Foundations of Computational Mathematics

, Volume 16, Issue 3, pp 677–721 | Cite as

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Article
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Abstract

This paper is concerned with the question of reconstructing a vector in a finite-dimensional complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well as an iterative algorithm that finds the least-square solution, which is robust in the presence of noise. We analyze its numerical performance by comparing it to the Cramer–Rao lower bound.

Keywords

Frame Phase retrieval Cramer–Rao lower bound  Phaseless reconstruction 

Mathematics Subject Classification

15A29 65H10 90C26 
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Notes

Acknowledgments

The author was partially supported by NSF under DMS-1109498 and DMS-1413249 Grants. The author thanks the Erwin Schrödinger Institute for the hospitality shown during the special workshop on “Phase Retrieval” in October 2012. Some of the results obtained here were presented at that workshop and later at the Workshop on “Phaseless Reconstruction”, UMD, February 2013. The author also thanks Bernhard Bodmann, Jameson Cahill, Martin Ehler, Boaz Nadler, Oren Raz and Yang Wang for fruitful discussions. He also thanks the anonymous referees for their helpful comments and careful reading of the first draft. Additionally he is grateful to Friedrich Philipp [29] for his comments and for pointing out several errors in the first draft, in particular the dimension of \({\mathcal {S}^{1,1}}\) is Lemma 3.11. Last but not least, the author thanks the anonymous referees for their comments and their patience with an earlier draft of this paper.

References

  1. 1.
    B. Alexeev, A. S. Bandeira, M. Fickus, D. G. Mixon, Phase retrieval with polarization, SIAM J. Imaging Sci., 7(1) (2014), 35–66.CrossRefMATHGoogle Scholar
  2. 2.
    B. Alexeev, J. Cahill, D. G. Mixon, Full spark frames, J. Fourier Anal. Appl. 18 (2012), 1167–1194.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    R. Balan, P. Casazza, D. Edidin, On signal reconstruction without phase, Appl. Comput. Harmon. Anal. 20 (2006), 345–356.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    R. Balan, B. Bodmann, P. Casazza, D. Edidin, Painless reconstruction from Magnitudes of Frame Coefficients, J. Fourier Anal. Applic., 15(4) (2009), 488–501.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    R. Balan, On Signal Reconstruction from Its Spectrogram, Proceedings of the CISS Conference, Princeton NJ, May 2010.Google Scholar
  6. 6.
    R. Balan, Reconstruction of Signals from Magnitudes of Frame Representations, arXiv submission arxiv:1207.1134
  7. 7.
    R. Balan and Y. Wang, Invertibility and Robustness of Phaseless Reconstruction, available online arxiv:1308.4718v1, Appl. Comp. Harm. Anal., accepted July 2014
  8. 8.
    A. S. Bandeira, J. Cahill, D. Mixon, A. A. Nelson, Saving phase: Injectivity and Stability for phase retrieval, arXiv submission , arxiv:1302.4618, Appl. Comp. Harm. Anal. 37(1) (2014), 106–125.
  9. 9.
    R.H. Bates and D. Mnyama, The status of practical fourier phase retrieval, in W.H. Hawkes, ed., Advances in Electronics and Electron Physics, 67 (1986), 1–64.Google Scholar
  10. 10.
    R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997.CrossRefMATHGoogle Scholar
  11. 11.
    B. G. Bodmann, private communications, October 2012 and March 2013.Google Scholar
  12. 12.
    B. G. Bodmann, N. Hammen, Stable Phase Retrieval with Low-Redundancy Frames, arXiv submission:1302.5487v1, Adv. Comput. Math., accepted 10 April 2014.Google Scholar
  13. 13.
    E. Candés, T. Strohmer, V. Voroninski, PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming, Communications in Pure and Applied Mathematics, vol. 66, 1241–1274 (2013).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    E. Candés, Y. Eldar, T. Strohmer, V. Voloninski, Phase Retrieval via Matrix Completion Problem, SIAM J. Imaging Sci., 6(1) (2013), 199–225.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    E. J. Candès, X. Li, Solving quadratic equations via PhaseLift when there are about as many equations as unknowns, available online arxiv:1208.6247, Found. of Comput. Math. 14(5) (2012), 1017-1026
  16. 16.
    P. Casazza, The art of frame theory, Taiwanese J. Math., 4(2) (2000), 129–202.MathSciNetMATHGoogle Scholar
  17. 17.
    A. Conca, D. Edidin, M. Hering, C. Vinzant, An algebraic characterization of injectivity in phase retrieval, Appl. Comput. Harmon. Anal. 38 (2015), 346–356.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    L. Demanet, P. Hand, Stable optimizationless recovery from phaseless linear measurements, available online, arxiv:1208.1803v1 J. Fourier Anal. Appl., 20(1) (2014), 199-221
  19. 19.
    Y. Eldar, S. Mendelson, Phase Retrieval: Stability and Recovery Guarantees, available online arxiv:1211.0872v1, Appl. Comp. Harm. Anal. 36(3) (2014), 473-494
  20. 20.
    P. R. Halmos, Finite-Dimensional Vector Space, reprint of the 2nd ed. published by Van Nostrand, Princeton, NJ, Springer-Verlag, 1974.Google Scholar
  21. 21.
    P. Hand, Conditions for Existence of Dual Certificates in Rank One Semidefinite Problems, available online arxiv:1303.1598v2, Commun. Math. Sci, 12(7) (2014), 1363–1378
  22. 22.
    M. H. Hayes, J. S. Lim, and A. V. Oppenheim, Signal Reconstruction from Phase and Magnitude, IEEE Trans. ASSP 28(6) (1980), 672–680.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    T. Heinosaari, L. Mazzarella, M. M. Wolf, Quantum Tomography under Prior Information, available online arxiv:1109.5478v1, Commun. Math. Phys. 318, 355-374 (2013)
  24. 24.
    S. M. Kay, Fundamentals of Statistical Signal Processing. I. Estimation Theory, Prentice Hall PTR, 18th Printing, 2010.Google Scholar
  25. 25.
    T.-Y. Lam, Algebraic Theory of Quadratic Forms, W A Benjamin, 1973.Google Scholar
  26. 26.
    F. Riesz and B. Sz. Nagy, Functional Analysis, Ungar Publications, 2nd Edition, New York, 1955.Google Scholar
  27. 27.
    H. Nawab, T. F. Quatieri, and J. S. Lim, Signal Reconstruction from the Short-Time Fourier Transform Magnitude, in Proceedings of ICASSP 1984.Google Scholar
  28. 28.
    B. Noble, Applied Linear Algebra, Prentice-Hall, Englewood Cliffs, NJ, 1969.MATHGoogle Scholar
  29. 29.
    F. Phillip, personal communication, Aug. 2014.Google Scholar
  30. 30.
    B. Simon, Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory, AMS Colloquium Publications, vol. 54, 2004.Google Scholar
  31. 31.
    O. Raz, N. Dudovich, O. Raz, Vectorial Phase Retrieval of 1-D Signals, IEEE Trans. on Signal Processing, 61(7) (2013), 1632–1643.MathSciNetCrossRefGoogle Scholar
  32. 32.
    I. Waldspurger, A. dAspremont, S. Mallat, Phase recovery, MaxCut and complex semidefinite programming, available online arxiv:1206.0102, Mathematical Programming, 149(1-2) (2015), 47-81

Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Center for Scientific Computation and Mathematical ModelingUniversity of Maryland, College ParkCollege ParkUSA

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