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Published online by Cambridge University Press:  21 April 2022

Alexander H. Barnett
Affiliation:
Flatiron Institute
Charles L. Epstein
Affiliation:
Flatiron Institute
Leslie Greengard
Affiliation:
Courant Institute
Jeremy Magland
Affiliation:
Flatiron Institute
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Chapter
Information
Geometry of the Phase Retrieval Problem
Graveyard of Algorithms
, pp. 300 - 304
Publisher: Cambridge University Press
Print publication year: 2022

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References

Alaifari, R., Daubechies, I., Grohs, P., and Yin, R. 2019. Stable phase retrieval in infinite dimensions. Found. Comput. Math., 19, 896900. https://link.springer.com/article/ 10.1007%2Fs10208–018-9399-7Google Scholar
Barmherzig, D. A., Sun, J., Candès, E. J., Lane, T. J., and Li, P.-N. 2019a. Dual-reference design for holographic phase retrieval. 13th International Conference on Sampling Theory and Applications (SampTA), pp. 14.Google Scholar
Barmherzig, D. A., Sun, J., Candès, E. J., Lane, T. J., and Li, P-N. 2019b. Holographic phase retrieval and optimal reference design. Inverse Probl., 35(9), 094001.CrossRefGoogle Scholar
Barnett, A., Epstein, C. L., Greengard, L., and Magland, J. 2020. Geometry of the Phase Retrieval Problem. Inverse Probl., 36(9), 094003.Google Scholar
Bauschke, H. H. and Borwein, J. M. 1993. On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal., 1(2), 185212.Google Scholar
Bauschke, H. H., and Borwein, J. M. 1996. On projection algorithms for solving convex feasibility problems. SIAM Rev., 38(3), 367426.CrossRefGoogle Scholar
Bauschke, H. H., Combettes, P. L., and Russell, L. D. 2002. Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization. J. Opt. Soc. Am. A, 19, 13341345.Google Scholar
Beinert, R. 2017. Non-negativity constraints in the one-dimensional discrete-time phase retrieval problem. Information and Inference: A Journal of the IMA, 6, 213224.Google Scholar
Borwein, J. M. 2012. Maximum entropy and feasibility methods for convex and nonconvex inverse problems. Optimization, 61(1), 133.Google Scholar
Borwein, J. M., and Sims, B. 2011. The Douglas–Rachford algorithm in the absence of convexity. In Bauschke, H., Burachik, R., Combettes, P., et al. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 93109. Springer Optimization and Its Applications, Vol. 49. New York: Springer.Google Scholar
Bruck, Yu. M., and Sodin, L. G. 1979. On the ambiguity of the image reconstruction problem. Opt. Commun., 30, 304308.Google Scholar
Cahill, J., Casazza, P. G., and Daubechies, I. 2016. Phase retrieval in infinite-dimensional Hilbert spaces. Trans. Amer. Math. Soc., Series B, 3, 6376.Google Scholar
Candès, E. J., Eldar, Y. C., Strohmer, T., and Voroninski, V. 2015a. Phase retrieval via matrix completion. SIAM Rev, 57, 225251.CrossRefGoogle Scholar
Candès, E. J., Li, X., and Soltanolkotabi, M. 2015b. Phase retrieval via Wirtinger flow. IEEE Trans. Information Theory, 61(4), 19852007.CrossRefGoogle Scholar
Chapman, H. N., Barty, A., Marchesini, S., et al. 2006. High-resolution ab initio three-dimensional x-ray diffraction microscopy. J. Opt. Soc. Am. A, 23, 11791200.Google Scholar
Conca, A., Edidin, D., Hering, M., and Vinzant, C. 2015. An algebraic characterization of injectivity in phase retrieval. Appl. Comput. Harmon. A., 38(2), 346356.CrossRefGoogle Scholar
