Abstract

A non-locally coded Fourier-transform ghost imaging (FGI) scheme and relevant coded phase retrieval method have been proposed to improve the image quality in ghost imaging. By inserting masks in the reference beam, the sample in the test beam is non-locally modulated, and coded Fourier-transform diffraction patterns of the sample are obtained via intensity correlation calculations between the two beams. Encoding and decoding procedures are incorporated in the phase retrieval process based on traditional hybrid input-output algorithm. Simulation and experiment results show that the spatial information of samples is successfully recovered from the coded diffraction patterns obtained using three masks, and the image quality is improved remarkably. If promoting this approach to x-ray FGI systems, it may realize high-resolution x-ray microscopy without increasing the sample’s radiation damage.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]

2018 (5)

X. Shi, N. Burdet, D. Batey, and I. Robinson, “Multi-modal ptychography: Recent developments and applications,” Applied Sciences 8, 1054 (2018).
[Crossref]

S. Han, H. Yu, X. Shen, H. Liu, W. Gong, and Z. Liu, “A review of ghost imaging via sparsity constraints,” Applied Sciences 8, 1379 (2018).
[Crossref]

A. X. Zhang, Y. H. He, L. A. Wu, L. M. Chen, and B. B. Wang, “Table-top x-ray ghost imaging with ultra-low radiation,” Optica 5, 374–377 (2018).
[Crossref]

T. Goldstein and C. Studer, “Phasemax: Convex phase retrieval via basis pursuit,” IEEE Transactions on Information Theory 64, 2675–2689 (2018).
[Crossref]

R. Zhu, H. Yu, R. Lu, Z. Tan, and S. Han, “Spatial multiplexing reconstruction for fourier-transform ghost imaging via sparsity constraints,” Optics Express 26, 2181–2190 (2018).
[Crossref] [PubMed]

2017 (4)

A. Schori and S. Shwartz, “X-ray ghost imaging with a laboratory source,” Optics Express 25, 14822–14828 (2017).
[Crossref]

V. Katkovnik, I. Shevkunov, N. V. Petrov, and K. Egiazarian, “Computational super-resolution phase retrieval from multiple phase-coded diffraction patterns: simulation study and experiments,” Optica 4, 786–794 (2017).
[Crossref]

Y. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Communications on Pure & Applied Mathematics 70, 739–747 (2017).
[Crossref]

H. S. Kang, C. K. Min, H. Heo, C. Kim, H. Yang, G. Kim, I. Nam, S. Y. Baek, H. J. Choi, and G. Mun, “Hard x-ray free-electron laser with femtosecond-scale timing jitter,” Nature Photonics 11, 708–714 (2017).
[Crossref]

2016 (5)

W. Gong, C. Zhao, H. Yu, M. Chen, W. Xu, and S. Han, “Three-dimensional ghost imaging lidar via sparsity constraint,” Scientific Rep 6, 26133 (2016).
[Crossref]

G. Wang and G. B. Giannakis, “Solving random systems of quadratic equations via truncated generalized gradient flow,” Advances in Neural Information Processing Systems 1050, 568–576 (2016).

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard x rays,” Physical Review Letters 117, 113901 (2016).
[Crossref]

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Physical Review Letters 117, 113902 (2016).
[Crossref] [PubMed]

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Scientific Rep 6, 25718 (2016).
[Crossref]

2015 (4)

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Applied & Computational Harmonic Analysis 39, 277–299 (2015).
[Crossref]

W. Gong and S. Han, “High-resolution far-field ghost imaging via sparsity constraint,” Scientific Rep 5, 9280 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Transactions on Information Theory 61, 1985–2007 (2015).
[Crossref]

P. Netrapalli, P. Jain, and S. Sanghavi, “Phase retrieval using alternating minimization,” IEEE Transactions on Signal Processing 63, 4814–4826 (2015).
[Crossref]

2014 (3)

X. Xu, E. Li, H. Yu, W. Gong, and S. Han, “Morphology separation in ghost imaging via sparsity constraint,” Optics Express 22, 14375–14381 (2014).
[Crossref]

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref]

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR : Efficient Phase Retrieval of Sparse Signals,” IEEE Transactions on Signal Processing 62, 928–938 (2014).
[Crossref]

2013 (4)

E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Communications on Pure & Applied Mathematics 66, 1241–1274 (2013).
[Crossref]

