Abstract

Simplicity and few finely tuned parameters are the main advantages of alternating projection (AP) methods, a fundamental class of phase retrieval (PR) methods in the optical imaging field. However, AP methods often suffer from low-quality imaging when few diffraction patterns are recorded. Regularized PR methods avoid this deficiency by using some proper regularization models, but many finely tuned parameters are needed. In this work, we propose a novel unified framework called constrained PR (ConPR), which brings the AP method and the regularization approach together. The proposed ConPR framework not only can recover high-quality images from few diffraction patterns, but also does not need fine-tuning of the parameters. Our proposed generalized constrained PR optimization model consists of a relation function term, a regularization term, and a measurement constraint. The measurement constraint ensures that the recovered image matches with the measurement, and the regularization term can impose some desirable properties on the recovered image. The relation function promotes the approximation of the two underlying variables. The sparsity under the block-matching and 3D filtering frame is incorporated into the proposed ConPR framework. The problem formulation consists of an image updating sub-problem and a constrained optimization sub-problem. The epigraph set of the data fidelity function is defined, and the constrained optimization sub-problem is solved via the epigraph concept. Diffraction imaging from one noisy coded diffraction pattern demonstrates the effectiveness of the proposed algorithm.

© 2018 Optical Society of America

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References

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  1. Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
    [Crossref]
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    [Crossref]
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    [Crossref]
  4. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  5. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [Crossref]
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  7. S. Mukherjee and C. S. Seelamantula, “Fienup algorithm with sparsity constraints: application to frequency-domain optical-coherence tomography,” IEEE Trans. Signal Process. 62, 4659–4672 (2014).
    [Crossref]
  8. S. Qin, X. Hu, and Q. Qin, “Compressed sensing phase retrieval with phase diversity,” Opt. Commun. 310, 193–198 (2014).
    [Crossref]
  9. B. S. Shi, Q. S. Lian, and S. Z. Chen, “Sparse representation utilizing tight frame for phase retrieval,” EURASIP J. Adv. Signal Process. 2015, 1–11 (2015).
    [Crossref]
  10. Q. S. Lian, B. S. Shi, and S. Z. Chen, “Transfer orthogonal sparsifying transform learning for phase retrieval,” Digital Signal Process. 62, 11–25 (2017).
    [Crossref]
  11. E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
    [Crossref]
  12. Y. X. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Commun. Pure Appl. Math. 70, 822–883 (2017).
    [Crossref]
  13. A. M. Tillmann, Y. C. Eldar, and J. M. Mairal, “DOLPHIn: dictionary learning for phase retrieval,” IEEE Trans. Signal Process. 64, 6485–6500 (2016).
    [Crossref]
  14. V. Katkovnik, “Phase retrieval from noisy data based on sparse approximation of object phase and amplitude,” arXiv: 1709.01071 (2017).
  15. H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
    [Crossref]
  16. E. Pauwels, A. Beck, Y. C. Eldar, and S. Sabach, “On Fienup methods for regularized phase retrieval,” arXiv: 1702.08339 (2017).
  17. K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
    [Crossref]
  18. A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
    [Crossref]
  19. E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
    [Crossref]
  20. M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).
    [Crossref]
  21. H. Chang and S. Marchesini, “Denoising Poisson phaseless measurements via orthogonal dictionary learning,” arXiv: 1612.08656 (2016).
  22. M. Tofighi, K. Kose, and A. E. Cetin, “Denoising using projections onto the epigraph set of convex cost functions,” in IEEE International Conference on Image Processing (2015), pp. 2709–2713.
  23. D. L. Donoho and J. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
    [Crossref]
  24. C. A. Metzler, A. Maleki, and R. G. Baraniuk, “BM3D-PRGAMP: compressive phase retrieval based on BM3D denoising,” in IEEE International Conference on Image Processing (ICIP) (IEEE, 2016), pp. 2504–2508.
  25. L. Zhang, L. Zhang, X. Mou, and D. Zhang, “FSIM: a feature similarity index for image quality assessment,” IEEE Trans. Image Process. 20, 2378–2386 (2011).
    [Crossref]

2017 (2)

Y. X. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Commun. Pure Appl. Math. 70, 822–883 (2017).
[Crossref]

Q. S. Lian, B. S. Shi, and S. Z. Chen, “Transfer orthogonal sparsifying transform learning for phase retrieval,” Digital Signal Process. 62, 11–25 (2017).
[Crossref]

