Abstract

A new method of phase retrieval from a single near-field diffraction image with a large Fresnel number is presented and discussed. This method requires only the oversampled diffraction pattern without any other information such as the object envelope. Moreover, we show that the combination with a fast computational method is possible when the linear oversampling ratio is an integer. Numerical simulations are also presented, showing that the method works well with noisy data.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662-1669 (1998).
    [CrossRef]
  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,” Nature (London) 400, 342-344 (1999).
    [CrossRef]
  3. 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]
  4. G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
    [CrossRef] [PubMed]
  5. D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843-843 (1952).
    [CrossRef]
  6. J. Miao, K. Hodgson, and D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. U.S.A. 98, 6641-6645 (2001).
    [CrossRef] [PubMed]
  7. I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71, 245302 (2005).
    [CrossRef]
  8. 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. U.S.A. 102, 15343-15346 (2005).
    [CrossRef] [PubMed]
  9. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).
  10. J. R. Fienup, “Phase retrieval algorithms: a comparison.” Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  11. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40-55 (2003).
    [CrossRef]
  12. Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101 (2003).
    [CrossRef]
  13. 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 (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. H. M. Quiney, K. A. Nugent, and A. G. Peele, “Iterative image reconstruction algorithms using wave-front intensity and phase variation,” Opt. Lett. 30, 1638-1640 (2005).
    [CrossRef] [PubMed]
  16. X. Xiao and Q. Shen, “Wave propagation and phase retrieval in Fresnel diffraction by a distorted-object approach,” Phys. Rev. B 72, 033103 (2005).
    [CrossRef]
  17. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101-104 (2006).
    [CrossRef]
  18. T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53-70 (2004).
    [CrossRef]
  19. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: Matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932-1941 (1995).
    [CrossRef]
  20. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2964 (1996).
    [CrossRef] [PubMed]
  21. D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33-40 (2002).
    [CrossRef] [PubMed]
  22. S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin,A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289-2302 (2003).
    [CrossRef] [PubMed]
  23. X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett. 30, 379-381 (2005).
    [CrossRef] [PubMed]
  24. M. R. Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. 73, 1434-1441 (1983).
    [CrossRef]
  25. A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19, 472-480 (2002).
    [CrossRef]
  26. P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
    [CrossRef]
  27. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2004).
  28. S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335-338 (1996).
    [CrossRef]
  29. K. D. Mielenz, “Computation of Fresnel integrals. II,” J. Res. Natl. Inst. Stand. Technol. 105, 589-590 (2000).
  30. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737-1746 (1993).
    [CrossRef] [PubMed]
  31. T. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49-58 (2003).
    [CrossRef]

2006 (2)

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101-104 (2006).
[CrossRef]

2005 (5)

H. M. Quiney, K. A. Nugent, and A. G. Peele, “Iterative image reconstruction algorithms using wave-front intensity and phase variation,” Opt. Lett. 30, 1638-1640 (2005).
[CrossRef] [PubMed]

X. Xiao and Q. Shen, “Wave propagation and phase retrieval in Fresnel diffraction by a distorted-object approach,” Phys. Rev. B 72, 033103 (2005).
[CrossRef]

I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71, 245302 (2005).
[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. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett. 30, 379-381 (2005).
[CrossRef] [PubMed]

2004 (1)

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53-70 (2004).
[CrossRef]

2003 (7)

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

Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101 (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 (2003).
[CrossRef]

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]

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin,A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

T. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49-58 (2003).
[CrossRef]

2002 (2)

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19, 472-480 (2002).
[CrossRef]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33-40 (2002).
[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. Miao, K. Hodgson, and D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. U.S.A. 98, 6641-6645 (2001).
[CrossRef] [PubMed]

2000 (1)

K. D. Mielenz, “Computation of Fresnel integrals. II,” J. Res. Natl. Inst. Stand. Technol. 105, 589-590 (2000).

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

1998 (1)

1996 (2)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335-338 (1996).
[CrossRef]

1995 (1)

1993 (1)

1983 (1)

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

1952 (1)

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843-843 (1952).
[CrossRef]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Bauer, G.

I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71, 245302 (2005).
[CrossRef]

Bauschke, H. H.

Beetz, T.

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. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

Bronnikov, A. V.

Cai, Z.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101-104 (2006).
[CrossRef]

Chapman, H. N.

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 (2003).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662-1669 (1998).
[CrossRef]

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

Combettes, P. L.

