zbMATH — the first resource for mathematics

Variational phase retrieval with globally convergent preconditioned proximal algorithm. (English) Zbl 1398.65072

65F22 Ill-posedness and regularization problems in numerical linear algebra
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
49N30 Problems with incomplete information (optimization)
49N45 Inverse problems in optimal control
PDF BibTeX Cite
Full Text: DOI
[1] H. Attouch, J. Bolte, P. Redont, and A. Soubeyran, Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-Łojasiewicz inequality, Math. Oper. Res., 35 (2010), pp. 438–457. · Zbl 1214.65036
[2] R. Balan, P. Casazza, and D. Edidin, On signal reconstruction without phase, Appl. Comput. Harmon. Anal., 20 (2006), pp. 345–356. · Zbl 1090.94006
[3] A. S. Bandeira, J. Cahill, D. G. Mixon, and A. A. Nelson, Saving phase: Injectivity and stability for phase retrieval, Appl. Comput. Harmon. Anal., 37 (2014), pp. 106–125. · Zbl 1305.90330
[4] A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MOS–SIAM Ser. Optim. 2, Mathematical Optimization Society, SIAM, Philadelphia, 2001, . · Zbl 0986.90032
[5] H. M. Berman, T. Battistuz, T. Bhat, W. F. Bluhm, P. E. Bourne, K. Burkhardt, Z. Feng, G. L. Gilliland, L. Iype, S. Jain, et al., The protein data bank, Acta Crystallogr. Sect. D Biol. Crystallogr., 58 (2002), pp. 899–907.
[6] J. Bolte, S. Sabach, and M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Math. Program., 146 (2014), pp. 459–494. · Zbl 1297.90125
[7] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2011), pp. 1–122. · Zbl 1229.90122
[8] T. T. Cai, X. Li, and Z. Ma, Optimal rates of convergence for noisy sparse phase retrieval via thresholded wirtinger flow, Ann. Statist., 44 (2016), pp. 2221–2251. · Zbl 1349.62019
[9] E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, Phase retrieval via matrix completion, SIAM J. Imaging Sci., 6 (2013), pp. 199–225, . · Zbl 1280.49052
[10] E. J. Candes, X. Li, and M. Soltanolkotabi, Phase retrieval from coded diffraction patterns, Appl. Comput. Harmon. Anal., 39 (2015), pp. 277–299. · Zbl 1329.78056
[11] E. J. Candes, X. Li, and M. Soltanolkotabi, Phase retrieval via Wirtinger flow: Theory and algorithms, IEEE Trans. Inform. Theory, 61 (2015), pp. 1985–2007. · Zbl 1359.94069
[12] H. Chang, Y. Lou, Y. Duan, and S. Marchesini, Total Variation Based Phase Retrieval for Poisson Noise Removal, preprint, CAM report 16-76, UCLA, Los Angeles, CA, 2016.
[13] 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 (2016), pp. A3672–A3695, . · Zbl 1352.49034
[14] H. Chang and S. Marchesini, Denoising Poisson Phaseless Measurements via Orthogonal Dictionary Learning, preprint, , 2016.
[15] H. Chang and S. Marchesini, A General Framework for Denoising Phaseless Diffraction Measurements, preprint, , 2016.
[16] P. Chen and A. Fannjiang, Fourier phase retrieval with a single mask by Douglas–Rachford algorithms, Appl. Comput. Harmon. Anal., (2016), . · Zbl 06858981
[17] P. Chen, A. Fannjiang, and G. R. Liu, Phase retrieval with one or two diffraction patterns by alternating projections of the null vector, J. Fourier Anal. Appl., (2017), . · Zbl 06897259
[18] Y. Chen and E. Candes, Solving random quadratic systems of equations is nearly as easy as solving linear systems, in Advances in Neural Information Processing Systems, Montreal, Canada, 2015, pp. 739–747.
[19] A. Conca, D. Edidin, M. Hering, and C. Vinzant, An algebraic characterization of injectivity in phase retrieval, Appl. Comput. Harmon. Anal., 38 (2015), pp. 346–356. · Zbl 1354.42003
[20] Y. Duan, C. Wu, Z.-F. Pang, and H. Chang, \(L^0\)-Regularized Variational Methods for Sparse Phase Retrieval, preprint, , 2016.
[21] Y. C. Eldar and S. Mendelson, Phase retrieval: Stability and recovery guarantees, Appl. Comput. Harmon. Anal., 36 (2014), pp. 473–494. · Zbl 06298184
[22] E. Esser, X. Zhang, and T. F. Chan, A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, SIAM J. Imaging Sci., 3 (2010), pp. 1015–1046, . · Zbl 1206.90117
[23] J. R. Fienup, Phase retrieval algorithms: A comparison, Appl. Optics, 21 (1982), pp. 2758–2769.
[24] R. W. Gerchberg, A practical algorithm for the determination of phase from image and diffraction plane pictures, Optik, 35 (1972), pp. 237–246.