2016 (3)

B. Leshem, R. Xu, Y. Dallal, J. Miao, B. Nadler, D. Oron, N. Dudovich, and O. Raz, “Direct single-shot phase retrieval from the diffraction pattern of separated objects,” Nat. Commun. 7, 10820 (2016).
[Crossref]

A. M. Tillmann, Y. C. Eldar, and J. M. Mairal, “DOLPHIn: dictionary learning for phase retrieval,” IEEE Trans. Signal Process. 64, 6485–6500 (2016).
[Crossref]

H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
[Crossref]

2015 (5)

B. S. Shi, Q. S. Lian, and S. Z. Chen, “Sparse representation utilizing tight frame for phase retrieval,” EURASIP J. Adv. Signal Process. 2015, 1–11 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

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

S. Birkholz, G. Steinmeyer, S. Koke, D. Gerth, S. Bürger, and B. Hofmann, “Phase retrieval via regularization in self-diffraction-based spectral interferometry,” J. Opt. Soc. Am. B 32, 983–992 (2015).
[Crossref]

2014 (2)

S. Mukherjee and C. S. Seelamantula, “Fienup algorithm with sparsity constraints: application to frequency-domain optical-coherence tomography,” IEEE Trans. Signal Process. 62, 4659–4672 (2014).
[Crossref]

S. Qin, X. Hu, and Q. Qin, “Compressed sensing phase retrieval with phase diversity,” Opt. Commun. 310, 193–198 (2014).
[Crossref]

2012 (1)

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[Crossref]

2011 (1)

L. Zhang, L. Zhang, X. Mou, and D. Zhang, “FSIM: a feature similarity index for image quality assessment,” IEEE Trans. Image Process. 20, 2378–2386 (2011).
[Crossref]

2007 (2)

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

2006 (1)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[Crossref]

1994 (1)

D. L. Donoho and J. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[Crossref]

1978 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Baraniuk, R. G.

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).
[Crossref]

C. A. Metzler, A. Maleki, and R. G. Baraniuk, “BM3D-PRGAMP: compressive phase retrieval based on BM3D denoising,” in IEEE International Conference on Image Processing (ICIP) (IEEE, 2016), pp. 2504–2508.

Beck, A.

E. Pauwels, A. Beck, Y. C. Eldar, and S. Sabach, “On Fienup methods for regularized phase retrieval,” arXiv: 1702.08339 (2017).

Birkholz, S.

Bürger, S.

Candes, E. J.

Y. X. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Commun. Pure Appl. Math. 70, 822–883 (2017).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

Cetin, A. E.

M. Tofighi, K. Kose, and A. E. Cetin, “Denoising using projections onto the epigraph set of convex cost functions,” in IEEE International Conference on Image Processing (2015), pp. 2709–2713.

Chang, H.

H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
[Crossref]

H. Chang and S. Marchesini, “Denoising Poisson phaseless measurements via orthogonal dictionary learning,” arXiv: 1612.08656 (2016).

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: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Chen, S. Z.

Q. S. Lian, B. S. Shi, and S. Z. Chen, “Transfer orthogonal sparsifying transform learning for phase retrieval,” Digital Signal Process. 62, 11–25 (2017).
[Crossref]

B. S. Shi, Q. S. Lian, and S. Z. Chen, “Sparse representation utilizing tight frame for phase retrieval,” EURASIP J. Adv. Signal Process. 2015, 1–11 (2015).
[Crossref]

Chen, Y. X.

Y. X. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Commun. Pure Appl. Math. 70, 822–883 (2017).
[Crossref]

Cohen, O.

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

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

Dallal, Y.

B. Leshem, R. Xu, Y. Dallal, J. Miao, B. Nadler, D. Oron, N. Dudovich, and O. Raz, “Direct single-shot phase retrieval from the diffraction pattern of separated objects,” Nat. Commun. 7, 10820 (2016).
[Crossref]

Danielyan, A.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[Crossref]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[Crossref]

D. L. Donoho and J. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[Crossref]

Dudovich, N.

