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The numerics of phase retrieval

Published online by Cambridge University Press:  30 November 2020

Albert Fannjiang  and
Thomas Strohmer
Albert Fannjiang
Affiliation:
Department of Mathematics, University of California Davis, Davis, CA95616, USA E-mail: [email protected]
Thomas Strohmer
Affiliation:
Department of Mathematics, University of California Davis, Davis, CA95616, USA E-mail: [email protected] Center for Data Science and Artificial Intelligence Research, University of California Davis, Davis, CA95616, USA E-mail: [email protected]
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Abstract

Phase retrieval, i.e. the problem of recovering a function from the squared magnitude of its Fourier transform, arises in many applications, such as X-ray crystallography, diffraction imaging, optics, quantum mechanics and astronomy. This problem has confounded engineers, physicists, and mathematicians for many decades. Recently, phase retrieval has seen a resurgence in research activity, ignited by new imaging modalities and novel mathematical concepts. As our scientific experiments produce larger and larger datasets and we aim for faster and faster throughput, it is becoming increasingly important to study the involved numerical algorithms in a systematic and principled manner. Indeed, the past decade has witnessed a surge in the systematic study of computational algorithms for phase retrieval. In this paper we will review these recent advances from a numerical viewpoint.

Type
Research Article
Information
Acta Numerica , Volume 29 , May 2020 , pp. 125 - 228
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES 8

Ahmed, A., Recht, B. and Romberg, J. (2013), ‘Blind deconvolution using convex programming’, IEEE Trans. Inform. Theory 60, 1711–1732.CrossRefGoogle Scholar
Appleby, D. M. (2005), ‘Symmetric informationally complete–positive operator valued measures and the extended Clifford group’, J. Math. Phys. 46, 052107.CrossRefGoogle Scholar
Appleby, M., Bengtsson, I., Flammia, S. and Goyeneche, D. (2019), ‘Tight frames, Hadamard matrices and Zauner’s conjecture’, J. Phys. A 52, 295301.CrossRefGoogle Scholar
Arridge, S., Maass, P., Öktem, O. and Schönlieb, C.-B. (2019), Solving inverse problems using data-driven models. In Acta Numerica, Vol. 28, Cambridge University Press, pp. 1–174.Google Scholar
Bahmani, S. and Romberg, J. (2016), ‘Phase retrieval meets statistical learning theory: A flexible convex relaxation’, Electron. J. Statist. 11, 5254–5281.CrossRefGoogle Scholar
Balan, R. (2010), On signal reconstruction from its spectrogram. In 2010 44th Annual Conference on Information Sciences and Systems (CISS), IEEE, pp. 1–4.CrossRefGoogle Scholar
Balan, R., Bodmann, B., Casazza, P. and Edidin, D. (2009), ‘Painless reconstruction from magnitudes of frame coefficients’, J. Fourier Anal. Appl. 15, 488–501.CrossRefGoogle Scholar
Balan, R., Casazza, P. and Edidin, D. (2006), ‘On signal reconstruction without phase’, Appl. Comput. Harmon. Anal. 20, 345–356.CrossRefGoogle Scholar
Balan, R., Casazza, P. and Edidin, D. (2007), ‘Equivalence of reconstruction from the absolute value of the frame coefficients to a sparse representation problem’, IEEE Signal. Process. Lett. 14, 341–343.CrossRefGoogle Scholar
Bandeira, A. S., Cahill, J., Mixon, D. G. and Nelson, A. A. (2014), ‘Saving phase: Injectivity and stability for phase retrieval’, Appl. Comput. Harmon. Anal. 37, 106–125.CrossRefGoogle Scholar
Barmherzig, D. A., Sun, J., Candès, E. J., Lane, T. and Li, P.-N. (2019 a), Dual-reference design for holographic coherent diffraction imaging. arXiv:1902.02492 Google Scholar
Barmherzig, D. A., Sun, J., Lane, T., Li, P.-N. and Candès, E. J. (2019 b), Holographic phase retrieval and reference design. arXiv:1901.06453 CrossRefGoogle Scholar
Bauschke, H. H., Combettes, P. L. and Luke, D. R. (2004), ‘Finding best approximation pairs relative to two closed convex sets in Hilbert spaces’, J. Approx. Theory 127, 178–192.CrossRefGoogle Scholar
Beck, C. and D’Andrea, R. (1998), Computational study and comparisons of LFT reducibility methods. In 1998 American Control Conference (ACC), IEEE, pp. 1013–1017.CrossRefGoogle Scholar
Becker, S. R., Candès, E. J. and Grant, M. C. (2011), ‘Templates for convex cone problems with applications to sparse signal recovery’, Math. Program. Comput. 3, 165.CrossRefGoogle Scholar
Beinert, R. and Plonka, G. (2017), ‘Sparse phase retrieval of one-dimensional signals by Prony’s method’, Front. Appl. Math. Statist. 3, 5.CrossRefGoogle Scholar
Bendory, T., Beinert, R. and Eldar, Y. C. (2017), Fourier phase retrieval: Uniqueness and algorithms. In Compressed Sensing and its Applications (Boche, H. et al., eds), Applied and Numerical Harmonic Analysis, Springer, pp. 55–91.CrossRefGoogle Scholar