J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” Journal of Applied Crystallography 46, 312–318 (2013).
[Crossref] [PubMed]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

H. Jiang, R. Xu, C.-C. Chen, W. Yang, J. Fan, X. Tao, C. Song, Y. Kohmura, T. Xiao, and Y. Wang, “Three-dimensional coherent x-ray diffraction imaging of molten iron in mantle olivine at nanoscale resolution,” Physical Review Letters 110, 205501 (2013).
[Crossref] [PubMed]

2011 (1)

H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. Deponte, and U. Weierstall, “Femtosecond x-ray protein nanocrystallography,” Nature 470, 73–77 (2011).
[Crossref] [PubMed]

2009 (3)

I. Robinson and R. Harder, “Coherent x-ray diffraction imaging of strain at the nanoscale,” Nature Materials 8, 291 (2009).
[Crossref]

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Physical Review Letters 102, 018101 (2009).
[Crossref]

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

2008 (2)

T. Pierre, D. Martin, M. Andreas, B. Oliver, D. Christian, and P. Franz, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[Crossref]

2007 (2)

C. C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 3009–3014 (2007).
[Crossref]

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75, 021803 (2007).
[Crossref]

2004 (2)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Physical Review Letters 93, 093602 (2004).
[Crossref]

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in x-ray diffraction,” Physical Review Letters 92, 093903 (2004).
[Crossref]

2002 (1)

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “‘Two-photon’coincidence imaging with a classical source,” Physical Review Letters 89, 113601 (2002).
[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, 342 (1999).
[Crossref]

1995 (1)

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Physical Review A 52, R3429 (1995).
[Crossref]

1982 (1)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Applied Optics 21, 2758–2769 (1982).
[Crossref] [PubMed]

Andreas, M.

T. Pierre, D. Martin, M. Andreas, B. Oliver, D. Christian, and P. Franz, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

Aquila, A.

H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. Deponte, and U. Weierstall, “Femtosecond x-ray protein nanocrystallography,” Nature 470, 73–77 (2011).
[Crossref] [PubMed]

Bache, M.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Physical Review Letters 93, 093602 (2004).
[Crossref]

Baek, S. Y.

H. S. Kang, C. K. Min, H. Heo, C. Kim, H. Yang, G. Kim, I. Nam, S. Y. Baek, H. J. Choi, and G. Mun, “Hard x-ray free-electron laser with femtosecond-scale timing jitter,” Nature Photonics 11, 708–714 (2017).
[Crossref]

Barty, A.

H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. Deponte, and U. Weierstall, “Femtosecond x-ray protein nanocrystallography,” Nature 470, 73–77 (2011).
[Crossref] [PubMed]

Batey, D.

X. Shi, N. Burdet, D. Batey, and I. Robinson, “Multi-modal ptychography: Recent developments and applications,” Applied Sciences 8, 1054 (2018).
[Crossref]

Batey, D. J.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref]

Beck, A.

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR : Efficient Phase Retrieval of Sparse Signals,” IEEE Transactions on Signal Processing 62, 928–938 (2014).
[Crossref]

Bennink, R. S.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “‘Two-photon’coincidence imaging with a classical source,” Physical Review Letters 89, 113601 (2002).
[Crossref]

Bentley, S. J.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “‘Two-photon’coincidence imaging with a classical source,” Physical Review Letters 89, 113601 (2002).
[Crossref]

Bowman, A.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Bowman, R.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Boyd, R. W.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “‘Two-photon’coincidence imaging with a classical source,” Physical Review Letters 89, 113601 (2002).
[Crossref]

Brambilla, E.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Physical Review Letters 93, 093602 (2004).
[Crossref]

Burdet, N.

X. Shi, N. Burdet, D. Batey, and I. Robinson, “Multi-modal ptychography: Recent developments and applications,” Applied Sciences 8, 1054 (2018).
[Crossref]

Candes, E. J.

Y. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Communications on Pure & Applied Mathematics 70, 739–747 (2017).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Transactions on Information Theory 61, 1985–2007 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Applied & Computational Harmonic Analysis 39, 277–299 (2015).
[Crossref]

E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Communications on Pure & Applied Mathematics 66, 1241–1274 (2013).
[Crossref]

Cantelli, V.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Physical Review Letters 117, 113902 (2016).
[Crossref] [PubMed]

Cevher, V.

