Abstract

The twin-image problem in phase retrieval is characterized by the simultaneous occurrence of features from the original object and its inversion about the origin (twin image). This problem can occur in reconstructions for which the object support is centrosymmetric or loose, and in severe cases it can greatly hinder image quality. In this paper we examine this problem and find that it arises when the retrieved Fourier-domain phase is divided into sets of regions, some of which reconstruct the object while others the twin. We examine sample reconstructions that present the twin-image problem to different extents and find that, even when the twin-image problem is not visually evident, it can exist in small regions of the retrieved Fourier phase. The reduced-support constraint approach is shown to be effective in escaping stagnation caused by the twin-image problem.

© 2012 Optical Society of America

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  1. R. A. Gonsalves and R. Childlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).
  2. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [CrossRef]
  3. G. R. Brady and J. R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express 14, 474–486 (2006).
    [CrossRef]
  4. M. R. Bolcar and J. R. Fienup, “Sub-aperture piston phase diversity for segmented and multi-aperture systems,” Appl. Opt. 48, A5–A12 (2009).
    [CrossRef]
  5. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef]
  6. J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14, 498–508 (2006).
    [CrossRef]
  7. M. Guizar-Sicairos, and J. R. Fienup, “Phase retrieval with Fourier-weighted projections,” J. Opt. Soc. Am. A 25, 701–709 (2008).
    [CrossRef]
  8. 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,” Nature 400, 342–344 (1999).
    [CrossRef]
  9. D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
    [CrossRef]
  10. H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).
  12. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef]
  13. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
    [CrossRef]
  14. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Hybrid projection-reflection method for phase retrieval,” J. Opt. Soc. Am. A 20, 1025–1034 (2003).
    [CrossRef]
  15. C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input—output method,” Phys. Rev. B 76, 064113 (2007).
    [CrossRef]
  16. R. H. T. Bates and D. G. H. Tan, “Fourier phase retrieval when the image is complex,” Proc. SPIE 0558, 54–59 (1985).
    [CrossRef]
  17. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  18. R. G. Paxman, J. R. Fienup, and J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” Proc. SPIE 0828, 184–189 (1987).
  19. J. R. Fienup, T. R. Crimmins, and W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,” J. Opt. Soc. Am. 72, 610–624 (1982).
    [CrossRef]
  20. T. R. Crimmins, J. R. Fienup, and B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 3–13 (1990).
    [CrossRef]
  21. J. R. Fienup, B. J. Thelen, M. F. Reiley, and R. G. Paxman, “3-D locator sets for opaque objects for phase retrieval,” Proc. SPIE 3170, 88–96 (1997).
    [CrossRef]
  22. M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express 15, 17592–17612 (2007).
    [CrossRef]
  23. Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  24. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  25. E. Lima, L. Wiegart, P. Pernot, M. Howells, J. Timmins, F. Zontone, and A. Madsen, “Cryogenic x-ray diffraction microscopy for biological samples,” Phys. Rev. Lett. 103, 198102 (2009).
    [CrossRef]
  26. H. Chapman, “X-ray imaging beyond the limits,” Nat. Mater. 8, 299–301 (2009).
    [CrossRef]
  27. 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]
  28. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
    [CrossRef]
  29. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
    [CrossRef]
  30. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
    [CrossRef]
  31. J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. 36, 8352–8357 (1997).
    [CrossRef]

2009

E. Lima, L. Wiegart, P. Pernot, M. Howells, J. Timmins, F. Zontone, and A. Madsen, “Cryogenic x-ray diffraction microscopy for biological samples,” Phys. Rev. Lett. 103, 198102 (2009).
[CrossRef]

H. Chapman, “X-ray imaging beyond the limits,” Nat. Mater. 8, 299–301 (2009).
[CrossRef]

M. R. Bolcar and J. R. Fienup, “Sub-aperture piston phase diversity for segmented and multi-aperture systems,” Appl. Opt. 48, A5–A12 (2009).
[CrossRef]

2008

2007

C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input—output method,” Phys. Rev. B 76, 064113 (2007).
[CrossRef]