Bian, L., Suo, J., Chung, J., Ou, X., Yang, C., Chen, F. and Dai, Q. (2016), ‘Fourier ptychographic reconstruction using Poisson maximum likelihood and truncated Wirtinger gradient’, Sci. Reports 6, 27384.Google Scholar
Bianchi, G., Segala, F. and Volčič, A. (2002), ‘The solution of the covariogram problem for plane ${\mathbf{\mathcal{C}}}_{+}^2$ convex bodies’, J. Diff. Geom . 60, 177–198.CrossRefGoogle Scholar
Bogan, M. et al. (2008), ‘Single particle X-ray diffractive imaging’, Nano Lett. 8, 310–316.CrossRefGoogle Scholar
Bruck, Y. and Sodin, L. (1979), ‘On the ambiguity of the image reconstruction problem’, Optics Commun. 30, 304–308.CrossRefGoogle Scholar
Cai, T. T., Li, X. and Ma, Z. (2016), ‘Optimal rates of convergence for noisy sparse phase retrieval via thresholded Wirtinger flow’, Ann. Statist. 44, 2221–2251.CrossRefGoogle Scholar
Candès, E. J. and Li, X. (2014), ‘Solving quadratic equations via PhaseLift when there are about as many equations as unknowns’, Found. Comput. Math. 14, 1017–1026.CrossRefGoogle Scholar
Candès, E. J. and Tao, T. (2006), ‘Near-optimal signal recovery from random projections: Universal encoding strategies’, IEEE Trans. Inform. Theory 52, 5406–5425.CrossRefGoogle Scholar
Candès, E. J., Eldar, Y. C., Strohmer, T. and Voroninski, V. (2013 a), ‘Phase retrieval via matrix completion’, SIAM J. Imaging Sci. 6, 199–225.Google Scholar
Candès, E. J., Li, X. and Soltanolkotabi, M. (2015), ‘Phase retrieval from coded diffraction patterns’, Appl. Comput. Harmon. Anal. 39, 277–299.CrossRefGoogle Scholar
Candès, E. J., Strohmer, T. and Voroninski, V. (2013 b), ‘PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming’, Commun. Pure Appl. Math. 66, 1241–1274.CrossRefGoogle Scholar
Chandra, R., Zhong, Z., Hontz, J., McCulloch, V., Studer, C. and Goldstein, T. (2017), PhasePack: A phase retrieval library. In 2017 51st Asilomar Conference on Signals, Systems, and Computers, pp. 1617–1621.CrossRefGoogle Scholar
Chang, H., Enfedaque, P. and Marchesini, S. (2019), ‘Blind ptychographic phase retrieval via convergent alternating direction method of multipliers’, SIAM J. Imaging Sci. 12, 153–185.CrossRefGoogle Scholar
Chapman, H. N., Hau-Riege, S. P., Bogan, M. J., Bajt, S., Barty, A., Boutet, S., Marchesini, S., Frank, M., Woods, B. W., Benner, W. H. et al. (2007), ‘Femtosecond time-delay X-ray holography’, Nature 448 (7154), 676–679.CrossRefGoogle Scholar
Chapman, H. N., Fromme, P., Barty, A., White, A. T., Kirian, R. A., Aquila, A., Hunter, M. S., Schulz, J., DePonte, D. P., Weierstall, U. et al. (2011), ‘Femtosecond X-ray protein nanocrystallography’, Nature 470 (7332), 73–77.CrossRefGoogle Scholar
Chen, C., Miao, J., Wang, C. and Lee, T. (2007), ‘Application of the optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input–output method (GHIO)’, Phys. Rev. B 76, 064113.CrossRefGoogle Scholar
Chen, P. and Fannjiang, A. (2018 a), ‘Coded aperture ptychography: Uniqueness and reconstruction’, Inverse Problems 34, 025003.CrossRefGoogle Scholar
Chen, P. and Fannjiang, A. (2018 b), ‘Fourier phase retrieval with a single mask by Douglas–Rachford algorithms’, Appl. Comput. Harmon. Anal. 44, 665–699.CrossRefGoogle Scholar
Chen, P., Fannjiang, A. and Liu, G.-R. (2017), ‘Phase retrieval by linear algebra’, SIAM J. Matrix Anal. Appl. 38, 854–868.CrossRefGoogle Scholar
Chen, P., Fannjiang, A. and Liu, G.-R. (2018), ‘Phase retrieval with one or two diffraction patterns by alternating projections with the null initialization’, J. Fourier Anal. Appl. 24, 719–758.CrossRefGoogle Scholar
Chen, Y. and Candès, E. J. (2017), ‘Solving random quadratic systems of equations is nearly as easy as solving linear systems’, Commun. Pure Appl. Math. 70, 822–883.Google Scholar
Chen, Y., Chi, Y., Fan, J. and Ma, C. (2019), ‘Gradient descent with random initialization: Fast global convergence for nonconvex phase retrieval’, Math. Program. 176, 5–37.CrossRefGoogle Scholar
Cimmino, G. (1938), ‘Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari’, La Ricerca Scientifica (Roma) 1, 326–333.Google Scholar
Conca, A., Edidin, D., Hering, M. and Vinzant, C. (2015), ‘An algebraic characterization of injectivity in phase retrieval’, Appl. Comput. Harmon. Anal. 38, 346–356.CrossRefGoogle Scholar
Corbett, J. (2006), ‘The Pauli problem, state reconstruction and quantum-real numbers’, Rep. Math. Phys. 57, 53–68.CrossRefGoogle Scholar
Dainty, J. C. and Fienup, J. R. (1987), Phase retrieval and image reconstruction for astronomy. In Image Recovery: Theory and Application (Stark, H., ed.), Academic Press, pp. 231–275.Google Scholar
Davenport, M. A. and Romberg, J. (2016), ‘An overview of low-rank matrix recovery from incomplete observations’, IEEE J. Selected Topics Signal Process. 10, 608–622.Google Scholar
