photon-counting X-ray imaging at kilohertz frame rates,” Nucl. Instrum. Meth. A576, 109–112 (2007).
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F. Livet, F. Bley, J. Mainville, R. Caudron, S. G. J. Mochrie, E. Geissler, G. Dolino, D. Abernathy, G. Grübel, and M. Sutton, “Using direct illumination CCDs as high resolution area detector for X-ray scattering,” Nucl. Instr. Meth. A451, 596–609 (2000).
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S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectroholography,” Nature432, 885–888 (2004).
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A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy120, 64–72 (2012).
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F. Livet, F. Bley, J. Mainville, R. Caudron, S. G. J. Mochrie, E. Geissler, G. Dolino, D. Abernathy, G. Grübel, and M. Sutton, “Using direct illumination CCDs as high resolution area detector for X-ray scattering,” Nucl. Instr. Meth. A451, 596–609 (2000).
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P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 1569 (2011).
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J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive X-ray optics by ptychographic coherent diffractive imaging,” Opt. Express19, 21333–21344 (2011).
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P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 1569 (2011).
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F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B82, 121415 (2010).
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Pfeiffer, F.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 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,” Science321, 379–382 (2008).
[CrossRef] [PubMed]

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, 34801 (2007).
[CrossRef]

Pohl, E.

C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad.13, 120–130 (2006).
[CrossRef]

Ponchut, C.

C. Ponchut, J. Clément, J.-M. Rigal, E. Papillon, J. Vallerga, D. LaMarra, and B. Mikulec, “Photon-counting X-ray imaging at kilohertz frame rates,” Nucl. Instrum. Meth. A576, 109–112 (2007).
[CrossRef]

Putkunz, C. T.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett.108, 73901 (2012).
[CrossRef]

Qian, J.

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv:optics (2011).

Reese, M.

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical constrasts in soft X-ray ptychography,” Phys. Rev. Lett.107, 208101 (2011).
[CrossRef] [PubMed]

Réfrégier, Ph.

Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuation to Information (Springer, 2004).

Rigal, J.-M.

C. Ponchut, J. Clément, J.-M. Rigal, E. Papillon, J. Vallerga, D. LaMarra, and B. Mikulec, “Photon-counting X-ray imaging at kilohertz frame rates,” Nucl. Instrum. Meth. A576, 109–112 (2007).
[CrossRef]

Roberts, A.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett.108, 73901 (2012).
[CrossRef]

Robinson, I.

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Cryst.A63, 36–42 (2007).

Rodenburg, J. M.

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy120, 64–72 (2012).
[CrossRef] [PubMed]

D. Claus, A. M. Maiden, F. Zhang, F. G. R. Sweeney, M. Humphry, H. Schluesener, and J. M. Rodenburg, “Quantitative phase contrast optimised cancerous cell differentiation via ptychography,” Opt. Express20, 9911– 9918 (2012).
[CrossRef] [PubMed]

M. J. Humphry, B. Kraus, A. C. Hurst, A. M. Maiden, and J. M. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun.3, 730 (2012).
[CrossRef] [PubMed]

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

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B82, 121415 (2010).
[CrossRef]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy109, 1256–1262 (2009).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy107, 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, 34801 (2007).
[CrossRef]

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

J. M. Rodenburg, “Ptychography and related diffracted imaging methods,” in “Advances in Imaging and Electron Physics,” 150, P. W. Hawkesed. (Elsevier, 2008), 87–184.
[CrossRef]

Roques, J.-P.

M. Allain and J.-P. Roques, “High resolution techniques for gamma-ray diffuse emission: application to INTEGRAL/SPI,” Astron. Astrophys.43, 1175–1187 (2006).
[CrossRef]

Rosenhahn, A.

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical constrasts in soft X-ray ptychography,” Phys. Rev. Lett.107, 208101 (2011).
[CrossRef] [PubMed]

Salditt, T.

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical constrasts in soft X-ray ptychography,” Phys. Rev. Lett.107, 208101 (2011).
[CrossRef] [PubMed]

Sarahan, M. C.

