Abstract

I propose the use of a simple real space constraint for iterative phase retrieval intended for diffractive imaging. The proposed constraint is a single parameter equal to the number of nonzero pixels in the image. This greatly simplifies the procedure to determine the constraint. A series of algorithms using this constraint can be easily deduced from existing algorithms, such as hybrid-input–output and difference map algorithms. The high efficiency of these algorithms is largely preserved, as confirmed by numerical studies. A concept widely used in control system theory, proportional-integral–derivative control, is shown to increase the execution speed of the proposed constraint significantly.

© 2006 Optical Society of America

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  1. J. C. H. Spence, M. Howells, L. D. Marks, and J. Miao, 'Lensless imaging: a workshop on 'new approaches to the phase problem for non-periodic objects',' Ultramicroscopy 90, 1-6 (2001).
    [CrossRef]
  2. J. W. 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 (London) 400, 342-344 (1999).
    [CrossRef]
  3. H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, 'Inversion of x-ray diffuse scattering to images using prepared objects,' Phys. Rev. B 67, 174114 (2003).
    [CrossRef]
  4. S. Eisebitt, M. Lorgen, W. Eberhardt, J. Luning, S. Andrews, and J. Stohr, 'Scalable approach for lensless imaging at x-ray wavelengths,' Appl. Phys. Lett. 84, 3373-3375 (2004).
    [CrossRef]
  5. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, 'Atomic resolution imaging of a carbon nanotube from diffraction intensities,' Science 300, 1419-1421 (2003).
    [CrossRef] [PubMed]
  6. I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, 'Reconstruction of the shapes of gold nanocrystals using coherent X-ray diffraction,' Phys. Rev. Lett. 87, 195505 (2001).
    [CrossRef] [PubMed]
  7. 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]
  8. R. W. Gerchberg and W. O. Saxton, 'Practical algorithm for determination of phase from image and diffraction plane pictures,' Optik (Stuttgart) 35, 237-246 (1972).
  9. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, 'X-ray image reconstruction from a diffraction pattern alone,' Phys. Rev. B 68, 140101(R) (2003).
    [CrossRef]
  10. V. Elser, 'Phase retrieval by iterated projections,' J. Opt. Soc. Am. A 20, 40-55 (2003).
    [CrossRef]
  11. G. Oszlanyi and A. Suto, 'Ab initio structure solution by charge flipping,' Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 134-141 (2004).
    [CrossRef]
  12. W. McBride, N. L. O'Leary, and L. J. Allen, 'Retrieval of a complex-valued object from its diffraction pattern,' Phys. Rev. Lett. 93, 233902 (2004).
    [CrossRef] [PubMed]
  13. H. M. L. Faulkner and J. M. Rodenburg, 'Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,' Phys. Rev. Lett. 93, 023903 (2004).
    [CrossRef] [PubMed]
  14. J. S. Wu, U. Weierstall, J. C. H. Spence, and C. T. Koch, 'Iterative phase retrieval without support,' Opt. Lett. 29, 2737-2739 (2004).
    [CrossRef] [PubMed]
  15. In reality, the histogram relies on a comparison operation. Ambiguity arises for comparison between complex values. This is the fundamental reason why histogram constraint or any constraint that relies on comparison is best suited for real-valued problems. For more information, see Section and Appendix .
  16. The histogram in this paper is more often called a cumulative histogram in the digital image processing community. A cumulative histogram is intertransformable to the normal sense histogram through derivative/integral operations and both are equivalent representations of the image's histogram.
  17. J. R. Fienup, 'Phase retrieval algorithms--a comparison,' Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  18. J. Miao, J. Kirz, and D. Sayre, 'The oversampling phasing method,' Acta Crystallogr., Sect. D: Biol. Crystallogr. 56, 1312-1315 (2000).
    [CrossRef]
  19. J. R. Fienup and C. C. Wackerman, 'Phase-retrieval stagnation problems and solutions,' J. Opt. Soc. Am. A 3, 1897-1907 (1986).
    [CrossRef]
  20. H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, 'Use of extended and prepared reference objects in experimental Fourier transform x-ray holography,' Appl. Phys. Lett. 85, 2454-2456 (2004).
    [CrossRef]
  21. S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, 'Lensless imaging of magnetic nanostructures by X-ray spectro-holography,' Nature (London) 432, 885-888 (2004).
    [CrossRef]
  22. K. R. Castleman, Digital Image Processing (Prentice Hall, 1979).

