October  2021, 15(5): 1051-1075. doi: 10.3934/ipi.2021028

Phase retrieval from Fourier measurements with masks

1. 

Department of Mathematics, Hangzhou Normal University, 2318 Yuhangtang Road, 311121

2. 

School of Mathematical Sciences, Zhejiang University, 38 Zheda Road, 310027, Hangzhou, China

* Corresponding author: Song Li

Received  November 2019 Revised  March 2021 Published  October 2021 Early access  May 2021

Fund Project: This work is supported by the key project of NSFC under grant number 11531013, 12071426, the fundamental research funds for the central universities (2020XZZX002-03) and the Zhejiang Provincial Natural Science Foundation grant (LQ19A010008)

This paper concerns the problem of phase retrieval from Fourier measurements with random masks. Here we focus on researching two kinds of random masks. Firstly, we utilize the Fourier measurements with real masks to estimate a general signal $ \mathit{\boldsymbol{x}}_0\in \mathbb{R}^d $ in noiseless case when $ d $ is even. It is demonstrated that $ O(\log^2d) $ real random masks are able to ensure accurate recovery of $ \mathit{\boldsymbol{x}}_0 $. Then we find that such real masks are not adaptable to reconstruct complex signals of even dimension. Subsequently, we prove that $ O(\log^4d) $ complex masks are enough to stably estimate a general signal $ \mathit{\boldsymbol{x}}_0\in \mathbb{C}^d $ under bounded noise interference, which extends E. Candès et al.'s work. Meanwhile, we establish tighter error estimations for real signals of even dimensions or complex signals of odd dimensions by using $ O(\log^2d) $ real masks. Finally, we intend to tackle with the noisy phase problem about an $ s $-sparse signal by a robust and efficient approach, namely, two-stage algorithm. Based on the stable guarantees for general signals, we show that the $ s $-sparse signal $ \mathit{\boldsymbol{x}}_0 $ can be stably recovered from composite measurements under near-optimal sample complexity up to a $ \log $ factor, namely, $ O(s\log(\frac{ed}{s})\log^4(s\log(\frac{ed}{s}))) $

Citation: Huiping Li, Song Li. Phase retrieval from Fourier measurements with masks. Inverse Problems & Imaging, 2021, 15 (5) : 1051-1075. doi: 10.3934/ipi.2021028
References:
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[2]

S. Bahmani and J. Romberg, Efficient compressive phase retrieval with constrained sensing vectors, IEEE Neural Information Processing Systems, 1 (2015), 523–531. Available from: https://dl.acm.org/doi/abs/10.5555/2969239.2969298. Google Scholar

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R. BalanB. G. BodmannP. G. Cassazza and D. Edidin, Painless reconstruction from magnitudes of frame coefficients, J. Fourier Anal. Appl., 15 (2009), 488-501.  doi: 10.1007/s00041-009-9065-1.  Google Scholar

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A. S. BandeiraJ. CahillD. G. Mixon and A. A. Nelson, Saving phase: Injectivity and stability for phase retrieval, Appl. Comput. Harmon. Anal., 37 (2014), 106-125.  doi: 10.1016/j.acha.2013.10.002.  Google Scholar

[5]

A. S. BandeiraY. Chen and D. G. Mixon, Phase retrieval from power spectra of masked signals, Inform. Inference: A Journal of the IMA, 3 (2014), 83-102.  doi: 10.1093/imaiai/iau002.  Google Scholar

[6]

T. T. CaiX. Li and Z. Ma, Optimal rates of convergence for noisy sparse phase retrieval via thresholded Wirtinger flow, Ann. Statist., 44 (2016), 2221-2251.  doi: 10.1214/16-AOS1443.  Google Scholar

[7]

E. J. Candès, The restricted isometry property and its implications for compressed sensing, Comptes Rendus Mathematique, 346 (2008), 589-592.  doi: 10.1016/j.crma.2008.03.014.  Google Scholar

[8]

E. J. CandèsY. C. EldarT. Strohmer and V. Voroninshi, Phase retrieval via matrix completion, SIAM J. Imaging Sci., 6 (2013), 199-225.  doi: 10.1137/110848074.  Google Scholar

[9]

E. J. Candès and X. Li, Solving quadratic equations via phaselift when there are about as many equations as unknowns, Found. Comput. Math., 149 (2013), 1-10.  doi: 10.1007/s10208-013-9162-z.  Google Scholar

