Phase retrieval via matrix completion.

*(English)*Zbl 1344.49057##### MSC:

49N45 | Inverse problems in optimal control |

49N30 | Problems with incomplete information (optimization) |

49M37 | Numerical methods based on nonlinear programming |

90C25 | Convex programming |

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

94A20 | Sampling theory in information and communication theory |

68U10 | Computing methodologies for image processing |

##### Keywords:

phase retrieval; matrix completion; convex optimization; trace-norm minimization; diffraction; Fourier transform
Full Text:
DOI

##### References:

[1] | R. Balan, B. Bodmann, P. G. Casazza, and D. Edidin, Painless reconstruction from magnitudes of frame coefficients, J. Fourier Anal. Appl., 15 (2009), pp. 488–501. · Zbl 1181.42032 |

[2] | R. Balan, P. G. Casazza, and D. Edidin, On signal reconstruction without noisy phase, Appl. Comput. Harmon. Anal., 20 (2006), pp. 345–356. · Zbl 1090.94006 |

[3] | H. H. Bauschke, P. L. Combettes, and D. R. Luke, Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization, J. Opt. Soc. Amer. A, 19 (2002), pp. 1334–1345. |

[4] | A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), pp. 183–202. · Zbl 1175.94009 |

[5] | C. Beck and R. D’Andrea, Computational study and comparisons of LFT reducibility methods, in Proceedings of the American Control Conference, 1998, pp. 1013–1017. |

[6] | S. Becker, E. J. Candès, and M. Grant, Templates for convex cone problems with applications to sparse signal recovery, Math. Program. Comput., 3 (2011), pp. 165–218. · Zbl 1257.90042 |

[7] | A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MOS-SIAM Ser. Optim. 2, SIAM, Philadelphia, 2001. · Zbl 0986.90032 |

[8] | G. Bianchi, F. Segala, and A. Volcic, The solution of the covariogram problem for plane \({\mathcal C}^2_+\) convex bodies, J. Differential Geom., 60 (2002), pp. 177–198. · Zbl 1047.52002 |

[9] | M. J. Bogan, W. H. Benner, S. Boutet, U. Rohner, M. Frank, A. Barty, M. M. Seibert, F. Maia, S. Marchesini, S. Bajt, B. Woods, V. Riot, S. P. Hau-Riege, M. Svenda, E. Marklund, E. Spiller, J. Hadju, and H. N. Chapman, Single particle X-ray diffractive imaging, Nano Lett., 8 (2008), pp. 310–316. |

[10] | L. M. Brègman, The method of successive projection for finding a common point of convex sets, Soviet Math. Dokl., 6 (1965), pp. 688–692. · Zbl 0142.16804 |

[11] | Y. M. Bruck and L. G. Sodin, On the ambiguity of the image reconstruction problem, Opt. Comm., 30 (1979), pp. 304–308. |

[12] | O. Bunk, A. Diaz, F. Pfeiffer, C. David, B. Schmitt, D. K. Satapathy, and J. F. van der Veen, Diffractive imaging for periodic samples: Retrieving one-dimensional concentration profiles across microfluidic channels, Acta Crystallogr. A, 63 (2007), pp. 306–314. |

[13] | E. J. Candès and B. Recht, Exact matrix completion via convex optimization, Found. Comput. Math., 9 (2009), pp. 717–772. · Zbl 1219.90124 |

[14] | E. Candès, T. Strohmer, and V. Voroninski, PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming, Comm. Pure Appl. Math., 66 (2013), pp. 1241–1274. · Zbl 1335.94013 |

[15] | E. J. Candès and T. Tao, The power of convex relaxation: Near-optimal matrix completion, IEEE Trans. Inform. Theory, 56 (2010), pp. 2053–2080. · Zbl 1366.15021 |

[16] | E. J. Candès, M. B. Wakin, and S. P. Boyd, Enhancing sparsity by reweighted \(l_1\) minimization, J. Fourier Anal. Appl., 14 (2008), pp. 877–905. · Zbl 1176.94014 |

[17] | A. Carballar and M. A. Muriel, Phase reconstruction from reflectivity in fiber Bragg gratings, J. Lightwave Technol., 15 (1997), pp. 1314–1322. |

[18] | A. Chai, M. Moscoso, and G. Papanicolaou, Array Imaging Using Intensity-only Measurements, Technical report, Stanford University, Stanford, CA, 2010. · Zbl 1207.78022 |

[19] | C.-C. Chen, J. Miao, C. W. Wang, and T. K. Lee, Application of the optimization technique to noncrystalline X-ray diffraction microscopy: Guided hybrid input-output method (GHIO), Phys. Rev. B, 76 (2007), 064113. |

[20] | J. V. Corbett, The Pauli problem, state reconstruction and quantum-real numbers, Rep. Math. Phys., 57 (2006), pp. 53–68. · Zbl 1110.81007 |