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy120, 64–72 (2012).
[CrossRef] [PubMed]

Sauer, K.

C. A. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process.5, 480–492 (1996).
[CrossRef] [PubMed]

Schirotzek, A.

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv:optics (2011).

Schlotter, W. F.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectroholography,” Nature432, 885–888 (2004).
[CrossRef] [PubMed]

Schluesener, H.

Schmitt, B.

C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad.13, 120–130 (2006).
[CrossRef]

Schneider, P.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010).
[CrossRef] [PubMed]

Scholten, R. E.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett.108, 73901 (2012).
[CrossRef]

Schulze-Briese, C.

C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad.13, 120–130 (2006).
[CrossRef]

Senkbeil, T.

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical constrasts in soft X-ray ptychography,” Phys. Rev. Lett.107, 208101 (2011).
[CrossRef] [PubMed]

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L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag.1, 113–122 (1982).
[CrossRef]

Stangl, J.

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 1569 (2011).
[CrossRef]

V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett.104, 165501 (2010).
[CrossRef] [PubMed]

Stöhr, J.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectroholography,” Nature432, 885–888 (2004).
[CrossRef] [PubMed]

Stuart, A.

M. G. Kendall and A. Stuart, The advanced theory of statistics2a (Griffin, 1963).

Sutton, M.

F. Livet, F. Bley, J. Mainville, R. Caudron, S. G. J. Mochrie, E. Geissler, G. Dolino, D. Abernathy, G. Grübel, and M. Sutton, “Using direct illumination CCDs as high resolution area detector for X-ray scattering,” Nucl. Instr. Meth. A451, 596–609 (2000).
[CrossRef]

Suzuki, M.

C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad.13, 120–130 (2006).
[CrossRef]

Sweeney, F.

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B82, 121415 (2010).
[CrossRef]

Sweeney, F. G. R.

Thibault, P.

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

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 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,” Science321, 379–382 (2008).
[CrossRef] [PubMed]

Tomizaki, T.

C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad.13, 120–130 (2006).
[CrossRef]

Toyokawa, H.

C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad.13, 120–130 (2006).
[CrossRef]

Tuckey, J. W.

M. F. Freeman and J. W. Tuckey, “Transformations related to the angular and the square root,” Ann. Math. Statist.21, 607–611 (1950).
[CrossRef]

Vallerga, J.

C. Ponchut, J. Clément, J.-M. Rigal, E. Papillon, J. Vallerga, D. LaMarra, and B. Mikulec, “Photon-counting X-ray imaging at kilohertz frame rates,” Nucl. Instrum. Meth. A576, 109–112 (2007).
[CrossRef]

Vardi, Y.

L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag.1, 113–122 (1982).
[CrossRef]

Vartanyants, I.

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Cryst.A63, 36–42 (2007).

Vila-Comamala, J.

Wackerman, C. C.

Wagner, A.

C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad.13, 120–130 (2006).
[CrossRef]

Wepf, R.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010).
[CrossRef] [PubMed]

Weyland, M.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett.108, 73901 (2012).
[CrossRef]

Williams, G.

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Cryst.A63, 36–42 (2007).

Yang, C.

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv:optics (2011).

Zhang, F.

Acta Cryst. (1)

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Cryst.A63, 36–42 (2007).

Adv. Phys. (1)

K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys.59, 1–100 (2010).
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M. F. Freeman and J. W. Tuckey, “Transformations related to the angular and the square root,” Ann. Math. Statist.21, 607–611 (1950).
[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]

Astron. Astrophys. (2)

L. Bouchet, “A comparative-study of deconvolution methods for gamma-ray spectra,” Astron. Astrophys.113, 167–183 (1995).

M. Allain and J.-P. Roques, “High resolution techniques for gamma-ray diffuse emission: application to INTEGRAL/SPI,” Astron. Astrophys.43, 1175–1187 (2006).
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J. F. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika35, 246–254 (1948).

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IEEE Trans. Med. Imag. (5)

K. Lange and R. Carson, “EM reconstruction algorithm for emission and transmission tomography,” IEEE Trans. Med. Imag.8, 306–316 (1984).