Crimmins, T. R. and Fienup, J. R. 1983. Uniqueness of phase retrieval for functions with sufficiently disconnected support. J. Opt. Soc. Am., 73, 218221.Google Scholar
Deutsch, F. 1985. Rate of convergence of the method of alternating projections. In Brosowski, B. and Deutsch, F. (eds) Parametric Optimization and Approximation (Oberwolfach, 1983), pp. 96107. Internat. Schriftenreihe Numer. Math., Vol. 72. Basel: Birkhäuser.Google Scholar
Dierolf, M., Menzel, A., Thibault, P., et al. 2010. Ptychographic X-ray computed tomography at the nanoscale. Nature, 467, 436439.Google Scholar
Elser, V. 2003. Phase retrieval by iterated projections. JOSA A, 20(1), 4055.CrossRefGoogle ScholarPubMed
Elser, V., Lan, , T-Y., and Bendory, T. 2018. Benchmark problems for phase retrieval. SIAM J. Imaging Sci., 11(4), 24292455.Google Scholar
Elser, V., Rankenburg, I., and Thibault, P. 2007. Searching with iterated maps. P. Natl. Acad. Sci. USA, 104(2), 418423.Google Scholar
Epstein, C. L. 2005. How well does the finite Fourier transform approximate the Fourier transform? Comm. Pure and App. Math., 58, 14211435.Google Scholar
Epstein, C. L. and Schotland, J. 2008. The bad truth about Laplace’s transform. SIAM Review, 50(3), 504520.Google Scholar
Fienup, J. R. 1978. Reconstruction of an object from the modulus of its Fourier transform. Opt. Lett., 3, 2729.Google Scholar
Fienup, J. R. 1982. Phase retrieval algorithms: A comparison. Appl. Opt., 21, 27582769.Google Scholar
Fienup, J. R. 1987. Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint. J. Opt. Soc. Am. A, 4, 118123.Google Scholar
Fienup, J. R. and Kowalczyk, A. M. 1990. Phase retrieval for a complex-valued object by using a low-resolution image. J. Opt. Soc. Am. A, 7(3), 450458.Google Scholar
Fienup, J. R. and Wackerman, C. C. 1986. Phase-retrieval stagnation problems and solutions. J. Opt. Soc. Am. A, 3, 18971907.Google Scholar
Fienup, J. R., Crimmins, T. R. and Holsztynski, W. 1982. Reconstruction of the support of an object from the support of its autocorrelation. J. Opt. Soc. Am., 72, 610624.CrossRefGoogle Scholar
Fienup, J. R., Crimmins, T. R. and Thelen, B. J. 1990. Improved bounds on object support from autocorrelation support and application to phase retrieval. J. Opt. Soc. Am. A, 7(1), 313.Google Scholar
Gerchberg, R. W. and Saxton, W. O. 1972. A practical algorithm for the determination of the phase from image and diffraction plane pictures. Optik, 35(2), 237246.Google Scholar
Gonsalves, R. A. 1976. Phase retrieval from modulus data. J. Opt. Soc. Am., 66(9), 961964.Google Scholar
Gravel, S. and Elser, V. 2008. Divide and concur: A general approach to constraint satisfaction. Phys. Rev. E, 78(3), 036706–5.Google Scholar
Griffiths, P. and Harris, J. 1978. Principles of Algebraic Geometry. New York: Wiley-Interscience.Google Scholar
Guizar-Sicairos, M. and Fienup, J. R. 2007. Holography with extended reference by autocorrelation linear differential operation. Opt. Express, 15(26), 1759217612.Google Scholar
Guizar-Sicairos, M. and Fienup, J. R. 2008. Direct image reconstruction from a Fourier intensity pattern using HERALDO. Opt. Lett., 33(22), 26682670.CrossRefGoogle ScholarPubMed
Guizar-Sicairos, M., Johnson, I., Diaz, A., et al. 2014. High-throughput ptychography using Eiger: Scanning X-ray nano-imaging of extended regions. Opt. Express, 22(12), 1485914870.Google Scholar