2004 (7)

S. Eisebitt, M. Lorgen, W. Eberhardt, J. Luning, S. Andrews, and J. Stohr, 'Scalable approach for lensless imaging at x-ray wavelengths,' Appl. Phys. Lett. 84, 3373-3375 (2004).
[CrossRef]

G. Oszlanyi and A. Suto, 'Ab initio structure solution by charge flipping,' Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 134-141 (2004).
[CrossRef]

W. McBride, N. L. O'Leary, and L. J. Allen, 'Retrieval of a complex-valued object from its diffraction pattern,' Phys. Rev. Lett. 93, 233902 (2004).
[CrossRef] [PubMed]

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

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, 'Use of extended and prepared reference objects in experimental Fourier transform x-ray holography,' Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, 'Lensless imaging of magnetic nanostructures by X-ray spectro-holography,' Nature (London) 432, 885-888 (2004).
[CrossRef]

J. S. Wu, U. Weierstall, J. C. H. Spence, and C. T. Koch, 'Iterative phase retrieval without support,' Opt. Lett. 29, 2737-2739 (2004).
[CrossRef] [PubMed]

2003 (4)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, 'X-ray image reconstruction from a diffraction pattern alone,' Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, 'Inversion of x-ray diffuse scattering to images using prepared objects,' Phys. Rev. B 67, 174114 (2003).
[CrossRef]

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

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, 'Atomic resolution imaging of a carbon nanotube from diffraction intensities,' Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

2001 (2)

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, 'Reconstruction of the shapes of gold nanocrystals using coherent X-ray diffraction,' Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

J. C. H. Spence, M. Howells, L. D. Marks, and J. Miao, 'Lensless imaging: a workshop on 'new approaches to the phase problem for non-periodic objects',' Ultramicroscopy 90, 1-6 (2001).
[CrossRef]

2000 (1)

J. Miao, J. Kirz, and D. Sayre, 'The oversampling phasing method,' Acta Crystallogr., Sect. D: Biol. Crystallogr. 56, 1312-1315 (2000).
[CrossRef]

1999 (1)

J. W. 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 (London) 400, 342-344 (1999).
[CrossRef]

1987 (1)

1986 (1)

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, 'Practical algorithm for determination of phase from image and diffraction plane pictures,' Optik (Stuttgart) 35, 237-246 (1972).

Allen, L. J.

W. McBride, N. L. O'Leary, and L. J. Allen, 'Retrieval of a complex-valued object from its diffraction pattern,' Phys. Rev. Lett. 93, 233902 (2004).
[CrossRef] [PubMed]

Andrews, S.

S. Eisebitt, M. Lorgen, W. Eberhardt, J. Luning, S. Andrews, and J. Stohr, 'Scalable approach for lensless imaging at x-ray wavelengths,' Appl. Phys. Lett. 84, 3373-3375 (2004).
[CrossRef]

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice Hall, 1979).

Chapman, H.

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, 'Inversion of x-ray diffuse scattering to images using prepared objects,' Phys. Rev. B 67, 174114 (2003).
[CrossRef]

Chapman, H. N.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, 'Use of extended and prepared reference objects in experimental Fourier transform x-ray holography,' Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, 'X-ray image reconstruction from a diffraction pattern alone,' Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Charalambous, P.

J. W. 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 (London) 400, 342-344 (1999).
[CrossRef]

Eberhardt, W.

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, 'Lensless imaging of magnetic nanostructures by X-ray spectro-holography,' Nature (London) 432, 885-888 (2004).
[CrossRef]

S. Eisebitt, M. Lorgen, W. Eberhardt, J. Luning, S. Andrews, and J. Stohr, 'Scalable approach for lensless imaging at x-ray wavelengths,' Appl. Phys. Lett. 84, 3373-3375 (2004).
[CrossRef]

Eisebitt, S.

S. Eisebitt, M. Lorgen, W. Eberhardt, J. Luning, S. Andrews, and J. Stohr, 'Scalable approach for lensless imaging at x-ray wavelengths,' Appl. Phys. Lett. 84, 3373-3375 (2004).
[CrossRef]

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, 'Lensless imaging of magnetic nanostructures by X-ray spectro-holography,' Nature (London) 432, 885-888 (2004).
[CrossRef]

Elser, V.

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] [PubMed]

Fienup, J. R.

Gao, M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, 'Atomic resolution imaging of a carbon nanotube from diffraction intensities,' Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, 'Practical algorithm for determination of phase from image and diffraction plane pictures,' Optik (Stuttgart) 35, 237-246 (1972).

