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

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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.

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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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Correspondence to Rima Alaifari.

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Communicated by Roman Vershynin.

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Alaifari, R., Daubechies, I., Grohs, P. et al. Reconstructing Real-Valued Functions from Unsigned Coefficients with Respect to Wavelet and Other Frames. J Fourier Anal Appl 23, 1480–1494 (2017). https://doi.org/10.1007/s00041-016-9513-7

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  • DOI: https://doi.org/10.1007/s00041-016-9513-7

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