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

Journal Article (Journal Article)

In this paper we consider the following problem of phase retrieval: given a collection of real-valued band-limited functions {ψλ}L2(Rd) that constitutes a semi-discrete frame, we ask whether any real-valued function f∈ L2(Rd) can be uniquely recovered from its unsigned convolutions { | f∗ ψλ| } λ∈Λ. We find that under some mild assumptions on the semi-discrete frame and if f has exponential decay at ∞, it suffices to know | f∗ ψλ| 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 L2(Rd) , 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.

Full Text

Duke Authors

Cited Authors

  • Alaifari, R; Daubechies, I; Grohs, P; Thakur, G

Published Date

  • December 1, 2017

Published In

Volume / Issue

  • 23 / 6

Start / End Page

  • 1480 - 1494

International Standard Serial Number (ISSN)

  • 1069-5869

Digital Object Identifier (DOI)

  • 10.1007/s00041-016-9513-7

Citation Source

  • Scopus