[10]

E. J. CandèsX. Li and M. Soltanolkotabi, Phase retrieval from coded diffraction patterns, Appl. Comput. Harmon. Anal., 39 (2015), 277-299.  doi: 10.1016/j.acha.2014.09.004.  Google Scholar

[11]

E. J. CandèsX. Li and M. Soltanolkotabi, Phase retrieval via Wirtinger Flow: Theory and algorithms, IEEE Trans. Inform. Theory, 61 (2015), 1985-2007.  doi: 10.1109/TIT.2015.2399924.  Google Scholar

[12]

E. J. Candès and Y. Plan, A probabilistic and RIP-less theory of compressed sensing, IEEE Trans. Inform. Theory, 57 (2011), 7235-7254.  doi: 10.1109/TIT.2011.2161794.  Google Scholar

[13]

E. J. CandèsT. Stromher and V. Voroninshi, PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming, Comm. Pure Appl. Math., 66 (2013), 1241-1274.  doi: 10.1002/cpa.21432.  Google Scholar

[14]

E. J. Candès and T. Tao, Near optimal signal recovery from random projections: Universal encoding strategies?, IEEE Trans. Inform. Theory, 52 (2006), 5406-5425.  doi: 10.1109/TIT.2006.885507.  Google Scholar

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E. J. Candès and T. Tao, The power of convex relaxation: Near-optimal matrix completion, IEEE Trans. Inform. Theory, 56 (2010), 2053-2080.  doi: 10.1109/TIT.2010.2044061.  Google Scholar

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A. Chai, M. Moscoso and G. Papanicolaou, Array imaging using intensity-only measurements, Inverse Probl., 27 (2011), 015005, 16 pp. doi: 10.1088/0266-5611/27/1/015005.  Google Scholar

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Y. Chen and E. J. Candès, Solving random quadratic systems of equations is nearly as easy as solving linear systems, Comm. Pure Appl. Math., 70 (2015), 739-747.  doi: 10.1002/cpa.21638.  Google Scholar

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J. C. Dainty and J. R. Fienup, Phase retrieval and image reconstruction for astronomy, in Image Recovery: Theory and Application, Academic Press, (1987), 231–275. Available from: https://www.researchgate.net/publication/247171131_Phase_retrieval_and_image_reconstruction_for_astronomy. Google Scholar

[20]

L. Demanet and P. Hand, Stable optimizationless recovery from phaseless linear measurements, J. Fourier Anal. Appl., 20 (2014), 199-221.  doi: 10.1007/s00041-013-9305-2.  Google Scholar

[21]

A. Fannjiang and W. Liao, Phase retrieval with random phase illumination, Inverse Problems, 28 (2012), 075008, 20 pp. doi: 10.1364/JOSAA.29.001847.  Google Scholar

[22]

J. R. Fienup, Phase retrieval algorithms: A comparison, Appl. Opt., 21 (1982), 2758-2769.  doi: 10.1364/AO.21.002758.  Google Scholar

[23]

R. W. Gerchberg and W. O. Saxton, A practical algorithm for the determination of phase from image and diffraction plane pictures, Optik, 35 (1972), 237–246. Available from: https://www.researchgate.net/publication/221725051_A_practical_algorithm_for_the_determination_of_phase_from_image_and_diffraction_plane_pictures. Google Scholar

[24]

D. Gross, Recovering low rank matrices from few coefficients in any basis, IEEE Trans. Inform. Theory, 57 (2011), 1548-1566.  doi: 10.1109/TIT.2011.2104999.  Google Scholar

[25]

D. GrossF. Krahmer and R. Kueng, A partial derandomization of PhaseLift using spherical designs, J. Fourier Anal. Appl., 21 (2015), 229-266.  doi: 10.1007/s00041-014-9361-2.  Google Scholar

[26]

D. GrossF. Krahmer and R. Kueng, Improved recovery guarantees for phase retrieval from coded diffraction patterns, Appl. Comput. Harmon. Anal., 42 (2017), 37-64.  doi: 10.1016/j.acha.2015.05.004.  Google Scholar

[27]

R. W. Harrison, Phase problem in crystallography, J. Opt. Soc. Amer. A, 10 (1993), 1046-1055.  doi: 10.1364/JOSAA.10.001046.  Google Scholar

