The generic crystallographic phase retrieval problem

TL;DR

This paper proves that sparse signals in a generic basis can be uniquely reconstructed from their power spectrum, with specific sparsity thresholds ensuring uniqueness in real and complex cases.

Contribution

It establishes new sparsity bounds under which signals are uniquely determined by their power spectrum, extending previous results to generic bases and complex signals.

Findings

Unique recovery for sparsity up to ~N/2 in real signals.

Universal recovery for sparsity up to ~N/4 in real signals.

Extended results to complex signals, generalizing earlier work.

Abstract

In this paper we consider the problem of recovering a signal $x \in \mathbb{R}^N$ from its power spectrum assuming that the signal is sparse with respect to a generic basis for $\mathbb{R}^N$. Our main result is that if the sparsity level is at most $\sim\! N/2$ in this basis then the generic sparse vector is uniquely determined up to sign from its power spectrum. We also prove that if the sparsity level is $\sim\! N/4$ then every sparse vector is determined up to sign from its power spectrum. Analogous results are also obtained for the power spectrum of a vector in $\mathbb{C}^N$ which extend earlier results of Wang and Xu \cite{arXiv:1310.0873}.