M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express 15, 17592–17612 (2007).
[CrossRef]

2006

2005

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[CrossRef]

2004

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]

2003

1999

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,” Nature 400, 342–344 (1999).
[CrossRef]

1997

J. R. Fienup, B. J. Thelen, M. F. Reiley, and R. G. Paxman, “3-D locator sets for opaque objects for phase retrieval,” Proc. SPIE 3170, 88–96 (1997).
[CrossRef]

J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. 36, 8352–8357 (1997).
[CrossRef]

1993

1990

1987

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[CrossRef]

R. G. Paxman, J. R. Fienup, and J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” Proc. SPIE 0828, 184–189 (1987).

1986

1985

R. H. T. Bates and D. G. H. Tan, “Fourier phase retrieval when the image is complex,” Proc. SPIE 0558, 54–59 (1985).
[CrossRef]

1982

1979

R. A. Gonsalves and R. Childlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

1978

Barty, A.

Bates, R. H. T.

R. H. T. Bates and D. G. H. Tan, “Fourier phase retrieval when the image is complex,” Proc. SPIE 0558, 54–59 (1985).
[CrossRef]

Bauschke, H. H.

Beetz, T.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[CrossRef]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[CrossRef]

Bolcar, M. R.

Brady, G. R.

Bruck, Y. M.

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Bunk, O.

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

Chapman, H.

H. Chapman, “X-ray imaging beyond the limits,” Nat. Mater. 8, 299–301 (2009).
[CrossRef]

Chapman, H. N.

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,” Nature 400, 342–344 (1999).
[CrossRef]

Chen, C.

C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input—output method,” Phys. Rev. B 76, 064113 (2007).
[CrossRef]

Childlaw, R.

R. A. Gonsalves and R. Childlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

Clinthorne, J. T.

R. G. Paxman, J. R. Fienup, and J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” Proc. SPIE 0828, 184–189 (1987).

Combettes, P. L.

Crimmins, T. R.

Cui, C.

David, C.

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

Dierolf, M.

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

Elser, V.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[CrossRef]

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[CrossRef]

Faulkner, H. M. L.

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]

Fienup, J. R.

M. R. Bolcar and J. R. Fienup, “Sub-aperture piston phase diversity for segmented and multi-aperture systems,” Appl. Opt. 48, A5–A12 (2009).
[CrossRef]

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

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
[CrossRef]

M. Guizar-Sicairos, and J. R. Fienup, “Phase retrieval with Fourier-weighted projections,” J. Opt. Soc. Am. A 25, 701–709 (2008).
[CrossRef]

M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express 15, 17592–17612 (2007).
[CrossRef]

G. R. Brady and J. R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express 14, 474–486 (2006).
[CrossRef]

J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14, 498–508 (2006).
[CrossRef]

J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. 36, 8352–8357 (1997).
[CrossRef]

J. R. Fienup, B. J. Thelen, M. F. Reiley, and R. G. Paxman, “3-D locator sets for opaque objects for phase retrieval,” Proc. SPIE 3170, 88–96 (1997).
[CrossRef]

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[CrossRef]

T. R. Crimmins, J. R. Fienup, and B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 3–13 (1990).
[CrossRef]

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[CrossRef]

R. G. Paxman, J. R. Fienup, and J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” Proc. SPIE 0828, 184–189 (1987).

J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef]

J. R. Fienup, T. R. Crimmins, and W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,” J. Opt. Soc. Am. 72, 610–624 (1982).
[CrossRef]

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef]

Gonsalves, R. A.

R. A. Gonsalves and R. Childlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

Guizar-Sicairos, M.

Hau-Riege, S. P.

He, H.

Holsztynski, W.

Howells, M.

E. Lima, L. Wiegart, P. Pernot, M. Howells, J. Timmins, F. Zontone, and A. Madsen, “Cryogenic x-ray diffraction microscopy for biological samples,” Phys. Rev. Lett. 103, 198102 (2009).
[CrossRef]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[CrossRef]

Howells, M. R.

Jacobsen, C.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[CrossRef]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[CrossRef]

Kirz, J.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[CrossRef]

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,” Nature 400, 342–344 (1999).
[CrossRef]

Lee, T. K.

