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

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.