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Davis, T.

de Jonge, M. D.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

Dhal, B. B.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

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. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

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

Fienup, J. R.

Gao, D.

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335-338 (1996).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2004).

Gureyev, T.

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53-70 (2004).
[CrossRef]

T. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49-58 (2003).
[CrossRef]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin,A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335-338 (1996).
[CrossRef]

Gureyev, T. E.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: Matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932-1941 (1995).
[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 (2003).
[CrossRef]

He, H.

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 (2003).
[CrossRef]

Hodgson, K.

J. Miao, K. Hodgson, and D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. U.S.A. 98, 6641-6645 (2001).
[CrossRef] [PubMed]

Howells, 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. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

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 (2003).
[CrossRef]

Ishikawa, T.

Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101 (2003).
[CrossRef]

Jacobsen, C.

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. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

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. U.S.A. 102, 15343-15346 (2005).
[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,” Nature (London) 400, 342-344 (1999).
[CrossRef]

Lima, E.

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. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

Liu, H.

Luke, D. R.

Marchesini, S.

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 (2003).
[CrossRef]

Mayo, S.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin,A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

McMahon, P. J.

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Metzger, H.

I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71, 245302 (2005).
[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. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

Miao, J.

Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101 (2003).
[CrossRef]

J. Miao, K. Hodgson, and D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. U.S.A. 98, 6641-6645 (2001).
[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,” Nature (London) 400, 342-344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662-1669 (1998).
[CrossRef]

Mielenz, K. D.

K. D. Mielenz, “Computation of Fresnel integrals. II,” J. Res. Natl. Inst. Stand. Technol. 105, 589-590 (2000).

Miller, P.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin,A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

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. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

Nishino, Y.

Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101 (2003).
[CrossRef]

Noy, A.

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 (2003).
[CrossRef]

Nugent, K. A.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101-104 (2006).
[CrossRef]

H. M. Quiney, K. A. Nugent, and A. G. Peele, “Iterative image reconstruction algorithms using wave-front intensity and phase variation,” Opt. Lett. 30, 1638-1640 (2005).
[CrossRef] [PubMed]

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: Matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932-1941 (1995).
[CrossRef]

Onken, J. D.

I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71, 245302 (2005).
[CrossRef]

Paganin, D.

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53-70 (2004).
[CrossRef]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin,A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Paterson, D.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101-104 (2006).
[CrossRef]

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Peele, A. G.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101-104 (2006).
[CrossRef]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

H. M. Quiney, K. A. Nugent, and A. G. Peele, “Iterative image reconstruction algorithms using wave-front intensity and phase variation,” Opt. Lett. 30, 1638-1640 (2005).
[CrossRef] [PubMed]

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Pfeifer, M. A.

I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71, 245302 (2005).
[CrossRef]

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]

Pfeiffer, F.

I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71, 245302 (2005).
[CrossRef]

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]

Pogany, A.

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53-70 (2004).
[CrossRef]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin,A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335-338 (1996).
[CrossRef]

Quiney, H. M.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101-104 (2006).
[CrossRef]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

H. M. Quiney, K. A. Nugent, and A. G. Peele, “Iterative image reconstruction algorithms using wave-front intensity and phase variation,” Opt. Lett. 30, 1638-1640 (2005).
[CrossRef] [PubMed]

Rau, C.

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Roberts, A.

Robinson, I. K.

I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71, 245302 (2005).
[CrossRef]

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]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

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. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

J. Miao, K. Hodgson, and D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. U.S.A. 98, 6641-6645 (2001).
[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,” Nature (London) 400, 342-344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662-1669 (1998).
[CrossRef]

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843-843 (1952).
[CrossRef]

Shapiro, 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. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

Shen, Q.

X. Xiao and Q. Shen, “Wave propagation and phase retrieval in Fresnel diffraction by a distorted-object approach,” Phys. Rev. B 72, 033103 (2005).
[CrossRef]

Snigirev, A.

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Spence, J. C. H.

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 (2003).
[CrossRef]

Stevenson, A.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin,A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335-338 (1996).
[CrossRef]

Teague, M. R.

Thibault, P.

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. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

Tran, C. Q.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

Vartanyants, I. A.

I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71, 245302 (2005).
[CrossRef]

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]

Weierstall, U.

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 (2003).
[CrossRef]

Weitkamp, T.

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

Wilkins, S.