C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input—output method,” Phys. Rev. B 76, 064113 (2007).
[CrossRef]

Lima, E.

E. Lima, L. Wiegart, P. Pernot, M. Howells, J. Timmins, F. Zontone, and A. Madsen, “Cryogenic x-ray diffraction microscopy for biological samples,” Phys. Rev. Lett. 103, 198102 (2009).
[CrossRef]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[CrossRef]

Luke, D. R.

Madsen, A.

E. Lima, L. Wiegart, P. Pernot, M. Howells, J. Timmins, F. Zontone, and A. Madsen, “Cryogenic x-ray diffraction microscopy for biological samples,” Phys. Rev. Lett. 103, 198102 (2009).
[CrossRef]

Marchesini, S.

Menzel, A.

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

Miao, H.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[CrossRef]

Miao, J.

C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input—output method,” Phys. Rev. B 76, 064113 (2007).
[CrossRef]

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,” Nature 400, 342–344 (1999).
[CrossRef]

Neiman, A. M.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[CrossRef]

Noy, A.

Paxman, R. G.

J. R. Fienup, B. J. Thelen, M. F. Reiley, and R. G. Paxman, “3-D locator sets for opaque objects for phase retrieval,” Proc. SPIE 3170, 88–96 (1997).
[CrossRef]

R. G. Paxman, J. R. Fienup, and J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” Proc. SPIE 0828, 184–189 (1987).

Pernot, P.

E. Lima, L. Wiegart, P. Pernot, M. Howells, J. Timmins, F. Zontone, and A. Madsen, “Cryogenic x-ray diffraction microscopy for biological samples,” Phys. Rev. Lett. 103, 198102 (2009).
[CrossRef]

Pfeiffer, F.

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

Reiley, M. F.

J. R. Fienup, B. J. Thelen, M. F. Reiley, and R. G. Paxman, “3-D locator sets for opaque objects for phase retrieval,” Proc. SPIE 3170, 88–96 (1997).
[CrossRef]

Rodenburg, J. M.

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]

Rosen, R.

Sayre, D.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[CrossRef]

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,” Nature 400, 342–344 (1999).
[CrossRef]

Shapiro, D.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[CrossRef]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[CrossRef]

Sodin, L. G.

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Spence, J. C. H.

Tan, D. G. H.

R. H. T. Bates and D. G. H. Tan, “Fourier phase retrieval when the image is complex,” Proc. SPIE 0558, 54–59 (1985).
[CrossRef]

Thelen, B. J.

J. R. Fienup, B. J. Thelen, M. F. Reiley, and R. G. Paxman, “3-D locator sets for opaque objects for phase retrieval,” Proc. SPIE 3170, 88–96 (1997).
[CrossRef]

T. R. Crimmins, J. R. Fienup, and B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 3–13 (1990).
[CrossRef]

Thibault, P.

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

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[CrossRef]

Thurman, S. T.

Timmins, J.

E. Lima, L. Wiegart, P. Pernot, M. Howells, J. Timmins, F. Zontone, and A. Madsen, “Cryogenic x-ray diffraction microscopy for biological samples,” Phys. Rev. Lett. 103, 198102 (2009).
[CrossRef]

Wackerman, C. C.

Wang, C. W.

C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input—output method,” Phys. Rev. B 76, 064113 (2007).
[CrossRef]

Weierstall, U.

Wiegart, L.

E. Lima, L. Wiegart, P. Pernot, M. Howells, J. Timmins, F. Zontone, and A. Madsen, “Cryogenic x-ray diffraction microscopy for biological samples,” Phys. Rev. Lett. 103, 198102 (2009).
[CrossRef]

Zontone, F.

E. Lima, L. Wiegart, P. Pernot, M. Howells, J. Timmins, F. Zontone, and A. Madsen, “Cryogenic x-ray diffraction microscopy for biological samples,” Phys. Rev. Lett. 103, 198102 (2009).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nat. Mater.