Hau-Riege, S.

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, 'Inversion of x-ray diffuse scattering to images using prepared objects,' Phys. Rev. B 67, 174114 (2003).
[CrossRef]

Hau-Riege, S. P.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, 'X-ray image reconstruction from a diffraction pattern alone,' Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

He, H.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, 'Use of extended and prepared reference objects in experimental Fourier transform x-ray holography,' Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, 'X-ray image reconstruction from a diffraction pattern alone,' Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, 'Inversion of x-ray diffuse scattering to images using prepared objects,' Phys. Rev. B 67, 174114 (2003).
[CrossRef]

Hellwig, O.

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, 'Lensless imaging of magnetic nanostructures by X-ray spectro-holography,' Nature (London) 432, 885-888 (2004).
[CrossRef]

Howells, M.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, 'Use of extended and prepared reference objects in experimental Fourier transform x-ray holography,' Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, 'Inversion of x-ray diffuse scattering to images using prepared objects,' Phys. Rev. B 67, 174114 (2003).
[CrossRef]

J. C. H. Spence, M. Howells, L. D. Marks, and J. Miao, 'Lensless imaging: a workshop on 'new approaches to the phase problem for non-periodic objects',' Ultramicroscopy 90, 1-6 (2001).
[CrossRef]

Howells, M. R.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, 'X-ray image reconstruction from a diffraction pattern alone,' Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Kirz, J.

J. Miao, J. Kirz, and D. Sayre, 'The oversampling phasing method,' Acta Crystallogr., Sect. D: Biol. Crystallogr. 56, 1312-1315 (2000).
[CrossRef]

J. W. 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 (London) 400, 342-344 (1999).
[CrossRef]

Koch, C. T.

Lorgen, M.

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, 'Lensless imaging of magnetic nanostructures by X-ray spectro-holography,' Nature (London) 432, 885-888 (2004).
[CrossRef]

S. Eisebitt, M. Lorgen, W. Eberhardt, J. Luning, S. Andrews, and J. Stohr, 'Scalable approach for lensless imaging at x-ray wavelengths,' Appl. Phys. Lett. 84, 3373-3375 (2004).
[CrossRef]

Luning, J.

S. Eisebitt, M. Lorgen, W. Eberhardt, J. Luning, S. Andrews, and J. Stohr, 'Scalable approach for lensless imaging at x-ray wavelengths,' Appl. Phys. Lett. 84, 3373-3375 (2004).
[CrossRef]

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, 'Lensless imaging of magnetic nanostructures by X-ray spectro-holography,' Nature (London) 432, 885-888 (2004).
[CrossRef]

Marchesini, S.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, 'Use of extended and prepared reference objects in experimental Fourier transform x-ray holography,' Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, 'X-ray image reconstruction from a diffraction pattern alone,' Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, 'Inversion of x-ray diffuse scattering to images using prepared objects,' Phys. Rev. B 67, 174114 (2003).
[CrossRef]

Marks, L. D.

J. C. H. Spence, M. Howells, L. D. Marks, and J. Miao, 'Lensless imaging: a workshop on 'new approaches to the phase problem for non-periodic objects',' Ultramicroscopy 90, 1-6 (2001).
[CrossRef]

McBride, W.

W. McBride, N. L. O'Leary, and L. J. Allen, 'Retrieval of a complex-valued object from its diffraction pattern,' Phys. Rev. Lett. 93, 233902 (2004).
[CrossRef] [PubMed]

Miao, J.

J. C. H. Spence, M. Howells, L. D. Marks, and J. Miao, 'Lensless imaging: a workshop on 'new approaches to the phase problem for non-periodic objects',' Ultramicroscopy 90, 1-6 (2001).
[CrossRef]

J. Miao, J. Kirz, and D. Sayre, 'The oversampling phasing method,' Acta Crystallogr., Sect. D: Biol. Crystallogr. 56, 1312-1315 (2000).
[CrossRef]

Miao, J. W.

J. W. 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 (London) 400, 342-344 (1999).
[CrossRef]

Nagahara, L. A.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, 'Atomic resolution imaging of a carbon nanotube from diffraction intensities,' Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Noy, A.

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, 'Inversion of x-ray diffuse scattering to images using prepared objects,' Phys. Rev. B 67, 174114 (2003).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, 'X-ray image reconstruction from a diffraction pattern alone,' Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

O'Leary, N. L.