L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag.1, 113–122 (1982).
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J. Opt. Soc. Am. (1)

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

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

J. Synchrotron Rad. (1)

C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad.13, 120–130 (2006).
[CrossRef]

Nat. Commun. (2)

M. J. Humphry, B. Kraus, A. C. Hurst, A. M. Maiden, and J. M. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun.3, 730 (2012).
[CrossRef] [PubMed]

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 1569 (2011).
[CrossRef]

Nature (2)

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectroholography,” Nature432, 885–888 (2004).
[CrossRef] [PubMed]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010).
[CrossRef] [PubMed]

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P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys.14, 063004 (2012).
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Nucl. Instr. Meth. A (1)

F. Livet, F. Bley, J. Mainville, R. Caudron, S. G. J. Mochrie, E. Geissler, G. Dolino, D. Abernathy, G. Grübel, and M. Sutton, “Using direct illumination CCDs as high resolution area detector for X-ray scattering,” Nucl. Instr. Meth. A451, 596–609 (2000).
[CrossRef]

Nucl. Instrum. Meth. A (1)

C. Ponchut, J. Clément, J.-M. Rigal, E. Papillon, J. Vallerga, D. LaMarra, and B. Mikulec, “Photon-counting X-ray imaging at kilohertz frame rates,” Nucl. Instrum. Meth. A576, 109–112 (2007).
[CrossRef]

Opt. Express (3)

Phys. Rev. B (2)

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B82, 121415 (2010).
[CrossRef]

P. Godard, M. Allain, and V. Chamard, “Imaging of highly inhomogeneous strain field in nanocrystals using x-ray Bragg ptychography: A numerical study,” Phys. Rev. B84, 144109 (2011).
[CrossRef]

Phys. Rev. Lett. (4)

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical constrasts in soft X-ray ptychography,” Phys. Rev. Lett.107, 208101 (2011).
[CrossRef] [PubMed]

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, 34801 (2007).
[CrossRef]

V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett.104, 165501 (2010).
[CrossRef] [PubMed]

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett.108, 73901 (2012).
[CrossRef]

Science (1)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science321, 379–382 (2008).
[CrossRef] [PubMed]

Ultramicroscopy (4)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy109, 1256–1262 (2009).
[CrossRef] [PubMed]

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy120, 64–72 (2012).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy107, 227–231 (2007).
[CrossRef]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009).
[CrossRef] [PubMed]

Other (16)

From [52, p. 339], one notes that the constraint defined by the data set in the DM strategy takes the form of Eq. (12b), suggesting that the data fluctuations are described by the Gaussian model defined in Sec. 2.2.

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv:optics (2011).

Since the three fitting functions ℒ𝒫, ℒ𝒢 and ℒℛ are equivalent w.r.t. a nil data, only the data such that ym,j ≠ 0 should be considered in order to discriminate the noise-models.

This non-monotonic behaviour of the relative error is standard when inverse problems (e.g., image restoration or tomographic reconstruction) are solved with gradient optimization technics, see for instance [38, Chap. 6].

Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuation to Information (Springer, 2004).

J. M. Rodenburg, “Ptychography and related diffracted imaging methods,” in “Advances in Imaging and Electron Physics,” 150, P. W. Hawkesed. (Elsevier, 2008), 87–184.
[CrossRef]

R. A. Fisher, Statistical Methods and Scientific Inference (Oliver & Boyd, 1956).

M. G. Kendall and A. Stuart, The advanced theory of statistics2a (Griffin, 1963).

Provided that the fluctuations in one measurement are accurately described by a Poisson PDF, then this PDF is defined by a single (positive) parameter that is the mean and the variance.

The original version of the PIE introduced by Rodenburg and Faulkner in [31] considers another definition for Dj.

By definition, the likelihood is the PDF of the noise model seen as a function of the unknown parameters ρ. In practice, the opposite of the logarithm of the likelihood is rather considered. However, the logarithm function being a monotonic increasing function, the minimiser of the neg-loglikelihood is also the maximiser of the likelihood, i.e., the ML estimator.

