Abstract

In both lensless Fourier transform holography (FTH) and coherent diffraction imaging (CDI), a beamstop is used to block strong intensities which exceed the limited dynamic range of the sensor, causing a loss in low-frequency information, making high quality reconstructions difficult or even impossible. In this paper, we show that an image can be recovered from high-frequencies alone, thereby overcoming the beamstop problem in both FTH and CDI. The only requirement is that the object is sparse in a known basis, a common property of most natural and manmade signals. The reconstruction method relies on compressed sensing (CS) techniques, which ensure signal recovery from incomplete measurements. Specifically, in FTH, we perform compressed sensing (CS) reconstruction of captured holograms and show that this method is applicable not only to standard FTH, but also multiple or extended reference FTH. For CDI, we propose a new phase retrieval procedure, which combines Fienup’s hybrid input-output (HIO) method and CS. Both numerical simulations and proof-of-principle experiments are shown to demonstrate the effectiveness and robustness of the proposed CS-based reconstructions in dealing with missing data in both FTH and CDI.

© 2015 Optical Society of America

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References

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2015 (1)

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

2013 (1)

R. N. Wilke, M. Vassholz, and T. Salditt, “Semi-transparent central stop in high-resolution X-ray ptychography using Kirkpatrick–Baez focusing,” Acta Crystallogr. A 69(5), 490–497 (2013).
[Crossref]

2012 (1)

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

2010 (3)

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010).
[Crossref]

X. Huang, J. Nelson, J. Steinbrener, J. Kirz, J. J. Turner, and C. Jacobsen, “Incorrect support and missing center tolerances of phasing algorithms,” Opt. Express 18(25), 26441–26449 (2010).
[Crossref] [PubMed]

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond x-ray holography using extended references,” Phys. Rev. Lett. 105(9), 093901 (2010).
[Crossref] [PubMed]

2009 (2)

2008 (3)

M. Guizar-Sicairos and J. R. Fienup, “Direct image reconstruction from a Fourier intensity pattern using HERALDO,” Opt. Lett. 33(22), 2668–2670 (2008).
[Crossref] [PubMed]

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

2007 (3)

M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express 15(26), 17592–17612 (2007).
[Crossref] [PubMed]

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23(3), 969–985 (2007).
[Crossref]

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

2006 (7)

E. J. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
[Crossref]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[Crossref]

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[Crossref]

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from X-ray diffraction data,” Acta Crystallogr. A 62(4), 248–261 (2006).
[Crossref] [PubMed]

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (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. U.S.A. 102(43), 15343–15346 (2005).
[Crossref] [PubMed]

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95(8), 085503 (2005).
[Crossref] [PubMed]

2004 (1)

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (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(14), 140101 (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(5624), 1419–1421 (2003).
[Crossref] [PubMed]

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(22), 220101 (2003).
[Crossref]

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67(17), 174104 (2003).
[Crossref]

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 400(6742), 342–344 (1999).
[Crossref]

1992 (2)

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier transform X-ray holograph,” Science 256(5059), 1009–1012 (1992).
[Crossref] [PubMed]

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1-4), 259–268 (1992).
[Crossref]

1988 (1)

1987 (1)

1982 (1)

1965 (1)

G. W. Stroke, “Lensless Fourier‐transform method for optical holography,” Appl. Phys. Lett. 6(10), 201–203 (1965).
[Crossref]

Anderson, E. H.

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67(17), 174104 (2003).
[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 holograph,” Science 256(5059), 1009–1012 (1992).
[Crossref] [PubMed]

Bajt, S.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

Barty, A.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006).
[Crossref]

Beetz, T.

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 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. U.S.A. 102(43), 15343–15346 (2005).
[Crossref] [PubMed]

Bogan, M. J.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

Boutet, S.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

Boutu, W.

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond x-ray holography using extended references,” Phys. Rev. Lett. 105(9), 093901 (2010).
[Crossref] [PubMed]

Bouzerdoum, A.

F. H. C. Tivive and A. Bouzerdoum, “A compressed sensing method for complex-valued signals with application to through-the-wall radar imaging,” in Pro. IEEE Int. Conf. Acoust. Speech Signal Process. (ICASSP)2144–2148 (2014).

Brady, D. J.

Bullkich, E.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Candès, E. J.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23(3), 969–985 (2007).
[Crossref]

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
[Crossref]

Carré, B.

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond x-ray holography using extended references,” Phys. Rev. Lett. 105(9), 093901 (2010).
[Crossref] [PubMed]

Cederquist, J. N.

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[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(14), 140101 (2003).
[Crossref]

Chapman, N.

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[Crossref]

Chen, K.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[Crossref]

Choi, K.