W. McBride, N. L. O'Leary, and L. J. Allen, 'Retrieval of a complex-valued object from its diffraction pattern,' Phys. Rev. Lett. 93, 233902 (2004).
[CrossRef] [PubMed]

Oszlanyi, G.

G. Oszlanyi and A. Suto, 'Ab initio structure solution by charge flipping,' Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 134-141 (2004).
[CrossRef]

Padmore, H. A.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, 'Use of extended and prepared reference objects in experimental Fourier transform x-ray holography,' Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

Pfeifer, M. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, 'Reconstruction of the shapes of gold nanocrystals using coherent X-ray diffraction,' Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Pitney, J. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, 'Reconstruction of the shapes of gold nanocrystals using coherent X-ray diffraction,' Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Robinson, I. K.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, 'Reconstruction of the shapes of gold nanocrystals using coherent X-ray diffraction,' Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

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] [PubMed]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, 'Practical algorithm for determination of phase from image and diffraction plane pictures,' Optik (Stuttgart) 35, 237-246 (1972).

Sayre, D.

J. Miao, J. Kirz, and D. Sayre, 'The oversampling phasing method,' Acta Crystallogr., Sect. D: Biol. Crystallogr. 56, 1312-1315 (2000).
[CrossRef]

J. W. 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 (London) 400, 342-344 (1999).
[CrossRef]

Schlotter, W. F.

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, 'Lensless imaging of magnetic nanostructures by X-ray spectro-holography,' Nature (London) 432, 885-888 (2004).
[CrossRef]

Spence, J. C. H.

J. S. Wu, U. Weierstall, J. C. H. Spence, and C. T. Koch, 'Iterative phase retrieval without support,' Opt. Lett. 29, 2737-2739 (2004).
[CrossRef] [PubMed]

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, 'Use of extended and prepared reference objects in experimental Fourier transform x-ray holography,' Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, 'X-ray image reconstruction from a diffraction pattern alone,' Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, 'Inversion of x-ray diffuse scattering to images using prepared objects,' Phys. Rev. B 67, 174114 (2003).
[CrossRef]

J. C. H. Spence, M. Howells, L. D. Marks, and J. Miao, 'Lensless imaging: a workshop on 'new approaches to the phase problem for non-periodic objects',' Ultramicroscopy 90, 1-6 (2001).
[CrossRef]

Stohr, J.

S. Eisebitt, M. Lorgen, W. Eberhardt, J. Luning, S. Andrews, and J. Stohr, 'Scalable approach for lensless imaging at x-ray wavelengths,' Appl. Phys. Lett. 84, 3373-3375 (2004).
[CrossRef]

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, 'Lensless imaging of magnetic nanostructures by X-ray spectro-holography,' Nature (London) 432, 885-888 (2004).
[CrossRef]

Suto, A.

G. Oszlanyi and A. Suto, 'Ab initio structure solution by charge flipping,' Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 134-141 (2004).
[CrossRef]

Vartanyants, I.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, 'Atomic resolution imaging of a carbon nanotube from diffraction intensities,' Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Vartanyants, I. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, 'Reconstruction of the shapes of gold nanocrystals using coherent X-ray diffraction,' Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Wackerman, C. C.

Weierstall, U.

J. S. Wu, U. Weierstall, J. C. H. Spence, and C. T. Koch, 'Iterative phase retrieval without support,' Opt. Lett. 29, 2737-2739 (2004).
[CrossRef] [PubMed]

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, 'Use of extended and prepared reference objects in experimental Fourier transform x-ray holography,' Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, 'X-ray image reconstruction from a diffraction pattern alone,' Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, 'Inversion of x-ray diffuse scattering to images using prepared objects,' Phys. Rev. B 67, 174114 (2003).
[CrossRef]

Williams, G. J.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, 'Reconstruction of the shapes of gold nanocrystals using coherent X-ray diffraction,' Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Wu, J. S.

Zhang, R.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, 'Atomic resolution imaging of a carbon nanotube from diffraction intensities,' Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Zuo, J. M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, 'Atomic resolution imaging of a carbon nanotube from diffraction intensities,' Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Acta Crystallogr., Sect. A: Found. Crystallogr. (1)

G. Oszlanyi and A. Suto, 'Ab initio structure solution by charge flipping,' Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 134-141 (2004).
[CrossRef]

Acta Crystallogr., Sect. D: Biol. Crystallogr. (1)

J. Miao, J. Kirz, and D. Sayre, 'The oversampling phasing method,' Acta Crystallogr., Sect. D: Biol. Crystallogr. 56, 1312-1315 (2000).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, 'Use of extended and prepared reference objects in experimental Fourier transform x-ray holography,' Appl. Phys. Lett. 85, 2454-2456 (2004).
[CrossRef]

