Abstract

In this work, issues in phase retrieval in the coherent diffractive imaging (CDI) technique, from discussion on parameters for setting up a CDI experiment to evaluation of the goodness of the final reconstruction, are discussed. The distribution of objects under study by CDI often cannot be cross-validated by another imaging technique. It is therefore important to make sure that the developed CDI procedure delivers an artifact-free object reconstruction. Critical issues that can lead to artifacts are presented and recipes on how to avoid them are provided.

© 2018 Optical Society of America

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2016 (2)

P. Kliuiev, T. Latychevskaia, J. Osterwalder, M. Hengsberger, and L. Castiglioni, “Application of iterative phase-retrieval algorithms to ARPES orbital tomography,” New J. Phys. 18, 093041 (2016).
[Crossref]

F. Soulez, E. Thiebaut, A. Schutz, A. Ferrari, F. Courbin, and M. Unser, “Proximity operators for phase retrieval,” Appl. Opt. 55, 7412–7421 (2016).
[Crossref]

2015 (3)

T. Latychevskaia, Y. Chushkin, F. Zontone, and H.-W. Fink, “Imaging outside the box: resolution enhancement in x-ray coherent diffraction imaging by extrapolation of diffraction patterns,” Appl. Phys. Lett. 107, 183102 (2015).
[Crossref]

T. Latychevskaia and H.-W. Fink, “Practical algorithms for simulation and reconstruction of digital in-line holograms,” Appl. Opt. 54, 2424–2434 (2015).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. W. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

2013 (1)

T. Latychevskaia and H.-W. Fink, “Coherent microscopy at resolution beyond diffraction limit using post-experimental data extrapolation,” Appl. Phys. Lett. 103, 204105 (2013).
[Crossref]

2012 (2)

2011 (2)

A. Tripathi, J. Mohanty, S. H. Dietze, O. G. Shpyrko, E. Shipton, E. E. Fullerton, S. S. Kim, and I. McNulty, “Dichroic coherent diffractive imaging,” Proc. Natl. Acad. Sci. USA 108, 13393–13398 (2011).
[Crossref]

T. Latychevskaia, J.-N. Longchamp, and H.-W. Fink, “Novel Fourier-domain constraint for fast phase retrieval in coherent diffraction imaging,” Opt. Express 19, 19330–19339 (2011).
[Crossref]

2010 (2)

J. Steinbrener, J. Nelson, X. J. Huang, S. Marchesini, D. Shapiro, J. J. Turner, and C. Jacobsen, “Data preparation and evaluation techniques for x-ray diffraction microscopy,” Opt. Express 18, 18598–18614 (2010).
[Crossref]

R. Harder, M. Liang, Y. Sun, Y. Xia, and I. K. Robinson, “Imaging of complex density in silver nanocubes by coherent x-ray diffraction,” New J. Phys. 12, 035019 (2010).
[Crossref]

2009 (1)

M. C. Newton, S. J. Leake, R. Harder, and I. K. Robinson, “Three-dimensional imaging of strain in a single zno nanorod,” Nat. Mater. 9, 120–124 (2009).
[Crossref]

2008 (2)

2007 (1)

P. Thibault and I. C. Rankenburg, “Optical diffraction microscopy in a teaching laboratory,” Am. J. Phys. 75, 827–832 (2007).
[Crossref]

2006 (2)

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. R. 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]

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
[Crossref]

2005 (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. USA 102, 15343–15346 (2005).
[Crossref]

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Prob. 21, 37–50 (2005).
[Crossref]

2004 (1)

G. Oszlanyi and A. Suto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. Sect. A 60, 134–141 (2004).
[Crossref]

2003 (1)

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]

2002 (1)

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[Crossref]

2001 (1)

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

2000 (1)

J. Miao and D. Sayre, “On possible extensions of x-ray crystallography through diffraction-pattern oversampling,” Acta Crystallogr. Sect. A 56, 596–605 (2000).
[Crossref]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

1998 (1)

1992 (1)

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier-transform x-ray holography,” Science 256, 1009–1012 (1992).
[Crossref]

1990 (1)

1988 (1)

J. J. Barton, “Photoelectron holography,” Phys. Rev. Lett. 61, 1356–1359 (1988).
[Crossref]

1987 (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 35, 237–246 (1972).

