Abstract

In this work, we demonstrate an improved method for iterative phase retrieval with application to coherent diffractive imaging. By introducing additional operations inside the support term of existing iterated projection algorithms, we demonstrate improved convergence speed, higher success rate and, in some cases, improved reconstruction quality. New algorithms take a particularly simple form with the introduction of a generalized projection-based reflector. Numerical simulations verify that these new algorithms surpass the current standards without adding complexity to the reconstruction process. Thus the introduction of this new class of algorithms offers a new array of methods for efficiently deconvolving intricate data.

© 2012 OSA

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    [CrossRef]
  2. C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
    [CrossRef]
  3. S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).
  4. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400, 342–344 (1999).
    [CrossRef]
  5. C. a. Larabell and K. a. Nugent, “Imaging cellular architecture with x-rays.” Curr. Opin. Struct. Biol.20, 623–631 (2010).
    [CrossRef] [PubMed]
  6. H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  18. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization,” J. Opt. Soc. Am. A vol. 19, 1334–1345 (2002).
    [CrossRef]
  19. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A20, 40–55 (2003).
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  20. H. H. Bauschke, P. L. Combettes, and D. Luke, “Finding best approximation pairs relative to two closed convex sets in Hilbert spaces,” J. Approx. Theory127, 178–192 (2004).
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  24. D. F. Gardner, B. Zhang, M. D. Seaberg, L. S. Martin, D. E. Adams, F. Salmassi, E. Gullikson, H. Kapteyn, and M. Murnane, “High numerical aperture reflection mode coherent diffraction microscopy using off-axis apertured illumination,” Opt. Express20, 19050 (2012).
    [CrossRef] [PubMed]
  25. A. DAlfonso, A. Morgan, A. Martin, H. Quiney, and L. Allen, “Fast deterministic approach to exit-wave reconstruction,” Phys. Rev. A85, 1–9 (2012).
  26. D. Luke, “Local linear convergence of approximate projections onto regularized sets,” Nonlinear Analysis: Theory, Methods & Applications (2011).
    [PubMed]

2012 (6)

J. Miao and R. Sandberg, “Coherent x-ray diffraction imaging,” J. Sel. Top. Quantum Electron.18, 399– 410 (2012).
[CrossRef]

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

D. J. Brady, M. E. Gehm, R. a. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography.” Nature486, 386–389 (2012).
[CrossRef] [PubMed]

D. F. Gardner, B. Zhang, M. D. Seaberg, L. S. Martin, D. E. Adams, F. Salmassi, E. Gullikson, H. Kapteyn, and M. Murnane, “High numerical aperture reflection mode coherent diffraction microscopy using off-axis apertured illumination,” Opt. Express20, 19050 (2012).
[CrossRef] [PubMed]

A. DAlfonso, A. Morgan, A. Martin, H. Quiney, and L. Allen, “Fast deterministic approach to exit-wave reconstruction,” Phys. Rev. A85, 1–9 (2012).

2011 (2)

2010 (3)

C. a. Larabell and K. a. Nugent, “Imaging cellular architecture with x-rays.” Curr. Opin. Struct. Biol.20, 623–631 (2010).
[CrossRef] [PubMed]

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
[CrossRef] [PubMed]

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 529 (2010).
[CrossRef] [PubMed]

2009 (1)

I. Robinson and R. Harder, “Coherent x-ray diffraction imaging of strain at the nanoscale,” Nature Mater.8, 291–298 (2009).
[CrossRef]

2007 (2)

S. Marchesini, “Invited article: a [corrected] unified evaluation of iterative projection algorithms for phase retrieval.” Rev. Sci. Instrum.78, 011301 (2007).
[CrossRef] [PubMed]

S. Marchesini, “Phase retrieval and saddle-point optimization.” J. Opt. Soc. Am. A24, 3289–3296 (2007).
[CrossRef]

2005 (1)

D. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl.21,37 (2005).
[CrossRef]

2004 (1)

H. H. Bauschke, P. L. Combettes, and D. Luke, “Finding best approximation pairs relative to two closed convex sets in Hilbert spaces,” J. Approx. Theory127, 178–192 (2004).
[CrossRef]

2003 (1)

2002 (1)

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400, 342–344 (1999).
[CrossRef]

1984 (1)

1982 (1)

1978 (1)

1972 (1)

R. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35 (1972).

1971 (1)

R. Gerchberg and W. O. Saxton, “Phase determination from Image and diffraction plane pictures in the electron microscope,” Optik34, 275–283 (1971).

Adam, R.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

Adams, D. E.

Aeschlimann, M.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

Allen, L.