B. Leshem, R. Xu, Y. Dallal, J. Miao, B. Nadler, D. Oron, N. Dudovich, and O. Raz, “Direct single-shot phase retrieval from the diffraction pattern of separated objects,” Nat. Commun. 7, 10820 (2016).
[Crossref]

Egiazarian, K.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

Eldar, Y. C.

A. M. Tillmann, Y. C. Eldar, and J. M. Mairal, “DOLPHIn: dictionary learning for phase retrieval,” IEEE Trans. Signal Process. 64, 6485–6500 (2016).
[Crossref]

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

E. Pauwels, A. Beck, Y. C. Eldar, and S. Sabach, “On Fienup methods for regularized phase retrieval,” arXiv: 1702.08339 (2017).

Fienup, J. R.

Foi, A.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gerth, D.

Hofmann, B.

Hu, X.

S. Qin, X. Hu, and Q. Qin, “Compressed sensing phase retrieval with phase diversity,” Opt. Commun. 310, 193–198 (2014).
[Crossref]

Johnstone, J. M.

D. L. Donoho and J. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[Crossref]

Katkovnik, V.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

V. Katkovnik, “Phase retrieval from noisy data based on sparse approximation of object phase and amplitude,” arXiv: 1709.01071 (2017).

Koke, S.

Kose, K.

M. Tofighi, K. Kose, and A. E. Cetin, “Denoising using projections onto the epigraph set of convex cost functions,” in IEEE International Conference on Image Processing (2015), pp. 2709–2713.

Leshem, B.

B. Leshem, R. Xu, Y. Dallal, J. Miao, B. Nadler, D. Oron, N. Dudovich, and O. Raz, “Direct single-shot phase retrieval from the diffraction pattern of separated objects,” Nat. Commun. 7, 10820 (2016).
[Crossref]

Li, X.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

Lian, Q. S.

Q. S. Lian, B. S. Shi, and S. Z. Chen, “Transfer orthogonal sparsifying transform learning for phase retrieval,” Digital Signal Process. 62, 11–25 (2017).
[Crossref]

B. S. Shi, Q. S. Lian, and S. Z. Chen, “Sparse representation utilizing tight frame for phase retrieval,” EURASIP J. Adv. Signal Process. 2015, 1–11 (2015).
[Crossref]

Lou, Y.

H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
[Crossref]

Mairal, J. M.

A. M. Tillmann, Y. C. Eldar, and J. M. Mairal, “DOLPHIn: dictionary learning for phase retrieval,” IEEE Trans. Signal Process. 64, 6485–6500 (2016).
[Crossref]

Maleki, A.

C. A. Metzler, A. Maleki, and R. G. Baraniuk, “BM3D-PRGAMP: compressive phase retrieval based on BM3D denoising,” in IEEE International Conference on Image Processing (ICIP) (IEEE, 2016), pp. 2504–2508.

Marchesini, S.

H. Chang and S. Marchesini, “Denoising Poisson phaseless measurements via orthogonal dictionary learning,” arXiv: 1612.08656 (2016).

Metzler, C. A.

C. A. Metzler, A. Maleki, and R. G. Baraniuk, “BM3D-PRGAMP: compressive phase retrieval based on BM3D denoising,” in IEEE International Conference on Image Processing (ICIP) (IEEE, 2016), pp. 2504–2508.

Miao, J.

B. Leshem, R. Xu, Y. Dallal, J. Miao, B. Nadler, D. Oron, N. Dudovich, and O. Raz, “Direct single-shot phase retrieval from the diffraction pattern of separated objects,” Nat. Commun. 7, 10820 (2016).
[Crossref]

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

Moravec, M. L.

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).
[Crossref]

Mou, X.

L. Zhang, L. Zhang, X. Mou, and D. Zhang, “FSIM: a feature similarity index for image quality assessment,” IEEE Trans. Image Process. 20, 2378–2386 (2011).
[Crossref]

Mukherjee, S.

S. Mukherjee and C. S. Seelamantula, “Fienup algorithm with sparsity constraints: application to frequency-domain optical-coherence tomography,” IEEE Trans. Signal Process. 62, 4659–4672 (2014).
[Crossref]

Nadler, B.

B. Leshem, R. Xu, Y. Dallal, J. Miao, B. Nadler, D. Oron, N. Dudovich, and O. Raz, “Direct single-shot phase retrieval from the diffraction pattern of separated objects,” Nat. Commun. 7, 10820 (2016).
[Crossref]

Ng, M. K.