[25] 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
[26] R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging, SIAM J. Imaging Sci., 8 (2015), pp. 426–457, . · Zbl 1320.90063
[27] M. Iwen, A. Viswanathan, and Y. Wang, Robust sparse phase retrieval made easy, Appl. Comput. Harmon. Anal., 42 (2017), pp. 135–142. · Zbl 1393.94274
[28] A. S. Lewis, D. R. Luke, and J. Malick, Local linear convergence for alternating and averaged nonconvex projections, Found. Comput. Math., 9 (2009), pp. 485–513. · Zbl 1169.49030
[29] G. Li and T. K. Pong, Calculus of the Exponent of Kurdyka-Łojasiewicz Inequality and Its Applications to Linear Convergence of First-Order Methods, preprint, , 2016.
[30] X. Li and V. Voroninski, Sparse signal recovery from quadratic measurements via convex programming, SIAM J. Math. Anal., 45 (2013), pp. 3019–3033, . · Zbl 1320.94023
[31] S. Loock and G. Plonka, Phase retrieval for Fresnel measurements using a shearlet sparsity constraint, Inverse Problems, 30 (2014), 055005. · Zbl 1293.42034
[32] D. R. Luke, Relaxed averaged alternating reflections for diffraction imaging, Inverse Problems, 21 (2005), pp. 37–50. · Zbl 1146.78008
[33] S. Marchesini, Invited article: A unified evaluation of iterative projection algorithms for phase retrieval, Rev. Sci. Instr., 78 (2007), 011301.
[34] S. Marchesini, Y.-C. Tu, and H.-T. Wu, Alternating projection, ptychographic imaging and phase synchronization, Appl. Comput. Harmon. Anal., 41 (2016), pp. 815–851. · Zbl 1388.94015
[35] M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, Compressive phase retrieval, in Wavelets XII, Proc. SPIE 6701, SPIE, Bellingham, WA, 2007, 670120.
[36] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren Math. Wiss. 330, Springer-Verlag, Berlin, 2006.
[37] S. Mukherjee and C. S. Seelamantula, Fienup algorithm with sparsity constraints: Application to frequency-domain optical-coherence tomography, IEEE Trans. Signal Process., 62 (2014), pp. 4659–4672. · Zbl 1394.94410
[38] J. Nocedal and S. Wright, Numerical Optimization, 2nd ed., Springer Ser. Oper. Res., Springer, New York, 1999, pp. 67–68. · Zbl 0930.65067
[39] H. Ohlsson, A. Yang, R. Dong, and S. Sastry, CPRL—an extension of compressive sensing to the phase retrieval problem, in Advances in Neural Information Processing Systems, Lake Tahoe, NV, 2012, pp. 1367–1375.
[40] J. Qian, C. Yang, A. Schirotzek, F. Maia, and S. Marchesini, Efficient algorithms for ptychographic phase retrieval, in Inverse Problems and Applcations, Contemp. Math. 615, AMS, Providence, RI, 2014, pp. 261–279. · Zbl 1329.65332
[41] L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259–268. · Zbl 0780.49028
[42] 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
[43] P. Schniter and S. Rangan, Compressive phase retrieval via generalized approximate message passing, IEEE Trans. Signal Process., 63 (2015), pp. 1043–1055. · Zbl 1394.94506
[44] Y. Shechtman, A. Beck, and Y. C. Eldar, GESPAR: Efficient phase retrieval of sparse signals, IEEE Trans. Signal Process., 62 (2014), pp. 928–938. · Zbl 1394.94522
[45] 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 (2015), pp. 87–109.
[46] J. Sun, Q. Qu, and J. Wright, A geometric analysis of phase retrieval, in Proceedings of the IEEE International Symposium on Information Theory (ISIT), IEEE, Washington, DC, 2016, pp. 2379–2383.
[47] P. Thibault and M. Guizar-Sicairos, Maximum-likelihood refinement for coherent diffractive imaging, New J. Phys., 14 (2012), 063004.
[48] A. M. Tillmann, Y. C. Eldar, and J. Mairal, DOLPHIn—Dictionary Learning for Phase Retrieval, preprint, , 2016. · Zbl 1414.94621
[49] T. Valkonen, A primal–dual hybrid gradient method for nonlinear operators with applications to MRI, Inverse Problems, 30 (2014), 055012. · Zbl 1310.47081
[50] I. Waldspurger, A. d’Aspremont, and S. Mallat, Phase recovery, MaxCut and complex semidefinite programming, Math. Program., 149 (2015), pp. 47–81, . · Zbl 1329.94018
[51] G. Wang, L. Zhang, G. B. Giannakis, M. Akçakaya, and J. Chen, Sparse Phase Retrieval via Truncated Amplitude Flow, preprint, , 2016.
[52] K. Wei, Solving systems of phaseless equations via Kaczmarz methods: A proof of concept study, Inverse Problems, 31 (2015), 125008. · Zbl 1332.65045
[53] Z. Wen, C. Yang, X. Liu, and S. Marchesini, Alternating direction methods for classical and ptychographic phase retrieval, Inverse Problems, 28 (2012), 115010. · Zbl 1254.78037
[54] C. Wu and X.-C. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3 (2010), pp. 300–339, . · Zbl 1206.90245
[55] C. Wu, J. Zhang, and X.-C. Tai, Augmented Lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Probl. Imaging, 5 (2011), pp. 237–261. · Zbl 1225.80013
[56] Z. Yang, C. Zhang, and L. Xie, Robust Compressive Phase Retrieval via l1 Minimization with Application to Image Reconstruction, , 2013.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.