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53-70 (2004).
[CrossRef]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin,A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335-338 (1996).
[CrossRef]

Williams, G. J.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71, 245302 (2005).
[CrossRef]

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, X.

Xiao, X.

X. Xiao and Q. Shen, “Wave propagation and phase retrieval in Fresnel diffraction by a distorted-object approach,” Phys. Rev. B 72, 033103 (2005).
[CrossRef]

Yan, A.

Zhong, Z.

I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71, 245302 (2005).
[CrossRef]

Acta Crystallogr. (1)

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843-843 (1952).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480-1482 (2003).
[CrossRef]

J. Microsc. (1)

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

J. Res. Natl. Inst. Stand. Technol. (1)

K. D. Mielenz, “Computation of Fresnel integrals. II,” J. Res. Natl. Inst. Stand. Technol. 105, 589-590 (2000).

Nat. Phys. (1)

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101-104 (2006).
[CrossRef]

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

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335-338 (1996).
[CrossRef]

Opt. Commun. (2)

T. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49-58 (2003).
[CrossRef]

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53-70 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Optik (Stuttgart) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Phys. Rev. B (4)

X. Xiao and Q. Shen, “Wave propagation and phase retrieval in Fresnel diffraction by a distorted-object approach,” Phys. Rev. B 72, 033103 (2005).
[CrossRef]

Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101 (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 (2003).
[CrossRef]

I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71, 245302 (2005).
[CrossRef]

Phys. Rev. Lett. (3)

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]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (2)

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. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

J. Miao, K. Hodgson, and D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. U.S.A. 98, 6641-6645 (2001).
[CrossRef] [PubMed]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2004).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Experimental setup for collecting the diffraction image when a point source is used.

Fig. 2
Fig. 2

Plot of T z ( s ) with N = 138 , η = 2 , N F = 250 .

Fig. 3
Fig. 3

Phase (a) and magnitude (b) of the test object, each characterized by 118 × 118   pixels . The phase varies from 0 to π.

Fig. 4
Fig. 4

Diffraction wave field modulus of the zero-padded test object ( 138 × 138   pixels ) generated using Eq. (15) with η = 2 , N F = 250 , and J = 257 . The image is noise free. It has the same area as the zero-padded object, with 276 × 276   pixels . A clear-cut boundary is included within the image.

Fig. 5
Fig. 5

Shown in panel (a) is the evolution of χ 2 (in a logarithmic scale) with the iteration number in the reconstructions from the diffraction image shown in Fig. 3 after Poisson noise was added using N F = 250 , 251 , 252 , 253 , 254 , 255 . Shown in panel (b) are the χ 2 values that the reconstructions converged to.

Fig. 6
Fig. 6

Shown in panels (a)–(h) are the reconstructed phase and magnitude in the test of N F = 250 ( N F orig = 182.79 ) using (i) N F = 250 [panels (a), (b)]; (ii) N F = 249 [panels (c), (d)]; (iii) N F = 251 [panels (e), (f)]; (iv) N F = 255 [panels (g), (h)]. The quality of the first three groups of results are much higher than that of the fourth group.

Fig. 7
Fig. 7

Shown in panels (a) and (b) are the reconstructed phase and magnitude in the test of N F = 1200 ( N F orig 877.38 ) using N F = 1200 . In panels (c) and (d) are the reconstructed phase and magnitude in the case of N F = 200 ( N F orig = 50 ) using N F = 200 .

