Skip to main content
SpringerLink
Log in

Frame dimension functions and phase retrievability

  • Original Paper
  • Published:
Advances in Operator Theory Aims and scope Submit manuscript

Abstract

The frame dimension function of a frame \({{\mathcal {F}}}= \{f_j\}_{j=1}^{n}\) for an n-dimensional Hilbert space H is the function \(d_{{{\mathcal {F}}}}(x) = \dim {\textrm{span}}\{ \langle x, f_{j}\rangle f_{j}: j=1,\ldots , N\}, 0\ne x\in H.\) It is known that \({{\mathcal {F}}}\) does phase retrieval for an n-dimensional real Hilbert space H if and only if \({\textrm{range}} (d_{{{\mathcal {F}}}}) = \{ n\}.\) This indicates that the range of the dimension function is one of the good candidates to measure the phase retrievability for an arbitrary frame. In this paper we investigate some structural properties for the range of the dimension function, and examine the connections among different exactness of a frame with respect to its PR-redundance, dimension function and range of the dimension function. A subset \(\Omega \) of \(\{1,\ldots , n\}\) containing n is attainable if \({\textrm{range}} (d_{{{\mathcal {F}}}}) = \Omega \) for some frame \({{\mathcal {F}}}.\) With the help of linearly connected frames, we show that, while not every \(\Omega \) is attainable, every (integer) interval containing n is always attainable by an n-linearly independent frame. Consequently, \({\textrm{range}}(d_{{{\mathcal {F}}}})\) is an interval for every generic frame for \({\mathbb {R}}\,^n.\) Additionally, we also discuss and post some questions related to the connections among ranges of the dimension functions, linearly connected frames and maximal phase retrievable subspaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balan, R.: Stability of phase retrievable frames. In: Proceedings of SPIE, Wavelets and Sparsity XV, 88580H (2013). Editors: Dimitri Van De Ville, Vivek K. Goyal, Manos Papadakis

  2. Balan, R., Casazza, P., Edidin, D.: On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20, 345–356 (2006)

    Article  MathSciNet  Google Scholar 

  3. Balan, R., Casazza, P.G., Edidin, D.: On signal reconstruction from the absolute value of the frame coefficients. In: Proceedings of SPIE, vol. 5914, p. 591415 (1–8) (2005). Editors: Manos Papadakis, Andrew F. Laine, Michael A. Unser.

  4. Balan, R., Casazza, P.G., Edidin, D.: Equivalence of reconstruction from the absolute value of the frame coefficients to a sparse representation problem. IEEE Signal Process. Lett. 14, 341–343 (2007)

    Article  Google Scholar 

  5. Balan, B., Bodmann, B.G., Casazza, P.G., Edidin, D.: Painless reconstruction from magnitudes of frame vectors. J. Fourier Anal. Appl. 15, 488–501 (2009)

    Article  MathSciNet  Google Scholar 

  6. Balan, R., Bodmann, B.G., Casazza, P.G., Edidin, D.: Fast algorithms for signal reconstruction without phase. In: Proceedings of SPIE-Wavelets XII, San Diego, vol. 6701, pp. 670111920–670111932 (2007), Editors: Dimitri Van De Ville, Vivek K. Goyal, Manos Papadakis.

  7. Balan, R., Wang, Y.: Invertibility and robustness of phaseless reconstruction. Appl. Comput. Harmon. Anal. 38, 469–488 (2015)

    Article  MathSciNet  Google Scholar 

  8. Bandeira, A.S., Cahill, J., Mixon, D.G., Nelson, A.A.: Saving phase: injectivity and stability for phase retrieval. Appl. Comput. Harmon. Anal. 37, 106–125 (2014)

    Article  MathSciNet  Google Scholar 

  9. Bodmann, B.G., Casazza, P.G., Edidin, D., Balan, R.: Frames for linear reconstruction without phase. In: CISS Meeting, Princeton, NJ (2008)

  10. Cahill, J., Casazza, P., Peterson, J., Woodland, L.: Phase retrieval by projections. Houst. J. Math. 42(2), 537–558 (2016)

    MathSciNet  Google Scholar 

  11. Cahill, J., Casazza, P., Peterson, J., Woodland, L.: Using projections for phase retrieval. In: Proceedings of SPIE, Optics and Photonics (2013). Editors: Dimitri Van De Ville, Vivek K. Goyal, Manos Papadakis

  12. Casazza, P.G., Woodland, L.M.: Phase retrieval by vectors and projections. In: Operator Methods in Wavelets, Tilings, and Frames. Contemporary Mathematics, vol. 626, pp. 1–17. American Mathematical Society, Providence (2014). Editors: V. Furst, K. Kornelson and E. Weber.

  13. Conca, A., Edidin, D., Hering, M., Vinzant, C.: An algebraic characterization of injectivity in phase retrieval. Appl. Comput. Harmon. Anal. 38, 346–356 (2015)

    Article  MathSciNet  Google Scholar 

  14. Edidin, D.: Projections and phase retrieval. Appl. Comput. Harmon. Anal. 42(2), 350–359 (2017)

    Article  MathSciNet  Google Scholar 

  15. Han, D., Pehlivan, S., Mohapatra, R.: Linearly connected sequences and spectrally optimal frames for erasures. J. Funct. Anal. 265, 2855–2876 (2013)

    Article  MathSciNet  Google Scholar 

  16. Han, D., Juste, T.: Phase-retrievable operator-valued frames and representations of quantum channels. Linear Algebra Appl. 579, 148–168 (2019)

    Article  MathSciNet  Google Scholar 

  17. Han, D., Juste, T., Li, T., Sun, W.: Frame phase-retrievability and exact phase-retrievable frames. J. Fourier Anal. Appl. 25, 3154–3173 (2019)

    Article  MathSciNet  Google Scholar 

  18. Wang, Y., Xu, Z.: Generalized phase retrieval: measurement number, matrix recovery and beyond. Appl. Comp. Harmon. Anal. 47, 423–446 (2019)

    Article  MathSciNet  Google Scholar 

  19. Wang, Y., Xu, Z.: Phase retrieval for sparse signals. Appl. Comput. Harmon. Anal. 37, 531–544 (2014)

    Article  MathSciNet  Google Scholar 

  20. Xu, Z.: The minimal measurement number for low-rank matrices recovery. Appl. Comput. Harmon. Anal. 44, 497–508 (2018) (accepted)

  21. Xu, Z.: The minimal measurement number problem in phase retrieval: a review of recent developments. J. Math. Res. Appl. 37, 40–46 (2017)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the referees very much for several constructive comments and suggestions that helped improve the presentation of this paper. Deguang Han is partially supported by the NSF grant DMS-2105038.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Deguang Han & Kai Liu

Authors
  1. Deguang Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kai Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Deguang Han.

Additional information

Communicated by Christopher Heil

Dedicated to Professor Chi-Kwong Li’s 65th Birthday.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, D., Liu, K. Frame dimension functions and phase retrievability. Adv. Oper. Theory 9, 31 (2024). https://doi.org/10.1007/s43036-024-00331-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s43036-024-00331-y

Keywords

Mathematics Subject Classification

Search

Navigation