Following [33], it is shown that a second order Taylor expansion around hm,j = ym,j of the Poissonian fitting function ℒ𝒫 leads to (10e).

M. Bertero and P. Boccacci, Introduction to inverse problems in imaging (Institute of Physics Publishing, 1998).
[CrossRef]

Since the condition hm,j > 0 is enforced if bm,j > 0 [cf. Eq. (1)], an arbitrary small background component can be introduced, hence allowing all the fitting functions and gradients to be well defined.

In the optimization literature, OS algorithms are also known as incremental gradient methods or block iterative methods, see for instance [42, Sec. 1.5.2 ] or [43] for details.

D. P. Bertsekas, Nonlinear programming, 2nd ed. (Athena Scientific, 1999).

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Figures (7)

Fig. 1
Fig. 1

The test object is a (support-limited) quadrant of a Fresnel zone-plate extending over 100×100 pixels within a 260×260 pixel image. The modulus (a) is 1 within the support of the object and the phase (b) ranges from 0 to 1.72 rad. The corresponding cross-sections are plotted along the 86-th column of the image. A real probe function (c) is chosen so that it extends over 58×58 pixels (full-width at half maximum) within a 100×100 pixel image, corresponding in an oversampling ratio of 1.7; the corresponding cross-section is plotted along the 50-th column of the image.

Fig. 2
Fig. 2

A ptychographical reconstruction illustrates the convergence behavior of the OS and the SG strategies. In these examples, the fitting function is ℒ�� given in (10c), such that the OS strategy corresponds to the standard PIE. Top line: for the OS (dashed line) and the SG (solid line), (a) evolution of the fitting function ℒ�� w.r.t. the iteration k for a noise-free data set (thick line) and an example of a noisy data set (thin line) with a maximum of 103 photons on the camera; (b) idem for the gradient norm ‖��(ρ(k))‖; (c) idem for the error Err(ρ(k)) defined by (25). Second and third lines: reconstruction from a noisy data set; with the OS strategy, (★) is the estimate that minimizes the error depicted in (c) and (□) is the estimate obtained after k =2000; with the SG strategy, (×) is the estimate after k =2000. The shown results correspond to a 180 × 180 window centered around the object central pixel. The respective color scales are indicated on the figure.

Fig. 3
Fig. 3

The average solution for each noise model evaluated over a series of 100 noisy data sets. The initial guess is the true object and the SG strategy is used for the optimization of the fitting function. For each noisy data set, no more than 103 photons impinge on the detector. The grey level scaling in each column shares the same linear scale. The shown results correspond to a 180 × 180 window centered around the object central pixel.

Fig. 4
Fig. 4

Evaluation of the standard deviation for each noise model computed over a series of 100 noisy data sets. The initial guess is the true object and the SG strategy is used for the optimization of the fitting function. For each noisy data set, no more than 103 photons impinge on the camera area.

Fig. 5
Fig. 5

The modulus (left) and the phase (right) that are obtained when high-frequency components are filtered out of the original object shown in Fig. 1.

Fig. 7
Fig. 7

The retrieved phase obtained by the minimization of ℒ�� with, either the hybrid strategy (right), or the SG strategy with the true object as initial guess (left).

Tables (2)

Tables Icon

Table 1 Figure of merit of each noise model. The l2-norms of the bias, standard deviation (STD) and mean-square error (MSE) as well as the error (Err) in the object plane for the fitting functions defined in section 2 are given. The SG strategy is used with the true object as initial guess.

Tables Icon

Table 2 The l2-norms of the bias, STD and MSE as well as Err in the object plane achieved by the fitting functions ℒ��, ℒ�� and ℒ when either the SG or the OS strategy is used with a free-space as initial guess. The results achieved by the DM and the hybrid method are also presented.