1965 (1)

G. W. Stroke, “Lensless Fourier-transform method for optical holography,” Appl. Phys. Lett. 6, 201–203 (1965).
[Crossref]

1964 (1)

E. M. Hofstetter, “Construction of time-limited functions specified autocorrelation functions,” Ultramicroscopy 10, 119–126 (1964).
[Crossref]

Anderson, E. H.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[Crossref]

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier-transform x-ray holography,” Science 256, 1009–1012 (1992).
[Crossref]

Aquila, A.

Bajt, S.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
[Crossref]

Barthelmess, M.

Barton, J. J.

J. J. Barton, “Photoelectron holography,” Phys. Rev. Lett. 61, 1356–1359 (1988).
[Crossref]

Barty, A.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
[Crossref]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. R. 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]

Beetz, T.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. R. 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]

Benner, H.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
[Crossref]

Bergh, M.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
[Crossref]

Bogan, M. J.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
[Crossref]

Bostedt, C.

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H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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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).
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H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. W. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
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T. Latychevskaia, Y. Chushkin, F. Zontone, and H.-W. Fink, “Imaging outside the box: resolution enhancement in x-ray coherent diffraction imaging by extrapolation of diffraction patterns,” Appl. Phys. Lett. 107, 183102 (2015).
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H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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A. Tripathi, J. Mohanty, S. H. Dietze, O. G. Shpyrko, E. Shipton, E. E. Fullerton, S. S. Kim, and I. McNulty, “Dichroic coherent diffractive imaging,” Proc. Natl. Acad. Sci. USA 108, 13393–13398 (2011).
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H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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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).
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Hau-Riege, S. S.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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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).
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Hodgson, K. O.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. R. 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).
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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).
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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, K. O. Hodgson, and D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images,” Proc. Natl. Acad. Sci. USA 98, 6641–6645 (2001).
[Crossref]

J. Miao and D. Sayre, “On possible extensions of x-ray crystallography through diffraction-pattern oversampling,” Acta Crystallogr. Sect. A 56, 596–605 (2000).
[Crossref]

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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).
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Schneider, J. R.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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Schutz, A.

Segev, M.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. W. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
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G. Oszlanyi and A. Suto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. Sect. A 60, 134–141 (2004).
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Szoke, A.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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P. Thibault and I. C. Rankenburg, “Optical diffraction microscopy in a teaching laboratory,” Am. J. Phys. 75, 827–832 (2007).
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H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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Tripathi, A.

A. Tripathi, J. Mohanty, S. H. Dietze, O. G. Shpyrko, E. Shipton, E. E. Fullerton, S. S. Kim, and I. McNulty, “Dichroic coherent diffractive imaging,” Proc. Natl. Acad. Sci. USA 108, 13393–13398 (2011).
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Tschentscher, T.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. R. 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).
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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).
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White, T. A.

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H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. S. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. V. der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006).
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Zontone, F.

T. Latychevskaia, Y. Chushkin, F. Zontone, and H.-W. Fink, “Imaging outside the box: resolution enhancement in x-ray coherent diffraction imaging by extrapolation of diffraction patterns,” Appl. Phys. Lett. 107, 183102 (2015).
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T. Latychevskaia, Y. Chushkin, F. Zontone, and H.-W. Fink, “Imaging outside the box: resolution enhancement in x-ray coherent diffraction imaging by extrapolation of diffraction patterns,” Appl. Phys. Lett. 107, 183102 (2015).
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Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. W. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
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Figures (16)