[28]

W. HuangK. A. Gallivan and X. Zhang, Solving PhaseLift by low-rank Riemannian optimization methods, Procedia Computer Science, 80 (2016), 1125-1134.   Google Scholar

[29]

M. J. HumphryB. KrausA. C. HurstA. M. Maiden and J. M. Rodenburg, Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging, Nature Communications, 3 (2012), 1-7.  doi: 10.1038/ncomms1733.  Google Scholar

[30]

M. IwenA. Viswanathan and Y Wang, Robust sparse phase retrieval made easy, Appl. Comput. Harmon. Anal., 42 (2017), 135-142.  doi: 10.1016/j.acha.2015.06.007.  Google Scholar

[31]

K. Jaganathan, Y. C. Eldar and B. Hassibi, Phase retrieval with masks using convex optimization, IEEE International Symposium on Information Theory, (2015), 1655–1659. Google Scholar

[32]

F. Kramher and Y.-K. Liu, Phase retrieval without small-ball probability assumptions, IEEE Trans. Inform. Theory, 64 (2018), 485-500.  doi: 10.1109/TIT.2017.2757520.  Google Scholar

[33]

R. KuengH. Rauhut and U. Terstiege, Low rank matrix recovery from rank one measurements, Appl. Comput. Harmon. Anal., 42 (2014), 88-116.  doi: 10.1016/j.acha.2015.07.007.  Google Scholar

[34]

X. Li and V. Voroninski, Sparse signal recovery from quadratic measurements via convex programming, SIAM J. Math. Anal., 45 (2013), 3019-3033.  doi: 10.1137/120893707.  Google Scholar

[35]

R. Millane, Phase retrieval in crystallography and optics, J. Opt. Soc. Amer. A, 7 (1990), 394-411.  doi: 10.1364/JOSAA.7.000394.  Google Scholar

[36]

M. L. MoravecJ. K. Romberg and R. G. Baraniuk, Compressive phase retrieval, Proceedings of SPIE, 6701 (2007), 6701201-67012011.  doi: 10.1117/12.736360.  Google Scholar

[37]

P. NetrapalliP. Jain and S. Sanghavi, Phase retrieval using alternating minimization, IEEE Trans. Signal Process, 63 (2015), 4814-4826.  doi: 10.1109/TSP.2015.2448516.  Google Scholar

[38]

H. Ohlsson, A. Yang, R. Dong and S. Sastry, CPRL–An extension of compressive sensing to the phase retrieval problem, IEEE Neural Information Processing Systems, (2012), 1367–1375. Available from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.244.9102&rep=rep1&type=pdf Google Scholar

[39]

R. Pedarsani, K. Lee and K. Ramchandran, Phasecode: Fast and efficient compressive phase retrieval based on sparse-graph codes, Allerton Conference on Communication, Control and Computing, (2014), 842–849. doi: 10.1109/ALLERTON.2014.7028542.  Google Scholar

[40]

B. RechtM. Fazel and P. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev., 52 (2010), 471-501.  doi: 10.1137/070697835.  Google Scholar

[41]

Y. ShechtmanA. Beck and Y. C. Eldar, GESPAR: Efficient phase retrieval of sparse signals, IEEE Trans. Signal Process, 62 (2014), 928-938.  doi: 10.1109/TSP.2013.2297687.  Google Scholar

[42]

Y. ShechtmanY. C. EldarO. CohenH. N. ChapmanJ. Miao and M. Segev, Phase retrieval with application to optical imaging: A contemporary overview, IEEE Signal Processing Mag., 32 (2015), 87-109.  doi: 10.1109/MSP.2014.2352673.  Google Scholar

[43]

Y. ShechtmanY. C. EldarA. Szameit and M. Segev, Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing, Optics Express, 19 (2011), 14807-14822.  doi: 10.1364/OE.19.014807.  Google Scholar

[44]

I. WaldspurgerA. d'Aspremont and S. Mallat, Phase recovery, maxcut and complex semidefinite programming, Math. Prog., 149 (2015), 47-81.  doi: 10.1007/s10107-013-0738-9.  Google Scholar

[45]

G. WangL. ZhangG. B. GiannakisM. Akcakaya and J. Chen, Sparse phase retrieval via truncated amplitude flow, IEEE Trans. Signal Process, 66 (2018), 479-491.  doi: 10.1109/TSP.2017.2771733.  Google Scholar