Equations (42)

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ψ j ( r ) : = p j ( r ) × ρ ( r )
ρ = { ρ n } n = 1 N with ρ n : = ρ ( r n ) ,
ψ j : = P j ρ
Ψ j : = W ψ j
h m , j : = 𝒜 | Ψ m , j | 2 + b m , j ( m , j )
f 𝒫 ( y ; ρ ) = j m e h m , j × [ h m , j ] y m , j y m , j ! .
y m , j 1 / 2 = h m , j 1 / 2 + ε m , j ( m , j )
f 𝒢 ( y ; ρ ) = j m ( 2 π σ 2 ) 1 / 2 exp ( 1 2 [ y m , j 1 / 2 h m , j 1 / 2 σ ] 2 ) .
y m , j = h m , j + ε m , j ( m , j )
f ( y ; ρ ) = ( m , j ) h m , j 0 ( 2 π h m , j ) 1 / 2 exp ( 1 2 [ y m , j h m , j h m , j 1 / 2 ] 2 ) .
f 𝒬 ( y ; ρ ) = ( m , j ) y m , j 0 ( 2 π y m , j ) 1 / 2 exp ( 1 2 [ y m , j h m , j y m , j 1 / 2 ] 2 ) .
ρ = arg min ρ N ( ρ )
: = log f
= j ; j
𝒫 ; j ( ρ ) : = m h m , j ( ρ ) y m , j log [ h m , j ( ρ ) ]
𝒢 ; j ( ρ ) : = m [ y m , j 1 / 2 h m , j 1 / 2 ( ρ ) ] 2
𝒬 ; j ( ρ ) : = 1 2 m y m , j 0 [ y m , j h m , j ( ρ ) y m , j 1 / 2 ] 2
; j ( ρ ) : = 𝒬 ; j ( ρ ) + m y m , j = 0 h m , j ( ρ ) .
: = j ; j
; j ( ρ ) = P j W [ Ψ j ( ρ ) Ψ ; j ( ρ ) ]
Ψ 𝒫 ; m , j : = Ψ m , j × y m , j h m , j
Ψ 𝒢 ; m , j : Ψ m , j × y m , j 1 / 2 h m , j 1 / 2
Ψ 𝒬 ; m , j : = { Ψ m , j × ( 2 h m , j y m , j ) if y m , j 0 , Ψ m , j otherwise ,
Ψ ; m , j : = { Ψ m , j × ( 2 h m , j y m , j ) if y m , j 0 , 0 otherwise .
j = 1 , , J ρ ( k , 0 ) : = ρ ( k ) ρ ( k , j ) : = ρ ( k , j 1 ) β D j × ; j ( ρ ( k ; j 1 ) ) ρ ( k + 1 ) : = ρ ( k , J )
; j 𝒢 ; j
D j = ( 1 / max m | p m , j | 2 ) × I
ρ ( k + 1 ) : = ρ ( k ) β × Λ 1 ( ρ ( k ) ) β > 0
Λ : = j P j P j + α α 0 .
n , BIAS ; n : = ρ ¯ ; n ρ ; n
n , ρ ¯ ; n : = ρ ; n e ι ϕ
ϕ = arg ( ρ ρ )
n , STD ; n : = | ρ ; n e ι ϕ ρ ¯ ; n | 2 1 / 2 .
n , MSE ; n : = | ρ ; n e ι ϕ ρ ; n | 2
= STD ; n 2 + | BIAS ; n | 2
Ψ 𝒫 ; j = Diag ( y m , j 1 / 2 h m , j 1 / 2 ) × Ψ 𝒢 ; j
Ψ ; j , m 2 ( h m , j 1 / 2 y m , j 1 / 2 ) × Ψ 𝒢 ; j , m
Err ( ρ k ) : = ρ ρ k e ι ϕ k ρ ( with ι 1 )
ϕ k = arg ρ k ρ .
h ( y ) h ( μ ) = ( y μ ) h ( μ ) + R
VAR ( h ( y ) ) = VAR ( h ( y ) h ( μ ) ) = VAR ( ( y μ ) h ( μ ) + R ) = VAR ( ( y μ ) h ( μ ) ) + VAR ( R ) + 2 ( ( y μ ) h ( μ ) R )
VAR ( h ( y ) ) ~ ( h ( μ ) ) 2 VAR ( y μ ) = ( h ( μ ) ) 2 VAR ( y ) = ( h ( μ ) f ( μ ) ) 2 .

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