A. DAlfonso, A. Morgan, A. Martin, H. Quiney, and L. Allen, “Fast deterministic approach to exit-wave reconstruction,” Phys. Rev. A85, 1–9 (2012).

Bauschke, H. H.

H. H. Bauschke, P. L. Combettes, and D. Luke, “Finding best approximation pairs relative to two closed convex sets in Hilbert spaces,” J. Approx. Theory127, 178–192 (2004).
[CrossRef]

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization,” J. Opt. Soc. Am. A vol. 19, 1334–1345 (2002).
[CrossRef]

Beerlink, A.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 529 (2010).
[CrossRef] [PubMed]

Brady, D. J.

D. J. Brady, M. E. Gehm, R. a. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography.” Nature486, 386–389 (2012).
[CrossRef] [PubMed]

Censor, Y.

Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications. (Oxford University Press: 1997).

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400, 342–344 (1999).
[CrossRef]

Chen, C.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
[CrossRef] [PubMed]

Chen, C.-C.

Combettes, P. L.

H. H. Bauschke, P. L. Combettes, and D. Luke, “Finding best approximation pairs relative to two closed convex sets in Hilbert spaces,” J. Approx. Theory127, 178–192 (2004).
[CrossRef]

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization,” J. Opt. Soc. Am. A vol. 19, 1334–1345 (2002).
[CrossRef]

DAlfonso, A.

A. DAlfonso, A. Morgan, A. Martin, H. Quiney, and L. Allen, “Fast deterministic approach to exit-wave reconstruction,” Phys. Rev. A85, 1–9 (2012).

Dierolf, M.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 529 (2010).
[CrossRef] [PubMed]

Eich, S.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

Elser, V.

Fahimian, B.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
[CrossRef] [PubMed]

Feller, S. D.

D. J. Brady, M. E. Gehm, R. a. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography.” Nature486, 386–389 (2012).
[CrossRef] [PubMed]

Fienup, J.

Gardner, D. F.

Gehm, M. E.

D. J. Brady, M. E. Gehm, R. a. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography.” Nature486, 386–389 (2012).
[CrossRef] [PubMed]

Gerchberg, R.

R. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35 (1972).

R. Gerchberg and W. O. Saxton, “Phase determination from Image and diffraction plane pictures in the electron microscope,” Optik34, 275–283 (1971).

Giewekemeyer, K.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 529 (2010).
[CrossRef] [PubMed]

Goebel, K.

K. Goebel, Topics in metric fixed point theory (Cambridge University Press, 1990).
[CrossRef]

Golish, D. R.

D. J. Brady, M. E. Gehm, R. a. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography.” Nature486, 386–389 (2012).
[CrossRef] [PubMed]

Granitzka, P.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

Grychtol, P.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

Gullikson, E.

Harder, R.

I. Robinson and R. Harder, “Coherent x-ray diffraction imaging of strain at the nanoscale,” Nature Mater.8, 291–298 (2009).
[CrossRef]

Jiang, H.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
[CrossRef] [PubMed]

Kalbfleisch, S.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 529 (2010).
[CrossRef] [PubMed]

Kapteyn, H.

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

D. F. Gardner, B. Zhang, M. D. Seaberg, L. S. Martin, D. E. Adams, F. Salmassi, E. Gullikson, H. Kapteyn, and M. Murnane, “High numerical aperture reflection mode coherent diffraction microscopy using off-axis apertured illumination,” Opt. Express20, 19050 (2012).
[CrossRef] [PubMed]

Kapteyn, H. C.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

M. D. Seaberg, D. E. Adams, E. L. Townsend, D. A. Raymondson, W. F. Schlotter, Y. Liu, C. S. Menoni, L. Rong, C.-C. Chen, J. Miao, H. C. Kapteyn, and M. M. Murnane, “Ultrahigh 22 nm resolution coherent diffractive imaging using a desktop 13 nm high harmonic source.” Opt. Express19, 22470–22479 (2011).
[CrossRef] [PubMed]

Kewish, C.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 529 (2010).
[CrossRef] [PubMed]

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400, 342–344 (1999).
[CrossRef]

Kittle, D. S.

D. J. Brady, M. E. Gehm, R. a. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography.” Nature486, 386–389 (2012).
[CrossRef] [PubMed]

La-O-Vorakiat, C.

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

Larabell, C. a.

C. a. Larabell and K. a. Nugent, “Imaging cellular architecture with x-rays.” Curr. Opin. Struct. Biol.20, 623–631 (2010).
[CrossRef] [PubMed]

Lee, T.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
[CrossRef] [PubMed]

Levi, A.