Demanet, L. and Hand, P. (2014), ‘Stable optimizationless recovery from phaseless linear measurements’, J. Fourier Anal. Appl. 20, 199–221.CrossRefGoogle Scholar
Dhifallah, O., Thrampoulidis, C. and Lu, Y. M. (2017), Phase retrieval via linear programming: Fundamental limits and algorithmic improvements. In 2017 55th Annual Allerton Conference on Communication , Control, and Computing, IEEE, pp. 1071–1077.Google Scholar
Dierolf, M., Menzel, A., Thibault, P., Schneider, P., Kewish, C. M., Wepf, R., Bunk, O. and Pfeiffer, F. (2010), ‘Ptychographic X-ray computed tomography at the nanoscale’, Nature 467 (7314), 436–439.CrossRefGoogle Scholar
Doelman, R., Thao, N. H. and Verhaegen, M. (2018), ‘Solving large-scale general phase retrieval problems via a sequence of convex relaxations’, J. Optical Soc. Amer. A 35, 1410–1419.CrossRefGoogle Scholar
Donoho, D. L. (2006), ‘Compressed sensing’, IEEE Trans. Inform. Theory 52, 1289–1306.CrossRefGoogle Scholar
Donoho, D. L., Maleki, A. and Montanari, A. (2010), Message passing algorithms for compressed sensing, I: Motivation and construction. In 2010 IEEE Information Theory Workshop on Information Theory (ITW 2010), IEEE, pp. 1–5.CrossRefGoogle Scholar
Drémeau, A. and Krzakala, F. (2015), Phase recovery from a Bayesian point of view: The variational approach. In 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, pp. 3661–3665.CrossRefGoogle Scholar
Duadi, H., Margalit, O., Mico, V., Rodrigo, J. A., Alieva, T., Garcia, J. and Zalevsky, Z. (2011), Digital holography and phase retrieval. In Holography, Research and Technologies (J. Rosen, ed.), InTech, pp. 407–420.CrossRefGoogle Scholar
Eldar, Y. C., Sidorenko, P., Mixon, D. G., Barel, S. and Cohen, O. (2014), ‘Sparse phase retrieval from short-time Fourier measurements’, IEEE Signal Process. Lett. 22, 638–642.CrossRefGoogle Scholar
Elser, V., Lan, T.-Y. and Bendory, T. (2018), ‘Benchmark problems for phase retrieval’, SIAM J. Imaging Sci. 11, 2429–2455.CrossRefGoogle Scholar
Fannjiang, A. (2012), ‘Absolute uniqueness of phase retrieval with random illumination’, Inverse Problems 28, 075008.CrossRefGoogle Scholar
Fannjiang, A. (2019), ‘Raster grid pathology and the cure’, Multiscale Model. Simul. 17, 973–995.CrossRefGoogle Scholar
Fannjiang, A. and Chen, P. (2020), ‘Blind ptychography: Uniqueness and ambiguities’, Inverse Problems 36, 045005.CrossRefGoogle Scholar
Fannjiang, A. and Zhang, Z. (2020), ‘Fixed point analysis of Douglas–Rachford splitting for ptychography and phase retrieval’, SIAM J. Imaging Sci. 13, 609–650.CrossRefGoogle Scholar
Faridian, A., Hopp, D., Pedrini, G., Eigenthaler, U., Hirscher, M. and Osten, W. (2010), ‘Nanoscale imaging using deep ultraviolet digital holographic microscopy’, Optics Express 18, 14159–14164.CrossRefGoogle Scholar
Faulkner, H. M. L. and Rodenburg, J. M. (2004), ‘Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm’, Phys. Rev. Lett. 93, 023903.Google Scholar
Faulkner, H. M. L. and Rodenburg, J. M. (2005), ‘Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy’, Ultramicroscopy 103, 153–164.CrossRefGoogle Scholar
Fienup, J. R. (1978), ‘Reconstruction of an object from the modulus of its Fourier transform’, Optics Lett. 3, 27–29.CrossRefGoogle Scholar
Fienup, J. R. (1982), ‘Phase retrieval algorithms: A comparison’, Appl. Optics 21, 2758–2768.CrossRefGoogle Scholar
Fienup, J. R. and Wackerman, C. C. (1986), ‘Phase-retrieval stagnation problems and solutions’, J. Optical Soc. Amer. A 3, 1897–1907.CrossRefGoogle Scholar
Fortin, M. and Glowinski, R. (2000), Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, Elsevier.Google Scholar
Foucart, S. and Rauhut, H. (2013), A Mathematical Introduction to Compressive Sensing, Springer.Google Scholar
Fuchs, C. A., Hoang, M. C. and Stacey, B. C. (2017), ‘The SIC question: History and state of play’, Axioms 6, 21.CrossRefGoogle Scholar
Gabor, D. (1947), Improvements in and relating to microscopy. Patent GB685286.Google Scholar
Gabor, D. (1948), ‘A new microscopic principle’, Nature 161, 777–778.CrossRefGoogle Scholar
Gabor, D., Stroke, G., Brumm, D., Funkhouser, A. and Labeyrie, A. (1965), ‘Reconstruction of phase objects by holography’, Nature 208 (5016), 1159–1162.CrossRefGoogle Scholar
Gerchberg, R. and Saxton, W. (1972), ‘A practical algorithm for the determination of phase from image and diffraction plane pictures’, Optik 35, 237–246.Google Scholar
Ghigo, E., Kartenbeck, J., Lien, P., Pelkmans, L., Capo, C., Mege, J.-L. and Raoult, D. (2008), ‘Ameobal pathogen mimivirus infects macrophages through phagocytosis’, PLoS Pathogens 4, e1000087.CrossRefGoogle Scholar
Giselsson, P. and Boyd, S. (2016), ‘Linear convergence and metric selection for Douglas–Rachford splitting and ADMM’, IEEE Trans. Automat. Control 62, 532–544.CrossRefGoogle Scholar