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Cohen-Hyams, T.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Cui, C.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006).
[Crossref]

Dana, H.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Donoho, D.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

Eberhardt, W.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[Crossref]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Eisebitt, S.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[Crossref]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Eldar, Y. C.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
[Crossref] [PubMed]

Elser, V.

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from X-ray diffraction data,” Acta Crystallogr. A 62(4), 248–261 (2006).
[Crossref] [PubMed]

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(43), 15343–15346 (2005).
[Crossref] [PubMed]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1-4), 259–268 (1992).
[Crossref]

Fienup, J. R.

Frank, M.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

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(5624), 1419–1421 (2003).
[Crossref] [PubMed]

Gauthier, D.

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond x-ray holography using extended references,” Phys. Rev. Lett. 105(9), 093901 (2010).
[Crossref] [PubMed]

Gazit, S.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
[Crossref] [PubMed]

Ge, X.

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond x-ray holography using extended references,” Phys. Rev. Lett. 105(9), 093901 (2010).
[Crossref] [PubMed]

Guizar-Sicairos, M.

Günther, C.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[Crossref]

Hajdu, J.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

Hau-Riege, S. P.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006).
[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(14), 140101 (2003).
[Crossref]

He, H.

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006).
[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(14), 140101 (2003).
[Crossref]

Hellwig, O.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[Crossref]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Hodgson, K. O.

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67(17), 174104 (2003).
[Crossref]

Horisaki, R.

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(43), 15343–15346 (2005).
[Crossref] [PubMed]

Howells, M. R.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006).
[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(14), 140101 (2003).
[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 holograph,” Science 256(5059), 1009–1012 (1992).
[Crossref] [PubMed]

Huang, X.

Ishikawa, T.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95(8), 085503 (2005).
[Crossref] [PubMed]

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(22), 220101 (2003).
[Crossref]

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67(17), 174104 (2003).
[Crossref]

Jacobsen, C.

X. Huang, J. Nelson, J. Steinbrener, J. Kirz, J. J. Turner, and C. Jacobsen, “Incorrect support and missing center tolerances of phasing algorithms,” Opt. Express 18(25), 26441–26449 (2010).
[Crossref] [PubMed]

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006).
[Crossref]

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from X-ray diffraction data,” Acta Crystallogr. A 62(4), 248–261 (2006).
[Crossref] [PubMed]

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(43), 15343–15346 (2005).
[Crossref] [PubMed]

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier transform X-ray holograph,” Science 256(5059), 1009–1012 (1992).
[Crossref] [PubMed]

Javidi, B.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010).
[Crossref]

Johnson, B.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95(8), 085503 (2005).
[Crossref] [PubMed]

Kern, D. P.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier transform X-ray holograph,” Science 256(5059), 1009–1012 (1992).
[Crossref] [PubMed]

Kirz, J.

X. Huang, J. Nelson, J. Steinbrener, J. Kirz, J. J. Turner, and C. Jacobsen, “Incorrect support and missing center tolerances of phasing algorithms,” Opt. Express 18(25), 26441–26449 (2010).
[Crossref] [PubMed]

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(43), 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 400(6742), 342–344 (1999).
[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 holograph,” Science 256(5059), 1009–1012 (1992).
[Crossref] [PubMed]

Kley, E. B.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Kohmura, Y.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95(8), 085503 (2005).
[Crossref] [PubMed]

Lee, J. Y.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

Lim, S.

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(43), 15343–15346 (2005).
[Crossref] [PubMed]

Lörgen, M.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Lüning, J.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[Crossref]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Lustig, M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Marchesini, S.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006).
[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(14), 140101 (2003).
[Crossref]

Marks, D. L.

Marron, J. C.

McNulty, I.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[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 holograph,” Science 256(5059), 1009–1012 (1992).
[Crossref] [PubMed]

Merdji, H.

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond x-ray holography using extended references,” Phys. Rev. Lett. 105(9), 093901 (2010).
[Crossref] [PubMed]

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(43), 15343–15346 (2005).
[Crossref] [PubMed]

Miao, J.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95(8), 085503 (2005).
[Crossref] [PubMed]

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67(17), 174104 (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(22), 220101 (2003).
[Crossref]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400(6742), 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(5624), 1419–1421 (2003).
[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(43), 15343–15346 (2005).
[Crossref] [PubMed]

Nelson, J.

Nishino, Y.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95(8), 085503 (2005).
[Crossref] [PubMed]

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(22), 220101 (2003).
[Crossref]

Noy, A.

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006).
[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(14), 140101 (2003).
[Crossref]

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1-4), 259–268 (1992).
[Crossref]

Osherovich, E.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Pauly, J. M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Paxman, R. G.

