Abstract

Iterative algorithms that reconstruct images from far-field x-ray diffraction data are plagued with convergence difficulties. An iterative image reconstruction algorithm is described that ameliorates these convergence difficulties through the use of diffraction data obtained with illumination modulated in both intensity and phase.

© 2005 Optical Society of America

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References

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  1. J. Miao, H. N. Chapman, J. Kirz, D. Sayre, and K. O. Hodgson, Annu. Rev. Biophys. Biomol. Struct. 33, 157 (2004).
    [CrossRef]
  2. J. Miao, K. O. Hodgson, and D. Sayre, Proc. Natl. Acad. Sci. U.S.A. 98, 6641 (2001).
    [CrossRef]
  3. R. Gerchberg and W. Saxton, Optik (Stuttgart) 35, 237 (1972).
  4. J. R. Fienup, Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  5. J. R. Fienup, Appl. Opt. 32, 1737 (1993).
    [CrossRef] [PubMed]
  6. V. Elser, J. Opt. Soc. Am. A 20, 40 (2003).
    [CrossRef]
  7. H. H. Bauschke, P. L. Combettes, and D. R. Luke, J. Opt. Soc. Am. A 19, 1334 (2002).
    [CrossRef]
  8. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
    [CrossRef]
  9. K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, Phys. Rev. Lett. 91, 203902 (2003).
    [CrossRef]
  10. J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, Appl. Opt. 32, 1747 (1993).
    [CrossRef] [PubMed]
  11. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), pp. 81–89.

2004 (1)

J. Miao, H. N. Chapman, J. Kirz, D. Sayre, and K. O. Hodgson, Annu. Rev. Biophys. Biomol. Struct. 33, 157 (2004).
[CrossRef]

2003 (3)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef]

V. Elser, J. Opt. Soc. Am. A 20, 40 (2003).
[CrossRef]

2002 (1)

2001 (1)

J. Miao, K. O. Hodgson, and D. Sayre, Proc. Natl. Acad. Sci. U.S.A. 98, 6641 (2001).
[CrossRef]

1993 (2)

1982 (1)

1972 (1)

R. Gerchberg and W. Saxton, Optik (Stuttgart) 35, 237 (1972).

Bauschke, H. H.

Chapman, H. N.

J. Miao, H. N. Chapman, J. Kirz, D. Sayre, and K. O. Hodgson, Annu. Rev. Biophys. Biomol. Struct. 33, 157 (2004).
[CrossRef]

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, Phys. Rev. Lett. 91, 203902 (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, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Combettes, P. L.

Elser, V.

Fienup, J. R.

Gerchberg, R.

R. Gerchberg and W. Saxton, Optik (Stuttgart) 35, 237 (1972).

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, 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, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Hodgson, K. O.

J. Miao, H. N. Chapman, J. Kirz, D. Sayre, and K. O. Hodgson, Annu. Rev. Biophys. Biomol. Struct. 33, 157 (2004).
[CrossRef]

J. Miao, K. O. Hodgson, and D. Sayre, Proc. Natl. Acad. Sci. U.S.A. 98, 6641 (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, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Kirz, J.

J. Miao, H. N. Chapman, J. Kirz, D. Sayre, and K. O. Hodgson, Annu. Rev. Biophys. Biomol. Struct. 33, 157 (2004).
[CrossRef]

Luke, D. R.

Mancuso, A. P.

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef]

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, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Marron, J. C.

Miao, J.

J. Miao, H. N. Chapman, J. Kirz, D. Sayre, and K. O. Hodgson, Annu. Rev. Biophys. Biomol. Struct. 33, 157 (2004).
[CrossRef]

J. Miao, K. O. Hodgson, and D. Sayre, Proc. Natl. Acad. Sci. U.S.A. 98, 6641 (2001).
[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, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Nugent, K. A.

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef]

Peele, A. G.

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), pp. 81–89.

Saxton, W.

R. Gerchberg and W. Saxton, Optik (Stuttgart) 35, 237 (1972).

Sayre, D.

J. Miao, H. N. Chapman, J. Kirz, D. Sayre, and K. O. Hodgson, Annu. Rev. Biophys. Biomol. Struct. 33, 157 (2004).
[CrossRef]

J. Miao, K. O. Hodgson, and D. Sayre, Proc. Natl. Acad. Sci. U.S.A. 98, 6641 (2001).
[CrossRef]

Schulz, T. J.

Seldin, J. H.

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, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), pp. 81–89.

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, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Annu. Rev. Biophys. Biomol. Struct. (1)

J. Miao, H. N. Chapman, J. Kirz, D. Sayre, and K. O. Hodgson, Annu. Rev. Biophys. Biomol. Struct. 33, 157 (2004).
[CrossRef]

Appl. Opt. (3)

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

Optik (Stuttgart) (1)

R. Gerchberg and W. Saxton, Optik (Stuttgart) 35, 237 (1972).

Phys. Rev. B (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

J. Miao, K. O. Hodgson, and D. Sayre, Proc. Natl. Acad. Sci. U.S.A. 98, 6641 (2001).
[CrossRef]

Other (1)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), pp. 81–89.

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Figures (1)

Fig. 1
Fig. 1

Test images used in numerical studies. Both images are 128 × 128   pixel gray-scale objects, which in practice are zero padded to 256 × 256   pixels to satisfy the oversampling requirement. The left-hand image was generated by assigning each pixel a random number, n on the interval 0 < n < 1 , and is denoted ξ random ( x ) . The right-hand image consists of a number of identical, irregularly shaped cells, the interiors of which are also generated by random pixel assignment. The cells are arranged on a regular but incompletely filled grid. This image is denoted ξ quasi - periodic ( x ) .

Tables (1)

Tables Icon

Table 1 Dependence of Number of Iterations m Required to Achieve an Image Reconstruction with a Mean-Square Error of 1.0 × 10 6 per Pixel on Fresnel Number N F ( s )

Equations (8)

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ξ ( k ) ( x ) = T ̂ ξ ( k 1 ) ( x ) , for k = 1 , 2 , ,
ξ rad ( x ) = exp ( μ r 2 ) ,
T ̂ μ = A exp ( μ r r 2 ) exp ( i μ i r 2 ) ,
T ̂ μ 1 = A 1 exp ( μ r r 2 ) exp ( i μ i r 2 ) ,
T ̂ FT , μ = T ̂ μ 1 T ̂ FT T ̂ μ ,
g μ ( r ) = exp ( μ * r 2 ) exp ( 2 μ r x 2 ) exp ( 2 μ * r x ) ξ o ( x ) ξ o ( x + r ) d x ,
g μ = i μ i ( r ) = exp ( i μ i r 2 ) exp ( i 2 μ i r x ) ξ o ( x ) ξ o ( x + r ) d x .
δ = 1 N 2 { i = 1 N j = 1 N [ ξ o ( x i j ) ξ o ( k ) ( x i j ) ] 2 } 1 2 .

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