Gladrow, J. (2019), Digital phase-only holography using deep conditional generative models. arXiv:1911.00904 Google Scholar
Glusker, J. P. (1984), ‘The Patterson function’, Trends Biochem. Sci. 9, 328–330.CrossRefGoogle Scholar
Godard, P., Allain, M., Chamard, V. and Rodenburg, J. (2012), ‘Noise models for low counting rate coherent diffraction imaging’, Optics Express 20, 25914–25934.CrossRefGoogle Scholar
Goldstein, T. and Studer, C. (2018), ‘PhaseMax: Convex phase retrieval via basis pursuit’, IEEE Trans. Inform. Theory 64, 2675–2689.CrossRefGoogle Scholar
Goodman, J. W. (2005), Introduction to Fourier Optics, Roberts & Company.Google Scholar
Gray, R. M. (2006), ‘Toeplitz and circulant matrices: A review’, Found . Trends Commun. Inform. Theory 2, 155–239.CrossRefGoogle Scholar
Gröchenig, K. (2001), Foundations of Time-Frequency Analysis, Birkhäuser.CrossRefGoogle Scholar
Grohs, P., Koppensteiner, S. and Rathmair, M. (2020), ‘Phase retrieval: Uniqueness and stability’, SIAM Rev. 62, 301–350.CrossRefGoogle Scholar
Gross, D. (2011), ‘Recovering low-rank matrices from few coefficients in any basis’, IEEE Trans. Inform. Theory 57, 1548–1566.CrossRefGoogle Scholar
Gross, D., Krahmer, F. and Kueng, R. (2015), ‘A partial derandomization of PhaseLift using spherical designs’, J. Fourier Anal. Appl. 21, 229–266.CrossRefGoogle Scholar
Gross, D., Krahmer, F. and Kueng, R. (2017), ‘Improved recovery guarantees for phase retrieval from coded diffraction patterns’, Appl. Comput. Harmon. Anal. 42, 37–64.Google Scholar
Guizar-Sicairos, M. and Fienup, J. R. (2007), ‘Holography with extended reference by autocorrelation linear differential operation’, Optics Express 15, 17592–17612.CrossRefGoogle Scholar
Haah, J., Harrow, A. W., Ji, Z., Wu, X. and Yu, N. (2017), ‘Sample-optimal tomography of quantum states’, IEEE Trans. Inform. Theory 63, 5628–5641.Google Scholar
Hand, P. (2017), ‘PhaseLift is robust to a constant fraction of arbitrary errors’, Appl. Comput. Harmon. Anal. 42, 550–562.CrossRefGoogle Scholar
Hand, P. and Voroninski, V. (2016), An elementary proof of convex phase retrieval in the natural parameter space via the linear program PhaseMax. arXiv:1611.03935 Google Scholar
Hand, P., Leong, O. and Voroninski, V. (2018), Phase retrieval under a generative prior. In Advances in Neural Information Processing Systems 31, Curran Associates, pp. 9136–9146.Google Scholar
Harrison, R. (1993), ‘Phase problem in crystallography’, J. Optical Soc. Amer. A 10, 1045–1055.CrossRefGoogle Scholar
Hauptman, H. A. (1997), Shake-and-bake: An algorithm for automatic solution ab initio of crystal structures. Methods Enzymol. 277, 3–13.CrossRefGoogle ScholarPubMed
Hayes, M. (1982), ‘The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform’, IEEE Trans. Acoust. Speech Signal Process. 30, 140–154.CrossRefGoogle Scholar
Heinosaari, T., Mazzarella, L. and Wolf, M. M. (2013), ‘Quantum tomography under prior information’, Commun. Math. Phys. 318, 355–374.CrossRefGoogle Scholar
Hoppe, W. (1969), ‘Beugung im inhomogenen Primärstrahlwellenfeld, I: Prinzip einer Phasenmessung von Elektronenbeugungsinterferenzen’, Acta Cryst. A 25, 495–501.CrossRefGoogle Scholar
Horisaki, R., Egami, R. and Tanida, J. (2016), ‘Single-shot phase imaging with randomized light (spiral)’, Optics Express 24, 3765–3773.CrossRefGoogle Scholar
Horstmeyer, R., Chen, R. Y., Ou, X., Ames, B., Tropp, J. A. and Yang, C. (2015), ‘Solving ptychography with a convex relaxation’, New J. Phys. 17, 053044.CrossRefGoogle Scholar
Horstmeyer, R., Chung, J., Ou, X., Zheng, G. and Yang, C. (2016), ‘Diffraction tomography with Fourier ptychography’, Optica 3, 827–835.CrossRefGoogle Scholar
Huang, W., Gallivan, K. A. and Zhang, X. (2017), ‘Solving PhaseLift by low-rank Riemannian optimization methods for complex semidefinite constraints’, SIAM J. Sci. Comput. 39, B840–B859.CrossRefGoogle Scholar
Hurt, N. (1989), Phase Retrieval and Zero Crossings, Kluwer.CrossRefGoogle Scholar
Iwen, M., Preskitt, B., Saab, R. and Viswanathan, A. (2016), Phase retrieval from local measurements: Improved robustness via eigenvector-based angular synchronization. arXiv:1612.01182 Google Scholar
Iwen, M., Viswanathan, A. and Wang, Y. (2017), ‘Robust sparse phase retrieval made easy’, Appl. Comput. Harmon. Anal. 42, 135–142.CrossRefGoogle Scholar
Jaganathan, K., Eldar, Y. and Hassibi, B. (2015), Phase retrieval with masks using convex optimization. In 2015 IEEE International Symposium on Information Theory (ISIT), IEEE, pp. 1655–1659.CrossRefGoogle Scholar
Jaganathan, K., Oymak, S. and Hassibi, B. (2017), ‘Sparse phase retrieval: Uniqueness guarantees and recovery algorithms’, IEEE Trans. Signal Process. 65, 2402–2410.CrossRefGoogle Scholar