Hayes, M. H. 1982. The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform. IEEE Trans. on Acoustics, Speech and Sig. Proc., 30, 140153.Google Scholar
Hesse, R. and Luke, D. R. 2013. Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems. SIAM J. Optimiz., 23(4), 23972419.Google Scholar
Huang, K., and Eldar, Y. C. 2017. Phase retrieval using a conjugate symmetric reference. International Conference on Sampling Theory and Applications (SampTA), Vol. 134, pp. 331335Google Scholar
Huang, K., Eldar, Y. C., and Sidiropoulos, N. D. 2016. Phase retrieval from 1D Fourier measurements: Convexity, uniqueness, and algorithms. IEEE T. Signal Proces., 64(23), 61056117.CrossRefGoogle Scholar
Jacobsen, C. 2019. X-ray Microscopy, 1st ed. Advances in Microscopy and Microanalysis. Cambridge: Cambridge University Press.Google Scholar
John, F. 1960. Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Commun. Pur. App. Math., 13(4), 551585.Google Scholar
Katznelson, Y. 1968. An Introduction to Harmonic Analysis. New York-London-Sydney: John Wiley & Sons.Google Scholar
Lane, R. G. 1991. Phase retrieval using conjugate gradient minimization. J. Mod. Optic., 38(9), 17971813.Google Scholar
Lee, J. M. 2018. Introduction to Riemannian Manifolds, 2nd ed. Graduate Texts in Mathematics, Vol. 176. New York: Springer.Google Scholar
Leshem, B., Raz, O., Jaffe, A., and Nadler, B. 2018. The discrete sign problem: Uniqueness, recovery algorithms and phase retrieval applications. Appl. Comput. Harmon. A., 45(3), 463485.CrossRefGoogle Scholar
Levin, E. and Bendory, T. 2019. A note on Douglas-Rachford, subgradients, and phase retrieval. arXiv: abs/1911.13179Google Scholar
Li, G. and Pong, T. K. 2016. Douglas–Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems. Math. Program., 159(1), 371401.Google Scholar
Lindstrom, S. B. and Sims, B. 2021. Survey: Sixty years of Douglas–Rachford. J. Aust. Math. Soc., 110(3), 333370.Google Scholar
Luke, D. R. 2005. Relaxed averaged alternating reflections for diffraction imaging. Inverse Probl., 21(1), 3750.Google Scholar
Luke, D. R. and Martins, A. L. 2020. Convergence analysis of iterative algorithms for phase retrieval. In Salditt, T., Egner, A., and Luke, D. R. (eds.) Nanoscale Photonic Imaging 583–601. Topics in Applied Physics, Vol. 134. Cham: Springer.Google Scholar
Marchesini, S. 2007. Phase retrieval and saddle-point optimization. J. Opt. Soc. Am. A, 24(10), 32893296.Google Scholar
Marchesini, S., Tu, Y.-C., and Wu, H.-T. 2016. Alternating projection, ptychographic imaging and phase synchronization. Appl. Comput. Harmon. A., 41(3), 815851.Google Scholar
Maretzke, S. and Hohage, T. 2016. Pinhole-CDI: Unique and Deterministic Phase Retrieval via Beam-Confinement. http://ip.math.uni-goettingen.de/data-smaretzke/poster coherence 2016-06 smaretzke.pdf.Google Scholar
Maretzke, S. and Hohage, T. 2017. Stability estimates for linearized near-field phase retrieval in X-ray phase constrast imaging. SIAM J. Appl. Math., 77, 384408.Google Scholar
Milnor, J. 1963. Morse Theory. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51. Princeton, NJ: Princeton University Press.Google Scholar
Nakajima, N. 1995. Phase Retrieval Using the Properties of Entire Functions. Advances in Imaging and Electron Physics, Vol. 93. New York: Elsevier.Google Scholar
Nakajima, N. and Asakura, T. 1985. A new approach to two-dimensional phase retrieval. Optica Acta: International Journal of Optics, 32(6), 647658.Google Scholar