S. Eisebitt, M. Lorgen, W. Eberhardt, J. Luning, S. Andrews, and J. Stohr, 'Scalable approach for lensless imaging at x-ray wavelengths,' Appl. Phys. Lett. 84, 3373-3375 (2004).
[CrossRef]

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

Nature (London) (2)

J. W. 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 (London) 400, 342-344 (1999).
[CrossRef]

S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, 'Lensless imaging of magnetic nanostructures by X-ray spectro-holography,' Nature (London) 432, 885-888 (2004).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

R. W. Gerchberg and W. O. Saxton, 'Practical algorithm for determination of phase from image and diffraction plane pictures,' Optik (Stuttgart) 35, 237-246 (1972).

Phys. Rev. B (2)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, 'X-ray image reconstruction from a diffraction pattern alone,' Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, 'Inversion of x-ray diffuse scattering to images using prepared objects,' Phys. Rev. B 67, 174114 (2003).
[CrossRef]

Phys. Rev. Lett. (3)

W. McBride, N. L. O'Leary, and L. J. Allen, 'Retrieval of a complex-valued object from its diffraction pattern,' Phys. Rev. Lett. 93, 233902 (2004).
[CrossRef] [PubMed]

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

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, 'Reconstruction of the shapes of gold nanocrystals using coherent X-ray diffraction,' Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Science (1)

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, 'Atomic resolution imaging of a carbon nanotube from diffraction intensities,' Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Ultramicroscopy (1)

J. C. H. Spence, M. Howells, L. D. Marks, and J. Miao, 'Lensless imaging: a workshop on 'new approaches to the phase problem for non-periodic objects',' Ultramicroscopy 90, 1-6 (2001).
[CrossRef]

Other (3)

In reality, the histogram relies on a comparison operation. Ambiguity arises for comparison between complex values. This is the fundamental reason why histogram constraint or any constraint that relies on comparison is best suited for real-valued problems. For more information, see Section and Appendix .

The histogram in this paper is more often called a cumulative histogram in the digital image processing community. A cumulative histogram is intertransformable to the normal sense histogram through derivative/integral operations and both are equivalent representations of the image's histogram.

K. R. Castleman, Digital Image Processing (Prentice Hall, 1979).

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Figures (2)

Fig. 1
Fig. 1

(a) Semilog plot of the descending sorted values versus pixel number (see text for details), which is the (cumulative) histogram of the real and positive ( 256 × 256 ) image in (b). Only the nonzero part is plotted. The histogram is normalized to the maximum value. The inset of (a) is a linear plot of the same histogram.

Fig. 2
Fig. 2

(a) Typical evolution of the R factors versus the elapsed iteration time for cases with (solid curve) and without (dashed curve) PID control (a starting value of T NNP = 0.5 is used). (b) Dynamically measured, unchanged pixel number [ NNP of Eq. (1)] versus iteration number for PID control. The dotted line is the NNP.

Tables (1)

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Table 1 Success Rate of Reconstructing a Real- and Positive-Valued Image [Fig. 1b] for Typical Algorithms a

Equations (10)

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T NNP n = T NNP c + P Δ N = T NNP c + P ( NNP NNP ) ,
P = { ( 1 T NNP c ) 2 ( N NNP ) if NNP > NNP 0 if NNP = NNP T NNP c 2 NNP if NNP < NNP } ,
g = π HIO [ π mod ( g ) ] = π mod ( g ) S + [ g β π mod ( g ) ] ( 1 S )
= π GS [ ( 1 + β ) π mod ( g ) g ] + [ g β π mod ( g ) ] ,
S i j = { 1 , if π mod ( g ) i j , the norm , is among the NNP largest 0 , otherwise } .
g = g + β { π 1 [ ( 1 + γ 2 ) π 2 ( g ) γ 2 g ] π 2 [ ( 1 + γ 1 ) π 1 ( g ) γ 1 g ] } ,
g = g + { π 1 [ 2 π 2 ( g ) g ] π 2 [ g ] } .
g = π CF [ π mod ( g ) ] = π mod ( g ) S π mod ( g ) ( 1 S ) .
S i j = { 1 , if Re [ π mod ( g ) ] i j > t 0 , otherwise } ;
R = ( u , v ) F cal ( u , v ) F obs ( u , v ) ( u , v ) F obs ( u , v ) ,

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