H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
[Crossref]

Oron, D.

B. Leshem, R. Xu, Y. Dallal, J. Miao, B. Nadler, D. Oron, N. Dudovich, and O. Raz, “Direct single-shot phase retrieval from the diffraction pattern of separated objects,” Nat. Commun. 7, 10820 (2016).
[Crossref]

Pauwels, E.

E. Pauwels, A. Beck, Y. C. Eldar, and S. Sabach, “On Fienup methods for regularized phase retrieval,” arXiv: 1702.08339 (2017).

Qin, Q.

S. Qin, X. Hu, and Q. Qin, “Compressed sensing phase retrieval with phase diversity,” Opt. Commun. 310, 193–198 (2014).
[Crossref]

Qin, S.

S. Qin, X. Hu, and Q. Qin, “Compressed sensing phase retrieval with phase diversity,” Opt. Commun. 310, 193–198 (2014).
[Crossref]

Raz, O.

B. Leshem, R. Xu, Y. Dallal, J. Miao, B. Nadler, D. Oron, N. Dudovich, and O. Raz, “Direct single-shot phase retrieval from the diffraction pattern of separated objects,” Nat. Commun. 7, 10820 (2016).
[Crossref]

Romberg, J. K.

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).
[Crossref]

Sabach, S.

E. Pauwels, A. Beck, Y. C. Eldar, and S. Sabach, “On Fienup methods for regularized phase retrieval,” arXiv: 1702.08339 (2017).

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Seelamantula, C. S.

S. Mukherjee and C. S. Seelamantula, “Fienup algorithm with sparsity constraints: application to frequency-domain optical-coherence tomography,” IEEE Trans. Signal Process. 62, 4659–4672 (2014).
[Crossref]

Segev, M.

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

Shi, B. S.

Q. S. Lian, B. S. Shi, and S. Z. Chen, “Transfer orthogonal sparsifying transform learning for phase retrieval,” Digital Signal Process. 62, 11–25 (2017).
[Crossref]

B. S. Shi, Q. S. Lian, and S. Z. Chen, “Sparse representation utilizing tight frame for phase retrieval,” EURASIP J. Adv. Signal Process. 2015, 1–11 (2015).
[Crossref]

Soltanolkotabi, M.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

Steinmeyer, G.

Tillmann, A. M.

A. M. Tillmann, Y. C. Eldar, and J. M. Mairal, “DOLPHIn: dictionary learning for phase retrieval,” IEEE Trans. Signal Process. 64, 6485–6500 (2016).
[Crossref]

Tofighi, M.

M. Tofighi, K. Kose, and A. E. Cetin, “Denoising using projections onto the epigraph set of convex cost functions,” in IEEE International Conference on Image Processing (2015), pp. 2709–2713.

Xu, R.

B. Leshem, R. Xu, Y. Dallal, J. Miao, B. Nadler, D. Oron, N. Dudovich, and O. Raz, “Direct single-shot phase retrieval from the diffraction pattern of separated objects,” Nat. Commun. 7, 10820 (2016).
[Crossref]

Zeng, T.

H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
[Crossref]

Zhang, D.

L. Zhang, L. Zhang, X. Mou, and D. Zhang, “FSIM: a feature similarity index for image quality assessment,” IEEE Trans. Image Process. 20, 2378–2386 (2011).
[Crossref]

Zhang, L.

L. Zhang, L. Zhang, X. Mou, and D. Zhang, “FSIM: a feature similarity index for image quality assessment,” IEEE Trans. Image Process. 20, 2378–2386 (2011).
[Crossref]

L. Zhang, L. Zhang, X. Mou, and D. Zhang, “FSIM: a feature similarity index for image quality assessment,” IEEE Trans. Image Process. 20, 2378–2386 (2011).
[Crossref]

Appl. Comput. Harmon. Anal. (1)

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

Biometrika (1)

D. L. Donoho and J. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[Crossref]

Commun. Pure Appl. Math. (1)

Y. X. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Commun. Pure Appl. Math. 70, 822–883 (2017).
[Crossref]

Digital Signal Process. (1)

Q. S. Lian, B. S. Shi, and S. Z. Chen, “Transfer orthogonal sparsifying transform learning for phase retrieval,” Digital Signal Process. 62, 11–25 (2017).
[Crossref]