A. Yurtsever, Y. P. Hsieh, and V. Cevher, “Scalable convex methods for phase retrieval,” in IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing(CAMSAP), (IEEE, 2015), pp. 381–384.

Chapman, H. N.

H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. Deponte, and U. Weierstall, “Femtosecond x-ray protein nanocrystallography,” Nature 470, 73–77 (2011).
[Crossref] [PubMed]

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, 342 (1999).
[Crossref]

Chen, C. C.

C. C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 3009–3014 (2007).
[Crossref]

Chen, C.-C.

J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” Journal of Applied Crystallography 46, 312–318 (2013).
[Crossref] [PubMed]

H. Jiang, R. Xu, C.-C. Chen, W. Yang, J. Fan, X. Tao, C. Song, Y. Kohmura, T. Xiao, and Y. Wang, “Three-dimensional coherent x-ray diffraction imaging of molten iron in mantle olivine at nanoscale resolution,” Physical Review Letters 110, 205501 (2013).
[Crossref] [PubMed]

Chen, L. M.

Chen, M.

W. Gong, C. Zhao, H. Yu, M. Chen, W. Xu, and S. Han, “Three-dimensional ghost imaging lidar via sparsity constraint,” Scientific Rep 6, 26133 (2016).
[Crossref]

Chen, Y.

Y. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Communications on Pure & Applied Mathematics 70, 739–747 (2017).
[Crossref]

Cheng, J.

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75, 021803 (2007).
[Crossref]

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in x-ray diffraction,” Physical Review Letters 92, 093903 (2004).
[Crossref]

Choi, H. J.

H. S. Kang, C. K. Min, H. Heo, C. Kim, H. Yang, G. Kim, I. Nam, S. Y. Baek, H. J. Choi, and G. Mun, “Hard x-ray free-electron laser with femtosecond-scale timing jitter,” Nature Photonics 11, 708–714 (2017).
[Crossref]

Christian, D.

T. Pierre, D. Martin, M. Andreas, B. Oliver, D. Christian, and P. Franz, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

Claus, D.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref]

Deponte, D. P.

H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. Deponte, and U. Weierstall, “Femtosecond x-ray protein nanocrystallography,” Nature 470, 73–77 (2011).
[Crossref] [PubMed]

Du, G.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard x rays,” Physical Review Letters 117, 113901 (2016).
[Crossref]

Edgar, M. P.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Egiazarian, K.

Eldar, Y. C.

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR : Efficient Phase Retrieval of Sparse Signals,” IEEE Transactions on Signal Processing 62, 928–938 (2014).
[Crossref]

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

Fan, J.

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W. Gong and S. Han, “High-resolution far-field ghost imaging via sparsity constraint,” Scientific Rep 5, 9280 (2015).
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X. Xu, E. Li, H. Yu, W. Gong, and S. Han, “Morphology separation in ghost imaging via sparsity constraint,” Optics Express 22, 14375–14381 (2014).
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H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. Deponte, and U. Weierstall, “Femtosecond x-ray protein nanocrystallography,” Nature 470, 73–77 (2011).
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H. S. Kang, C. K. Min, H. Heo, C. Kim, H. Yang, G. Kim, I. Nam, S. Y. Baek, H. J. Choi, and G. Mun, “Hard x-ray free-electron laser with femtosecond-scale timing jitter,” Nature Photonics 11, 708–714 (2017).
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H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. Deponte, and U. Weierstall, “Femtosecond x-ray protein nanocrystallography,” Nature 470, 73–77 (2011).
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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, 342 (1999).
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H. Jiang, R. Xu, C.-C. Chen, W. Yang, J. Fan, X. Tao, C. Song, Y. Kohmura, T. Xiao, and Y. Wang, “Three-dimensional coherent x-ray diffraction imaging of molten iron in mantle olivine at nanoscale resolution,” Physical Review Letters 110, 205501 (2013).
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C. C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 3009–3014 (2007).
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Li, E.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Scientific Rep 6, 25718 (2016).
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X. Xu, E. Li, H. Yu, W. Gong, and S. Han, “Morphology separation in ghost imaging via sparsity constraint,” Optics Express 22, 14375–14381 (2014).
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Li, X.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Transactions on Information Theory 61, 1985–2007 (2015).
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E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Applied & Computational Harmonic Analysis 39, 277–299 (2015).
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S. Han, H. Yu, X. Shen, H. Liu, W. Gong, and Z. Liu, “A review of ghost imaging via sparsity constraints,” Applied Sciences 8, 1379 (2018).
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M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75, 021803 (2007).
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Liu, Y.