Liu, Y.

Lu, C.-h.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
[CrossRef] [PubMed]

Luke, D.

D. Luke, “Local linear convergence of approximate projections onto regularized sets,” Nonlinear Analysis: Theory, Methods & Applications (2011).
[PubMed]

D. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl.21,37 (2005).
[CrossRef]

H. H. Bauschke, P. L. Combettes, and D. Luke, “Finding best approximation pairs relative to two closed convex sets in Hilbert spaces,” J. Approx. Theory127, 178–192 (2004).
[CrossRef]

Luke, D. R.

Marchesini, S.

S. Marchesini, “Invited article: a [corrected] unified evaluation of iterative projection algorithms for phase retrieval.” Rev. Sci. Instrum.78, 011301 (2007).
[CrossRef] [PubMed]

S. Marchesini, “Phase retrieval and saddle-point optimization.” J. Opt. Soc. Am. A24, 3289–3296 (2007).
[CrossRef]

Marks, D. L.

D. J. Brady, M. E. Gehm, R. a. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography.” Nature486, 386–389 (2012).
[CrossRef] [PubMed]

Martin, A.

A. DAlfonso, A. Morgan, A. Martin, H. Quiney, and L. Allen, “Fast deterministic approach to exit-wave reconstruction,” Phys. Rev. A85, 1–9 (2012).

Martin, L. S.

Mathias, S.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

Menoni, C. S.

Miao, J.

J. Miao and R. Sandberg, “Coherent x-ray diffraction imaging,” J. Sel. Top. Quantum Electron.18, 399– 410 (2012).
[CrossRef]

M. D. Seaberg, D. E. Adams, E. L. Townsend, D. A. Raymondson, W. F. Schlotter, Y. Liu, C. S. Menoni, L. Rong, C.-C. Chen, J. Miao, H. C. Kapteyn, and M. M. Murnane, “Ultrahigh 22 nm resolution coherent diffractive imaging using a desktop 13 nm high harmonic source.” Opt. Express19, 22470–22479 (2011).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400, 342–344 (1999).
[CrossRef]

Morgan, A.

A. DAlfonso, A. Morgan, A. Martin, H. Quiney, and L. Allen, “Fast deterministic approach to exit-wave reconstruction,” Phys. Rev. A85, 1–9 (2012).

Murnane, M.

D. F. Gardner, B. Zhang, M. D. Seaberg, L. S. Martin, D. E. Adams, F. Salmassi, E. Gullikson, H. Kapteyn, and M. Murnane, “High numerical aperture reflection mode coherent diffraction microscopy using off-axis apertured illumination,” Opt. Express20, 19050 (2012).
[CrossRef] [PubMed]

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

Murnane, M. M.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

M. D. Seaberg, D. E. Adams, E. L. Townsend, D. A. Raymondson, W. F. Schlotter, Y. Liu, C. S. Menoni, L. Rong, C.-C. Chen, J. Miao, H. C. Kapteyn, and M. M. Murnane, “Ultrahigh 22 nm resolution coherent diffractive imaging using a desktop 13 nm high harmonic source.” Opt. Express19, 22470–22479 (2011).
[CrossRef] [PubMed]

Nakashima, A.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
[CrossRef] [PubMed]

Nembach, H.

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

Nembach, H. T.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

Nugent, K. a.

C. a. Larabell and K. a. Nugent, “Imaging cellular architecture with x-rays.” Curr. Opin. Struct. Biol.20, 623–631 (2010).
[CrossRef] [PubMed]

Pfeiffer, F.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 529 (2010).
[CrossRef] [PubMed]

Quiney, H.

A. DAlfonso, A. Morgan, A. Martin, H. Quiney, and L. Allen, “Fast deterministic approach to exit-wave reconstruction,” Phys. Rev. A85, 1–9 (2012).

Raines, K.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
[CrossRef] [PubMed]

Raymondson, D. A.

Robinson, I.

I. Robinson and R. Harder, “Coherent x-ray diffraction imaging of strain at the nanoscale,” Nature Mater.8, 291–298 (2009).
[CrossRef]

Rong, L.

Salditt, T.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 529 (2010).
[CrossRef] [PubMed]

Salmassi, F.

Sandberg, R.

J. Miao and R. Sandberg, “Coherent x-ray diffraction imaging,” J. Sel. Top. Quantum Electron.18, 399– 410 (2012).
[CrossRef]

Saxton, W. O.

R. Gerchberg and W. O. Saxton, “Phase determination from Image and diffraction plane pictures in the electron microscope,” Optik34, 275–283 (1971).

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400, 342–344 (1999).
[CrossRef]

Schlotter, W. F.