H. Chapman, “X-ray imaging beyond the limits,” Nat. Mater. 8, 299–301 (2009).
[CrossRef]

Nature

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,” Nature 400, 342–344 (1999).
[CrossRef]

Opt. Commun.

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Supplementary Material (4)

» Media 1: MOV (4357 KB)     
» Media 2: MOV (4369 KB)     
» Media 3: MOV (3925 KB)     
» Media 4: MOV (2985 KB)     

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Figures (12)

Fig. 1.
Fig. 1.

Object (a) magnitude and (b) phase; Fourier (c) magnitude and (d) phase. Because of its large dynamic range, the magnitude is displayed in (c) raised to the 1/5 power. Phase is shown from 0.5 to 0.5 rad in (b) and from π to π in (d).

Fig. 2.
Fig. 2.

Image reconstruction exhibiting a pronounced twin-image problem. (a) The stagnated image magnitude; the Fourier phase errors with respect to (c) the upright ideal image, f(x,y), and (d) the twin image, f*(x,y), and (b) regions of the Fourier domain that reconstruct either the upright image (bright red), the twin image (blue), or neither (black) after 1000 iterations. Phase error is shown from π to π radians in (c) and (d). The evolution of the regions with iteration number is shown in Media 1.

Fig. 3.
Fig. 3.

Fourier filtering of the ideal image and the reconstruction shown in Fig. 2(a) using the Fourier regions shown in Fig. 2(b). Filtering of the (a) ideal image and (b) reconstruction using the bright red Fourier regions. Filtering of the (c) ideal image and (d) reconstruction using the blue Fourier regions.

Fig. 4.
Fig. 4.

(a) Support error for the reconstruction shown in Fig. 2(a) versus iteration number. (b) NRMSE of the reconstruction with respect to the ideal image versus iteration number. (c) Energy fraction of the reconstruction versus iteration number for the Fourier regions that reconstruct the upright image (solid curve) and the twin image (dashed curve). The sum of the two energy fractions is indicated by a dotted curve in (c).

Fig. 5.
Fig. 5.

Same as Fig. 2 but for an image reconstruction exhibiting a moderate twin-image problem. The evolution of the regions with iteration number is shown in Media 2.

Fig. 6.
Fig. 6.

Same as Fig. 4, but for the reconstruction shown in Fig. 5(a).

Fig. 7.
Fig. 7.

Same as Fig. 3, but for the reconstructed image shown in Fig. 5(a).

Fig. 8.
Fig. 8.

Same as Fig. 2 but for an image reconstruction where the twin-image problem cannot be detected visually. The evolution of the regions with iteration number is shown in Media 3.

Fig. 9.
Fig. 9.

Same as Fig. 4 but for the reconstruction shown in Fig. 8(a).

Fig. 10.
Fig. 10.

(a) Stagnated reconstruction after 200 iterations and (b) its Fourier regions. (c) Reconstruction after 10 iterations using the reduced-support method and (d) its Fourier regions. (e) Reconstruction after reinstating the original support constraint and performing 10 more iterations and (f) its Fourier regions. Regions of the far field that reconstruct either the upright or twin image are shown in bright red or blue, respectively, in (b), (d), and (f).

Fig. 11.
Fig. 11.

Same as Fig. 5, from same starting guess but using the reduced-support method in iteration 200. The evolution of the regions with iteration number is shown in Media 4.

Fig. 12.
Fig. 12.

Same as Fig. 4, but for the reconstruction shown in Fig. 11, using the reduced-support method. The final support error was E=2.1×103.

Equations (6)

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F(u,v)=|F(u,v)|exp[iϕ(u,v)]=1MNx,yf(x,y)exp[i2π(uxM+vyN)],
f(x,y)=0,(x,y)S,
F*(u,v)=|F(u,v)|exp[iϕ(u,v)].
ε2=minα,x0,y0u,v|αG(u,v)exp[i2π(ux0M+vy0N)]F(u,v)|2u,v|F(u,v)|2,
ε2=minα,x0,y0x,y|αg(xx0,yy0)f(x,y)|2x,y|f(x,y)|2,
E2=(x,y)S|g(x,y)|2x,y|g(x,y)|2,

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