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75, 021803 (2007).
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S. Han, H. Yu, X. Shen, H. Liu, W. Gong, and Z. Liu, “A review of ghost imaging via sparsity constraints,” Applied Sciences 8, 1379 (2018).
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Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Scientific Rep 6, 25718 (2016).
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R. Zhu, H. Yu, R. Lu, Z. Tan, and S. Han, “Spatial multiplexing reconstruction for fourier-transform ghost imaging via sparsity constraints,” Optics Express 26, 2181–2190 (2018).
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H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard x rays,” Physical Review Letters 117, 113901 (2016).
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A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Physical Review Letters 93, 093602 (2004).
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Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Physical Review Letters 102, 018101 (2009).
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T. Pierre, D. Martin, M. Andreas, B. Oliver, D. Christian, and P. Franz, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
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J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” Journal of Applied Crystallography 46, 312–318 (2013).
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C. C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 3009–3014 (2007).
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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, 342 (1999).
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H. S. Kang, C. K. Min, H. Heo, C. Kim, H. Yang, G. Kim, I. Nam, S. Y. Baek, H. J. Choi, and G. Mun, “Hard x-ray free-electron laser with femtosecond-scale timing jitter,” Nature Photonics 11, 708–714 (2017).
[Crossref]

Mun, G.

H. S. Kang, C. K. Min, H. Heo, C. Kim, H. Yang, G. Kim, I. Nam, S. Y. Baek, H. J. Choi, and G. Mun, “Hard x-ray free-electron laser with femtosecond-scale timing jitter,” Nature Photonics 11, 708–714 (2017).
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H. S. Kang, C. K. Min, H. Heo, C. Kim, H. Yang, G. Kim, I. Nam, S. Y. Baek, H. J. Choi, and G. Mun, “Hard x-ray free-electron laser with femtosecond-scale timing jitter,” Nature Photonics 11, 708–714 (2017).
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P. Netrapalli, P. Jain, and S. Sanghavi, “Phase retrieval using alternating minimization,” IEEE Transactions on Signal Processing 63, 4814–4826 (2015).
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Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Physical Review Letters 102, 018101 (2009).
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T. Pierre, D. Martin, M. Andreas, B. Oliver, D. Christian, and P. Franz, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
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B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
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T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Physical Review A 52, R3429 (1995).
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D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Physical Review Letters 117, 113902 (2016).
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I. Robinson and R. Harder, “Coherent x-ray diffraction imaging of strain at the nanoscale,” Nature Materials 8, 291 (2009).
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J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” Journal of Applied Crystallography 46, 312–318 (2013).
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Sanghavi, S.

P. Netrapalli, P. Jain, and S. Sanghavi, “Phase retrieval using alternating minimization,” IEEE Transactions on Signal Processing 63, 4814–4826 (2015).
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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, 342 (1999).
[Crossref]

Scheel, M.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Physical Review Letters 117, 113902 (2016).
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A. Schori and S. Shwartz, “X-ray ghost imaging with a laboratory source,” Optics Express 25, 14822–14828 (2017).
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H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. Deponte, and U. Weierstall, “Femtosecond x-ray protein nanocrystallography,” Nature 470, 73–77 (2011).
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S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Optics Express 17, 23920–23946 (2009).
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T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Physical Review A 52, R3429 (1995).
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S. Han, H. Yu, X. Shen, H. Liu, W. Gong, and Z. Liu, “A review of ghost imaging via sparsity constraints,” Applied Sciences 8, 1379 (2018).
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Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Scientific Rep 6, 25718 (2016).
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M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75, 021803 (2007).
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Shevkunov, I.

Shi, X.

X. Shi, N. Burdet, D. Batey, and I. Robinson, “Multi-modal ptychography: Recent developments and applications,” Applied Sciences 8, 1054 (2018).
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T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Physical Review A 52, R3429 (1995).
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A. Schori and S. Shwartz, “X-ray ghost imaging with a laboratory source,” Optics Express 25, 14822–14828 (2017).
[Crossref]

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E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Transactions on Information Theory 61, 1985–2007 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Applied & Computational Harmonic Analysis 39, 277–299 (2015).
[Crossref]

Song, C.