[46]

G. ZhengR. Horstmeyer and C. Yang, Wide-field, high-resolution Fourier pty-chographic microscopy, Nature Photonics, 7 (2013), 739-745.  doi: 10.1038/nphoton.2013.187.  Google Scholar

show all references

References:
[1]

B. Adcock and A. C. Hansen, Generalized sampling and infinite-dimensional compressed sensing, Found. Comput. Math., 16 (2016), 1263-1323.  doi: 10.1007/s10208-015-9276-6.  Google Scholar

[2]

S. Bahmani and J. Romberg, Efficient compressive phase retrieval with constrained sensing vectors, IEEE Neural Information Processing Systems, 1 (2015), 523–531. Available from: https://dl.acm.org/doi/abs/10.5555/2969239.2969298. Google Scholar

[3]

R. BalanB. G. BodmannP. G. Cassazza and D. Edidin, Painless reconstruction from magnitudes of frame coefficients, J. Fourier Anal. Appl., 15 (2009), 488-501.  doi: 10.1007/s00041-009-9065-1.  Google Scholar

[4]

A. S. BandeiraJ. CahillD. G. Mixon and A. A. Nelson, Saving phase: Injectivity and stability for phase retrieval, Appl. Comput. Harmon. Anal., 37 (2014), 106-125.  doi: 10.1016/j.acha.2013.10.002.  Google Scholar

[5]

A. S. BandeiraY. Chen and D. G. Mixon, Phase retrieval from power spectra of masked signals, Inform. Inference: A Journal of the IMA, 3 (2014), 83-102.  doi: 10.1093/imaiai/iau002.  Google Scholar

[6]

T. T. CaiX. Li and Z. Ma, Optimal rates of convergence for noisy sparse phase retrieval via thresholded Wirtinger flow, Ann. Statist., 44 (2016), 2221-2251.  doi: 10.1214/16-AOS1443.  Google Scholar

[7]

E. J. Candès, The restricted isometry property and its implications for compressed sensing, Comptes Rendus Mathematique, 346 (2008), 589-592.  doi: 10.1016/j.crma.2008.03.014.  Google Scholar

[8]

E. J. CandèsY. C. EldarT. Strohmer and V. Voroninshi, Phase retrieval via matrix completion, SIAM J. Imaging Sci., 6 (2013), 199-225.  doi: 10.1137/110848074.  Google Scholar

[9]

E. J. Candès and X. Li, Solving quadratic equations via phaselift when there are about as many equations as unknowns, Found. Comput. Math., 149 (2013), 1-10.  doi: 10.1007/s10208-013-9162-z.  Google Scholar

[10]

E. J. CandèsX. Li and M. Soltanolkotabi, Phase retrieval from coded diffraction patterns, Appl. Comput. Harmon. Anal., 39 (2015), 277-299.  doi: 10.1016/j.acha.2014.09.004.  Google Scholar

[11]

E. J. CandèsX. Li and M. Soltanolkotabi, Phase retrieval via Wirtinger Flow: Theory and algorithms, IEEE Trans. Inform. Theory, 61 (2015), 1985-2007.  doi: 10.1109/TIT.2015.2399924.  Google Scholar

[12]

E. J. Candès and Y. Plan, A probabilistic and RIP-less theory of compressed sensing, IEEE Trans. Inform. Theory, 57 (2011), 7235-7254.  doi: 10.1109/TIT.2011.2161794.  Google Scholar

[13]

E. J. CandèsT. Stromher and V. Voroninshi, PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming, Comm. Pure Appl. Math., 66 (2013), 1241-1274.  doi: 10.1002/cpa.21432.  Google Scholar

[14]

E. J. Candès and T. Tao, Near optimal signal recovery from random projections: Universal encoding strategies?, IEEE Trans. Inform. Theory, 52 (2006), 5406-5425.  doi: 10.1109/TIT.2006.885507.  Google Scholar

[15]

E. J. Candès and T. Tao, The power of convex relaxation: Near-optimal matrix completion, IEEE Trans. Inform. Theory, 56 (2010), 2053-2080.  doi: 10.1109/TIT.2010.2044061.  Google Scholar

[16]

