Reconstructing real-valued functions from unsigned coefficients with respect to wavelet and other frames

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 \({\{|f \ast \psi_\lambda|\}_{\lambda \in \Lambda}}\). We find that under some mild assumptions on the semi-discrete frame and if \(f\) has exponential decay at \(\infty\), it suffices to know \(|f \ast \psi_\lambda|\) 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.