Jeong, H. and Güntürk, C. S. (2017), Convergence of the randomized Kaczmarz method for phase retrieval. arXiv:1706.10291 Google Scholar
Johnson, I., Jefimovs, K., Bunk, O., David, C., Dierolf, M., Gray, J., Renker, D. and Pfeiffer, F. (2008), ‘Coherent diffractive imaging using phase front modifications’, Phys. Rev. Lett. 100, 155503.CrossRefGoogle Scholar
Johnston, S. F. (2005), ‘From white elephant to Nobel Prize: Dennis Gabor’s wavefront reconstruction’, Hist. Stud. Phys. Biol. Sci. 36, 35–70.CrossRefGoogle Scholar
Jung, P., Krahmer, F. and Stöger, D. (2017), ‘Blind demixing and deconvolution at near-optimal rate’, IEEE Trans. Inform. Theory 64, 704–727.CrossRefGoogle Scholar
Kaczmarz, S. (1937), ‘Angenäherte Auflösung von Systemen linearer Gleichungen’, Bull. Internat. Acad. Pol. Sci. Lett. Ser. A 35, 355–357.Google Scholar
Kikuta, S., Aoki, S., Kosaki, S. and Kohra, K. (1972), ‘X-ray holography of lensless Fourier-transform type’, Optics Commun. 5, 86–89.CrossRefGoogle Scholar
Kim, K.-S. and Chung, S.-Y. (2019), ‘Fourier phase retrieval with extended support estimation via deep neural network’, IEEE Signal Process. Lett. 26, 1506–1510.CrossRefGoogle Scholar
Klibanov, M., Sacks, P. and Tikhonravov, A. (1995), ‘The phase retrieval problem’, Inverse Problems 11, 1–28.CrossRefGoogle Scholar
Konijnenberg, A., Coene, W. and Urbach, H. (2018), ‘Model-independent noise-robust extension of ptychography’, Optics Express 26, 5857–5874.CrossRefGoogle Scholar
Krahmer, F. and Stöger, D. (2019), Complex phase retrieval from subgaussian measurements. arXiv:1906.08385 Google Scholar
Kueng, R., Rauhut, H. and Terstiege, U. (2017), ‘Low rank matrix recovery from rank one measurements’, Appl. Comput. Harmon. Anal. 42, 88–116.CrossRefGoogle Scholar
Kueng, R., Zhu, H. and Gross, D. (2016), Low rank matrix recovery from Clifford orbits. arXiv:1610.08070 Google Scholar
Kumar, S. and Deen, M. J. (2014), Fiber Optic Communications: Fundamentals and Applications, Wiley.CrossRefGoogle Scholar
Latychevskaia, T. (2019), ‘Iterative phase retrieval for digital holography: Tutorial’, J. Optical Soc. Amer. A 36, D31–D40.CrossRefGoogle Scholar
Latychevskaia, T. and Fink, H.-W. (2015), ‘Practical algorithms for simulation and reconstruction of digital in-line holograms’, Appl. Optics 54, 2424–2434.CrossRefGoogle Scholar
Latychevskaia, T., Longchamp, J.-N. and Fink, H.-W. (2012), ‘When holography meets coherent diffraction imaging’, Optics Express 20, 28871–28892.CrossRefGoogle Scholar
Li, H., Schwab, J., Antholzer, S. and Haltmeier, M. (2018), NETT: Solving inverse problems with deep neural networks. arXiv:1803.00092 Google Scholar
Li, J. and Zhou, T. (2017), ‘On relaxed averaged alternating reflections (RAAR) algorithm for phase retrieval with structured illumination’, Inverse Problems 33, 025012.CrossRefGoogle Scholar
Li, X. and Voroninski, V. (2013), ‘Sparse signal recovery from quadratic measurements via convex programming’, SIAM J. Math. Anal. 45, 3019–3033.CrossRefGoogle Scholar
Li, X., Ling, S., Strohmer, T. and Wei, K. (2019), ‘Rapid, robust, and reliable blind deconvolution via nonconvex optimization’, Appl. Comput. Harmon. Anal. 47, 893–934.CrossRefGoogle Scholar
Li, Y., Lee, K. and Bresler, Y. (2016), ‘Identifiability in blind deconvolution with subspace or sparsity constraints’, IEEE Trans. Inform. Theory 62, 4266–4275.CrossRefGoogle Scholar
Ling, S. and Strohmer, T. (2015), ‘Self-calibration and biconvex compressive sensing’, Inverse Problems 31, 115002.CrossRefGoogle Scholar
Ling, S. and Strohmer, T. (2017), ‘Blind deconvolution meets blind demixing: Algorithms and performance bounds’, IEEE Trans. Inform. Theory 63, 4497–4520.CrossRefGoogle Scholar
Ling, S. and Strohmer, T. (2019), ‘Regularized gradient descent: A non-convex recipe for fast joint blind deconvolution and demixing’, Inform . Inference 8, 1–49.CrossRefGoogle Scholar
Liu, Y. et al. (2008), ‘Phase retrieval in x-ray imaging based on using structured illumination’, Phys. Rev. A 78, 023817.CrossRefGoogle Scholar
Loewen, E. and Popov, E. (1997), Diffraction Gratings and Applications, Marcel Dekker.Google Scholar
Loh, N., Bogan, M. J., Elser, V., Barty, A., Boutet, S., Bajt, S., Hajdu, J., Ekeberg, T., Maia, F. R., Schulz, J. et al. (2010), ‘Cryptotomography: Reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns’, Phys. Rev. Lett. 104, 225501.CrossRefGoogle Scholar
Longchamp, J.-N., Rauschenbach, S., Abb, S., Escher, C., Latychevskaia, T., Kern, K. and Fink, H.-W. (2017), ‘Imaging proteins at the single-molecule level’, Proc. Nat. Acad. Sci. 114, 1474–1479.CrossRefGoogle Scholar
Lu, Y. M. and Li, G. (2017), Phase transitions of spectral initialization for high-dimensional nonconvex estimation. arXiv:1702.06435 CrossRefGoogle Scholar