Fig. 1.
Fig. 1. Geometrical arrangement in coherent diffractive imaging. (a) General scheme. (b) Illustration to the vector definitions.
Fig. 2.
Fig. 2. Schematics of a general algorithm for iterative phase retrieval.
Fig. 3.
Fig. 3. Effects of oversampling ratio. (a) Sample distribution and (b) the reconstruction obtained from its diffraction pattern where the oversampling condition was not fulfilled along K x -dimension. The reconstruction was obtained by applying the shrinkwrap algorithm for 2000 iterations; 10 reconstructions with the smallest errors were selected, aligned, and averaged; the number of pixels was 256 × 256 pixels. (c) Sample distribution consisting of four objects placed together in the center and (d) its reconstruction obtained from the diffraction pattern with the parameters as described in the Appendix A. (e) Sample distribution consisting of four objects scattered over the entire imaged area and (f) reconstruction obtained from its diffraction pattern. The oversampling ratio was σ = 4 . The reconstruction was obtained by applying the shrinkwrap algorithm for 2000 iterations; one reconstruction with the least error was selected; the number of pixels was 256 × 256 pixels. Note that because the shrinkwrap algorithm begins with obtaining support from cross-correlation of the object, one part of the sample will always show up in the center of the reconstruction.
Fig. 4.
Fig. 4. Effects of oversampling ratio. (a)–(c) Illustration of different oversampling ratios; the total number of pixels is 256 × 256 pixels, and the object size is (a)  128 × 128 pixels ( σ = 2 ), (b)  64 × 64 pixels ( σ = 4 ), and (c)  32 × 32 pixels ( σ = 8 ). (d) Original object sampled with 64 × 64 pixels and (e) reconstruction obtained from its diffraction pattern. The diffraction pattern was simulated and reconstructed as described in the Appendix A. (f),(g) Error as a function of the iteration number at different oversampling ratios. (f) The object size is constant and amounts to 64 × 64 pixels; the total sample area size is 128 × 128 pixels ( σ = 2 ), 256 × 256 pixels ( σ = 4 ), and 512 × 512 pixels ( σ = 8 ). (g) The total size remains constant and amounts to 256 × 256 pixels; the object size is 128 × 128 pixels ( σ = 2 ), 64 × 64 pixels ( σ = 4 ), and 32 × 32 pixels ( σ = 8 ). The parameters of the reconstruction procedure are provided in the Appendix A. The shown error curves are the result of averaging over the 10 error curves corresponding to the 10 selected reconstructions with the smallest errors.
Fig. 5.
Fig. 5. Effects of the intensity dynamic range of the detector. (a) (b) Fragments of diffraction patterns when the detecting system has the intensity dynamic range of (a) 20 bit and (b) 16 bit; 50 × 50 pixel left bottom fragments of 256 × 256 pixel diffraction patterns are shown, the intensity scalebar in gray levels. (c),(d) Corresponding reconstructions obtained from (c) 20 bit and (d) 16 bit intensity range diffraction patterns. The parameters of the diffraction pattern and the reconstruction procedure are provided in the Appendix A.
Fig. 6.
Fig. 6. Effect of the intensity distribution of the incident wavefront on the reconstructed sample distribution. (a),(b) Intensity distributions of the diffraction patterns simulated with an incident wavefront of (a) a constant amplitude and (b) amplitude in form of a Gaussian distribution with a standard deviation of 20 pixels; 50 × 50 pixel left bottom fragments of 256 × 256 pixel diffraction patterns are shown. (c) Reconstruction obtained from the diffraction pattern simulated with the incident wavefront with a Gaussian-distributed amplitude. The parameters of the diffraction pattern and the reconstruction procedure are provided in the Appendix A.
Fig. 7.
Fig. 7. Effect of noise in diffraction pattern. (a),(b)  50 × 50 pixel left bottom fragments of 256 × 256 pixel diffraction patterns with signal-to-noise ratio (SNR) of (a)  SNR = 10 and (b)  SNR = 5 are shown. (c),(d) Reconstructions obtained from diffraction pattern with (c)  SNR = 10 and (d)  SNR = 10 . (e) Reconstruction with the least error obtained from diffraction pattern with SNR = 5 . The parameters of the diffraction patterns and the reconstruction procedure are provided in the Appendix A.
Fig. 8.
Fig. 8. Effect of distortions in diffraction pattern. (a) Original diffraction pattern. (b),(c) illustrations to Z distortion in the upper right quadrant of the diffraction pattern: the variation in the Z distance of the detecting system ranges from 0.1 m to 0.11 m. (d) Corresponding reconstruction obtained from the diffraction pattern with the Z distortion. (e) Illustration of lateral distortions in the upper right quadrant, not drawn to scale. (f) Reconstruction obtained from the diffraction pattern where the coordinates of the detected intensities are shifted in the upper right quadrant by a factor 1.05 along both X and Y directions. The diffraction pattern is simulated for a wavelength of 500 nm, Z = 0.1 m, and sample area size of 1    mm × 1    mm . The parameters of the reconstruction procedure are described in the Appendix A.
