Journal of Fourier Analysis and Applications

, Volume 23, Issue 6, pp 1480–1494 | Cite as

Reconstructing Real-Valued Functions from Unsigned Coefficients with Respect to Wavelet and Other Frames

  • Rima Alaifari
  • Ingrid Daubechies
  • Philipp Grohs
  • Gaurav Thakur
Article

Abstract

In this paper we consider the following problem of phase retrieval: given a collection of real-valued band-limited functions \(\{\psi _{\lambda }\}_{\lambda \in \Lambda }\subset L^2(\mathbb {R}^d)\) that constitutes a semi-discrete frame, we ask whether any real-valued function \(f \in L^2(\mathbb {R}^d)\) can be uniquely recovered from its unsigned convolutions \({\{|f *\psi _\lambda |\}_{\lambda \in \Lambda }}\). We find that under some mild assumptions on the semi-discrete frame and if f has exponential decay at \(\infty \), it suffices to know \(|f *\psi _\lambda |\) on suitably fine lattices to uniquely determine f (up to a global sign factor). We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of \(L^2(\mathbb {R}^d)\), \(d=1,2\), we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.

Keywords

Phase retrieval Semi-discrete frames Injectivity Stability Sampling 

Mathematics Subject Classification

42C15 49N45 94A12 94A20 

Notes

Acknowledgements

R.A. is supported by an ETH Postdoctoral Fellowship. R.A., I.D. and P.G. would like to thank the Mathematisches Forschungsinstitut Oberwolfach (MFO). The authors also give their thanks to the anonymous referee for the detailed and useful comments and for inspiring our example given after Proposition 1.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Rima Alaifari
    • 1
  • Ingrid Daubechies
    • 2
  • Philipp Grohs
    • 1
    • 3
  • Gaurav Thakur
    • 4
  1. 1.Department of MathematicsETH ZürichZurichSwitzerland
  2. 2.Department of MathematicsDuke UniversityDurhamUSA
  3. 3.Faculty of MathematicsUniversity of ViennaViennaAustria
  4. 4.INTECH Investment ManagementPrincetonUSA

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