Luke, D. R. (2004), ‘Relaxed averaged alternating reflections for diffraction imaging’, Inverse Problems 21, 37–50.CrossRefGoogle Scholar
Luke, D. R. (2008), ‘Finding best approximation pairs relative to a convex and prox-regular set in a Hilbert space’, SIAM J. Optim. 19, 714–739.CrossRefGoogle Scholar
Luke, D. R. (2017), ‘Phase retrieval, what’s new?’, SIAG/OPT Views News 25, 1–5.Google Scholar
Luke, D. R., Burke, J. V. and Lyon, R. G. (2002), ‘Optical wavefront reconstruction: Theory and numerical methods’, SIAM Rev. 44, 169–224.CrossRefGoogle Scholar
Luo, W., Alghamdi, W. and Lu, Y. M. (2019), ‘Optimal spectral initialization for signal recovery with applications to phase retrieval’, IEEE Trans. Signal Process. 67, 2347–2356.CrossRefGoogle Scholar
Ma, C., Wang, K., Chi, Y. and Chen, Y. (2020), ‘Implicit regularization in nonconvex statistical estimation: Gradient descent converges linearly for phase retrieval, matrix completion, and blind deconvolution’, Found. Comput. Math. 20, 451–632.CrossRefGoogle Scholar
Maiden, A. M. and Rodenburg, J. M. (2009), ‘An improved ptychographical phase retrieval algorithm for diffractive imaging’, Ultramicroscopy 109, 1256–1262.CrossRefGoogle Scholar
Maiden, A., Johnson, D. and Li, P. (2017), ‘Further improvements to the ptychographical iterative engine’, Optica 4, 736–745.CrossRefGoogle Scholar
Maiden, A. M., Morrison, G. R., Kaulich, B., Gianoncelli, A. and Rodenburg, J. M. (2013), ‘Soft X-ray spectromicroscopy using ptychography with randomly phased illumination’, Nature Commun. 4, 1–6.CrossRefGoogle Scholar
Marchesini, S. (2007), ‘A unified evaluation of iterative projection algorithms for phase retrieval’, Rev. Sci. Inst. 78, 011301.CrossRefGoogle Scholar
Marchesini, S. and Sakdinawat, A. (2019), ‘Shaping coherent x-rays with binary optics’, Optics Express 27, 907–917.CrossRefGoogle Scholar
Marchesini, S., Krishnan, H., Daurer, B. J., Shapiro, D. A., Perciano, T., Sethian, J. A. and Maia, F. R. (2016), ‘SHARP: A distributed GPU-based ptychographic solver’, J. Appl. Crystallogr. 49, 1245–1252.CrossRefGoogle Scholar
Mesbahi, M. and Papavassilopoulos, G. P. (1997), ‘On the rank minimization problem over a positive semidefinite linear matrix inequality’, IEEE Trans. Automat. Control 42, 239–243.CrossRefGoogle Scholar
Metzler, C. A., Maleki, A. and Baraniuk, R. G. (2016), BM3D-PRGAMP: Compressive phase retrieval based on BM3D denoising. In 2016 IEEE International Conference on Image Processing (ICIP), IEEE, pp. 2504–2508.CrossRefGoogle Scholar
Metzler, C. A., Schniter, P., Veeraraghavan, A. and Baraniuk, R. G. (2018), prDeep: Robust phase retrieval with a flexible deep network. arXiv:1803.00212 Google Scholar
Metzler, C. A., Sharma, M. K., Nagesh, S., Baraniuk, R. G., Cossairt, O. and Veeraraghavan, A. (2017), Coherent inverse scattering via transmission matrices: Efficient phase retrieval algorithms and a public dataset. In 2017 IEEE International Conference on Computational Photography (ICCP), IEEE, pp. 1–16.CrossRefGoogle Scholar
Miao, J., Chapman, H. N. and Sayre, D. (1997), ‘Image reconstruction from the oversampled diffraction pattern’, Microsc. Microanal. 3 (suppl. 2), 1155–1156.CrossRefGoogle Scholar
Miao, J., Charalambous, P., Kirz, J. and Sayre, D. (1999), ‘Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens’, Nature 400 (6742), 342.CrossRefGoogle Scholar
Miao, J., Ishikawa, T., Shen, Q. and Earnest, T. (2008), ‘Extending X-ray crystallography to allow the imaging of noncrystalline materials, cells and single protein complexes’, Annu. Rev. Phys. Chem. 59, 387–410.CrossRefGoogle Scholar
Miao, J., Kirz, J. and Sayre, D. (2000), ‘The oversampling phasing method’, Acta Crystallogr. Sect. D 56, 1312–1315.CrossRefGoogle Scholar
Miao, J., Sayre, D. and Chapman, H. (1998), ‘Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects’, J. Optical Soc. Amer. A 15, 1662–1669.CrossRefGoogle Scholar
Millane, R. (1990), ‘Phase retrieval in crystallography and optics’, J. Optical Soc. Amer. A. 7, 394–411.CrossRefGoogle Scholar
Millane, R. (2006), Recent advances in phase retrieval. In Image Reconstruction from Incomplete Data IV (Bones, P., Fiddy, M. and Millane, R., eds), Vol. 6316 of Proc. SPIE, pp. 139–149.Google Scholar
Misell, D. (1973), ‘A method for the solution of the phase problem in electron microscopy’, J. Phys. D 6, L6–L9.CrossRefGoogle Scholar
Mondelli, M. and Montanari, A. (2019), ‘Fundamental limits of weak recovery with applications to phase retrieval’, Found. Comput. Math. 19, 703–773.CrossRefGoogle Scholar
Monteiro, R. D. (1997), ‘Primal–dual path-following algorithms for semidefinite programming’, SIAM J. Optim. 7, 663–678.CrossRefGoogle Scholar
Nawab, S., Quatieri, T. and Lim, J. (1983), ‘Signal reconstruction from short-time Fourier transform magnitude’, IEEE Trans. Acoust. Speech Signal Process. 31, 986–998.CrossRefGoogle Scholar