[21] | J. C. Dainty and J. R. Fienup, Phase Retrieval and Image Reconstruction for Astronomy, in Image Recovery: Theory and Application, H. Stark, ed., Academic Press, New York, 1987, pp. 231–275. |

[22] | M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, Ptychographic X-ray computed tomography at the nanoscale, Nature, 467 (2010), pp. 436–439. |

[23] | H. Duadi, O. Margalit, V. Mico, J. A. Rodrigo, T. Alieva, J. Garcia, and Z. Zalevsky, Digital holography and phase retrieval, in Holography, Research and Technologies, J. Rosen, ed., InTech, 2011; available online at http://www.intechopen.com/books/holography-research-and-technologies/digital-holography-and-phase-retrieval. |

[24] | A. Faridian, D. Hopp, G. Pedrini, U. Eigenthaler, M. Hirscher, and W. Osten, Nanoscale imaging using deep ultraviolet digital holographic microscopy, Opt. Express, 18 (2010), pp. 14159–14164. |

[25] | M. Fazel, Matrix Rank Minimization with Applications, Ph.D. thesis, Stanford University, Stanford, CA, 2002. |

[26] | M. Fazel, H. Hindi, and S. Boyd, Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices, in Proceedings of the American Control Conference, 2003, pp. 2156–2162. |

[27] | J. R. Fienup, Reconstruction of an object from the modulus of its Fourier transform, Opt. Lett., 3 (1978), pp. 27–29. |

[28] | J. R. Fienup, Phase retrieval algorithms: A comparison, Appl. Opt., 21 (1982), pp. 2758–2768. |

[29] | J. Finkelstein, Pure-state informationally complete and “really” complete measurements, Phys. Rev. A (3), 70 (2004), 052107. · Zbl 1227.81029 |

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

[31] | M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM, 42 (1995), pp. 1115–1145. · Zbl 0885.68088 |

[32] | L. Gubin, B. Polyak, and E. Raik, The method of projections for finding the common point of convex sets, USSR Comput. Math. Math. Phys., 7 (1967), pp. 1–24. |

[33] | R. W. Harrison, Phase problem in crystallography, J. Opt. Soc. Amer. A, 10 (1993), pp. 1045–1055. |

[34] | H. Hauptman, The direct methods of X-ray crystallography, Science, 233 (1986), pp. 178–183. |

[35] | M. Hayes, The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform, IEEE Trans. Acoust. Speech Signal Process., 30 (1982), pp. 140–154. · Zbl 0563.65084 |

[36] | T. Heinosaari, L. Mazzarella, and M. M. Wolf, Quantum Tomography under Prior Information, preprint, http://arxiv.org/abs/1109.5478arXiv:1109.5478v1 [quant-ph], 2011. · Zbl 1263.81102 |

[37] | N. Hurt, Phase Retrieval and Zero Crossings, Kluwer Academic Publishers, Norwell, MA, 1989. · Zbl 0687.94001 |

[38] | E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, Coherent detection in optical fiber systems, Opt. Express, 16 (2008), pp. 753–791. |

[39] | Y. Ivankovski and D. Mendlovic, High-rate long-distance fiber-optic communication based on advanced modulation techniques, Appl. Opt., 38 (1999), pp. 5533–5540. |

[40] | I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, Coherent diffractive imaging using phase front modifications, Phys. Rev. Lett., 100 (2008), 155503. |

[41] | J. Kiefer, Sequential minimax search for a maximum, Proc. Amer. Math. Soc., 4 (1953), pp. 502–506. · Zbl 0050.35702 |

[42] | T. Kleine-Ostmann and T. Nagatsuma, A review on terahertz communications research, J. Infrared Milli. Terah. Waves, 32 (2011), pp. 143–171. |

[43] | M. V. Klibanov, P. E. Sacks, and A. V. Tikhonravov, The phase retrieval problem, Inverse Problems, 11 (1995), p. 1–28. · Zbl 0821.35150 |

[44] | A. Levi and H. Stark, Restoration from phase and magnitude by generalized projections, in Image Recovery: Theory and Application, H. Stark, ed., Academic Press, San Diego, CA, 1987, pp. 277–320. |

[45] | Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkens, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phase retrieval in x-ray imaging based on using structured illumination, Phys. Rev. A, 78 (2008), 023817. |

[46] | E. G. Loewen and E. Popov, Diffraction Gratings and Applications, Marcel Dekker, New York, 1997. |

[47] | Y. Lu and M. Vetterli, Sparse spectral factorization: Unicity and reconstruction algorithms, in Proceedings of the 36th International Conference on Acoustics, Speech and Signal Processing (ICASSP), Prague, Czech Republic, 2011. |

[48] | D. R. Luke, Finding best approximation pairs relative to a convex and prox-regular set in a Hilbert space, SIAM J. Optim., 19 (2008), pp. 714–739. · Zbl 1169.65056 |