H. Jiang, R. Xu, C.-C. Chen, W. Yang, J. Fan, X. Tao, C. Song, Y. Kohmura, T. Xiao, and Y. Wang, “Three-dimensional coherent x-ray diffraction imaging of molten iron in mantle olivine at nanoscale resolution,” Physical Review Letters 110, 205501 (2013).
[Crossref] [PubMed]

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T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Physical Review A 52, R3429 (1995).
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E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Communications on Pure & Applied Mathematics 66, 1241–1274 (2013).
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Studer, C.

T. Goldstein and C. Studer, “Phasemax: Convex phase retrieval via basis pursuit,” IEEE Transactions on Information Theory 64, 2675–2689 (2018).
[Crossref]

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B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Szameit, A.

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

Takahashi, Y.

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Physical Review Letters 102, 018101 (2009).
[Crossref]

Tan, S.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Scientific Rep 6, 25718 (2016).
[Crossref]

Tan, Z.

R. Zhu, H. Yu, R. Lu, Z. Tan, and S. Han, “Spatial multiplexing reconstruction for fourier-transform ghost imaging via sparsity constraints,” Optics Express 26, 2181–2190 (2018).
[Crossref] [PubMed]

Tao, X.

H. Jiang, R. Xu, C.-C. Chen, W. Yang, J. Fan, X. Tao, C. Song, Y. Kohmura, T. Xiao, and Y. Wang, “Three-dimensional coherent x-ray diffraction imaging of molten iron in mantle olivine at nanoscale resolution,” Physical Review Letters 110, 205501 (2013).
[Crossref] [PubMed]

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B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

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E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Communications on Pure & Applied Mathematics 66, 1241–1274 (2013).
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Wang, B. B.

Wang, C. W.

C. C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 3009–3014 (2007).
[Crossref]

Wang, G.

G. Wang and G. B. Giannakis, “Solving random systems of quadratic equations via truncated generalized gradient flow,” Advances in Neural Information Processing Systems 1050, 568–576 (2016).

Wang, Y.

H. Jiang, R. Xu, C.-C. Chen, W. Yang, J. Fan, X. Tao, C. Song, Y. Kohmura, T. Xiao, and Y. Wang, “Three-dimensional coherent x-ray diffraction imaging of molten iron in mantle olivine at nanoscale resolution,” Physical Review Letters 110, 205501 (2013).
[Crossref] [PubMed]

Wei, Q.

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75, 021803 (2007).
[Crossref]

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H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. Deponte, and U. Weierstall, “Femtosecond x-ray protein nanocrystallography,” Nature 470, 73–77 (2011).
[Crossref] [PubMed]

Welsh, S.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

White, T. A.

H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. Deponte, and U. Weierstall, “Femtosecond x-ray protein nanocrystallography,” Nature 470, 73–77 (2011).
[Crossref] [PubMed]

Wu, J.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Scientific Rep 6, 25718 (2016).
[Crossref]

Wu, L. A.

Xiao, T.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard x rays,” Physical Review Letters 117, 113901 (2016).
[Crossref]

H. Jiang, R. Xu, C.-C. Chen, W. Yang, J. Fan, X. Tao, C. Song, Y. Kohmura, T. Xiao, and Y. Wang, “Three-dimensional coherent x-ray diffraction imaging of molten iron in mantle olivine at nanoscale resolution,” Physical Review Letters 110, 205501 (2013).
[Crossref] [PubMed]

Xie, H.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard x rays,” Physical Review Letters 117, 113901 (2016).
[Crossref]

Xu, R.

J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” Journal of Applied Crystallography 46, 312–318 (2013).
[Crossref] [PubMed]

H. Jiang, R. Xu, C.-C. Chen, W. Yang, J. Fan, X. Tao, C. Song, Y. Kohmura, T. Xiao, and Y. Wang, “Three-dimensional coherent x-ray diffraction imaging of molten iron in mantle olivine at nanoscale resolution,” Physical Review Letters 110, 205501 (2013).
[Crossref] [PubMed]

Xu, W.

W. Gong, C. Zhao, H. Yu, M. Chen, W. Xu, and S. Han, “Three-dimensional ghost imaging lidar via sparsity constraint,” Scientific Rep 6, 26133 (2016).
[Crossref]

Xu, X.