Schneider, C.

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

Schneider, C. M.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

Seaberg, M. D.

Shaw, J.

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

Shaw, J. M.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

Siemens, M. E.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

Silva, T.

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

Silva, T. J.

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

Song, C.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
[CrossRef] [PubMed]

Stack, R. a.

D. J. Brady, M. E. Gehm, R. a. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography.” Nature486, 386–389 (2012).
[CrossRef] [PubMed]

Stark, H.

Teale, C.

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

Thibault, P.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 529 (2010).
[CrossRef] [PubMed]

Townsend, E. L.

Turgut, E.

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

Urano, J.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
[CrossRef] [PubMed]

Vera, E. M.

D. J. Brady, M. E. Gehm, R. a. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography.” Nature486, 386–389 (2012).
[CrossRef] [PubMed]

Xu, R.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
[CrossRef] [PubMed]

Zenios, S. A.

Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications. (Oxford University Press: 1997).

Zhang, B.

Appl. Opt. (1)

Curr. Opin. Struct. Biol. (1)

C. a. Larabell and K. a. Nugent, “Imaging cellular architecture with x-rays.” Curr. Opin. Struct. Biol.20, 623–631 (2010).
[CrossRef] [PubMed]

Inverse Probl. (1)

D. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl.21,37 (2005).
[CrossRef]

J. Approx. Theory (1)

H. H. Bauschke, P. L. Combettes, and D. Luke, “Finding best approximation pairs relative to two closed convex sets in Hilbert spaces,” J. Approx. Theory127, 178–192 (2004).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Sel. Top. Quantum Electron. (1)

J. Miao and R. Sandberg, “Coherent x-ray diffraction imaging,” J. Sel. Top. Quantum Electron.18, 399– 410 (2012).
[CrossRef]

Nature (2)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400, 342–344 (1999).
[CrossRef]

D. J. Brady, M. E. Gehm, R. a. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography.” Nature486, 386–389 (2012).
[CrossRef] [PubMed]

Nature Mater. (1)

I. Robinson and R. Harder, “Coherent x-ray diffraction imaging of strain at the nanoscale,” Nature Mater.8, 291–298 (2009).
[CrossRef]

Nonlinear Analysis: Theory, Methods & Applications (1)

D. Luke, “Local linear convergence of approximate projections onto regularized sets,” Nonlinear Analysis: Theory, Methods & Applications (2011).
[PubMed]

Opt. Express (2)

Opt. lett. (1)

Optik (2)

R. Gerchberg and W. O. Saxton, “Phase determination from Image and diffraction plane pictures in the electron microscope,” Optik34, 275–283 (1971).

R. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35 (1972).

Phys. Rev. A (1)

A. DAlfonso, A. Morgan, A. Martin, H. Quiney, and L. Allen, “Fast deterministic approach to exit-wave reconstruction,” Phys. Rev. A85, 1–9 (2012).

Phys. Rev. X (1)

C. La-O-Vorakiat, E. Turgut, C. Teale, H. Kapteyn, M. Murnane, S. Mathias, M. Aeschlimann, C. Schneider, J. Shaw, H. Nembach, and T. Silva, “Ultrafast demagnetization measurements using extreme ultraviolet light: comparison of electronic and magnetic contributions,” Phys. Rev. X2, 1–7 (2012).
[CrossRef]

Proc. Nat. Acad. Sci. USA (3)

S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Probing the timescale of the exchange interaction in a ferromagnetic alloy.” Proc. Nat. Acad. Sci. USA pp. 1–6 (2012).

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C.-h. Lu, T. Lee, A. Nakashima, and J. Urano, and Others, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 11234 (2010).
[CrossRef] [PubMed]

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA107, 529 (2010).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

S. Marchesini, “Invited article: a [corrected] unified evaluation of iterative projection algorithms for phase retrieval.” Rev. Sci. Instrum.78, 011301 (2007).
[CrossRef] [PubMed]

Other (2)

K. Goebel, Topics in metric fixed point theory (Cambridge University Press, 1990).
[CrossRef]

Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications. (Oxford University Press: 1997).

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Figures (4)

Fig. 1
Fig. 1

Generalized reflector, where �� is the set around which the guess u is being reflected. For γ < 1 the reflector is a relaxation parameter, γ = 1 recovers the standard reflector and γ > 1 chooses the depth of reflection.

Fig. 2
Fig. 2

(a) The object used for reconstruction. (b) Simulated diffraction pattern in the presence of noise raised to the quarter power for visibility. (c) The phase profile added to reconstructions when the non-negativity constraint was omitted. (d) Support used for reconstruction when the non-negativity constraint could be used. A similar but tighter support (not shown) was used in simulations without nonnegativity.