Fig. 9.
Fig. 9. Effect of centring of diffraction pattern. Reconstructions obtained from a diffraction pattern, with oversampling ratio σ = 4 , which was off-centered by (a) 1 pixel in the v direction and (b) 1 pixel in both v and w directions. (c) Reconstructions obtained from a diffraction pattern with oversampling ratio σ = 8 , which was mis-centered by 1 pixel in v direction. The parameters of the reconstruction procedure are described in the Appendix A.
Fig. 10.
Fig. 10. Effect of a constant background added to diffraction pattern. Reconstructions obtained from the diffraction pattern when a constant of (c)  10 6 , (d)  10 5 , and (e)  10 4 of the diffraction pattern maximal intensity is added to the diffraction pattern. The parameters of the diffraction pattern and the reconstruction procedure are provided in the Appendix A.
Fig. 11.
Fig. 11. Auto-correlation of the object. (a) Original object and (b) its diffraction pattern shown in an inverted logarithmic intensity scale. (c) Amplitude of the Fourier transform of the diffraction pattern shown in (b). The obtained distribution consists of four centro-symmetric images of the original object.
Fig. 12.
Fig. 12. Effect of pixels with artifact values in diffraction pattern. To mimic the experimental situation, the values of pixels at randomly distributed positions are set to wrong values. (a)–(d) Effect of “dead” pixels with zero values. (a)–(c) Reconstructions obtained from diffraction patterns where the number of pixels with values of zero is (a) 0.001, (b) 0.002, and (c) 0.005 of the total number of pixels. (d) Reconstruction obtained by the HIO algorithm where at each iteration, the values at the pixels with missing values were replaced by the iterated amplitudes | G k ( v , w ) | . (e)–(h) Effect of saturated (“bright”) pixels with values of 0.1 of the intensity maximum in the diffraction pattern. (e)–(g) Reconstructions obtained from diffraction patterns where the number of pixels with artifact values is (e) 0.001, (f) 0.002, and (g) 0.005 of the total number of pixels. (h) Reconstruction obtained by the HIO algorithm where at each iteration, the pixels with artifact values were replaced by the iterated amplitudes | G k ( v , w ) | . The parameters of the diffraction patterns and the reconstruction procedure are described in the Appendix A.
Fig. 13.
Fig. 13. Recovery of binary objects. (a) Binary object created from “Lena” image. (b) Error as a function of iteration number for diffraction patterns of the non-binary and binary “Lena” image, noise-free and with SNR = 10 . The shown error curves are the result of averaging over the 10 error curves corresponding to the 10 selected reconstructions with the smallest errors. The parameters of the diffraction pattern and the reconstruction procedure are provided in the Appendix A.
Fig. 14.
Fig. 14. Obtaining low-resolution reconstruction. (a) Original 256 × 256 pixels diffraction pattern, oversampling ratio σ = 4 . (b) Reconstruction obtained from the central part of 128 × 128 pixel of the original diffraction pattern as indicated by the dashed square in (a). The obtained object reconstruction has a size of 32 × 32 pixels. The reconstruction was obtained by applying the HIO algorithm with the tight support in the form of a squared patch of 32 × 32 pixels and real and positivity constraint for 2000 iterations; 10 reconstructions with the smallest errors were selected, aligned, and averaged.
Fig. 15.
Fig. 15. Phase retrieval transfer function (PRTF) for a noise-free diffraction pattern and for diffraction patterns with SNR = 10 and SNR = 5 . The parameters of the diffraction patterns and the reconstruction procedure are provided in the Appendix A.
Fig. 16.
Fig. 16. Wrapping effect. Diffraction on a square aperture is considered. (a) Diffraction pattern calculated as an analytical solution to the problem of diffraction on a square aperture, calculated for 256 × 256 pixels (DP1) and 512 × 512 pixels (DP2). (b) Diffraction on a square aperture calculated by FFT for 256 × 256 pixels (DP3). The spectrum that exceeds region 256 × 256 pixels is wrapped around the edges and shows up inside the 256 × 256 pixels window. (c) Difference between the diffraction patterns simulated as an analytical solution and by FFT, calculated as ( DP 3 DP 1 ) / max ( DP 1 ) . It is apparent that the difference is maximal at the edges. (d) Intensity profiles through the center of diffraction patterns shown in (b), demonstrating the difference in intensity values at the coordinate of v = 128 pixel (wrapping).