A. Chai, M. Moscoso and G. Papanicolaou, Array imaging using intensity-only measurements, Inverse Probl., 27 (2011), 015005, 16 pp. doi: 10.1088/0266-5611/27/1/015005.  Google Scholar

[17]

Y. Chen and E. J. Candès, Solving random quadratic systems of equations is nearly as easy as solving linear systems, Comm. Pure Appl. Math., 70 (2015), 739-747.  doi: 10.1002/cpa.21638.  Google Scholar

[18]

J. V. Corbett, The Pauli problem, state reconstruction and quantum-real numbers, Rep. Math. Phys., 57 (2006), 53-68.  doi: 10.1016/S0034-4877(06)80008-X.  Google Scholar

[19]

J. C. Dainty and J. R. Fienup, Phase retrieval and image reconstruction for astronomy, in Image Recovery: Theory and Application, Academic Press, (1987), 231–275. Available from: https://www.researchgate.net/publication/247171131_Phase_retrieval_and_image_reconstruction_for_astronomy. Google Scholar

[20]

L. Demanet and P. Hand, Stable optimizationless recovery from phaseless linear measurements, J. Fourier Anal. Appl., 20 (2014), 199-221.  doi: 10.1007/s00041-013-9305-2.  Google Scholar

[21]

A. Fannjiang and W. Liao, Phase retrieval with random phase illumination, Inverse Problems, 28 (2012), 075008, 20 pp. doi: 10.1364/JOSAA.29.001847.  Google Scholar

[22]

J. R. Fienup, Phase retrieval algorithms: A comparison, Appl. Opt., 21 (1982), 2758-2769.  doi: 10.1364/AO.21.002758.  Google Scholar

[23]

R. W. Gerchberg and W. O. Saxton, A practical algorithm for the determination of phase from image and diffraction plane pictures, Optik, 35 (1972), 237–246. Available from: https://www.researchgate.net/publication/221725051_A_practical_algorithm_for_the_determination_of_phase_from_image_and_diffraction_plane_pictures. Google Scholar

[24]

D. Gross, Recovering low rank matrices from few coefficients in any basis, IEEE Trans. Inform. Theory, 57 (2011), 1548-1566.  doi: 10.1109/TIT.2011.2104999.  Google Scholar

[25]

D. GrossF. Krahmer and R. Kueng, A partial derandomization of PhaseLift using spherical designs, J. Fourier Anal. Appl., 21 (2015), 229-266.  doi: 10.1007/s00041-014-9361-2.  Google Scholar

[26]

D. GrossF. Krahmer and R. Kueng, Improved recovery guarantees for phase retrieval from coded diffraction patterns, Appl. Comput. Harmon. Anal., 42 (2017), 37-64.  doi: 10.1016/j.acha.2015.05.004.  Google Scholar

[27]

R. W. Harrison, Phase problem in crystallography, J. Opt. Soc. Amer. A, 10 (1993), 1046-1055.  doi: 10.1364/JOSAA.10.001046.  Google Scholar

[28]

W. HuangK. A. Gallivan and X. Zhang, Solving PhaseLift by low-rank Riemannian optimization methods, Procedia Computer Science, 80 (2016), 1125-1134.   Google Scholar

[29]

M. J. HumphryB. KrausA. C. HurstA. M. Maiden and J. M. Rodenburg, Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging, Nature Communications, 3 (2012), 1-7.  doi: 10.1038/ncomms1733.  Google Scholar

[30]

M. IwenA. Viswanathan and Y Wang, Robust sparse phase retrieval made easy, Appl. Comput. Harmon. Anal., 42 (2017), 135-142.  doi: 10.1016/j.acha.2015.06.007.  Google Scholar

[31]

K. Jaganathan, Y. C. Eldar and B. Hassibi, Phase retrieval with masks using convex optimization, IEEE International Symposium on Information Theory, (2015), 1655–1659. Google Scholar

[32]

F. Kramher and Y.-K. Liu, Phase retrieval without small-ball probability assumptions, IEEE Trans. Inform. Theory, 64 (2018), 485-500.  doi: 10.1109/TIT.2017.2757520.  Google Scholar

[33]

R. KuengH. Rauhut and U. Terstiege, Low rank matrix recovery from rank one measurements, Appl. Comput. Harmon. Anal., 42 (2014), 88-116.  doi: 10.1016/j.acha.2015.07.007.  Google Scholar

[34]

