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

TL;DR

This paper investigates the problem of reconstructing real-valued functions from unsigned wavelet and frame coefficients, establishing conditions for unique recovery and stability, with practical oversampling solutions for certain frames.

Contribution

It introduces new conditions under which real-valued functions can be uniquely reconstructed from unsigned frame coefficients, including stability analysis and oversampling strategies.

Findings

Unique reconstruction from unsigned coefficients is possible under mild assumptions.

Exponential decay of functions ensures recoverability from sampled unsigned convolutions.

Oversampling enables practical reconstruction for specific frame examples.

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.