Equations (21)

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U ( R ) = i λ exp ( i k z ) o ( r ) exp ( i k | R r | ) | R r | d r ,
| R r | ( Z z ) + ( X x ) 2 + ( Y y ) 2 2 Z .
U ( R ) i λ Z exp ( i k Z ) exp [ i π λ Z ( X 2 + Y 2 ) ] × [ o ( x , y , z ) d z ] exp [ 2 π i λ Z ( x X + y Y ) ] d x d y ,
U ( X , Y ) i λ Z exp ( i k Z ) exp [ i π λ Z ( X 2 + Y 2 ) ] × U 0 ( x , y ) exp [ 2 π i λ Z ( x X + y Y ) ] d x d y .
| R r | R R r R .
U ( K ) = i λ R exp ( i k R ) exp ( i k z ) o ( r ) exp ( i k K r ) d r .
U ( K x , K y ) = i λ R exp ( i k R ) exp ( i k z ) o ( x , y , z ) × exp [ i k ( x K x + y K y ) ] exp ( i k z 1 K x 2 K y 2 ) d x d y d z .
U ( K x , K y ) = i λ R exp ( i k R ) [ o ( x , y , z ) d z ] × exp [ i k ( x K x + y K y ) ] d x d y ,
I ( v , w ) = | F ( v , w ) | 2 = | f ( x , y ) exp ( i ( x v + y w ) ) d x d y | 2 ,
| F ( u ) | = | r = 0 N 1 f ( r ) exp ( i u r ) | .
σ 0 = total pixel number unknown - valued pixel number .
σ > 2 1 / n ,
( i )    G k ( v , w ) = | G k ( v , w ) | exp [ i φ k ( v , w ) ] = F [ g k ( x , y ) ] , ( ii )    G k ( v , w ) = | F ( v , w ) | exp [ i φ k ( v , w ) ] ( iii )    g k ( x , y ) = | g k ( x , y ) | exp [ i θ k ( x , y ) ] = F 1 [ G k ( v , w ) ] , ( iv )    g k + 1 ( x , y ) = { g k ( x , y ) , if ( x , y ) γ 0 , if ( x , y ) γ ,
( iv )    g k + 1 ( x , y ) = { g k ( x , y ) , if ( x , y ) γ g k ( x , y ) β g k ( x , y ) , if ( x , y ) γ ,
Error F k = { N 2 v , w [ | G k ( v , w ) | | F ( v , w ) | ] 2 v , w | F ( v , w ) | 2 } 1 / 2 .
Error f k = { N 2 x , y γ | g k ( x , y ) | 2 x , y γ | g k ( x , y ) | 2 } 1 / 2 .
F q = p = 0 N 1 f p exp ( 2 π i p q / N ) ,
Δ v Δ x = 2 π N .
I sym ( v , w ) = I ( v , w ) + I ( u , w ) 2 .
PRTF ( u ) = Γ ( u ) | F ( u ) | ,
Γ ( u ) = 1 M m = 1 M Γ m ( u ) = 1 M m = 1 M FT [ g m ( x , y ) ] = 1 M FT [ m = 1 M g m ( x , y ) ] ,

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