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

E ( R ; z ) = E ( r ; 0 ) exp [ i π ( R r ) 2 z λ ] d r ,
4 z 3 λ π [ ( R r ) 4 ] max .
E ( R ; z ) exp [ i π R 2 ( z λ ) ] = E ( r ; 0 ) exp [ i π r 2 ( z λ ) ] exp [ 2 π i R r ( z λ ) ] d r .
E ( R ; z ) exp [ i π R 2 z λ ( 1 M 1 ) ] = E ( r ; 0 ) H ( r , z 0 ) exp [ i π ( R M r ) 2 ( z M ) λ ] d r ,
E ( p ; z ) = m = N 2 N 2 1 E ( m ; 0 ) × 1 2 1 2 d α exp [ i π d 2 z λ ( δ d p m α ) 2 ] .
E ( p ; z ) = m = N 2 N 2 1 E ( m ; 0 ) T z ( p η m ) ,
T z ( s ) = 1 2 1 2 d α exp [ i π ( N 2 4 N F ) 1 ( η 1 s α ) 2 ] ,
D = p = N O 2 N O 2 1 I est ( p ; z ) I exp ( p ; z ) 2 ,
I est ( p ; z ) = E est ( p ; z ) = m = N 2 N 2 1 E est ( m ; 0 ) T z ( p η m ) .
{ Re E k + 1 est ( m ; 0 ) = Re E k est ( m ; 0 ) ϵ G k r ( m ; 0 ) Im E k + 1 est ( m ; 0 ) = Im E k est ( m ; 0 ) ϵ G k i ( m ; 0 ) } .
{ G k r ( m ) = D k Re E k est ( m ; 0 ) = 2 Re G k ( m ) G k i ( m ) = D k Im E k est ( m ; 0 ) = 2 Im G k ( m ) } ,
G k ( m ) = p = N O 2 N O 2 1 Δ k * ( p ; z ) T z ( p η m ) ,
Δ k ( p ; z ) = [ 1 I exp ( p ; z ) I k est ( p ; z ) ] E k est ( p ; z ) .
E ( p + j J ; z ) = m = N 2 N 2 1 E ( m ; 0 ) T z ( p η m + j J ) .
I exp ( p ; z ; J ) = [ 1 J j J 2 E ( p + j J ; z ) 2 ] 1 2 ,
I exp ( p ; z ) = lim J I exp ( p ; z ; J ) .
χ 2 = D p I exp ( p ; z ) 2 .
E ( p ; z ) = m = N 2 N 2 1 E ( m ; 0 ) T z ( p O m ) ,
E pad ( m ; 0 ) = { E ( m ; 0 ) , for m = N 2 , , N 2 1 0 , for m = N , , N 2 1 , N 2 , , N 1 } .
E ̃ ( u ; 0 ) = m = N N 1 E pad ( m ; 0 ) exp [ 2 π i ( m u ) 2 N ]
E ̃ ( u ; 0 ) = E ̃ ( u ± 2 N ; 0 ) ,
E pad ( m ; 0 ) = 1 ( 2 N ) u = N N 1 E ̃ ( u ; 0 ) exp [ 2 π i ( m u ) 2 N ] = 1 O 1 ( 2 N ) u = N O N O 1 E ̃ ( u ; 0 ) exp [ 2 π i ( m u ) 2 N ] .
T z ( s ) = 1 ( 2 N O ) u = N O N O 1 T ̃ z ( u ) exp [ 2 π i ( s u ) 2 N O ] .
E ( p ; z ) = m = N N 1 1 O 1 ( 2 N ) u = N O N O 1 E ̃ ( u ; 0 ) exp [ 2 π i ( m u ) 2 N ] × 1 ( 2 N O ) u = N O N O 1 T ̃ z ( u ) exp { 2 π i [ ( p O m ) u ] 2 N O } = 1 O 1 ( 2 N O ) u = N O N O 1 u = N O N O 1 E ̃ ( u ; 0 ) T ̃ z ( u ) exp [ 2 π i ( p u ) 2 N O ] × 1 ( 2 N ) m = N N 1 exp { 2 π i [ m ( u u ) ] 2 N } = 1 O 1 ( 2 N O ) u = N O N O 1 E ̃ ( u ; 0 ) T ̃ z ( u ) exp [ 2 π i ( p u ) 2 N O ] .
G ( m ) = p = N O 2 N O 2 1 T z ( p O m ) Δ * ( p ; z ) ,
Δ pad ( p ; z ) = { Δ ( p ; z ) , for p = N O 2 , , N O 2 1 0 , for p = N O , , N O 2 1 , N O 2 , , N O 1 }
Δ pad ( p ; z ) = 1 ( 2 N O ) u = N O N O 1 Δ ̃ ( u ; z ) exp [ 2 π i ( u p ) 2 N O ] .
G ̃ ( u ) = 1 O k 1 = k 1 min k 1 max T ̃ z ( u + 2 N k 1 ) Δ ̃ * ( u + 2 N k 1 ; z ) ,
( k 1 min , k 1 max ) = { ( ( O 1 ) 2 , O 2 ) , for u = N , , 1 ( O 2 , ( O 1 ) 2 ) , for u = 0 , , N 1 } ,
G ( m ) = 1 ( 2 N ) u = N N 1 G ̃ ( u ) exp [ 2 π i ( m u ) 2 N ] ,

Metrics