Rick, R.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[Crossref]

Risbud, S. H.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95(8), 085503 (2005).
[Crossref] [PubMed]

Rivenson, Y.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010).
[Crossref]

Romberg, J.

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23(3), 969–985 (2007).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
[Crossref]

Rosen, R.

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1-4), 259–268 (1992).
[Crossref]

Sakdinawat, A. E.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

Salditt, T.

R. N. Wilke, M. Vassholz, and T. Salditt, “Semi-transparent central stop in high-resolution X-ray ptychography using Kirkpatrick–Baez focusing,” Acta Crystallogr. A 69(5), 490–497 (2013).
[Crossref]

Sayre, D.

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from X-ray diffraction data,” Acta Crystallogr. A 62(4), 248–261 (2006).
[Crossref] [PubMed]

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(43), 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 400(6742), 342–344 (1999).
[Crossref]

Scherz, A.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[Crossref]

Schlotter, W. F.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[Crossref]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Segev, M.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
[Crossref] [PubMed]

Seibert, M. M.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

Shaevitz, J. W.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

Shapiro, D.

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from X-ray diffraction data,” Acta Crystallogr. A 62(4), 248–261 (2006).
[Crossref] [PubMed]

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 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. U.S.A. 102(43), 15343–15346 (2005).
[Crossref] [PubMed]

Shapiro, D. A.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

Shechtman, Y.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Shoham, S.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Sidorenko, P.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Song, C.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95(8), 085503 (2005).
[Crossref] [PubMed]

Spence, J. C. H.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006).
[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(14), 140101 (2003).
[Crossref]

Steinbrener, J.

Steiner, S.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Stern, A.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010).
[Crossref]

Stöhr, J.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[Crossref]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Stroke, G. W.

G. W. Stroke, “Lensless Fourier‐transform method for optical holography,” Appl. Phys. Lett. 6(10), 201–203 (1965).
[Crossref]

Szameit, A.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
[Crossref] [PubMed]

Szöke, A.

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

Tao, T.

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
[Crossref]

Thibault, P.

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from X-ray diffraction data,” Acta Crystallogr. A 62(4), 248–261 (2006).
[Crossref] [PubMed]

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(43), 15343–15346 (2005).
[Crossref] [PubMed]

Tivive, F. H. C.

F. H. C. Tivive and A. Bouzerdoum, “A compressed sensing method for complex-valued signals with application to through-the-wall radar imaging,” in Pro. IEEE Int. Conf. Acoust. Speech Signal Process. (ICASSP)2144–2148 (2014).

Turner, J. J.

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(5624), 1419–1421 (2003).
[Crossref] [PubMed]

Vassholz, M.

R. N. Wilke, M. Vassholz, and T. Salditt, “Semi-transparent central stop in high-resolution X-ray ptychography using Kirkpatrick–Baez focusing,” Acta Crystallogr. A 69(5), 490–497 (2013).
[Crossref]

Wakin, M. B.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

Weierstall, U.

N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006).
[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(14), 140101 (2003).
[Crossref]

Wilke, R. N.

R. N. Wilke, M. Vassholz, and T. Salditt, “Semi-transparent central stop in high-resolution X-ray ptychography using Kirkpatrick–Baez focusing,” Acta Crystallogr. A 69(5), 490–497 (2013).
[Crossref]

Yavneh, I.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

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(5624), 1419–1421 (2003).
[Crossref] [PubMed]

Zibulevsky, M.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[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(5624), 1419–1421 (2003).
[Crossref] [PubMed]

Acta Crystallogr. A (2)

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from X-ray diffraction data,” Acta Crystallogr. A 62(4), 248–261 (2006).
[Crossref] [PubMed]

R. N. Wilke, M. Vassholz, and T. Salditt, “Semi-transparent central stop in high-resolution X-ray ptychography using Kirkpatrick–Baez focusing,” Acta Crystallogr. A 69(5), 490–497 (2013).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

G. W. Stroke, “Lensless Fourier‐transform method for optical holography,” Appl. Phys. Lett. 6(10), 201–203 (1965).
[Crossref]

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, C. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89(16), 163112 (2006).
[Crossref]

Commun. Pure Appl. Math. (1)

E. J. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
[Crossref]

IEEE Signal Process. Mag. (2)

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

IEEE Trans. Inf. Theory (3)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[Crossref]

Inverse Probl. (1)

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23(3), 969–985 (2007).
[Crossref]

J. Disp. Technol. (1)

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010).
[Crossref]

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

Magn. Reson. Med. (1)

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Nat. Mater. (1)

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).
[Crossref] [PubMed]

Nat. Photonics (1)