Nesterov, Y. (2004), Introductory Lectures on Convex Optimization: A Basic Course, Vol. 87 of Applied Optimization, Kluwer.CrossRefGoogle Scholar
Neutze, R., Wouts, R., Van der Spoel, D., Weckert, E. and Hajdu, J. (2000), ‘Potential for biomolecular imaging with femtosecond X-ray pulses’, Nature 406, 752–757.CrossRefGoogle Scholar
Ohlsson, H., Yang, A. Y., Dong, R. and Sastry, S. S. (2012), ‘Compressive phase retrieval from squared output measurements via semidefinite programming’, IFAC Proceedings 45, 89–94.Google Scholar
Paris, M. and Řeháček, J., eds (2004), Quantum State Estimation, Vol. 649 of Lecture Notes in Physics, Springer.CrossRefGoogle Scholar
Peng, X., Ruane, G. J., Quadrelli, M. B. and Swartzlander, G. A. (2017), ‘Randomized apertures: High resolution imaging in far field’, Optics Express 25, 18296–18313.CrossRefGoogle Scholar
Pfander, G. E. and Salanevich, P. (2019), ‘Robust phase retrieval algorithm for time-frequency structured measurements’, SIAM J. Imaging Sci. 12, 736–761.CrossRefGoogle Scholar
Pfeiffer, F. (2018), ‘X-ray ptychography’, Nature Photonics 12, 9–17.CrossRefGoogle Scholar
Pohl, V., Yang, F. and Boche, H. (2015), ‘Phase retrieval from low-rate samples’, Sampl. Theory Signal Image Process. 14, 71–99.Google Scholar
Qian, J., Yang, C., Schirotzek, A., Maia, F. and Marchesini, S. (2014), Efficient algorithms for ptychographic phase retrieval. In Inverse Problems and Applications, Vol. 615 of Contemporary Mathematics, American Mathematical Society, pp. 261–280.CrossRefGoogle Scholar
Rauhut, H., Schneider, R. and Stojanac, Ž. (2017), ‘Low rank tensor recovery via iterative hard thresholding’, Linear Algebra Appl. 523, 220–262.CrossRefGoogle Scholar
Raz, O., Leshem, B., Miao, J., Nadler, B., Oron, D. and Dudovich, N. (2014), ‘Direct phase retrieval in double blind Fourier holography’, Optics Express 22, 24935–24950.CrossRefGoogle Scholar
Recht, B., Fazel, M. and Parrilo, P. A. (2010), ‘Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization’, SIAM Rev. 52, 471–501.CrossRefGoogle Scholar
Reichenbach, H. (1944), Philosophic Foundations of Quantum Mechanics, University of California Press.Google Scholar
Rivenson, Y., Zhang, Y., Günaydın, H., Teng, D. and Ozcan, A. (2018), ‘Phase recovery and holographic image reconstruction using deep learning in neural networks’, Light Sci. Appl. 7, 17141–17141.CrossRefGoogle Scholar
Rodenburg, J. M. (2008), ‘Ptychography and related diffractive imaging methods’, Adv. Imaging Electron Phys. 150, 87–184.CrossRefGoogle Scholar
Rodenburg, J. M. and Faulkner, H. M. (2004), ‘A phase retrieval algorithm for shifting illumination’, Appl. Phys. Lett. 85, 4795–4797.CrossRefGoogle Scholar
Saliba, M., Bosgra, J., Parsons, A., Wagner, U., Rau, C. and Thibault, P. (2016), ‘Novel methods for hard X-ray holographic lensless imaging’, Microsc. Microanal. 22, 110–111.CrossRefGoogle Scholar
Saliba, M., Latychevskaia, T., Longchamp, J. and Fink, H. (2012), ‘Fourier transform holography: A lensless non-destructive imaging technique’, Microsc. Microanal. 18, 564–565.CrossRefGoogle Scholar
Sanz, J. (1985), ‘Mathematical considerations for the problem of Fourier transform phase retrieval from magnitude’, SIAM J. Appl. Math. 45, 651–664.CrossRefGoogle Scholar
Scapin, G. (2006), ‘Structural biology and drug discovery’, Current Pharmaceut. Design 12, 2087–2097.CrossRefGoogle Scholar
Schniter, P. and Rangan, S. (2014), ‘Compressive phase retrieval via generalized approximate message passing’, IEEE Trans. Signal Process. 63, 1043–1055.CrossRefGoogle Scholar
Schniter, P. and Rangan, S. (2015), A message-passing approach to phase retrieval of sparse signals. In Excursions in Harmonic Analysis 4 (Balan, R. et al., eds), Applied and Numerical Harmonic Analysis, Springer, pp. 177–204.CrossRefGoogle Scholar
Schwarz, H. A. (1870), ‘Über einen Grenzübergang durch alternierendes Verfahren’, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 15, 272–286.Google Scholar
Scott, A. J. and Grassl, M. (2010), ‘Symmetric informationally complete positive-operator-valued measures: A new computer study’, J. Math. Phys. 51, 042203.CrossRefGoogle Scholar
Seaberg, M. H., d’Aspremont, A. and Turner, J. J. (2015), ‘Coherent diffractive imaging using randomly coded masks’, Appl. Phys. Lett. 107, 231103.CrossRefGoogle Scholar
Seelamantula, C. S., Pavillon, N., Depeursinge, C. and Unser, M. (2011), ‘Exact complex-wave reconstruction in digital holography’, J. Optical Soc. Amer. A 28, 983–992.CrossRefGoogle Scholar
Shechtman, Y., Beck, A. and Eldar, Y. C. (2014), ‘GESPAR: Efficient phase retrieval of sparse signals’, IEEE Trans. Signal Process. 62, 928–938.CrossRefGoogle Scholar
