Advances in Computational Mathematics

, Volume 41, Issue 2, pp 317–331 | Cite as

Stable phase retrieval with low-redundancy frames

  • Bernhard G. BodmannEmail author
  • Nathaniel Hammen


We investigate the recovery of vectors from magnitudes of frame coefficients when the frames have a low redundancy, meaning a small number of frame vectors compared to the dimension of the Hilbert space. We first show that for complex vectors in d dimensions, 4d−4 suitably chosen frame vectors are sufficient to uniquely determine each signal, up to an overall unimodular constant, from the magnitudes of its frame coefficients. Then we discuss the effect of noise and show that 8d−4 frame vectors provide a stable recovery if part of the frame coefficients is bounded away from zero. In this regime, perturbing the magnitudes of the frame coefficients by noise that is sufficiently small results in a recovery error that is at most proportional to the noise level.


Magnitude measurements Trigonometric polynomials Roots of complex polynomials Newton’s identitites 

Mathematics Subject Classifications (2010)

15A29 42C15 30C15 


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  1. 1.
    Akutowicz, E. J.: On the determination of the phase of a Fourier integral, I. Trans. Am. Math. Soc. 83, 179–192 (1956)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Alexeev, B., Bandeira, A. S., Fickus, M., Mixon, D. G.: Phase retrieval with polarization. SIAM J. Imaging Sci 7, 35–66 (2014)CrossRefzbMATHGoogle Scholar
  3. 3.
    Balan, R.: Reconstruction of signals from magnitudes of redundant representations, preprint. arxiv:1207.1134
  4. 4.
    Bodmann, B. G., Casazza, P. G., Edidin, D., Balan, R.: Fast algorithms for signal reconstruction without phase, Proc. SPIE 6701, Wavelets XII (2007) 67011LGoogle Scholar
  5. 5.
    Balan, R., Bodmann, B. G., Casazza, P. G., Edidin, D.: Painless reconstruction from magnitudes of frame coefficients. J. Fourier Anal. Appl. 15, 488–501 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Balan, R., Casazza, P., Edidin, D.: On signal reconstruction without phase. Appl. Comp. Harmon. Anal. 20, 345–356 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bandeira, A. S., Cahill, J., Mixon, D. G., Nelson, A. A.: Saving phase: Injectivity and stability for phase retrieval. Appl. Comput. Harmon. Anal., to appear (2014)Google Scholar
  8. 8.
    Bunk, O., Diaz, A., Pfeiffer, F., David, C., Schmitt, B., Satapathy, D. K., van der Veen, J. F.: Diffractive imaging for periodic samples: retrieving one-dimensional concentration profiles across microfluidic channels. Acta. Cryst. A63, 306–314 (2007)CrossRefGoogle Scholar
  9. 9.
    Candès, E. J., Eldar, Y., Strohmer, T., Voroninski, V.: Phase retrieval via matrix completion. SIAM J. Imaging Sci. 6, 199–225 (2014)CrossRefGoogle Scholar
  10. 10.
    Candès, E. J., Li, X.: Solving quadratic equations via PhaseLift when there are about as many equations as unknowns. Found. Comput. Math., to appear (2014)Google Scholar
  11. 11.
    Candès, E. J., Strohmer, T., Voroninski, V.: PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming. Commun. Pure Appl. Math. 66, 1241–1274 (2013)CrossRefzbMATHGoogle Scholar
  12. 12.
    Demanet, L., Hand, P.: Stable optimizationless recovery from phaseless linear measurements. J. Fourier Anal. Appl., to appear (2014)Google Scholar
  13. 13.
    Eldar, Y. C., Mendelson, S.: Phase retrieval: Stability and recovery guarantees. Appl. Comput. Harmon. Anal. 36, 473–494 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Fienup, J. R.: Phase retrieval algorithms: a comparison. Appl. Opt. 21(15), 2758–2769 (1982)CrossRefGoogle Scholar
  15. 15.
    Finkelstein, J.: Pure-state informationally complete and “really” complete measurements. Phys. Rev. A 70, 052107 (2004)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Heinosaari, T., Mazzarella, L., Wolf, M. M.: Quantum tomography under prior information. Commun. Math. Phys. 318, 355–374 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Jaming, P.: Uniqueness results for the phase retrieval problem of fractional Fourier transforms of variable order. Appl. Comput. Harmon. Anal., in press (2014)Google Scholar
  18. 18.
    Milgram, R. J., Immersing projective spaces. Ann. Math. 85, 473–482 (1967)CrossRefzbMATHGoogle Scholar
  19. 19.
    Mukherjee, A.: Embedding complex projective spaces in Euclidean space. Bull. London Math. Soc. 13, 323–324 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Steer, B.: On the embedding of projective spaces in Euclidean space. Proc. London Math. Soc. 21, 489–501 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Waldspurger, I., d’Aspremont, A., Mallat, S.: Phase recovery, maxCut and complex semidefinite programming, preprint (2014)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.651 Philip G. Hoffman Hall, Mathematics DepartmentUniversity of HoustonHoustonUSA

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