Fig. 3
Fig. 3

(a) Diffraction pattern data raised to the quarter power for visibility. (b) Reconstructed object with dashed line showing the boundary of the support, S used for both GIF-RAAR and RAAR. The inset shows a traditional microscope image of the test pattern for comparison (Thorlabs R3L3S1N - Negative 1951 USAF Test Target, 3″ × 3″). (c) Relative error as a function of iteration number for both GIF-RAAR and RAAR respectively.

Fig. 4
Fig. 4

Definition of relevant quantities for convex analysis, considering two sets in ℝ2 for visualization. A and B are convex sets in ℒ, E and F are the points in A and B closest to B and A, respectively. The gap vector g is effectively the shortest distance between A and B.

Tables (2)

Tables Icon

Table 1 Reconstruction quality, convergence speed and success rate without the use of the non-negativity constraint.

Tables Icon

Table 2 Reconstruction quality, convergence speed and success rate with the use of the non-negativity constraint.

Equations (38)

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find u S M
P M ( u ) = 1 v ^ v ^ ( ξ ) = { m ( ξ ) u ( ξ ) | u ( ξ ) | if u ( ξ ) 0 m ( ξ ) otherwise ,
( x N ) ( P S ( u ) ) ( x ) = { u ( x ) if x D 0 otherwise .
R C γ ( 1 + γ ) P C γ I
( x N ) u n + 1 ( x ) = { ( P M ( u n ) ) ( x ) if x D β n u n + ( 1 2 β n ) ( P M ( u n ) ) ( x ) otherwise ,
u n + 1 = ( 1 2 β n ( R S R M + I ) + ( 1 β n ) P M ) ( u n ) ,
u n + 1 = ( 1 2 β n ( 2 P S ( R M ) R M + I ) ( 1 + β n ) P M ) ( u n ) .
u n + 1 = ( 1 2 β n ( 2 P S ( R M α ( I P M ) ) R M + I ) ( 1 + β n ) P M ) ( u n ) ,
u n + 1 = 1 2 β n ( R S R M γ + I ) u n + 1 2 β n ( R M γ R M ) u n + ( 1 β n P M ) u n ,
u n + 1 = ( 1 D [ P M β ( 1 P M ) ] + 1 C D [ β I + ( I 2 β ) P M ] + 1 D [ R M ( I P M ) ] ) ( u n ) .
u n + 1 ( x ) = { ( ( P M ( u n ) β ( I P M ) u n ) ( x ) if x D and ( R M ( I P M ) ) u n ( x ) 0 β n u n ( x ) + ( 1 2 β n ) ( P M ( u n ) ) ( x ) otherwise .
u n + 1 ( x ) = { ( R M β n ( u n ) ) ( x ) if x D and ( R M I + P M ) u n ( x ) 0 β n u n ( x ) + ( 1 2 β n ) ( P M ) ( u n ) ( x ) otherwise .
T * γ 1 2 ( R A R B γ + I ) + 1 2 ( R B γ R B )
T * γ = P A R A γ P B + I ,
𝒱 ( T * γ , β ) β T * γ + ( 1 β ) P B
u F β 1 β g
u = P B u β 1 β g
( u ) u T * γ u = P A R B γ u + f ,
β T * γ u + ( 1 β ) P B u = u ,
u T * γ u = ( 1 β ) β ( P B u u ) = 1 β β y .
P A R B γ u = f + 1 β β y .
( a A ) a P A R B γ u , R B γ u P A R B γ u 0 ,
( a f ) 1 β β y , ( γ + 1 β β ) y 0 .
a + f , y + 1 β β y 2 0 .
b f , y 0
b f , y + a + f , y + 1 β β y 2 0 ,
b a , y 1 β β y 2 0 .
( n ) g n , y 1 β β y 2 .
y β 1 β g .
y = β 1 β ( P A R B γ u f ) .
y = β 1 β P A R B γ u f .
y = β 1 β P A R B γ u f β 1 β g
y = β 1 β g ,
u = F β 1 β g
N A : x { { u : a A , a x , u 0 } x A otherwise
P B u = P B ( f β 1 β g ) = f R B γ u = f + γ β 1 β g
P A R B γ u = P A ( f + γ β 1 β g ) = P A ( e + ( 1 + γ β 1 β ) g ) = e
𝒱 ( T * γ , β ) = β T * γ u + ( 1 β ) P B u = β ( I P B + P A R B γ ) u + ( 1 β ) P B u = β ( u g ) + ( 1 β ) f = u

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