Nakajima, N. and Asakura, T. 1986. Two-dimensional phase retrieval using the logarithmic Hilbert transform and the estimation technique of zero information. J. Phys. D: Appl. Phys., 19(3), 319331.CrossRefGoogle Scholar
Osherovich, E. 2011. Numerical methods for phase retrieval. arXiv:1203.4756v1 [physics.optics]. PhD Thesis, Technion.Google Scholar
Pfeiffer, F. 2018. X-ray ptychography. Nat. Photonics, 12, 917.CrossRefGoogle Scholar
Pham, M., Yin, P., Rana, A., Osher, S., and Miao, J. 2019. Generalized proximal smoothing for phase retrieval. Microsc. Microanal., 25(S2), 118119.Google Scholar
Phan, H. M. 2016. Linear convergence of the Douglas-Rachford method for two closed sets. Optimization, 65(2), 369385.CrossRefGoogle Scholar
Podorov, S. G., Pavlov, K. M., and Paganin, D. M. 2007. A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging. Opt. Express, 15(16), 99549962.Google Scholar
Rodenburg, J. M., Hurst, A. C., Cullis, A. G., et al. 2007. Hard-X-ray lensless imaging of extended objects. Phys. Rev. Lett., 98(Jan), 034801.Google Scholar
Sanz, J. L. C. 1985. Mathematical considerations for the problem of Fourier transform phase retrieval from magnitude. SIAM J. Appl. Math., 45, 651664.Google Scholar
Seldin, J. H. and Fienup, J. R. 1990. Numerical investigation of the uniqueness of phase retrieval. J. Opt. Soc. Am. A, 7(3), 412427.Google Scholar
Shechtman, Y., Eldar, Y. C., Szameit, A., and Segev, M. 2011. Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing. Opt. Express, 19(16), 1480714822.Google Scholar
Sidorenko, P., Kfir, O., Shechtman, Y., et al. 2015. Sparsity-based super-resolved coherent diffraction imaging of one-dimensional objects. Nat. Commun., 6(1), 8209.Google Scholar
Spivak, M. 1965. Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus. New York and Amsterdam: W. A. Benjamin, Inc.Google Scholar
Spivak, M. 1979. A Comprehensive Introduction to Differential Geometry. Vol. I. 2nd ed. Wilmington, CA: Publish or Perish, Inc.Google Scholar
Stein, E. M. and Shakarchi, R. 2003a. Complex Analysis. Princeton Lectures in Analysis, Vol. 2. Princeton, NJ: Princeton University Press.Google Scholar
Stein, E. M. and Shakarchi, R. 2003b. Fourier Analysis. Princeton Lectures in Analysis, Vol. 1. Princeton, NJ: Princeton University Press.Google Scholar
Taylor, M. E. 1996. Partial Differential Equations. II. Applied Mathematical Sciences, Vol. 116. New York: Springer-Verlag.Google Scholar
Thibault, P., Dierolf, M., Menzel, A., et al. 2008. High-resolution scanning X-ray diffraction microscopy. Science, 321, 379382.Google Scholar
von Neumann, J. 1950. Functional Operators, Vol. II. Reprint of 1933 lecture notes. Princeton, NJ: Princeton University Press.Google Scholar
Watson, G. N. 1922. A Treatise on the Theory of Bessel Functions. Cambridge: Cambridge University Press.Google Scholar
Weinberger, S. 2004. On the topological social choice model. J. Econom. Theory, 115(2), 377384.Google Scholar
Wu, S.-P., Boyd, S. P., and Vandenberghe, L. 1996. FIR filter design via semidefinite programming and spectral factorization. IEEE Conference on Decision and Control, 1, 271276.Google Scholar
Yasir, P. A. A. and Ivan, J. S. 2016. Phase estimation using phase gradients obtained through Hilbert transform. J. Opt. Soc. Am. A, 33(10), 20102019.Google Scholar

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