X. Xu, E. Li, H. Yu, W. Gong, and S. Han, “Morphology separation in ghost imaging via sparsity constraint,” Optics Express 22, 14375–14381 (2014).
[Crossref]

Yang, H.

H. S. Kang, C. K. Min, H. Heo, C. Kim, H. Yang, G. Kim, I. Nam, S. Y. Baek, H. J. Choi, and G. Mun, “Hard x-ray free-electron laser with femtosecond-scale timing jitter,” Nature Photonics 11, 708–714 (2017).
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Figures (10)

Fig. 1
Fig. 1 Scheme for non-locally coded Fourier-transform ghost imaging(FGI).
Fig. 2
Fig. 2 Map for the sample. (a) is the amplitude distribution, (b) is the phase distribution.
Fig. 3
Fig. 3 Empirical probability of success based on 50 random trials for each kind of mask distribution.
Fig. 4
Fig. 4 The relationship between RMSE and SNR (a) for different mask numbers with all masks conforming zeros mean binary distribution, (b) for different mask numbers with all masks conforming non-zeros mean binary distribution, (c) for different mask distributions with three masks used in simulation. The curves representing uniform random, zeros mean quaternary and zeros mean binary are almost coincident.
Fig. 5
Fig. 5 Recovered amplitude and phase results under different mask feature sizes. The first row are the phase distributions of the masks, the second row are the corresponding recovered amplitude results, and the third row are the corresponding recovered phase results. The mask feature size is (a) 1 × 1 pixels, (b) 8 × 8 pixels, (c) and (d) 64 × 64 pixels. Six masks are used in the last column, while three masks are used in the other columns.
Fig. 6
Fig. 6 Convergence for different support looseness. γ is the area ratio of support region to sample size.
Fig. 7
Fig. 7 Coded Fourier-transform diffraction patterns obtained from the second-order intensity correlation calculations. (a), (b) and (c) were the results of different subsets of the random phase plate, labeled as P 1 , P 2 and P3.
Fig. 8
Fig. 8 (a) Sample’s photograph by optical microscopy, (b) recovered result of traditional HIO algorithm, (c)-(d) recovered results from one coded diffraction patterns, (e)-(g) recovered results from two coded diffraction patterns, (h) recovered results from all three coded diffraction patterns.
Fig. 9
Fig. 9 Recovery for different support sizes. (a) 1 × 1 m m 2, (b) 1.5 × 1.5 m m 2, (c) 2 × 2 m m 2.
Fig. 10
Fig. 10 Recovered results in the case of low SNR. The number of coded diffraction patterns used in recovery is (a)1 (b)2 (c)3.

Equations (15)

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Δ I r ( x r ) Δ I t ( x t ) = I r ( x r ) I t ( x t ) I r ( x r ) I t ( x t ) = | d x 1 d x 2 G ( x 1 , x 2 ) h r * ( x 1 , x r ) h t ( x 2 , x t ) | 2 ,
G ( x 1 , x 2 ) = δ ( x 1 , x 2 ) .
h t ( x 2 , x t ) d x exp  { i π λ d 1 ( x 2 x ) 2 } t ( x ) exp  { i π λ d 2 ( x x t ) 2 } ,
h r ( x 1 , x r ) d x exp  { i π λ d 1 ( x 1 x ) 2 } s ( x ) exp  { i π λ d 2 ( x x r ) 2 } ,
Δ I r ( x r ) Δ I t ( x t = 0 ) | d x s * ( x ) t ( x ) exp  { i 2 π x x r λ d 2 } | 2 | F { s * ( x ) t ( x ) } | 2 ,
Δ I r ( x r ) Δ I t ( x t = 0 ) | d x t ( x ) exp  { i 2 π x x r λ d 2 } | 2 | F { t ( x ) } | 2 .
f = φ n , l s l *
F = F { f } = A exp  ( i ϕ )
f = F 1 { G l exp  ( i ϕ ) }
φ n , l   ' = f / s l *
φ n , l + 1 = φ n , l   ' S + ( φ n , l   ' β φ n , l ) ( 1 S )
s l = { 1   with prob .1 / 4 i   with prob .1 / 4 1   with prob .1 / 4 i   with prob .1 / 4
s l = { 1   with prob .1 / 2 1   with prob .1 / 2
s l = { 1   with prob .1 / 2 i   with prob .1 / 2
RMSE = ( | φ n , L | | t | ) 2 ( | t | 2 ) .

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