The generic crystallographic phase retrieval problem

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}.

20 pages