EURASIP J. Adv. Signal Process. (1)

B. S. Shi, Q. S. Lian, and S. Z. Chen, “Sparse representation utilizing tight frame for phase retrieval,” EURASIP J. Adv. Signal Process. 2015, 1–11 (2015).
[Crossref]

IEEE Signal Process. Mag. (1)

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

IEEE Trans. Image Process. (3)

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Figures (6)

Fig. 1.
Fig. 1. Original images. Top row (left to right): Lena, Barbara, hill, boat; bottom row (left to right): couple, fingerprint, acinar cell, chromaffin cell.
Fig. 2.
Fig. 2. Comparison with the previous algorithms in the case of one diffraction pattern. Top: the plot of PSNR versus SNR; bottom: the plot of FSIM versus SNR. In each case, results are the average PSNR or FSIM values of the eight testing images.
Fig. 3.
Fig. 3. Lena images recovered by the five PR algorithms at SNR=20  dB. For comparing clearly, the parts of the images are presented. From left to right and top to bottom: the original image, the image recovered by WF (PSNR=10.93  dB, FSIM=0.8967), DOLPHIn (PSN R=21.36  dB, FSIM=0.9819), BM3D-prGAMP (PSNR=31.94  dB, FSIM=0.9972), SPAR (PSNR=31.82  dB, FSIM=0.9966), and ConPR (PSNR=33.51  dB, FSIM=0.9985).
Fig. 4.
Fig. 4. Hill images recovered by the five PR algorithms at SNR=25  dB. From left to right and top to bottom: the original image, the image recovered by WF (PSNR=10.94  dB, FSIM=0.9129), DOLPHIn (PSNR=24.04  dB, FSIM=0.9921), BM3D-prGAMP (PSNR=31.97  dB, FSIM=0.9984), SPAR (PSNR=31.89  dB, FSIM=0.9980), and ConPR (PSNR=33.73  dB, FSIM=0.9992).
Fig. 5.
Fig. 5. Acinar cell images recovered by the five PR algorithms at SNR=25  dB. From left to right and top to bottom: the original image, the image recovered by WF (PSNR=13.16  dB, FSIM=0.9891), DOLPHIn (PSNR=17.19  dB, FSIM=0.9960), BM3D-prGAMP (PSNR=24.39  dB, FSIM=0.9994), SPAR (PSNR=25.80  dB, FSIM=0.9995), and ConPR (PSNR=27.36  dB, FSIM=0.9997).
Fig. 6.
Fig. 6. Convergence curves of the proposed ConPR algorithm for recovering the testing images at (a) SNR=15  dB and (b) SNR=20  dB.

Tables (2)

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Algorithm 1 Constrained Phase Retrieval (ConPR) Algorithm

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Table 1. Comparison of the Imaging Time (s) Consumed by the PR Algorithmsa

Equations (24)

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y=G[ϕ(x)],
find    xMS,
minxIM(x)+IS(x).
minx,zIM(z)+IS(x)+xz22,
minxyϕ(x)22+λR(x),
minx,zIM(z)+xz22+λR(x).
minx,zxz22+λR(x)s.t.  f(z)ε.
x(t)=argminx{xz(t1)22+λR(x)},
z(t)=argminz{x(t)z22}s.t.  f(z)ε.
α=ΨBM3Dx.
x^=ϕBM3Dα^,
minx,zxz22+λΨBM3Dx0s.t.  yϕ(z)22ε.
x(t)=argminx{xz(t1)22+λΨBM3Dx0},
z(t)=argminz{x(t)z22}s.t.  yϕ(z)22ε.
x(t)=ϕBM3DTτ(ΨBM3Dz(t1)).
Cerror={z_=[zTq]T  :  qf(z)}.
z_(t)=proj(u_(t))=argminz_Cerror{u_(t)z_22},
Ax=F(Ix),AHc=1MI¯(FHc).
[vjTf(vj)]T,vj=vj1f(vj1)f(vj1)22+1f(vj1).
f(vj1)=real{AH[Avj1(|Avj1|2y)]},
proj(u_(t))=[vJTf(vJ)]T,
y=|F(Ix)|2+n.
SNR=10log10[y¯22/y¯y22]dB,
PSNR=20log10[512/wx2]dB,

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