S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel x-ray holography,” Nat. Photonics 2(9), 560–563 (2008).
[Crossref]

Nature (2)

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[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 400(6742), 342–344 (1999).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. B (3)

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(14), 140101 (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(22), 220101 (2003).
[Crossref]

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67(17), 174104 (2003).
[Crossref]

Phys. Rev. Lett. (2)

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond x-ray holography using extended references,” Phys. Rev. Lett. 105(9), 093901 (2010).
[Crossref] [PubMed]

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of GaN quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95(8), 085503 (2005).
[Crossref] [PubMed]

Physica D (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1-4), 259–268 (1992).
[Crossref]

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

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Figures (10)

Fig. 1
Fig. 1 Beam stop in (a) CDI and (b) FTH, causing central low-frequencies information to be lost in the diffraction pattern.
Fig. 2
Fig. 2 (a) Object, (b) The Fourier spectrum of the object, the black rectangle mimics the beam stop, (c) conventional reconstruction by taking the inverse Fourier transform of the incomplete measurement, (d) CS reconstruction.
Fig. 3
Fig. 3 Image reconstruction from a single hologram with missing central data in multiple references FTH. (a) Sample mask contains an object and 2 references. (b) Simulated hologram with 41 × 41 pixel region missing in the center. (c) Inverse FT of (b) by setting unmeasured region to zero. (d) CS reconstruction from (b).
Fig. 4
Fig. 4 Image reconstruction from a single hologram with missing central data in HERALDO. (a) The sample mask contains an object and a thin slit reference. (b) Simulated processed hologram with 41 × 41 pixels region missing in the center. (c) Inverse FT of (b) by setting unmeasured region to zero. (d) CS reconstruction.
Fig. 5
Fig. 5 Left: Reconstructed images with different missing area sizes for 35db noise, using IFT and CS algorithms. Right: The SNR of reconstructed images is plotted against missing area sizes using IFT and CS methods. Plots for different noise levels are shown in different colors.
Fig. 6
Fig. 6 Top row: FTH experimental setup. L1: collimator, O: object mask: containing an object and a point reference, L2: Fourier transforming lens. Bottom row: FTH real experiment. (a) Captured hologram, the red rectangle denotes the missing data area of 41 × 41 pixels. (b) IFT reconstruction from the complete hologram. (c) IFT reconstruction from incomplete hologram. (d) CS reconstruction from incomplete hologram.
Fig. 7
Fig. 7 Reconstruction of a diffraction pattern with missing central data. (a) An original 128 × 128 pixels image is embedded in a 512 × 512 pixels array. (b) The diffraction intensity pattern. The data at the central 31 × 31 pixels are removed. (c) HIO reconstruction after 2000 iterations (d) reconstruction after 2000 iterations using proposed HIO + CS algorithm.
Fig. 8
Fig. 8 Left: Reconstruction error vs. iteration number of the missing data region (black) and high frequency region (blue) for conventional HIO. Reconstruction error of the missing data region (red) for the proposed HIO + CS algorithm. Right: The block diagram of the proposed HIO + CS method.
Fig. 9
Fig. 9 Left: reconstructed images with different missing area sizes using HIO and HIO + CS algorithms (Only central 200 × 200 pixels are shown out of 512 × 512). Right: The SNR of reconstructed images is plotted against missing area sizes using HIO and HIO + CS algorithms. Plots for different noise levels are shown in different colors.
Fig. 10
Fig. 10 Top row: CDI experimental setup. L1: collimator, S: aperture stop, O: object, L2: Fourier transforming lens. Bottom row: CDI experiment. (a) Captured diffraction pattern with red rectangle showing the removed part. (b) HIO reconstruction after 2000 iterations with complete data (c) HIO reconstruction after 2000 iterations with missing data (d) HIO + CS Reconstruction (2000 HIO iterations total).

Equations (14)

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g=Φf,
f= i=1 N α i ψ i =Ψα ,
α = min α α l 1  subject to g=Φf=ΦΨα,
α = min α α l 1  subject to  gΦΨα l 2 ε,
O H =HFo,
o = min o o TV  subject to  O H =HFo,
f( r )=o( r )+δ( r r 0 ).
I=F A f f ,
A f f ( r )= A δδ ( r )+ A oo ( r )+ o ( r r 0 )+o( r+ r 0 ).
I H =HF A f f ,
F 1 I H =( F 1 H ) A f f ,
A f f = min A f f A f f TV  subject to  I H =HF A f f .
G k+1 ={ G k+1 ,                for( fx,fy )B | G |exp( i ϕ k+1 ),  for( fx,fy )B.
g k+1 ={ g k+1 ,          for( x,y )C g k β g k+1 ,  for( x,y )C.

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