[49] | D. R. Luke, J. V. Burke, and R. G. Lyon, Optical wavefront reconstruction: Theory and numerical methods, SIAM Rev., 44 (2002), pp. 169–224. · Zbl 1094.78004 |

[50] | S. Marchesini, Phase retrieval and saddle-point optimization, J. Opt. Soc. Amer. A, 24 (2007), pp. 3289–3296. |

[51] | S. Marchesini, A unified evaluation of iterative projection algorithms for phase retrieval, Rev. Sci. Instrum., 78 (2007), 011301. |

[52] | S. Marchesini, Ab Initio Compressive Phase Retrieval, preprint, http://arxiv.org/abs/0809.2006arXiv:0809.2006v1 [physics.optics], 2008. |

[53] | M. Mesbahi and G. P. Papavassilopoulos, On the rank minimization problem over a positive semidefinite linear matrix inequality, IEEE Trans. Automat. Control, 42 (1997), pp. 239–243. · Zbl 0872.93029 |

[54] | J. Miao, H. N. Chapman, and D. Sayre, Image reconstruction from the oversampled diffraction pattern, Microscopy Microanalysis, 3 (Suppl. 2) (1997), pp. 1155–1156. |

[55] | J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, Extending X-ray crystallography to allow the imaging of noncrystalline materials, cells and single protein complexes, Annu. Rev. Phys. Chem., 59 (2008), pp. 387–410. |

[56] | J. Miao, J. Kirz, and D. Sayre, The oversampling phasing method, Acta Cryst., D56 (2000), pp. 1312–1315. |

[57] | J. Miao, D. Sayre, and H. N. Chapman, Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects, J. Opt. Soc. Amer. A, 15 (1998), pp. 1662–1669. |

[58] | R. P. Millane, Phase retrieval in crystallography and optics, J. Opt. Soc. Amer. A., 7 (1990), pp. 394–411. |

[59] | R. P. Millane, Recent advances in phase retrieval, in Image Reconstruction from Incomplete Data IV, P. J. Bones, M. A. Fiddy, and R. P. Millane, eds., Proc. SPIE 6316, SPIE, Bellingham, WA, 2006, 63160E. |

[60] | D. L. Misell, A method for the solution of the phase problem in electron microscopy, J. Phys. D: Appl. Phys., 6 (1973), pp. L6–L9. |

[61] | Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Appl. Optim. 87, Kluwer Academic Publishers, Boston, 2004. · Zbl 1086.90045 |

[62] | K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Manusco, Unique phase recovery for nonperiodic objects, Phys. Rev. Lett., 91 (2003), 203902. |

[63] | L. Rabiner and B. H. Juang, Fundamentals of Speech Recognition, Signal Processing Series, Prentice-Hall, Upper Saddle River, NJ, 1993. · Zbl 0762.62036 |

[64] | B. Recht, M. Fazel, and P. A. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev., 52 (2010), pp. 471–501. · Zbl 1198.90321 |

[65] | H. Reichenbach, Philosophic Foundations of Quantum Mechanics, University of California Press, Berkeley, CA, 1944. |

[66] | J. M. Rodenburg, Ptychography and related diffractive imaging methods, Adv. Imaging Electron Phys., 150 (2008), pp. 87–184. |

[67] | J. L. C. Sanz, Mathematical considerations for the problem of Fourier transform phase retrieval from magnitude, SIAM J. Appl. Math., 45 (1985), pp. 651–664. · Zbl 0569.42009 |

[68] | G. Scapin, Structural biology and drug discovery, Curr. Pharm. Des., 12 (2006), pp. 2087–2097. |

[69] | Y. Shechtman, Y. C. Eldar, A. Szameit, and M. Segev, Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing, Opt. Express, 19 (2011), pp. 14807–14822. |

[70] | P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, Probe retrieval in ptychographic coherent diffractive imaging, Ultramicroscopy, 109 (2009), pp. 338–343. |

[71] | D. P. Varn, G. S. Canright, and J. P. Crutchfield, Discovering planar disorder in close-packed structures from x-ray diffraction: Beyond the fault model, Phys. Rev. B, 66 (2002), 174110. |

[72] | A. Vogt, Position and momentum distributions do not determine the quantum mechanical state, in Mathematical Foundations of Quantum Theory, A. R. Marlow, ed., Academic Press, New York, 1978, pp. 365–372. |

[73] | J. von Neumann, Functional Operators, Vol. II, Ann. Math. Stud. 22, Princeton University Press, Princeton, NJ, 1950. |

[74] | A. Walther, The question of phase retrieval in optics, Optical Acta, 10 (1963), pp. 41–49. |

[75] | D. C. Youla, Mathematical theory of image restoration by the method of convex projections, in Image Recovery: Theory and Application, H. Stark, ed., Academic Press, San Diego, CA, 1987, pp. 29–77. |

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