X. Li and V. Voroninski, Sparse signal recovery from quadratic measurements via convex programming, SIAM J. Math. Anal., 45 (2013), 3019-3033.  doi: 10.1137/120893707.  Google Scholar

[35]

R. Millane, Phase retrieval in crystallography and optics, J. Opt. Soc. Amer. A, 7 (1990), 394-411.  doi: 10.1364/JOSAA.7.000394.  Google Scholar

[36]

M. L. MoravecJ. K. Romberg and R. G. Baraniuk, Compressive phase retrieval, Proceedings of SPIE, 6701 (2007), 6701201-67012011.  doi: 10.1117/12.736360.  Google Scholar

[37]

P. NetrapalliP. Jain and S. Sanghavi, Phase retrieval using alternating minimization, IEEE Trans. Signal Process, 63 (2015), 4814-4826.  doi: 10.1109/TSP.2015.2448516.  Google Scholar

[38]

H. Ohlsson, A. Yang, R. Dong and S. Sastry, CPRL–An extension of compressive sensing to the phase retrieval problem, IEEE Neural Information Processing Systems, (2012), 1367–1375. Available from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.244.9102&rep=rep1&type=pdf Google Scholar

[39]

R. Pedarsani, K. Lee and K. Ramchandran, Phasecode: Fast and efficient compressive phase retrieval based on sparse-graph codes, Allerton Conference on Communication, Control and Computing, (2014), 842–849. doi: 10.1109/ALLERTON.2014.7028542.  Google Scholar

[40]

B. RechtM. Fazel and P. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev., 52 (2010), 471-501.  doi: 10.1137/070697835.  Google Scholar

[41]

Y. ShechtmanA. Beck and Y. C. Eldar, GESPAR: Efficient phase retrieval of sparse signals, IEEE Trans. Signal Process, 62 (2014), 928-938.  doi: 10.1109/TSP.2013.2297687.  Google Scholar

[42]

Y. ShechtmanY. C. EldarO. CohenH. N. ChapmanJ. Miao and M. Segev, Phase retrieval with application to optical imaging: A contemporary overview, IEEE Signal Processing Mag., 32 (2015), 87-109.  doi: 10.1109/MSP.2014.2352673.  Google Scholar

[43]

Y. ShechtmanY. C. EldarA. Szameit and M. Segev, Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing, Optics Express, 19 (2011), 14807-14822.  doi: 10.1364/OE.19.014807.  Google Scholar

[44]

I. WaldspurgerA. d'Aspremont and S. Mallat, Phase recovery, maxcut and complex semidefinite programming, Math. Prog., 149 (2015), 47-81.  doi: 10.1007/s10107-013-0738-9.  Google Scholar

[45]

G. WangL. ZhangG. B. GiannakisM. Akcakaya and J. Chen, Sparse phase retrieval via truncated amplitude flow, IEEE Trans. Signal Process, 66 (2018), 479-491.  doi: 10.1109/TSP.2017.2771733.  Google Scholar

[46]

G. ZhengR. Horstmeyer and C. Yang, Wide-field, high-resolution Fourier pty-chographic microscopy, Nature Photonics, 7 (2013), 739-745.  doi: 10.1038/nphoton.2013.187.  Google Scholar

Table 1.  The Number of Masks Needed for Each Case
complex random masks (4) real random masks (3)
$ \boldsymbol{x}_0\in\mathbb{C}^{d}/\mathbb{R}^{d} $, $ d $ is odd $ O(\log^4d) $ $ O(\log^2d) $
$ \boldsymbol{x}_0\in\mathbb{C}^{d} $, $ d $ is even $ O(\log^4d) $ not uniquely recovered
$ \boldsymbol{x}_0\in\mathbb{R}^{d} $, $ d $ is even $ O(\log^4d) $ $ O(\log^2d) $
complex random masks (4) real random masks (3)
$ \boldsymbol{x}_0\in\mathbb{C}^{d}/\mathbb{R}^{d} $, $ d $ is odd $ O(\log^4d) $ $ O(\log^2d) $
$ \boldsymbol{x}_0\in\mathbb{C}^{d} $, $ d $ is even $ O(\log^4d) $ not uniquely recovered
$ \boldsymbol{x}_0\in\mathbb{R}^{d} $, $ d $ is even $ O(\log^4d) $ $ O(\log^2d) $
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