Shechtman, Y., Eldar, Y. C., Cohen, O., Chapman, H. N., Miao, J. and Segev, M. (2015), ‘Phase retrieval with application to optical imaging: A contemporary overview’, IEEE Signal Process. Mag. 32, 87–109.CrossRefGoogle Scholar
Singer, A. (2019), Mathematics for cryo-electron microscopy. In Proceedings of the International Congress of Mathematicians (ICM 2018), World Scientific, pp. 3995–4014.CrossRefGoogle Scholar
Strohmer, T. and Heath, R. (2003), ‘Grassmannian frames with applications to coding and communications’, Appl. Comput. Harmon. Anal. 14, 257–275.CrossRefGoogle Scholar
Strohmer, T. and Vershynin, R. (2009), ‘A randomized Kaczmarz algorithm with exponential convergence’, J. Fourier Anal. Appl. 15, 262.CrossRefGoogle Scholar
Sun, J., Qu, Q. and Wright, J. (2018), ‘A geometric analysis of phase retrieval’, Found. Comput. Math. 18, 1131–1198.Google Scholar
Sun, R. and Luo, Z.-Q. (2016), ‘Guaranteed matrix completion via non-convex factorization’, IEEE Trans. Inform. Theory 62, 6535–6579.CrossRefGoogle Scholar
Tan, Y. S. and Vershynin, R. (2019), ‘Phase retrieval via randomized Kaczmarz: Theoretical guarantees’, Inf. Inference 8, 97–123.CrossRefGoogle Scholar
Thibault, P. and Guizar-Sicairos, M. (2012), ‘Maximum-likelihood refinement for coherent diffractive imaging’, New J. Phys. 14, 063004.CrossRefGoogle Scholar
Thibault, P., Dierolf, M., Bunk, O., Menzel, A. and Pfeiffer, F. (2009), ‘Probe retrieval in ptychographic coherent diffractive imaging’, Ultramicroscopy 109, 338–343.CrossRefGoogle Scholar
Thibault, P., Dierolf, M., Menzel, A., Bunk, O., David, C. and Pfeiffer, F. (2008), ‘High-resolution scanning x-ray diffraction microscopy’, Science 321(5887), 379–382.CrossRefGoogle Scholar
Tillmann, A. M., Eldar, Y. C. and Mairal, J. (2016), ‘DOLPHIn: Dictionary learning for phase retrieval’, IEEE Trans. Signal Process. 64, 6485–6500.CrossRefGoogle Scholar
Toh, K.-C., Todd, M. J. and Tütüncü, R. H. (1999), ‘SDPT3: A MATLAB software package for semidefinite programming, version 1.3’, Optim. Methods Softw. 11, 545–581.CrossRefGoogle Scholar
Tu, S., Boczar, R., Simchowitz, M., Soltanolkotabi, M. and Recht, B. (2015), Low-rank solutions of linear matrix equations via Procrustes flow. arXiv:1507.03566 Google Scholar
Vinzant, C. (2015), A small frame and a certificate of its injectivity. In 2015 International Conference on Sampling Theory and Applications (SampTA), IEEE, pp. 197–200.CrossRefGoogle Scholar
von Neumann, J. (1950), Functional Operators: Measures and Integrals, Vol. 1, Princeton University Press.Google Scholar
Waldspurger, I., d’Aspremont, A. and Mallat, S. (2015), ‘Phase recovery, MaxCut and complex semidefinite programming’, Math. Program. 149, 47–81.CrossRefGoogle Scholar
Walther, A. (1963), ‘The question of phase retrieval in optics’, Optica Acta 10, 41–49.CrossRefGoogle Scholar
Wang, G., Giannakis, G. B. and Eldar, Y. C. (2018), ‘Solving systems of random quadratic equations via truncated amplitude flow’, IEEE Trans. Inform. Theory 64, 773–794.CrossRefGoogle Scholar
Wang, G., Zhang, L., Giannakis, G. B., Akçakaya, M. and Chen, J. (2017), ‘Sparse phase retrieval via truncated amplitude flow’, IEEE Trans. Signal Process. 66, 479–491.CrossRefGoogle Scholar
Wei, K. (2015), ‘Solving systems of phaseless equations via Kaczmarz methods: A proof of concept study’, Inverse Problems 31, 125008.CrossRefGoogle Scholar
Wiener, N. (1932), ‘Tauberian theorems’, Ann. of Math. (2) 33, 1–100.CrossRefGoogle Scholar
Yeh, L.-H., Dong, J., Zhong, J., Tian, L., Chen, M., Tang, G., Soltanolkotabi, M. and Waller, L. (2015), ‘Experimental robustness of Fourier ptychography phase retrieval algorithms’, Optics Express 23, 33214–33240.CrossRefGoogle Scholar
Yuan, Z., Wang, H. and Wang, Q. (2019), ‘Phase retrieval via sparse Wirtinger flow’, J. Comput. Appl. Math. 355, 162–173.CrossRefGoogle Scholar
Zauner, G. (1999), Quantendesigns: Grundzüge einer nichtkommutativen Designtheorie. PhD thesis, Universität Wien.Google Scholar
Zhang, F., Chen, B., Morrison, G. R., Vila-Comamala, J., Guizar-Sicairos, M. and Robinson, I. K. (2016), ‘Phase retrieval by coherent modulation imaging’, Nature Commun. 7, 1–8.CrossRefGoogle Scholar
Zhang, G., Guan, T., Shen, Z., Wang, X., Hu, T., Wang, D., He, Y. and Xie, N. (2018), ‘Fast phase retrieval in off-axis digital holographic microscopy through deep learning’, Optics Express 26, 19388–19405.CrossRefGoogle Scholar
Zhang, Y., Song, P. and Dai, Q. (2017), ‘Fourier ptychographic microscopy using a generalized Anscombe transform approximation of the mixed Poisson–Gaussian likelihood’, Optics Express 25, 168–179.CrossRefGoogle Scholar
Zuo, C., Sun, J. and Chen, Q. (2016), ‘Adaptive step-size strategy for noise-robust Fourier ptychographic microscopy’, Optics Express 24, 20724–20744.CrossRefGoogle Scholar